The Golden Age of Theoretical Physics: Selected Essays

The angular motion produced by molecular motion thus decreases much more rapidly than the translational motion. For particles of radius P = .5/i in water at 17°C, Einstein calculated an average turn of about 100° of arc in one second, but only 11 seconds of arc in one second for P = 0.5 mm. Equation (72) cannot be valid for arbitrarily small times because the velocity of change of a as a result of irregular thermal processes,

H^J-

(77)

becomes infinity. Einstein noted: 'The reason is that we have implicitly assumed in our development that the events during the time t are to be looked upon as

Albert Einstein

and Marian von Smoluchowski

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171

phenomena independent of the events in the time preceding. But this assumption becomes harder to justify the smaller the time t is chosen.' 61 Einstein derived a limit for the time interval t, for which Eq. (77) is still valid, from the following consideration. At t = 0, the instantaneous velocity change is (30 = ^ . Assuming that /?, at any subsequent instant, depends only on the mobility B, then =

- ^

(78)

D>

where p is defined such that i/x.B2 is the kinetic energy corresponding to the change. Integrating Eq. (78), 0 = poe-t/liB . (79) Thus for t > /J,B the motions are independent and Eq. (72) holds. For particles of lfj. in diameter in water at room temperature, t must be larger than 1 0 - 7 sec. 5.9. Concerning

the experiments

of

Svedberg

In 1906 Teodor Svedberg published two papers, reporting on the empirical observations of Brownian motions of colloidal particles. 62 They stimulated Einstein to present a simplified account of his theory. 63 In the first note, submitted in January 1907, Einstein summarized the results he had obtained in his two earlier papers in order 'to facilitate for physicists who handle the subject experimentally the interpretation of their observations as well as the comparison of the latter with the theory.' 64 He calculated for Svedberg's objects, the colloidal platinum particles of mass m = 2.5 x 10-15<7, the average kinetic energy, at room temperature, of

/=

jZRT

„

cm

/on,

The velocity of these particles of radius P is damped, in a fluid with viscosity A;, according to the law CUV

m = — = -GnkPv. at

(81)

Thus, after a time 0 , 0

=

0.434 x 6nkP ' ^82' the velocity will drop to 10% of its original value. For Svedberg's particles, P = 2.5 X 10~ 6 cm, and © = 3 x 1 0 - 7 sec. Since, however, the particles obtain new impulses from the surrounding molecules, a certain average velocity will remain independent of viscous friction. These impulses are statistically independent. The average length of the path of the suspended particle in time r » 0 is given by RT 1 A2=V^/ N 3wkP ' A/A2

and the average velocity of its motion is -^p-5-, which is greater the smaller the time of observation T (;§> 0 ) is. Einstein concluded: 'Since an observer operating with

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definite means of observation in a definite manner can never perceive the actual path traversed in an arbitrarily small time, a certain mean velocity will always appear to him as an instantaneous velocity. But it is clear that the velocity thus ascertained corresponds to no objective property of the motion under investigation — at least, if the theory corresponds to the facts.' 65 5.10. Elementary

theory of Brownian

motion

In the second note Einstein stressed the underlying ideas of his theory of Brownian motion. 66 This theory consists of two parts. First, a diffusion process is studied by which particles, which are much larger in size than the molecules of the surrounding fluid, pass through the liquid. The diffusion coefficient in this example contains no quantities which depend on the nature of the solution except the viscosity and the radius of the 'dissolved' particles. The origin of the diffusion process has then to be looked for in the irregular thermal motion of the solute molecules, and the difusion coefficient related to this motion.

A cylinder Z is separated by a movable semipermeable wall into two parts A and B. If the concentration of the solution is higher on the side A, one has to exert a force on the wall to keep it in place. 67 Now Einstein considered a one-dimensional motion along the i-axis. The force on a volume dx (assuming the cross section of the cylinder to be unity), at the position between x and x + dx from one end of the cylinder, is given by the pressure difference p — p\ i.e., K = P-P dx

P -P

dx

dp dx

(83)

Since the osmotic pressure is p = RTv,

(84)

where v is the number of 'dissolved gram-atoms' per unit volume,

K

dx

(85)

The solvent of viscosity k exerts a resistance to the motion of a particle with velocity v. In the case of spherical suspended particles of radius P, the force becomes

Albert Einstein and Marian von Smoluchowski ...

173

proportional to the radius and the viscosity, and K

QnkPN

-D»

(86)

dx '

with N 6irkP' where vv is the amount of the substance in gram-molecules transported through the cross section of the cylinder, and D is the diffusion coefficient which depends on the viscosity k and the radius P of the molecules. According to the molecular theory of heat, the molecules of a fluid are in irregular motion by means of which a uniform concentration is gradually achieved. If one knows the position of all the solute molecules at times t and t + r, where r is so small that the concentration has not changed appreciably, then one can say that the molecules have changed (in time r ) their original positions by displacements Ai, A 2 , . . . of arbitrary sign, which are statistically independent of each other, provided the solution is dilute. If all molecules changed their positions by A, half by + A and the other half by - A , then \y\A particles would pass the cross section from left to right and 5v^A from right to left, where v\ and 1/2 are the concentrations of the left- and right-hand side respectively. Thus the total diffusion from the left to the right side is 1.,

2

A

l.2l'l-l/2

.

^ - ^ = 2

A

1A2^I/

- 5 - = - 2

A

S -

/o»,\ (87)

Equation (87) follows from the definition of the concentration difference. Hence the total amount of the substance (expressed in gram-molecules) passing through the cross section in time r is given by Eq. (87) or by D=\—

or

A = V2Dy/r.

(88)

2 T

A is more accurately to be taken as the square root of the mean square translation of the solute molecule in time r. It is not proportional to the time because these translations (average lengths of paths) do not add. With Eq. (86) for D, the radius of the suspended particles may be calculated from the diffusion coefficient. For a sugar solution one obtains reasonable radii of the molecules, in agreement with the value determined from the volume of solid sugar and Avogadro's number N. 5.11. Another

determination

of the elementary

quantum

In a paper entitled 'On the Range of Validity of the Laws Concerning Thermodynamic Equilibrium and... the Possibility of a New Determination of the Elementary Quanta,' Einstein proposed another experiment from which one could derive molecular dimensions. 68 He began by noting that 'The state of a physical system can be determined in the thermodynamic sense by parameters A, fx, etc. (e.g., the reading of a thermometer, the length or volume of a body, the quantity of substance of a certain type in a certain phase). If the system does not interact with others,

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which we assume, then, according to thermodynamics, equilibrium for those values Ao, Mo, etc. of the parameters exists which are determined by the fact that the entropy of the system is a maximum. According to the molecular theory of heat, this is not exactly but only approximately correct. The parameter A, in this theory, does not possess a constant value even at temperature equilibrium, but fluctuates irregularly; however, it deviates significantly from Ao only rarely.' 69 Einstein continued: 'The theoretical study of the statistical law to which these fluctuations are subject, at first sight, requires definite assumptions concerning the molecular conception to be applied. This, however, is not the case. It is sufficient to apply the well-known Boltzmann relation which combines the entropy S with the statistical probability of a state, i.e., S = j^ log W.,7° Einstein considered a state described by the parameter A, very close to its equilibrium value Ao,

e = A - Ao ,

-J- < 1.

(89)

Ao The state A can be reached from Ao by a reversible path. For systems with constant energy E, the work A necessary to reach the state A is compensated by a loss of heat Q at constant temperature,

I dE- [TdS = - JTdS =-T(S

RT W - So) = - — l o g - .

(90)

From Eq. (90), the probability of the state with the parameter value A is given by W = W0e-&A,

(91)

or, more accurately, for a state between A and A + dX, dW = const.e - T&- A dA.

(92)

A = ae2,

(93)

dW = c o n s t e * 0 " 2 d s ,

(92a)

Assuming that then and the average work required to take the equilibrium state Ao to the state Ao + V s1 is A = \% = i o - 1 6 r .

(94)

Einstein applied this result to the discharge of a condenser of capacity C. If the average voltage, taking into consideration the fluctuations, is designated by fp2, then A=\C?.

(95)

Albert Einstein

and Marian von Smoluchowski

...

175

In the case of an air condenser consisting of 30 plates of 100 cm 2 area each, with the distance of 1 mm between the plates, the capacity is 5 x 10 3 c.g.s. units, and the fluctuation voltage at ordinary temperature, from Eqs. (94) and (95), is given by _ :10~ 1 6 volts.

(96)

By rotating the plates, the capacity can be changed to 10 c.g.s. units. Then the potential difference obtained from the change of capacity by a factor of 500 is of the order of V ^ = 10- 6 x ^ 9 . = 5 x i o - 4 l t (97) v 10 ' Einstein then proposed the following experiment. A tuning condenser of a maximum capacity of 5,000 c.g.s. units should be discharged. After its capacity is reduced to, say, 10 c.g.s. the molecular potential difference, Eq. (97), should be measurable. He concluded: 'Thereby a phenomenon related to the Brownian motion will be found in the domain of electricity that could be used for the determination of N [Avogadro's number].' 71 6. Smoluchowski's Theory of Brownian Motion Marian von Smoluchowski had been interested in the problem of Brownian motion at least since 1900.39 In Section 2 we discussed Smoluchowski's work of 1904 on density fluctuations. 9 He reported his work on Brownian motion to the Cracow Academy on 9 July 1906, and submitted his paper 'The Kinetic Theory of Brownian Molecular Motion and the Suspensions' in September 1906.72 Thus it appeared over one year after Einstein's theory of Brownian motion. Concerning Smoluchowski's work, Arnold Sommerfeld remarked: 'Smoluchowski solved the riddles of Brownian molecular motion in 1906, independently of and simultaneously with Einstein. That Boltzmann himself, who had after all laid the foundation for all that had not drawn this most obvious consequence, is rather astonishing, as Einstein remarked in a discussion. It is tragic that the proof of atomistic theory, which removed all doubts and silenced its enemies, did not become ripe before the year of Boltzmann's death. Each of the two derivations, by Smoluchowski and Einstein, has its particular advantages. Smoluchowski lets us cast a deeper glance into the mechanism of impacts; the dependence on temperature, the size of the particles, and viscosity, is derived correctly, just the numerical coefficient remains uncertain because of the difficulties of averaging. Einstein obtains the final result in one bold action, without losing time with the details of the process. Einstein's numerical coefficient, which deviates by a factor 27/64 from Smoluchowski's, has been proved beyond doubt. The whole theory has been confirmed since then, especially by Perrin, most exactly and completely. Smoluchowski returned to this subject several times [Ref. 73], as well with respect to the practical aspects because of the measurements of Ehrenhaft, Millikan and Brillouin, the range of validity of Stokes' law, and the organization of the subject within the general principles of statistics.' 74

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The Golden Age of Theoretical Physics

In his first memoir on Brownian motion, Smoluchowski at once took note of Einstein's work. He noted: 'The highly controversial question about the nature of the movements, discovered by the botanist Robert Brown in 1827, which occur in microscopically small particles suspended in fluids, has been brought to attention again recently by two theoretical papers of Einstein. 44 ' 46 The conclusions reported in them agree completely with certain results which I had obtained several years ago by pursuing an entirely different line of thought, and which I have considered since then as an important argument for the kinetic [-theoretic] nature of this phenomenon. Although I have not yet been able to test experimentally the consequences of these concepts, as I intended to do originally, I have finally decided to publish them now in order to contribute to the clarification of views concerning this interesting topic, especially since my method seems to me to be more direct and simpler, and therefore more convincing, than Einstein's.' 75 6.1. The experimental

background

of Smoluchowski's

work

Since he had himself not had the opportunity of experimenting extensively, Smoluchowski reviewed critically the empirical facts about Brownian motion. He noted the following principal features: (a) The Brownian motion is a general phenomenon, and it occurs with all kinds of substances, including gas bubbles. (b) The velocity of motion increases with the decreasing size of the suspended particles. For instance, the experiments of Exner gave for particles of diameter 10~ 4 cm velocities of the order of 10~ 4 cm/sec. 37 (c) A higher viscosity of the fluid damps Brownian motion. (d) The movements never cease for all kinds of suspensions, and do not change character with time. For instance, Cantoni had observed this in one probe for one year. 76 (e) The movements do not depend on external parameters that do not affect the viscosity of the fluid. These facts helped to eliminate some of the earlier 'explanations' like the effect of convection currents and radiometer forces,77 as well as the hypothesis of stimulation by repulsive forces78 or electrical forces.79 In the latter case the particles should stick together. Also the capillary forces80 could not be effective in all cases. Thus the kinetic motion of molecules remained as the only source of Brownian motion. Earlier attempts to explain Brownian motion in accordance with the kinetic theory had already been made by Wiener 28 and Cantoni. 29,76 But since the observable suspended particles are rather large, Nageli expected that their velocities would remain invisible because the molecules of the fluid move irregularly and the impact momentum transfers should cancel out on the average. 30 However, as Smoluchowski remarked correctly: 'This is the same error in thinking as if a gambler believed that he could never lose a larger amount than what the stake for a single throw would be.' 8 1 He decided to pursue this analogy further.

Albert Einstein

and Marian von Smoluchowski

...

177

If one assumes equal probability for a winning or a losing throw, the mean deviation from the average value zero for many throws, say n, is given by

f.

m

The same fact applies to the momentum transfer to a particle suspended in a liquid which is hit by 10 20 molecules (in water) per second. It could obtain velocities of 102 to 104 cm/sec. However, it does not happen so because as the velocities increase it becomes less probable that the impacts would further contribute to them. Moreover, according to the Maxwell-Boltzmann kinetic theory, the average kinetic energy would obey the equipartition theorem, and ^particle — ^-molecule

V •'"part.

hence a typical velocity will be of the order of 0.4 cm/sec. On the other hand, F. Exner had observed average velocities of 3 x 1 0 - 4 cm/sec. 37 This is due to the fact that a particle moves along a zig-zag path very often in a second, and one just observes a macroscopic path with a much lower velocity. 6.2. The path of a suspended

particle

In order to calculate the motion of a suspended particle Smoluchowski assumed, as a first approximation, that the particle has a constant average magnitude C but an arbitrary direction of the velocity. The suspended particles (mass M) are scattered by molecules (mass m) and are deflected by an average angle e given by 3m c 3C £= - ——= .

/.,„„\ (100)

Smoluchowski now considered two cases: (i) The ratio between the radius, R, of the particles and the mean free path, A, of the molecules is very small, i.e., | « i ,

(ioi)

in which case the suspended particle does not influence the motions of the molecules; (ii) the radius of the suspended particles is very large, |

» 1.

(102)

In the first case one considers n scattering processes, in each of which the suspended particle is deflected a bit form its original direction. Then the mean free path of the particle, A/A 2 , can be determined from the average path I of the particle between two impacts and the average angle e. It is given by

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The Golden Age of Theoretical

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where 1 — 6 = cose for n6 -C 1, or lO.rt

n5 » 1.

(104)

An interpolating formula is valid for the intermediate ranges, nd ~ 1. In the range of validity of Eq. (104), i.e., for many impacts n, we substitute S = j £ 2 and obtain for the path A in one second (I = ^ ) , A = f ^ f .

(105)

Thus a path A of a suspended particle does not depend at all on its mass, because the larger its mass the smaller its velocity and the longer it remains undeflected. The result, as expressed in Eqs. (103), (104) or (105), would not be changed very much if one replaces the (average) velocity C by the actual velocity distribution. In the second case of a large suspended particle in a dense fluid with small mean free path of molecules, one cannot treat the consecutive scattering processes as being statistically independent. Since the neighboring fluid molecules block the change of direction of the particles, A is increased compared to its value in Eq. (105). The actual calculation being rather tedious, Smoluchowski tried as an appoximation the exponential damping of the initial velocity of motion C of the particle in a viscous fluid, given by V = Ce-t/T , (106) where r is given by the ratio of the mass M to the coefficient of resistance, S, which the particle encounters in the fluid,

r=f.

(107)

Since the kinetic energy of the particle remains constant on the average, it obtains a velocity perpendicular to its original motion so that the average velocity in the direction of motion also remains unchanged. The motion of the suspended particle with mass M resembles that of a molecule with mean free path A = CT, where A is the length of a straight path. From the theory of diffusion Smoluchowski found that the straight path A of this 'equivalent molecule' in one second can be put equal to ^ nr-

^

/2M

2m

A = CV27 = C y ' — = c ^ / — ,

.„„„.

(108)

where c and m refer to the molecules of the fluid and S to the resistance which a suspended particle experiences in the fluid, all expressed in appropriate units in order to account for the dimensions in Eq. (108). In order to test to the validity of Eq. (108), one considers particles of very small radii, i.e., case (i). Smoluchowski did not use Stokes' formula for the resistance coefficient 5, Eq. (40), but calculated it directly from the number of impacts between the suspended particles and the molecules, obtaining S = —R2pc o

= -ran, o

(109)

Albert Einstein and Marian von Smoluchowski ...

179

with n = NR2nc, where p is the density of the fluid molecules, m their mass, and c their velocity. Hence, from Eq. (108), A

= c VVnn'

(H0)

which, up to a factor ^Wi is identical with Eq. (105). For large radii, where Stokes' law makes sense, Smoluchowski took Eq. (105) as the correct expression, and obtained

A = _L C /

2rn =

_1_£\M

(in)

Smoluchowski noted that 'this result, obtained in a completely different manner, is almost identical with the one derived by Einstein [Eq. (52)]. The only difference is the numerical factor which, compared to Einstein's, is smaller by t / g j . ' 8 2 Actually, Einstein had not treated the case of particles which are small compared to the mean free path of the fluid molecules, to which Stokes' law does not apply. 'Without discussing here the very reasonable method followed by Einstein,' Smoluchowski remarked, 'I would like to point out that it rests on an indirect consideration which does not always seem to be convincing — e.g., the application of the laws of osmotic pressure to these particles and the calculation of their diffusion velocity, or the application pf Boltzmann's theorem (on the statistical distribution of the states of systems which are subject to forces due to potentials) to the frictional resistance experienced by a particle [of mass] Af.'83 'In any case,' continued Smoluchowski, 'the agreement between these two different methods [his own and Einstein's], which provide a direct perception of the mechanism of these phenomena, is a satisfactory result for both investigations. The difference between the numerical factors can be explained on account of the different simplifying assumptions employed (e.g., the constant C), and really plays no role.' 83 6.3. Application

to gases and

liquids

Smoluchowski derived the consequences from Eqs. (104) and (111). In the case of gases, he found

for large ^ , and

for small j j , with k = A c ^ p . For air at normal pressure, with particles of radius R = 5 x 10~ 5 cm, the average path in one second is given by A = 1.4 x 1 0 - 3 cm. For particles which are slightly smaller than these, gravitation is 'balanced' by Brownian motion. 84 In liquids the mean free path A is so small that the radii of the suspended particles are always large in comparison with it; thus Stokes' law applies. However,

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The Golden Age of Theoretical Physics

capillary forces and deviations from a spherical shape of the particles have to be taken into account in this case. Smoluchowski concluded that the following features of Brownian motion could be explained by his theory: (i) The motion does not depend on the particle mass M. (ii) The velocity of the particles increases with decreasing particle radius, (iii) The temperature dependence, observed by F. Exner, 37 is well reproduced, (iv) In high viscosity fluids only a small motion exists. He noted, however, that 'A thorough verification of the theory would, of course, require more extensive and accurate measurements; but as far as we know now, we are justified in considering Brownian motion as an obvious proof of our molecular-kinetic hypotheses.' 85 6.4. The effect of external

forces

Several years later, in 1913, Smoluchowski extended his theory to include the effect of external forces on the suspended particles explicitly.86 As an example, he treated the case of a force free distribution of a displacement between x and x + dx (xo being the mean position) by the probability W0(x)dx = —L=e-(*-Xof'ADtdx,

(114)

2VnDt which, when the forces exerted by the walls are included, is given by W(x)dx = _ L = [ e - ( * - * ° ) 2 / 4 D t

2

+ e{x+X0)

/iDt]dx

( n 5 )

Thus the force tends to make the differences X — XQ smaller. After a very long time one obtains the old equation (114) without any forces. If the gravitational force is considered as the external field, the statistical displacement and the displacement due to the gravitational force have to be included. Another case of interest is that of an elastic force, which is proportional to the displacement, and Smoluchowski dealt with it. 87 Let the restoring force be X = —a(x — x0),

(116)

and let the particle be moved back toward xo, in a short time r, by an amount XQ(3T. Then, 2 W(x)dx = i _ (117) e-[x-xo(i-0T)] /4*Tdx For a very long time t = nr (n very large) W(x)dx =

1

2V^D^ /WT[1

l

•

2

y/\ + (1 - /3r2) + . . . + ( ! _ / f r ) " -

1

• e -[*-*o(i-/3i-r] 2

+ (1 - (3T)2 + ••• + ( ! - /3T) 2 "- 1 ]

^C-.TT,

(118)

Albert Einstein

and Marian von Smoluchowski

...

181

with the mean square fluctuation £2 = l i m ^ = ^ . (119) t-*oo 0 Smoluchowski's theory was used extensively in various applications, such as the Brownian motion of a galvanometer mirror. 88 The experimentalists found Smoluchowski's treatment very appealing because he took into account the experimental situation carefully; this was probably because he was himself a good experimentalist. 7. The Proofs of Molecular Reality In one of the earliest experimental tests of Einstein's theory of Brownian motion, Teodor Svedberg studied the Brownian motion of suspended particles which floated by in the view of his microscope. 62 Another early experiment was carried out by M. Seddig, who determined the temperature dependence of A2 and found approximate agreement with the formulas of Einstein and Smoluchowski.89 Felix Ehrenhaft studied the Brownian motion in gases, finding agreement with the prediction of Smoluchowski when the particle radius was small with respect to the mean free path of the gas molecules.90 Maurice de Broglie, on the other hand, measured the mobility of charged particles from their Brownian motion. 91 The elementary charge of the particles could then be calculated and from it, with Faraday's constant, the Avogadro number N. Other experiments were carried out by E. Weiss92 and H. Fletcher. 93 Fletcher found that the pressure dependence of A2 in gases could be reproduced in accordance with the Einstein-Smoluchowski formulas. The most careful experiments which gave support to the molecular hypothesis were performed, however, by Jean Perrin (1870-1942) and his co-workers. 94 At the first Solvay Conference on Physics in Brussels in 1911, Jean Perrin reported on 'the proofs of molecular reality' and gave a detailed account of the experimental evidence in favor of the molecular hypothesis. 95 7.1. Jean Perrin

and the molecular

dimensions

At the first Solvay Conference in 1911, Jean Perrin declared: 'As long as one concentrated on operating with invisible molecules, it was still possible to believe in the strict validity of Carnot's principle, since one could still deny the real existence of molecules. Today, however, this narrow formulation is at variance with a phenomenon which is directly observable.' 96 According to Clausius and Maxwell, one obtains for the surface of N molecules in kinetic theory 4nNR2 = j^= , (120) where L is the average free path, v the volume of N particles, and R the radius of the molecules, assumed to be spherical. The left-hand side of Eq. (120) can be determined with the help of the real gas equation of van der Waals, l*R3N

= \ ,

(121)

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where | represents the volume of N gas molecules in closest packing. From these two relations one can obtain the diameter 2R of the gas molecules and the Avogadro number N. For argon one obtains N = 6.2 x 10 23 molecules, and for chloroform N = 2.0 x 10 23 molecules. Argon is clearly closer to the assumptions of the theory. As a first approximation, the kinetic theory of gases should also apply to emulsions. Perrin tested this empirically by checking the barometric formula of Loschmidt for colloidal particles. The weight and dimensions of the particles were determined by means of several independent methods and the frequency of the particles in a cylinder could be counted as a function of their height above the bottom of the vessel. The Avogadro number given by RT log -?.

N

-WiE^)

(122)

(where g is the acceleration due to gravity, h the height, V the volume of the particles, and A — <S the relative density of the particles) was found experimentally to lie between 6.0 x 10 23 and 7.0 x 10 23 . Perrin remarked: 'On account of this result one can hardly deny the objective reality of molecules. At the same time their motion becomes visible. The Brownian motion is a good representation of it or, more accurately, it is itself a molecular motion.' 97

7.2. Perrin and Brownian

motion

Jean Perrin had taken a keen interest in the atomic and molecular hypothesis since 1901, and before the decade was over he had become one of the chief spokesmen of atomism. 98 He was strongly opposed to the 'energetic' point of view of Ostwald and Mach." In a lecture at the Societe de Philosophic in 1906, Perrin took note of the criticism of molecular hypotheses by declaring: 'Briefly, we are going to show that sufficiently careful observation reveals that at every instant, in a mass of fluid, there is an irregular spontaneous agitation which cannot be reconciled with Carnot's principle except just on the condition of admitting that his principle has the probabilistic character suggested to us by molecular hypotheses.' 100 The demonstration which Perrin used was the concept of osmotic pressure, which he explained in the same way as Einstein had done in his 1905 paper. 44 Again, in an article written about the same time, Perrin gave a qualitative discussion of Brownian motion and cited it as an evidence of molecular motion. 101 On 9 March 1908, Paul Langevin presented an account of Einstein's theory of Brownian motion to the Academie des Sciences in Paris, 102 from which Perrin learned about the 'current status' of the theory. His goal immediately became 'to apply to it the test of experiment.' 103 In the first experiment, Perrin determined Avogadro's number from the barometric formula (see Section 7.1). For the determination of the mass of the colloidal particles, he used Stokes' law. One week after Perrin's first report, Victor Henri reported on a cinematographic study of Brownian motion, which seemed to contradict Einstein's displacement formula, and Henri suggested that the application of Stokes' law might be responsible for the deviation. 104 A controversy thus arose about the

Albert Einstein and Marian von Smoluchowski .,.

183

validity of Stokes' law for suspended particles of small size. Perrin checked the law and found it to be valid even for particles of diameter of the order of 0.1/i. 105 Now Perrin turned his attention to the Brownian motion itself.106 In a note presented on 5 October 1908, l o r he reviewed four methods of determining Avogadro's number: (i) from the kinetic theory (see Section 7.1); from the determination of the elementary charge; (iii) from Planck's theory of heat radiation; (iv) from Brownian motion itself. The value of N was now 7.1 x 10 23 and the electronic charge was 4.1 x 1 0 - 1 0 esu. During late 1908 and early 1909, Perrin and his co-workers checked Einstein's theory in elaborate detail. 108 In his memoir on 'The Brownian Motion and Molecular Reality,' Perrin declared that ' . . . the measurements that I am going to summarize leave no doubt of the formula proposed by Einstein.' 109 In his report to the first Solvay Conference on Physics in 1911, Jean Perrin summarized the latest results in detail. 95 The principal steps which had been applied by him and his co-workers were: (1) The law of errors could be applied to Brownian motion; (2) Avogadro's number could be estimated in a variety of ways. Concerning the first point, he had checked the proportionality of the mean square root of the displacement with the square root of time. 'Although this result is beyond any doubt,' Perrin said, 'I had it checked quantitatively. From a large number of measurements it really follows that the mean square of displacements of a particle in two minutes is four times larger than in half a minute.' 110 Perrin did not consider the previous experiments on Brownian motion really conclusive. For instance, Svedberg's measurements 62 had disagreed quantitatively with Einstein, and Perrin noted: 'In the studies of Svedberg on Brownian motion the lengths regarded as displacements are six to seven times too large, which cannot be considered, if regarded as having been correctly defined, as indicating any progress, especially if one takes into account Smoluchowski's contribution to the subject. Svedberg believed, and it is much more serious, that Brownian motion would become oscillatory for ultramicroscopic particles.' 111 Perrin remarked further: 'When I succeeded in obtaining particles with an accurately measurable radius, I was in a position to perform absolute measurements and check Einstein's theory exactly and completely. Above all, I studied whether the quotient [jr3^k] is independent of the emulsion used and identical with the value of A^ already found.' 112 And, as a footnote to this remark he added: 'This seemed doubtful at first because a cinematographic record prepared by V. Henri seemed to be very unfavorable to Einstein's theory. I was at that time surprised how easily many physicists, who otherwise believed completely in the kinetic theory, were led to assume that unjustified hypotheses were contained in Einstein's calculation. This demonstrated to me how little trust we accord to theories which are instruments of discovery rather than actual demonstrations.' 11 Jean Perrin and his co-workers [Chaudesaigues, Bjerrum, Dabrowski] studied emulsions in which the masses varied up to a factor of 1.5 x 10 4 and the viscosity of the fluids up to a factor of 125. 'Nevertheless, the quotient remained close to 70 for different kinds of fluids and particles, just as in the case of the

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The Golden Age of Theoretical Physics

distribution with height [barometric formula],' reported Perrin. 'This remarkable agreement proves the rigorous validity of Einstein's formula and brilliantly confirms the molecular theory.' 113 The diffusion coefficient D is given by Einstein's Eq. (43). In his doctoral thesis, Einstein had obtained another relation between the radius P and the Avogadro number N:45 N spheres of total volume v contribute to the viscosity of a fluid, and change it from k to k', k' = k(l \

+ 2.5^)

,

vJ

(123)

3

Actually, in Einstein's thesis the numerical factor 1, instead of 2.5, and appeared, and Bancelin, a collaborator of Perrin's, found the correction. 114 'We informed Einstein about this deviation; and then he discovered an error in his calculation and found the correct result [see Ref. 45], which now agrees well with Bancelin's measurements,' Perrin reported. 115 This established the validity of the application of Stokes' law. Another test of Stokes' law was made by Leon Brillouin in Perrin's laboratory. He observed diffusion directly and determined the coefficient D. The calculated N according to the Einstein-Stokes formulas was in excellent agreement with other determinations. 116 In the preceding discussion of Perrin's experimental verification of the theory of Brownian motion, mostly Einstein's name has occurred in connection with the theory. In his report to the first Solvay Conference, 95 Perrin referred time and again to the 'Einstein-Smoluchowski' theory of Brownian motion. 117 7.3. Correction

to Einstein's

displacement

formula

The experiments had demonstrated a satisfactory quantitative agreement with the theory developed by Einstein and Smoluchowski. Nevertheless some theoretical criticism of Einstein's theory was raised in the years following 1905. We have already mentioned one point, i.e., the question of the validity of the use of Stokes' law in Einstein's theory. This problem was first noted and treated correctly by Smoluchowski.72 In the experiments on Brownian motion in fluids, Stokes' law was always correctly applied. Another question concerned Einstein's Eq. (52), which related the average displacement of the Brownian particle to the time and the diffusion coefficient. The essential point used by Einstein in his derivation was the small, fictitious time interval r. As Fiirth noted, 'The introduction of this time interval r forms a weak point in Einstein's argument, since it is not previously established that such a time interval can be assumed at all. for it might well be the case that, in the observed interval of time, there was a definite dependence of the motion of the particle on the initial state.' 1 1 8 In order to derive Einstein's formula, one often refers to the Langevin equation which relates the change of velocity of the particle to a random and rapidly

Albert Einstein

and Marian von Smoluchowski

...

185

changing force mP (with average P = 0) and a damping mechanism (expressed by the constant /?), £^ = - / 3 u + P ( i ) .

(124)

This equation leads to Einstein's result immediately if the particle motions at subsequent times are assumed to be really independent. But for very small time intervals, as Einstein himself noted, 46 a correlation exists, given by P(h)P(t2)

=
(125)

In order to integrate Eq. (124), with corrections of the form (125), one proceeds, as was done by Ornstein, 119 and calculates the value of A2 for a given average UQ. Its solution is u = uoe-V + e-V

f e^P(£)d£, Jo

(126)

and the displacement A is given by A = x - xo = ^ ( 1 - e " * ) + f e - " " f e « • P ( £ R . P Jo Jo

(127)

By squaring and averaging, and evaluating the double integrals, one obtains VU°

= ^

+^ d "

e

^ )

2

+ ^ 3 ( " 3 + 4 e " ^ - e^),

(128)

withr1 = 2 a i . Again, by averaging over UQ, and taking UQ — ^ , one obtains F=^j(/ft-l-e-'»).

(129)

R. Fiirth, 120 independently, obtained the result of Eq. (129) which applies for all times, in particular for large t when the Einstein term dominates. For very short times, the motion becomes linear but not reversible, because of the introduction of/3. Equation (129) may be generalized further by including other forces in the Langevin equation (128). The distribution function / then does not satisfy Einstein's simple diffusion equation (48), but the more general Fokker-Planck equation, 121

7.4. The conceptual

impact

Wilhelm Ostwald had been among the most distinguished and outspoken critics of the atomic hypothesis, even as late as 1906. However, by 1909 his opinion was completely changed and he noted: 'I have convinced myself that we have recently come into possession of experimental proof of the discrete grainy nature of matter, for

186

The Golden Age of Theoretical

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which the atomic hypothesis had vainly sought for centuries, even millennia.' 122 The experimental work of J.J. Thomson on 'the isolation and counting of gas ions,' on the one hand, and of J. Perrin concerning 'the agreement of Brownian movements with the predictions of the kinetic hypothesis,' on the other, 'now justifies even the most cautious scientist in speaking of the experimental proof of the atomic nature of space-filling matter.' 1 2 2 Among the leading scientists, Ernst Mach was still not convinced. But the course of the events and the rapid progress of atomic physics made his opinions in this regard irrelevant. 8. The Fluctuations and Opalescence The scattering of light by small particles had been studied experimentally by John Tyndall, 123 and was explained by Lord Rayleigh afterwards. 124 Assuming the small particles to be spherical, Rayleigh obtained the result that the total scattered intensity is proportional to the inverse fourth power of the wavelength of the scattered radiation, and also that it increases with the sixth power of the radius of the scattering particles. He also concluded that no scattered intensity is observed in the direction of vibration of the incident wave. Rayleigh's theory explained the blue color of the sky, and further refinements made direct comparison with experimental results possible. 125 Thus the phenomenon of the scattering of light by particles suspended in liquids and gases, or by gas molecules themselves, could be explained. However, one still did not know whether fluids with no suspended particles also showed a Tyndall effect. This question was settled experimentally by M. Le Blanc and W. Kongro in 1913, who found that all fluids, even as pure as possible, show a scattering of light. 126 Several years earlier, however, in December 1907, Smoluchowski proposed the theory of this phenomenon, which was further improved by Einstein in 1910. and was tested experimentally in the early 1920s. 8.1. Smoluchowski

and the theory of critical

opalescence

In December 1906, Marian von Smoluchowski presented a paper to the Cracow Academy on the 'Molecular Kinetic Theory of the Opalescence of Gases at the Critical Point, and Some Related Phenomena.' 127 Smoluchowski first referred to the success which the recent theory of Brownian motion had obtained; he noted the experimental studies of Svedberg and Ehrenhaft and, in particular, the early studies of Zsigmondy on colloids. Zsigmondy, in his memoir on 'Zur Erkenntnis der KoUoide,1 had reported that colloidal gold particles of diameter 6 x 1 0 - 7 cm could move about 10/i in 1/6 to 1/8 of a second. Smoluchowski estimated that, according to his theory, particles of diameter 1 0 - 4 cm would be displaced around 8.8/i. He then returned to the ideas on fluctuations which he had developed in his 1904 contribution to the Boltzmann-Festschrift.9,11 By applying general principles Smoluchowski concluded that the fluctuations would become more apparent at the critical point, and in order to observe the effect of density fluctuations he proposed to examine the phenomena of opalescence.

Albert Einstein and Marian von Smoluchowski

...

187

8.1.1. Density fluctuations and the slope of the isotherm The density fluctuations are given by Eq. (8). In order to obtain a more general treatment, Smoluchowski considered a cylinder of volume VQ filled with q grams of gas and connected to a reservoir of pressure po- The probability of the pressure fluctuation is given by the Boltzmann-Einstein formula, W{v)dv = be'^dv

,

(131)

where <j> = —q Jv (p — po)dv. Since the amount of the gas remains constant, the volume will change to v because of the change of pressure. The expression for h is given by

<132»

* = p5f and the number of molecules enclosed in the volume is u= ^ . P

(133)

Equation (131) will be replaced by W{v)dv = be+^

Op-po)dvdv,

(134)

where the integral in the exponent indicates the compression energy or the area between the coordinates po and vo and the isotherm To in the py-diagram. Now, for small deviations from the normal state VQ, the expansion gives

/

-4(l)0<

w

and from Eq. (134), W{6)dS = be-ad

,

(136)

with a

= -57H&(&)o = 2 H 7 ^ > a n d P = -hl% = the compressibility. The average deviation from the normal density is found to be -s=

1 y/om

/2*W V

VK

Thus the accidental fluctuations are smaller the larger is the number of molecules and the steeper the isotherm g£ rises. At | 2 = o, the gas crosses a boundary of stability.

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8.1.2. Specific volume at the critical point The most probable value for the specific volume v is given by the minimum of the exponent in Eq. (134), and the least probable by its maximum. The extrema are given by

ny-

• p0)dv

=0

or

(I38)

p = po •

For the van der Waals equation, in addition to Eq. (138), at the critical point ^

- ^

= 0

(139)

W(S)dd = be-rSi,

(140)

and one obtains with 7 = - - ^ . ^ ( ^ )

(141)

7

U

RTclit. 4! WJcHt.

^

The average deviation from the normal density becomes

" V7 r ( i ) " V7

V^

(142)

with (|^?)crit. = -g^s — - x i l r " ' t l from *^ e v a n ^ e r Waals equation. Equation (142) is valid for any gas, and depends only on the number v of the molecules. Smoluchowski illustrated the difference between the normal and the critical case with the following example: a cube of linear dimension 0.6^i, under normal conditions, would contain 9 x 106 gas molecules, and according to Eq. (8) a density fluctuation S = 2.7 x 10~ 4 would occur. If ethyl ether fills the same volume at the critical point, then it has 5.4 x 108 molecules and, according to Eq. (142), 6 = 7.2 x 1(T 3 , which is of the order of 1%. 8.1.3. Observation of critical opalescence According to Rayleigh's theory, in a medium of volume V containing TV small suspended particles per unit volume the intensity of the incident light of wavelength A is decreased, and the effective absorption coefficient is given by h=^NV>^,

(143)

where e is the percentage by which the refractive index of the suspended particles is larger than that of the medium. Whereas e appears linearly in the refractive index of the medium, and therefore drops out on averaging, its effect on the absorption coefficient, Eq. (143), persists and increases at the critical point. The phenomenon of marked opalescence at the critical point was first observed by M. Altschul 128 and K. v. Wesendock. 129 M. Travers and F. Usher 130 and S. Young 131 reported further results. Smoluchowski considered F. Donnan's hypothesis, 132 that

Albert Einstein

and Marian von Smoluchowski

...

189

the formation of droplets at the critical point might explain opalescence, rather improbable. Similar opalescence phenomena were observed in solutions of certain pairs of fluids in the neighborhood of the critical point. Below the critical temperature they coexist as two different phases, but above Tc the surface disappears. When the system is cooled down from values above Tc, a blue light is observed and the phases become separated. 133 Even in 1903, D. Konowalow proposed to explain this phenomenon as being due to dust particles suspended in the liquid. 134 8.2. The Boltzmann

principle

and Einstein's

theory of

opalescence

Einstein 'completed' Smoluchowski's theory, in particular by calculating the amount of light scattered through opalescence. 135 From Boltzmann's definition of the entropy one finds that observable deviations from the state of equilibrium do occur with some probability when the work required to change the equilibrium state into the new state is of the order of the kinetic energy of a monatomic gas molecule. If this work is sufficient to produce in volumes of the order of (wavelength) 3 a considerable deviation from the mean density, then opalescence should occur. Boltzmann's principle is given by the equation R S = — log W = const.,

(144)

where R is the gas constant, N the number of molecules in a gram-molecule, S the entropy, and W the probability of the state which has entropy S. The probability in Boltzmann's principle is usually equated to the number of possible complexions in which one can realize a state in the sense of the molecular theory. Einstein pointed out that 'In order to calculate W one needs a complete theory (e.g., the molecular-mechanical theory) of the system in question. Therefore it is questionable whether the Boltzmann principle by itself makes any sense without a complete molecular-mechanical theory or a theory which represents the elementary process completely. Considered from the phenomenological viewpoint alone, i.e., without a fundamental theory, Eq. (144) appears to be without content. 'Boltzmann's principle, however, obtains a meaning independent of any fundamental theory if one assumes and generalizes from molecular-kinetics the theorem that the irreversibility of physical processes is only an apparent one.' 136 If one considers a closed system which, in the course of time, assumes all states consistent with a given energy, then the probability of a state is denned by the ratio of the time spent in this state and the ('enormously long') time for which the system is observed. Following Boltzmann, one may attribute apparent irreversibility to the fact that the system assumes the states of greater probability whenever it finds itself in a state of relatively small probability. From this definition of probability the Boltzmann principle can be derived because entropy and probability never decrease. Given any function of the entropy which never decreases (i.e., one which has a positive derivative with respect to the entropy) and the fact that entropies add while the probabilities are multiplied, one obtains the desired result.

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The Golden Age of Theoretical Physics

It should be added, however, that one cannot talk about the probability of a state but only about the probability of a domain in phase space. For an infinitesimal domain this probability should become infinitely small. Since, however, logW is multiplied by the very small factor j£, one can choose the domain Gw in phase space in such a way that -^ log ^¥- (where G is an observable region of phase space) become very small; Eq. (144) would then be sufficiently exact. 8.2.1. Deviations from thermodynamic

equilibrium

From Eq. (144), it follows that W = const. e%s .

(145)

At equilibrium, the entropy SQ is a maximum and W(So) = e%s° » 1,

(146)

and, in general, for a state with entropy S, W{S) = e%(s~s°K

(147)

The probability that the state variables A i , . . . , An lie between A and A i + d A i , . . . , An and An + dXn, is then given by dW = c # < s - 5 o > / ( A i , . . . , An)dAi • • • d\n .

(148)

The parameters A i , . . . , An do not determine the system completely because the energy, depending on them, has already been fixed. / is a function of A i , . . . , A n , and is of the order of 1. One can expand S and / near the maximum and obtain

(149)

s = 5 o2- ^ V A ' + and

/ = /o + £ v ( £ ) + - with A^x' = Ao/i - An . The double sum in the expansion for S in Eq. (149) is positive because of the entropy maximum. Transforming the symmetric matrix S^ to the diagonal form, one finds from Eq. (149) dW = const. e ~ & £ S " A * " + " (f0 + V* ^ \

• A„» • dXx

^-^ a\vii

d\n„ )

(150)

J

Since / is a slowly vanishing function of the new variable differences Xvni the exponential in Eq. (150) depends strongly on Xv» because of the large factor ^ , and one can replace / by /o close to the equilibrium, obtaining dW = const. e - # ( s - 5 o > d A i • • • dXn .

(151)

Albert Einstein

and Marian von Smoluchowski

... 191

The states in the neighborhood of the equilibrium can be reached by a reversible path from the equilibrium state. At constant To, the change of energy is compensated by a loss of heat, 0 = dU = dA + T0dS or S-S0 = ±-A, (152) Jo where A is the work done in taking the system from one state of equilibrium to another. From Eqs. (150), (151) and (152), one obtains dW = const, e - ^ ^ ^ ! • • • d\n = const. e~^5^au^dXi---dXn,

(153)

indicating that A can be developed in a series A, having the form A + \ X] ^i/A^ for av > 0. Hence the average work Av associated with the fluctuation parameter is 1

T>rp

^ = 2°^ = W8.2.2. Density

(154)

fluctuations

Let po denote the average density of a homogeneous substance, such as the average density of a component of a binary mixture. On account of the irregularity of thermal motion, the density p at a point in the fluid will be different from po. Einstein considered the fluid enclosed in a cube ( 0 < x < X , 0 < y < L , 0 < z < L ) and denned the density fluctuation term A by p = po + A

(155)

and used a Fourier series, A = V^ Bp„T cos 2np—— cos 2ira— cos 27TT— . ' J 2L/ ZLi ILi

(156)

One cannot sum to infinitely large p,
*-i.£j.A+iU?;.A+-' with tp(po) = 0 and f Adr = 0.

(l58)

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The Golden Age of Theoretical

Physics

Hence, for the fluctuation energy (or work) one has

*-(% L

3 3

+ 1

*%)l**

^

v

u

2

B

(159)

= l6 a^2_ ^' par

with
Yv

(160)

J***" = ~W•

The statistical properties are thus reduced to the thermodynamical function ip, which is the energy related to the change in specific volume. Equation (160) is valid only if -Q^ is not too small. It disappears at the critical point and, with Smoluchowski, one has to consider higher orders of A. 1 3 7 8.2.3. Density fluctuations and light Let the medium be characterized by a refractive index g (= yfi, where e = dielectric constant) with respect to monochromatic light, e can be expressed as a function of the density,

£ = e0+(p-J

A = e 0 +£',

(161)

where e' is infinitesimally small. According to Maxwell's equations, the electric vector E obeys the equations e 92E c2 df2

A E - grad • divE

(162)

and div(eE) = 0. Einstein now wrote as an Ansatz E - E 0 + e,

(163)

where e is a weak opalescence field. From Eq. (162) one obtains SW-Ae

= -J£-d^-

grad

• dlV

E

( 1 6 4 )

and div(e'Eo) + div(£ 0 e) = 0. Since div E = 0 and grad £Q — 0, one obtains div e = — E 0 grad e'

(165)

Albert Einstein

and Marian von Smoluchowski

. . . 193

and ^ f j f - Ae = ~^!e2^^L + j - -grad{E 0 grad e'} — a. The vector a is a known potential, and e can be expressed as a function of a as dr{a}t„-5

^

(166)

r

AKJ

where r is the difference of dr to the point under consideration, and V = - £ - , the velocity of light in the medium. The integration in Eq. (166) extends over the whole volume, which could be considered as a cube of dimension £(-C L, considered earlier). Let the incident wave be given by E 0 = A cos 27m ( t - ^ J ,

(167)

where n is the unit vector (components a, 0,7) along the wave normal and r is the radial vector (components x, y, z). Let the point of observation be at a distance D (:» l) along the x-axis. Then

G=

(* =t°-v)-

tto /<*>*+**

(168)

Introducing the expression for a from Eq. (165), one obtains ex=0, e6y --J-[£>(£d2Eo^ ~ 4irDc*J \

+

^)dT J '

dt*

e

> = "4^5? J e {

-3

) dT-

(169)

With the help of Eqs. (156) and (161), one obtains -A M ! ^ ! r p

c

4TTDC2 dp

j

(170)

par

where JP
=

Ay y j tipar £> p'a'T1 J par >> p'a1 T'

= AlY,WrJU,

(171)

because of the independence of the Fourier coefficients. The averaging over the time affects only the factor JpaT. Finally, inserting for B%aT from Eq. (160) one obtains

JV 0 U J (4^)2 2H ' where $ is the volume through which the radiation passes.

(1?2)

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The Golden Age of Theoretical

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Einstein rewrote Eq. (172) as the ratio j&, where Je is the intensity of the opalescent light at a distance D from the point where it is created,

^ " ' " I T W (wcos*'

(173)

where 6 is the angle between the electric vector of the incident light and the plane normal to the observed opalescent light. Equation (173) may be integrated over all directions 9. The absorption coefficient a is denned by Je-Jo

= Jee~a5,

(174)

where S is the thickness of a layer, and e~aS is the damping factor for the intensity. Thus

1 RTov(^)2

(2n\4

a=

^-N-^r{-x)

(175)

•

Einstein noted: 'It is important that the principal result of our investigation [given by Eq. (173)] permits an exact determination of the constant N.'138 He applied this result to the special case of a homogeneous substance at the critical point. 8.2.4. Homogeneous substance near the critical point Consider a homogeneous substance for which r/> = / pdv

(isothermal process).

(176)

According to the Clausius-Mosotti-Lorentz relation, v = const.,

(177)

and one obtains

z - -N- .*») y~) v^cos e-

<178)

For an ideal gas e » l , and

Jo

=

RT0(e-l)2

f2n\*

$

je ir-7-lxJ (4W

2

(179)

This formula is identical to the result obtained by Lord Rayleigh to explain the blue of the sky. 139 The second application of Eq. (173) is to a mixture of two incompressible fluids in the neighborhood of the critical point. Let one of the components be designated by the specific volume v, and let ij){y) be the work necessary to take the specific volume at the equilibrium state along reversible path at constant temperature to another specific volume. The ratio of the mass of the second component to the first

Albert Einstein and Marian von Smoluchowski

...

195

is as k : 1 The mixture is in equilibrium with its own saturated vapor of pressure p. Let the volume of the first component in the gaseous phase be v', of the second v", and let p' and p" be the partial pressures, respectively. Thus,

(180)

{•^)0-M*lMy-' and

fde

dk

where M" = the molecular weight of the second component. And Jo _ M" u ( t ) 2 /2TT\ 4 $ Je N ^(lo^gfcP',) \ \ J (4?rD)2

2

9.

(181)

Equation (181) determines fully the opalescent properties of binary fluid mixtures if their vapors can be regarded as ideal gases and if one is not too close to the critical point. Einstein concluded his theoretical investigation with the remark: 'A quantitative experimental study of the phenomena discussed here would be of great interest. Because, on the one hand, it would be valuable to know whether Boltzmann's principle really describes the phenomena in question correctly; on the other hand, one could obtain from such investigations the exact values for the number AT.'140 9. Experimental Tests Smoluchowski's theory of density fluctuations was extended by W.H. Keesom, 141 who also derived a formula similar to Eq. (178), and tested it experimentally with Kamerlingh Onnes. 142 They studied the behavior of ethyline near the critical temperature of 11.8° C, and obtained a value of N = 75 x 10 22 ± 15%. Einstein's formula, Eq. (181), had not been checked experimentally until the time when Jean Perrin gave his report at the first Solvay Conference in 1911. 95 However, the experimental results of Perrin and co-workers seemed to confirm Rayleigh's theory of the blue of the sky [which also follows from Einstein's theory, Eq. (179)]. 143 Later checks of Einstein's theory revealed that it was not completely correct. For instance, it follows from his theory that light emitted perpendicular to the primary beam should be linearly polarized, which is at variance with empirical facts. Gans discovered the reason for this discrepancy in 1923. 144 One has also to assume fluctuating anisotropics in the medium, which are caused by the double refraction of the fluid. The double refraction, on the other hand, is produced by electrical fields, and the polarization of the Tyndall light (opalescence) is decreased by this effect. Equation (178) has, therefore, to be corrected by a factor which takes into account the anisotropy (which is related to the Kerr constant). This multiplicative factor can take on values up to 4 in the case of chlorene-benzene.

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The Golden Age of Theoretical

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The experiments by C.V. Raman and K.R. Ramanathan 1 4 5 with C 0 2 at high pressures, by R. Venkateswaran with petane, 146 and by C.V. Raman and K.S. Rao with many other substances, 147 yielded reasonable values for the Avogadro number N. Raman and Ramanathan 1 4 5 and Gans 144 also completed Einstein's theory of the mixture of fluids, Eq. (181). The first observations had been made by R. Fiirth 148 with phenol-water mixtures. Later measurements by W.H. Martin and S. Lehman 1 4 9 were discussed by Raman and Ramanathan 1 4 5 and found to be in reasonable agreement with the theory. The observations of CS2-acetone mixtures by J.C.K. Row 150 were found to be in very good agreement with theory. Appendix Einstein's Doctoral Thesis: 'A N e w Determination of Molecular Dimensions' In Spring 1905 Albert Einstein presented his doctoral thesis entitled 'A New Determination of Molecular Dimensions' to the University of Zurich. It was published the following year in Annalen der Physik.45 In 1911 Einstein published a correction to this paper because 'Some weeks ago Mr. Bacelin, who carried out an experimental study on the viscosity of suspensions on behalf of [Jean] Perrin, wrote to me in a letter that the coefficient of viscosity of suspensions was according to his results larger than due to the formula in §2 of my paper. I had Mr. Hopf check my calculations and he in fact found an error of computation which falsifies the result considerably.' 151 The thesis examiners were Professors Alfred Kleiner and Heinrich Burkhardt. Kleiner, in his recommendation of Einstein's work, said: 'The ideas and calculations which have been treated here belong to the [more] difficult aspects of hydrodynamics, which could only have been attempted by one who possesses an understanding of and practice in the treatment of mathematical and physical problems.' 152 And Burkhardt remarked: 'The way of treating [the subject] shows a thorough mastery of the mathematical methods concerned.' 153 1. Can the Molecular Dimensions B e Determined from Liquids? The kinetic theory of gases, in the hands of Boltzmann and van der Waals, led to the first determination of molecular dimensions. Einstein thought (in 1905) that the same ideas could not be applied to liquids, because the kinetic theory — as applied to liquids — was still very incomplete. One could, however, determine the size of the molecules of a solute in an undissociated solution from the viscosity of the pure solvent and the solution, and the rate of diffusion of the solute into the solvent. Einstein decided to solve the equations of diffusion under the condition that the volume of the solute is large in comparison with the molecules of the solvent. In this case, the hydrodynamic equations could be applied to the motion of the solvent in the neighborhood of a molecule, ignoring the molecular structure of the solvent altogether.

Albert Einstein

and Marian von Smoluchowski

...

197

2. Hydrodynamics of a Liquid Around a Very Small Sphere The large solute molecules in question are considered as rigid spheres. The liquid has a viscosity k and the velocity components u, v, and w at the space-time point (x, y, z, t). Einstein studied the flow of the liquid in the neighborhood of a reference point x, y, z, so that in a Taylor expansion x - x0 = £,

y - y0 = 77,

z - z0 = C,

(1)

only the linear terms appear. The motion of the liquid, in general, can be of three kinds: (i) Parallel displacement of all the particles of the liquid; (ii) A 'rigid' rotation of the liquid without change of the relative positions of all particles; (iii) A dilatation in three perpendicular directions, expressed by the equations uo = A£,

v0 = Br/,

WQ

= cC

(2)

and A + B + C = 0, because of the incompressibility of the fluid. Einstein now considered a rigid sphere of radius P in a domain G, such that the volume of the sphere is much smaller than G. The rigid sphere is also assumed to move slowly, so that its kinetic energy and that of the liquid may be neglected. Moreover, the velocity of an element of the surface of the sphere is assumed to coincide with the velocity of the neighboring fluid. The sphere thus follows all motions of the fluid. However, it influences the dilatation motion (iii) in its neighborhood. From the symmetry of the equation of motion (2) it follows that, in a spherical coordinate system p, <j>, 6, with the origin at the center of the particle, u = v = w = 0 at

p= y/£2 + 7]2 + C2 > 0 .

(3)

For infinitely large £, 77, ( (i.e., far away from the body), u, v, and w have the undisturbed values given by Eq. (2). Thus u = A £ + ui, v — B j] + v\, to = c C +

VJI

,

(4)

with linip-voo ui,vi,wi —> 0. Now, citing G. Kirchhoff's Lectures on Mechanics, Einstein applied the hydrostatic pressure p to the problem: -£- = kAu,

-~- = kAv,

and

-£- = kAw,

(5)

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The Golden Age of Theoretical Physics

where A = g - + ^

+ $ , and div u = 0,

u(u, v, w).

The velocities are given by the solutions of Eqs. (5); hence the quantities u\, i>i, and w\ must also satisfy them. The solutions of Eqs. (4) and (5) are then given by

* - - I " - { A ^ + s ^ + c ^ } + —• • 3

jr

at,

V = BTJ- -P3B-L 3 - ^ , p dr]

..CC-'i-C^.W,

(6)

where

Equation (6) gives the only solutions of Eqs. (5) consistent with the boundary conditions given by Eqs. (3) and (4) Einstein now proceeded to calculate the energy transformed into heat (per unit time) within the liquid when pressure is exerted on the boundaries, say spherical, of the liquid. If the components of the pressure exerted on the surface of the sphere of radius R(R^> P) are Xn, Yn,Zn, then the energy W per unit time (equal to the mechanical work done on the liquid) is given by

w

J

(Xnu + Ynv + Znw)ds,

•/surface /surface

where

Yn = -(Y^+Y^+Y^) V P Zn = — [ Zt

V

, P

1- Zn

p

P

P)

h Z{ - J , P)

(7)

Albert Einstein

and Marian von Smoluchowski

...

199

and

„, dv

„

, Idm v = X< = - k ( -

Yn=p-2k-,

Z,

Z<=P-2k-,

X„ = Y, = - ^ -

+

du\ - y

+

- j .

From a straightforward calculation Einstein obtained W = ^irR3k62 - ^irP3k62

= 262k (v + | ) ,

(8)

with 62 = A2 + B2 + C2 and ^-R3 = V, §TTP 3 = $ . If the suspended sphere were not present ( $ = 0), then the energy used up in the volume V would be W = 2S2kV. On account of the presence of the sphere, the energy used up is diminished by 52k. 3. Viscosity Coefficient in a Liquid C o n t a i n i n g R a n d o m l y D i s t r i b u t e d Spheres A large number (n per unit volume) of small spheres are distributed in the domain G, their total volume being small compared to G. If an arbitrary dilatation (Eq. (2)) is considered at point (x, y, z), it will be affected by the presence of a sphere at that point. Since many spheres may be present with their centers at (av,2/i/,z„), their contributions to the velocity components u, v, w (Eq. (6)) will have to be added up. Neglecting higher-order terms, one obtains u = Ax - £

| \ ^ U M l

+ Brit + CO/PI

+ ••-,

v^By-Y,

{ yfi^Ml

+ Brit + CQ,)IPI

+ • •• ,

w

= Cz - E {15 C " ( A ^ 2 + B rf+ c $)ifi+• • •'

(9)

with £„ = x — xv,r\v = y — yu, and <^v = z — zu. The total energy also becomes a sum of the individual energies, W = 262k{l+Tf)=262k{l

+l),

(10)

where

(11)

200

The Golden Age of Theoretical Physics

with

+

*-(£L-* (£L-*-£(feL-«'-*>The last equation follows since, for x = y = z = 0, u» = -

~

V K M

2 r

^ " " "

- — > ,

(12)

where r„ = ^/x£ + yl + z%, and the summation extends over the large sphere of radius R in which the small spheres are equally distributed such that one can integrate, A* = A — n I •7—^-dxudyudzu J oxv = A-n(±*P^A.

(13) 2

Similar expressions follow for the coefficients B and C, and for 5* one obtains J*2 =A*2+B*2+C*2

=52(l-2
(14)

The coefficient of viscosity may be determined from Eq. (10) via

W* = 2 C T V = 26*2k (l + 1 ^ .

(15)

Thus ** = k (l + \^j

.

(16)

Hence in a fluid in which small rigid spheres are suspended, the coefficient of internal friction is increased by a fraction which is 2.5 times the total volume of the spheres suspended in a unit volume, provided that the total volume is small. As an example, Einstein chose the solution of sugar molecules in water. He calculated the relative volume of the molecules for a 1% aqueous sugar solution from the ratio k*/k to be 2.5 (p = 0.245

for

0.01 gm of sugar.

(17)

Thus one gram of sugar has the same effect as 0.98 cm 3 of rigid spheres in a liter. One gram of crystalline sugar has a volume of 0.61 cm 3 . The one and a half times larger volume might be due to the fact that water molecules gather round the surface of the sugar molecules. Diffusion

of undissociated

molecules

in a fluid

If the force K acts on a rigid sphere of radius P it will move in a fluid of viscosity k with the velocity

Albert Einstein and Marian von Smoluchowski

...

201

The force K due to the osmotic pressure p is given by v

pNdx

'

where (-e-)iV is the number of actual molecules in a unit volume. Since in a dilute solution the osmotic pressure is given by P=-pT, (20) m the amount of substance transported through a unit area per unit time is given by

•*— B S- "i,h "-Osm-

<2I>

At this point Einstein remarked: 'In this calculation osmotic pressure is treated as a force acting on the individual molecules, which evidently does not correspond with the conceptions of the kinetic-molecular theory, since, according to the latter, the osmotic pressure in the case under discussion must be thought of as a virtual force only. However, this difficulty vanishes if we reflect that (dynamic) equilibrium with the (virtual) osmotic forces, which correspond to the differences in concentration of the solution, can be established by the aid of a numerically equal force acting on the single molecules in the opposite direction.' 154 Einstein thus added and subtracted the external force Px and — Px per unit mass, equal to the osmotic pressure, then retaining a force equal to the osmotic pressure. Using N p —= — (m = molecular weight), (22) n m Einstein obtained from Eq. (16)

ATP3 =

J L m / ^ \ IOTT p V k

J

•

v

'

Thus NP3 is determined from the ratio of the viscosities, and NP from the diffusion coefficient; and from these, Einstein obtained the radius P of the suspended molecules and Avogadro's number. With the latest values of the diffusion coefficient of sugar in water, which became available to him in January 1906, Einstein determined that P = 0.99 x 10~ 7 cm and N = 2.1 x 10 23 . He corrected these in 1911 to give P = 0.49 x 1 0 - 7 cm and n = 6.56 x 10 23 . However, in the conclusion of his thesis he had already maintained that ' JV must show itself to be independent of the nature of the solvent, of the solute and of temperature, if our theory is to correspond with the facts.' 155 5. The Importance of Einstein's Thesis The hydrodynamical theory of the viscosity in suspensions had its origin in Einstein's doctoral thesis. The significance of this theory is two-fold. First, it provided a generalization of the familiar Stokes problem and, therefore, was of interest as a

202

The Golden Age of Theoretical

Physics

question of theoretical hydrodynamics. Second, t h e results h a d i m p o r t a n t applications for t h e viscosity of a solution a n d t h e size a n d form of t h e particles dissolved. T h e latter point was recognized only much later. Einstein's theory was extended in 1922 by G.B. Jeffery to t h e case of elliptic suspended particles, where t h e r o t a t i o n also exerts a t o r q u e on t h e particles. 1 5 6 T h i s torque must b e balanced by a torque caused from t h e p r i m a r y flow of t h e fluid. In any case, a definite angular velocity between t h e fluid a n d t h e ellipsoid has t o be t a k e n into account. Wall effects generally increase the viscosity in Eq. (16), a n d the same is t r u e for t h e concentration of spheres in certain regions. 1 5 7 T h e goal of Einstein's thesis h a d been to calculate molecular dimensions, b u t he h a d also succeeded in opening other unexplored regions of classical physics. References and Notes 1. For historical details of the founding of statistical thermodynamics, see Jagdish Mehra, 'Einstein and the Foundation of Statistical Mechanics,' Physica 79A, No. 5, 447-477 (1975). 2. E. Mach, Die Mechanik in ihrer Entwicklung, Leipzig 1883; translated into English as The Science of Mechanics, Chicago 1883. See pp. 588-589 of the latter. 3. L. Boltzmann, Lectures on Gas Theory, Berkeley and Los Angeles, 1964, p. 318. (English translation by S.G. Brush of Boltzmann's Vorlesungen iiber Gastheorie, J.A. Barth (Leipzig); Part I, 1896; Part II, 1898. Reprinted 1910-1912.) 4. J. Willard Gibbs, Elementary Principles in Statistical Mechanics, Yale University Press 1902; Dover Edition 1960, p. 72, Eq. (205). 5. J.W. Gibbs, Ref. 4, p. 74. 6. S. Arrhenius, Conferences sur quelques theme choisis de la chimie physique pure et applique, faite a I'Universite de Paris du 6 au 13 Mars 1911, Paris 1912, p. 12. 7. Robert Brown, 'A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies,' Edinburgh New Philosophical Journal 5, 358-371 (1828); Philosophical Magazine 4, 161-173 (1828); Poggendorf's Annalen der Physik 14, 294-313 (1828). Also published at several other places. 8. Who was Marian von Smoluchowski, who, together with the most distinguished scientists of the world, was invited in 1903 to contribute to the Boltzmann-Festschrift? Smoluchowski was born on 28 May 1872 in Vorder-Briihl near Vienna, Austria, the son of a senior official in the chancellery of Emperor Franz-Joseph. He spent his youth in Vienna, but since his family was Polish he also became acquainted with Polish cultural traditions at home. From 1880 to 1885 he attended secondary school, the famous Collegium Theresianum. Smoluchowski's interest in the natural sciences was awakened at the Theresianum by his teacher A. Hofler. He studied at the University of Vienna from 1870 onwards under the direction of J. Stefan and Franz Exner, obtaining his doctorate lsub auspiciis imperratoris' in 1895 with a thesis on 'Akustische Untersuchungen iiber die Elastizitat weicher Korper.' Although Boltzmann returned to Vienna from Munich in 1894, after the death of Stefan, Smoluchowski did not come in close contact with him. With F. Hasenohrl, however, who had also attended the Theresianum, Smoluchowski was bound in great friendship, and shared interests in music, mountaineering and skiing with him. After his doctorate Smoluchowski left Austria for studies abroad. From 1895 to 1896 he worked with G. Lippmann in Paris on heat radiation, from 1896 to 1897 with Lord Kelvin in Glasgow on radioactivity, and also for a period during 1897 with

Albert Einstein

and Marian von Smoluchowski

...

203

E. Warburg in Berlin. After his return to Vienna he became a Privatdozent at the University in 1898, but left for Lemberg (Lwow) the following year. Smoluchowski became an extraordinary (or associate) professor at the University of Lemberg in 1900 and a full professor in 1903. During the succeeding years he gained recognition and renown, and in 1913 he was invited as a professor of experimental physics at the Jagellonian University in Cracow. He became a member of the Cracow Academy. Upon Hasenohrl's death on the battlefield in 1915, S. Meyer, F . Exner, and his old teacher A. Hofler sought to persuade the University of Vienna to invite Smoluchowski as his successor, but nationalism prevailed with the other members of the selection committee and the offer was not made. Smoluchowski would have been happy to return to Vienna in a chair whose occupants had been Stefan, Boltzmann, and Hasenohrl. But soon thereafter, the University of Warsaw invited him to head a laboratory that was designed to expand into a large research center, and to take over the courses in theoretical physics. But Cracow wanted to retain him, and showed it by electing him Rector of the University in 1917. He died in Cracow of dysentery the same year on 5 September, at the age of forty-five. Smoluchowski's reputation could have been established on the basis of his article in the Boltzmann-Festschrift alone, where he not only postulated the existence of density fluctuations but also stated his concept of entropy. He developed the latter conception in his great lectures at the Congress of Natural Scientists (Naturforscherversammlung) in Miinster (1912) and the Wolfskehl Congress on kinetic gas theory organized by David Hilbert (1913). Smoluchowski pursued his ideas on density fluctuations in papers on the opalescence of gases near the critical point, in which he combined his conceptions with Lord Rayleigh's theory of the blue of the sky. Smoluchowski placed great emphasis on the comparison of theoretical results with experiment. He had worked on the theory of Brownian motion already since 1900, but he refrained from publishing his results before they could be confirmed by experiment. Einstein, in his papers of 1905 and 1906, published the theory of Brownian motion. It was then that Smoluchowski published his results in 1906, based on a different approach. A comparison of the approaches of Einstein and Smoluchowski to the same problem sheds some light on their different personalities. Smoluchowski's scientific approach was akin to Boltzmann's; he knew the experimental details completely and sought to elaborate mathematical methods. Einstein was not quite certain about experimental details or the existence of phenomena about which he drew far-reaching conclusions from his statistical thermodynamics. He wrote simple and lapidary sentences, alternating between inspired guesswork and keen grasp. Of Smoluchowski, Sommerfeld said: 'Nobody who came close to him could scarcely escape the charm of his esthetically refined character. His presentations in writing and lecturing demonstrated artistic fascination... [and] his lectures were perfect in form.' (A. Sommerfeld, Phys. Zeit. 18, 533-539 (1917), p. 534) Einstein and Smoluchowski developed a personal acquaintance in which, quite probably, Paul and Tatiana Ehrenfest played the role of intermediaries. In March 1912 Einstein wrote to Smoluchowski inviting him to stay with him in Prague during his visit to see Einstein at Easter. (A. Teske, Ref. 10, p. 178) He wrote to Smoluchowski again in the same year, telling him about his research in general relativity. In his obituary notice of Smoluchowski, Einstein wrote: 'Everybody who knew Smoluchowski personally liked him not only as an enthusiastic researcher but as a noble, sensitive, and generous man. The world catastrophe of the last [few] years [i.e., World War I] aroused in him no other feeling than nameless pain about the crudeness of man and the damage to our cultural development. Fate has cut off his fruitful work as

204

9. 10. 11. 12. 13. 14. 15. 16.

17. 18.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29.

30.

31. 32. 33. 34. 35.

The Colden Age of Theoretical Physics researcher and teacher too early; we shall, however, continue to esteem his example and his works.' (A. Einstein, Naturwissenschaften 5, 107-108 (1917)) M. v. Smoluchowski, ' Uber Unregelmassigkeiten in der Verteilung von Gasmolekiilen und deren Einfluss auf Entropie und Zustandsgleichung,' Ref. 11, pp. 626-641. Armin Teske, Marian Smoluchowski, Cracow/Warsaw, 1955, pp. 209-210. Festschrift Ludwig Boltzmann, gewidmet zum sechzigsten Geburtstage 20 Februar 1904, Leipzig, 1904. M. Smoluchowski, Ref. 9, p. 626. M. Smoluchowski, Ref. 9, pp. 627-628. M. Smoluchowski, Ref. 9, p. 630. M. Smoluchowski, Ref. 9, p. 641. M. Smoluchowki, 'Studien uber Molekularstatistk von Emulsionnen und deren Zusammenhang mit der Brownschen Bewegung,' Sitz. Ber. Kais. Akad. Wiss. (Wieri), Math. naturwiss. Kl. 123, Ila, 2381-2405 (1914). T. Svedberg, Z. Phys. Chem. 77, 147 (1911). Max Born, 'Einstein's Statistical Theories,' in P.A. Schilpp (Ed.), Albert Einstein: Philosopher-Scientist, Library of Living Philosophers, 1949; Harper Torchbooks, 1959. See Vol. I, p. 165, Harper Torchbooks edition. A. Einstein, 'Zur allgemeinen molekularen Theorie der Warme,' Annalen der Physik 14, 354-362 (1904). A. Einstein, Ref. 19, p. 360. A. Einstein, Ref. 19, pp. 360-361. A. Einstein, Ref. 19, pp. 361-362. See S.G. Brush, Archive for Hist. Exact Sci. 5, 1 (1968), p. 3. G.W. Muncke, 'Uber Robert Browns microscopische Beobachtungen,' Poggendorf Annalen der Physik 17, 159-176 (1829). Bence Jones, The Life and Letters of Faraday, London: Longmans, Green &: Co., 1870, Vol. 1, p. 403. Quoted in Ref. 23. Robert Brown, 'Additional Remarks on Active Molecules,' Edinburgh Journal of Sciences 1, 314-320 (1829); Phil. Mag. 6, 161-166 (1830). J. Regnauld, 'Etudes relatives au phenomene designe le nom du mouvement Brownien,' Journ. Pharm. Chim. [3] 34, 141 (1858). C. Wiener, 'Erklarung des atomistischen Wesens des fliissigen Korperzustandes und Bestatigung desselben durch die sogennanten Molecularbewegungen,' Poggendorff Annalen der Physik 118, 79-94 (1863). G. Cantoni, 'Su alcune condizioni fisiche dell'affinita, e sul moto browniano,' Reale Istituto Lombardo di scienze e lettere (Milano) Rendiconti [2] 1, 56-67 (1867). K. Nageli, Uber die Bewegungen kleinster Korperchen,' Sitz. Ber. Bayr. Akad. Wiss., Math. Phys. Kl. 9, 389-453 (1979). See also J. Delsaux, Monthly Microscopic Journal 18, 1-6 (1877). J. Loschmidt, 'Zur Grosse der Luftmolecule,' Sitz. Ber. Kaiserl. Akad. Wiss. Kl. II 52, 395-413 (1865). William Ramsay, 'On Brownian or Pedetic Motion,' Proc. Bristol Naturalists' Society 3, 299-302 (1882). William Ramsay, 'Pedetic Motion in Relation to Coolloidal Solutions,' Chemical News 65, 90-91 (1892) ; Proc. Chem. Soc. 8, 17-19 (1894). L. Gouy, 'Note sur le mouvement brownien,' Journal de Physique [2] 7, 561-564 (1888). Jules Henri Poincare, 'The Principles of Mathematical Physics,' in Congress of Arts and Science, Universal Exposition, St. Louis, 1904 (Boston & New York: Houghton, Mifflin & Co., 1905), Vol. I, pp. 604-622, p. 610.

Albert Einstein

and Marian von Smoluchowski

...

205

In 1900 L. Bachelier presented his doctoral thesis to the Faculte des Sciences de Paris. It dealt with the theory of speculation in the stock market. Bachelier developed a theory of stochastic processes rather similar to the theory of Brownian motion. Poincare was one of his examiners and the thesis was dedicated to him. In the 'Seconde These: Propositions donnees par la Faculte,' the problem assigned to Bachelier was: 'Resistance d'une masse liquide indefinie pourvue de frottements interieurs, regis par les formules de Navier, aux petits mouvements varies de translation d'une sphere solide, immergee dans cette masse et adherent a la couche fluide qui la touche.' (L. Bachelier's thesis, 'Theorie de la Speculation,' was published by Gauthier Villars, Paris, 1900.) 36. Bodaszewski, Dingler's Polytech. Journ. 239, 325 (1881). 37. F.M. Exner, 'Notiz zu Browns Molecularbewegung,' Annalen der Physik 12, 843-847 (1900). 38. S. Exner, Sitz. Ber. Kais. Akad. Wiss. (Wien) 56, 116 (1867). 39. In reply to a letter of Smoluchowski, F. Exner wrote (dated Charlottenburg, 12 November 1900): 'As far as molecular motion is concerned, I am very glad that you consider it worthy of theoretical investigation because I have always heard that one didn't know what to do with it. I racked my brains in vain last year in trying to understand it. As for the diagrams, they do no longer exist; they had been scratched into smoked glass plates and fixed. I can only draw for you from memory the approximate 'Krix-Krax' [sketch] The molecular motion can be easily seen through a microscope of [even] small magnification.' 40. A. Einstein, 'Autobiographical Notes,' in Albert Einstein: Philosopher-Scientist, Ref. 18, pp. 1-95. See pp. 46-49. 41. A. Einstein, Kinetische Theorie des Wdremegleichgewichtes und des zweiten Hauptsatzes, Annalen der Physik 9, 417-433 (1902). 42. A. Einstein, 'Theorie der Grundlagen der Thermodynamik,' Annalen der Physik 1 1 , 170-187 (1903). 43. A. Einstein, 'Uber einen die Erzeugung und Verwandlung des Lichtes betreffended heuristischen Gesichtspunkt,' Annalen der Physik 17, 132-148 (1905). 44. A. Einstein, 'On the Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular-Kinetic Theory of Heat,' Annalen der Physik 17, 549-560 (1905). (Received 11 May 1905) This is the title, in English translation, of Einstein's 1905 paper on Brownian motion. This paper, together with Refs. 42-45, is contained in A. Einstein, Investigations on the Theory of Brownian Movement, edited with notes by R. Fiirth, translated into English by A.D. Cowper, Dover Publications 1956. 45. A. Einstein, 'A New Determination of Molecular Dimensions,' Annalen der Physik 19, 289-306 (1906). Corrections, ibid. 591-592 (1911). For a discussion of this paper, which was Einstein's doctoral dissertation at the University of Zurich, see the Appendix. 46. A. Einstein, 'On the Theory of the Brownian Movement,' Annalen der Physik 19, 371-381 (1906). 47. A. Einstein, 'Theoretical Observations on the Brownian Motion,' Zeit. f. Elektrochemie 13, 41-42 (1907). 48. 'The Elementary Theory of the Brownian Motion,' Zeit. f. Elektrochemie 14, 235-239 (1908). 49. A. Einstein, Ref. 44, English translation, pp. 1-2. 50. A. Einstein, ibid., p. 9. 51. A. Einstein, ibid., pp. 10-11. 52. The Stokes law is not necessarily satisfied. If the particles in liquids or gases are large enough in comparison with the molecules, then the assumption is all right. In

206

53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67.

68.

69. 70. 71.

72. 73.

74. 75. 76. 77. 78. 79. 80. 81.

The Golden Age of Theoretical Physics gas, however, with large mean free paths, a correction must be applied, which was studied theoretically by E. Cunningham (Proc. Roy. Soc. London A 8 3 , 357 (1919)) and F. Zener (Phys. Zeit. 20, 546 (1919)), and experimentally by Knudsen and Weber (Ann. d. Phys. 36, 981 (1911)), McKeehan (Phys. Zeit. 12, 707 (1911), E. Meyer and W. Gerlach (Elster-Geitel Festschrift, Vieweg, p. 196)) and others. The application of Stokes' law to discontinuous motions seems to be verified on the average. (See the Notes by Fiirth, Ref. 44, pp. 92-96.) A. Einstein, Ref. 44, pp. 12-13. A. Einstein, ibid., p. 16. A. Einstein, Ref. 46. See R. Fiirth, Ref. 44, p. 19. This is the case if one considers, for instance, a as a spatial displacement or angle of rotation, because of the homogeneity of space. Equation (56) goes back to Boltzmann, who used it often in his Gastheorie. A. Einstein, Ref. 46, See R. Fiirth, Ref. 44, p. 23. A. Einstein, ibid., p. 26. A similar Ansatz was made around the same time by Walther Nernst and others in the problem of electrical conduction through electrolytes. A. Einstein, Ref. 46, p. 34, in Ref. 44. T. Svedberg, Zeit. f. Elektrochemie 12, 853-860, 909-910 (1906). A. Einstein, Ref. 47. A. Einstein, Ref. 47; see R. Fiirth, Ref. 44, p. 63. A. Einstein, ibid., p. 67. A. Einstein, Ref. 48. If the force is removed, the wall will move far enough so that the osmotic pressures on both sides balance. The osmotic pressure is, therefore, responsible for the diffusion process. A. Einstein,' Giilitigkeitsgrenze des Satzes vom thermodynamischer Gleichgewicht und die Moglichkeit einer neuen Bestimmung der Elementarquanta,' Annalen der Physik 22, 559-572 (1907). A. Einstein, Ref. 68, p. 569. A. Einstein, Ref. 68, p. 572. A wide range of electrical Brownian motion experiments has been performed. (See, for example, the review by E. Kappeler, Naturwissenschaften 27, 649-658, 666-670 (1939), for earlier literature. In February 1908, Einstein proposed a 'new electrostatic method to measure small electrical quantities,' developed jointly with his wife Mileva and Paul Habicht, for the measurement of small potential differences of the order of 5 x 10~ 4 volts. M. von Smoluchowski, lZur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,' Annalen der Physik 2 1 , 756-780 (1906). W. Ostwald's Klassiker der Exakten Wissenschaften, No. 207: Abhandlungen iiber die Brownsche Bewegung und verwandte Erscheinungen von M. v. Smoluchowski, A. Einstein, u.a. A. Sommerfeld, Phys. Zeit. 18, 533-539 (1917), p. 535. M. Smoluchowski, Ref. 72; contained in Ref. 73, p. 1. G. Cantoni, Nuovo Cimento 27, 156 (1867). F. Kolacek, Beibl. Ann. d. Phys. 13, 877 (1889); G. Quincke, Naturforscher Versammlunq Dusseldorf, 1898, p. 28, Beibl. Ann. d. Phys. 23, 934 (1898). Meade Bache, Proc. Amer. Phil. Soc. 33 (1894); Chem. News 7 1 , 47 (1895). Jevons, Proc. Manchester Soc. 9, 78 (1869). G. van der Mensbrugghe, Poggendorf's Annalen der Physik 138, 323 (1869). M. Smoluchowski, Ref. 72 (Ref. 73, p. 7).

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207

82. M. Smoluchowski, Ref. 72 (Ref. 73, p. 16). In order to compare Eqs. ( I l l ) and (52), one has to put t = 1 sec, £jf = :n§-, and A = ^ | . 83. M. Smoluchowski, ibid., pp. 16-17. 84. Smoluchowski referred to the observations of Bodaszewski (Beibl. Ann. d. Phys. 8, 488 (1883); Dingler's Journ. 239, 325 (1882)) and O. Lehmann (Molekularphysik 2, p. 5) that could be interpreted according to his theory; however, no accurate measurements were available at that time. 85. M. Smoluchowski, Ref. 72; (Ref. 73, p. 20). 86. M. Smoluchowski, 'Einige Beispiele Brownscher Molekularbewegung unter Einfluss ausserer Krafte,' Bulletin International de VAcademic des Sciences de Cracovie, Mat. naturw. Klasse A, 418-434 (1913). 87. M. Smoluchowski, Phys. Zeit. 13, 1069 (1912). 88. See, for instance, the review by Kappler cited in Ref. 71. 89. M. Seddig, Ber. naturwiss. Ges. Marburg, 1907, p. 182; Phys. Zeit. 9, 465 (1908). 90. F. Ehrenhaft, Sitz. Ber. Kais. Akad. Wiss. (Wien) 116, 1139 (1907). 91. M. de Broglie, Compt. rend. 148, 1163, 1315 (1909); 154, 112 (1912). 92. E. Weiss, Phys. Zeit. 12, 630 (1911). 93. H. Fletcher, Phys. Rev. 33, 81 (1911). 94. J. Perrin, Compt. rend. 146, 967 (1908); 147, 530, 594 (1908). Perrin's first full report can be found in Ann. de. Chimie et de Physique 18, 1 (1909). 95. J. Perrin, 'Les Preuves de la Realite Moleculaire,' in La Theorie du Rayonnement et les Quanta, Rapports et discussions du premier Conseil Solvay (1911), Gauthier Villars, Paris 1912, pp. 153-250. 96. J. Perrin, ibid., p. 165. 97. J. Perrin, ibid., p. 184. 98. In 1901, in a contribution to a symposium on molecular hypotheses in Paris, Perrin proposed a model of atomic structure, with a positively-charged 'sun' (nucleus) surrounded by many negatively charged 'planets,' the periods of rotation of the latter corresponding to the wavelength in the emission spectrum. (J. Perrin, 'La structure nucleo-planetaires des atomes,' Revue Scientifique 15, 449 (1901).) 99. J. Perrin, Traite de Chimie Physique, Les Principes, Gauthier-Villars, Paris, 1903. 100. J. Perrin, 'Le contenu essentiel des principe de la thermodynamique,' Bull. Soc. Philosophic 6, 81-111 (1906). Quoted in S.G. Brush, Ref. 23. 101. J. Perrin, 'La discontinuite de la matiere,' Revue du Mois 1, 323-343 (1906). 102. P. Langevin, Sur la theorie du mouvement brownien,'1 Comptes Rendus Acad. Sci. Paris 146, 530-533 (1908). 103. J. Perrin, Atoms (translated by D. Hammick), Constable & Co., London, Second English Edition, 1923, p. 114. 104. V. Henri, 'Etudes cinematographique des mouvements brownien,' Compt. Rendus Acad. Sci. Paris 146, 1024-1026 (1908). Presented on 18 May 1908. 105. J. Perrin, 'La lois de Stokes et le mouvement brownien,' Compt. Rendus Acad. Sci. Paris 147, 475-476 (1908). Presented on 7 September 1908. 106. J. Perrin, 'L'origine du mouvement brownien,' Compt. Rendus Acad. Sci. Paris 147, 530-532; presented on 21 September 1908. 107. J. Perrin, 'Grandeur des molecules et charge de 1'electron,' Compt. Rendus Akad. Sci. Paris 147, 594-596 (1908). Presented on 5 October 1908. 108. Chaudesaiges, 'Le mouvement brownien et la formule d'Einstein,' Compt. Rendus Acad. Sci. Paris 147, 1044-1046 (30 November 1908); J. Perrin, 'Le mouvement brownien de rotation,' ibid. 149, 549 (1909); Perrin and Dabrowski, 'Mouvement brownien et constantes moleculaire,' ibid. 149, 477-479 (1909). 109. J. Perrin, Annales de Chimie et de Physique 18, 1-14 (1909); Oeuvres, 171-239, p. 214; Atoms, p. 121.

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110. 111. 112. 113. 114. 115. 116. 117.

J. Perrin, Ref. 95, p. 192. J. Perrin, Ref. 95, p. 203. J. Perrin, Ref. 95, pp. 204-205. J. Perrin, Ref. 95, pp. 206-207. Bancelin, Comp. Rend. Acad. Sci. Paris, 191 (1910). J. Perrin, Ref. 95, p. 211. L. Brillouin, Annates de Chimie Phys. 27, 412 (1912). Jean Perrin's pioneering work almost exhausted the research on the experimental determination of N, the Avogadro number, and the verification of the EinsteinSmoluchowski theory of Brownian motion. Note should, however, be taken of the later investigations of T. Svedberg and K. Inouye [Arch. f. Kemi 4, No. 19 (1911). A controversy arose from the work of R. Ettenreich [Sitz. Ber. Kais. Akad. Wiss. (Wien) 121, 1163 (1912)], who found a smaller value of N, i.e., larger A 2 , for horizontal motion. But in his case, Stoles' formula had to be corrected, because it is valid only when the hydrodynamical equation holds. R. Fiirth, Note No. (8), Ref. 44 [Dover edition], p. 97. L.S. Ornstein, Proc. Amsterdam Acad. 2 1 , 96 (1919). R. Fiirth, Zeit. f. Phys. 2, 244 (1920). A. Fokker, Dissertation, Leiden, 1913; M. Planck, Sitz. Ber. Preuss. Akad. Wiss. 324 (1927); M. Smoluchowski, Phys. Zeit. 17, 557 (1916). W. Ostwald, Grundriss der allgemeinen Chemi: Leipzig, Verlag von Wilhelm Engelmann, 4th edition, 1909, see Preface. J. Tyndall, Proc. Roy. Soc. (London) 17, 223 (1868). J.W. Strutt (Lord Rayleigh), Phil. Mag. 4 1 , 107, 274, 447 (1871); 44, 28 (1897); 47, 375 (1899). G. Mie, Annalen der Physik 25, 377 (1908). M. Le Blanc and W. Kongro, Zeit. f. Elektrochemi 19, 794 (1913); W. Kongro, Z. phys. Chemi 87, 257 (1914). M. Smoluchowski, Acad, des Sci. de Cracovie, December 1907; Annalen der Physik 25, 205 (1908). M. Altschul, Zeit. phys. Chemie 1 1 , 578 (1893). K. v. Wesendonck, Naturwiss. Rundschau 9, 210 (1894); 22, 145 (1907). M. Travers and F. Usher, Zeit. phys. Chemie 57, 365 (1906). S. Young, Proc. Roy. Soc. (London) 78, 262 (1906). F. Donnan, Chem. News 20, 139 (1904). In 1913 Donnan wrote to Smoluchowski: 'There is no doubt that extremely important progress in science has been brought about by you, Einstein, Perrin and Svedberg. The progress is due to the explanation of fluctuations and the molecular character of many phenomena.' (See A. Teske, Ref. 10, Chapter 5.) V. Rothmund, Zeit. phys. Chemie 26, 433 (1898); I. Priedlander, Zeit. phys. Chemie 38, 385 (1901). D. Konowalox, Annalen der Physik 10, 360 (1903). A. Einstein, 'Theorie der Opalescenz von homegenen Fliissigkeiten und Fussigketsgemischen in der Nahe des kritschen Zustandes,' Annalen der Physik 33, 1275-1298 (1910). A. Einstein, Ref. 135, p. 1276. M. Smoluchowski, Ref. 127, p. 215. A. Einstein, Ref. 135, p. 1294. Lord Rayleigh, Phil. Mag. 47, 375 (1899). A. Einstein, Ref. 135, p. 1298. W.H. Keesom, Annalen der Physik 35, 591 (1911).

118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132.

133. 134. 135.

136. 137. 138. 139. 140. 141.

Physics

Albert Einstein

and Marian von Smoluchowski

...

209

142. H. Kamerlingh Onnes and W.H. Keesom, Comm. from the Phys. Lab. of Leiden, 1908. 143. J. Perrin, Ref. 95, pp. 221-227. 144. R. Gans, 'Lichtzerstreuung,' in Wien-Harms' Handbuch der Experimentalphysik 19, Leipzig: 1928. 145. C.V. Raman and K.R. Ramanathan, Proc. Roy. Soc. (London) A 1 0 4 , 357 (1923). 146. R. Venkateswaran, Trans. Chem. Soc. 121, 2655 (1922). 147. C.V. Raman and K.S. Rao, Phil. Mag. 45, 632 (1923). 148. R. Fiirth, Site. Ber. Kais. Akad. Wiss. (Wien) H a 124, 577 (1915). 149. W.H. Martoin and S. Lehman, J. Phys. Chem. 26, 75 (1922). 150. J.C.K. Row, Phys. Rev. 78 (1928). 151. Correction to Ref. 45, Annalen der Physik 34, 591-592 (1911), p. 591. See. Eq. (123) of our paper. 152. A. Kleiner's note of recommendation concerning Einstein's thesis, quoted in M. Fliickiger, Albert Einstein in Bern: Paul Haupt, Bern 1961, p. 70. 153. H. Burkhardt, also Ref. 152, pp. 70-71. 154. Ref. 45, English translation in Fiirth, Ref. 44, pp. 58-59. 155. Ref. 44, Fiirth, p. 60. 156. G.B. Jeffery, Proc. Roy. Soc. (London) A102, 161 (1922). 157. E. Guth, 'On the Hydrodynamical Theory of the Viscosity of Suspensions,' Proc. 5th Int. Congress of Applied Mechanics, 1938.

7 T h e Historical Origins of t h e Special Theory of Relativity* Historical Preliminaries The development of the principle of relativity is one of the most exciting topics in the history of modern physics. On the surface, the situation with this principle looks perfectly simple. One can even cite one scientific paper — submitted from Berne, Switzerland, in the second half of June 1905 (and received by Annalen der Physik on 30 June 1905) — by Albert Einstein, an unknown Technical Expert, Third Class, at the Swiss Patent Office (or The Office for Intellectual Property) in Berne; to this paper has often been assigned the full glory for having created the special relativity theory (SRT). Albert Einstein was the more sympathetic figure, since he was an outsider in the world of established science. After obtaining his Diplom from the ETH (the Swiss Federal Institute of Technology) in Zurich, Einstein had made attempts to obtain a position there (and elsewhere) as an assistant, but none of the professors was sufficiently interested in his work or appreciated it enough to engage him. After going through a precarious existence for over two years, during which he survived by undertaking jobs as a substitute teacher in Winterthur and Schaffhausen, his friend and former classmate at the ETH — Marcel Grossmann — through his father, helped him in obtaining the position as a patent examiner, for which some knowledge of electromagnetic theory was required. He had been interested in electromagnetic theory since his days at the ETH, and even before that. At the Patent Office, Einstein did his work, examining the applications for This lecture was presented extempore at the Albert Einstein Centennial Symposium at The International Solvay Institutes in Brussels, Belgium, early in 1979, where it was tape-recorded, and then repeated at the Einstein Symposium at the University of Houston, which I organized, and given again at SUNY, Stony Brook, New York, and at the University of California a t Irvine, and at several other institutions, during 1979-1980. It draws on numerous studies of Einstein's work on special relativity, and a detailed version of this essay, complete with references and footnotes, will be published elsewhere. I have learned much from the two-volume work on A History of the Theories of Aether and Electricity by Sir Edmund Whittaker on the work of Lorentz and Poincare, the writings of Russell McCormmach on the work of Lorentz, the writings of H.M. Schwartz on the analysis of Poincare's papers, my discussions with Charles >Scribner Jr. on Einstein, and my discussions, friendship, and collaboration with G.H. Keshwani. I am deeply indebated to Eugene Wigner, Freeman Dyson, C.N. Yang, Richard Feynman and Julian Schwinger, for conversations on the items treated in this essay.

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new patents, and trying to find loopholes in them, in order to be able to reject most of the submitted claims. In the evenings, he often met with a couple of friends to discuss the important aspects of science and philosophy, and together they studied the great scientific classics. Einstein — a self-made man and independent thinker — would all of a sudden appear like a comet on the horizon of physics in 1905 and brighten the world with three great gifts: the theory of molecular-kinetic theory of motion, leading to the final proof of the existence of atoms and molecules; the hypothesis of light-quanta, throwing new light on the concept of Planck's quantum of action; and ideas on the electrodynamics of moving bodies. Strangely enough, the innocent-sounding last paper of about thirty pages, became the most exciting of them all in the following discussions in scientific literature. It seems that Einstein had intended only a slight improvement of the theory as developed by James Clerk Maxwell and Heinrich Hertz, but he introduced a revolution of thought by his 'relativistic kinematics.' Even the nonscientific public was touched by this revolution, which made its way into the writings of philosophers, and hardly anything else was so widely misquoted and abused as the theory of relativity, which, according to more perspicacious physicists should better be called 'the theory of a viewpoint.' In the beginning there was Maxwell's electromagnetic theory. Then there arose the desire to extend it to embrace within its framework the electrodynamic phenomena taking place in moving media. This problem became entangled with another one, namely to describe matter. An ingenious proposal was made to develop the basis of the description of nature on two concepts: the concept of the charged particle and the concept of the luminiferous ether, which, ever since the days of Rene Descartes, Robert Hooke, and Christiaan Huygens had been postulated as the allpervading material medium in which optical disturbances were propagated. The charged particles, which became known as the electrons, served to explain many phenomena including those of propagation of light in matter. It is of interest to note that even Newton, although himself a leading proponent of the corpuscular theory of light, did not wish to exclude the postulate of a material ether from his considerations in connection with optical and gravitational phenomena. Then, until about the middle of the 18th century, the majority of physicists — following Leibniz and Huygens — favored some kind of contact action, which they imagined as taking place through the intermediary of a universal medium, which was capable of sustaining longitudinal vibrations of light analogous to the elastic vibrations associated with sound waves. The luminferous ether came into favor again in the 19th century with the advent of the refined undulatory theories of light of Thomas Young and Augustin Fresnel. The recognition of the fact that the vibrations of light are transverse led to the introduction of the perfectly rigid elastic model for the ether; the general theory of elasticity was forged towards the middle of the 19th century by Navier, Cauchy, Stokes and others. It was Gabriel Stokes who sought to find an acceptable explanation for the apparently contrary attributes one was forced to assign to the ether: the absolute rigidity of its substance

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and complete penetrability through it of all ponderable bodies. However, it seems that the conviction of the necessity of the ether postulate was so overwhelming, that its paradoxical attributes did not cause undue concern. Augustin Presnel's successful explanation (1818) of Francois Arago's experiment on the refraction of stellar light in prisms — in terms of the partial dragging of the ether by moving bodies, the velocity of drag being the fraction (1 — 1/n 2 ) of the velocity of the body relative to the otherwise absolutely stationary ether (n being the index of refraction of the transparent body when at rest in the ether) — gave a special boost to the conception of the ether. This hypothesis was not only especially successful in explaining the failure to detect any effect of the Earth's motion through the ether in Arago's experiment, but also in a number of related experiments, such as that of Armand Fizeau (1851) on the interference of two light beams sent in opposite directions through water flowing in a tube, which was later on verified by more refined experiments by Michelson and Morley (1886) and by Zeeman (19141927); and that of Airy (1871) on the aberration of light moving through telescopes filled with water. It was in this kind of ether-dominated scientific climate that the investigators of that period attacked the challenging problems presented by one of the grandest theoretical structures of the 19th century: Maxwell's theory of the electromagnetic field. Both Maxwell himself and his immediate successors considered the postulate of a universal material medium subject to the laws of continuum mechanics, and serving as the seat of all electromagnetic actions, as an integral part of the general theory of electrodynamics. How else, it was felt, could energy and momentum be propagated in the presence of a field? Consequently, it was generally held that it should be possible to detect the motion of the Earth relative to the reference frame determined by the universal medium. However, certain effects demanded a strange property of the moving electrons; the negative result of the experiments (such as Michelson-Morley's) to detect the absolute motion of objects against the ether called for a contraction of the linear dimension. Then even the time scale had to be changed. A mathematician came to the assistance of Hendrik Antoon Lorentz, who had done most of the gigantic work, to understand the implications of his results far better: Henri Poincare called for a new mechanics in which the velocity of light should play a dominant role. Finally, he discovered that the mechano-electrodynamics of Lorentz did the job. However, at the same time Albert Einstein came out with his relativistic kinematics, claiming that it was not so much the dynamics that should be modified than the concepts of length and time. So all of a sudden, two theories existed at the same time — both yielding the same empirical results, one rather involved and mixed and the other a simpler one: on the one side stood the seasoned relativistic dynamics of the electron and, on the other, the relativistic kinematics. It turned out that the latter was more practical to extend, and it won the race in the future. But the younger, fresher theory was based on whatever the primitive precursor of electron's dynamics had been. No Einstein could follow without Lorentz, unless he

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was Lorentz and Einstein rolled into one person. Yet Lorentz did exist, and Einstein became his more accomplished successor. Who had established the principle of relativity? By 1953, when Edmund Whittaker published the second volume of his History of the Theories of Aether and Electricity, there had been no question and less discussion about the historical origins of the Special Relativity Theory. Whittaker argued that Lorentz and Poincare did so, since their scheme of relativistic electron dynamics had been completed even before Einstein submitted his paper. And Whittaker, eloquently and thoroughly (with a full panoply of mathematical details) supported his view. The simplified, indeed simplistic, historical picture which characterized the intellectual evolution of the origins of special relativity, stated that the Michelson-Morley experiments had obtained a null result in an attempt to measure the relative motion of the Earth and the ether, and then Albert Einstein, in his great paper of 1905 'On the Electrodynamics of Moving Bodies' killed the 'ether' in one fell swoop — the young St. George of the 20th century dispensing the imaginary dragon which had enchanted or bedevilled the scientific psyche of the previous centuries. In this simplistic picture, in between the mention of the Michelson-Morley experiment and Einstein's heroic supremacy over the discordant myth, there occurred a feeble mention of the Lorentz-FitzGerald attempts to explain the 'contraction' of length by means of an ad hoc hypothesis and Lorentz's derivation of the transformations that go by his name in the context of the electrodynamics of moving bodies. With Einstein's work everything fell into place, and the gifted technicians of the new science could build ever more complicated structures on the new edifice. Now Whittaker, in the book just mentioned, wrote a chapter entitled 'The Relativity Theory of Poincare and Lorentz,' thus throwing a fly in the historical ointment. At one point in this chapter, he casually wrote: 'In the autumn of the same year [1905] Einstein published a paper which set the relativity theory of Poincare and Lorentz with some amplifications, and which attracted much attention.' Whittaker's thesis made Max Born, Einstein's old friend and colleague, very unhappy. Since Born at that time held the Tait Chair of Natural Philosophy at Edinburgh, where Whittaker also lived, he took great pains to convince Whittaker about the error of his ways — and in letters to Einstein he maintained that he had no truck with the stubborn Scotsman. Einstein did not give a hoot about what Whittaker thought — and as the St. George grown old that he now was — conscious, though half-mockingly of being the 'chosen among the chosen people,' he knew that his place in the mythology of science was so secure as not to merit even half a frown at the impertinence of the wayward Whittaker. Born, a lesser man but a loyal friend, soon took the opportunity of commenting briefly on the origins of the special theory of relativity in an address to the conference on the 50th anniversary of Special Relativity, held in Berne, Switzerland, soon after Einstein's death in 1955, What the audience in Berne did not know was the effect which Whittaker's views had on Born, and that he was responding essentially to a private discomfort. As for Whittaker, who, in his youth, had been a friend and admirer of both Lorentz

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and Poincare, also wrote Einstein's obituary notice in the Biographical Memoirs of Fellows of the Royal Society in 1955, and in a reference to the special theory he mentioned that 'Einstein adopted Poincare's Principle of Relativity.' In completing the mathematical and physical structure of special relativity, the work of Lorentz, Poincare and Einstein, was of fundamental importance — and each brought the theory nearer to completion. In studying their work, one must carefully distinguish between their respective mathematical formalisms and their physical ideas because of deceptive similarities both in their equations and the words used. Each of these three great men had different viewpoints and relativities. We must be most careful in analyzing Henri Poincare's contributions because of his somewhat inconsistent position concerning the transition between classical and relativistic physics. Whittaker failed to appreciate the physical approach of Einstein and gave Poincare most credit for supplying the principle of relativity. Moreover, he misinterpreted Poincare's achievements in relativity as being mainly physical, and those of Lorentz as being mainly mathematical. Generally, insufficient credit has been given to Poincare for his mathematical contributions to relativity. This is regrettable because this part of his work has retained its unchanged significance, while some of his physical contributions must be revised after the advances of his successors. Otherwise, Poincare is known to physicists usually in reference to Poincare stresses and the Poincare group (a name, incidentally, conferred on this structure by Eugene Paul Wigner in 1939; this was historical justice, for Poincare himself had named the Lorentz group in 1906). Frequently one also tends to misinterpret the work of Albert Einstein, by ascribing to him many results which others had obtained first. For instance, the mass-increase formula,

m = m0 11

(*)

J]

had first been introduced by Lorentz in 1904 for electrons and other particles. It was Max Planck in 1906 and 1908, and Hermann Minkowski in 1908, who first obtained the equations of the dynamics of a particle in special relativity. In 1905, Einstein considered m0c2 I 1

j )

_ m

oc2

(2)

as the kinetic energy of an electron and showed that

m=4

(3)

cz was approximately valid for a charged particle radiating electromagnetic energy. While Einstein had assumed it to be true in general, it was Lorentz who in 1911 showed that in general all types of energies must be included in E. In 1908, Minkowski completed the relativistic treatment of the electrodynamics of matter in general (including magnetization).

The Historical Origins of the Special Theory of Relativity

215

We should also know exactly which marvelous simplifications and contributions Einstein's genius made in special relativity. The name relativity can be misleading. It emphasizes the reality of the relative and perhaps implies that 'all views are relative.'' However, the basis and applications of the theory of relativity are characterized by phenomena that do not change according to relative observers. Indeed, the postulates of relativity state that the laws of physics and the speed of light do not change for relatively moving reference systems (being covariant and invariant, respectively). Let us now examine in some detail as to how the theory of special relativity developed historically. It developed from attempts to develop a satisfactory electrodynamics of moving bodies. Newtonian mechanics was known to be Galilean invariant, while the laws of electrodynamics were not. Since Maxwell's unification of optics and electrodynamics, the luminiferous ether assumed even greater importance than it had before as the medium of propagation of light waves. The ether's rest-frame, in which light moves with speed c and Maxwelll's equations are valid, was thought to be an absolute rest-frame. Experiments failed to detect this restframe. They did not show that modifications of the laws of electrodynamics were required for moving systems. Attempts were therefore made at a theoretical explanation by a study of the equations of electrodynamics, by a search for principles to explain the experiments, and by attempts to revise existing laws. Maxwell's equations were developed in classical physics. Therefore, it is surprising that they are not Galilean invariant. (The name 'Galilean transformation' was introduced by Phillip Frank in 1911, Annalen der Physik 34, 825, — a year before he became Einstein's successor in Prague.) The Lorentz transformations could have been derived by Maxwell, but required several more decades — Maxwell having died at the age of 48 in 1879, the year in which Einstein was born. The efforts of many physicists, notably Lorentz, Poincare, Einstein, Planck and Minkowski solved the problems. Newtonian mechanics was revised, while electrodynamics was left unchanged and shown to be Lorentz invariant. Other major results were the rejection of absolute time and the ether, constancy of the speed of light (it being also the maximum signal velocity attainable) and the connection between mass and energy. Lorentz, Larmor and Planck An early forerunner of the Lorentz transformations is found in the work of Woldemar Voigt, which was published in the Proceedings of the Royal Gottingen Academy in 1887. Voigt studied the Doppler principle for waves propagating with speed c in an elastic medium. He obtained the invariance of the wave equation with respect to coordinate transformations, equivalent to the modern Lorentz transformations multiplied by (1 — v'2/c2)~^ (with c, the speed of light, v, the uniform speed of the reference frame). Not being familiar with Voigt's work, Lorentz (in 1892-1895) started to develop his own electron theory of matter. In contrast to Maxwell's phenomenological theory, Lorentz's electron theory explained many macroscopic

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phenomena of electrodynamics by the microscopic behavior of electrons (and atoms) and their interactions with the field. From the start of his professional career, Hendrik Antoon Lorentz, began to unify the structure of physics. In this effort, he attempted to unite the ideas of Augustin Fresnel on the interaction of ether and matter with Maxwell's description of electromagnetic phenomena and the 'atomic view of electricity' of Weber and Clausius. Lorentz's doctoral dissertation, completed in 1875, which dealt with the refraction of light, was actually the first link in this lifelong study. Lorentz's thesis, in a way, decisively settled the dispute over the nature of the ether. By showing that it was impossible to suppress the longitudinal wave that was created in any model of the ether that was an elastic solid, Lorentz argued for an electromagnetic ether whose properties were largely undefined, except that it transmitted electromagnetic phenomena. Three years later, in 1878, Lorentz published a work on dispersion in which he made a radical departure from his predecessors. In this paper, Lorentz assumed that the ether was the same in and out of matter — that it was simply not affected by matter. According to Lorentz, light waves moving through the ether caused the ions in matter to oscillate, thereby creating innumerable small waves that through interference with themselves and with the incident radiation were changed in such a way that the velocity appeared to depend on the frequency of light, thereby causing the phenomenon of dispersion. As a result of this analysis, Lorentz predicted a relationship between the density of a dielectric medium and its index of refraction. Most significantly, however, the analysis was done by assuming that the ether was absolutely fixed. In his electron theory of matter (1892-95), Lorentz tried to show the invariance of form of the equations of electrodynamics in uniformly moving frames, for this would have meant agreement with experiments showing the equivalence of all such frames. By transforming fields, coordinates, and time, Lorentz had the transformations to the first order in v/c, and this accounted for experiments not involving more than the first power of v/c. This meant a change in the origin of time, without a contraction of the time scale, but the physical implications were not recognized — for the change was regarded as just a mathematical convenience. In order to account for the Michelson-Morley experiments involving v2/c2, Lorentz assumed a contraction of moving bodies in their direction of motion by a factor (l + 2t/ 2 /c 2 )

in

1892

(4)

in

1895.

(5)

and a factor (l-v2/c2)-i

In 1899 Lorentz even considered a change in time scale in a transformation. (In 1889, George F. FitzGerald In Dublin had the same view concerning the explanation of contraction of length in the Michelson-Morley experiment, and he sent

The Historical Origins of the Special Theory of Relativity

217

a note to Science, the AAAS journal, but he himself did not know whether his note had been published. Lorentz learned from an account of Sir Oliver Lodge on 'Aberration Problem' about FitzGerald's ideas and wrote to him, and since then this phenomenon has been referred to as the Lorentz-FitzGerald contraction.) In 1900, Joseph Larmor, in his electron theory of matter had already used the 'Lorentz' transformation (for x, y, z, t only) in his Adams Prize Essay, entitled Aether and Matter, these equations being, x' =

E*(X

— vt)

y' = y;

z = z,

t' =ei{t-v£(x-vt)/c2}

(6)

where £ = 7 2.

7=(1

_t;2/c3)-J.

The time part reduced to the familiar

As the electromagnetic fields were not properly transformed, Larmor obtained invariance only to the order v2 / c 2 . Larmor was the first person to derive the LorentzFitzGerald contraction from the transformation equations. By further developing his electron theory, Lorentz in 1904, proved the invariance of Maxwell's equations in free space under the transformation: x' = l*y(x — vt) y' -ly t, =

or

;

z — Iz

It _ l-yvjx - vt) 7 c2

f = *

7

(*-^),

where I is a function of velocity, which Lorentz showed to be equal to one. The invariance of Maxwell's equations including charges and currents was only approximate in Lorentz's treatment, because he incorrectly transformed the velocity, current and charge density. Neither Lorentz nor Poincare used this kinematics to derive the LorentzFitzGerald contraction. For them, the contraction of moving bodies was a physical fact. It remained for Einstein to show that it is not physical but a kinematical appearance, and that it is mutual with respect to the two frames, depending only on relative motion.

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Lorentz explained contraction by the assumption that molecular forces change in motion as electrostatic forces do, for the shape of the body depends on molecular forces. Assuming also that electrons become flattened ellipsoids in motion, Lorentz derived the variation of mass with velocity, Eq. (1). However, Lorentz's assumptions were not satisfactory for they just covered up the difficulties but did not remove them. Lorentz did not use a principle of relativity, although he indicated the advantage of using some general fundamental postulate. Thus, starting from Maxwell's equations, Lorentz established their invariance of form and, therefore, the invariance of the laws of electrodynamics with respect to moving reference frames, e.g. the motion of the Earth. Lorentz made assumptions about the structure (shape, charge distribution, etc.) of the electron. Using the fundamental equations, he obtained F'=

* 2 (1,7,7)F

a' = Z(7 3 ,7 2 ,7 2 )a

(9)

and for the force and acceleration transformed to the instantaneous (i.e. primed) restframe of the electron. From this, it followed that 1 1N

1

m ' =1 if - 3 , - , - \

V7

3

(10)

7 7/

where m = m is the rest-mass and m the mass in the laboratory frame. Then, 7711=77710,

m||=73m0,

(11)

for the masses, transverse and longitudinal with respect to the velocity, which meant a tensor mass, as the force and acceleration are, in general, in different directions. (The terms 'longitudinal' and 'transverse' mass were introduced by Abraham in 1903.) Although others, such as Larmor and Voigt, considered a tensor mass before Lorentz, their results were not correct. But Einstein made no assumptions about the structure of the electron, and thus he freed the theory from assumptions regarding the theory of matter. Finally, in 1906, Max Planck, obtained the usual formulae for the mass without assumptions about the electron structure. He also completed the theory of dynamics in special relativity by generalizing Newton's second law and obtaining the Hamiltonian and Lagrangian equations for a particle. The arguments by means of which Planck derived his expressions for the kinetic energy and momentum of a particle in relativity theory were felt to be not entirely satisfactory. Three years later, Gilbert N. Lewis and Richard C. Tolman gave a proof of very different character. In 1908,

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219

Planck gave a general definition of relativistic momentum and mass-energy. By introducing the notion of 'proper time,' he obtained the realtivistically invariant form of his equations of dynamics. Also, in 1908, G.N. Lewis proved by means of the theory of radiation pressure, that a body which absorbs radiant energy increases in mass according to the relation dE = c 2 dm and affirmed that the mass of a body is a direct measure of its total energy according to the relation E = mc2. Lorentz had also generalized his mass-velocity relation (of 1904) for the electron to all particles, by saying that 'the proper relation between the forces and accelerations will exist... if we suppose that the masses of all particles are influenced by a translation in the same degree as the electromagnetic mass of the electrons.' But Einstein, in 1905, justified it by a better argument. He noted 'that these results as to the mass are also valid for ponderable material points, because a ponderable material point can be made into an electron by the addition of an electric charge, as small as desired.' Henri Poincare Poincare's main contribution to relativity was his mathematical work of 1905, briefly outlined in an article of 5 June (Comptes Rendus 140, 1504 (1905)) and published fully in a 47-page memoir in January 1906 in Rendiconti del Circulo Matematico di Palermo 2 1 , pp. 129-176 (1906) (presented on 23 July 1905); it was entitled 'Sur la dynamique de 1'electron,' and was dated July 1905. His 1905 memoir was completed just before Einstein's work (which was dated 30 June and published in the glorious Volume 17 of Annalen der Physik). In this work Poincare analyzed the transformation properties of many physical quantities, gave the complete treatment of the covariance of Maxwell's equations, introduced four-vectors and Poincare stresses, and established the group character of the Lorentz transformations, and introduced the name 'Lorentz group.' Poincare's physical contributions to relativity included his work on, and enunciation of, the two postulates of relativity (including the name Principle of Relativity), his treatment of time, electromagnetic momentum, mass-energy, and gravitation. Already in 1902, Poincare had recognized the importance of relativity of displacement, i.e. the homogeneity of space. He had started to put forward the principle of relativity for unifirmly moving system in 1895 (in a contribution to the volume L'Eclairage Electrique). He named this principle as the 'principle of relativity' in 1902. In his work Science et Hypothe'se; he postulated it in 1904 (in Bulletin des Science Mathematique), and declared it publicly in a lecture in 1904. The lecture was delivered before the Section of Applied Mathematics at the International Congress of Arts and Sciences on the occasion of the World Exposition held in St. Louis, Missouri. The Congress was organized to celebrate the great progress in all fields in the 19th century, including the opening of the American West, of which

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St. Louis was the Gateway. This Congress undertook to summarize the entire panorama of the arts and sciences, and there were great festivities-intellectual and otherwise. This was also the occasion when two brothers from Boston introduced to the world for the first time that unique culinary phenomenon: the American hamburger! Among the prominent representatives of European science who attended this Congress were: Henri Poincare, Ludwig Boltzmann and Paul Langevin. Poincare spoke on ''The present and the future of mathematical physics' He listed six fundamental principles on which physics is based: conservation of energy, principle of dissipation of mechanical energy, principle of action and reaction, principle of relativity, conservation of mass, and the principle of least action. Now, 'according to the principle of relativity,' Poincare said, 'the laws of physical phenomena must be the same for a fixed observer as for an observer who has a uniform motion of translation; so that we have not, and cannot have any means of discerning whether we are, or are not, involved in such movement.' And, in 1905, Poincare declared: 'It seems that this impossibility of experimentally demonstrating the absolute motion of the Earth is a general law of nature; we are led to admit this law, which we shall call the postulate of relativity, and admit it without restriction.' By this principle Poincare meant that the laws in uniformly moving frames have the same form, and that one cannot detect the absolute motion because of the compensatory effects of the motion, but that an absolute frame is conceptually distinguishable. Not only was his framework not only fully relativistic, but a more complicated and less general formalism resulted in contrast to Einstein's (two universal postulates). Poincare had also started to recognize the principle of the constancy of the velocity of light in 1898. However, he did not mean this principle in Einstein's sense, for his speed of light did not appear as a universal constant, but as a result of compensating effects of motion on length and time. Poincare also preceded Einstein in his further treatment of the concept of time in which he utilized the constancy of the speed of light for synchronization of clocks by observers communicating by light signals. In 1900, he defined the 'local time' of Lorentz t' = t - vx/c2 , (12) as the time measured by clocks synchronized in this way. In 1904 he amplified his ideas on 'local time' to include the motion of observers, and concluded that: 'The clocks synchronized in this way do not indicate what one can call the local time, so that one lags behind the other. This hardly matters, for we have no means of perceiving it. All the phenomena in a given frame will be retarded, but they will be so equally, and the observer will not perceive it, for his clock lags too; thus, as the principle of relativity requires, he will not have any means of knowing whether he is at absolute rest or in absolute motion.'

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Again, Poincare was inconsistent, for he stated that there is no absolute time and simultaniety of two events, yet elsewhere he distinguished the absolute system at rest with respect to the ether as evidenced by his words true and local time, instead of using quotation marks as did Einstein in 1905. However, with all his inconsistencies, Poincare showed tremendous insight. As he commented in 1904: 'From all these results, if they will be confirmed, will start a completely new mechanics, characterized by this fact that no speed could exceed that of light (for bodies would oppose a growing inertia to the causes tending to accelerate their movement, and this inertia would become infinite when one would approach the speed of light.' He continued: 'For an observer, carried along in translation, of which he is aware, no apparent speed could exceed that of light; that would be a contradiction, if one did not remember that the observer would not be using the same watches as a fixed observer, but those indicating 'local time.' Poincare conclusion in his 1904 article was: 'Perhaps we must also construct a new mechanics, which we can only dimly see now, where the inertia growing with speed, the speed of light would become an impassable limit. The common mechanics, as more simple, will stay as a first approximation as it would be valid for speeds not too great, so that one would find again the old mechanics under the new one.' In 1900, Poincare had introduced the momentum of the electromagnetic field, in his units, as G = fExBdV.

(13)

Then he generalized Newton's third law for electromagnetic phenomena by obtaining the law of conservation of momentum, ^]mv+

ExB

dV = constant.

(14)

Poincare stated without proof that the angular momentum of the field obeys a conservation equation analogous to that of energy and momentum. Thus he restored agreement between Lorentz's theory and Newton's third law, which the theory had seemed to violate. It is interesting to note that his further interpretation of the results led to a connection between electromagnetic mass and energy, but without proof. Poincare regarded the electromagnetic energy as a fictitious (not indestructible) fluid of mass J/c2, with J = B2/8w + 2nc2E2 = Poincare's energy of the

field,

(15)

which is equivalent to E = mc2 for the electromagnetic case. However, none of the general implications were recognized by Poincare. One of his examples was the following: 'The electromagnetic energy behaving as a fluid endowed with inertia, one must conclude that if any device after having produced electromagnetic energy,

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sends it by radiation, this device will recoil as a cannon recoils when it launches a projectile.' Poincare was the first to study Newton's law of gravitation in his special theory of relativity. In his 1905 memoir, he used the term 'gravitational waves,' for he considered the propagation of gravitational effects with speed c. Poincare's relativity required the Lorentz invariance of the laws of gravitation and prohibited a speed of propagation of gravity greater than that of light. Poincare modified Newton's law by constructing functions of invariants of the Lorentz group and requiring agreement with the old law for D < c (where v is the speed of the attracted body at the time of arrival of the gravitational wave). The derivation from Newton's law contains terms of the order of v2/c2 only. This is barely measurable, but it gives some modification for the motion of the planet Mercury, although not enough to account for the anomalous precession. Later on, others also considered the law of force instead of using differential equations of the field and a new geometry; only Einstein finally succeeded in solving the problem. Albert Einstein Almost simultaneously with Poincare's work, Einstein in 1905, at the age of 26, with superior clarity and simplicity, derived all the main previous results and some new ones by stating two postulates: that of the relativity or equivalence of all inertial frames of reference, and the constancy of the speed of light. Einstein's postulates were completely general, so that the concepts of ether and its 'true time' disappeared. It just made no sense to speak of a medium for the propagation of light in vacuum, contrary to the case of sound. No experiments could detect the ether, so why keep it? This is what Einstein decided to do. As he pointed out in his paper on the general theory of relativity in 1916, the theory of special relativity does not depart from classical mechanics through the postulate of the constancy of light in vacuum. It is from this principle that there follow, in the well-known way, the relativity of simultaniety, the Lorentz transformations, and the related laws for the behavior of moving bodies and clocks. Einstein derived the Lorentz transformations from the postulate of the constancy of the velocity of light alone, along with the considerations of symmetry, linearity, and homogeniety. He proved for a spherical wave of light, the covariance of, x2 = c2t2 ,

(16)

in all systems, and noted that the Lorentz transformations could have been derived more simply from this fact. Einstein applied the transformations to moving rods and obtained Lorentz contraction. It is evident that to him the Lorentz contraction was an apparently kinematical consequence, and not a complicated actual physical contraction as it had been for Lorentz. In his work on contraction, Einstein expressed the interesting view that, 'For v = c, all moving bodies viewed from the "stationary" system shrink into plain figures.

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For velocities greater than that of light our considerations become meaningless. However,' he said, 'we shall find that the velocity of light in our theory plays the role physically of an infinitely great velocity.' Poincare had already realized this upper limit in 1904. Einstein also applied the Lorentz transformation to moving clocks and obtained the dilatation of time. He even considered a 'twin problem.' Einstein demonstrated the invariance of Maxwell's equations, as Poincare had also done, but he did not consider macroscopic matter described by the four electromagnetic field vectors in general. (Therefore, one might say that the title of his paper, i.e. 'On the Electrodynamics of Moving Bodies' was inconsistent. Einstein did not enter upon the general structure of equations for ponderable bodies but confined himself to the equations of the isolated electron.) He derived the formulas for aberration and the relativstic Doppler principle, including the important transverse shift. He obtained important results on mass and energy. The kinetic energy of an electron or any ponderable mass point was given as, m0c2 = [ ( l - u 2 / c 2 ) - * - l ] ,

(17)

and, in his second article of 1905 on special relativity, Einstein proved that the relation, Am = AE/c2 , (18) holds approximately for an electron in a process involving a decrease in electromagnetic energy by radiation. He then assumed that this relation held in general, and declared that: 'The mass of a body is a measure of its energy content....' Einstein's work of 1905 was characterized by great and ingenious simplicity, both physically and mathematically. Starting from a set of two simple assumptions, Einstein achieved a simple formalism and derived all his results — whereas Lorentz and Poincare started by assuming electrons, ether, Maxwell's equations, and some fragmentary hypotheses. Einstein derived the Lorentz transformations, while Lorentz and Poincare had assumed them in order to obtain the invariance of the electromagnetic equations. Einstein's principal new results were the formulas for the relativistic transformation of velocities, aberration and Doppler shift, and the relativistic kinetic energy of a particle. Einstein overlooked the difficulties with accelerated frames of reference, and this provided a simplification without which he could not have achieved a simple relativity theory. This simplification was later on removed in the general theory. (See Essay 8.) In his first simple theory, Einstein did not consider a theory of electrons, for the task was too complicated. Indeed, it required several more decades to approach a solution classically. We must further note the universality of Einstein's special relativity. Lorentz and Poincare based their theory on electrodynamics, but Einstein's theory was based on general assumptions — which was more appropriate, as he felt that Maxwell's equations were only approximate and were to be replaced later by

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some (quantum) theory. The importance of Einstein's postulational approach was that it freed relativity from its electrodynamic basis, and made it more universal. There is no experimental decision between Lorentz's and Einstein's theories, but Einstein's is simpler, more universal, and it predicted many new results. Einstein's audacity was also impressive. His great passion for the physical explanation of the laws of nature resulted in his abandoning ether and absolute time, this radically modifying long-established Newtonian space-time. Thus he was the first true relativist. It would be unfair to neglect Einstein's predecessors, who prepared his way, or his successors who carried on his work forward; Poincare undoubtedly approached Einstein closer than Lorentz or Larmor. Lorentz had many of the results of Einstein, but his physical principles were not relativistic. Poincare perfected Lorentz's mathematical results and extended the mathematics and physics of relativity. In this sense, Poincare was Einstein's immediate predecessor. Einstein, Poincare and Lorentz An interesting historical question arises: Was Einstein aware of Poincare's work? He himself denied it. He also denied the knowledge of Lorentz's 1904 paper. Einstein introduced the novel key phrase, 'the principle of relativity,' in the very beginning of his 1905 paper — exactly in the words used by Poincare earlier in his Science and Hypothesis and in his lecture at St. Louis. Should this be regarded as a marvelous coincidence? 'Relativity' was a little used word then in physics, and Einstein used it without explanation. Einstein was aware of Poincare's earlier works. He had fully absorbed Poincare's Science and Hypothesis, having read this book in the company of his friends at their Olympia Academy in Berne: Conrad Habicht and Maurrice Solovine. It is reasonable to assume that Einstein learned of the principle of relativity from Poincare. In his fundamental paper of 1905 on special relativity, Einstein made use of the method of synchronization of clocks within a system exactly as Poincare had done earlier in 1904 — and his postulate that the velocity of light in vacuum cannot be surpassed was not different from Poincare's — though, of course, he arrived at different results from these premises. Evidently, Einstein had great respect for Poincare, but he never referred to him in the context of relativity — and vice versa. However, at one place he adopted a curiously reverent attitude towards Poincare. In his reply to the criticisms of various contributors to the volume Albert Einstein — Philosopher-Scientist, edited by Paul Arthur Schilpp in 1949, Einstein put forward the arguments about the question, whether geometry can be considered verifiable or falsifiable from the physical point of view, in the form of a discussion between Poincare and Hans Reichenbach. However, after proceeding in this manner for a while in a purely scientific spirit, Einstein seems to have discarded this volunatarily chosen pair in the imaginary debate with the sudden remark: 'The conversation cannot be continued in this fashion because the respect of the present writer for Poincare's superiority as thinker and author does not permit it; in what follows, therefore, an anonymous non-positivist is substituted

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for Poincare.' Poincare had briefly surfaced in Einstein's consciousness after several decades, only to be discarded peremptorily. As to the question whether Einstein was aware of Lorentz's 1904 memoir? Einstein had published seven papers, all of them in Annalen der Physik, one of the most prestigious journals of the day, during the period 1901-1905, before publishing his paper 'On the Electrodynamics of Moving Media' in the same journal. He was therefore definitely in close contact with the current ideas in physics, but there was a conspicuous peculiarity about this memoir on special relativity. As Max Born said, 'The striking point is that it contains not a single reference to the previous literature. It gives the impression of quite a new venture. But this, of course, is not true.' Only Born said it in 1956, a year after Einstein's passing. In his paper, Einstein went through the derivation of the same formulas as Lorentz had done. Incidentally, Einstein got an incorrect expression for the transverse mass, m 0 / ( l - v2/c2) instead of m 0 / ( l - v2/c2)* . (19) However, for the longitudinal mass he gave the correct expression, mo/(l - v2/c2f'2

(20)

Lorentz had given correct expressions for both masses. Let us now mention the question whether Einstein knew of Lorentz's memoir of 1904. There is no definite evidence, but there is one significant remark in Einstein's 1905 paper to the effect: 'Since, as follows from the theorem of addition of velocities, the vector (ux,uy, uz) is nothing else than the velocity of the electric charge, measured in the system K, we have the proof that, on the basis of our kinematic principles, the electrodynamic foundation of Lorentz's theory of moving bodies is in agreement with the principle of relativity.' Einstein did not say exactly which theory or memoir of Lorentz's he meant. But it is evident that the theory in question was in accord with the principle of relativity — and this on the basis of Einstein's kinematical principles. It should be noted that Einstein was not claiming any significant advance but only an agreement. Now, none of Lorentz's theories that he published before his 1904 memoir was in agreement with the principle of relativity, but the basic Lorentz transformation equations of his 1904 memoir were strictly in accord with the principle of relativity. Is it possible that Einstein could have had in mind any other theory of Lorentz than the one proposed by him in 1904? It just seems to be very strange. (By the way, Einstein also knew Max Abraham's 1903 work in which Abraham had introduced the terms 'longitudinal' and 'transverse' mass, but he did not refer to it either.) Einstein's remark, which we have already quoted, was exactly the advance that Einstein had made over Lorentz's work. Was this union of Lorentz's equations and the principle of relativity not a marriage of the ideas of Lorentz and Poincare? Einstein had solemnized this marriage, but we do not know why, as priest, he chose not to make it public? Of course, Einstein's work cannot be regarded as merely the bringing together of two previous works. Since it

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appeared so soon after the two, and he struck out in so many new directions that it is quite probable that he was only looking for some confirmation of his audacious ideas before publishing them. Now what did Poincare, Lorentz, and Einstein themselves say about the authorship of the theory of special relativity? Poincare lived for about seven years after the publication of Einstein's paper, but nowhere did he credit Einstein with the discovery of relativity. On the contrary, he stated often that Hendrik Antoon Lorentz was the discoverer. And the reason for this, in Poincare's view, was that Lorentz, in 1904, had been the first to use the Lorentz transformation equations to explain the null results undertaken to discover the absolute motion of the Earth. Indeed the Lorentz transformation equations form the vehicle in which the principle of relativity advances and establishes its dominion in the world of physics wherever high velocities are encountered. Of course, Poincare was well aware of Einstein's work, and in 1911 he spoke in glowing terms about Einstein when he was asked to give his opinion concerning Einstein's work for an appointment at the ETH in Zurich, but nowhere did he attribute the theory of relativity to Einstein. Indeed, in his Wolfskehl lectures in Gottingen in April 1909, at the invitation of David Hilbert, Poincare spoke on 'La mecanique nouvelle,' i.e. on relativity, but he did not mention the name of Einstein — although he did mention other names, and seemed to be recording Lorentz's work. While Poincare considered Lorentz as the discoverer, Lorentz himself never made any claim to the discovery — however, he sometimes credited Poincare with the discovery, but elsewhere solely Einstein. Commenting on the contributions of Poincare to mathematical physics, he said in 1914: 'It was these considerations which I published in 1904 that inspired Poincare to write his memoir on the dynamics of the electron, in which he attached my name to the transformation I spoke about. I must say, however, that the same transformation occurred already in an article of Monsieur Voigt, and that I did not draw from this device all the conclusions possible. Indeed, for certain physical quantities that enter into the formulas, I did not indicate the best possible [i.e. correct] transformation. This was done by Poincare and later by Einstein and [Hermann] Minkowski.' And, a little later on, Lorentz remarked: 'Poincare, on the contrary, obtained the perfect invariance of the equations of electrodynamics, and that he formulated the "principle of relativity," a phrase which he [Poincare] was the first to employ.' On many occasions later on, Lorentz mentioned Einstein's name only. For example, at a conference on the Michelson-Morley experiment in 1927, where Michelson, Miller and Kennedy, were present, Lorentz said: ' . . . I introduced the conception of local time... but I never thought that this had anything to do with real time. This real time for me was still represented by the older classical notion of absolute time There existed for me only one true time. I considered my time transformation only as a heuristic working hypothesis. So, the theory of relativity is really solely Einstein's work.

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'Asked if I consider this contraction as a real one, I should answer, "Yes." It is as real as anything we can observe.' Up to the end — Lorentz died in 1928 — he was reluctant to give up this conception of absolute space and time. As for Einstein, he did not bother with such niceties as did Poincare and Lorentz. In a letter to Max Born, written just two months before his own death on 18 April 1955, Einstein remarked that 'the theory of relativity, if we regard its development in retrospect, was ripe for discovery in 1905 My work of 1905 was independent of Poincare and Lorentz.' The saucer of cream belonged to him alone. As we have tried to show, Einstein knew both of Lorentz and Poincare before he wrote his 1905 paper on the theory of special relativity — but since he had baked enough of a cake there himself, he wanted to eat it, too — and that also by himself alone. Finally, let me make a few remarks about Hermann Minkowski. Although Einstein rejected the notion of the ether or an absolute frame of reference, derived the Lorentz transformation equations, which had been posited ad hoc by Lorentz earlier, and established some new results in his paper of 1905, there exists evidence that he had not yet fully dominated the theory he propunded. The man who turned the theory from its beginnings into clarity of mathematical representation and physical interpretation was Hermann Minkowski. Minkowski sought his inspiration neither in Lorentz nor in Einstein, but he closely traced and extended Poincare's thoughts, even though he was not fully aware of Poincare's work in this domain. Minkowski's conception of a four-dimensional continuum and its invariant metric stemmed from purely geometrical roots. Indeed it was Minkowski who introduced tensors into the theory of relativity. So revolutionary was Minkwoski's presentation and analysis of the theory that Einstein remarked: 'Since the mathematicians have now invaded the theory of relativity, I do not understand it myself anymore!' Minkowski was the first true mathematician of the special theory of relativity, just as Hilbert, seven years later, would become the first mathematician of the general relativity theory. What is remarkable is that according to Minkowski's presentation, the 'notion of relativity' became less significant. The key notion now became the 'invariants,' which remained the same no matter what the frame of reference in which they were measured. The simplest of these invariants was the four-dimensional metric of space-time. A particle in uniform translational motion describes an invariant four-dimensional 'world-line,' an element of which at the 'world-point' (x,y,z,t) in any reference frame is: dr = l/c{c2dt2 - dx2 - dy2 -dz2)i

,

(21)

being physically the 'proper' time measured by a clock stationary at the particle itself. Whatever be the frame of reference, dr remains the same. Now, dr is not a total differential. Thus, for two world-points connected by two different world-lines, f dr, the integral of the proper time, does not have the same value. However, the world-lines of two particles in uniform motion can intersect at no more than one point. Minkowski drew attention to the absolute elements which

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are preserved in the flux of space-time, which do not change no matter what the frame of reference might be. Indeed, without Minkowski's ideas, the development of general relativity theory would not have been possible — certainly not as rapidly as it did. It is interesting to note what Minkowski himself thought of the origin of the ideas associated with relativity theory. 'The validity of the world postulate, I like to think,' he declared, 'is the true nucleus of an electromagnetic image of the world, which, discovered by Lorentz, and further revealed by Einstein, now lies open in the full light of day.' It was Hermann Minkowski who had ushered in the sunshine.

8 T h e Historical Origins of t h e General Theory of Relativity* From 1907 to 1925, Albert Einstein dominated the development of the general relativity theory of gravitation. His own work on the theory of gravitation from 1912 to 1916 had the drama of high adventure. David Hilbert, fascinated by Einstein's work on relativity and Gustav Mie's work on electrodynamics, decided to construct a unified field theory of matter. In two communications to the Gottingen Academy (on 20 November 1915 and 23 December 1916), Hilbert developed his theory of the foundations of physics. In the first of these communications, he derived the field equations of gravitation and the conditions governing them. Hilbert's work, although inspired by Einstein, was independent of and simultaneous with Einstein's derivation of the field equations and, in certain essential respects, went beyond Einstein's. Einstein, at the time, was very critical of the efforts of Mie, Hilbert and Weyl to construct a unified field theory of gravitation and electromagnetism. After 1925, such a programme became his primary mission. In this paper, we have given an account of the intellectual struggles of this fascinating period when the modern theory of gravitation was created.

Contents 1. Introduction 2. Steps Towards a New Theory of Gravitation 2.1. The Principles of Equivalence and Relativity 2.2. Abraham's Objections 2.3. Nordstrom's Theory 2.4. Einstein and Grossmann: Search for the Mathematical Tools of General Relativity 2.5. The Foundation of General Relativity 3. Physics and Geometry This investigation developed from a question which Professor Eugene Wigner asked me in November 1971 about Hilbert's contribution to the equations of general relativity, and my reply to him. See Acknowledgments and Appendix. Dedicated to P.A.M. Dirac on his 70th birthday and published in Jagdish Mehra (ed.): The Physicist's Conception of Nature, D. Reidel Publishing Co., Dordrecht, Holland/Boston, USA, 1973, pp. 92-178, under the title 'Einstein, Hilbert, and the Theory of Gravitation.' The contents of this essay were presented as lectures at many universities in Europe and the USA, also as the Provost's Lecture at Rice University, Houston, Texas, at the invitation of Solomon Bochner, and at institutions such as CERN, Geneva, and UNESCO, Paris.

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3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.

The Role of Axiomatization in Hilbert's Work Concepts of Space and Geometry Space-Time and Invariants Mie's Field Theory of Matter Hilbert's Communication to the Gottingen Academy Axioms of Hilbert's Theory of Matter Hilbert's Fundamental Theorem, Energy Conservation and the World Function 4. Completion of the Theory of Gravitation 4.1. General Covariance and the Equivalence Principle 4.2. Theory of the Gravitational Field 4.3. Consequences and Applications of the Theory of Gravitation 4.4. The Cosmological Term 5. Physical Interpretation of General Covariance 5.1. Axiom of Physical Interpret ability 5.2. Restrictions due to Causality 5.3. Hilbert's Conclusion 6. Hamilton's Principle and the Conservation Laws 6.1. Lorentz and the Hamilton Principle in Einstein's Theory 6.2. Einstein and the Hamilton Principle 7. General Covariance and Conservation Laws 7.1. Energy Conservation and the Theory of Invariants 7.2. Identities from General Covariance 7.3. Bianchi Identities and Hilbert's Constraint Conditions 8. Extension of General Relativity 8.1. Generalization of Riemannian Geometry 8.2. Weyl's Unified Theory 8.3. Five-Dimensional Relativity 9. 'Ein Bleibender Kern' — An Enduring Core References and Notes Acknowledgements Appendix

1. Introduction The modern theory of gravitation, more than any other theory in physics, is connected with the name of a single personality, that of Albert Einstein. Almost immediately after he had published his work on the theory of special relativity, Einstein restlessly moved on to incorporate the effects of gravitation in a consistent theoretical framework. Indeed this problem occupied him so much that he essentially abandoned quantum theory for many years, a field to which he had made singular contributions. A letter to Sommerfeld, dated 29 October 1912, gives evidence of the situation. Einstein wrote:

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Dear Colleague: Your kind note only adds more to my predicament. But I assure you that with respect to the quantum [theory] I have nothing new to say that should arouse interest I am now exclusively occupied with the problem of gravitation and hope, with the help of a local mathematician friend [Marcel Grossmann], to overcome all the difficulties. One thing is certain, however, that never in Me have I been quite so tormented. A great respect for mathematics has been instilled within me, the subtler aspects of which, in my stupidity, I regarded until now as pure luxury.* Against this problem [of gravitation], the original problem of the theory if relativity is child's play. Abraham's new theory, t so far as I can see, is indeed logically correct, but otherwise a monstrous embarrassment. The hitherto existing relativity theory is certainly not as wrong as Abraham seems to think.1 Almost all the physicists of the first rank were absorbed at that time in an effort to understand the quantum problem which the work of Einstein, more than even Planck's, had brought to the forefront. Einstein had pushed the quantum problem into the background in his own work, having 'nothing new to say' about it for the time being, as he wrote to Sommerfeld. He was fully absorbed in 'generalizing' the relativity theory to include gravitation. Still, he did not develop his ideas on gravitation in isolation or in a vacuum of response from eminent physicists. Contrary to the lore which has become fashionable among the 'relativists' nowadays, Einstein had the full benefit of the cut and thrust of scientific debate with some first-rate minds in the development of his theory of gravitation. Indeed there were even those who made valuable, even crucial, contributions to this theory. It is convenient, for the sake of discussion, to single out three distinct stages in the early development of general relativity, each of them brief in its duration. During the first period, from 1907 to 1914, Einstein attempted to formulate and sharpen the principle of equivalence, and engaged in numerous discussions, especially with Abraham and Nordstrom. The second period, from 1912 to 1914 which overlapped with the first, was one in which Einstein and his friend Grossmann incorporated into the new theory differential geometry and other great mathematical structures of the nineteenth century. During the third period, 1915-16, the field equations of gravitation were formulated by Einstein, were independently derived by Hilbert, and the mathematical structure of the equations was clarified. We shall discuss this lively development which culminated in the establishment of the general relativity theory of gravitation. 2. S t e p s Towards a N e w T h e o r y of G r a v i t a t i o n 2 . 1 . The principles

of equivalence

and

relativity

As soon as the so-called theory of special relativity was formulated, Einstein sought for ways of 'generalizing' it. In his article on 'The Principle of Relativity and Its Consequences,' which he wrote in 1907 for the Jahrbuch der Radioaktivitdt und "Writing to the physicist Paul Hertz at about this time, Einstein said that he had been 'a mathematical ignoramus until I reached this shore.' [I am grateful to Rudolf Hertz, t h e son of Paul Hertz, for sending me a photocopy of this letter.] t\Ve shall discuss Abraham's work later on in its proper context.

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Elektronik, he already suggested the equivalence between a gravitational field and an accelerated frame of reference.2 Two points from this article are of great interest in the present context. First, the strong expression which Einstein gave to the assumption of proportionality between the inertial and gravitational masses of a body. The proportionality between the inertial and gravitational masses holds for all bodies without exception, with the [experimental] accuracy achieved thus far, so that we may assume its general validity until proved otherwise.3 He then took a stand with respect to an extension of the special theory: Is it conceivable, that the principle of relativity also holds for systems which accelerate relative to each other? This is, of course, not the place for a detailed discussion of this question. But since this question must occur to anyone who has followed the applications of the relativity principle made hitherto, I shall not fail to mention it here.4 Einstein considered two systems S i and £2, o n e a t rest in a field of constant gravity, and the other undergoing a uniform acceleration with respect to the first. The laws of physics then do not allow one to make any distinction between the two systems. He stated: Hence, on the basis of our actual experience, we have no reason to suppose that the systems Ei and £2 can be distinguished from each other in any way. We shall therefore assume complete physical equivalence between the gravitational field and the corresponding acceleration of the reference system. This assumption extends the principle of relativity to the case of the uniformly accelerated translational motion of the reference system. The heuristic value of the assumption lies in the fact that it allows a homogeneous gravitational field to be replaced by a uniformly accelerated reference system, a fact which, to a certain extent, makes it amenable to theoretical treatment.5 Einstein then considered the properties of space and time in a uniformly accelerated frame of reference and found corrections only of the second order to the spatial shapes of bodies, but 'the process which occurs in watches — indeed any process, in general — takes place faster the larger is the gravitational potential present at the space point in question.' 6 From this he derived the result 'that the light coming from the surface of the sun has a wavelength which is larger by two parts in a million than the one produced by the same atoms on earth.' 7 Einstein further concluded that light beams, which do not travel along the axis of the acceleration, are bent by the gravitational field, leading to a decrease of the velocity of light which is proportional to the gravitational potential. 8 Einstein's paper of 1907 may be regarded, certainly in the published literature, as the explicit starting point of the theory of general relativity, in which several of the assumptions and results to be justified later on were already present. With respect to the two central points of his theory — the equivalence of the inertial and gravitational masses and the principle of general relativity of uniformly accelerated frames of reference — Einstein would later on turn out to be rigid and

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uncompromising in all discussions with his competitors, and both of these ideas were present in his memoir of 1907.9 At this time Einstein's hypothesis of the light quantum was barely two years old, and he was deeply involved in the problems of radiation and the quantum theory. These problems would occupy him fully for another four years before he would return to work on general relativity in 1911. 10 In a lecture to the Naturforschende Gesellschaft on 16 January 1911, shortly before he left Zurich to take the chair of physics at the German University of Prague, Einstein presented a synopsis of his ideas on special relativity. 11 The chair of physics in Prague, which had been occupied before him by Ernst Mach, proved to be very stimulating for the progress of general relativity. In June, 1911, Einstein submitted a paper to the Annalen der Physik, improving his earlier statements of 1907 on the action of gravitation on the propagation of light. 12 What was new in this derivation in which he again denied the existence of an absolute acceleration? Einstein showed that, from the equivalence of an accelerated frame of reference with a corresponding rest frame including a static homogeneous field of gravitation, the identity of inertial and gravitational masses can be derived. Again the velocity of light turns out to be c{4>) = c (l + ^

,

(1)

where <j> is the {negative) gravitational potential. Hence 'the principle of the constancy of the speed of light, according to this theory, is not valid in the same manner as it serves to establish the ordinary [special] theory of relativity.' 13 With the help of Huygens' principle and his theory of gravitation Einstein derived the important consequence, that 'a light ray passing close to the sun would suffer a bending of 0.83 seconds of arc.' 14 2.2. Abraham's

objections

It was already at this early stage in the development of general relativity that Max Abraham entered into the discussion. 15 In a series of papers, Abraham formulated a theory of gravitation, starting from Einstein's consideration of a velocity of light dependent on the gravitational potential. 16 He assumed the existence of a potential <j> [on the (four-dimensional) gradient of which the ten components of the (symmetric) gravitational tensor would depend]. 17 In his calculation, the dependence of the velocity of light c on the gravitational potential turned out to be,

£ . _ £ = *_*,,

(2)

which agreed with Einstein's expression (1) in the first order. On considering the attraction of two masses, Abraham found that the gain of kinetic energy when the masses approached each other was only one-half of the

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loss in potential energy, leading to the conclusion that the gravitational field also contains energy. 18 He derived an expression for this field energy density, (3)

7 being the constant of gravitation. This expression is essentially positive definite, due to the fact that the plus sign appears in the last term within the curly brackets. The reason for this change of sign lay in a new Ansatz for the 'hydrostatic pressure' of the gravitational field.19 The non-covariance under Lorentz transformations did not bother Abraham, since he found that invariance missing even in infinitesimal regions. 20 One of the consequences of this theory was the dependence of the rest mass mo, on the velocity of light M

m

(4)

°W = WJ'

which agreed with Einstein's energy-mass formula in the first approximation. 21 Newton's law of gravitation did not, however, follow strictly from this theory, and had to be replaced by

where B A

jm =

2c2

then, in general 22

£«'•

(5

»

Abraham's theory had certain peculiarities which were quite unusual from the 'relativistic' point of view. First, the absolute value of the potential entered through Eq. (2). 23 Second, his theory did not throw any light on the problem of matter, emphasizing rather the essential disparity between the dynamics of the electron and the theory of the gravitation. 24 These features brought Abraham immediately into conflict with Einstein whose physical reasoning and mental attitude were against them. Abraham had produced a theory which, in Einstein's view [as he noted in his letter to Sommerfeld quoted earlier 1 ], was 'logically correct, but otherwise a monstrous embarrassment.' In a paper on 'The Velocity of Light and the Statics of the Gravitational Field,' Einstein criticized Abraham's theory and summarized his own point of view. 25 His main argument was that 'Abraham's system of equations cannot be brought into accord with the equivalence hypothesis, quite apart from the fact that his treatment of space and time could not be upheld from a purely formal mathematical point of view.' 26 Einstein again dealt with the equivalence between the static gravitational

The Historical Origins of the General Theory of Relativity

235

field and a uniformly accelerated frame of reference, suggesting the following relationship between the second derivatives of the velocity of light and the gravitational source term, Ac = jcg, (6) where 7 is the universal gravitational constant, 7 = Ank/(? and g the mass density. 27 He improved upon this equation in a later work, replacing it by

A"7{^ + ^

}

(6')

where the second term expresses the energy density of the gravitational field.28 In the case of small velocities, v being the gravitational potential. The usual gravitational constant 7 is therefore no longer a universal constant. 29 Having deduced the conservation of energy and momentum for a particle in the static gravitational field, Einstein commented on the space-time problem: The space-time problem seems to me to be as follows. If one restricts oneself to a region of constant gravitational potential, the laws of nature would then assume an especially simple and invariant form, provided one refers them to that manifold of space-time systems which are related to each other by the Lorentz transformations of constant c. If one does not restrict oneself to regions of constant c, then the manifold of equivalent systems as well as the manifold of transformations which leave the laws of nature unaltered would become larger, and because of that the laws would become more complicated.30 In his next paper, Einstein discussed the influence of a static gravitational field on electromagnetic and thermal processes. 28 Similar to Abraham, he found [see Eq. (4), for example] that some of the conventional expressions have to be multiplied by the varying light velocity c to form 'scalars.' For instance, the energy density of the electromagnetic field is given by

i(^ 2 + ^ 2 )-^|(
(7)

or the genuine temperature T is given by T = cT*,

(8)

T* being the usual temperature. Einstein's clarification gave rise to a strong and lively discussion between him and Abraham. 31 In his note on 'Relativity and Gravitation, Reply to an Observation of Mr. A. Einstein,' Abraham insisted on the validity of the Lorentz invariance in infinitesimal regions, emphasizing the existence of an absolute system in which the gravitational field is quasi-static. 32 Einstein gave further clarification of the theory of relativity, since Abraham had concluded the 'breakdown' of that theory. 33 Abraham had also rejected any relativistic space-time concept, including the general spacetime transformations. 34 These criticisms had to be answered, and Einstein did so

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very clearly by stating the two assumptions of special relativity, namely (i) the principle of relativity, claiming that no absolute frame of reference should exist, and (ii) the constancy of the velocity of light. The latter has to be given up if non-inertial frames are admitted, in agreement with the first principle. Abraham was not very impressed by Einstein's answer, and he repeated his doubts in a second note. 35 He accepted the equivalence of inertial and gravitational masses, as established by the experiments of Eotvos, 36 but did not wish to imprison this result into the equivalence hypothesis concerning the frames of reference. As for Einstein he proceeded to establish his claims within a few years. 37 2.3. Nordstrom's

theory

Abraham had certainly stimulated Einstein to strengthen his efforts. In the same year, 1912, a series of papers on the relativity principle and gravitation was published, which led to a more friendly and fruitful discussion with Einstein. The author of these papers was Gunnar Nordstrom of Helsingfors, who proposed in them the first consistent theory of gravitation. 38 Nordstrom started from the question whether one could construct a consistent theory of gravitation in which the principle of constancy of the velocity of light would be maintained, since the Abraham-Einstein discussions seemed to have shown the difficulties of the whole idea of relativity if one allowed for a variable velocity of light. 39 With Abraham, Nordstrom introduced a gravitational potential • # = 4nkg0 (9) where A; denotes the gravitational 'constant' and QQ the rest-density of matter. Then the constancy of the velocity of light c can be maintained provided one allows for a change of mass with the gravitational potential, m = m0e9^c2,

g = Ank.

(10)

In his second note, Nordstrom supplemented Eq. (9) with equations for the gravitational force, K

* = -9Qo££-,

(^ = 0,1,2,3).

(11)

The system of Eqs. (9) and (11) would thus form a complete basis for the theory of gravitation. The gravitational factor g, however, does not turn out to be constant if the principle of equivalence of gravitational and inertial masses is to be preserved. 40 The rest density, according to Laue, is defined as eo

= -^Trr^

(12)

where TM„ = the matter tensor. The dependence of g on the gravitational potential is then given by

The Historical Origins of the General Theory of Relativity

237

which guarantees the numerical equivalence between the inertial mass m = mo 1 + ^ ( 0 -

h)

(14)

and the gravitational mass

Mg = Jg{<j>)eod3x,

(15)

to a certain extent, which we shall discuss later. The role of the gravitational potential ^o is to define the scale between gravitation and the other laws of nature. Nordstrom's theory was the first attempt to provide a consistent approach to the problem of gravitation, a fact which was acknowledged by Einstein himself.41 In a lecture on 'The Present Status of the Problem of Gravitation,' delivered at the Eighty-fifth Conference of Natural Scientists in Vienna, in 1913, Einstein discussed the hypotheses and theories of gravitation. He listed the general postulates which may be valid in a theory of gravitation: 42 (i) energy-momentum conservation is satisfied; (ii) gravitational and inertial masses of closed systems are equal; (iii) special relativity is valid; (iv) the laws of nature do not depend on the absolute value of the gravitational potential. In discussing these, Einstein remarked postulate (i) should be generally accepted, while postulate (iii) might not be valid except in regions of constant gravitational potential. The second postulate, he thought, ought to be included because of the experimental evidence, while postulate (iv) could only be justified by the simplicity which the laws of nature obtain once it is applied. Nordstrom's theory is derived from a scalar potential and satisfies these four postulates. 43 Among the theories of gravitation which should be regarded as the 'most natural,' Einstein mentioned Nordstrom's theory and his own new theory worked out in collaboration with Grossmann. He derived Nordstrom's theory by assuming Hamilton's principle for a massive (point) particle and invariance under Lorentz transformations, and discussed the two consequences which follow from this theory: (a) light does not get deflected in a gravitational field; (b) the inertia of a particle is determined by the scalar potential <j>, or the larger the masses which are accumulated in the vicinity of a particle the smaller becomes its inertia. Nordstrom's theory had already been discussed in critical detail by Einstein and Fokker, especially its basis on the scalar potential. 44 It had been established that the line element ds depends on (f> according to the relation ds2 = <j>2(dx2 + dy2 + dz2 - c2dt2),

(16)

the differential equation for the gravitational potential taking the form cf>n<j> = kTiTliU,

(17)

where k is a constant and T^ is the general energy(-momentum) tensor. In his lecture at Vienna, Einstein summarized the situation as follows:45 We can sum up by saying that Nordstrom's scalar theory adheres to the postulate of the constancy of the velocity of light and satisfies all the requirements which can be placed

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upon a theory of gravitation, based upon the present experimental knowledge. The only unsatisfactory thing is the fact that according to this theory the inertia of a body seems to be affected, although not produced, by other bodies. Because, according to this theory, the inertia of a body increases the farther the other bodies are removed from it. 46 ' 47 Einstein's main point was that he wished to stick to his four postulates, mentioned earlier, the more so because, with Grossmann, he was on the way to developing another consistent theory of gravitation, using the invariant theory of the absolute differential calculus. He had already stated in the work with Fokker: We note that only the application of the invariant theory of the absolute differential calculus would provide us a clear insight into the formal content of Nordstrom's theory. Furthermore, this method makes it possible for us to determine the influence of the gravitational field on arbitrary physical phenomena, as is to be expected from Nordstrom's theory, without invoking new hypotheses. The relationship of the Nordstrom theory to that of Einstein-Grossmann thus becomes completely clear. Finally, the role which the Riemann-Christoffel differential tensor plays in this [present] investigation suggests the idea, that it might also open the way for deriving the EinsteinGrossmann equations for gravitation independent of physical assumptions. The proof of the existence or non-existence of such a connection would signify an important theoretical advance.48 In other words, Einstein was now seeking to become a mathematician for the sake of the theory of gravitation. 2.4. Einstein and Grossmann: search for the tools of general relativity

mathematical

Einstein returned from Prague to Zurich in the summer of 1912. He had by then already formulated the fundamental physical principles of the general relativity theory of gravitation, and was now searching for their mathematical structure. At the E.T.H. [the Swiss Federal Institute of Technology in Zurich], where he now returned as professor of theoretical physics, Einstein met again his old friend and former fellow student Marcel Grossmann, who was now a professor of mathematics and his colleague. With Grossmann, and under his guidance, Einstein studied the mathematical literature, especially the theory of invariants and the absolute differential calculus of Christoffel, Ricci, Levi-Civita and others. Einstein developed the mathematical structure of his theory jointly with Grossmann, and in his celebrated paper on the general theory of relativity in 1916 he acknowledged the help which his friend had given him. It was Grossmann's help which had, Einstein said, 'spared me not only the study of the relevant mathematical literature, but who [Grossmann] also assisted me in searching for the field equations of gravitation.' 49 This study of the mathematical literature and the search for the proper mathematical tools led to several joint papers with Grossmann during Einstein's all too brief stay in Zurich. These papers contained the first attempts toward a generalized theory of relativity, using the new mathematical tools, and gave full expression to Einstein's earlier physical insights. Although the actual outcome of these studies, which went into

The Historical Origins of the General Theory of Relativity

239

the final theory, seems to be small, their 'psychological value,' to use Einstein's own phrase, was most important. 50 Three papers resulted from the collaboration of Einstein and Grossmann, and Einstein reported on their results in his lecture at Vienna in the fall of 1913. 51 In a fourth paper, 52 written by Einstein alone more than a year after he had moved from Zurich to his position at the Prussian Academy in Berlin, Einstein classified and summarized the achievements of the search up to that time. 53 The work of Einstein and Grossmann seems to have been a highly technical struggle with the formal possibilities offered by the absolute differential calculus. It did not have the simplicity and persuasive elegance of the later theory, mainly because of the fact that Einstein had to find the way of translating the physical postulates into mathematics. Another fact which made Einstein's search a sort of a comedy of errors was his great ability to give, for any mathematical result, a good looking physical 'reason,' and he was very quick in providing such 'explanations.' This capacity of Einstein's to invent physical explanations, as we shall see later on, was the main reason why he missed the right solution during the period between 1912 and 1914, a fact which was lamented by him later on. 50 And the evidence suggests that it was not Einstein's collaborator, the mathematician Grossmann, who was responsible for the delay in the creation of a consistent theory of gravitation. The first two joint papers of Einstein and Grossmann, and Einstein's lecture at Vienna, came at about the same time, and we shall consider them together. 54 After stating the physical postulates, which we have discussed above, Einstein went on to discuss the consequences which would follow from the general invariance of the line element ds, given by ds2 = g^dx^dx" , (18) [where summation is understood over the repeated indices (Einstein convention)] where the symmetric tensor g^ represents the metric dependent on space-time. The ten quantities g^ therefore characterize the gravitational field, replacing the scalar potential of the Newtonian theory and determining the behaviour of clocks and rods. 55 Now the line element, defined by Eq. (18), also determines a Riemannian geometry, and the question arises about the relation between its invariants and covariants, on the one hand, and the mathematical and the physical quantities, on the other. Grossmann, who searched through the mathematical literature in order to assist Einstein, also discussed the first of these questions. 56 In dealing with tensor algebra and analysis, Grossmann introduced the notions of covariant and contravariant tensors of any rank, according to Ricci and Levi-Civita, and of divergence and gradients, using the Christoffel symbols, ri*,ve = \(d9»»/dxe

+ dgllvldxv-dgve/dx>i)

and

Y$e = g**ro,ve.

(19)

The so-called Riemann-Christoffel tensor of the fourth rank plays an important role in differential geometry, and is written out as

We* = dr^/dx? - dT$jd*' + r£„r«, - r^r-,.

(20)

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G r o s s m a n n commented u p o n t h e i m p o r t a n c e of tensors as follows: The overriding significance of these structures [tensors] for the differential geometry of a manifold, defined by its line element, makes it a priori probable, that these general differential tensors could also be important for the problem of the differential equations of gravitational field. [The identical vanishing of the tensor Rtpa provides the necessary and sufficient condition that the differential form can be transformed to the form £,, ± (dx*2)2, where the choice of signs depends on the signature of the manifold.] Indeed, it is possible to specify a covariant differential tensor R^„, of second rank and second order, which might enter those equations, namely Rfj.v = Rfigu • (21) This in itself shows that this tensor, in the special case of an infinitely weak gravitational field, would not reduce to the expression A<£. We must therefore leave open the question as to what extent in the general theory of the differential tensors linked with the gravitational field is related to the problem of the equations of gravitation. Such a connection must exist, insofar as the equations of gravitation allow arbitrary substitutions, because already in this case it seems to be excluded to obtain differential equations of the second order. If, on the other hand, it could be established that the equations of gravitation only admit a certain group of transformations, then one could understand why the differential tensors yielded by the general theory would not be acceptable. As mentioned in the physical part [written by Einstein], as yet we are not in a position to discuss this question. 57 Einstein continued in t h e a d d e n d u m t o their joint p a p e r : In writing up this paper, we have regarded it as a shortcoming that it has not been possible to write down equations for the gravitational field that are covariant generally, i.e. for arbitrary substitutions. Subsequently, I found, however, that equations which determine the g^v uniquely from 0M„ [the energy tensor], and which are uniquely covariant, cannot exist at all, as the following proof shows. 58 These q u o t a t i o n s from t h e 'workshop' of general relativity are i m p o r t a n t because they indicate t h a t Grossmann h a d stressed t h e i m p o r t a n c e of t h e R i e m a n n Christoffel tensor to Einstein, but the latter failed, on account of his 'physical reasoning,' to use it as the fundamental concept. From t h e procedure employed by Einstein, one can see how this mistake came a b o u t . He s t a r t e d from t h e generalized line element, Eq. (18), a n d studied the m o t i o n of a material point in t h e field of gravitation according t o t h e equation Si

{/*}-o. dsl^O,

(22)

from which t h e equation of motion follows d2x>*

_ u dxe dx"

,

s

Einstein could now reformulate t h e laws of physics in t h e presence of a gravitational field, given by the metric g^. T h e energy tensor plays an i m p o r t a n t role in w h a t he called t h e 'laws of material p h e n o m e n a . ' 5 9 Now, on generalization of the

The Historical Origins of the General Theory of Relativity

241

energy tensor (the special relativistic quantity) in the presence of a gravitation field in which the g^ are not constant, the newly formed quantity T^u is not conserved, and one finds

*rc+S>=0.

(24)

v dxv ' Here the new Va represents the energy tensor of the gravitational field, and Einstein motivated its existence by saying: 'It thus represents the physical fact that in a gravitational field the impulse and energy of a material system vary with the time, in that the gravitational field transports energy and momentum to the material system.' 59 The fact that a gravitational energy-momentum tensor exists, has important consequences: 'Since we demand the validity of the conservation laws, we restrict the reference frame to the corresponding extent, and thereby renounce the setting up of the equations of gravitation in general covariant form.' 60 Thus it seemed that the whole programme of general relativity had broken down, as was correctly pointed out to Einstein after his Vienna lecture by the probings of Mie and Reissner. 61 What error had really crept into the framework of the theory?

In generalizing the Poisson equation (9), which can be written as a tensor equation by a generalization of Newton's theory, Einstein looked for an expression of the form 7 V = r M" -

(where 9^ =Tflv),

(25)

where T^, is a covariant tensor of the second rank, 7 a constant and 0MJ/ or TM„ the covariant energy tensor. In analogy with the Newton-Poisson law, one postulates with Einstein that Eqs. (25) are of the second order. 62 ' 63 The difficult question now arose about the tensor generalization of (p. Einstein appended the following remarks to the addendum in his joint paper with Grossmann. 64 It must, however, be emphasized that it seems to be impossible, under this assumption [that the equations are of second order] to find a differential expression r M „ which is a generalization of A(j> and turns out to be a tensor under arbitrary transformations. It cannot, however, be denied a priori that the final exact equations of gravitation can be of higher than the second order. Thus there always exists the possibility that perfectly exact differential equations of gravitation might be covariant with respect to arbitrary substitutions. The attempt at a discussion of such possibilities would, however, be premature in view of the present status of our knowledge about the physical properties of the gravitational field. We are thus restricted to [the equations of] the second order, and must therefore abstain from setting up equations of gravitation which would turn out to be covariant with respect to arbitrary transformations. Moreover, it must be emphasized that we have no clue regarding a general covariance of the equations of gravitation. The idea was the following: If the metric <7M„ were determined entirely by the energy tensor where 6^ or TM„, and all general transformations were admitted, then a contradiction would arise. The minimum transformation group, under which the quantity Y^ is permitted to be covariant, is a group of linear transformations. Einstein and Grossmann proposed that

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The Golden Age of Theoretical

1

Physics

~ ^dx"\9

+

29

V

9

9

dxPj

dx« dx0

9

9Te

dx"dxP

49 9 dx* dx? •

(2t>j

The equations which determine the gravitational field could then be simplified to A""( ff ) = 7(0'" / + A'"')>

(27)

with

v=

,r

"-e)-£sM'* %r)- ''"

re

dg»T dgve

dxa

dxP

and

^7A - yg g

dxa dxl3

2s

g ^

QxP

j.

The sum of the 'energy tensors' of matter and that of the gravitational field satisfy the conservation law, Eq. (24). Einstein and Grossmann, in their third paper of 1914, drew consequences from the existence of the energy tensor of the gravitational field, £M„ (#*" = A'"'). 5 1 They 'showed' that one could not have generally covariant equations (25), which determine the characteristic metric g/iU, thereby improving their work of 1913. Taking the derivative of Aau, and summing over u, one obtains four conditions

B

^e^{^~"a"9^)=°-

W

The equations of gravitation (27) are covariant with respect to all 'justified' transformations of the coordinate system, i.e. those which satisfy the conditions (28). In order to interpret these conditions, one considers a variational principle,

5J = S I Hdr = 0,

(29)

with

where H can be regarded as the Hamiltonian of the gravitational field. The allowed coordinate system is then 'chosen such that, for the fixed boundary values of the coordinates and their first derivatives (considered in an arbitrary coordinate system), the integral J approaches a maximum.' 65 One could, therefore, say that 'since the conditions Ba = 0, with the help of which we have restricted the coordinate systems, are the immediate consequence of equations of gravitation, our considerations show that the covariance of the equations is as far-reaching as imaginable.' 66 The fourth and final paper of the series in 1914, dealing with the results of the Einstein-Grossmann collaboration, 52 which Einstein communicated (in his own

The Historical Origins of the General Theory of Relativity

243

name alone) to the Prussian Academy, did not add anything new to the theory except some mathematical clarifications.67 Einstein, by that time, had mastered the mathematical tools himself, and he did not need his friend any more. 68 Yet, the theory of Einstein and Grossmann was not Einstein's final theory. 2.5. The foundation

of general

relativity

In the spring of 1914 Einstein moved from Zurich to Berlin to take up his appointment as a member of the Prussian Academy and director of the research institute of physics at the Kaiser-Wilhelm-Gesellschaft. 69 The new position provided him with full freedom for his research work, an activity for which Planck and his colleagues had estimated him so highly. Having succeeded sufficiently in his first attempt to generalize relativity, or at least in making a plan of its generalization, Einstein was not so sure if he could do much more at that time, and remarked to his friend and former fellow student Louis Kollros on the occasion of his farewell from Zurich: 'The Germans are betting on me as a prize hen; I am myself not sure whether I am going to lay another egg.' 70 Lay a few golden eggs he still would in Berlin in both the fields of quantum theory and general relativity. The latter was completed in Berlin soon after his arrival there, as far as the problem of gravitation was concerned. 71 Before Einstein went to Berlin, what he and Grossmann had achieved was an Entwurf— a plan or outline of the shape which, they thought, the theory of general relativity would take. In this initial sketch, the mathematical ideas had played a smaller role than one would have expected. 'The general theory of invariants acted as an obstacle. The direct way turned out to be the only one practicable,' Einstein wrote to his friend Michele Besso in March 1914.72 During most of the year 1914 Einstein was almost content with the results of his labours with Grossmann, achieved in 1913. Early in 1915, however, he was again at work, starting by drawing the experimental consequences which the theory might have. On 12 February 1915, he wrote to Michele Besso: 73 I am able to inform you about two beautiful scientific matters: (1) Gravitation. Red-shift of spectral lines — the spectroscopic double stars have the same average velocity at the radius of vision. The masses of the stars are obtained from the periodic Doppler fluctuation of the lines. The component due to the heavier star should have a mean red-shift of the spectral lines relative to the other. This has been verified. Since it is possible to estimate the radii of the stars (from their spectral type), it has even provided a satisfactory quantitative verification of the theory.' [The other scientific matter which Einstein reported to Besso was the experimental verification of his Ampere-molecular current hypothesis.] At three consecutive sessions of the Prussian Academy — on 4, 11, and 18 November 1915 — Einstein presented his new theory. Opening his first contribution on the 'general theory of relativity,' he remarked: During the past [several] years, I have tried hard to construct a general relativity theory on the assumption of the relativity [also] of non-uniform motions. I actually thought that I had discovered the only law of gravitation corresponding analogously to the postulate of general relativity, and sought to demonstrate the necessity of exactly this solution in

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a memoir published last year in these proceedings. [See Sitzungsberichte, 1914, pp. 10661077, Ref. 59.] A fresh examination showed me that the necessity [of the solution] from the method proposed there cannot be proved at all; that this seemed to be so was due to an error. The postulate of relativity, to the extent that I have invoked it there, is always satisfied if one takes Hamilton's principle as the basis; in fact, however, it does not provide any means of determining the Hamiltonian function H of the gravitational field. It actually affects the equation restricting the choice of H [Eq. (28)] in no other way, except that H should be invariant with respect to Unear transformations, a requirement which is of no use in accomplishing the relativity of accelerated [motions]. Moreover, the choice [for the Hamiltonian] indicated [by Eq. (29)] is in no way determined [by Eq. (28)]. For these reasons I lost all faith in the field equations which I had set up, and began to look for a path which restrict the possibilities in a natural manner. Thus I reverted to the requirement of the general covariance of the field equations, from which I had departed with a heavy heart in the first place three years ago when I worked together with my friend Grossmann. In fact, at that time we had already come quite close to the solution of the problem given in the following. Just as the special relativity theory is founded upon the postulate that its equations must be covariant with respect to Unear, orthogonal transformations, the theory developed here is based on the postulate of the covariance of all systems of equations with respect to the transformations of the substitution determinant 1. The fascination of this theory would hardly leave anybody who has reaUy grasped it. It represents a real triumph of the method of the general differential calculus founded by Gauss, Riemann, Christoffel, Ricci and Levi-Civita. 74 T h e restriction on general transformations, with t h e substitution d e t e r m i n a n t 1, simplifies t h e formal foundations of t h e theory of general relativity considerably. For instance, it allows one t o relate covariant a n d contravariant indices much more easily. T h e most important change in a t t i t u d e , however, was t o express t h e problem of gravitation not by the artificial forms introduced earlier, b u t by t h e R i e m a n n Christoffel tensor as suggested by G r o s s m a n n already in 1913. 7 5 Gravitation is connected with a tensor of t h e second rank, a n d Einstein, therefore, now chose t h e reduced Riemann-Christoffel tensor Rfiv = R^QU •

(30)

He split this tensor into two terms,

where

R^ = ~-Q^- + n,r;e

(3i)

a n d R'^ vanishes wherever v^—0 = 1In considering t h e conservation law for t h e energy, Einstein h a d earlier assumed t h e components of t h e gravitational field t o b e described by

The Historical Origins of the General Theory of Relativity

but now he found the Christoffel symbol r j j this purpose.

245

to be a more a d e q u a t e expression for

T h e most i m p o r a n t question concerning the generalization of the Poisson equation h a d now been answered by t h e Ansatz76 R»v = -jT^

.

(33)

These equations can be written in the Hamiltonian form with J?=:«rT^lt

a )

(34)

and t h e conservation of energy is satisfied with

-£x{T°+t°)=0

(35)

where

Here again, t h e 'energy tensor' of t h e gravitational field (which, however, is a tensor only with respect to linear transformations) occurs. 7 7 T h e reason for its existence is the curvature of space, which does not allow for a conservation law of T$ alone. Now t h e Ansatz (33) brings a problem along with it, since it d e m a n d s t h a t IJ=0,

(36)

for V~~£ — 1- Hence there seem t o exist conditions which restrict t h e allowed coordinate frames. 7 8 Already in a previous note, in fact, Einstein proposed t o take T£ = 0, since t h e electromagnetic energy-momentum tensor shows this property. 7 9 He s t a t e d explicitly: We must remember, however, that according to our knowledge, matter is not to be conceived as (primitively) given and physically simple [entity]. And it is not just a few who hope that [the conception of] matter can be reduced to purely electromagnetic phenomena which, however, would follow a more complete theory than Maxwell's electrodynamics. Let us suppose for a moment that in such a complete electrodynamics the scalar of the energy tensor would also disappear! Would then the result just mentioned above prove that matter cannot be constructed with the help of this theory? I expect to be able to contradict this. Because it is indeed quite possible that gravitational fields constitute an essential part of matter to which pertains the expression given above. Hence T% can be apparently positive for the entire structure (matter + gravitational field), while in reality only (T^ +1%) is positive and T£ vanishes everywhere. In the following, we shall assume that the condition T% = 0 is actually fulfilled in general.80 O n e can t h e n fix the coordinates by assuming t h a t yf^g = 1. In his contribution of 25 November 1915, 8 1 his fourth on the new approach during t h a t November, Einstein went a n o t h e r step further, realizing t h a t t h e assumption of t h e zero energy

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The Golden Age of Theoretical Physics

scalar is not necessary, if t h e equations of gravitation would b e changed t o R„v = -7 (T^

-

^ T )

,

(37)

where T = T£. T h e reason for this new Ansatz is t h a t now t h e tensors of energy, m a t t e r a n d t h a t of the gravitational field occur always in t h e same way in all equations. Einstein, therefore, concluded: 8 1 With that, the general relativity theory is finally completed as a logical structure. In its most general formulation, which turns the space-time coordinates into parameters devoid of physical significance, the principle of relativity leads, with conclusive necessity, to quite a definite theory of gravitation which can explain the perihelion motion of Mercury. 82 On the other hand, the postulate of general relativity reveals to us nothing more about the character of the other natural phenomena than what the theory of special relativity had already done. The opinion which I recently expressed in this regard [in these Proceedings] was wrong. 82 Every physical theory conforming to special relativity can, with the help of the absolute differential calculus, be presented in the framework of the theory of general relativity, without the latter offering a criterion for the validity of that theory. 83 Einstein was filled with t h e joy and excitement of achievement, when he wrote to Besso again on 10 December 1915: 7 3 The boldest dreams have been realized. General covariance. Perihelion motion of Mercury, wonderfully exact. The latter is perfectly determined from the astronomical point of view, because the determination of the masses of the inner planets by Newcomb was based upon periodic perturbations (not the secular ones). This time [the most natural mathematical assumptions] at hand turned out to be correct. Grossmann and I had thought that the conservation laws would not be satisfied, and Newton's law would not emerge in the first approximation.... Einstein sent his new p a p e r s t o his friend Michele Besso, a n d w a n t e d him to read t h e m . 8 4 T h e winter of 1915 a n d the spring of 1916 was p e r h a p s t h e period of greatest scientific contentment and happiness in Einstein's life. 85 T w o questions arise immediately with respect to Einstein's derivation of the equations of gravitation. First, how cogent a n d necessary is t h e derivation of Eq. (37)? Second, how independent of gravitation are t h e other laws of n a t u r e ? Hilbert took a stand with respect t o b o t h of these questions. First, he derived the equations of gravitation, E q . (37), from a set of axioms. Second, he showed t h a t the same set of axioms also leads to conditions for t h e laws of n a t u r e , in particular electromagnetism, which cause m a t t e r to exist. 3. P h y s i c s a n d G e o m e t r y 3 . 1 . The role

of axiomatization

in Hilbert's

work

To David Hilbert, one of t h e greatest m a t h e m a t i c i a n s of the twentieth century, axiomatization signified a guiding principle of the s t r u c t u r e of m a t h e m a t i c s and

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the mathematical laws of physics. That Hilbert's approach towards the foundations of mathematics was too optimistic, and essentially unfeasible, was established by Godel's theorem more than a decade before Hilbert died. In his own work, Hilbert systematized and unified numerous branches of mathematics: theory of invariants (1885-1893), theory of algebraic number fields (1893-1899), theory of analytic functions and variational problems (1899-1909), and the theory of linear integral equations (1902-1912). Hilbert's work on linear integral equations extended the pioneering work of Predholm, and led to the introduction of the celebrated 'Hilbert space,' which would provide the fundamental basis of the mathematical formulation of quantum mechanics. 86 The prelude which determined the second half of Hilbert's mathematical life, certainly from 1912 onwards, was stated in his famous book Die Grundlagen der Geometrie, which first appeared in 1899. 87 Hilbert worked on the axiomatic foundations of geometry almost until 1909. Much later, after 1922, he turned to a systematic axiomatization of mathematics, including mathematical logic. In between Hilbert sought to apply his axiomatization methods to selected fields at the frontiers of physics, having declared that 'physics is much too difficult for the physicists.' 8 — The study of minimum principles on which physics could be based had occupied Hilbert even before he formally undertook the axiomatization of physics in 1912, as his earlier papers indicate. Hilbert had lectured on physics in Gottingen since 1902. After the death of his friend Hermann Minkowski in 1909, he lectured more regularly on physics from 1910 to 1918. Minkowski had introduced him to several fields in physics. 88 Minkowski's death in January 1909 left him very lonely and sad — and the axiomatization of physics seemed to be an active and loving tribute to his memory. Hilbert's efforts picked up slowly, and beginning in 1912 they really took on. Hilbert had always been interested in the fundamental problems of physics, but now he was really inspired. In 1913 he organized the 'Gottinger Gaswoche' which brought together Planck, Debye, Nernst, Smoluchowski, Sommerfeld, Lorentz, and several others to lecture on and discuss the problems of the kinetic and quantum theories. 89 Another indication of his strong and continuing interest in physics was the fact that he had already established, several years before the First World War, a special position for an assistant who would work with him in physics, a position which he normally offered to Sommerfeld's students. 90 Starting in 1912, numerous students wrote dissertations on various problems of theoretical physics under Hilbert. 91 The subjects which Hilbert chose for his work in physics were characteristic of his mathematical temperment. In his early lectures he dealt with the foundations of mechanics, using Gauss' minimal principle as the fundamental axiom. 92 Other than mechanics, Hilbert found it quite desirable to pursue the axiomatization of thermodynamics and kinetic theory, radiation theory, electricity and gravitation (including their unification), and later on quantum theory. Under Hilbert's inspiration at Gottingen, Caratheodory sought to axiomatize thermodynamics, 93 and Paul Hertz discussed its 'mechanical' foundations. 94

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One of the important problems in the programme of axiomatization of physics was the examination of the consistency of the axioms and their independence, and Hilbert's fields of research were determined by it. Hilbert first turned to the mathematical foundations of kinetic gas theory. 95 He then dealt with the general formulation of Kirchhoff's laws of radiation without referring either to the classical wave theory or the quantum theory. 96 Inspired by the success of the special theory of relativity, and the progress of Einstein's work on general relativity and the theory of gravitation, Hilbert also entered upon research in this frontier of physics, making original and important contributions. 97 It is well known that Hilbert vigorously stimulated research on the mathematical foundations of quantum mechanics, in particular through his collaborator Von Neumann. The new atomic mechanics had its first breakthrough (matrix mechanics) in Gottingen, and Heisenberg lectured on it in Hilbert's seminar already in the fall of 1925. 98 Hilbert not only guided Jordan in his work on the transformation theory, 99 but entered the field himself, first lecturing on the foundations of quantum mechanics at Gottingen, and then publishing a paper jointly with Von Neumann and Nordheim. 100 This paper also initiated Von Neumann's work on the foundations of quantum mechanics, leading to his famous book on this subject. 101 For the purpose of this study, we shall deal only with Hilbert's search for and foundation of a unified theory of natural phenomena, based upon a unification of electricity and gravitation in a single mathematical scheme. Three principal ingredients determined Hilbert's research on this subject: (i) the mathematical inheritance of the nineteenth century, in particular the theories of space and the concept of invariance in geometry and dynamics; (ii) the ideas of Gustav Mie concerning the theory of matter; and finally, (iii) Einstein's formulation of the theory of gravitation. A review of the status of these 'ingredients' is of interest before discussing Hilbert's contributions to the 'foundations of physics.' 97 Einstein was strongly critical of Hilbert's work but others, such as Weyl, tried to follow it and improve upon it by bringing in other concepts of their own. Ironically, it was Einstein, who had pooh-poohed Hilbert's programme of a unified theory of matter, who would himself spend the last three decades of his life on this search along lines similar to those of Hilbert's — and that too during a period when discoveries of new particles and forces had rendered the entire approach doubtful. 3.2. Concepts

of space and

geometry

The concept of space has played an important role since the earliest times. The Greek philosophers and mathematicians, the medieval Christian thinkers and the men of science of the Renaissance had their notions of space. Newton finally decided upon 'absolute space' as the theatre of natural phenomena, and Kant made it into an a priori of experience. 102 The decisive progress in the ideas of space and geometry was, however, initiated by the great mathematicians of the early 19th century, especially Gauss, Lobatchevski and Bolyai, who first discovered the existence of

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non-Euclidean geometries. 103 In his survey, known as the ' Hannoversche Gradmessung1 [the Hanover Survey] Gauss even attempted to determine as to which geometry of space would be valid in nature. Although his attempt was not successful, in that it did not prove (nor did it disprove) that the geometry of space was compatible with Euclidean geometry, he wrote to Olbers in 1817: I became more and more convinced that the necessity of our geometry [i.e., Euclidean] cannot be demonstrated, at least neither by, nor for, the human intellect. In some future life, perhaps, we may have other ideas about the nature of space which, at present, are inaccessible to us. Geometry, therefore, has to be ranked until such time not with arithmetic, which is of purely aprioristic nature, but with mechanics.104 It was Riemann who developed Gauss' ideas. He introduced his conception in his famous 'trial' lecture [Probevorlesung] ' On the hypotheses which lie at the basis of geometry,''105 which he delivered at Gottingen on 10 June 1854. He took note of the fact that he could not base the concepts of a new geometry on any available wealth of the existing mathematical literature, but could only refer to Gauss' 'second memoir on Biquadratic Residues in the Gottingen Gelehrte Anzeiger, and in his Jubileebook, and some philosophical researches of Herbart.' 105a Riemann introduced the general line element ds, n

ds2 = YJ 9^udxlldxv

(38)*

in the n-dimensional space, and the concept of the geodesic in curved space. His investigations led to the so-called Riemannian geometry which can be characterized by two axioms: 106 Axiom I. The differential coefficient ds/dt at a given point on the curve depends only on the derivatives dx^/dt at this point and not on higher derivatives or on the behaviour of the curve elsewhere. Axiom II. ds/dt is the square root of a quadratic form of the dx^/dt: ds

dx» dxv

/ a

Tt = i ^-dT-

.„„. M

< >

Axiom II, in fact, implies Axiom I. The important parameters g^, are called the metric of the space, i.e. they determine whether it is Euclidean or not. Riemann concluded his lecture with the remark: 'The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space This leads us into the domain of another science, of physics, into which the object of this work does not allow us to go today.' 107 Clifford not only translated Riemann's paper into English, but also tried to extend his ideas to a 'Space Theory of Matter.' 108 He proposed the thesis that the 'variation of curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal.' 109 Other mathematicians *In modern, notation, Eq. (38) is written as Eq. (18).

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followed much less the physical speculations of Riemann, than seeking to work out the scheme of his geometry. Another line of mathematical development by men like Beltrami, Christoffel, and Lipschitz, placed emphasis on certain aspects of differential geometry, making connection with invariance problems. Christoffel influenced Bianchi and Ricci in their development of the absolute differential calculus for the theory of invariants of quadratic differential forms. Levi-Civita, a disciple of Ricci's, carried on the programme further, and in 1917 introduced the remarkably useful concept of parallel displacement. The French school later on contributed to differential geometry through Elie Cartan. Rather than going into the details of these mathematical developments, let us mention an important extension of the geometry of space, namely, the introduction of the time as another coordinate of physical space by Hermann Minkowski. On 21 December 1907 Minkowski submitted a lengthy memoir entitled 'Die Grundgleichungen fur die elektromagnetischen Vorgange in bewegten Korpern' to the Gottingen Academy. 110 He discussed a new formulation of the laws of physics, especially mechanics and electrodynamics, on the basis of the work of Lorentz, Poincare and, more particularly, his former Zurich student Einstein who had 'most acutely' expressed the 'new concept of time.' 1 1 1 The important point of Minkowski's considerations was that he treated the time on an equal footing with the space coordinates and invented the notion of the four-dimensional space-time continuum.112 Minkowski's new formal and intuitive space-time geometry opened the path for the later researches of Einstein, leading to the creation of general relativity on the basis of Riemannian geometry. It is remarkable that at about the same time as Minkowski, Hargreaves quite independently suggested the use of space-time vectors. 113 His concept of using spacetime as a unified geometrical concept was, however, only indirect in that by organizing space and time derivatives together the corresponding equations gained a much simpler form. In fact, both Hargreaves and Poincare had used the equivalence of space and time coordinates in the mathematical forms which they had employed, 113 without giving it the full physical and intuitive emphasis that characterized the approach of Minkowski. It may be of interest to note, however, that the first scientist, who suggested a connection between the space and time dimensions in modern science, seems to have been the physiologist Fechner in the 1840's. 114

3.3. Space-time

and

invariants

On 3 June 1868, Hermann von Helmholtz submitted to the Gottingen Academy of Sciences an extended version of his lecture %'On the facts which lie at the basis of geometry.' 115 He introduced his memoir with the remark: My researches on spatial perception in the field of view have motivated me to undertake the investigation of the problem of the origin and nature of our perceptions of space in general. In this regard, the question which impressed upon me, and which obviously belongs in the

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domain of the exact sciences, was chiefly this: How much objectively valid meaning do the axioms of geometry have? How much, on the other hand, depends only on definitions, or deductions from definitions, or on the form of the representation?116 Helmholtz, who had worked, independently of Riemann on similar ideas, now extended the latter's conception. 117 He discussed in particular that hypothesis of Riemann, according to which the line element is equal to the square root of a homogeneous function of the second degree, as in Eq. (39). This hypothesis, Helmholtz claimed, could be derived on the basis of more general assumptions. He started by considering the motions of a solid body in space. The 'translation invariance' yields \n(n — 1) equations in an n-dimensional space. In addition there might be the possibility of congruence under rotations, leading to further equations. All of these equations can be formulated as differential expressions if applied to infinitesimally small coordinate differences. The important point, which he could demonstrate, was that there exists a quantity of second degree in the differentials which is invariant under all the symmetry transformations. Together with the two other postulates, that space should have three dimensions and that it should extend to infinity, Helmholtz regarded the foundations of geometry as being fully laid. The ideas of Helmholtz were both conservative and progressive at the same time. On the one hand, he limited himself to infinite spaces of three dimensions; on the other, he stressed the use of symmetry transformations and invariants of a geometry which became a turning point for the later development. The concept of an invariant had been introduced by Sylvester in 1851. 118 The theory of invariants was systematically developed on the basis of Grassman's theory of 'Stufen,' from which the tensor calculus also arose. 119 The next step was to determine the invariants of quadratic forms and their classifications, to which Jacobi, in particular, contributed. Felix Klein created a new approach to geometry when he announced his 'Erlanger Prograrnm' in 1872. 120 In this programme he considered a geometry as being given by a 'manifold on which a transformation group is defined,' and he looked for the invariant theory with respect to the group.' 1 2 1 Beginning in 1872 Sophus Lie, who had influenced Klein, developed the theory of continuous groups. In his lectures in Gottingen on the development of mathematics in the nineteenth century, Klein declared that it was only a small step from his Erlangen Programme to the theory of general relativity. 122 The theory of invariants of a symmetry group, which was so very much present already in Helmholtz' lecture, became familiar to the physicists and mathematicians through the work of Klein and Lie. Jacobi had in fact already found a connection between the Euclidean invariance of the mechanical Lagrangian and the conservation laws for linear and angular momenta. 123 Finally Hamel, unaware of the work of his predecessors, proceeded systematically to exploit the connection between symmetry transformations and conserved quantities in mechanics. 124 Hamel extensively used Lie's work on continuous groups, 125 deriving the result that the coefficients in the Lagrange equations are constants

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if the related infinitesimal transformations generate a symmetry group, and if the coefficients are constants then the conservation laws are satisfied.126 A generalization of Hamel's work, without making any reference to it, however, was made by Herglotz in 1911. 127 He discussed the ten-parameter Poincare group 128 of 'motions' in the four-dimensional space, and related its symmetries to ten general integrals, among which the most familiar are the energy, momentum and angular momentum. The other three conserved quantities correspond to a generalization of the center-of-mass integrals in classical mechanics. As we have already mentioned, Hilbert himself knew much about invariants. Indeed, by solving the fundamental problems of the theory of invariants Hilbert had killed the subject. 129 In addition, he had with him a young expert from the mathematical centre of the theory of invariants by the name of Emmy Noether. Emmy, daughter of Max Noether, came from Erlangen where Klein had developed his famous 'programme,' where Clebsch had worked and Gordan had been the 'king of invariants.' In Gottingen around 1915, where Hilbert, at this stage in his distinguished career, was buried in his studies on the foundations of physics and general relativity, where Klein whose great mathematical programme was being carried forward by the 'relativists' (and who, in his latter years, had again become interested in current research), Emmy Noether's work became very important. 1293 Even before Klein knew about Noether's work, he had requested Engel, the old collaborator of Sophus Lie, to derive all the ten known integrals of the nonrelativistic mechanical n-body system 'in the spirit of Lie.' 130 But the most general and systematic treatment was already available in Emmy Noether's work, which Klein communicated to the Gottingen Academy only in 1918, and which became celebrated later on. 131 The important generalization in her work lay in the fact that she studied both finite continuous groups, i.e. those that depend continuously on g parameters e, as well as infinite continuous groups depending on g continuous functions. Noether also dealt with mixed groups, in the sense defined, and established two theorems in her fundamental paper: 132 (1) If the variational integral is invariant under a finite continuous group Ge, then g linearly independent combinations of the Lagrangian expressions become divergences, and vice versa. This theorem is also valid in the limit of infinitely many parameters. (2) If the variational integral is invariant under an infinite continuous group C?ooe, in which arbitrary functions and their derivatives up to the cr-th order appear, then g identical relations are satisfied between the Lagrangian expressions and their derivatives up to the order a. The inverse of this theorem is also true. Emmy Noether established the most general equivalence between the invariance under symmetry transformations of the classical equations of motion of the action integral, from which they can be derived, and the conservation of certain quantities which generate the group in the sense of Lie. Her work was carried on, or explained better, in terms of physics by Bessel-Hagen. 133 Noether's theorem obtains a much

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simpler form in quantum theory due to the fact that the 'complicated' Poisson bracket of classical theory is replaced by the 'simple' commutator. 134 3.4. Mie's field theory of

matter

Gustav Mie made the first attempt to construct a complete theory of matter in the twentieth century. 135 In particular, he wanted 'to explain the existence of an indivisible electron and to relate the phenomenon of gravitation to the existence of matter.' 1 3 6 Three assumptions formed the basis of his theory: (i) Electric and magnetic fields exist inside the electrons as well. 137 (ii) The principle of (special) relativity is valid throughout. (iii) 'The hitherto known states of the ether, namely the electric field, the magnetic field, the electric charge, and the charge current are entirely sufficient to describe all phenomena in the material world.' 138 Concerning these fundamental assumptions, Mie remarked that the second one is valid beyond doubt (!), and the third can only be justified by its success. As regards his concept of matter, Mie thought that the atoms consist of a number of electrons bound together by a dilute positive charge as in the Thomson model. Moreover, the atoms are surrounded by atmospheres which, since the whole atom is neutral, do not cause electric but gravitational fields. Mie thought that the enormous intensity of the [electromagnetic] field creates spatial concentrations of charge which we signify as electrons, to which the ordinary Maxwell equations do not apply. He further remarked: The electron in my theory does not occupy a well-defined portion of space in ether. Rather, it consists of a 'nucleus' which goes over continuously into an 'atmosphere' of electric charge. The 'atmosphere' extends to infinity, becoming extremely thin in the neighbourhood of the 'nucleus,' so that one cannot detect it experimentally in any way.139 Prom his three assumptions, Mie was led to a generalization of the equation for the ether. This extension of the Maxwell-Lorentz theory was determined on the basis of the validity of the principle of energy conservation and the existence of a localizable energy. Mie did not quite realize how much freedom was left to him, and how many more assumptions were built into his derivation than was necessary. The Hamiltonian H played an essential role in Mie's theory, where H depends on what he called 'quantities of magnitude,' @ (electric displacement), 3*1? (magnetic field), Q (charge density) and / (electric current). Prom this function H, one obtains the 'quantities of intensity,' £ (the electric field), 38 (the magnetic induction), and the corresponding 'charge' e and 'current' i, by taking the derivative with respect to the correlated 'quantities of magnitude.' Or, inversely, one may start with a Hamiltonian, $ of the 'quantities of intensity' and derive the 'quantities of magnitude' by proper differentiation. Apart from this general structure, the Hamiltonian H should be invariant under Lorentz transformations, consistent with the assumption (ii). In addition, Mie simplified the form of the Hamiltonian by

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using the hypothesis: H =

H(a,p),

with
and

p

= g2-j?>2.

(40)

The detailed results which Mie obtained on the basis of his assumptions are both complicated and incomplete, and we shall only indicate certain features of his conclusions. He claimed, in particular, to have shown that it was possible to have stable electrons in the sense in which he had conceived them, with the electromagnetic and gravitational fields having the following properties: (i) In the interior of an electron § and £P, as well as <j> and g, must have opposite signs. (ii) The gravitational field must be represented by a four-vector. (hi) The inertial and gravitational masses of a particle are identical only if there are no motions in the interior of the particle. The mass ratio and the gravitational constant thus turn out to be functions of the temperature, decreasing with increasing values of the latter. Mie's theory obviously ran into serious difficulties. Some of these are connected with the fact that no one has succeeded in deriving solutions for static electrons in which the charge is 'quantized.' In fact, all the trial solutions arc consistent with an arbitrary choice of the total charge. Even more serious is the fact that Mie's theory depends on an absolute potential. The equations of motion do not remain unchanged if one replaces the potential <j> by <j> + const., due to the fact that the skew symmetric tensor, dxi

dxk '

in the interior of the electron depends on the potential Ak explicitly. A material particle will therefore not be able to exist in a constant external potential field.140 Mie's theory suffers from having been made much too early. Thus it could not properly recognize the existence of the energy quantum or forces other than the electromagnetic and gravitational ones. 141 Moreover, his approach to gravitation was very 'conventional' and would soon be overthrown by the new revolutionary ideas of Einstein and Hilbert. 3.5. Hubert's

communication

to the Gottingen

Academy

What we have discussed in the previous sections represents, briefly, the state of affairs when Hilbert, in 1915, wrote his first note on the foundations of physics. During the first World War Hilbert remained fully absorbed in physics. In the summer of 1914 Debye had joined the faculty of the University of Gottingen and organized, at Hilbert's request, a seminar on the structure of matter. Alfred Lande was Hilbert's assistant for physics at that time, and he kept Hilbert informed about the developments in quantum theory, but he left in December 1914 to join the

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Red Cross. 142 At this time, when only a few students attended the Hilbert-Debye seminars, Hilbert became fascinated with the new ideas of Gustav Mie and Einstein, and he sought to find a connection between them. Einstein visited Gottingen at Hilbert's invitation and presented his ideas there. In a letter to Arnold Sommerfeld, dated 15 July 1915, Einstein wrote: 'I had the great joy of seeing in Gottingen that everything [about the theory of relativity] is understood to the last detail. With Hilbert I am just enraptured. An important man!' 143 Hilbert was no less fascinated by his encounter with Einstein, and he devoted the following several months to the study of gravitation and electromagnetism in the search for a unified field theory of matter. At about the same time when Einstein presented [on 4, 11, 18 and 25 November 1915] a series of communications on the general relativity theory of gravitation to the Prussian Academy, Hilbert presented [on 20 November 1915] a derivation of Einstein's equations of gravitation to the Royal Academy of Sciences in Gottingen. Hilbert's communication appeared in the third issue of the Proceedings of the Gottingen Academy for 1915, and in his published version he referred to all of Einstein's communications of November 1915 to the Prussian Academy. There can be no doubt that Hilbert's derivation was entirely independent. The dates of the papers involved, and remarks made in these papers by both Einstein and Hilbert, are quite clear. In the first three contributions, up to November 18, Einstein still used the gravitational equation (33) without the term involving the trace. Only in the last paper, read on November 25, Einstein gave the form of Eq. (37) to the equations, saying that 'the field equations for vacuum, on which I have based the explanation of the perihelion motion of Mercury, remain unaffected by this modification.' 144 This would indicate that he had found the new improved equations between 18 November, when he gave the 'explanation' of the perihelion motion of Mercury, and 25 November, when he presented his new note on 'The Field Equations of Gravitation.' Hilbert reported his own derivation of the equations of gravitation on 20 November, a few days ahead of Einstein, certainly in the public domain. In his published paper Hilbert, realizing this situation, pointed out: 'It seems to me that the differential equations of gravitation so realized [by me] are in agreement with the beautiful theory of general relativity proposed by Einstein in his later [25 November 1915] memoir.' 145 Hilbert had indicated quite correctly that he shared, at the very least, with Einstein, the derivation of the field equations of gravitation. Indeed, Hilbert's was more like a derivation than Einstein's, which was based more on the simplicity of the Ansatz than on a full grasp of the implications of the structure of the equations. Hilbert's achievement in deriving the field equations of gravitation is not generally known, even among 'general relativists,' and seldom cited in the literature. 146 Of course, there never was any quarrel over priority between Hilbert and Einstein, who admired one another deeply. 147 Hilbert once remarked: 'Every boy in the streets of Gottingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians.' 148 Or in a

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public lecture: 'Do you know why Einstein said the most original and profound things about space and time in our generation? Because he learned nothing at all about the philosophy and mathematics of time and space.' 149 It was also Hilbert who recommended the award of the third Bolyai Prize in 1915 to Einstein 'for the high mathematical spirit behind all his achievements.' 150 3.6. Axioms

of Hilbert's

theory of

matter

Hilbert had studied the work of Abraham, Nordstrom, Mie and Einstein very thoroughly. He tried now, using his powerful mathematical tools including the variational calculus and the theory of invariants, to construct a consistent theory of matter on axiomatic grounds. In particular he was attracted by the 'giant propositions' of Einstein and the 'deep ideas and original concepts' on which Mie had built his electrodynamics. 151 He thus proposed: 'Following the axiomatic method, in fact from two simple axioms, I would like to propose a new system of the basic equations of physics. They are of ideal beauty and, I believe, they contain the solution of the problems of Einstein and Mie at the same time.' 152 Starting from four 'world parameters' x,i(fj, = 1,...,4), characterizing the world points in the space-time continuum, Hilbert assumed that the following quantities determine the processes at x11 (zeroth axiom): (i) the ten gravitational potentials g^ of Einstein, exhibiting the form of a symmetric tensor under general transformations of xM; (ii) the four electromagnetic potentials A^ forming a vector with respect to general transformations. Now the physical processes are determined by two axioms: 153 Axiom I (Mie's axiom of the world-function): The laws of physics are determined by the world-function H, containing the arguments 9\ivi

9IMU,Q -

and AfifAfi^g

~ dxe _dAfl

-

9tiV,QtJ

dxedx"

(e>
(42)

such that the variation of the integral J HJgdAx,

( 5 = det| 5M1/ |,

d4x = dx°dx1dx2dx3)

(43)

vanishes for any of the 14 potentials g^, and A^. Axiom II (axiom of general invariance): The world-function H is an invariant with respect to an arbitrary transformation of the coordinates x^. Concerning this axiom Hilbert remarked: Mie had already stated the requirement of orthogonal invariance. In the above-mentioned Axiom II, Einstein's fundamental and basic idea of general invariance finds its simplest expression, although with Einstein the Hamilton principle plays only an auxiliary role. His H functions are in no sense general invariants, nor do they contain the electric potentials.154

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Hilbert correctly noted the fact that Einstein's first publication of November 1915 still contained restrictions upon the allowed transformations. It seems that the necessity of the general invariance of all physical laws first became clearer to Hilbert than to Einstein. Of course, Hilbert had good reasons for that. Besides being a mathematician who was always interested in utmost generality and greatest possible elegance, he had an important argument for such an assumption; he had discovered a mathematical theorem which made the connection between the laws of gravitation and of electrodynamics possible. 155 3.7. Hilbert's fundamental theorem, and the world function

energy

conservation,

(i) Hilbert's Theorem Hilbert formulated his fundamental theorem, on which he based his other derivations, in the following form: Let J be an invariant under arbitrary transformations of the four world-parameter, comprising n quantities and their derivatives. Prom the variation sfj^gd4x = 0,

(44)

with respect to the n quantities, n Lagrange (variational) equations can be formed, such that in this invariant system of n differential equations for the n quantities four of them are always a consequence of the remaining n — 4, in the sense that between the n differential equations and their total derivatives four combinations, independent from each other, are always identically satisfied.156 This theorem of Hilbert's is a consequence of the second part of Noether's theorem. In fact, Emmy Noether had been in close contact with Hilbert in those days. 157 The consequences of the application of this theorem, together with the axioms I and II, are of great interest. From Hamilton's principle, and the form of H in accordance with axiom I, Hilbert derived 14 Lagrangian equations for the 14 potentials. Ten of these equations contain the variations with respect to the gravitational potentials g^v and are called the equations of gravitation; while four stem from the variation with respect to the electrodynamic potential A^, and give rise to the generalized Maxwell equations. However, Hilbert said: Because of the theorem stated, the four equations [which describe the generalized Maxwell equations] can be regarded as a consequence of the equations [of gravitation]. On account of this mathematical theorem, we can therefore immediately make the assertion that, in the sense indicated, the electrodynamic phenomena are the consequences of gravitation. In the recognition of this fact, I discern the simple and very surprising solution of the problem of Riemann, who was the first to investigate theoretically the connection between gravitation and light.158 Hilbert's theorem thus led him to prove the connection between the laws of gravitation and the laws of electrodynamics. This result was not quite in agreement with the spirit of Einstein, who had explicitly stated that general relativity

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does not impose any restriction on the laws of physics, gravitation excluded. But even Einstein agreed in his paper of 1916 in the Annalen der Physik that the field equations of gravitation 'completely provide the equations of material processes if the latter can be characterized by four independent differential equations.' 159 More light was thrown on this problem by Hermann Weyl who stressed the concept of gauge groups, of which the general transformations were an example. (ii) Energy

Conservation

From axioms I and II Hilbert derived the energy-momentum vector in the form e

A _ Hp umxx

x LAx „ Ax -a„A -bx-c -d ,

(45)

with ,A

dH

w

^ r = p r + (r^ e " + r^p^) and dxK '

PK

{g^va-gi",vva--9V°Pl),

=

P

and

6A = B > ^ , dH

A

and

d dH dH

\

x_

r*

dH dAKtX

and

dH

1 ( d dJgH 2Jg\dx»dA^

PK,

d^gH dA^P

\ ")•

This energy vector, although it still depends on an arbitrary vector p", satisfies identically the invariant energy conservation equation

Hilbert's derivation clearly used the fact that his world function H was a general invariant. The question now was, how the energy theorem, Eq. (46), had anything to do with the conservation of energy in Einstein's theory? Hilbert's approach was equivalent to Einstein's theory [as presented on 25 November 1915 (Ref. 83)], and Felix Klein would later on take up the crucial question of energy conservation in Einstein's theory.

The Historical Origins of the General Theory of Relativity 259 (iii) The Form of the World Function Hilbert took over Einstein's postulate that the equations of gravitation should depend only on the second derivatives of the potentials (in axiom I). He then concluded (Axiom III) that H must be of the form H = R + L,

(47)

where

R=

sTR^

is the invariant deriving from the Riemann tensor, the curvature, and where

(48)

'-K'-^-sO-

Rfiv is the contracted Riemann-Christoffel tensor of Eq. (21). For the invariant L Hilbert proved the fact, that dL

dL + *T-=0.

(49)

leading to the result that the derivatives of the electrodynamic vector A^ occur only in the form £iu> ~ Afj,v — Av
V9 (R»» - \R9»J) = - ^

•

(51)*

•Hilbert wrote his Eqs. (51) in the equivalent form R/iv ~ —Rg^v — —Tixu ,

(1)

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The Golden Age of Theoretical Physics

Under t h e condition t h a t t h e energy m o m e n t u m tensor of m a t t e r is completely given by t h e electrodynamic function L, t h a t is dL T*> = ^dg^' 7 .

(52)

E q u a t i o n s (51) t u r n out t o b e identical with Einstein's equations (37). In his p u b lished p a p e r on ' T h e Foundations of Physics,' Hilbert noted t h e agreement between his equations a n d those formulated by Einstein in his theory. 1 4 5 T h e i m p o r t a n c e of Hilbert's result lies in t h e fact t h a t t h e equations of electrodynamics immediately follow from t h e variational principle, by t a k i n g t h e variation with respect t o t h e electromagnetic potentials, or symbolically, [y/gL]h=0,

(53)

once t h e equations of gravitation (51) are given. T h i s is t h e consequence of t h e Hilbert-Noether theorem. As Hilbert noted, ' This is the precise mathematical expression of the general assertion made above about the character[-istic feature] of electrodynamics as a consequence of gravitation.'162 In fact, Hilbert regarded his derivation of Einstein's equations of gravitation, a n d generalization of t h e equations of electrodynamics of Mie, as a t r i u m p h of t h e axiomatic m e t h o d . He remarked: As one can see, the few simple assumptions stated in the axioms I and II are sufficient, by analogous interpretation, for the construction of the theory. With [the help of] these [axioms], not only are our conceptions of space, time and motion reconstructed in a fundamental way, in the sense put forward by Einstein, but I am also convinced that through the basic equations set forth here the hitherto concealed phenomena inside the atom would find the closest explanation. In particular the reduction, in general, of all physical constants to mathematical constants must be possible, because it is only then that the possibility would come closer that, in principle, from physics would arise a science of the type of geometry — surely [it would be] the greatest glory of the axiomatic method which, as we have seen here, is the most powerful tool that analysis, namely variational calculus and invariant theory, can employ. 163 Hilbert was obviously fascinated by the possibility t h a t a unified theory of gravitation a n d electromagnetism would explain t h e existence of elementary q u a n t a [of m a t t e r a n d radiation], a possibility which Einstein would pursue t h e rest of his life. thus defining T^.u by the equation d

^ l^r

L

rnT = VaT^.

Now multiplying (1) by g**" and summing over (i and u, one obtains R-2R=~T

or

R = T.

(2)

Substituting this into (1), one obtains R,xv = —Tfn, + -Tg^v which is Einstein's equation (37) for 7 = 1.

(3)

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But in 1916, a n d during the following several years, Einstein did not t h i n k much of such a possibility. To Rudolf H u m m , a student from Hilbert's circle in Gottingen, had gone t o Berlin in the spring of 1917 to hear Einstein's lectures a n d t o talk to him, Einstein confided some of his thoughts at t h a t time. O n a visit which he paid to Einstein in May 1917, H u m m noted: He [Einstein] is a bad calculator, he said; he rather works conceptually. He does not seem to believe to be right what we are doing in Gottingen. He himself has never thought so formalistically. His imagination is firmly tied to reality. He is very careful, and entirely a physicist. He does not rush immediately to generalize as we do in Gottingen. He explains this [attitude] by saying that he bad to rid himself of his prejudices very deliberately. That's why he did not grasp straight away how a general invariance could exist. Rather, he had to come to this view step by step, which subsequently seemed to be very plausible indeed. But before that, he had real aversion to it because the quantities employed there — the curvature tensors — had seemed to him very unclear. I [Humm] suggested to him that he had taken recourse to the quantum theory in order to modify the theory of gravitation, while Hilbert on the other hand wanted to derive the quantum theory from the theory of gravitation. At this he made a roguish grin. This would not do, he said, even though the theory of gravitation was more general. But [in his view] the ideas of relativity could not yield anything more than gravitation. 164 T w o questions are now important. First, the response of Einstein t o Hilbert's first memoir on the foundations of physics. Einstein m a d e only a few direct or indirect comments on this, referring t o it in particular in his celebrated p a p e r on the theory of general relativity in the Annalen der Physik in t h e early spring of 1916, which we shall discuss next. We shall then t u r n to Hilbert's second memoir on the foundations of physics, in which he went beyond Einstein in t h e physical interpretation of t h e new theory, before we deal with t h e second vital question: W h a t h a d Hilbert's theory t o do with Einstein's? Were t h e y in any sense equivalent? 4. C o m p l e t i o n of t h e T h e o r y of Gravitation On 20 March 1916, Albert Einstein's p a p e r on ' T h e foundations of t h e theory of general relativity' arrived at t h e Annalen der Physik.165 A long p a p e r of m o r e t h a n fifty printed pages, it established the theory of gravitation within t h e framework of general relativity, and gave a complete account of t h e fundamental principles and t h e most i m p o r t a n t consequences. 4 . 1 . General

covariance

and the equivalence

principle

General relativity signified a generalization of Einstein's special theory of 1905, because it now postulated t h a t 'the laws of physics must b e so formulated t h a t they remain valid with respect to frames of reference in arbitrary m o t i o n . ' 1 6 6 [Einstein thought t h a t this generalization would satisfy Mach's critique of mechanics.] An important consequence of this postulate, Einstein noted, would be t h a t 'In the theory of relativity, t h e magnitudes of space and t i m e cannot be defined such t h a t

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the differences of the space coordinates can be measured directly with a measuring rod, and that of time with a normal clock.' 167 The introduction of a frame of reference only simplifies the description of coincidences, on which the laws of physics can be built. Of course, one has to say how the experiments which deal with events in space and time have to be described. This connection, in Einstein's theory, was made through the line element in which the metric g^u entered, and the important generalization became: If the new space-time coordinates x1 • • • x 4 are introduced by means of an arbitrary substitution, then in the new system the g^v are no longer constants but functions of space-time. At the same time, the motion of a free mass point in the new coordinates becomes curvilinear, rather than uniform, whereby the law of motion becomes independent of the nature of the moving mass point. We shall interpret such motion as taking place under the influence of a gravitational field. We note that the appearance of a gravitational field is tied together with the variation of g^ with respect to space-time. Also in the general case, where we cannot ensure the validity of special relativity in a finite region by an appropriate choice of the coordinates, we will have to hold on to the view that the gar describe the gravitational field. In the theory of general relativity, gravitation plays an exceptional role above all other forces, in particular the electromagnetic. This is so because the ten functions gaT which represent the gravitational field also determine the metrical properties of the fourdimensional space.168 Although the last remark might seem to pinpoint the difference in the points of view of Einstein and Hilbert, it is quite consistent with Hilbert's results. In fact Hilbert had clearly stated that once the equations of gravitation are given, and thus also the metric, important consequences for the laws of electromagnetism would follow. That Einstein did not like this conclusion, he made amply clear later on. It is perhaps worthwhile at this point to mention the discussions about the physical meaning of general relativity, the principle of equivalence in particular, which Einstein had with Friedrich Kottler and Erich Kretschmann, soon after the publication of his paper on general relativity in 1916. Kottler raised the question whether Einstein, in his final formulation of general relativity, had given up the real principle of equivalence. 169 Einstein insisted that this was not so and, in any case, it depended on what one understood by the principle of equivalence. Insofar as he was concerned, this principle was the very 'basis of my theory.' 170 The essential point about the equivalence principle, in Einstein's view, is its dynamical nature. By no means can one assert that the gravitational field can be thought of as being purely kinematical. A 'kinematic, non-dynamical formulation of gravitation' is not possible. By merely transforming from a Galilean system to another through the acceleration transformations, we do not learn about arbitrary gravitational fields, rather only about those particular ones which must also satisfy the same law as all gravitational fields. This is again only another formulation of the equivalence principle (especially, in its application to gravitation). A theory of gravitation violates the equivalence principle in a sense, as I understand it, only if the equations of gravitation are not satisfied in any reference frame which moves nonuniformly relative to a Galilean frame of reference. It is quite evident that this criticism

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cannot be leveled against my theory with the general covariant equations, for in this case the equations are valid with respect to every frame of reference. The requirement of general covariance of the equations includes that of the equivalence principle as a special case. Kottler's other questions referred to how fundamental were the various concepts that had been used in Einstein's theory, in particular the 'energy tensor of the gravitational field.' These, Einstein replied, were convenient names rather than deep concepts; the only real concept in general relativity, according to him, was general covariance. Kretschmann's attempts were directed towards another question related to the equivalence principle. 172 He started from the observation that all physical laws could be brought into a general covariant form by mathematical re-arrangement. He thought: It must be possible, however, to ascribe some significance to the postulates of realtivity other than the purely formal mathematical one. For only when such a significance is present could one explain, for instance, the obvious and generally recognized impossibility of adapting the notion of the rigid body (which, more than anything else, is so easy to characterize by means of purely topological features) to Einstein's original theory of relativity.173 Kretschmann sought to sharpen the physical and conceptual content of relativity by considering an example in four-dimensional geometry and attempting to bring it into agreement with a generally valid formulation of an arbitrary postulate of relativity. 173 He emphasized that: A system of physical laws of fulfills the relativity postulate of a group (G) of invariant transformations if, and only if, for every representation of arbitrary form of precisely these laws of the system, the physically valid (and thereby distinguishable, in principle, from all others by observation) inertial systems required for its representation form such a large set (under all the possible physical conditions permitted by the laws), that the set of transformations relating them comprises the group G in some fashion or is identical with it.17i According to this statement, the validity of the relativity postulate for a system of physical laws is independent of the form of its mathematical expression, and is determined entirely by its physical content. Einstein's relativity postulate was weaker than this, Kretschmann insisted, and only stated that all coordinate frames were possible. The theory of general relativity, he said, did not satisfy any sharply defined principle. 'In a physical sense, Einstein's theory does not satisfy any postulate of relativity at all; in its content it is a perfectly "absolutist" theory.' 175 In his reply, re-stating the principles of the general relativity theory, Einstein stressed the heuristic value of general covariance. 176 He maintained that the theory did in fact lead to a new equation for the gravitational field, and that was significant. 4.2. Theory of the gravitational

field

After a lengthy section on a treatment of the mathematical tools in which, for instance, he introduced the sum convention [i.e., sum always over two equal indices

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in different positions 1 7 7 ], Einstein presented the theory of t h e gravitational field in his famous p a p e r of 1916. He had already, in t h e m a t h e m a t i c a l section, argued t h a t g could not vanish: Should V ~ S vanish at a spot in the four-dimensional continuum, then it signifies that a finite coordinate volume there corresponds to an infinitesimal 'natural' volume. This, however, may not be so anywhere, and therefore the sign of g cannot change. Following special relativity, we shall assume that g always has a finite negative value. This represents a hypothesis about the physical nature of the continuum under consideration and at the same time a determination of the choice of coordinates. If, however, y/^g is always positive and finite, then it is obvious that a posteriori the choice of coordinates can be made such that this quantity is equal to l. 1 7 8 T h e equations t h e n become much more simplified. Einstein developed t h e theory in two steps. First, he set u p t h e equations of gravitation without the presence of m a t t e r . We distinguish in the following between the 'gravitational field' and matter in the sense that everything outside the gravitational field is regarded as 'matter' — i.e. not only 'matter' in the ordinary sense, but also [including] the electromagnetic field.1 9 In t h e case of special relativity, i.e. g^ being constants, t h e Riemann-Christoffel tensor given b y Eq. (20) is identically zero. T h e generalization of this situation clearly requires Rp, = Rleu=0.

(54)

Einstein s t a t e d : It must be pointed out that only a minimum arbitrariness is connected with these equations. For, other than R^, there is no tensor of the second rank connected with it which can be constructed from the g^v and their derivatives, which does not contain higher than second order derivatives, and is linear in them. 1 8 0 A n d he added, That these equations, which follow from the requirement of general relativity in a strictly mathematical way and, together with the equations of motion, yield Newton's law of attraction in the first approximation and the explanation of the excess (i.e. the excess remaining after the perturbation correction has been included) perihelion motion of Mercury (discovered by Leverrier) in the second approximation, should in my opinion be convincing regarding the correctness of the theory. 181 Einstein t h e n derived t h e field equations from a Hamiltonian principle, using t h e Hamilton function

H = g^T^TL.

(55)

T h e energy of t h e gravitational field is t h u s conserved, or

&"° with

<56>

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Although i" is not a tensor, the conservation equation is valid in all systems with y/^g = 1, and therefore the energy and momentum of the gravitational field are conserved. The generalization of the equations of gravitation (54) starts with the assumption of a symmetric energy tensor Tap.182 Since Eq. (54) can be written as

£zV'^) = ^(p-\rJ) with * = *£,

(57)

one immediately obtains the equations of gravitation in the presence of matter by replacing it through

l&W'W

= -7 i% + TZ)-\s°(t + T)

(58)

Equation (58) is equivalent to the Hilbert-Einstein equation (37). Concerning this 'derivation,' Einstein remarked: We must admit that the introduction of the energy tensor of matter by means of the postulate of relativity alone is not justified. For this reason, we have derived it above from the requirement that the energy of the gravitational field should act [i.e. gravitate] in the same way as every other form of energy. The strongest reason for the choice of the above equations [Eqs. (58)] is that they yield the conservation relations for the components of the total energy (energy and momentum) as a consequence.183 The energy momentum conservation can thus be obtained directly from the field equations (58). The occurrence of the gravitational energy term in the conservation equation is equivalent to Ffpa

d£ = -T"^-

( 59 )

This condition expresses the influence of the energy of the gravitational field, and Einstein remarked, referring to Hilbert's four auxiliary conditions: Thus the field equations of gravitation contain four simultaneous conditions, which material processes must satisfy. They yield the equations of a material process completely, provided the latter can be characterized by four independent differential equations.184 Immediately afterwards, however, Einstein stated that all laws of physics could be generalized into a covariant form without restricting the possibilities offered by special covariance. This fact explains why it is not necessary to introduce special assumptions about the nature of matter (in a restricted sense). In particular, the question may remain open whether or not the theory of the electromagnetic and gravitational fields together provides a sufficient basis for the theory of matter. The postulate of general relativity, in principle, cannot teach us anything about it. Only the further development of the theory will show

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whether the principles of electromagnetism and gravitation together can accomplish what the former [electromagnetism] alone cannot.185 4.3. Consequences

and applications

of the theory

of

gravitation

Einstein generalized Euler's equations of hydrodynamics and Maxwell's equations of electrodynamics, without changing the structure of the latter. 186 His treatment stands in sharp contrast to Hilbert's, for whom his own derivation of Mie's equations of electrodynamics from the equations of gravitation, in the limiting case of special relativity, meant a particular triumph. For this, Einstein criticized Hilbert rather sharply. In a postcard to Ehrenfest he wrote: I do not like Hilbert's formulation. It [Hilbert's formulation] is needlessly specialized and, as far as matter is concerned, unnecessarily complicated. It is not honest (= Gaussian) in design, [and reflects] the pretension of a superman by a camouflage of techniques.187 Einstein's triumph was not the field theory of matter but the correct theory of gravitation, which led to Newton's theory and the bending of light in the first approximation and to the rotation of the perihelion of Mercury in the second. The approximation which reproduced Newton's law of gravitation was that the metric g^v becomes Minkowskian in the limit of spatial infinity, and that the term dxA/ds is large compared to the other quantities [dx1/ds, etc.]. The equation of motion of a massive particle, in a static gravitational field, thus becomes dV _

1 dg00

~dW-~2~dxT'

.

(* = 1 ' 2 ' 3 ) >

( 6 °)

in agreement with Newton's result. Only the potential #00 plays a role in this approximation and the Poisson equation becomes Agoo = 1Q, (7 = 8nk/c2 = 1.87 x 10 _27 c.g.s. units).

(61)

For the bending of light by the sun, Einstein found 1.7 seconds of arc. 188 The calculation for the rotation of the perihelion of Mercury was based upon the vacuum Eqs. (54). Making an Ansatz of radial symmetry, one finds in the first order 9ke = ~6kt - a ^ r , 9ko=9ok, 1

Poo = 1

(4 = 1,2,3),

( A , / = 1,2,3), (62)

a

• r However, in the second order, the motion of a planet of small mass deviates from the Kepler-Newton laws. The orbit obtains, in the direction of the motion, a small rotation of magnitude

The Historical Origins of the General Theory of Relatimty

267

where a is the semi-major axis, e the eccentricity of the orbit, T the period of the planet, and c the velocity of light. 188 Soon after the publication of his memoir of 1916 on the foundations of the theory of general relativity, Einstein began to seek new conceptual developments of the theory. One of the first such developments was the introduction, in a note on the approximation integration method for the field equations of gravitation, of the concept of gravitational waves. 189 Prom the Ansatz Qnv = -r)»v + AM„ ,

(64)

Einstein stated that Vn =''722 = ?733 = - 1 = 1744;

Viiv=0

for

p±v

where, in first approximation, h^ is small compared to 1. According to a proposal of de Sitter, the first order calculation is simplified by not choosing coordinate systems with yf—g = 1. Choosing new small parameters defined by Tfw — "^1/

(65)

„ VlivV ftgo 1

the equations of gravitation become cPl^v dxa2

d2-yua dx»dxa

&lpa. dxvdxa

<927Q/3 = tlv a dx dxP

2jT„u,

(66)

and reduce to V

dx°dxP

7

""

(67)

if ("681 v dx" ' ' 190 The latter restriction can be imposed because of general covariance. This reasoning of Einstein is exactly the same as Hilbert's. Prom Eq. (67), one finds that gravitational fields propagate with the velocity of light, and the potentials /iM„ or 7^,, can be calculated from the T^, |^or^(»o,ito>za,t-r)> 7 /

"~

(69)

27T

where dVo,= dxodyodzo. Since T^v is of the second order in h^, Ansatz and evaluating carefully, one finds

using a wave

where a

ei T iaeo = 0,

(70)

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and one obtains a solution of the problem with (a!22 - 033)

o

2

+«23

(71)

Waves with 0:22 — 0:23 and a\z equal to zero are spurious. Considering the total emission of gravitational radiation from a body, Einstein found that it is always finite, and in conclusion he remarked that, 'Already in the earlier treatment, we had emphasized that the final result (requiring an energy loss from bodies) of this consideration must arouse doubts about the general validity of the theory. It appears that an improved quantum theory would also bring about a modification of the theory of gravitation.' 191 4.4. The cosmological

term

Einstein's second extension of the theory of gravitation was his introduction of the so-called cosmological term or constant. 192 Starting from the consideration that Poisson's equation does not entirely replace Newton's theory of gravitation if one does not add the condition that the potential becomes finite in the limit of spatial infinity, Einstein looked for similar conditions in the theory of relativity. This condition, however, leads to difficulties even in Newton's theory, because the density of matter then has to be zero at infinity and stars could escape into the space beyond. A possibility of avoiding this conclusion would be to assume a very high value of the potential at infinite distances. 'This would be a feasible way, if the form of the gravitational potential were not determined by the celestial bodies themselves,' Einstein thought. 193 If equilibrium is assumed, with the presence of a finite potential difference between the centre and the infinity of space, the ratio of the densities must be finite; given a zero density at infinity, there must follow a zero density everywhere else. Einstein then noted: These difficulties would hardly be overcome within the framework of Newtonian theory. One might ask whether they could be avoided by a modification of Newton's theory. Here we shall propose a way for which no claim is made to be taken seriously by itself, but it serves to bring the following into better focus. Replace Poisson's equation by, A-X<j> = 4-Kkg,

(72)

where X denotes a universal constant. Let go be the (uniform) mass density, then o = — ^ - 0 0

(73)

is a solution [of Eq. (72)].194 Einstein thus introduced a finite average density of matter. The real density distribution leads to accumulations of g, and over and above the constant potential 4>Q a Newtonian potential due to 4nkg is added. Space thus constructed could extend to infinity, but this would not be true in general relativity if the equations

The Historical Origins of the General Theory of Relativity 269 of gravitation were modified by the 'cosmological term.' Einstein's reasoning was that, 'In a logically consistent theory of relativity, there can be no inertia relative to space, but only a mutual inertia of masses. Hence if a mass is removed spatially far enough from all other masses of the universe, then its inertia must go to zero.' 195 An infinite space does not meet this Machian condition, Einstein concluded, after computations of models with Grommer, because one cannot find suitable boundary values at infinity. But if the universe could be imagined as a spatially closed but unbounded continuum, boundary conditions of that kind would be totally unnecessary. I will show below that both the postulate of general relativity, and the fact that the velocities of fixed stars axe insignificant, are consistent with the hypothesis of a spatially closed, unbounded universe. However, in order to carry through this fundamental idea, it is necessary to make a general modification of the field equationa.196 Einstein considered a spherical space with constant curvature and radius a, leading to the metric 9u = -&u + -5—r~2~,—T~<—2\ a* — (xf + x% + x%)>

(&>^ = 1>2>3)>

goo - 1

(6 = 1,2,3).

and

5*0 = 0,

(74)

This metric is compatible, not with the equations of gravitation (37), but with the generalized equations [i.e., those including the cosmological term], Rtiv - ^3tiv = —7 ( Tpv - -xQiivT \

(75)

For sufficiently small A, this field equation is in any case also consistent with the empirical facts pertaining to the solar system. It also satisfies the conservation of momentum and energy, because one obtains Eq. [(75)] instead of Eq. [(37)] if, in Hamilton's principle, in the place of the scalar of Riemann's tensor one uses this scalar together with an additional universal constant. In this way [i.e. using Hamilton's principle] the conservation laws are assured. That the field Eqs. [75] are consistent with our notions concerning field and 197 matter will be shown below In Hilbert's formulation, the field Eqs. (75) could be obtained very easily, just by replacing R by R + A in Eq. (47). For the cosmological term A, Einstein obtained the expression A = i .

(76)

He was now ready to sum up these considerations: The theoretical conception of the real world corresponding to our ideas would therefore be the following. The curvature of space varies with the temporal and spatial distribution of matter, but it can be roughly approximated by a sphere. In any case, this notion is logically consistent, and from the standpoint of general relativity the most obvious; whether it is also valid in the light of current astronomical knowledge will not be discussed

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here. In order to achieve this logically consistent formulation, we had to introduce a new generalization of the gravitational field equations, which however is still not justified by our actual knowledge of gravitation. It should be emphasized, however, that a positive curvature of space by the presence in it of matter also comes about if the additional term [universal constant] is not introduced. The latter is only necessary in order to allow for a quasi-static distribution of matter, which corresponds to the fact of the insignificant velocities of fixed stars. 1 9 8 In any case, Einstein was very content with general relativity, a n d he felt really fulfilled as confirmations of his predictions were obtained, t h e most important p e r h a p s being t h e measurements of t h e deflection of light from t h e sun which were carried out during the solar eclipse expedition of 1919 led by Eddington. W h e n Eddington's cable with the results of t h e expedition arrived, Use Rosenthal-Schneider was with Einstein and he showed it t o her. At her joy t h a t the results agreed with his theory, 'He remarked totally without emotion, "I knew t h a t t h e theory was correct." A n d when I told h i m what if there had been n o verification of his prediction, he replied, "I would have felt sorry for the dear Lord! T h e theory is, of course, all right."'199 5. P h y s i c a l I n t e r p r e t a t i o n o f G e n e r a l C o v a r i a n c e In his p a p e r of 1916 on general relativity, Einstein sought to interpret t h e meaning of general covariance. A n important postulate of all physical laws is their causal structure, a n d a causal 'law has significance for t h e empirical world only if observable facts alone a p p e a r as causes a n d effects.' 2 0 0 T h e p o s t u l a t e of general covariance takes away from space and t i m e the last vestige of physical objectivity All of our assertions concerning space-time always amount to the measurements of space-time coincidences. If, for instance, the physical process is only the motion of a material point, then in effect nothing more could be observed than the encounters of two or three such points. The results of our measurements are in fact nothing but the statement of such encounters of material points of our apparatus with other material points, or coincidences respectively between the pointers of clocks, dials, and other point-events under observation, occurring at the same place and time. 201 Writing t o Besso on 31 October 1916, after a visit t o Holland, Einstein elaborated these points further: The objective significance of space and time lies in the first instance in that the fourdimensional continuum is hyperbolic, in such a way that originating from every point there are spatial and temporal line elements, i.e. those for which ds2 > 0 and ds2 < 0 respectively. The coordinates x11 per se do not have spatial or temporal character. In order to uphold usage, those systems are to be preferred for which goo{dx°f

> 0,

gnidx1)2

+ 2gi2dx1dx2

+ ••• + g33(dx3)2

< 0.

However, an objective justification of this kind is not involved in this choice. Hence the 'spatial' or 'temporal' character is real. But nature has not made one coordinate temporal and other spatial. 2 0 2

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It was exactly on this basis that Hilbert, in his axiomatic approach to the foundations of physics, developed his own physical interpretation of covariance. 203 5.1. Axiom

of physical

interpretability

In his second memoir on 'The foundations of physics,' presented to the Gottingen Academy on 23 December 1916, Hilbert developed further the axiomatic foundations of general relativity by emphasizing 'some more general questions of a logical and physical character.' 204 He considered the Riemannian metric g^ in x — t space and formulated, what he later on called his Axiom IV, the space-time axiom. He asserted: 'The quadratic form

G(xl,x2,xz,xA)

=

Y,9^xv

is such that, in its representation as a sum of four linear quadratic forms of x^, there always appear three square [terms] with the positive and one square [term] with the negative sign.' 205 The quadratic form, given above, therefore gives the metric of a pseudo-geometry in the four-dimensional world of x11. Hilbert then denned the space- and time-lines, calling the segment of a curve x^ = x^ip), for which

a 'space-line,' with the length

A=

/*f(¥)-

(re)

and a segment of the curve, where

G

(Hr)<°

<79>

a 'time-line' with the proper time [i.e. the time as measured by an observer moving on this world-line],

/ * / For a 'null line,' therefore,

He remarked: In order to elucidate this notion of pseudo-geometry, let us imagine two ideal measuring devices: the measuring string, by means of which we can measure the length A of any

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path; and, second, the light-clock by means of which we can determine the proper time of any time line. The measuring string reads null and the light-clock stands still along every null line; whereas the string cannot be used along the time line, the clock cannot be used along the space line.206 With either of these devices it should be possible to compute the values of g^ as a function of x*1, if only one introduces a certain space-time frame of reference. It is, moreover, straightforward to axiomatize the pseudo-geometry suitably.

5.2. Restrictions

due to causality

The above considerations have important consequences for the problem of causality, as Hilbert pointed out: Until now we have considered all coordinate systems x^ , which arise from any arbitrary transformation, as equivalent. This arbitrariness must be restricted as soon as we insist that two world points on the same time-line can be related to each other by cause and effect, and therefore it should be impossible to transform such world points into simultaneous world points. By designating x° as the proper time coordinate, we define: For a proper space-time coordinate system, not only g ^ 0 always, but the four inequalities ffll

012

021

922

ffii > 0,

>o,

9ii

912

gi3

921

922

923

g3i

932

333

> 0,

goo < 0

(82)

are also always satisfied. A transformation which takes one such space-time coordinate system into another space-time coordinate system is called a proper space-time transformation.207 Hilbert's treatment was so straightforward and classical that Pauli took over an entire section of this paper in his review article on the theory of relativity. 208 Reichenbach did the same in his axiomatization of general relativity. 209 In his memoir, Hilbert went on to discuss the usual principle of causality, namely the postulate that from the knowledge of the physical quantities and their appropriate derivatives in the present, they can also be determined in the future. In general relativity this principle seemed to be violated, but it could be restored in the following way: 'From a knowledge of the 14 potentials g^v^A^ at a given moment, one can necessarily and without ambiguity infer all facts of interest about them at a later time, if only they have a physical meaning.' 210 By introducing physically meaningful results Hilbert sharpened the formulation of the causality principle, thereby completing the axiomatic structure of the new physics: As I have pointed out, physics is a four-dimensional pseudo-geometry, whose metric g,,v is related to the electromagnetic quantities, that is matter, by the basic equations (4) and (5) of my first communication. With this realization, a long-standing geometrical problem can be readily resolved: namely, the problem whether, and in which sense, Euclidean geometry (which is only known from mathematics to be a logically consistent structure) has validity in reality.

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273

The old physics, with its notion of absolute time, assumed the axioms of Euclidean geometry and made them the basis of every physical theory. Even Gauss hardly differed from this [approach]. He constructed hypothetically a non-Euclidean physics, retaining the concept of an absolute time, and waiving only the axiom of parallels of Euchdean geometry. The measurement of the [inner] angles of a triangle of large dimensions convinced him of the invalidity of this non-Euclidean physics. The new physics of Einstein's general relativity principle takes an entirely different approach to geometry. It does not rely upon either the Euchdean or any other geometry in order to deduce the actual laws of physics. Instead, the new theory of physics at once yields, from one and the same Hamilton principle, the geometrical and physical laws [i.e. the fundamental equation (4) and (5) of Hilbert's first memoir]. These equations show how the metric gM„ which also gives the mathematical expression for the physical phenomenon of gravitation, is connected with the values of the electrodynamic potentials A^. The Euchdean geometry is as foreign to modernphysics as action-at-a-distance. In so far as the theory of relativity rejects it [Euchdean geometry] as the general basis of physics, it [the theory of relativity] teaches us that geometry and physics are equal in rank, comprising one science resting upon a common foundation. 211 5 . 3 . Hilbert's

conclusion

Hilbert concluded t h a t t h e only solution of the equations of m a t t e r (i.e. the equations of gravitation and electricity), which is consistent with Minkowskian geometry, is one in which all sources of m a t t e r a n d electricity are removed, because from t h e equations of gravitation in this metric it follows t h a t

^ = 0 .

(83)

T h i s is b o t h a necessary a n d a sufficient condition. It is impossible to obtain a regular metric, which at the same time is not pseudo-Euclidean and yet corresponds to a universe without electricity, by the variation of the metric of pseudo-Euclidean geometry under the assumptions I and II. [The first assumption being that the variation from the pseudo-Euclidean geometry does not depend on the time variable, and the second that the variations should behave regularly at infinity.] 212 Central to Hilbert's conclusion is t h e assumption t h a t t h e metric <7M„ remains regular. Hilbert emphasized his belief t h a t 'it is only t h e regular solutions of t h e fundamental equations of physics t h a t represent reality immediately., 2 1 3 He also believed t h a t t h e solutions which were not regular did provide a n i m p o r t a n t m a t h e m a t ical tool in approximating characteristic regular solutions, citing t h e Schwarzschild solution of Einstein's equations, which are singular at r = 0, as an example. 2 1 4 W i t h these considerations Hilbert concluded his t r e a t m e n t of t h e foundations of physics, a work whose significance in the development of the general relativity theory of gravitation has seldom been appreciated. 6. H a m i l t o n ' s P r i n c i p l e a n d t h e C o n s e r v a t i o n L a w s In his celebrated Encyclopaedia article on the theory of relativity, Pauli explicitly acknowledged Hilbert's formulation of the law of gravitation, Eq. (37), a n d remarked:

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At the same time as Einstein, and independently, Hilbert formulated the generally covariant field equations. His presentation, though, would not seem to be acceptable to physicists, for two reasons. First, the existence of a variational principle is introduced as an axiom. Secondly, of more importance, the field equations are not derived for an arbitrary system of matter, but specifically based on Mie's theory of matter.215 Pauli's comments about Hilbert's formulation, in which he had introduced the variational principle as an axiom, merely reflected the prevailing taste of the day. During the late nineteen-tens and early twenties [Pauli's article was published in 1921], several prominent theoretical physicists shared a belief that a Hamiltonian principle should not necessarily be used to describe the laws of physics. The main reason for this feeling was that there existed great difficulties at that time with respect to the description of the laws of microscopic physics. Although many of these difficulties were removed after the creation in 1925 of a consistent theory of atomic phenomena, there has again been a resurgence of doubt whether the laws of high energy physics can be formulated by using a variational principle. In the macroscopic domain, however, with which, according to expert scientific opinion, general relativity seems to deal principally, Hilbert's procedure of assuming a Hamiltonian principle as a fundamental axiom is now regarded as being well justified. 216 The question persisted, of course, whether the two theories, Hilbert's and Einstein's, were identical. Lorentz and Einstein dealt with this question initially, and the proof was finally completed by Klein. 6.1. Lorentz

and the Hamilton

principle

in Einstein's

theory

In a letter to Lorentz, dated 16 August 1914, Einstein discussed several details of the theory of general relativity which he had developed with Grossmann, in particular the meaning of energy conservation. 217 Already in 1915, before the new developments of the theory were published by Einstein and Hilbert, Lorentz studied the problem of the conservation of energy in a note on 'Hamilton's principle in Einstein's theory of relativity,' which he submitted to the Royal Dutch Academy of Sciences.217* Lorentz thought that the discussion of some aspects of Einstein's theory of gravitation would perhaps gain in simplicity and clarity, if it were based on a principle similar to that of Hamilton's, so much so indeed that Hamilton's name might properly be attached to it. Now that Einstein's theory is available, we can easily determine how this variational principle should be formulated for different kinds of systems and for the gravitational field itself.218 Einstein had already shown that one could derive the equation of motion of a material point in the gravitational field from the Lagrangian L = - T O — = -my/g^vPv" where v" =

dx^jdt.

(84)

The Historical Origins of the General Theory of Relativity

275

Equation (84) could be generalized to describe continuous matter if mv^ were replaced by QV11, and L expresses a Lagrangian density. Lorentz then derived the equations of motion under a given force, and finally turned to the motion of a charged particle in electromagnetic and gravitational fields. From the theory of electrons two sets of equations arise. Lorentz considered one of these sets as defining the mathematical description of the system, and derived the second set from Hamilton's principle. By varying the electromagnetic potentials, he obtained Maxwell's equations and the laws of conservation of energy and momentum. The equations of gravitation could be derived from the Hamiltonian, including Einstein's function H defined by Eq. (29), and varying it with respect to the gravitational potentials. Lorentz found full agreement with the results of Einstein, including the fact that the total energy and momentum are only conserved in systems of adapted coordinates, as defined by Eq. (28). Shortly after the appearance of Hilbert's first memoir on the foundations of physics in November 1915, and Einstein's papers of November 1915 and his memoir on general relativity early in 1916, Lorentz published a second note on 'Einstein's theory of gravitation.' 219 He wrote: In the pursuit of his important researches on gravitation Einstein has recently attained the goal which he had constantly kept in sight: he has succeeded in establishing equations whose form is not changed by an arbitrarily chosen change of the coordinate system. Shortly afterwards [sic], working out an idea that had already been expressed in one of Einstein's papers [sic], Hilbert has demonstrated the use that may be made of a variation law, which may be regarded as Hamilton's principle in a suitably generalized form. With these results, the theory of general relativity may be said to have taken a definitive form, although much remains to be done still in its further development and application to special problems. It would also be desirable to present the fundamental ideas in as simple a form as possible.220 Lorentz took the general function H, the 'principal function,' to consist of 'three parts, of which the first relates to the material (mass) points, the second to the electromagnetic field, and the third to the gravitational field itself.'221 For the first part, he used the function L, Eq. (84), giving the kinetic term, and it turned out to be invariant under general transformations. Passing on to the second part, Lorentz remarked that 'the mathematical expression for this part was communicated to me by Einstein in our correspondence. It is also to be found in Hilbert's paper, where it is remarked that the quantity in question may be regarded as the measure of the curvature of the four-dimensional generalization to which [the expression for the line element] relates. Here we shall talk only about the interpretation of this quantity.' 222 Again the term K, which Hilbert had used, could be interpreted with the help of Gauss, and it did not change with coordinate transformations. That part of the principal function which is due to the curvature is given by H3 = +-

f Rdi},

(85)

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where 7 is the gravitational constant, and d&l a four-dimensional volume element. The derivation of the electromagnetic part took rather long, but again the generalization of the usual expression was obtained, containing the sum of S2 and JF2, i.e. the scalars of the electric and the magnetic fields in suitable form. 223 In his third and fourth communications on this subject, Lorentz derived the Hilbert-Einstein field equations, in particular Eq. (37), by a variation of the gravitational potential for the two cases, namely the TM„ being due to the electromagnetic or the mechanical part respectively. Altogether Lorentz had produced a complete proof of the equivalence of Einstein's inductive and Hilbert's deductive methods, treating all the delicate points clearly and in detail. 6.2. Einstein

and the Hamilton

principle

Nevertheless, Einstein returned to a discussion of this problem at the session of the Prussian Academy on 26 October 1916. 224 He obviously felt that he should give the connection between Hilbert's formulation and his own, a still simpler form, and also take a stand against Hilbert's attempt to incorporate Mie. He did not waste any time in guarding his own stake in this problem. Lorentz and Hilbert [note the order of these names] have recently succeeded in presenting the theory of general relativity in an especially clear form, by deriving its equations from a single variational principle. We shall also follow the same method in this work. In doing so, I shall attempt to present the fundamental connections [between Hamilton's principle and general relativity] as clearly and generally as possible, as fax as the standpoint of general relativity permits. In particular, especially in contrast to Hilbert, as few restrictive assumptions as possible about the constitution of matter will be made. On the other hand, contrary to my own previous paper, the choice of the coordinate system shall remain perfectly free.225 Einstein reformulated the Hamiltonian function for the variation by letting it depend only on the first derivatives of g^ and A^. Again, the Hamiltonian function had to be split into a gravitational part and all the rest, by assumption, otherwise the equations of gravitation, etc., would not follow. He easily derived the conservation of energy and momentum, including both the matter tensor TM„ and the field energy tensor t^u. From the latter equations, and the field equations of gravitation, there would follow the four equations

£±
2 dx

""

=0

(86) K

'

for the energy-momentum tensor of matter. These are essentially Hilbert's four conditions, which Einstein did not want to link with Mie's equations of electrodynamics. First, he did not like the latter, and second, because he was not convinced that electrodynamics alone could provide a complete theory of matter. Einstein changed this attitude in his later years, but at the time when Hilbert and Weyl professed interest in the electromagnetic theory of matter, he was strongly opposed to it.

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277

Clearly, Einstein's conservation law for the energy tensor and Hilbert's four conditions for the energy tensor of matter are closely linked. We shall pursue this connection in some detail in the next section, thereby seeking to make a little more evident the meaning of energy conservation in general relativity. Let us mention here only that the so-called energy conservation law to which Einstein and Lorentz addressed themselves appears more like the definition of the energy tensor of gravitation ty.v than a physical law. The fact, that it was possible to find a tMI/ which depends only on the gravitational fields and its derivatives, might be considered important. But that is not really surprising, since the total principal function H had been assumed to consist of two parts, one of which contained entirely the gravitational term R. 7. General Covariance and Conservation Laws After strenuous efforts, having followed several dead-ends, Einstein had succeeded in establishing a consistent theory of general relativity. The mathematicians, whose illustrious ancestors had laid its mathematical foundation, happily took up the development and further interpretation of the new theory. Some disagreement arose on a basic question concerning its physical interpretation. Einstein, in this case, stuck to a more formal approach, putting a formal energy conservation law in the forefront. The reason is to be found in his earlier procedure, in which full covariance was rather doubtful [see Sec. II.4]. Hilbert, on the other hand, tried to exploit covariance physically. Finally it was Felix Klein, the author of the Erlangen Programme and grand old master of geometry, who, in his last important work, completed the synthesis of the views of Einstein and Hilbert — the synthesis, in a sense, of the theory of relativity and the theory of invariants. In 1917, he published two memoirs dealing with 'energy-momentum conservation' in general relativity. Shortly before this, in an exchange of letters with Hilbert, Klein clarified the meaning of the four conditions which Hilbert had discovered in this context. What escaped Klein was the fact that there existed an even more elegant connection with the equations of general geometries, the Bianchi identities. 7.1. Energy conservation

and the theory of

invariants

In an extensive memoir 226 on 'the differential laws for the conservation of momentum and energy in Einstein's theory of gravitation,' which Klein communicated to the Gottingen Academy on 19 July 1918, he attempted to unify the different forms of the differential laws for the conservation of energy and momentum that could be found in the papers of Einstein, Lorentz and Hilbert. The important tool which he used in doing this was the theory of invariants. 227 Under the assumptions of Einstein, Hilbert, and Lorentz, the variational integral could be split into two terms, of which only one contains the curvature scalar R. Klein proceeded in three steps: first he considered J? as a function of g^u and its first and second derivatives; then he drew conclusions from the fact that R is an

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invariant under generally covariant transformations; finally he used the fact that R is an invariant of specific form and content. From this procedure, there followed a set of differential relations which could be interpreted in physical terms. From the fact that R is a function of g'"' and its first and second derivatives, Klein derived the equation aia>-g?> = 2-5g-,

(r = l , . . . , 4 )

(87)

where @^ = VgR^

and

«rr* = Vflt^,

and

g^ =

- ^ .

Here ^ / i s a mixed tensor under linear transformations. But due to the fact that R is generally invariant, one finds that the divergence oiSft + fy is generally covariant. Altogether Klein deduced 152 equations containing M and ^ and their derivatives, but not all of them were independent. One specific set reproduced three vector equations,

E ^ +^r=o,

(88a)

and

d [sr? + ±w°)

0,

(88b)

and

which, in this sequence, are identical with Hilbert's four conditions, and with Lorentz' and Einstein's conservation laws. The second set essentially reproduced Hilbert's energy conservation law, Eq. (46). But it is physically equivalent to Lorentz' law (88b), and merely means that an equation for the gravitational field is introduced. Nevertheless, Klein preferred Hilbert's complicated energy law, Eq. (46); it is complicated because the vector eA has such a tricky structure. However, it [Hilbert's energy conservation law] has the advantage that it not only makes a simple [invariant-theoretic] assertion, but also the quantity eA appearing in it can be readily characterized invariantly. It [Hilbert's energy conservation law] contains the auxiliary vector pT, and besides it is independent of the T^, R^ and their differential coefficient.229, Felix Klein's work is closely connected with the studies of Emmy Noether concerning the relations between symmetry properties and conservation laws. 229 Conservation laws can be stated in two forms, i.e. the differential and integral forms. Klein turned to the latter in his second note on 'the integral form of the conservation laws and the theory of the spatially closed universe.' 230 Concerning the physics underlying his considerations, Klein referred to the papers of Einstein 192 and de Sitter. 231 Expressing his motivation, Klein noted: First of all, I wish to express my viewpoint about the integral form of the conservation laws, which Einstein had preferred to formulate in differential form. In this connection, I shall

The Historical Origins of the General Theory of Relativity

279

treat Einstein's theory of the spatially closed but unbounded universe and its modification by de Sitter. My goal is to state the mathematical connections as clearly as posssible, while touching upon the physical question only briefly. I feel a certain satisfaction in this, since [it will be seen that my] old ideas of 1871-72 attain decisive validity. [Here Klein was referring to his Erlangen Programme.] The reader may decide for himself the extent to which progress has been made.232 In some general remarks, Klein stated the physical concepts of vectors etc., which have meaning only if one refers to a certain group: There are as many different kinds of theories of relativity as there are [mathematical] groups. This statement is perhaps in contradiction to the much publicized current discussion about Einstein's general theories, but it is not significantly different from Einstein's own detailed arguments, to which I attach great importance. Einstein's papers, upon which I shall comment here, indeed demonstrate that at times, although not systematically, he [Einstein] resorts to the same conceptual freedom which I have proposed in my Erlangen programme.233 Klein then proceeded to discuss the generalizations of Einstein's closed spherical universe, namely elliptical universes, and their related symmetry groups. These universes are based on the use of cosmological constants A, which change the original Einsteinian energy tensor of the gravitation field to

£=£ + ^ , or U? = U? + \6°.

(89)

But otherwise the considerations leading to Eq. (88) remain unchanged. It further turned out that the closed Einstein universes would lead to an integral vector whose first three (spatial) components are zero. This is easily understood since a static universe of fixed radius had been chosen. The fourth component was interpreted as the total energy of the universe. Now the amazing result of an explicit calculation was that the two differential conservation laws, Eqs. (88b) and (88c), do not yield the same total 'gravitational' energy for a finite universe. Klein's results indicated that one had to be very careful in applying the law of energy conservation in general relativity. One might again stress the view which was expressed by Hilbert, that energy conservation is not necessarily a characteritic of general relativity. By energy conservation, Hilbert obviously meant well defined terms based upon physical considerations. This brings us to a discussion of the point whether one should prefer Hilbert's formulation based on constraints, or Einstein's concept of energy conservation, in general relativity. 7.2. Identities

from general

covariance

Felix Klein's considerations, which we have discussed in the foregoing, were rather general, resulting in the conclusion that both Hilbert's constraint conditions for the energy tensor of matter and Einstein's conservation of energy momentum followed

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from general covariance. An earlier note of Klein, which was both complete and general, already contained a specific discussion of the main points. 234 Discussing the variation of Hilbert's two Hamiltonian integrals / x = IRdQ,

and I2=fLdQ,

(90)

where dil = y/gdx1 ---dx0,

(dn = dAx),

Klein derived fourteen field equations, namely the ten Eqs. (37) for the gravitational field, and the four equations for the 'matter field.' The latter equations followed from the former in the case of general covariance. 235 Klein then discussed Hilbert's energy-conservation law, Eq. (46), and concluded that it would follow from the fourteen field equations as a mathematical identity. Hence, it 'can obviously not be considered analogous to the law of conservation of energy, as it exists in ordinary mechanics. For, if in ordinary mechanics we write that d{T + U) dt

0,

(91)

then [our understanding is that] this relation is not a mathematical identity, but rather a consequence of the differential equations of mechanics.' 236 Klein also admitted that Einstein's conservation law could hardly be considered as a law in the usual sense. 237 Hilbert very much agreed with Klein's conclusions, and in a letter to Klein, which Klein submitted with his own note to the Gottingen Academy, Hilbert expressed this agreement. 238 He asserted, moreover, that 'in general relativity, i.e. in the case of the general invariance of Hamilton's function, equations for the energy (which correspond to the energy equations in orthogonally invariant theories) do not exist at all. In fact, I regard this situation as typical for general relativity.' 239 After discussing energy conservation in orthogonal invariant theories, Hilbert shared the view that in a generally invariant theory one could not derive the conservation law in the same fashion. 'In general relativity, instead of the missing energy equations there are just the four overdetermined Lagrange equations, as expressed by the four identities. Conversely, the energy equations of the orthogonally invariant theories appear as the residue of those four identities of the theory of gravitation.' 240 In his reply, Klein went further in the discussion of the identities. 241 Starting from the point of view that the two terms, R and L, in the Hamiltonian function of Hilbert, should be separated, and discussing the general covariance of both terms, Klein found two sets of four constraints. In addition to these eight equations, he found the fourteen Lagrangian equations which exist for the fields, say the gravitational and the electric potentials. Klein showed that from the four constraints governing the L term there still follow the conservation laws, but they do not preserve their separate (physical) meaning any more, since they can be reduced to the four (Hilbert) conditions of constraint for matter.

The Historical Origins of the General Theory of Relativity 281

7.3. Bianchi

identities

and Hubert's

constraint

conditions

On the basis of the foregoing discussion, it may be justified to describe Klein's studies of the conservation laws and invariants in general relativity as the completion of the latter theory. They also stressed the importance of the conditions which Hilbert had placed on the energy tensor of matter. Since these conditions are obtained only with the help of the field equations of gravitation (37) [i.e. when the latter are satisfied], we might as well consider them as conditions for the curvature tensor Ra^x or, more precisely, for the contracted tensor i?M„ which enters the equations of gravitation. Now Hilbert's statement means simply this: once general covariance is satisfied, there exist only four differential identities satisfied by i?M„. What are these identities? It is remarkable that great mathematicians and geometers like Hilbert and Klein did not see the connection between the four identities of Hilbert and the well-known identities of Bianchi. 242 The latter read R\uA;r + R\xr;v

+ R%ru;X = 0 ,

(94)

where ';' denotes covariant differentiation with respect to the coordinates. The Bianchi identities are valid not only for Riemannian, but also for more general affme geometries. Contracting the indices a = u, one obtains i?/iA;r ~ R"fiXr;* ~ Rpr^X = 0 .

(95)

If we multiply Eqs. (95) with g^T and contract again, we arrive at

2^Rt-\g\RJ

=0-

(96)

These four equations may be interpreted either as the four identities of Hilbert or Einstein's energy conservation laws, if one connects them with the equations of gravitation (37). Thus Hilbert had actually discovered, with his identities, a part of the Bianchi identities which can be derived from purely geometrical considerations. It should also be worthwhile to relate some of the remaining 148 identities, which Klein found in his work of 1918, to the other consequences of the Bianchi identities. The overlap certainly is very large if not complete. 243 We might actually venture the guess that we owe Hilbert's memoirs on the foundations of physics to the fact that he did not recognize his four identities as a consequence of the Bianchi identities. If he had been aware of the connection, he might have quite possibly slipped over their content without trying to extract the essential physical meaning. But it was the physical consequences which led Hilbert to his valuable contributions to the theory of general relativity. In a certain sense, we could apply to Hilbert himself the words which be used about Einstein: that the reason why he made progress in the theory of relativity was that he did not know everything about the mathematics of space and time before he set out to make discoveries in it.

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8. Extension of General Relativity Einstein's theory of general relativity incorporated gravitation entirely. Concerning matter itself, it was less complete, as Einstein's controversy with Hilbert showed. With respect to electromagnetism, in particular, Einstein had left everything as it was, but the unsatisfactory status of his assertions was felt more and more by other scientists. The first, who stepped forward, already in 1918, was Hermann Weyl. Weyl proposed nothing less than a new geometry and a unified theory of gravitation and electricity. Another attempt was started by Kaluza in 1921, and revived later on by Oskar Klein in 1926. Finally Einstein, soon after completing the first phase of his discussions with Bohr about the new quantum theory, moved into the direction of a unified field theory of matter, and spent the rest of his life trying to cope with it — entertaining various interesting, if not entirely successful, attempts. The underlying mathematics, which generalized differential geometry, was developed by Levi-Civita in 1917. 8.1. Generalization

of Riemannian

geometry

Riemannian geometry rests essentially on the consequences derived from the line element ds2, in which the metric tensor appears. From the metrical tensor is constructed the Riemann-Christoffel tensor, and from the latter the invariants which play such an important role in general relativity. But one could even go further in geometry, endowing the three-index Christoffel symbol T with full meaning. This was done by Levi-Civita, and extended by Weyl who, at the starting point of his study, stated that: Inspired by the weighty inferences of Einstein's theory to examine the mathematical foundations anew, the present writer made the discovery that Riemann's geometry goes only half way towards attaining the ideal of a pure infinitesimal geometry. It still remains to eradicate the last element of geometry 'at-a-distance,' a remnant of its Euclidean past. Riemann assumes that it is possible to compare the lengths of two line elements at different points of space, too; it is not permissible to use comparisons at a distance in an 'infinitely near' geometry. One principle alone is allowable; by this a division of length is transferable from one point to that infinitely adjacent to it.244 The initiative of developing the mathematical extensions was provided by Einstein's theory of gravitation itself. Levi-Civita responded first. 245 Then Hessenberg followed, who considered space as formed by a great number of small elements cemented together by parallel transports. 246 The idea behind these attempts was to make the notion of covariant differentiation equivalent to a parallel transport of vectors in curved space. This can be done with the help of the geodesic components Vtv as, for instance, in the case of a vector a^ or 6M (97) b

— —— -I- T^ be

The Historical Origins of the General Theory of Relativity

283

The Christoffel symbol then expresses the infinitesimal transport of the line element in an 'affinely connected manifold.' This theory was completed in Weyl's dissertation on a 'pure infinitesimal geometry.' 247 In a pure infinitesimal geometry of the n-dimensional continuum each set of coordinates xl • • • xn refers to a different point. The connection between the different points is expressed by continuous functions possessing at least continuous first derivatives and a non-zero Jacobian. One can now define an infinitesimal displacement for a given point in space. Since the notion of vectors, tensors, etc., is a priori meaningful only at a point in a consistent differential geometry, one has to look for its meaning in the entire space. This can be done if one assumes that the space forms an 'affinely related' manifold, or the fact that for any point P there exists a neighbourhood such that every vector at P can be transformed by a parallel displacement into a vector at P' (where P' is in that neighbourhood). The coefficient in front of this displacement is the Christoffel symbol T. A 'metrical manifold' is finally obtained only if the line element at any point P can be compared with respect to the lengths.

8.2. Weyl's unified theory What has been achieved by this generalization of geometry? Obviously, it is now possible only to compare lengths measured at one and the same point. Moreover, it is no longer the <7M„ that make sense now or can be determined by measurements, but rather their ratios or the relations between them. If we denote the length of a short rod by I, then at different points the measuring rod is changed infinitesimally as

where the parameter t appears in a curve connecting the infinitesimally close points P and P'. For the affinely related manifolds Weyl obtained, cU/> = (jf^dx* .

(99)

Riemannian geometry is obtained if cj>u = 0. The Christoffel symbols in the generalized theory read

I W = r £ ™ n + \{gM4>* + 9aAe - gM •

(ioo)

Here the absolute values of the g^v seem to be fixed, which need not always be the case. The invariance of all geometrical relations must exist under the gauge transformation 9'^.u = x9nv

a n d ^ = - - —

.

(101)

Weyl could now proceed to interpret the metric vector <j>a, which determines the lengths in accordance with Eq. (99), as being identical with the electromagnetic fourpotential, apart from a numerical factor. 248 In the presence of the electromagnetic fields Ffu,, this geometry will therefore deviate from Riemann's.

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This idea of Weyl's was very original and attractive, because it made electrodynamics as much a consequence of geometry as Einstein had made gravitation. But some difficulties arose immediately, and Einstein pointed out that in Weyl's theory spectral lines with definite frequencies could not exist. 249 To this, Weyl replied that since the g^v and 0M were not measurable, it made the connection between ideal mathematical quantities and the physical quantities rather loose but suggestive. One should, however, note that the existence of the quantum of action might very well be at the basis of this failure of Weyl's theory, just as its consequences for Einstein's theory of gravitation have not yet been ascertained. In any case, Einstein greatly admired Weyl for his mathematics, though not for his physics, as he repeatedly expressed in letters. 250 A little later on, Eddington considered even more general geometries than Weyl's. Even Einstein considered these geometries for a while, but left the field very soon. 251 After quantum mechanics was created, Weyl noted that his own theory had really no physical basis. 8.3. Five-dimensional

relativity

Whereas Weyl had introduced the variation of the measure ds2 and linked it to the existence of electromagnetic potentials, Kaluza remained in the realm of metrical geometries and sought to include the electromagnetic field by extending the dimensions of the universe. 252 In the five-dimensional geometry the field theories of gravitation and electrodynamics could be unified in a very simple way. The line element could now be written as da = y/Yikdxidxk

,

(102)

where X j . • • j X 3 X 6 the space coordinates, and Tik are the 15 covariant components of a five-dimensional symmetric tensor related to the usual four-dimensional ones and the electromagnetic potentials, according to the fact that the conventional space coordinates and the metric should not depend on x°. Hence we put foo = a, and dA = dx° + ^dxl; ds2 = (rik - M ° < A dxidxk (104) loo \ loo / become the infinitesimal invariants, the sum of which gives the invariant da2. One finds that the four Tot transform like the covariant components of a four-vector, and the Ansatz that Toi = aP4>i, (1 = 1,2,3,4) (105) where <j>i is the electrodynamic potential, can therefore be used. Einstein's equations of gravitation as well as the equations of electrodynamics (in covariant form) can be derived from the variational principle SJ = 6 f dx°dx1dx2dx3dx*V-TP

=0

(106)

The Historical Origins of the General Theory of Relativity

285

where P is the curvature corresponding to the metric tensor Yik, and \a(32=1.

(107)

One might have sought to generalize the covariant or invariant expression for P in order to obtain more general equations, but Oscar Klein, who extended Kaluza's work, was not primarily interested in this problem; he rather hoped to incorporate quantum effects in the five-dimensional theory. 253 Since the motions in the fifth dimension are not perceptible in ordinary experiments, one could just as well average over the entire motion; the quantum of action could thus follow from a periodicity in the fifth dimension, and the conjugate momentum p could be fixed to have the two values +e and —e. Then the free Hamiltonian of a charged particle, say an electron in an electromagnetic field, would read H = ^-TikPiPk

,

(108)

and the usual Hamiltonian formalism would apply. Klein attempted in various ways to relate the quantities of the five-dimensional theory to the wave function in quantum theory. One such attempt was to write the electrodynamical potential <& of an electron as $ = -x° + S(x1,x2,x3,x4),

(109)

and to obtain the periodicity of <& from a periodicity in the fifth coordinate; this was possible because one could relate — $ to the transformed coordinate x°.25i Another possibility which Klein proposed was to try 'to relate the fifteenth quantity Too with the wave function U, which characterizes matter, in order to achieve a formal unity between matter and field.'255 Several years later, Veblen and Hoffmann would mathematically replace the Kaluza-Klein five-dimensional theory by a four-dimensional projective geometry. 256 9. 'Ein Bleibender

Kern1 — an Enduring Core

The creation of the theory of relativity in the first two decades of this century drastically changed the inherited physical concepts of space and time. The vision of a unified theory of matter first appeared in the work of Mie, Hilbert, Weyl and Kaluza. But, at the same time, another revision of classical theory had begun which also affected the understanding of the notions of space and time; it led in the middle twenties to a consistent theory of the microscopic processes. Oskar Klein's early attempts to unite the two lines of development failed, and we are not much closer now to a unified theory of matter. In the late nineteen-twenties already, certainly since Dirac's theory of the electron, it became clear that electromagnetism was not the only force which had to be united with gravitation. In quick succession new forces and fields were discovered, and the existence of a massive particle with

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no charge at all destroyed t h e hopes of ever achieving a n electromagnetic theory of m a t t e r . It seems t h a t Einstein's remarks concerning Hilbert's theory h a d been prophetic. He had said: To me Hilbert's Ansatz about matter appears to be childish, just like an infant who is unaware of the pitfalls of the real world In any case, one cannot accept the mixture [i.e. Hilbert's] of well-founded considerations arising from the postulate of general relativity and unfounded, risky hypotheses about the structure of the electron [or matter in general]. I am the first to admit that the discovery of the proper hypothesis, or the Hamilton function, of the structure of the electron is one of the most important tasks of the current theory. The 'axiomatic method,' however, can be of little use [in this]. 257 B u t strangely enough, Hilbert did not seem t o be very impressed by Einstein's negative j u d g e m e n t . In 1924, he published a n almost unaltered imited version of his two memoirs on the foundations of physics, including the notes he h a d written to Felix Klein. 2 5 8 He again stressed the fact t h a t Mie h a d first d e m o n s t r a t e d the manner in which one could describe m a t t e r by a unified field theory. Now, in 1924, he wrote: As I have pointed out in my first memoir, the tremendous problems and the structure of ideas of general relativity, developed by Einstein, find their simplest and most natural expression as well as, in a formal sense, a systematic generalization and completion, in the method proposed by Mie. Since the publication of my first memoir, significant publications have appeared on this subject. Let me mention only the brilliant and profound investigations of Weyl and the always new formulations and thoughtful contributions of Einstein. However, on the one hand, Weyl in the development of his ideas is finally led to the equations proposed by me; and on the other, even Einstein, in spite of his repeated efforts to start from various different premises, in his latest publications finally returns directly to the equations proposed in my theory. I am quite sure that the theory as developed here by me contains an enduring core, and provides a framework within which there is sufficient margin for the future development of physics, pursuing the ideal of a unified field theory. In any case, it is also of epistemological interest to note in which way the few and simple assumptions as expressed in axioms I, II, III and IV, formulated by me, are sufficient for the construction of the complete theory. Whether in fact the ideal of a pure unified field theory is a definitive one, or if certain extensions and modifications of it will be necessary, in order to obtain in particular the theoretical justification for the existence of the electron and the proton, and the consistent formulation of the laws valid in the interior of an atom — to answer this is the task of the future. 259 Ironically, it was exactly this hope — t h e realization of the ideal of a unified field theory — t h e hopes of Mie, Hilbert, a n d Weyl, which he h a d brushed aside with a confidence bordering on contempt, t h a t Einstein, in outspoken contrast to the developments in q u a n t u m theory, entertained later on. Such hope was already latent in a n o t e o n 'Do gravitational fields play a n essential role in t h e structure of elementary particles?,' which h e communicated to t h e P r u s s i a n Academy on 10 April 1919. 2 6 0 He acknowledged t h a t the theories of Mie, Hilbert a n d Weyl were deep and beautiful mathematically, even t h o u g h they led to unsatisfactory physical

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results. He thought t h a t he knew t h e first steps which might lead t o a solution of t h e problem. 2 6 1 Starting from t h e observation t h a t if gravitational fields were essential for t h e s t r u c t u r e of m a t t e r , t h e field Eqs. (37) could not be accepted in t h e limit of constant g^, he said t h a t , 'Material particles could not exist, if g^ were constant. If, therefore, one considers t h e possibility t h a t gravitation, among other forces, plays a p a r t in the s t r u c t u r e of corpuscles, t h e n Eq. [(37)] could not be regarded as valid.' 2 6 2 Only if one would stick to Eq. (37) would one have t o use Mic's theory. 2 6 3 However, t h e inclusion of t h e cosmological constant A would already change the equations of gravitation (37). As a new Ansatz, instead of these equations, he proposed t h e new equations

Rv ~ \a»»R = -TT/U/,

(T^

= \g„vFapFaP - F^F^A

,

(110)

with t h e electromagnetic energy tensor T^. E q u a t i o n (37) multiplied by g M ", would t h e n hold, because g^T^ — 0. In addition, Einstein assumed t h e four equations

av dxo

a*„e +

-dxT

+

a$eM ~dx^-°'

getting 12 independent equations for 16 variables g^ Eqs. (110) as Rut/ — j9nvR

) = — 7 Tui/ — —9u.v(R 47

(111)

and Q^.

— RQ)

He t h e n rewrote

(112)

Since R is related to A, the cosmological constant appeared only as a constant of inte- gration in the new theory. He concluded by saying t h a t : The above-mentioned considerations indicate the possibility of a theoretical explanation on the basis of the gravitational and electromagnetic fields alone, without the introduction of additional hypothetical terms as in Mie's theory. This possibility appears to be particularly promising, since it makes it unnecessary to introduce the special constant A, which would otherwise be needed to solve the cosmological problem. On the other hand, there remains a peculiar difficulty. Namely, if we consider Eq. [(110)] in the spherically symmetric, static case, then we obtain one equation too few for the determination of gM„ and FMW, so that every spherically symmetric distribution of electricity can exist in equilibrium. Thus the problem of the structure of elementary quanta cannot be solved only on the basis of the equations given above. 264 Again a n d again, Einstein r e t u r n e d to the question whether one could obtain the q u a n t u m of action from a n over-determination of t h e field variables. 2 6 5 T h i s was also t h e motivation behind his later moves to abolish the q u a n t a by means of the unified field theory, on which he h a d started to work seriously beginning in 1928, after his efforts t o disprove t h e consistency of q u a n t u m mechanics had failed. O n 5 J a n u a r y 1929, he wrote to his friend Michele Besso: But the most wonderful thing is that, what I have brooded on and calculated all these days and nights, is now completed and in front of me, condensed to a mere seven pages under

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the title 'unified field theory.' This might appear old-fashioned, and my dear colleagues as well as you, my friend, will right away stick their tongues out as far as possible. Because, Planck's constant h does not appear in these equations. However, as soon as the limitations of the [current] statistical craze [i.e. quantum mechanics] are clearly seen, one will retreat repentantly to a space-time approach and then these equations [of mine] will become the starting point [of the new theory]. I have discovered a geometry which is characterized, not only by a Riemannian metric, but also by distant-parallelism. The latter was thus far generally felt to be a characteristic of Euclidean [geometry], and the simplest field equations of such a manifold lead to the well-known laws of electricity and gravitation. Even the equations Rik = 0, in spite of their success, now belong to the junk pile. I shall not forget to send you the reprints. If [on reading] you don't stick out your tongue, then you must be a hypocrite, because I know you! 266 Michele Besso did not, we can be sure, 'stick his tongue o u t ' at w h a t Einstein wrote t o him. He was Einstein's old friend and admirer, a n d his sounding-board for scientific ideas ever since the early days of special relativity in Berne, Switzerland. Einstein continued to develop various aspects a n d formulations of his unified field theory t o t h e e n d of his life, without ever really coming close t o his great goal. In these efforts he was often alone, and t h e scientific problems on which he worked were n o longer central t o t h e interests of a more p r a g m a t i c age. H e continued t o publish his reflections in respected scientific journals, a n d a t times they were reported in newspapers. No one ever stuck his tongue out a t w h a t Einstein h a d to say in scientific m a t t e r s , although his hopes and a t t e m p t s t o create a unified theory of physics caused a few raised eyebrows and aroused some melancholy a b o u t t h e lost and better-to-be-forgotten visions of heroic times. Dirac has called Einstein's theory of gravitation 'probably t h e greatest scientific discovery t h a t was ever m a d e . ' 2 6 7 Discussing t h e difficulty of u n d e r s t a n d i n g t h e origin a n d m e a n i n g of this discovery, Chandrasekhar has remarked t h a t , 'It a p p e a r s t h a t only by a mixture of physical reasonableness, m a t h e m a t i c a l simplicity, a n d aesthetic sensibility can one arrive at Einstein's field equations. T h e general theory of relativity is in fact a n example of ' t h e power of speculative t h o u g h t . ' 2 6 8 Hilbert had played a decisive role in determining the form of t h e equations of t h e general relativity theory of gravitation a n d t h e strict conditions to which t h e y are subject. In t h e Einstein-Hilbert equations of 1915, there shall forever remain a n enduring essence, 'a p e r m a n e n t core,' which gave such satisfaction t o Hilbert a n d brought fulfilment to Einstein. T h e unification of gravitation and electricity into a single field theory of all physical p h e n o m e n a , however, did not succeed. As Pauli said of these a t t e m p t s , 'No m a n shall join what God h a t h p u t asunder.' 2 6 9 References and Notes 1. Prom Einstein-Sommerfeld Briefwechsel, edited by A. Hermann, Schwabe and Co., Basel/Stuttgart 1968, p. 26. Writing to Hilbert on 1 November 1912 Sommerfeld said, 'My writing to Einstein was in vain Einstein is obviously so deeply immersed in [the theory of] gravitation, that he is deaf against everything else.' (p. 27).

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2. A. Einstein,' Uber das Relativitatsprinzip und die aus demselben gezogenen Folgerungen,' Jahrbuch der Radioaktivitat und Elektronik 4, 411-461 (1907); 5 (1908). Actually, Einstein never considered the principle of relativity as confined to inertial frames alone, certainly not since he had derived the dependence of the inertial mass of a system on its motion or its energy in his paper '1st die Tragheit eines Korpers von seinem Energieinhalt abhdngig?,' in Ann. Phys. (Leipzig) 18, 639-641 (1905). In May 1906, he submitted another paper on this subject, lDas Prinzip von der Erhaltung der Schwerpunktsbewegung und die Tragheit der Energie,' Ann. Phys. (Leipzig) 20, 627-633 (1906), in which he proved the theorem that 'either one must give up [Newton's] first law of mechanics, according to which a body originally at rest, and not subject to external forces, cannot have a translational motion, or assume that the inertia of a body, according to the laws proposed [by us], depends on its energy content.' (I.e., p. 633).

3. 4. 5. 6. 7. 8. 9.

Einstein analyzed his equivalence postulate of energy and inertial mass in a paper entitled ' Uber die von Relativitatsprinzip geforderte Tragheit der Energie,' Ann. Phys. (Leipzig) 23, 371-384 (1907). To whichever (simple) system he applied it, the results were consistent with the principles of mechanics such as energy conservation, etc. This was the state of affairs when he wrote, at the request of J. Stark, the article in the Jahrbuch. In this article, Einstein dealt with the optical prerequisites, the principle of the constancy of the velocity of light and its kinematic consequences. In part II of the article, he dealt with the transformations in electrodynamics. The mechanics of the electron and the possible tests of the experimental consequences, in particular Kaufmann's results (Ann. Phys. 19 (1906)) were discussed in part III. In the following section (IV), devoted to the mechanics and thermodynamics of systems, he stated the important result, that 'the law of the conservation of mass for an individual system, according to our result, is valid only if its energy remains constant; it is then equivalent to the law of conservation of energy. However, the changes to which the mass of a physical system is subject, in ordinary physical phenomena, remains undetectably small. For instance, the decrease in the mass of a system, which loses 1000 calories (c.g.s.) of its energy, amounts to 4.6 x 1 0 - 4 gm. In the radioactive decay of a substance tremendous amounts of energy are set free; even so the change of mass involved in such a process is not large enough to be observed.' (Jahrbuch der Radioaktivitat und Elektronik 4 (1907), p. 442). As Planck had noted, in the case of radium the loss in weight was beyond the reach of experimental accuracy which could be obtained in 1906. Finally, in section V, Einstein discussed the relation of the relativity principle to gravitation. A. Einstein, Jahrbuch der Radioaktivitat und Elektronik 4 (1907), pp. 443-444. A. Einstein, Jahrbuch 4 (1907), Ref. 2, p. 454. A. Einstein, Jahrbuch 4 (1907), Ref. 2, p. 454. A. Einstein, Jahrbuch 4 (1907), Ref. 2, pp. 458-459. A. Einstein, Jahrbuch 4 (1907), Ref. 2, p. 459. A. Einstein, Jahrbuch 4 (1907), Ref. 2, p. 461. Another development had started with the electromagnetic considerations of Henri Poincare as early as 1905. In his remarkable paper 'Sur la dynamique de V electron' which was received by the editor of Rendiconti del Circolo Mathematico di Palermo on 23 July 1905, and was published in Vol. 2 1 , pp. 129-176 (1906), Poincare not only studied, almost at the same time as Einstein (whose article 'Zur Elektrodynamik bewegter Korper' arrived at the Annalen der Physik in Leipzig on 30 June 1905) the consequences of the Lorentz contraction, but he also referred to the problem of

290

The Golden Age of Theoretical Physics gravitation. Poincare had first used the phrase 'principle of relativity' in his lecture on 'The present and future of mathematical physics' at the International Congress of Arts and Sciences in St Louis in 1904 (see H. Poincare, Oeuvres, Vol. 9, p. 248, Gauthier-Villars, Paris, 1954); and it occurred again on p. 129 of the communication under discussion. In the introduction to this paper, he wrote, 'If we accept the postulate of relativity, we shall find that among the laws of gravitation and the laws of electromagnetics there exists a common number. It is the velocity of light. We shall find that it occurs in all forces, of whatever origin, and it can only be explained in two ways: (1) Either there exists nothing in the universe that is not of electromagnetic origin; (2) or, this quantity, which is common to all physical phenomena, appears only because it relates to our methods of measurements.' (H. Poincare, Rendiconti, 1906, pp. 131-132). Poincare considered Lorentz contraction, and tried to derive the Lorentz covariant action principles. Let us also mention here that it was Poincare who first gave the name Lorentz group. In Section 9 of his paper (pp. 166-175), he finally dealt with the 'hypotheses about gravitation.' He stated: 'Lorentz was thus obliged to complete his hypothesis by supposing that forces of all possible origins, in particular gravitation, are subject to a translation (or, if one prefers, subject to the Lorentz transformation) in the same way as electromagnetic forces.' {I.e., p. 166) Poincare then studied the consequences of a Lorentz invariant theory of gravitation, finding a correction to Newton's law of gravitation, which is of the order of v2/c2. The next paper in this connection, R. Hargreaves' 'Integral forms and their connection with physical equations' {Trans. Camb. Phil. Soc. 2 1 , 107-122 (1908)), did not refer to gravitation specifically, but to the 'four-dimensional formulation' of the laws of physics; 'We are concerned here with the variation of integral forms, or more specially with their invariance, under the action of an operator which is an extension of the hydrodynamical operator in Euler's equation. In the integral forms temporal terms are admitted, i.e. terms containing dt as well as the differentials of coordinates, and it appears that the forms have special properties when the temporal terms are associated with the non-temporal in a definite way. These associated terms are significant quantities which include vector and scalar products as particular cases.' (Ref. 113, I.e., p. 107) The importance of this paper partly lay in the fact that Hargreaves' result was connected with the studies of H. Bateman. On 11 March 1909, Bateman read a paper on 'The Transformation of the Electrodynamical Equations' before the London Mathematical Society {Proc. Lond. Math. Soc. (2) 8, 223-264 (1910)). He said: 'The object of the present paper is to find all the transformations for which the electrodynamical equation are invariant. In the case of the simpler equations of the theory of electrons, it is proved that the transformations belong to a certain group which is isomorphic with the group of conformal transformations of a space of four dimensions. It is assumed, however, that the transformation is such that the total charge on a system of particles is unaltered.' {I.e., p. 224). What Bateman required was that the equation ds2 = 0 should be invariant, rather than just the line element. From the study of the propagation of luminous disturbances, described by the above equations, he concluded that a more general Ansatz for ds2 had to be chosen, than Minkowski's assumption that the coefficients g^ depend on the space-time components, since light does not travel with uniform velocity in media with varying dispersion. Although Bateman's motivation was entirely different from Einstein's, and moreover he dealt with the electromagnetic problem alone, he was led to a conclusion similar to Einstein's. Einstein

The Historical Origins of the General Theory of Relativity

10.

11. 12.

13. 14. 15.

16.

17.

291

obviously did not know about Bateman's work of 1912. (Also see Bateman's later paper 'The Electromagnetic Vectors,' Phys. Rev. 12, 459-481 (1918).) There exist two papers dealing with the electrodynamics of moving bodies, which appeared almost simultaneously with Minkowski's work, Ref. 110, which Einstein wrote with his first student J. Laub: Uber die elektromagnetischen Grundgleichungen fiir bewegte Korper,' Ann. Phys. (Leipzig) 26, 532-540 (1908), with errata in Ann. Phys. (Leipzig) 27, 232 (1908) and 28, 445-447 (1909); 'Uber die im elektromagnetischen Felde aufruhende Korper ausgeiibten ponderomotischen Krdfte,' Ann. Phys. (Leipzig) 26, 541-550 (1908). Einstein wrote another note on this topic, 'Bemerkung zu der Arbeit von D. Mirimanoff 'Uber die Grundgleichungen...,' Ann. Phys. (Leipzig) 28, 885-888 (1909). A. Einstein, 'Die Relativitatstheorie,' Vierteljahresschr. Naturforsch. Ges. Zurich, 56, 1-14 (1911). A. Einstein,' Uber den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes,' Ann. Phys. 35, 898-908 (1911). He said, 'In a paper published already three years ago, I tried to answer the question whether the propagation of light is affected by gravity. [See Ref. 2] Here I shall return to this question again for two reasons: first, because I am no longer satisfied with my [earlier] treatment; second, and more important, because in retrospect I realize that one of the most significant consequences of this question can be tested experimentally. One finds that indeed, according to the theory which I shall present, light rays which pass close to the sun should be bent by its gravitational field giving rise to an apparent increase, of almost one second of arc, in the angular distance of a fixed star which appears to be in the vicinity of the sun.' A. Einstein, Ref. 12, p. 906. A. Einstein, Ref. 12, p. 908. Max Abraham was a very gifted physicist, noted for his work in the field of electrodynamics, in particular electron theory. He had been a brilliant student of Max Planck, under whom he took his doctorate. In 1902 he published his first memoir on 'Dynamik des Elektrons,' Nachr. Ges. Wiss. Gott. pp. 20-41 (1902), in which he developed the so-called 'rigid' electron model, by which he hoped to account for the stability of the electron. Abraham obtained a velocity dependence for the mass of the electron, different from the result of Lorentz and Einstein, but consistent with the early experiments of Kaufrnann. Another feature of Abraham's theory was that he stuck to an absolute coordinate system. (See e.g. Phys. Z. 5, 576-579 (1904).) Only in the latter would the electromagnetic waves propagate with the velocity c in all directions. M. Abraham, 'Zur Theorie der Gravitation,' Phys. Z. 13, 1-4, 176 (1912); 'Das Elementargesetz der Gravitation,' Phys. Z. 13, 4-5 (1912); 'Der freie Fall,' Phys. Z. 13, 310-311 (1912); 'Die Erhaltung der Energie und der Materie im Schwerkraft Felde,' Phys. Z. 13, 311-314 (1912); 'Das Gravitatonsfeld,' Phys. Z. 13, 793-797 (1912). In the first paper of this series, he remarked, 'In a paper published recently, A. Einstein [Ref. 12] put forth the hypothesis that the velocity of light (c) depends on the gravitational potential (). In this note, I shall develop a theory of gravity, which satisfies the principle of relativity. Furthermore, I shall derive a relationship between c and 4>, which in its first approximation, is equivalent to Einstein's. This [my] theory ascribes to the densities of energy and energy current values in the gravitational field that are different from those accepted at present.' (Phys. Z. 13, 1.) They actually do so on bilinear expressions in the gradients. (See M. Abraham, Phys. Z. 13, 3.)

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18. M. Abraham, Ref. 16, p. 3, 19. M. Abraham, Ref. 16, p. 4. 20. M. Abraham, Phys. Z. 13, 793-794. 'I have just now resorted to the language of relativity. Still, we shall see that [the theory of relativity] is not in accord with the views concerning gravity as presented here, if only because the axiom of the constancy of light velocity is renounced here. In my previous papers on gravitation, I have attempted to preserve, at least in infinitesimal [domains], the invariance with respect to the Lorentz transformations. However, I am now convinced that my equations of motion for a material point can by no means be brought into accord with the principles of analytical mechanics. On the other hand, Einstein's equivalence hypothesis has been shown to be equally untenable. In order to develop the new theory of gravitation, I shall, therefore, prefer not to deal with the space-time problem here at all.' 21. M. Abraham, Phys. Z. 13, p. 796. 22. M. Abraham, Phys. Z. 13, p. 797. 23. Abraham derived from it an upper limit for the mass of a star which, however, turned out to be rather large (m < 10 8 m 0 ) {Phys. Z. 13, 311.) 24. Abraham concluded his last paper (Phys. Z. 13, 797 (1912)) with the remark: 'The theory of gravitation developed here is completely independent of the unresolved problems of the dynamics of the electron. Apart from the basic assumption that the gravitational field is completely determined by [specifying] the velocity of light [everywhere], our theory is based upon (i) the validity of the expressions for the gravitational tensor [(la)-(ld) Ref. 16], (ii) the principles of mechanics, i.e. conservation of energy and momentum and the Lagrange equations, and finally, (iii) the hypothesis of the proportionality of gravity and energy.' 25. A. Einstein, Ann. Phys. (Leipzig) 38, 355-369 (1912). 26. A. Einstein, Ref. 25, p. 355. 27. A. Einstein, Ref. 25, p. 360. 28. A. Einstein, 'Zur Theorie des statischen Gravitationfeldes,' Ann. Phys. (Leipzig) 38, 443-458 (1912), in particular p. 457. 29. A. Einstein, Ref. 25, p. 362. 30. A. Einstein, Ref. 25, pp. 368-369. 31. Abraham was very critical and argumentative, but he did strive for clarity. In their obituary of Max Abraham, Born and Von Laue wrote: 'Clarity was the essence of his nature, both in matters of mind and heart. Clarity was evident in all his writings and we believe that it meant more to him than the discovery of new phenomena. Whenever he found it wanting, he would always make a sharp, even exaggerated criticism.' [Phys. Z. 24, 49-53 (1953).] This habit of sharp criticism got Abraham into many difficulties, and this former brilliant disciple of Max Planck did not gain a stable position for a long time. After eight years as Privatdozent in Gottingen (1900-1908), he went to the University of Illinois in 1908 as a professor of physics, and left after six months. In 1909 he accepted a professorship at the University of Milan, but during the war he was inducted into service and worked on telegraphy for the Telefunken Company. Finally, when he was appointed to the chair of theoretical mechanics at Aachen, Abraham became severely ill and, after several months of great suffering, died on 16 September 1922. 32. M. Abraham, Ann. Phys. (Leipzig) 38, 1056-1058 (1912). 33. A. Einstein, Ann. Phys. (Leipzig) 38, 1059-1064 (1912). Abraham wrote: 'Einstein's theory of relativity has exercised a fascinating influence, especially on the younger mathematical physicists, which threatens to retard a healthy development

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of theoretical physics. It was evident to the most sober critics that this theory could never lead to a complete worldview, if it cannot even integrate within its system the most important and omnipresent (natural) force of gravity. The failure of the efforts directed toward this has led to a crisis of relativity theory. Already a year ago, by assuming the dependence of the velocity of light on the gravitational potential, Einstein abandoned the essential postulate of his earlier theory, namely the constancy of the velocity of light. [See our Ref. 12]. Now, in a paper published recently [our Refs. 25, 28], he has also renounced the requirement of the invariance of the equations of motion under Lorentz transformations, thereby dealing the theory of relativity a coup de grace. Anyone who, like this author, had warned against [listening to] the singing of the sirens of this theory, would greet with satisfaction [the fact] that the originator of this theory is himself convinced of its untenability.' [Ref. 32, p. 1056.] 34. Abraham continued: 'Indeed, any relativistic space-time formulation which is expressed through the relationships between the space-time parameters of [two coordinate systems] S and £ ' would be untenable. I shall certainly not consider such a space-time formulation. As I have pointed out elsewhere, it seems to me that an absolutist [rather than a relativistic] theory is needed. If one system is raised above all others, in which the gravitational field is static or quasi-static, then it is permissible to call a motion with respect to this system as being "absolute." In the old actionat-a-distance mechanics, it was possible to assume that the very remote masses of the fixed stars, or an imaginary [Neumann] "body a" would determine an inertial frame of reference. Such an idea is foreign to the new mechanics based on the concept of local interaction. However, it would be consistent with the new mechanics if the omnipresent field of gravity itself provides the absolute frame of reference for the motion of bodies. (If one so chooses, one may consider this as an argument for the "existence of the ether.") 'Einstein's starting point, in his recent investigation, is a heuristic hypothesis which he calls the equivalence principle, and which requires a space—time correspondence of two frames of reference in acceleration with respect to each other. But even in this new form, it does not seem to be possible to carry through a consistent relativistic space-time description. Thus even this latest theory of Einstein's rests on shaky foundations.' [Ref. 32, pp. 1057-1058.] 35. M. Abraham, 'Nochmals Relativitdt und Gravitation, Bemerkungen zu A. Einsteins Erwiderung,' Ann. Phys. (Leipzig) 39, 444-448 (1912). He repeated his conviction that the theory of relativity belonged to the past: 'The components of yesterday's [sic] relativity theory, which depend on the rigorous and general validity of its entire concept, are Einstein's kinematics and the space-time definition tied with it. It was just this relativistic concept of space and time which had endowed the relativity theory of the past [sic] with a certain philosophical lustre which is now fading. To be sure, in his reply [to all this], Mr. Einstein conjures up in the distance the fata morgana of a new all-embracing space-time definition of gravitation. He is already claiming credit [today] for a [non-existent] theory of relativity of the future, and is inviting the assistance of the experts for its discovery. Until this payment is received [i.e. this discovery is made], this claim [of Einstein's] will remain, in the account of relativity, on the debit side of the ledger.' In a letter to L. Hopf, written in the autumn of 1913, Einstein remarked: 'It is proceeding splendidly with gravitation [his work with Grossman]. Unless I am entirely mistaken [and he was!], I have obtained the general equations. As you may have read recently in the Physikalischen Zeitschrift, Abraham has slaughtered me and relativity theory altogether in two fierce assaults, and has at the same time presented the only

294

36.

37. 38.

39.

The Golden Age of Theoretical Physics correct theory of gravitation (a 'nostrification' of my results) — a magnificent steed, which only lacks three legs. He also maintains that it was Robert Mayer who first recognized the mass of the energy.' (C. Seelig, Albert Einstein, Europa Verlag, Zurich (1954), p. 171.) R.v. Eotvos, Math. u. Natw. Ber. Ungarn, 8, 65 (1890); improved later on by R. Eotvos, D. Pekar, and E. Fekete, Trans. XVI. Allgem. Konf. der int. Erdvermessung (1909), and D. Peker, Naturwiss. 7, 327 (1919). See his reply, in A. Einstein, Ann. Phys. (Leipzig) 39, 704 (1912). G. Nordstrom, ' Reiativitatsprinzip und Gravitation,'1 Phys. Z. 13, 1126-1129 (1912); ' Trage und schwere Masse in der Relativitatsmechanik,' Ann. Phys. (Leipzig) 40, 857878 (1912); 'Zur Theorie der Gravitation vom Standpunkt der Relativitatsprinzips,' Ann. Phys. (Leipzig) 42, 533-554 (1913); 'Die Fallgesetze und Planetenbewegungen in der Relativitatstheorie,' Ann. Phys. (Leipzig) 4 3 , 1101-1110 (1914). G. Nordstrom, Phys. Z. 13, 1126. 'Einstein's hypothesis of the dependence of the velocity of light c on the gravitational potential leads, as can be seen from the discussions between Einstein and Abraham, to considerable difficulties in connection with the principle of relativity. Thus one wonders whether it might be possible to replace Einstein's hypothesis by another, which would leave c constant and yet modify the theory of gravitation in such a way that, in accordance with the principle of relativity, gravitational and inertial masses would be identical.' In an addendum to the first paper, Nordstrom noted: 'Prom a letter from Professor Einstein I have learned that he had already previously considered the possibility [proposed by me, as quoted above] of treating gravitational phenomena in a simple way. However, he came to the conclusion that the consequences of such a theory would not correspond to reality. With the help of a simple example, he has shown that according to this theory a rotating system in a gravitational field will be accelerated less than a non-rotating one. In fact, this conclusion is really not questionable, because the difference [between the accelerations] is too insignificant to contradict experience. Still, this conclusion shows that my theory is not consistent with Einstein's equivalence hypothesis. According to the latter, an unaccelerated inertial system in a homogeneous gravitational field is equivalent to an accelerated inertial system in gravity-free space. However, in my view, this circumstance alone is not sufficient to abandon [my] theory. Although Einstein's hypothesis is most ingenious, it does lead to great difficulties. It is therefore desirable to pursue other approaches as well to [the problem of] gravitation, and I would like to contribute to these in my note.' (Ref. 39, p. 1129.)

40. In his third paper, Nordstrom placed certain conditions on the variability of the gravitational factor g. He said, 'Every indefiniteness in the theory, noted above, can be removed by a very plausible assumption, for which I am indebted to Laue and Einstein. Laue has shown that Einstein's equivalence principle (although, not in its fully generality) can be maintained by suitably defining the rest mass density of matter In addition, we shall see that Einstein's equivalence principle requires a quite definite dependence of the gravitational factor g on the gravitational potential <£...' [G. Nordstrom, Ann. Phys. 42, 533-534.] 41. Nordstrom had gone to Zurich to meet Einstein (and Laue), and he submitted his third paper (Ann. Phys. 42, 533 (1913)) from there on 24 July 1913. At that time, therefore, he was in close contact with Einstein. 42. A. Einstein, Phys. Z. 14, 1249-1266 (1913), in particular p. 1250. 43. The postulate (ii), which states the equivalence of masses, is also satisfied if the gravitational potential does not change significantly in the system considered.

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44. A. Einstein and A.D. Fokker, 'Die Nordstromische Gravitationstheorie von Standpunkte des absoluten Differentialkalkuls,' Ann. Phys. (Leipzig) 44, 321-328 (1914). In this paper, they stated: 'The gravitational field is determined by the ten quantities g^v • In the Einstein-Grossmann theory, ten formally similar equations are specified for these ten quantities. Nordstrom's theory, on the other hand, is based on the assumption that it is possible to satisfy the principle of the constancy of the velocity of light by suitably choosing the frame of reference. We shall see [in this note] that this [Nordstrom's assumption] is equivalent to reducing the ten quantities g^v to a single quantity <j>2 by an appropriate choice of the frame of reference.' (p. 324.) 45. A. Einstein, Phys. Z. 14, 1254 (1913). 46. Nordstrom's theory has been revived during the past few years by the work of Brans and Dicke in an attempt to give a certain scalar admixture to Einstein's tensor theory, which seems to be consistent with recent experiments on the bending of light from the sun. C.A. Brans and R.H. Dicke, Phys. Rev. 124, 925 (1961). 47. More important, for the current situation, is the discussion following Einstein's lecture in Vienna, in which Mie pointed out the connection between Abraham's theory and that of Nordstrom (Phys. Z. 14 (1913) 1262). 'I would like to supplement Einstein's interesting comments with some remarks on the historical development of the theory, since Einstein has referred to them only in passing. Nordstrom's theory is tied together with the investigations of Abraham. I consider it essential that it should be mentioned here that Abraham was the first who formulated rather reasonable equations of gravitation. While previously one always tried (there are indeed many older theories of gravitation) to represent the gravitational field analogous to the electromagnetic field, Abraham discovered a new possibility. It is impossible to bring the older attempts into accord with the relativity principle; for if the principle of the equivalence of inertia! and gravitational masses is sufficiently accurately satisfied, then the gravitational field cannot be represented by a six-vector. It is for this reason that Abraham first of all formulated a theory with a scalar gravitational potential. However, he made the mistake that he set g [which is the density go in Eq. (9)] identical with the density of inertial mass Naturally, the relativity principle cannot be satisfied in this way. 'Nordstrom then improved this theory by substituting for g a quantity which is invariant with respect to the Lorentz transformation. At about the same time [as Nordstrom], I also formulated a theory of gravitation. However, my theory is embedded in an extensive work on the theory of matter, and that's why my investigations have escaped [the notice of] Mr Einstein.' At the last remark, Einstein protested, 'No, no.' and replied, 'I have not spoken about Mr Mie's theory because the equivalence of the inertial and gravitational masses has not been worked through rigorously in it. It would, of course, have been illogical if, having proceeded on the basis of certain postulates, I had not held on to them. I agree that I have not studied Mie's theory as carefully as I should have, but it was the farthest from my thoughts to disparage it if I have not referred to it in this context.' {Phys. Z. 14, 1263.) Einstein then dealt with the connection between the ideas of Abraham and Nordstrom. 'As far as Nordstrom's theory is concerned, I cannot really say that it was Abraham who first proposed the way which has been suggested by Nordstrom. For Abraham's theory is based upon the fact that the velocity of light is variable and that, to a certain extent, it represents a measure of the gravitational potential. Nevertheless, he employs the form of the usual theory of relativity, which leads him to

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48. 49.

50.

51.

52.

53.

The Golden Age of Theoretical Physics a contradictory, hybrid position. It is indeed so objectionable, that to me his theory seems to be quite untenable.' Mie admitted these deficiencies [of Abraham's theory], but insisted that Nordstrom had started from Abraham's equations. Upon which Einstein remarked that, 'Yes, psychologically it is indeed so, but not logically, for Nordstrom's theory is fundamentally different from Abraham's.' (I.e., p. 1263). A. Einstein, Ref. 44, p. 328. A. Einstein, Ann. Phys. (Leipzig) 49, 769 (1916). Einstein's statements about the value of this collaboration differ at different places. Sommerfeld, for instance, had suggested to Einstein not to emphasize Grossmann's role too much, because 'he will claim half of this share [of credit] in the creation of general relativity.' To which Einstein replied, 'Grossmann will never raise a claim to be considered as the codiscover [of general relativity]. He only helped me in getting oriented with the mathematical literature, but he did not contribute [sic] anything material to the [ideas and] results.' (Letter from Einstein to Sommerfeld, dated 17 July 1915, quoted from Einstein-Sommerfeld Briefwechsel (ed. by A. Hermann), Schwabe and Co., Basel and Stuttgart (1968), p. 30.) [See our Ref. 1.] In 1933, Einstein remarked about this collaboration with Grossmann: 'I worked on these problems from 1912 to 1914 with my friend Grossmann. We found that the mathematical methods for solving the first question [translation of the laws into general relativity] was already waiting for us in the absolute differential calculus of Ricci and Levi-Civita. As to the second problem [law of gravitation], we soon recognized that the methods for doing this had long ago been worked out by Riemann (curvature tensor). Already two years before the final publication of the general theory of relativity, we had considered the correct field equations of gravitation, but we failed to recognize that they were physically applicable.' (A. Einstein, 'The Origins of General Theory of Relativity,' G.A. Gibson Foundation Lecture, University of Glasgow, 20 June 1933, Glasgow (1933), pp. 10-11.) A. Einstein and M. Grossmann, 'Entwurf einer verallgemeinerten Relativitatstheorie,' Z. Math. u. Phys. 62, 225-261 (1913) ('/. Physikalischer Tiei.' A. Einstein; lII. Mathematischer Tiel': M. Grossmann); A. Einstein, 'Physikalische Grundlagen einer Gravitationstheorie,' M. Grossmann, 'Mathematische Begriffsbildungen zur Gravitationstheorie,' Vierteljahresschr. Naturforsch. Ges. Zurich 58, 284-290, 291-297 (1913); A. Einstein and M. Grossmann, 'Kovarianzeigenschaften der Feldgleichungen der auf die verallgemeinerte Relativitatstheorie gegriindeten Gravitationstheorie,' Z. Math. u. Phys. 63, 215-225 (1914). See also our Ref. 47. A. Einstein, 'Die formale Grundlage der allgemeinen Relativitatstheorie,' Sitz. Ber. Kgl. Preuss. Akad. Wiss., Math.-Phys. Kl. (1914), 1030-1085, submitted on 29 October 1914, and read on 19 November 1914. In the introduction Einstein wrote: 'During the past few years, I have perfected a generalization of the theory of relativity, in cooperation with my friend Grossmann in [certain] parts. A colourful mixture of physical and mathematical requirements has been employed as a heuristic device in these investigations. For this reason, it is not easy to have an overview of and to characterize this theory from a formal mathematical point of view. In this [new] work, I have attempted to fill this gap. In particular, we succeeded in obtaining the equations of the gravitational field in a strictly covariant-theoretic way. In order to assist the reader in obtaining a complete understanding of the theory without recourse to the study of other purely mathematical treatises, I have tried to give simple derivations of the basic principles of the absolute differential calculus — parts of which, I believe, are new. (Ref. 52, p. 1030.)

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54. Einstein's lectures at the Naturforschende Gesellschaft, Zurich, were given on the occasion of the annual meeting of the Schweizerische naturforschende Gesellschaft, at Frauenfeld on 9 September 1913. The lecture at the Congress of Natural Scientists in Vienna was given in November 1913. The final version of the 'Entwurf' paper was written after his lectures at both of these conferences. 55. See the Frauenfeld lecture, 1913, the second paper in Ref. 51, p. 286. 56. In his lecture at Frauenfeld [the second paper in Ref. 51, p. 292], Grossmann explained this point of departure. 'The fundamental mathematical concept of Einstein's theory of gravitation, which is to characterize a gravitational field by means of a quadratic differential form with variable coefficients, necessitates a generalization of the definitions and methods of vector analysis. Of fundamental significance for this purpose is Christoffel's famous treatise, ' Uber die Transformation der homogenen Differentialausdriicke zweiten Grades7 [J. fur Math. 70 (1869)], as well as Ricci and LeviCivita's Methodes de Calcul differentiel absolu et leurs applications [Math. Ann. 54 (1901)], which developed it further. In the latter treatise, the authors developed methods which allowed one to write the differential equations of mathematical physics in an invariant [i.e. independent of coordinate systems] form. The new developments in vector analysis shed even more light on the advantages of [employing] such a general invariant-theoretic treatment. Such a treatment at once provides a complete system of vector-analytical definitions which have been introduced [gradually] by Minkowski, Sommerfeld, Laue, and others, for the description of the four-dimensional world of the relativity theory.' 57. 58. 59. 60. 61.

62. 63. 64. 65. 66. 67.

M. Grossmann, first paper in Ref. 51, Mathematischer Teil, p. 257. A. Einstein, first paper in Ref. 51, p. 260. A. Einstein, S.B. Preuss. Akad. Wiss. (1914), see pp. 1054-1057. A. Einstein, Phys. Z. 14, 1258 (1913). Phys. Z. 14, 1262-1266 (1913); see also Phys. Z. 15, 108-110 (1914) and Phys. Z. 15, 115, 169-176, 176-180 (1914). After Einstein's lecture, Mie remarked: 'It seems to be that in his work, Einstein has postulated the most intriguing principle of a general relativity. However, this principle has not been rigorously complied with in the theory which he has presented...' (Phys. Z. 14, 1264). A little later, he went on to say that, 'The generalization of the relativity principle, in the form in which it has been expressed in Einstein's paper, applies only to linear transformations, and thus has absolutely nothing to do with accelerated motions.' (Phys. Z. 15, 176 (1914).) As Einstein remarked correctly, there is no justification for this assumption. (See e.g., Z. Math. u. Phys. 62, 233 (1913).) A. Einstein, the first paper in Ref. 51, Z. Math. u. Phys. 62 (1913), pp. 233-234. First paper in Ref. 51. A. Einstein and M. Grossmann, Z. Math. u. Phys. 6 3 , 221 (1914). A. Einstein and M. Grossmann, Z. Math. u. Phys. 6 3 (1914), see p. 225. We refer to various remarks in this paper concerning the meaning of the tensors appearing in the derivation and the formulation of the equations. [Ref. 52, 'Die formale Grundlage der allgemeinen Relativitatstheorie,' Sitz. Ber. Kgl. Preuss. Akad., Math. Phys. Kl. (1914), 1030-1085.] In an article entitled 'Die JVeue Mechanik,' M. Abraham discussed the EinsteinGrossmann theory. [Rivista di Scienze 15, 8-27 (1914)]: 'With the help of the method of the 'absolute differential calculus' developed by Ricci and Levi-Civita, the authors of the 'Entwurf' have also succeeded in writing the electromagnetic and dynamical equations in a form which, at least in the infinitesimal case [i.e. differential form] satisfies the relativity requirement [of the general transformation]. However, it is the

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gravitational field, the integration of which into relativity theory seems to be the goal of the 'Entwurf,' does not yield to the proposed scheme. The differential equations of the gravitational field, proposed by the authors, are not invariant with respect to the general space-time transformations, i.e. a system of mutually gravitating masses in nonuniform or rotational motion is, in general, not equivalent to a system at rest. 'It is true that the dynamics becomes relativistic if one includes the forces of rotation along with gravitation. However, the gravitational field of a rotating system is then no longer identical with that of the non-rotating system. It is really of no avail to seek to develop a theory of gravitation which would satisfy the requirement of general relativity; even if such a mathematical theory exists, it would be physically meaningless. Every theory of relativity, be it the special theory of 1905 or the general theory of 1913, founders at the rock of gravity. The relativistic ideas are obviously not broad enough to provide a framework for a comprehensive worldview.' (M. Abraham, l.c, p. 25.) In writing to his friend Besso, Einstein showed some respect for Abraham's remarks. 'The community of physicists has shown little interest in [my] work on gravitation. Abraham probably understands it best. In the 'Scienza,' he has been screaming loudly against all relativity, but with a good grasp. I shall visit Lorentz in Spring, and expect to discuss all this with him. He takes great interest in these matters, as does Langevin. Laue does not quite seem to accept [my] principal ideas, nor does Planck, only perhaps Sommerfeld.' [A. Einstein to M. Besso, from Zurich, towards the end of 1913.] Abraham wrote a sober review of the new gravitational theories in Jahrbuch der Radioaktivitat 1 1 , 470 (1914), showing a deep understanding of the physical principles and mathematical structure. (Since the completion of this paper in early September 1972, the correspondence between Einstein and Besso has been published: Albert Einstein, Michele Besso, Correspondance, 1903-1955 (edited by Pierre Speziali), Hermann, Paris (1972). All the quotations from the letters of Einstein and Besso may be identified in this book from the dates which I have given in my references. The English translations of the quotations, used in this paper, are mine.) 68. With considerable confidence, Einstein wrote in this paper (Ref. 52): 'The laws governing these differential forms have been obtained by Christoffel, Ricci and LeviCivita. I would like to present here an especially simple derivation of these, which [also] appears to me to be new.' 69. The invitation to the Prussian Academy was initiated by the leading Berlin physicists Max Planck, Walther Nernst, Heinrich Rubens, and Emil Warburg. In particular, they stressed his research on relativity in a petition to the Prussian Minister of Education: 'His name is primarily known through his famous treatise on "The Electrodynamics of Moving Bodies" (1905), on the Principle of Relativity, according to which the contradiction between the previously accepted Lorentz' theory of the light ether at rest and the subsequently (experimentally) established independence of the electrodynamic-optical phenomena in terrestrial bodies from the movement of the earth, is radically explained by the fact that an observer moving with the earth uses a time coordinate different from that of a stationary observer in a heliocentric system. 'The revolutionary consequences of this new interpretation of the concept of time have repercussions on the whole of physics, and above all on mechanics and epistemology. They [the consequences] later found, through the work of the mathematician Minkowski, a formulation which gives the whole system of physics a new unification

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inasmuch as the time dimension enters the stage on completely equal terms with the three space dimensions. Although this idea of Einstein's has proved itself so fundamental for the development of physical principles, the application of it still lies for the moment at the frontier of the measurable...' (See C. Seelig, Albert Einstein [Ref. 35], pp. 173-174.) 70. C. Seelig, Ref. 35, p . 178. 71. There are various accounts of the times which Einstein spent in Berlin. It is often asserted that Einstein was lonely in Berlin and did not receive the proper attention to which he was entitled. (See, e.g., P. Frank, Einstein: His Life and Times, New York (1965), pp. 114-115.) The creation of general relativity was surely something to which very few contributed, and in which the most important theoretician in Berlin, Max Planck, did not actively participate. Planck was much more concerned with the quantum problem, taking a point of view to which Einstein himself would subscribe later on. Nevertheless, Einstein and his work received increasing attention in Berlin, and he was able to write to his old friend Michele Besso (after he had received and refused invitations to chairs of theoretical physics at Zurich University and E.T.H.): 'I must say that it has been very hard for me to come to a decision about it. It goes without saying that the general conditions there [in Switzerland] have greater appeal for me. But if you could only see how beautiful are the relationships which have developed between my colleagues (particularly Planck) and myself, and how welcome all of them have made me feel, and continue to do so, and if you could just imagine how my work has progressed mainly on account of the understanding which I have found for it here, then you will realize that I could not really turn my back on this place.' [Letter, dated Berlin, 8 September, 1918. See Correspondance, Ref. 67.] 72. Letter to Besso, dated Zurich, March 1914. See Correspondance, Ref. 73.]. 73. Albert Einstein, Michele Besso, Correspondance, 1903-1955 (edited by Pierre Speziali), Hermann, Paris (1972). 74. A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. pp. 778-786 (1915). 75. Einstein remarked explicitly: 'Mathematics teaches us that all of these covariants can be derived from the Riemann-Christoffel tensor of the fourth rank In the problem of gravitation, we are particularly interested in the tensors of the second rank, which can be constructed from these tensors of the fourth rank and the g^ by inner multiplication.' (Ref. 74, p. 781. Compare with our earlier quotations from Grossmann's paper.) 76. Einstein said that on the basis of his considerations 'the field equations of gravitation may be written in the form since we already know that these equations are covariant with respect to arbitrary transformations of the determinant 1.' (Ref. 74, p. 783.) 77. After his lecture at Vienna in 1913, Einstein had a discussion with Reissner, who asked for the meaning of this 'energy:' 'Einstein has talked about the bending of electromagnetic radiation in a gravitational field. I wonder if Einstein would tell us something about the elementary question concerning the effect of gravity on its own mass, i.e. the mass of the energy of the gravitational field. How can one better understand, or show mathematically, that the static energy of a pure gravitational field, although it possesses inertia and weight, does not possess the other attributes of ponderable matter, i.e. its kinetic behaviour [such as free fall or collapse]?' (Phys. Z. 14, 1265). It was a perceptive question, in response to which, Einstein made some comments, but Reissner (either then or afterwards) did not have the feeling that an answer had been given (I.e., p. 1265). Actually Einstein had completely

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78. 79. 80. 81. 82.

83. 84.

85.

86. 87.

87a.

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misunderstood Reissner's question. He returned to deal with it in considerable detail and quite satisfactorily in Phys. Z. 15, 108-110 (1914). A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. p. 785 (1915). A. Einstein, 'Zur allgemeinen Relativitatstheorie (Nachtrag),' Sitz. Ber. Preuss. Akad. Wiss. pp. 799-801 (1915). A. Einstein, Ref. 79, pp. 799-800. A. Einstein, 'Die Feldgleichungen der Gravitation,' Sitz. Ber. Preuss. Akad. Wiss. pp. 844-847 (1915). Einstein had explained this result, obtained empirically in his second note, submitted on 18 November 1915, entitled 'Erklarung der Perihelbewegung, des Merkur aus der allgemeinen Relativitatstheorie,' Sitz. Ber. Preuss. Akad. Wiss. pp. 831-839 (1915). Ref. 81, p. 847. He wrote to Besso again on 21 December 1915: 'Read the papers [i.e. Einstein's]! The final release from misery has been obtained. What pleases me most is the agreement with the perihelion motion of Mercury Even Planck now begins to take the thing seriously, although he still resists it a bit. But he is a splendid human being.' [See Correspondance, Ref. 73.] Writing again to Besso on 3 January 1916, he said: 'The great success with [the theory of] gravitation pleases me extraordinarily. I have seriously in mind to write a book in the near future on the special and general theory of relativity...' [See Correspondance, Ref. 73.] See R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York (1953). D. Hilbert, Die Grundlagen der Geometrie (Leipzig, 1899; 9th edition, Stuttgart, 1962). 'This means that it is still being read, and obviously by more people than read Hilbert's other works. It has gradually been modernized, but few readers realize that foundations of geometry as a field has developed more rapidly than [Die] Grundlagen der Geometrie as a sequence of reeditions, and that Hilbert's book is now a historical document rather than a basis of modern research or teaching.' — Hans Preudenthal in his biographical sketch of David Hilbert in the Dictionary of Scientific Biography, Charles Scribner's Sons, New York (1972). Hermann Weyl wrote: 'Hilbert is the champion of axiomatics. The axiomatic attitude seemed to him one of universal significance, not only for mathematics but for all sciences. His investigations in the field of physics are conceived in the axiomatic spirit. In his lectures he liked to illustrate the method by examples taken from biology, economics, and so on. The modern epistemological interpretation of science has been profoundly influenced by him. Sometimes when he praised the axiomatic method he seemed to imply that it was destined to obliterate completely the constructive or genetic method.' [See H. Weyl, Gesammelte Abhandlungen, Vol. 4, Springer Verlag, New York 1968, pp. 162-163.] In his famous address, Mathematische Probleme, which Hilbert delivered before the International Congress of Mathematicians at Paris in 1900, he stressed the importance of great concrete fruitful problems. He posed and discussed twenty-three unsolved problems. His sixth problem dealt with the axiomatization of physics. [See D. Hilbert, Mathematische Probleme, Gott. Nachr. 1900; see also D. Hilbert, Axiomatisches Denken, Math. Ann. 36, 405-415 (1918). For a discussion of Hilbert's twenty-three problems and their influence on the development of mathematics, see L. Bieberbach, Die Naturwissenschaften 18, 1101-1111 (1930).]

88. In his obituary of Minkowski, Hilbert wrote: 'During the period he was at Bonn [18921894], Minkowski would spend his vacations regularly in Konigsberg, where his family

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lived, and where he met Hurwitz and me almost daily. Mostly we went for walks in the surroundings of Konigsberg. Once, on Christmas 1890, Minkowski remained in Bonn. When I tried to persuade him to come to Konigsberg, he.described himself in a funny letter to me as one who was physically fully contaminated, and in need of a ten-day quarantine, before Hurwitz and I would allow him to go on our walks in the mathematically pure Konigsberg.' [See D. Hilbert, Gesammelte Abhandlungen, Vol. 3, Berlin (1935), p. 355.] Hilbert continued, 'For more than six years [from fall 1902 to January 1909, when Minkowski died] we, his closest mathematical colleagues, took our "mathematical" walk every Thursday punctually at 3 o'clock on Hainberg, including the last Thursday before his death, when he especially spiritedly told us about the latest progress on his electrodynamic researches...' [D. Hilbert, I.e. p. 364.] Hilbert's collaboration with Minkowski was noted by Blumenthal: 'It was with Minkowski that Hilbert first started intensive reading (something which he avoided) of classical theoretical physics.' (Lebensgeschichte, in D. Hilbert, Gesammelte Abhandlungen, Vol. 3, p. 417.) 89. See Vortrdge uber die Kinetische Theorie der Materie und der Elektrizitat, B.G. Teubner, Berlin (1914); Hilbert wrote the foreword to these lectures. 90. O. Blumenthal, Lebensgeschichte, I.e., p. 417 [see Ref. 88]. 91. L. Foppl, 'Stabile Anordnungen von Elektronen im Atom' (1 March 1912), J. reine angew. Math. 141, 251-302 (1912); H. Bolza, 'Anwendung der Theorie der Integralgleichungen auf die Elektronentheorie und die Theorie der verdiinnten Gase' (2 July 1913); B. Baule, ' Theoretische Behandlung der Erscheinungen in verdiinnten Gasen,' (18 February 1914), Ann. Phys. (Leipzig) 44 (1914); K. Schellenberg, 'Anwendung der Integralgleichungen auf die Theorie der Elektrolyse,' (24 June 1914), Ann. Phys. 47 (1915); H. Kneser, 'Untersuchungen zur Quantentheorie,' (2 March 1921), Math. Ann. 84, 277-302 (1921). (See D. Hilbert, Gesammelte Abhandlungen, Vol. 3 , p. 433.) 92. G. Hamel, who had worked on his thesis, ' Uber die Geometrien in welchen die Geraden die Kurzesten sind' (24 June 1901), under Hilbert, continued his attempts to establish, beginning in 1909, the axiomatic foundations of mechanics along the lines of Hilbert. See G. Hamel, 'Uber die Grundlagen der Mechanik,' Math. Ann. 66, 350-397 (1909); 'Uber Raum, Zeit und Kraft als apriorische Formen der Mechanik, J.B. deutsche. Math. Ver. 18, 357-385 (1909); 'Uber ein Prinzip der Befreiung bei Lagrange,' J.B. deutsche Math. Ver. 25, 60-65 (1916); Lehrbuch der elementaren Mechanik, Leipzig (1912); 'Die Axiome der Mechanik,' in H. Geiger and K. Scheel, Handbuch der Physik Vol. V, Berlin (1927). See also G. Hamel, 'Die LagrangeEulerschen Gleichungen der Mechanik,' Z. Math. u. Phys. 50, 1-57 (1904); and 'Uber die virtuellen Verschiebungen in der Mechanik,' Math. Ann. 59, 416-434 (1905); and G. Herglotz, 'Uber die Mechanik des deformierbaren Korpers von Standpunkte der Relativitatstheorie,' Ann. Phys. (Leipzig) 36, 493-533 (1911). 93. C. Caratheodory, 'Untersuchungen Uber die Grundlagen der Thermodynamik,' Math. Ann. 67, 355 (1909). 94. P. Hertz, ' Uber die mechanischen Grundlagen der Thermodynamik,' Ann. Phys. (Leipzig) 3 3 , 225-274, 537-552 (1910). See also Hertz' article on statistical mechanics in Weber and Gans Repertorium der Physik, Vol. 1/2, Leipzig (1916). Hertz was close to Hilbert in 1912. 95. D. Hilbert, 'Begriindung der Kinetischen Gastheorie,' Math. Ann. 7 1 , 562 (1912). 96. D. Hilbert, 'Zur Begriindung der elementaren Strahlungstheorie,' I, II, III: Nachr. Ges. Wiss. Gott. (1912) p. 773 (Phys. Z. 13, 1056 (1912); Nachr. Ges. Wiss. Gott. (1913), p. 409 (Phys. Z. 14, 592 (1913)); Nachr. Ges. Wiss. Gott. (1914), p. 275 (Phys. Z. 15, 878 (1914))).

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97. D. Hilbert, 'Die Grundlagen der Physikl, II,' Nachr. Ges. Wiss. Gott. (1915), p. 395; (1917), p. 201. He later on united these two communications into a single article with the same title, Math. Ann. 92, 1 (1924). 98. W. Heisenberg, Z. Phys. 3 3 , 879-893 (1925); Math. Ann. 95, 683-705 (1925/26). 99. P. Jordan, ' Uber kanonische Transformationen in der Quantenmechanik,' Z. Phys. 37, 383-386 (1926); 38, 513-517 (1926); 'Uber eine neue Begriindung der Quantenmechanik,' Z. Phys. 40,809-838 (1927). Jordan proved that his formal transformation theory contains not only Schrodinger's wave mechanics, but also the Born-Wiener operator calculus and Dirac's g-number algorithm as special cases. 100. D. Hilbert, J. Von Neumann, and L. Nordheim, Math. Ann. 98, 1-30 (1928). 101. J. von Neumann, Grundlagen der Quantenmechanik, Springer, Berlin (1931) (English translation by R.T. Beyer, Princeton (1955)). 102. See, e.g., M. Jammer, Concepts of Space, Harvard University Press (1954); Harper Torchbooks, New York (I960). 103. See F. Klein's Vorlesungen iiber die Entwicklung der Mathematik im 19 Jahrhundert, Vol. I, Berlin (1926), reprinted by Chelsea Publ. Co., New York (1966), pp. 57-60. Also E.T. Bell, The Development of Mathematics, McGraw-Hill, New York (1945), 2nd edition, pp. 329-330. Gauss had thought about the possibility of non-Euclidean geometry since his student days, and he discussed the axiom of parallels with his friend Wolfgang Bolyai. In various letters he expressed the opinion that Euclidean geometry should not apply to nature. Two non-mathematicians, F.C. Schweikart and his nephew F.A. Taurinus, worked on this problem, and the former published a paper on 'astral geometry.' Gauss was aware of these attempts, and one knows that he had himself constructed a non-Euclidean geometry but not published it, because he feared, as he wrote to Bessel, 'the clamour of the Boeotians.' Then Wolfgang Bolyai's son Johann published his 'absolute geometry' in 1831, on which he had worked since 1820. Although Gauss acknowledged this 'genius of the first rank' privately, he neither discussed it publicly nor published anything about it, which made Johann Bolyai very bitter. At the same time, beginning in 1826, Nikolai Lobatchevski submitted his memoirs to the Kasan Academy. Beginning with his first paper on 'imaginary' or 'pan-geometry' in 1829, Lobatchevski continued publishing until 1855. In this case, Gauss praised the author openly; he had got to know the author and his work personally, and he had him elected to the Gottingen Academy in 1842. 104. Letter to H.W.M. Olbers, dated Gottingen, 28 April 1817, reprinted in Werke, Vol. 8, p. 177; translated in M. Jammer, Ref. 102, p. 145. Gauss had something to do with the method of triangulation ever since 1816, although the actual work was carried out in 1821-23 and 1828-1844. He reported on the results of the measurement of the record triangle with vertices on the mountain tops Hohenhagen, Brocken, and Inselberg, the distances being 69, 85, and 107 km respectively. [Gauss' 1827 paper on curved surfaces did not report on the results of the measurement of the triangle HBI. The results are to be found in his Werke, Vol. 4, p. 449. I am grateful to Prof. B.L. van der Waerden for this information.] 105. Printed in German, for the first time in Abhand. Ges. d. Wiss. zu Gottingen 13, 1867, and then in Riemanns Werke (edited by R. Dedekind and H. Weber, 1st edition 1876, 2nd edition 1892): translated into English by W.K. Clifford, Nature 8, 14-17, 36, 37, reprinted in W.K. Clifford's Mathematical Papers, London (1882), p. 54. Riemann had proposed, as he was required to do, three topics from which the faculty could choose any one and notify him about three days before the 'trial' lecture [Probevorlesung]. It was Gauss who, being especially interested in Riemann's

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105a.

106. 107. 108.

109.

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ideas, selected the topic which Riemann had listed in the third and last place, almost as an afterthought. In fact, he was not really prepared to give this lecture, but this disadvantage developed into a remarkably historic expansion of his ideas. For the mathematical and technical formulation, see his contribution for the Paris prize 1861, reprinted in Riemann's Ges. Math. Werke, Dover, New York, p. 391. Hermann Weyl published an edition of Riemann's Habilitation thesis with a detailed introduction, notes, and commentary. It is a most valuable guide to Riemann's geometrical thinking. See B. Riemann, Uber die Hypothesen, welche der Geometrie zu Grunde liegen, neu herausgegeben und erlautert von H. Weyl (Julius Springer, Berlin, 1919)]. See Clifford's Mathematical Papers, p. 55. [See Ref. 105] About Herbart's influence, Riemann was more explicit in remarks which were published later on in his Werke, pp. 507-508: 'My principal work consists in a new formulation of the known laws of nature, i.e. expressing them by means of other fundamental notions, whereby the use of experimental data concerning the interaction between heat, light, magnetism and electricity would make it possible to investigate their inter-connections. I was led to this by a study of the works of Newton and Euler, on the one hand, and Herbart, on the other. As for the latter, I could almost completely agree with the results of the earliest investigations of Herbart (which are presented in his doctoral and inaugural dissertations of 22 and 23 October 1802). I did, however, differ from the course of his later speculations concerning his natural philosophy and principles of psychology which refer to it [natural philosophy]. The author is a follower of Herbart in psychology and epistemology. However, he cannot subscribe to Herbart's natural philosophy and the metaphysical disciplines based upon it.' We have checked the 'Habilitationsthesen' (reprinted in J.F. Herbart's Samtliche Werke, Vol. I, Scientia Verlag, Aalen (1964), pp. 277-278) and quote thesis V (23 October 1802) (p. 278): lSpatii et temporis cogitationem quod e mente nostra ejicere non possumus, hoc non probat, eas cogitationes natura nobis insitas esse. Qui in hac Kantianae rationis parte latet error, totum Aollit systema.' Another part of Herbart's writings which seems to bear on Riemann's interests, mentions the word 'quantum of the surface,' which reappeared in Riemann's inaugural lecture, is contained in Nachschrift zur zweyten Auflage, Pestalozzi's Idee eines ABC der Anschauung, 1802 and 1804, reprinted in Samtliche Werke I (see p. 254): 'Perhaps the content of a surface, considered in a pure sense, is a totally Platonic idea, which would destroy all forms including the perceptual. It is this notion which defines the pure quantum of the extent of the surface, quite apart from the fact whether this quantum may appear in a round or rectangular shape, whichever one pleases.' Concerning Gauss' paper, we refer to New General Investigations of Curved Surfaces (1825) and General Investigations of Curved Surfaces (1827); it is the latter memoir which Riemann had in mind. Both of these papers have been translated into English, and appeared as General Investigations of Curved Surfaces, New York (1965). [For Herbart's influence on Riemann's ideas, see Bertrand Russell, An Essay on the Foundations of Geometry, Dover edition, New York (1956), Chapter 1.] See W. Pauli, Theory of Relativity, Pergamon Press, Oxford (1958), p. 35. W.K. Clifford's translation, Ref. 105, p. 69. W.K. Clifford, On the Space Theory of Matter, read 21 February 1870; abstract printed in Camb. Phil. Soc. Proc. 2 (1876), reprinted in W.K. Clifford's Mathematical Papers, pp. 21-22. W.K. Clifford, I.e. p. 22. Other indications of Clifford's ideas may be found in his posthumously published book The Common Sense of the Exact Sciences, London (1885), in particular, p. 225: 'We may conceive our space to have everywhere a nearly

304

110. 111. 112.

113.

114.

The Golden Age of Theoretical Physics uniform curvature, but that slight variations of the curvature may occur from point to point, and themselves vary with the time. These variations of the curvature with the time may produce effects which we not unnaturally attribute to physical causes independent of the geometry of our space. We may even go as far as to assign to this variation of the curvature "what really happens in that phenomenon which we term the motion of matter."' H. Minkowski, Nachr. Ges. Wiss. Gott., pp. 53-112 (1908). H. Minkowski, Ref. 110, p. 55. H. Minkowski, Ref. 110, pp. 65-66. See also his lecture at the 80th Naturforscherversammlung (Congress of Natural Scientists) in Cologne on 'Raum und Zeit' printed in Phys. Z. 10, 104-111 (1909), where he gave a most illuminating exposition of the concepts of space and time and introduced the notions of 'world vectors' and 'world postulate.' R. Hargreaves, 'Integral forms and their connection with physical equations,' Trans. Camb. Phil. Soc. 2 1 , 107-122 (1908). In fact, H. Poincare already used id as the fourth coordinate x 4 in his memoir 'Sur la dynamique de Velectron,' Palermo Rend, 21 (1906), which was submitted on 23 June 1905. G. Fechner, 'Der Raum hat vier Dimensionen,' in Vier Paradora, Leopold Voss, Leipzig (1846). Fechner wrote it under the pseudonym 'Dr Mises.' Actually, D'Alembert suggested thinking of time as a fourth dimension in his article 'Dimension' in the Encyclopedic Lagrange, in studying the reduction of quadratic forms to standard forms, casually introduced forms in n variables. He, too, used time as a fourth dimension in his Mecanique analytique (1788) and in his Theorie des fonctions analytique (1797). He says in the latter work/Thus we may regard mechanics as a geometry of four dimensions and analytical mechanics as an extension of analytical geometry.' Lagrange's work put the three spatial coordinates and the fourth one representing time on the same footing. Further, George Green in his paper of 1828 on potential theory did not hesitate to consider potential problems in n dimensions; he says of the theory, 'It is no longer confined, as it was, to the three dimensions of space.' Cauchy emphasized the concept of n-dimensional space [Compt. Rend. Acad. Sci. Paris 24, 885-87 (1847)], and Grassmann considered the concept of n-dimensional geometry in a note published in 1845. [See M. Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York (1972), pp. 1029-1030.]

115. H. von Helmholtz, ' Uber die Tatsachen die der Geometric zugrundeliegen,' Nachr. Ges. Wiss. Gott, pp. 193-221 (1863). 116. H. von Helmholtz, Ref. 115, p. 193. 117. H. von Helmholtz, Ref. 115, pp. 194-195. Riemann's 'trial' lecture [Probevorlesung] was published in the Abhandl. kgl. Ges. Wiss. Gott. 13 (1867), although it had appeared as a pamphlet in 1854. 118. J.J. Sylvester, Collected Works, Cambridge (1940), p. 198, originally printed as 'On the general theory of associated algebraical forms,' Cambridge and Dublin Math. J . 4 (1851). In a further paper 'On the principles of the calculus of forms,' ibid. 7 (1852), he introduced the expressions 'cogradient' and 'contragradient.' Sylvester himself became interested in this field through Arthur Cayley, a London barrister, who had studied in Cambridge, and graduated with the highest distinction in 1841, and became professor in Cambridge only in 1863. Cayley, starting from the literature, in particular the contributions of Jacobi, 'created the algebraic geometry' (F. Klein, Entw. d. Math. I, 148, Ref. 103). In 1858 Cayley introduced the concept of matrices which had been contained implicitly in the work of Grassmann. Cayley

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told Tait in 1894 as to what had led to matrices 'I certainly did not get the notion of a matrix in any way through the quaternions; it was either directly from that of a determinant, or as a convenient mode of expression of the equations x = ax + by y = ex + dy

119.

120.

121. 122.

123.

124.

125.

(See E.T. Bell, The Development of Mathematics, McGraw-Hill, New York (1945), 2nd. ed., p. 205.) With G. Salmon and J.J. Sylvester, who went to Johns Hopkins University in 1876, and started the development of the science of pure mathematics in the United States, Cayley formed the very influential British school. He stimulated Hermite in France and cooperated with Clebsch and Gordan in Germany. H.G. Grassmann, Die lineare Ausdehnungslehre, ein neuer Zweig der Mathematik, Berlin (1844); Die Ausdehnungslehre, vollstdndig und in strenger Form bearbeitet (1862). In 1872, R.F.A. Clebsch included the 'Stufen' of Grassmann in the theory of invariants in his paper ' Uber eine Fundamental Aufgabe der Invariantentheorie,' Abh. Gott. Ges. Wiss. 17 (1872). The word 'tensor' was first used by W. Voigt in 1898, in connection with the elasticity of crystals. F. Klein, ' Vergleichende Betrachtungen iiber neuere geometrische Forschungen,' Erlangen, appeared in December, 1872; reprinted in Math. Ann. 4 3 (1893) and in Ges. Abh., Vol. I, p. 460; See F. Klein, Vorlesungen, Ref. 103, Vol. II, p. 28. See p. 7 of Klein's Erlangen Programme, or Ref. 103, p. 28. See Ref. 103, Vol. II, p. 28. In his Erlangen Programme, Klein had stated: 'We develop the theory of invariants with respect to groups.' Later on he remarked [Ref. 103, Vol. II, p. 28], 'Instead of this, one says: 'the theory of the relationships which are invariant relative to the group;' then from this to the phrase theory of relativity there is only one step, which the modern physicists, in their domain, set as a general goal.' C.G.J. Jacobi, ' Vorlesungen iiber Dynamik,' held in 1842-43, first edited in 1866 by Clebsch, and contained in the Supplement to ' Werke,' Berlin (1884). Later J.R. Schiitz, Nachr. Ges. Wiss. Gott. (1897), p. 110, derived the energy conservation from the symmetry principle. In fact, Jacobi was the first person who derived the 10 integrals of the mechanical equations of motion, (partly) by using the infinitesimal transformations contained in the Euclidean group. Jacobi worked in Konigsberg from 1826 to 1843 as a great, powerful and influential teacher, contributing to nearly all fields of mathematics — elliptic functions, mechanics, differential equations and the theory of variations. His method was more inductive and impulsive than rigorous, as his famous statement indicates: 'Gentlemen: We have no time [or patience] for Gauss-like rigour.' (See F. Klein, Entwicklung I, Ref. 103, p. 114.) Jacobi's fundamental contribution to the theory of determinants, we have already mentioned above. His influence stretched far beyond Prussia and Germany. Hermite and Liouville acknowledged him as their teacher, and the same was true of Cayley in England. G. Hamel, 'Die Lagrange-Eulerschen Gleichungen der Mechanik,' Z. Math. u. Phys. 50, 1-57 (1904) (inaugural lecture); See also 'Uber die virtuellen Verschiebungen in der Mechanik,' Math. Ann. 59, 416-434 (1904). S. Lie and F. Engel, Theorie der Transformationsgruppen, Teubner, Leipzig (188893); and S. Lie and G. Scheffers, Kontinuierlichen Gruppen, Teubner, Leipzig (1893). It should be mentioned that Sophus Lie, the friend and inspiring colleague of Felix Klein, did not possess the latter's systematic approach, and was only persuaded with effort, by Klein, with the help of Engel, to write the laborious masterpiece in three

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The Golden Age of Theoretical Physics volumes. At a much later time, after Lie's work had been propagated by his book, Bell made the judgement: 'Engel's triumph in organizing Lie seems to have been a mistake. But it was not fatal. Three hundred leaden volumes could not have crushed and buried a talent like Lie's. His ideas had done their enduring work long before all the freshness and ferment were systematized out of them. As for Klein's part, in this comedy of well-intentioned errors, it is only fair to record that he was actuated by the least inexcusable of all excuses for meddling with another man's life: the sincere and unselfish desire to make a friend do something distasteful for his own supposed good.' (E.T. Bell, The Development of Mathematics, 2nd edition, New York (1945), p. 442.) G. Hamel, Ref. 124, p. 12. G. Herglotz, ' Uber die Mechanik des deformierbaren Korpers von Standpunkte der Relativitatstheorie,' Ann. Phys. (Leipzig) 36, p. 1911. The name 'Poincare group' was first used by E.P. Wigner in: 'On Unitary Representations of the Inhomogeneous Lorentz Group,' Annals of Math. 40, 149 (1939). 'His papers on the theory of invariants had the unexpected effect of withering, as it were overnight, a discipline which so far had stood in full bloom.' [H. Weyl, Gesammelte Abhandlungen Vol. 4, Springer-Verlag, New York 1968, p. 124.]

129a. The first mention of Emmy Noether in this context occurs in Felix Klein's 'Zu Hilberts erster Note uber den Grundlagen der Physik,' Nachr. Ges. Wiss. Gott, Math. Phys. kl., 469 (1917), especially on p. 476: 'When I spoke to Miss Noether recently about my result concerning your [Hibert's] energy vector, she informed me that she was able to derive the same result already a year ago on the basis of your note (i.e. not from the simplified calculation in me memoir Nr. 4). At that time she wrote a manuscript, which she showed me. She had brought it to the decisive stage as I recently showed before the Mathematical Society.' Klein's statements occur in a letter to Hilbert (which he submitted to the Gottingen Academy on January 25, 1918.) In his reply, part of which Klein also mentioned in his note, Hilbert said: 'Emmy Noether, whom I had invited already a year ago to assist me in the clarification of the analytical questions concerning my energy principle, found at that time that the energy components, which I had stated (as well as those of Einstein) formally by means of the Lagrangian differential equations [Equations (4) and (5) of my first communication], could be reformulated, without resorting to the Lagrangian equations, as expressions whose divergence vanishes identically.' (I.e., p. 477) 130. F. Engel, ' Uber die zehn allgemeinen Integrate der klassischen Mechanik,' Nachr. Ges. Wiss. Gott, pp. 270-275 (1916), communicated by Klein on 11 November 1916. Engel stated in his introduction [I.e., p. 270]: 'At our last meeting you brought to my attention a paper by Herglotz [See Ref. 127]. He has treated a mechanical problem which is consistent with the ten component Lorentz group, and has derived the ten known integrals of the problem from the infinitesimal transformations of this group (entirely in the sense of Lie). In this connection, you had remarked that from these integrals one can obtain the ten known integrals of the ordinary n-body problem, by taking the limit and letting the velocity of light go to infinity. It is now only left to show how one can obtain, in the sense of Lie, the latter integrals from the infinitesimal transformations of a ten-component group belonging to the n-body problem, which, unfortunately, has been called the Galilean group. It is because it deals only with the integral of the [kinetic energy] and its second moments, since the zero-th and first moments are quite obvious.' 131. E. Noether, 'Invariante Variationsprobleme' (dedicated to F. Klein on the fiftieth anniversary of his doctorate), Nachr. Ges. Wiss. Gott, Math. Phys. Kl. pp. 235-257

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132. 133. 134.

135. 136. 137.

138. 139. 140.

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(1918), communicated on 26 July 1918. Emmy, daughter of the mathematician Max Noether from Erlangen, came to stay in Gottingen in 1916, although she had visited there earlier. The connections of Gottingen with Erlangen are obvious: Felix Klein had come to Gottingen from there and had kept up his contacts with Gordan and Max Noether (who became a professor of mathematics in Erlangen in 1875 and stayed there until his death in 1921). Max Noether 'played an important role in the development of the theory of algebraic functions as the chief representative of the algebraic geometrical school.' [H. Weyl, 'Emmy Noether,' Scripta Math. 3, 201-220 (1935), reprinted in Gesammelte Abhandlungen 3, 425-444; see p. 425.] When Gordan retired in 1910, Erhard Schmidt, one of Hilbert's students, brought his master's spirit to Erlangen, and so did Ernst Fischer who followed Schmidt there in 1912. At least since 1913, Weyl recalled, Emmy Noether had visited Gottingen occasionally, and both Hilbert and Klein welcomed her collaboration. In fact, Hilbert tried to get her through with her habilitation (i.e. installed as a Privatdozent in the University of Gottingen) already during World War I, but he did not succeed before 1919, because the philosophers and historians opposed her candidacy on the ground that she was a woman. The faculty members argued that since there existed no faculty rest rooms for ladies, this would cause difficulties for a woman lecturer. Upon which Hilbert remarked: 'I do not see that the sex of a candidate is an argument against her admission as Privatdozent. After all, we are a university and not a bathing establishment.' (H. Weyl, I.e., p. 431). In 1922 she was given the courtesy title, without official salary, of an extraordinary (i.e. associate) professor. She stayed in Gottingen until 1933, where many worked with her. She had to leave Germany in 1933, and she became a professor of mathematics at Bryn Mawr College, also lecturing at times at the Institute for Advanced Study in Princeton. She died on 14 April 1935. See also B.L. van der Waerden, 'Nachruf auf Emmy Noether,' Math. Annalen, pp. 469-476 (1935), and Auguste Dieck, 'Emmy Noether,' Birkhauser, Basel, 1970. E. Noether, Ref. 131, pp. 238-239. E. Bessel-Hagen, 'Uber die Erhaltungssatze der Elektrodynamik,' Math. Ann. 84, 254 (1921). The generalization of Noether's Theorem to quantum mechanics was made by E.P. Wigner in his paper ' Uber die Erhaltungssatze in der Quantenmechanik,' Nachr. Ges. Wiss. Gott, p. 375 (1927). Gustav Mie, 'Grundlagen einer Theorie der Materie (I),' Ann. Phys. (Leipzig) 37, 511-534 (1912); (II), 39, 1-40 (1912); (III), 40, 1-66 (1913). G. Mie, I.e. (I), p. 511. This is in contrast to the model, for instance, of Lorentz who thought of an electron, with its charge distributed on the surface of its shell. Poincare had already pointed out that the Lorentz electron model suffered from many difficulties. G. Mie, I.e. (I), p. 513. G. Mie, I.e. (I), p. 512. 'This, to us, seems to constitute a very weighty argument against Mie's theory,' Pauli remarked in 1921. (W. Pauli, 'Relativitatstheorie' in Encyklopadie der math. Wiss. V/19, Leipzig (1921); Theory of Relativity, Pergamon Press, Oxford (1958), p. 192, see Ref. 106). Although Mie did not succeed in discovering the appropriate 'world function' which could account for the existence, asymmetry, and stability of the proton and the electron, his investigations inspired the work of M. Born and L. Infeld on 'nonlinear electrodynamics' which corresponded entirely with Mie's programme. [See

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The Golden Age of Theoretical Physics M. Born's review article on the Born-Infeld non-linear electrodynamics in Ann. Inst. H. Poincare 7, 155 (1937), which gives an explicit discussion of the relation of this work to Mie.] G. Mie, Section 35 of (III), in Ref. 135, entitled 'Das Plancksche Wirkungsquantum,' does not represent a step in this direction. Constance Reid, Hilbert, Springer, New York (1970), pp. 140-141. Einstein-Sommerfeld Briefwechsel (see Ref. 1), p. 30. A. Einstein, S.B. Preuss. Akad. Wiss. (1915), p. 844. D. Hilbert, 'Die Grundlagen der Physik,' Nachr. Kgl. Ges. Wiss. Gott. pp. 395-407 (1915), especially p. 405. In an article, 'Contribution to the History of Einstein's Geometry as a Branch of Physics,' [Published in Relativity, Plenum Press (1970)], Eugene Guth wrote [p. 183]: 'Apparently, Weyl, in the first textbook of general relativity [H. Weyl, Raum, Zeit, Materie, Springer, Berlin (first edition 1918)], originated the story that the field equations of general relativity have been established also by Hilbert simultaneously and independently of Einstein. This remark was taken over also by Pauli in his well-known article on relativity [Ref. 140] and later on by several other textbook writers. The remark of Weyl and Pauli does not correspond with the historical truth. [My emphasis.] In the first place these two experts did not look up carefully the references in Hilbert's paper. Otherwise, they would have noticed that Hilbert quotes all of Einstein's communications of November 1915, in the Proceedings of the Berlin Academy. In particular, he quotes the paper on p. 844 which contains the final form of the GR [General Relativity] field equations.' Guth goes on to say [pp. 183-184] that, 'It is interesting, however, how this 'myth' has arisen. For the following remarks I am indebted to Professor P.P. Ewald who was, in 1915, Hilbert's 'assistant for physics.' His story is reinforced by the recently published exchange of letters between Einstein and Sommerfeld. [See our Ref. 1] In a letter dated 15 July 1915, Einstein says about his visit in Gottingen that he had great joy there; everything he said was understood in detail. He was very enthusiastic about Hilbert. Clearly Einstein gave a talk in Gottingen [at Hilbert's invitation, these were the well-known Wolfskehl lectures]. Hilbert, whose absentmindedness was legendary, started to think about the problem on the basis of what Einstein said, who at that time did not have yet the correct form of his field equations. I heard from Ewald that Hilbert, very likely in the fall of 1915, gave a talk in Gottingen presenting the correct equations without referring to Einstein. However, by that time, Einstein must have had also the correct equations, certainly before Hilbert did. Sommerfeld heard about Hilbert's talk in Gottingen and suggested that Hilbert write a letter apologizing to Einstein, which Hilbert, of course did. At any rate, he never claimed having been the independent discoverer of the field equations of G.R. [General Relativity].' We have not been able to find any evidence that would substantiate these remarks.* There does not exist any trace of a letter of apology from Hilbert to Einstein. The reason for that is quite simple: Hilbert did not need to apologize at all. As for Weyl and Pauli, they had read the literature quite carefully and concluded that Hilbert had discovered the field equations of gravitation [Equations (37)] simultaneously with and independently of Einstein. Hilbert's derivation [of Equations (37)] was announced on 20 November 1915 in Gottingen, a few days before Einstein presented his formulation to the Prussian Academy in Berlin (on 25 November 1915). Hilbert was able to refer in his published paper in the Gottinger Nachrichten to all four of

*See Note 270 added in proof.

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146.

147.

148. 149.

150.

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Einstein's communications of November 1915 because his paper, just as Einstein's communications, was published only several weeks later, and he could include these in the footnotes. (See footnotes on p. 395 and p. 405 of Ref. 145, and records of the meetings of the Konigliche Gesellschaft der Wissenschaften, Gottingen.) Hilbert was clearly inspired by Einstein's work of 1913-14 on gravitation, and Mie's work of 1912-13 on electrodynamics, to develop his unified theory of electromagnetism, and acknowledged this fact clearly and unequivocally. But his derivation (of Equations (37)) was completely independent of Einstein's approach. He used an axiomatic method and employed the variational principle. In fact, Einstein did not like Hilbert's approach. (See our References 149, 187, 203, 242, and 257.) Instead of an apology for his first communication, Hilbert wrote a second memoir on the foundations of physics, and presented it to the Gottingen Academy on 23 December 1916. That Hilbert had been guided by the intuitive physical reasoning of Einstein, he repeated often enough. That he considered Einstein as the principal architect of general relativity, he confirmed by his often expressed admiration for Einstein and by his nomination of Einstein for the third Bolyai prize in 1915 'for the high mathematical spirit behind all his achievements.' [The first and second recipients of the Bolyai prize had been Poincare and Hilbert respectively. Einstein did not receive this award.] Hilbert was justly proud of his contribution to the foundations of physics and his derivation of the field equations of gravitation. By 1920, Einstein's dominance over the entire range of questions of general relativity was such that people were forgetting what Hilbert had contributed. That is why, in 1924, he again published a treatise which consolidated the principal contents of his communications of 1915 and 1916 with additional comments. He felt that his contributions contained 'an enduring core.' [See Refs. 147 and 258.] Hilbert's contributions to the theory of gravitation are not mentioned in most textbooks or review articles on general relativity. There are, however, a few exceptions: e.g., H. Weyl, Space, Time, Matter (Dover, New York, 1952, based on the fourth German edition), H. Weyl, 'Zu David Hilberts siebzigstem Geburtstag,' Die Naturwissenschaften 20, 57-58 (1932), and H. Weyl, l50 Jahre Relativitatstheorie,' Die Naturwissenschaften 38, 73-83 (1950), W. Pauli, Theory of Relativity, p. 161, J.W. Anderson, Principles of Relativity Physics, Academic Press, New York (1967), p. 344. Only P. Jordan went so far as to call the equations of gravitation in the vaccum, the Einstein-Hilbert equations. (Schwerkraft und Weltall, Braunschweig (1952), p. 61.) In an essay in honour of Hilbert's 60th birthday, Max Born wrote: 'This remarkable coincidence, however, never gave rise to any controversy about priority between the two men. Rather, their correspondence, which had originated in an exchange of scientific views, developed into more personal and friendly communications. Hilbert always remained aware of the fact that the great principal physical idea was Einstein's, and he expressed it in numerous lectures and memoirs... (Naturwiss. 10, 88-93 (1922), in particular p. 92, reprinted in M. Born, Ausgewahlte Abhandlungen, Vol. II, pp. 584-598, see p. 595.) C. Reid, Hilbert, Springer, New York (1970), p. 142. Einstein once remarked, 'The people in Gottingen sometimes strike me, not as if they want to help one formulate something clearly, but as if they only want to show us physicists how much brighter they are than we.' (Reid, I.e., p. 142.) C. Reid, Hilbert, p. 142.

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151. 'Einstein has posed immense problems, has brought forth profound thoughts and unique conceptions, and has invented ingenious methods for dealing with them. Mie was able to construct his electrodynamics [on the basis of these], and they have opened up new avenues for the investigation of the foundations of physics.' (D. Hilbert, see Ref. 97, 'Die Grundlagen der Physik,' p. 395.) 152. D. Hilbert, Ref. 97, I, p. 395. 153. D. Hilbert, Ref. 97, I, p. 396. 154. D. Hilbert, Ref. 97, I, p. 396. 155. 'The leitmotiv for the construction of my theory has been the following mathematical theorem, the proof of which I shall give at another place.' (D. Hilbert, Ref. 97, I, p. 396.) 156. D. Hilbert, Ref. 97, I, p. 397. 157. See the remarks in Section III.3. Emmy Noether had given the proof in its most general form. Hilbert gave a complete proof for a special case (which he called Theorem 2) of the general theorem. 158. D. Hilbert, Ref. 97, I, pp. 397-398. 159. A. Einstein, Ann. Phys. (Leipzig) 49, 769 (1916), p. 810. 160. D. Hilbert, Ref. 97, I, p. 403. 161. D. Hilbert, Ref. 97, I, p. 404. 162. D. Hilbert, Ref. 97, I, p. 406. 163. D. Hilbert, Ref. 97, I, p. 407. 164. C. Seelig, Albert Einstein [Ref. 35], p. 188. 165. A. Einstein, Ann. Phys. (Leipzig) 49, 769-822 (1916). 166. A. Einstein, Ref. 165, p. 772. 167. A. Einstein, Ref. 165, p. 775. 168. A. Einstein, Ref. 165, p. 779. 169. F. Kottler, Ann. Phys. (Leipzig) 50, 955 (1916). Kottler had already given, as early as 1912, the expression for the electromagnetic field equations in a generally covariant form (S.B. Akad. Wiss. Wien 121, 1659 (1912)), and Einstein recognized his contribution. 'Among the papers which deal critically with the general theory of relativity, those of Kottler are particularly remarkable, because this expert has really penetrated into the spirit of the theory.' [A. Einstein, ' Uber Friedrich Kottlers Abhandlung 'Uber Einsteins Aquivalenzhypothese und die Gravitation,' Ann. Phys. (Leipzig) 5 1 , 639-642 (1916).] 170. A. Einstein, Ann. Phys. (Leipzig) 5 1 , 639 (1916). 171. A. Einstein, Ann. Phys. (Leipzig) 5 1 , 639 (1916), esp. pp. 640-641. 172. E. Kretschmann ' Uber den physikalischen Sinn der Relativitdtspostulate, A. Einsteins neue und seine urspriingliche Relativitdtstheorie,'1 Ann. Phys. (Leipzig) 53, 575-614 (1917). 173. E. Kretschmann, Ref. 172, p. 576. 174. E. Kretschmann, Ref. 172, p. 584. 175. E. Kretschmann, Ref. 172, p. 610. 176. A. Einstein, Ann. Phys. (Leipzig) 55, 241-244 (1918). He praised the author (Kretschmann) for his sharp wit, but did not find the sharper form of the equivalence principle either valuable or desirable. 177. A. Einstein, Ann. Phys. (Leipzig) 49, 769 (1916), esp. p. 781. 178. A. Einstein, Ref. 177, p. 789. 179. A. Einstein, Ref. 177, pp. 802-803.

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180. A. Einstein, Ref. 177, pp. 803-804. The formal proof of this statement was given by H. Vermeil, Nachr. Ges. Wiss. Gottingen (1917) 334; see also H. Vermeil, Math. Ann. 79 (1918) 289. 181. A. Einstein, Ref. 177, p. 804. 182. 'The special theory of relativity has led to the result that the inertial mass is nothing but the energy, which can be expressed mathematically by means of a symmetrical tensor of the second rank, the energy tensor. Hence, we shall have to introduce an energy tensor Taa in general relativity also, having the same mixed character as the components taa of the gravitational field, but belonging to a mixed symmetrical covariant tensor.' (Ref. 177, p. 807.) 183. A. Einstein, Ref. 177, p. 808. 184. A. Einstein, Ref. 177, p. 810. 185. A. Einstein, Ref. 177, pp. 810-811. 186. Einstein had already done this in a paper entitled 'Eine neue formale Deutung der Maxwellschen Feldgleichungen der Elektrodynamik,' Sitz. Ber. Preuss. Akad. Wiss. (1916), pp. 184-187. (Submitted and read on 3 February 1916.) 187. C. Seelig, Ref. 35, p. 199. 188. This value had already been calculated in his communication of 18 November 1915, dealing with the motion of the perihelion of Mercury. 189. A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (1916), pp. 688-696. See also the later paper, 'Uber Gravitationswellen,' Sitz. Ber. Preuss. Akad. Wiss. (1918), pp. 154-167, in which Einstein gave a more lucid presentation and corrected a previous error. 190. Einstein noted [similar to what Hilbert thought], 'At first sight, it seems strange that, in addition to the ten equations [Equations (66)] for the ten functions 7 M „, there should be four others which can be written arbitrarily along with them, without over-determination. The justification of this procedure is the following. The equations [of gravitation] are covariant with respect to arbitrary substitutions, i.e. they are satisfied for an arbitrary choice of the coordinate systems. If a new coordinate system is introduced, then the g^ of the new system would depend upon the four arbitrary functions which define the transformation of the coordinates. These four functions can be so chosen that the g^,, of the new system would satisfy four arbitrarily prescribed relations. The latter are assumed to be chosen such that they transform into the equations [(68)] in the case of the approximation of interest to us. Thus the latter equations signify the condition chosen by us according to which the coordinate system has to be selected. By means of the equations [(68)], one can now obtain the simple equations [(67)] instead of [(66)].' (A. Einstein, Ref. 189, (1918), pp. 155-156.) 191. A. Einstein, Ref. 189 (1918), p. 164. 192. A. Emsteiu,' Kosmologische Betrachtungen zur allgemeinen Relativitatstheorie,' Sitz. Ber. Preuss. Akad. Wiss. (1917), pp. 142-152 (submitted and read on 8 February 1917). In a letter dated 5 December 1916, Michele Besso thanked Einstein for the papers which he had sent him, and asked him rather vaguely about the role of gravitational energy. Einstein replied to him during the same month (December 1916), and said: 'Now concerning A = 1/R. Quite apart from the fact whether it holds or not, the question here is not one of great scientific significance.... If I choose the g^, to be Galilean at a certain location, and continue in a most appropriate manner, then [we ask ourselves] how do g^ depend on space and time at an extremely large distance? Is it possible to arrange things in such a way that the g^v shall be determined by matter alone, in accordance with the principle of relativity?... The permanence of the universe requires that motion (centrifugal forces) should oppose [gravitational]

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collapse. Such is the case with our solar system. But this is possible only if one lets the average density of matter suitably approach zero at infinity, or else infinite differences of potential would arise. 'Such a description would be unsatisfactory even in Newtonian theory. [The problem of the rarification of matter and the dissipation of energy at infinity.] It is even less satisfactory in relativity, because relativity does not satisfy inertia. The latter would be mainly determined at [spatial] infinity by the g^, and to a much lesser extent by the interaction with the other masses. I find such a description unbearable. The only solution which I can think of is the hypothesis of the closedness of space, the consistency of which I have already demonstrated. 'Of course, I do not really believe that the universe is in a statistical-mechanical equilibrium, even though I have [on occasion] treated it so. Because then, all the stars will coalesce together (if only a finite volume were at our disposal). However, a deeper examination shows that statistics can be used for the treatment of such questions which concern me. One can, however, also do without the statistical considerations. Certainly, the infinitely large differences of potential would give rise to very large stellar velocities, which should have been observable for a long time. Small potential difference, together with an infinite expansion of the universe, requires an empty space at infinity (constancy of the g^ at infinity with an appropriate choice of the coordinates), in contradiction to any sensible interpretation of relativity. Only the closedness of the universe can get rid of this dilemma, and in any case it is also suggested by the fact that the curvature has the same sign everywhere because the energy density is nowhere negative, as we know. 'The A introduced here for the first time has nothing to do with the previous one. The fact that an additional term +\gpV on the left side of the field equation does not interfere with its tensor character, had escaped my notice. My new approach, however, seems to require a nonvanishing A, which would give rise to a non-vanishing average density of matter Qo- A count of fixed stars leads to an order of magnitude for £>o of 1 0 - 2 2 gm/cm 3 , corresponding to a radius of the universe R = 10 7 light years, whereas the distance of the remotest fixed stars has been estimated 10 4 light years. You and Dallenbach should read [my] paper; you will have fun.' [See Correspondance, Ref. 73.] A. Einstein, S.B. Preuss. Akad. Wiss. (1917), p. 143. A. Einstein, Ref. 193, pp. 143-144. A. Einstein, Ref. 193, p. 145. A. Einstein, Ref. 193, p. 148. A. Einstein, Ref. 193, p. 151. A. Einstein, Ref. 193, p. 152. At first Einstein was very satisfied with his cosmological considerations. In a letter to Besso, dated 9 March 1917, he wrote: 'You will have received the "cosmological considerations." It provides at least a proof that my relativity [theory] can lead to a system without contradiction. Till now one had always to be afraid that the "infinite" would harbour inconsistencies which could not be resolved. Unfortunately there is little promise that these views can be checked in reality. If we take into consideration the astronomical results on the density of distribution of stars, we obtain an order of magnitude estimate of R = 10 7 light years, whereas visibility extends to R = 10 4 light years. Incidentally, the question arises whether we should not be able to see stars which lie close enough to our antipode. They should have a negative parallax. One must not forget, however, that the curvature of space is irregular, and light rays travel through an inhomogeneous medium.' [See Ref. 73.]

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Writing to Besso again on 20 August 1918, he said: 'Either the universe has a centre, has a vanishing density everywhere, empty at infinity where all the thermal energy is gradually lost as radiation; or, all the points are equivalent on the average, and the mean density is everywhere the same. In either case, one needs a hypothetical constant A, which specifies the particular mean density of matter consistent with equilibrium. One perceives at once that the second possibility is more satisfactory, especially since it implies a finite size for the universe. Since the universe is unique, there is no essential difference between considering 7 as a constant which is peculiar to a law of nature or as a constant of integration.' [See Ref. 73.] Later on, the cosmological constant had ups and downs, even in the esteem of Einstein. In the early nineteen-twenties he became interested in the mathematically more general affine geometry, close to the approaches of Weyl and Eddington, but he returned to his old equations in 1925. 'I had to reject my endeavour which followed Eddington's ideas. In any case, I am now convinced that unfortunately nothing is to be gained from the Weyl-Eddington complex of ideas. I regard the equation

199.

200. 201. 202. 203.

as the best (electromagnetic) [equation] that we have today. There are nine equations for fourteen quantities g^v and 7 ^ . New calculations seem to indicate that they yield the motion of the electrons. But it appears to be doubtful whether the quanta have a place in all this.' [Letter to Besso, dated 25 December 1925, I.e. in Ref. 73] Use Rosenthal-Schneider, 'Erinnerung an Gesprache mit Einstein,' manuscript dated 23 July 1957. [Einstein Archive, Princeton, N.J.] Singularly lacking are any eye-witness accounts of Einstein's tremendous absorption in the problems of general relativity during 1915-16. A charming glimpse is contained, however, in Charles Chaplin's My Autobiography (The Bodley Head, London 1964). Chaplin recalls a dinner at his home in California in 1926, at which Einstein, Mrs Einstein, and two other friends of Chaplin were present. At dinner Mrs Einstein 'told me the story of the morning he conceived the theory of relativity.' She related: 'The Doctor [i.e. Einstein] came down in his dressing-gown as usual for breakfast but he hardly touched a thing. I thought something was wrong, so I asked what was troubling him. 'Darling,' he said, 'I have a wonderful idea.' And after drinking his coffee, he went to the piano and started playing. Now and again he would stop, making a few notes then repeat: 'I've got a wonderful idea, a marvellous idea!' I said: 'Then for goodness' sake tell me what it is, don't keep me in suspense.' He said: 'It's difficult, I still have to work it out.' Mrs Einstein told Mr Chaplin that Einstein continued playing the piano and making notes for about half an hour, then went upstairs to his study, telling her that he did not wish to be disturbed, and remained there for two weeks. 'Each day I sent him up his meals,' she said, and in the evening he would walk a little for exercise, then return to his work again.' 'Eventually,' Mrs Einstein said, 'he came down from his study looking very pale. 'That's it,' he told me, wearily putting two sheets of paper on the table. And that was his theory of relativity.' A. Einstein, Ann. Phys. (Leipzig) 49, 771. A. Einstein, Ref. 200, p. 776. Letter from Einstein to M. Besso, dated Berlin, 31 October 1916. [See Ref. 73] Einstein expressed his prejudice against Hilbert's axiomatic method. 'In this paper, it is not my goal to present the theory of general relativity as a system, logically

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206. 207. 208. 209.

210. 211. 212. 213. 214. 215. 216.

217. 217a.

218. 219.

220. 221. 222.

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as simple as possible, based on a minimum of axioms. Rather, my motivation is to develop this theory in such a way that the reader would sense the psychological simplicity of our approach, and the basic assumptions would be verified as much as possible by our empirical knowledge.' (A. Einstein, Ref. 200, p. 777.) D. Hilbert, Nachr. Ges. Wiss. Gott., Math. Phys. Kl (1917), 53-76, in particular p. 53. D. Hilbert, 'Die Grundlagen der Physik,' Math. Ann. 92, 1-32 (1924). This is a slightly condensed version of Hilbert's two memoirs of 1915 and 1916, which were published in the Proceedings of the Gottingen Academy. The third axiom is the one which expresses H as a sum of a gravitational term K and an electrical term L. Axiom IV was stated in this paper (p. 11). [This paper was reprinted in Hilbert's Gesammelte Abhandlungen, Vol. 3, pp. 258-289.] D. Hilbert, Nachr. Gott. (1917), p. 54. D. Hilbert, Nachr. Gott. (1917), p. 57. W. Pauli, I.e. in Ref. 106, Section 22, 'Reality relations,' English edition, pp. 62-64. H. Reichenbach, Axiomatik der relativistischen Raum-Zeit-Lehre, Vieweg, Braunschweig (1965); English translation by M. Reichenbach, University of California Press, Berkeley (1969), pp. 179-181. It seems to be quite remarkable that Reichenbach ignored Hilbert's system of axioms completely. D. Hilbert, Nachr. Gott. (1917), p. 61. D. Hilbert, Nachr. Gott. (1917), pp. 63-64. D. Hilbert, Nachr. Gott. (1917), p. 66. D. Hilbert, Nachr. Gott. (1917), p. 70. Einstein oscillated between the opinions whether he should take the singular solution as the one corresponding to the real particle (electron, etc.) or not. W. Pauli, Theory of Relativity, p. 145, footnote 277 (see Ref. 106). Note, e.g., the statement in J.L. Anderson, Principles of Relativity Physics: 'Today most physicists would be not only willing to accept as axiomatic the existence of a variational principle but would also be loath to accept any dynamical equations that were not derivable from such a principle.' (J.L. Anderson, I.e., p. 344 (see Ref. 146).) Einstein-Lorentz correspondence, The Hague, The Netherlands. H.A. Lorentz, Versl. K. Akad. Wetensch. Amsterdam 23,1073 (1915); Proc. K. Akad. Amsterdam 19, 751 (1915); reprinted in H.A. Lorentz, Collected Papers, Vol. 5, The Hague (1937), pp. 229-245. H.A. Lorentz, Ref. 217a, p. 1073, or Collected Papers, Vol. 5, p. 229. H.A. Lorentz, Versl. K. Akad. Wetensch. Amsterdam 24, 1389, 1759; 25, 468, 1380 (1916); Proc. K. Akad. Amsterdam 19, 1341, 1354, 20, 2, 20 (1916), reprinted in Collected Papers, Vol. 5, pp. 246-313. H.A. Lorentz, Collected Papers, Vol. 5, p. 246. H.A. Lorentz, Ref. 220, p. 246. H.A. Lorentz, Ref. 220, p. 251. In his remarks Lorentz seems to convey the impression that everything had been done by Einstein, which, as we have discussed, was not entirely so. The fact was that Einstein was in close contact with Leiden since the time his friend Paul Ehrenfest succeeded Lorentz there. Einstein always expressed the feeling that he had found the greatest understanding in Holland. For instance, in a letter to Besso, written either at the end of 1913 or early in 1914, Einstein remarked: 'I shall visit Lorentz in Spring in order to discuss all these matters with him [i.e. general relativity]. He has great interest in this, just as Langevin.' [See Correspondance, Ref. 73.]

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Einstein also enjoyed continuous and warm personal contact with Lorentz during and after the first World War, at a time when relations between the scientists of Germany and those of other nations were rather limited. In the exchange of correspondence between the two men, several letters are concerned with the attitudes of Planck, Nernst, and others with whom Lorentz obviously did not have any exchange during that time. It was, therefore, quite natural, and further enhanced by the occasional visits of Einstein to Leiden, that Lorentz should regard almost everything in general relativity as coming from Einstein. One must also consider that Lorentz at that time, in 1915 and 1916, was beginning to depend much more on personal discussions for information about new developments rather than on his own study of the literature. 223. This part is contained in the second paper in H.A. Lorentz, Versl. K. Akad. Wetensch. Amsterdam 24, 1759 (1916). 224. A. Einstein, 'Hamiltonsches Prinzip und allgemeine Relativitatstheorie,' Sitz. Ber. Preuss. Akad. Wiss. (1916), pp. 1111-1116. 225. A. Einstein, Ref. 224, p. 1111. 226. F. Klein, Nachr. Ges. Wiss. Gott, Math. Phys. Kl. (1918), pp. 171-189. 227. He proudly noted: 'As one can see, there is nothing more to be really calculated in the integrals, except to make the obvious use of the elementary formulas of the classical variational calculation.' (F. Klein, Ref. 226, p. 172) 228. F . Klein, Ref. 226, p. 185. 229. In conclusion he noted: 'I must not forget to thank Miss Noether for her active assistance in my recent work. She has worked out in full generality the mathematical ideas which I have used in an attempt to relate the integral Ii [= j Kdx\ • • • dx^] with the physical problems, and she will publish them soon in these Proceedings.' [F. Klein, Ref. 226, p. 189.] 230. F. Klein, Nachr. Ges. Wiss. Gott. (1918), pp. 394-423 (submitted and read on 6 December 1918). 231. W. de Sitter, Proc. Acad. Sci. Amsterdam 19, 1217; 20, 229 (1917). 232. F. Klein, Nacht. Gott. (1918), Ref. 226, pp. 394-395. 233. F. Klein, Ref. 232, p. 399. 234. F . Klein, 'Zu Hilberts erster Note iiber die Grundlagen der Physik,' Nachr. Ges. Wiss. Gott (1917), pp. 469-489 (read on 25 January 1917). Klein's aim was to simplify Hilbert's calculations of the variational problem and obtain 'a clearer insight into the significance of the conservation laws.' [F. Klein, I.e., p. 469]. 235. Klein also favoured Hilbert's generalization of Maxwell's equations. 'It is also clear now how the old theory of Maxwell's is related to the new theory as a limiting case. If Maxwell's theory is written in terms of the general curvilinear coordinates Xi • • • X4, we always encounter ds , whose Riemannian curvature vanishes identically. On the other hand a is taken to be 0 [where a is proportional to the gravitational constant]. With this, the ten equations [Equations (37)] are satisfied automatically; the Q^ of the electromagnetic field are no longer subject to any restrictions, and just Maxwell's equations remain.' (Ref. 234, pp. 473-474). 236. F. Klein, Ref. 234, p. 475. 237. 'Because of all this I can hardly believe that it is appropriate to regard the arbitrarily chosen quantities tva as the energy-components of the gravitational field.' (Ref. 234, p. 477.) 238. D. Hilbert, Nachr. Gott. (1917), p. 477-480. 239. D. Hilbert, Ref. 238, p. 477. Hilbert also mentioned that Emmy Noether, whose help he had requested in his study of the question of energy conservation over a year

316

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243.

244.

245. 246. 247. 248.

The Golden Age of Theoretical Physics previously, had, in fact, studied the question thoroughly and arrived at Klein's conclusions. Klein had also seen Noether's proof after writing down his own derivation. (Ref. 234, p. 476.) D. Hilbert, Ref. 238, p. 480. F. Klein, Ref. 234, pp. 481-482. L. Bianchi, Mem. della Societa Italiana delle Scienze (3a) 11 (1897). Felix Klein quoted Bianchi in his Development of Mathematics in the 19th Century, Vol. II, on p. 149 [see Ref. 103], but he never referred to the Bianchi identities in the relevant papers. The English translation of W. Pauli's Encyclopaedia article on relativity was published early in 1958. In the English edition, Pauli appended a large number of supplementary notes. I had opportunity of talking to Pauli in Spring 1958, among other things, about this book on relativity. I told him that the original article was already so perfect that it was surprising that he should have thought of adding supplementary notes. He replied with a wide grin, 'You know, it was not all that perfect. I had not even mentioned the Bianchi identities.' This he had done now in Note 7, and discussed the identities in detail pp. 212-214 [see Ref. 106]. It is remarkable that when Einstein fully saw the advantages of the derivation of the field equations from a variational principle, he no longer recognized that this was precisely why Hilbert had pursued this approach. Einstein wrote: 'The derivation of the field equations from a variational principle has the advantage that the compatibility of the resulting system of equations is assured and that the identities connected with the general covariance, the 'Bianchi identities,' as well as the conservation laws result in a systematic manner.' [A. Einstein, The Meaning of Relativity, Fifth Edition, Princeton (1955), Appendix II, p. 154.] Although Einstein developed this remark to show how this could be done, he no longer remembered the fact that this is indeed what Hilbert had done in his first communication on the foundations of physics. It might not be complete due to the fact that not all of Klein's 152 conditions or identities seem to be independent. Also, he derived them for a special matter system. On the other hand, the number of 256 Bianchi identities would surely be reduced in a Riemannian geometry. Finally one must ask whether the Bianchi identities express the fact of general covariance completely, or whether there are further consequences from it. Without going into these details, let us just note the fact that there is considerable overlap between the identities derived by Bianchi and those derived by Klein on invariant-theoretic grounds. H. Weyl, Raum-Zeit-Materie, Leipzig (1918); the fourth edition (1921) was translated as Space-Time-Matter, Methuen, London (1922). Dover edition, New York (1952). The quotations are from the Dover edition, p. 102. T. Levi-Civita, Rend. del. Circ. Mat. di Palermo 42, 173 (1917). G. Hessenberg, ' Vektorielle Begriindung der Differentialgeometrie,' Math. Am. 78, 187 (1917). H. Weyl, Math. Z. 2, 384 (1918). See also Weyl, Sitz. Ber. Preuss. Akad. Wiss. (1918), p. 465. This follows by integrating the Equation (101), and noting that I is a constant throughout the space if ^^dx^ is a total differential, which means that the expressions F vanish.

^

= ^

dx»

-

^ i

dxv

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249. A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (1918), p. 478. 250. On receiving the proof sheets of Weyl's book (Raum, Zeit, Materie, Ref. 244), Einstein wrote to him on 8 March 1918: 'It is like a classical symphony. Each word is related to the whole and the design of the work is grandiose. The magnificent use of the infinitesimal parallel displacement from vectors to the deduction of the Riemannian tensors! How naturally it all falls into place! And now you have given birth to something which I could never have expected: the construction of Maxwell's equations. I am naturally very delighted with you and with your handling of the subject...' (C. Seelig, Ref. 35, p. 200.) But already on 31 June 1918, Einstein was remonstrating with Weyl: 'Could one really accuse the Good Lord of being inconsequential if he rejected the opportunity discovered by you for harmonizing the physical world? I do not think so. If he had made the world to your specification, dear Weyl, I would have gone to Him and said reproachfully: 'Dear Lord, if it did not lie within Thy power to give an objective meaning to the congruence of infinitely small rigid bodies, so that when they are removed from each other one cannot say if they are congruent or not, why hast Thou, O Inconceivable, not disdained to bequeath to the angle this property or that of similarity?"... But since the Good Lord noticed long before the development of theoretical physics that he could not come to terms with the opinions of the world, He does as He likes.' (C. Seelig, Ref. 35, p. 201.) Writing to Ehrenfest, he expressed himself again about Weyl's theory: 'A paper by Pauli on Weyl's theory already shows the consequence of the initial fallacy of this theory: in general, static solutions do not exist for non-vanishing potentials. It is quite incomprehensible to me that Weyl himself and all others cannot immediately sense that the theory is contrary to experience.' (Letter to Ehrenfest, dated 4 December 1919, reprinted in C. Seelig, Ref. 35, p. 202.) And in a letter to Besso, dated 26 July 1920, he wrote: 'Of course, I have been totally convinced from the very beginning of the incorrectness of Weyl's theory. Most [experimental] facts are against it, and none in its favour.' [See Correspondance, Ref. 73] 251. A.S. Eddington, Proc. Roy. Soc. (London) A99, 104 (1921); A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. A 2 3 , 32, 76, 137. 252. T. Kaluza, Sitz. Ber. Preuss. Akad. Wiss. (1921), p. 27. This theory was extended by O. Klein in several papers: Z. Phys. 37, 895 (1926); Z. Phys. 36,188 (1927), J. Phys. Rad. 8, 242 (1927); Ark. Mat. Ast. Fys. 34, No. 1 (1946). 253. O. Klein, Z. Phys. 37, 903 (1926). 254. To conclude, he said: 'As in de Broglie's work, our approach is also based upon the motivation that the analogy between mechanics and optics (which is evident in the Hamiltonian method) leads to a deeper understanding of quantum phenomena. That this analogy has real physical significance is shown by the similarity of the conditions for the stationary states of atomic systems and the interference phenomena in optics. Concepts such as point charge and mass point now appear to be strange even in classical field physics. The hypothesis has often been used that material particles are special solutions of the field equations which govern the gravitational and electromagnetic fields. It is only a short step to relate the analogy mentioned above to this idea. For, according to this hypothesis, it does not seem at all strange that the motion of material particles shows similarities with the propagation of waves. The analogy in question is, however, incomplete so long as one considers wave propagation in four dimensions only. This becomes already evident in the varying velocities of material particles. If one imagines the observed motion as a projection on the space-time of a

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256. 257. 258. 259. 260. 261.

262. 263. 264. 265.

The Golden Age of Theoretical Physics propagating wave (taking place in a five-dimensional space), then a complete analogy can be made. Mathematically, it means that the Hamilton-Jacobi equation cannot be understood as a set of characteristics of a four-dimensional wave equation, but rather of a five-dimensional one. We are led to Kaluza's theory in this way. 'Although the introduction of a fifth dimension into our physical picture might initially appear rather strange, such a radical modification of the geometrical basis of the field equations is suggested by quantum theory. For, as we know, it appears less and less likely, that quantum phenomena do not lend themselves to a unified space-time description. In contrast, the possibility of expressing these phenomena by a system of five-dimensional field equations should not be ruled out right away. (Footnote: Professor Bohr has made such remarks on numerous occasions, and they have had a decisive influence on our work.)' (Ref. 253, p. 906.) O. Klein. Z. Phys. 46, 207 (1927). Oscar Klein remarked immediately that his efforts were not successful. In a footnote in the same paper (pp. 190-191), he said: 'In the following I have given a much belated but complete exposition of my efforts on the five-dimensional form of the theory of relativity which (since, in the meantime, various researchers have been occupied with this subject) hardly offers any new mathematical results. Still I hope that it would arouse some interest because it does differ from the other endeavours in this field, including the author's, both in form as well as the physical point of view. In particular, I consider it no longer possible, by introducing the fifth dimension, to be able to bring into harmony the deviations introduced by the quantum theory with the space-time description of the classical theory.' O. Veblen and D. Hoffman, 'Projective Relativity,' Phys. Rev. 36, 810 (1930). Letter to H. Weyl, on 23 November 1916 (quoted from C. Seelig, Ref. 35, p. 200). D. Hilbert, 'Die Grundlagen der Physik,' Math. Ann. 92, 1-32 (1924), reprinted in D. Hilbert, Gesammelte Abhandlungen, Berlin (1935), pp. 258-289. D. Hilbert, pp. 1-2 (Ges. Abh. 3, pp. 258-259). A. Einstein, Sitz. Ber. Preuss. Akad. Wiss (1919), pp. 348-356. 'The theoreticians have tried very hard to devise a theory which would account for the electricity in equilibrium which constitutes an electron. In particular, G. Mie has investigated this question in great d e p t h . . . . However beautifuUy wrought its formal construction by Mie, Hilbert, and Weyl may have been, this theory has not yielded satisfactory physical results. On the one hand, the abundance of possibilities is discouraging; on the other, the additional terms [in the theory] could not be put simply enough to lead to a satisfactory solution.' (Ref. 260, p. 349.) A. Einstein, Ref. 260, p. 350. A. Einstein, Ref. 260, p. 351. A. Einstein, Ref. 260, pp. 355-356. 'The idea which I am grappling with concerns the understanding of quantum phenomena and it is this: over-determination of the laws by having more differential equations than field variables. In this way the inarbitrariness of the initial conditions could be handled, without renouncing field theory. This approach could well be quite wrong, but it must be attempted and, anyhow, it is logically possible. The equation of motion of material points (electrons) would be given up completely, and the kinetic content of the latter would be determined along with the field laws. I shall send you the preliminary work on this problem as soon as it is printed. The mathematics [in it] is enormously difficult, and the connection with what can be experienced becomes ever more indirect. But it is still a logical possibility, to do justice to reality without sacrificium intellectus.' (Letter to M. Besso, dated 5 January 1924.) [See Correspondance, in Ref. 73.]

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266. A. Einstein to M. Besso, dated Berlin, 5 January 1929 [see Ref. 73]. The paper in question was 'Einheitliche Feldtheorie,' Sitz. Ber. Preuss. Akad. Wiss (1929), pp. 2-7. 267. P.A.M. Dirac, Lecture on 'Methods in Theoretical Physics,' in the series From A Life of Physics, Trieste, Italy (June, 1968). 268. S. Chandrasekhar, 'On the Derivation of Einstein's Field Equations,' Am. J. Phys. 40, 224 (1972). Chandrasekhar has argued that no rigorous derivation of Einstein's equations exists. The basic assumption of Einstein's 1915 theory was that the spacetime is a quasi-Riemannian manifold endowed with a symmetric metric. Assuming that the Riemann-Christoffel tensor vanishes identically in the absence of a gravitational field, the simplest generalization in the presence of matter are the EinsteinHilbert equations (37). The constant 7 is derived by taking the limit to Newton's theory, and the matter tensor is generalized from special relativity. This was Einstein's 'derivation.' The other derivation, followed more closely by Hilbert, started from a variational principle. The question is how to 'derive' the Lagrangian or the Hamiltonian. Assuming it to be essentially proportional to the scalar R, one finds the equations for the vacuum, which had been 'guessed' or 'assumed' by Einstein. If matter is present one has to add a term 7 M , where M denotes the 'scalar' of matter, in order to obtain the Einstein-Hilbert equations by a variational procedure. It must still be argued whether one should replace R by R + A in the variation function, thus introducing the cosmological constant. 269. Professor Markus Fierz has pointed out to me that this remark originated with Hermann Weyl, who, more than Pauli, was used to quoting statements from the Bible or his own variations thereof. 'Pauli did enjoy repeating this remark without claiming authorship of it.' [Private communication] 270. [Note added in proof. To be read in conjunction with Ref. 14S\ Ewald was Hilbert's assistant during 1912-13, not in 1915 as mentioned in the quotation in Ref. 145. In a letter, dated 9 September 1973, Professor P.P. Ewald has informed me that, 'At the time the Hilbert and Einstein papers of 1915 appeared, I was with the German armies in Russia as a 'Field-X-ray Mechanic' and I did not learn of the General Relativity developments before my return to civilian life in 1918.1 then had so many problems before me in the rapidly developing field of X-ray crystallography that I took only a slight interest in the more general problems of physics. I cannot remember ever having had a conversation of the type ascribed to me [in Ref. 145]' I am very grateful to Professor Ewald for this clarification.

Acknowledgements Not long ago I published a small piece* dealing with a charming little essay on ' t h e s t a t e of ether in magnetic fields,' which t h e sixteen-year-old Einstein h a d written while he was awaiting admission t o the E.T.H. in Zurich. This p a p e r sought to trace t h e continuity between Einstein's early interest in electrodynamics a n d his later work o n t h e special a n d general relativity theories. O n reading this p a p e r , Professor Eugene Wigner asked me whether David Hilbert h a d not independently discovered the field equations of gravitation. !* His impression from his stay in G o t t i n g e n (where Wigner h a d been Hilbert's assistant for one year in t h e late nineteen-twenties) "Albert Einstein's "First" Paper,' Science Today (April 1971); 'Albert Einsteins erste wissenschaftliche Arbeit,' Physikalische Blatter 9, 385-391 (1971). tSee Appendix.

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was that Hilbert had indeed done so, and he asked me if it was true. I replied to Professor Wigner about Hilbert's contribution to the theory of gravitation.* He kindly encouraged me to expand my account to deal with the intricate and exciting details of the early years in the formulation of the general relativity theory of gravitation. This is what I have sought to do in this study. As work on this project developed, my friend Dr Helmut Rechenberg became interested in the arguments that were central to the investigation, and we discussed this subject together in great detail. He was collaborating with me in my work on the history of quantum theory, and we were both very interested in Hilbert's contributions to physics, especially his interest in atomic mechanics. Dr Rechenberg's assistance in an examination of primary sources, and in independently verifying my conclusions (such as Hilbert's rediscovery of the Bianchi identities), has been invaluable. I am grateful to him for his interest, support, and help in all aspects of the preliminary stages of this investigation. Professor Lothar Frommhold went over the translations of numerous original German quotations with me, and I am indebted to him for this assistance. Several persons read a draft manuscript of this paper. I am grateful for the invaluable comments of Professors Markus Fierz, Peter Havas, Pascual Jordan and Andrzej Trautman. I am especially indebted to Professors Res Jost, B.L. van der Waerden, and Arthur Wightman whose painstaking reading and detailed comments have helped to improve this study at many places. Professors S. Chandrasekhar, Robert Dicke, P.A.M. Dirac, Ivar Waller, John Wheeler, and Eugene Wigner also read the manuscript and encouraged me throughout, and I am grateful for their support. For any errors of fact or interpretation that remain, I alone am responsible. Appendix I. Letter from Eugene Wigner to Jagdish Mehra: 8 Ober Road Princeton, New Jersey 08540 November 29, 1971 Dear Jagdish: Just a line to thank you for your article Albert Einsteins erste wissenschaftliche Arbeit. I enjoyed it very much indeed. There is though one question in my mind which I would like to ask you. I was under the impression that, simultaneously with Einstein, Hilbert also found the now accepted equations of general relativity. Is this correct? If so, is there a reason no one seems to mention this now? I realize that the basic idea was due to Einstein but it is interesting that, even after the promulgation of the basic idea, it took a rather *See Appendix.

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long time to find the correct equations incorporating that idea — even though both Einstein and Hilbert seem to have worked on it. It was good to see you in Amherst and I wish to thank you again for your visit. Sincerely, Eugene (Wigner)

II. Reply to Eugene Wigner from Jagdish Mehra: 7606 Rustling Road Austin, Texas 78712 12 December 1971 Dear Eugene: Thank you very much for your kind note. I am happy to learn that you liked the article on Einstein's 'first' paper. Einstein had embarked quite seriously, already in 1907, on the considerations that led to his general theory and the theory of gravitation, beginning with the equivalence of inertial and gravitational mass. After four years of serious meditation, he wrote (in 1911) on the influence of gravity on light. In 1912, he discussed the velocity of light in a gravitational field (Ann. d. Phys. 38,443(1912)). With Marcel Grossmann (his fellow student from the early days at the E.T.H. in Zurich, who was now his colleague at the E.T.H. during the very brief period Einstein spent there after coming from Prague and before going to Berlin), Einstein summed up his ideas on the general theory of relativity and the theory of gravitation in 'Entwurf einer Verallgemeinerten Relativitatstheorie und eine Theorie der Gravitation' (Z. Math. u. Phys. 62, 225 (1913)). (By the way, it was Grossmann's father who had assisted Einstein in obtaining the job at the Patentamt in Bern. Later on, in discussing the origin of his 1913 paper with Marcel Grossmann, Einstein insisted that the entire inspiration and scheme of that paper were his own, and that Grossmann had merely helped him in carrying out the mathematical programme envisaged by him.) In 1914, Einstein wrote on the formal foundations of general relativity theory (Sitz. Ber. Preuss. Akad. Wiss. 1030, (1914)), gave several lectures on the problem of gravitation and relativity, and published a paper with M. Grossmann on the general covariance properties of the field equations of the theory of gravitation (Z. Math. u. Phys. 63, 215 (1914)). During 1915, he published new ideas on general relativity, especially the application to astronomy with the explanation of the perihelion of Mercury. It was during 1915 that David Hilbert became actively interested in the general theory of relativity, aroused essentially by Einstein's statements of 1914 on its formal foundations. At the end of 1915, Hilbert presented to the Gottinger Gesellschaft der

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Wissenschaften his first Mitteilung iiber die Grundlagen der Physik, and followed it with his Zweite Mitteilung at the close of 1916. Both of these papers held great excitement for Felix Klein, who was led to a more compact formulation of Hilbert's calculations with the help of the old and trusted method of Lie's infinitesimal transformations. In 1924, Hilbert himself gave the final, simplified formulation of his two earlier Mitteilungen on the foundations of general relativity. (Math. Ann. 92, 1-32 (1924).) Einstein's theory of gravitation is founded upon the tensor equation R«p ~ ga&R = 8n(G/c4)Ta/3 . These ten equations have between them four identities, four equations that really put no condition at all upon the geometry because they are automatically fulfilled whether these (above) equations are satisfied: Ra/3

_

ga/3Rj}

=

8

(G/c4)T^_

This was the feature of the Einstein equations that was discussed by Hilbert (20 November 1915): Since the ten differential equations leave a freedom of four arbitrary functions in the solution, they cannot be entirely independent, but must have four inner relations. These relations are a consequence of the Bianchi identities for the full Riemannian tensor, and are generally (?) known in the form of the condition that the Ricci tensor be divergenceless. Therefore, there are only six net equations for the ten quantities gap. This circumstance at first drove Einstein away from the final form (25 November 1915) of the field equations (Preuss. Akad. Wiss. Berlin; Sitzb. 844 (1915)), so that his preliminary publication (Preuss. Akad. Wiss. Berlin, Sitzb. 778 (1915)) put forward equations of the form (11 November 1915), R^

=

4n(G/c4)T^.

Hilbert (Gesell. Wiss. Gottingen, Nachrichten, 395 (1915)) pointed out that the equations set up to determine the gap would be quite unacceptable if they were really expected to determine them. What is really relevant, and what should be determined is the geometry and curvature of space-time. What coordinates we use to describe the geometry should be immaterial. If the equations were completely to determine the ten gap, they would tell us not only the geometry but also, unhappily, the coordinates in terms of which that geometry is expressed. However, we know that only the interval between one event and another (between one crossing of world lines and another, for example) has any real significance: ds2 =

gapdxadxP.

How one draws coordinate surfaces through space-time is a matter of paper work and bookkeeping, and has nothing to do with real physics.

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In his celebrated paper on 'Die Grundlage der allgemeinen Relativitatstheorie' (Ann. d. Phys. 49, 769 (1916)), Einstein mentioned (p. 810): 'Die Feldgleichungen der Gravitation enthalten also gleichzeitig vier Bedingungen, welchen der materielle Vorgang zu geniigen hat. Sie liefern die Gleichungen des materiellen Vorganges vollstandig, wenn letzterer durch vier voneinander unabhangige Differentialgleichungen charakterisierbar ist' At the end of this sentence, without pursuing this point further in detail, Einstein gave reference to Hilbert's work in a footnote, referring especially (and only) to p. 3 of Hilbert's 1915 paper in the Gottinger Nachrichten. Hilbert, in retrospect, could not have been satisfied by this weak reference to his work. In a sense, Einstein had 'appropriated' Hilbert's contribution to the gravitational field equations as a march of his own ideas — or so it would seem from the reading of his 1916 Ann. d. Phys. paper on the foundations of general relativity. In his article on 'Die Grundlagen der Physik' (Math. Ann. 92, 1-32 (1924)), in which Hilbert brought together his two earlier communications on general relativity, he wrote: 'Die gewaltigen Problemstellungen und Gedankenbildungen der allgemeinen Relativitatstheorie von EINSTEIN finden nun, wie ich in meiner ersten Mitteilung ausgefiihrt habe, auf dem von MIE betretenen Wege ihren einfachsten und natiirlichsten Ausdruck und zugleich in formaler Hinsicht eine systematische Erganzung und Abrundung. 'Seit der Veroffentlichung meiner ersten Mitteilung sind bedeutsame Abhandlungen iiber diesen Gegenstand erschienen: ich erwahne nur die gldnzenden und tiefsinnigen Untersuchungen von WEYL und die an immer neuen Ansatzen und Gedanken reichen Mitteilungen von EINSTEIN. Indes sowohl WEYL gibt spaterhin seinem Entwicklungsgange eine solche Wendung, dass er auf die von mir aufgestellten Gleichungen ebenfalls gelangt, und anderseits auch EINSTEIN, obwohl wiederholt von abweichenden und unter sich verschiedenen Ansatzen ausgehend, kehrt schliesslich in seinen letzten Publikationen geradenwegs zu den Gleichungen meiner Theorie zuriick.' [My emphasis.] 'Ich glaube sicher, dass die hier von mir entwickelte Theorie einen bleibenden Kern [my emphasis] enthalt und einen Rahmen schafft, innerhalb dessen fur den kiinftigen Aufbau der Physik im Sinne eines feldtheoretischen Einheitsideals geniigender Spielraum da ist. Auch ist es auf jeden Fall von erkenntnis — theoretischem Interesse, zu sehen, wie die wenigen einfachen in den Axiomen (I-IV) von mir ausgesprochenen Annahmen der ganzen Theorie geniigend sind.'1* Einstein did not entirely (or at all!) favour the developments of general relativity pursued by Weyl. By 1924 Hilbert had decided that, in spite of what Einstein and Weyl had wrought, his own contributions 'einen bleibenden Kern enthalten.' Hilbert's own goal, even in his earlier researches, had been to build a unified theory of electromagnetism and gravitation based upon the work of Mie. To this end, Hilbert made use of Hamilton's principle, applying it to an arbitrary function of *Hilbert was very pleased with the geometrical abstraction and its beauty in Einstein's theory. He recommended Einstein in 1915 for the third Bolyai Prize 'for the high mathematical spirit behind all his achievements.' This prize had been awarded first to Poincare and then to Hilbert.

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the gravitational potential and an electromagnetic four-vector, invariant under an arbitrary transformation of the coordinates. On the basis of a remarkable property of the variational equations, four equations are obtained, connecting the electromagnetic and gravitational quantities. The undeclared goal of Hilbert's two communications of 1915 and 1916 was to build a field theory of atomic nuclei and electrons, something in which he also did not succeed. What did still remain, of course, even as late as 1924 (or, for that matter, today) was lein bleibender Kern1 — the fact that Hilbert had provided the form of the Einstein field equations for gravitation, as well as the inner conditions by which they were determined. Your question is therefore quite to the point, and I hope I have provided the historical background. Hilbert had indeed provided the rigorous mathematical arguments for the form which the gravitational field equations were to obtain. Einstein absorbed these arguments in his great paper of 1916, just as he absorbed other things on several occasions. Hilbert had the vision of a unified field theory before Weyl and Einstein, and in this respect he was also the earlier failure. He had the dream of a universal law accounting both for the structure of the cosmos as a whole and of all atomic nuclei. As for the form of the gravitational equations in which Einstein finally presented them in his 1916 Ann. d. Phys. paper on the foundations of general relativity, and which Hilbert had derived in his Gottingen Nachrichten Mitteilung, the dates for the priority are instructive. Einstein presented his communications of 1915 to the Berlin Academy on November 4, 11, 18, and 25 respectively, while Hilbert's first Mitteilung to the Gottingen Gesellschaft was on November 20, 1915. Hilbert's communication contained the result which Einstein used later.* You ask: 'Is there a reason no one seems to mention this now?' — that is, Hilbert's work on the foundations of general relativity and the theory of gravitation. The only reason, probably, is that people do not read the literature. Another reason is the sociology of science, the question of the cat and the cream. Einstein was the big cat of relativity, and the whole saucer of its cream belonged to him by right and legend, or so most people assume! Still, I am very glad that you have raised the question. By the way, the mathematical tradition of Gottingen from Gauss to Hilbert formed an important background of the work on quantum mechanics done there. It might perhaps be appropriate to discuss Hilbert's contributions to physics. I have got a copy of the notes of Hilbert's lectures on the mathematical foundations of quantum mechanics, and they make fascinating reading. I expect to use these to discuss Hilbert's interest in the structure of matter. It might amuse you to know that the first problem of physics which Hilbert decided to work on seriously was the problem of the kinetic theory. He dealt with the problem of collisions in the Maxwell-Boltzmann distribution function, and this 'Einstein used Hilbert's result in his 1916 paper (Received 29 March 1916).

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was the occasion to apply the theory of integral equations that he had developed to the problems of physics. A fact that is perhaps not known is that David Enskog attended Hilbert's lectures in Gottingen in 1912/13 and wrote his thesis in Uppsala in 1917. Chapman became familiar with Enskog's work only several years after he had done his own work independently. Enskog did not mention in his thesis that he had benefited from Hilbert's lectures, although the latter served as a point of departure for his work on the kinetic theory of non-uniform gases. Once it occurred to me, on the basis of the formal similarity between Hilbert's and Enskog's work, to explore whether Enskog had come in contact with Hilbert. I found Enskog's name, after some search, among those who had taken Hilbert's course on the kinetic theory. With my best regards, Sincerely yours, Jagdish (Mehra) Professor Eugene Wigner 8 Ober Road Princeton, New Jersey 08540

9 Albert Einstein and the Origin of Light-Quantum Theory* On 13 December 1900, one day before Max Planck presented the derivation of his radiation law in Berlin, Albert Einstein in Zurich signed a paper on 'Folgerungen aus den Capillaritdtsercheinungen' ('Inferences Drawn from Capillary Phenomena') and submitted it to Annalen der Physik (Einstein, 1901). In the following years Einstein regularly submitted further contributions to the Annalen, dealing with problems of the foundations of thermodynamics and kinetic theory. Finally, j n _ 1905 he derived two important consequences from the latter. One, he claimed that particles suspended in a fluid must execute an irregular motion even if the fluid is at rest (Einstein, 1905c). The other consequence concerned the nature of radiation; it led him to expound a new, as he said, 'heuristic point of view' (Einstein, 1905b). Einstein described it in the introduction of his paper with the following words: According to the assumption considered here, when a light ray starting from a point is propagated, the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta, localized in space, which move without being divided and which can be absorbed or emitted only as a whole. (Einstein, 1905b, p. 133) With this hypothesis of the existence of localized energy quanta in radiation Einstein wanted to explain certain phenomena connected with the creation and conversion of light, especially photoluminescence, the production of cathode rays by ultraviolet light and the observed blackbody radiation. He thus placed himself in contrast to Max Planck, who had based his derivation of the energy distribution in blackbody radiation entirely on Maxwell's electromagnetic theory of light. How did he come to such a conclusion? In order to answer this question we shall briefly outline the background of the young Einstein. Albert Einstein was born on 14 March 1879 in Ulm, Wiirttemberg, Southern Germany, the son of the Jewish merchant Hermann Einstein and his wife Pauline, Lecture delivered on the occasion of the Einstein Centennial Celebration, 1979, at SUNY, Stony Brook, and Cornell University, Ithaca, New York, University of Houston, Houston, Texas, and at Tel Aviv Unversity, Israel (November 1981). Enlarged version published in The Historical Development of Quantum Theory (with Helmut Rechenberg, Springer-Verlag New York, 1982, pp. 59-83). 326

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nee Koch. 1 The family moved to Munich the following year, where Hermann Einstein opened a small electrotechnical workshop together with his brother Jakob. In Munich Einstein received his early education; in 1885 he became a pupil in a Catholic elementary school and, from 1889, he attended the Luitpold-Gymnasium.2 Several months after the family had moved to Milan in 1984, Einstein left the Gymnasium; he first joined the family in Italy and was then sent to Switzerland to complete his education. He finished high school in Aarau and began to study mathematics and physics at the Eidgenossische Technische Hochschule (E.T.H. or the Swiss Federal Institute of Technology) in Zurich. Following his examination for the Diplom (actually the physics teacher's certificate) in summer 1900, Einstein did not succeed in obtaining the position of as assistant in a university in order to be able to pursue scientific research. Instead of embarking upon a scientific career he had to accept various short-term teaching jobs until he joined the Eidgenossisches Amt fur geistiges Eigentum (Swiss Patent Office) in Berne in June 1902.3 This secure position allowed him to marry Mileva Marie, a former fellow student, and to found a family.4 It also offered him enough spare time to work on problems of theoretical physics in which he had become interested. Already at the Gymnasium, Einstein had been attracted by certain questions of mathematics and physics. Thus, for example, he had learned Euclidean geometry from a book and the general methods of science from a series of popular books; he had also become acquainted with the principles of differential and integral calculus. 5 'I had also already studied some theoretical physics when, at the age of 17,1 entered the Polytechnic Institute of Zurich as a student of mathematics and physics,' he recalled later (Einstein, 1949, p. 15). At the Zurich Polytechnic, Einstein had not been a particularly good student. As he confessed in his Autobiographical Notes: J

T h e Einsteins had been settled for centuries in the Swabian region between t h e Danube and t h e Lake of Constance, especially in the town of Buchau where Hermann Einstein was born in 1847. Hermann married Pauline Koch of Cannstadt (close to Stuttgart) in 1876; she was the daughter of a grain merchant and court purveyor of Stuttgart. In 1877 the Einstein couple moved to Ulm. 2 Albert Einstein's sister, Maria (Maja), was born in Munich on 18 November 1881. She would later on join her brother in Switzerland and marry one of his friends, Paul Winteler. She also joined Einstein in the United States and died there on 25 June 1951. 3 Einstein obtained the position at the Patent Office on the recommendation of the father of Marcel Grossmann, his friend and former fellow student. His provisory initial appointment as 'Expert III. Class,' made on 23 June 1902, was confirmed in 1904; in 1906 he was promoted to 'Expert II. Class.' Einstein became a Swiss citizen (citizen of Zurich) on 21 February 1901; he had already renounced his Wiirttemberg citizenship in early 1896. 4 Mileva Marie, born in 1875 near Novy Sad, Serbia, had also studied mathematics and physics at the Zurich Polytechnic (E.T.H.) and had graduated in 1901. T h e Einsteins had two sons, Hans Albert (1904-1973) and Eduard (1910-1965). Later, when Albert Einstein moved to Berlin in spring 1914, Mileva remained in Zurich with the children. They were divorced in 1919; Einstein then married Elsa Lowenthal, the niece of his mother. For further details of Einstein's life see one of his biographies, such as Carl Seelig (1954), Ronald W. Clark (1971), or Banesh Hoffmann (1972). 5 I n his old age Einstein still remembered the details of his early reading (see Einstein, 1949, p. 14). In some of his reading — e.g., to study Aaron Bernstein's Naturwissenschaftliche Volksbiichei Einstein was stimulated by Max Talmey, a young medical student at the University of Munich and a friend of the Einstein family (see Clark, 1972, pp. 33-34).

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There I had excellent teachers (for example, [Adolf] Hurwitz, [Hermann] Minkowski), so that I really could have gotten a sound mathematical education. However, I worked most of the time in the physical laboratory, fascinated by the direct contact with experiment. The balance of the time I used in the main in order to study at home the works of Kirchhoff, Helmholtz, Hertz, etc. (Einstein, 1949, p. 15)6 As a result, he obtained most of his education in theoretical physics from books written by the foremost scientists; these included, besides the ones already mentioned, Ludwig Boltzmarm, Hendrik Lorentz, James Clerk Maxwell and Henri Poincare. 7 However, there was one exception. In his last semester Einstein attended Hermann Minkowski's lectures on capillarity. 'That,' he is reported to have remarked, 'is the first lecture [course] on mathematical physics which we have heard at the Polytechnic]' (Kollross, 1956, p. 21). 8 The lectures stimulated Einstein to work on capillarity himself and he made it the subject of his first scientific paper (Einstein, 1901). By capillarity one denotes a special form of energy connected with the shape and position of the surface of fluids; thus, for instance, it determines the increase in the level of a fluid in a thin capillary tube. During the nineteenth century many scientists had concerned themselves with the problem of capillarity, among them Thomas Young, Pierre-Simon de Laplace, Carl Friedrich Gauss, James Clerk Maxwell, Josiah Willard Gibbs, Johannes Diderik van der Waals and Henri Poincare. 9 The existence of capillarity had been derived from the hypothesis of cohesive forces exerted by the molecules of a fluid on each other (Laplace, 1806). As a consequence one can obtain, from an experimental determination of the capillarity of a fluid, information about the intermolecular forces. Exactly this possibility interested Einstein in his first paper, which he entitled 'Inferences Drawn from Capillary Phenomena' (Einstein, 1901, p. 513). He assumed, 'guided by the analogy to the gravitational force' (Einstein, 1901, p. 515), that a universal function of the intermolecular distance determined the attraction between any two molecules, and he suggested the In his later scientific career Einstein continued to take an interest in direct contact with experimental observations. Among his published papers quite a few dealt with experimental questions, notably the joint papers with Wander Johannes de Haas on the experimental proof of Ampere's molecular currents (Einstein and de Haas, 1915; de Haas, 1916; Einstein, 1916b). Moreover, experimental verification always played an important role in his theoretical work. ^The professor of physics at E.T.H., Heinrich Friedrich Weber, did not lecture, for instance, on Maxwell's electrodynamics. Since Weber did not have a particularly high esteem for Einstein, he did not offer him, after the examination, a position as an assistant. (See Kollros, 1956, pp. 21-22.) 8 It is of interest t o note that Einstein did not learn much mathematics from Minkowski. Instead, he enjoyed Karl Friedrich Geiser's lectures on infinitesimal geometry (see Einstein, 1956, p. 11). Otherwise, during my student years I was not much interested in higher mathematics [he recalled in his Autobiographische Skizze, and added:] Mistakenly it seemed to me that it was a field with so many branches that one could easily squander one's entire energy in one of its remote parts. Furthermore, I thought in my innocence that it sufficed for a physicist to understand elementary mathematical concepts clearly in order to be able to apply them, and that the remainder consisted of subtleties that were fruitless for a physicist. (Einstein, 1956, p. 11) 9 See Hermann Minkowski's review article on capillarity in the Encyklopddie der mathematischen Wissenschaften (Minkowski, 1907).

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possibilities of deriving the parameters entering into this universal function in the case of given molecules from the data on fluids. He established, in particular, a relation between the energy necessary to create a unit area of surface oi" a fluid and the parameters characterizing the cohesive forces. By analyzing the data from various fluids, Einstein found fair agreement between his theory and experiment. He concluded: 'In summing up we can say that our fundamental assumption has stood the test: To each atom there corresponds a molecular field of attraction, which is independent of the temperature and independent of the type of chemical binding between atoms.' (Einstein, 1901, p. 523) 10 The study of intermolecular forces continued to occupy Einstein. 'I am now almost sure,' he wrote in April 1901 to Marcel Grossman, his friend from E.T.H., 'that my theory of the power of attraction can be extended to gases and that the characteristic constants for nearly all elements could be specified without undue difficulty.' And he added: It is a magnificent feeling to recognize the unity of a complex of phenomena which appear to be things quite apart from the direct visible truth. (Einstein to Grossmann, April 1901; quoted in Seelig, 1956, p. 53) His next paper, however, did not deal with the intermolecular forces in gases, but with forces existing at the boundary between solid and fluid matter. Einstein submitted it in April 1902, again to Annalen der Physih, he sought to explain in it the difference of the electric potential between metals and dissociated solutions of their salts (Einstein, 1902a). He calculated the potential difference by using the same assumption concerning the nature of the intermolecular forces as in his previous paper. Then he suggested a procedure for determining the parameters of the cohesive forces from observations of the potential difference. But the procedure seemed to offer little promise for a practical application, and Einstein concluded: Finally I wish to apologize for the fact that I have sketched here only a poor plan for a laborious investigation, without contributing personally to the experimental solution; however, I am not in a position to do so. Still this paper will achieve its goal, if it stimulates a scientist to attack the problem of molecular forces from the aspect indicated above. (Einstein, 1902a, p. 814) Einstein recognized that the theories he had attempted to construct so far did not bring him closer to his 'major aim to find facts which would guarantee as much as possible the existence of atoms of finite size' (Einstein, 1949, p. 47). He, therefore, extended his theoretical tools: thus far he had based his approach to the problems of physics only on mechanics and thermodynamics, now he would also apply the methods of the kinetic theory of matter. Within less than two months after the paper on the thermodynamic theory of potential difference (Einstein, 1902a), he 10

I t should be noted that the cohesive forces were assumed to be different from gravitational forces, having identical structure for all molecules; that is, they were taken to be forces depending only on (the same) inverse power of the distance between molecules with different coefficients for different types of molecules.

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submitted a new paper to Annalen der Physik dealing with the kinetic theory of heat equilibrium and the second law of thermodynamics (Einstein, 1902b). Two further papers, in which he developed the kinetic theory of heat, followed in the next two years (Einstein, 1903, 1904). They not only closed the gaps, which according to Einstein existed in the previous theories of Maxwell and Boltzmann, but also prepared the ground for the proof of 'the existence of atoms of finite size' (see Mehra, 1975a). 11 Statistical considerations had been introduced into the kinetic theory of gases by James Clerk Maxwell (1860) and Ludwig Boltzmann (1868, 1872, 1877). The main reason for their introduction was the recognition of the fact that purely dynamical arguments did not suffice for establishing the thermodynamic behaviour of gases. Boltzmann and Gibbs then wrote the first monographs on the new kinetic theory of matter (Boltzmann, 1895c, 1898; Gibbs, 1902), which Gibbs called 'statistical mechanics.' Thus the theory was incorporated into classical theoretical physics as a standard subject at the beginning of the twentieth century. Still it caused many controversial discussions and its value was doubted by many scientists, for it lacked a physical basis; in particular, there existed no definite proof of the molecular structure of matter, although a large variety of phenomena could be explained consistently on the basis of kinetic theory. The fundamental difficulty lay in the fact that the number of molecules in any given volume of matter appeared to be so inconceivably large that it appeared hopeless to find any direct evidence for the existence of molecules. Gradually the methods for determining that number began to be clarified; thus, for example, the number of molecules per gram-atom derived from the cross sections of gas molecules, on the one hand, and from Planck's radiation theory, on the other, agreed well, as both methods yielded the value of about 6 x 10 23 . However, all methods so far involved intermediate steps or additional assumptions, hence they could not provide immediate proof of the existence of molecules. It was in the midst of this situation that Einstein started his work on kinetic theory. A peculiar point concerning Einstein's endeavours was the fact that, as he confessed later, he was 'not acquainted with the investigations of Boltzmann and Gibbs, which had appeared earlier and actually exhausted the subject' (Einstein, 1949, p. 47). 12 11 Einstein mentioned the goal of his investigation in the introduction of his first paper on statistical thermodynamics in the following words: However great may have been the successes of the kinetic theory of heat in the field of gas theory, until now mechanics has not been able to provide an adequate foundation for the general theory of heat. The reason is that it has not yet been possible to derive the theorems concerning thermal equilibrium and the second law by using the mechanical equations and the probability calculus alone, though the theories of Maxwell and Boltzmann have come quite close to this goal. The aim of the following considerations is to fill this gap. (Einstein, 1902b, p. 417) 12 Since, already in his first paper on the subject, Einstein quoted only Boltzmann's lectures on gas theory (Einstein, 1902b, p. 420, footnote 1), it may well be that he did not know the details of the foundations of kinetic theory which Boltzmann had discussed in his original papers. On the other hand, Gibbs' book on statistical mechanics, which appeared in 1902, became available

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In his first paper on statistical thermodynamics Einstein presented what he called a 'kinetic theory' of two fundamental laws of thermodynamics: (i) the law that between any systems which are brought into contact a thermal equilibrium will be established, and (ii) the second law of thermodynamics (Einstein, 1902b). Einstein considered two mechanical systems described by momentum and position variables, and he assumed that on the masses of each system two kinds of forces may act: forces which can be derived from a potential (such as gravitation) and forces which cannot (such as the ones exerted by the fixed walls enclosing the system); he assumed that the latter changed fast in time and were responsible for the transfer of heat to the system. Then he considered N identical systems having energy values between E and E + SE, and studied their distribution among possible states, that is, among the regions of the multidimensional phase space. By applying Liouville's theorem Einstein obtained the equation dN = Af

dpi-dpi---

dqn, ,

(1)

JJga

for the number dN of systems whose variables, p\,..., qn (where the p's and q's represent momentum and position variables of the system, respectively), lie in that infinitesimal region of phase space g which is consistent with the allowed energy values. (A denotes a quantity independent of the p's and q's.) By regarding each of the N systems as being in a (stationary) interaction with a large system — the so-called heat reservoir — Einstein derived from Eq. (1) an expression for the probability dW of a state (which is characterized by the fact that the 2ra dynamical variables assume values between pi a n d p i + d p i , P2 andp2+dp2, • • • ,pn and pn+dpn, qi and q\ 4- dqn, q2 and q + dq2, ...,qn and qn + dqn, respectively), i.e., dW = A exp(-2/i£) dpi • • • dqn ,

(2)

where A denotes a constant. The factor h in the exponent represented a positive quantity depending only on the state of the heat reservoir, hence Einstein called it the 'temperature function' (Einstein, 1902b, p. 425). 13 He also showed that two systems, each having the temperature function h, may be joined into one system having the same h; that is, if two isolated systems possess the same temperature, the combined system will also do so. Further, he established between h and the absolute temperature T the equation i

= «T,

(3)

where K denotes a universal constant. Having prepared this ground, Einstein approached the second task, namely, to establish a mechanical definition of the entropy to Einstein only later in the German translation of 1905. (The young Einstein read publications in German and French, but not very well in English.) 13 Evidently, the value of the constant A entering Eqs. (1) and (2) can be found by integrating dW over all states consistent with an energy between E and E + 5E. T h e integration must yield the result A • J exp(—2hE) dpi • • • dqn = 1.

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of a system at temperature T. He found, in particular, that the quotient dQ/T, where dQ denotes the heat transferred to a system at temperature T, represented a total differential whose integration yielded the expression E t S = — + 2K In / exp(-2/i£) dpi • • • dqn

(4)

for the entropy of a system with energy E. (The phase-space integration of the righthand side of Eq. (4) extends over all possible values of the dynamical variables.) Einstein concluded his paper by drawing attention to two specific points. First, in the derivation of the expression for dQ/T he did not have to require that the forces, which he had introduced in order to carry out the adiabatic changes of the systems, really exist in nature; thus, for example, one was allowed to use the socalled semipermeable membranes separating two types of gas molecules without being concerned about their experimental verification. This fact confirmed a generalization of the second law of thermodynamics, which Einstein had already noted earlier: 'One remains in agreement with experience if one applies the second law of thermodynamics to physical mixtures, on the individual components of which conservative forces are acting' (Einstein, 1902a, p. 799). 14 The second point consisted in the observation that in the definition of the entropy, Eq. (4), only the total energy E of the system entered and not its separation into potential and kinetic parts. 'This fact leads us to suspect,' argued Einstein, 'that our results are much more general than the mechanical representation employed here' (Einstein, 1902b, p. 433). He returned to this point in detail in his next paper on the subject, where he answered in the affirmative the question 'whether the kinetic theory is really necessary for the derivation of the fundamental laws of thermodynamics [i.e., the law of temperature equilibrium and the second law], or whether perhaps assumptions of a general type may be sufficient for this purpose' (Einstein, 1903, p. 170). Einstein gave the new work the title lEine Theorie der Grundlagen der Thermodynamik' ('A Theory of the Foundations of Thermodynamics') in order to emphasize the independence of his procedure from mechanical models. The systems he now considered were most general ones; the only restriction was that they could be described by n dynamical variables (including both position and momentum variables) whose equations of motion led to just one constant of motion, the energy E. He assumed further that each system should, after a sufficient period of time, be in a stationary state which is defined by the fact that the dynamical variables, say pi,...,pn, always assume the same values with the same frequency. In other words, in a stationary state the variables of the system do change; however, they stay in a given volume of the phase space for a given, constant fraction of time. This concept of a stationary state yielded a new expression for the probability of a system to assume a particular state, say the one in which the dynamical variables 14 The extended version of the second law of thermodynamics allowed Einstein to replace real processes by idealized ones involving auxiliary forces: the results thus obtained remained valid in the real cases provided they did not depend on the auxiliary forces.

Albert Einstein

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have the values between p\ and p\ + dpi,..., and pn and pn + dpn, respectively. Einstein found In W = const. — / e In e dpi • • • dpn , (5) where the quantity e (which may depend on the variables, pi,...,pn, and on time t) represented the distribution of the systems — i.e., dN = e dpi • • • dpn. By assuming that the evolution of the N systems proceeds in such a way that the probability W, Eq. (5), increases in the course of time, Einstein proved the second law of thermodynamics (Einstein, 1903, Sections 8 and 9). Einstein completed his foundation of the molecular theory of heat in a third paper, submitted to Annalen der Physik at the end of March 1904 (Einstein, 1904). It served two purposes. One was to show that the definition of entropy, Eq. (4), agreed with the ones given by previous authors, notably Boltzmann and Planck; the other to provide a new interpretation of the universal constant K entering into the relation between the temperature function h and the absolute temperature T, Eq. (3). The first goal was easily achieved by rewriting the right-hand side of Eq. (4) in the form S = 2K\D[U(E)]

.

(6)

The energy function ui(E) was defined with the help of the equation rE+SE

u(E)8E=

/

dpi ••-elfin,

(6a)

JE

whose right-hand side denotes the volume of phase space assumed by the variables of the systems under consideration, which have energies between E and E+6E. The definition of entropy, Eq. (6), was now identical with what Planck had called 'Gibbs' third entropy definition,' in his article for the Boltzmann Festschrift (Planck, 1904, p. 119). 15 One possibility of obtaining a physical interpretation of the universal constant K could be achieved very easily by introducing systems consisting of massparticles; then the average kinetic energy of each particle was computed to be 3KT, and the quantity 2/c had to be taken equal to the ratio R/No, where R is the l^Gibbs' expression, apart from a multiplicative factor (which was Boltzmann's constant), was dlnV/dE, representing the derivative of the phase-space volume with respect to the energy E of the systems. Evidently, u(E) could be identified with dlnV/dE. T h e Festschrift Ludwig Boltzmann gewidmet zum sechzigsten Geburtstag, 20. Februar 1904 was published as a supplementary volume of Annalen der Physik early in 1904. It contained contributions by many authors: Max Planck's paper on the mechanical significance of temperature and entropy (Planck, 1904), Max Abraham's paper on radiation pressure and blackbody radiation (Abraham, 1904), Marian von Smoluchowski's paper on the influence of the irregularities of the distribution of molecules on entropy (Smoluchowski, 1904), and Walther Nernst's paper on chemical equilibrium and temperature gradient (Nernst, 1904). These and other articles did interest Einstein, and it is likely that he obtained valuable information from the Festschrift. Indeed, he reviewed several articles of the Festschrift for the Beibldtter zu den Annalen der Physik; his reviews appeared in its issue No. 5 of 1 March 1905 (see, e.g., Einstein, 1905a). During 1905 Einstein wrote further reviews of papers on thermodynamics and statistical mechanics for the Beibldtter. However, he stopped writing these reviews in the following year, when he contributed only t h e review of Planck's book on heat radiation (Einstein, 1906e).

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universal gas constant and No the number of molecules per gram-atom; in other words, the relation

2«=-£=*

(7)

iV0

existed between Einstein's n and Planck's universal constant k (called the Boltzmann constant). However, Einstein did not content himself with this interpretation, but continued to search for a 'general significance' ('allgemeine Bedeutung,' Einstein, 1904, p. 359) of K, that is, for an interpretation which did not depend on the representation of the system as a mechanical system of gas molecules. He, therefore, started from the equation for dW, the probability that a stationary system of temperature T has an energy value between E and E + dE, i.e., dW = Cexp(--^yu(E)dE,

(8)

where the constant C may be determined by integrating over all energies (/0°° C exp(-E/2K,T)-w(E) dE = 1). With the help of dW he calculated the average energy E(= /0°° CEexp(-E/2nT) • u>(E) dE). Evidently, the equation

Jo /o

\E - E) exp ( ~ ^ )

• <E) dE = 0

(9)

was also satisfied. From it, by differentiating with respect to the temperature T, Einstein obtained the result 2KT2^=W-(Ef = 71.

(10)

He commented on Eq. (10) as follows: In general the actual value E of the energy deviates from [the average value] E by a certain quantity, which we call 'energy fluctuation'; we put E = E + e • • •. The quantity e2 is then a measure of the thermal stability of the system; the larger e2 is, the smaller is its stability. The absolute constant K therefore determines the thermal stability of the systems. The last relation [Eq. (10)] is interesting, because it contains none of the quantities which recall the assumptions on which the [kinetic] theory is based. (Einstein, 1904, p. 360) Einstein immediately sought to confirm his new interpretation of the constant K. 'The equation just derived [Eq. (10)] would allow an exact determination of the universal constant K,' Einstein noted, 'if it were possible to determine the average value of the square of the energy fluctuation of a given system; this determination, however, cannot be performed with the present state of our knowledge' (Einstein, 1904, p. 360). The difficulty was, of course, that in the available molecular systems, say gases, the energy fluctuations were extremely small because of the large numbers of molecules present in any volume; hence they could not be observed. 16 On the 16

Marian von Smoluchowski, in his article for the Boltzmann Festschrift, studied another kind of fluctuations in gases: the fluctuations of density. He found that in the smallest microscopically observable volumes one would obtain deviations from the average density of one per mille (Smoluchowski, 1904, p. 628).

Albert Einstein and the Origin of Light-Quantum Theory 335

other hand, Eq. (10) was applicable to any system which exhibited thermodynamic behavior, especially to heat radiation. And Einstein saw the opportunity of checking the applicability of Eq. (10) to this system. 'Prom our experience we may surmise that only in one type of physical systems — if at all — an [observable] energy fluctuation will exist,' he argued. 'This is the empty space filled with heat radiation' (Einstein, 1904, p. 361). He knew, of course, that the linear dimension of any given volume was very large in comparison with the wavelengths of the observed heat radiation; for example, the wavelength emitted by a body having a temperature equal to that of the sun's surface (i.e., 6000°K) is of the order of a fraction of onemillionth of a metre. As a consequence, the amount of the energy fluctuation would be small in comparison with the average energy of radiation. 'However, if the space volume containing the radiation has the linear dimensions of a wavelength,' Einstein pointed out, 'then the energy fluctuation will have the same order of magnitude as the radiation energy contained in the space volume of radiation' (Einstein, 1904, p. 361). He then proposed to test this conclusion by the following consideration. In any volume v containing blackbody radiation of temperature T, the average energy E is, according to Stefan-Boltzmann's law, proportional to T 4 times the volume. The particular volume, v, in which the fluctuation term e2 is equal to the square of the average energy, (E)2, may then be obtained with the help of Eq. (10). By taking the third root of this volume, Einstein arrived at a length (in millimetres) determined by the equation

r» = i<&^ =

Q

-f,

(ID

where 'const.' denoted the constant in Stefan-Boltzmann's law, which Einstein wrote as E = const. • v • T*. Now, from Wien's displacement law, Eq. (4), the wavelength of the radiation maximum at temperature T followed from the equation .

_ 0.293

-Vax — —j,— •

(12)

One recognizes [Einstein concluded] that not only the type of temperature dependence but also the magnitude of Amax can be determined correctly with the help of the general molecular [kinetic] theory of heat; and I believe that this agreement [between the theoretical results derived from radiation theory and kinetic theory, respectively] cannot — in view of the great generality of our assumptions — be attributed to mere chance. (Einstein, 1904, p. 362) Einstein was thrilled by the result contained in Eq. (12) and its interpretation on the basis of kinetic theory. 'I have discovered the simplest possible way of the relationship between the size of the elementary units of matter and the wavelength of radiation,' he wrote to his friend Conrad Habicht about two weeks after sending off the paper to Annalen der Physik (Einstein to Habicht, April 1904; quoted in Seelig, 1956, p. 62). Still he did not receive much recognition for his papers on the foundation of statistical thermodynamics, because most of what he achieved had been done already by previous authors. As Max Born remarked later:

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In fact, Einstein's paper [Einstein, 1902b] is a rediscovery of all essential features of statistical mechanics and obviously written in total ignorance of the fact that the whole matter had been thoroughly treated by Gibbs a year before (1901) . . . . Einstein's method is essentially identical with Gibbs' theory of canonical assemblies. In a second paper, of the following year, entitled 'Eine Theorie der Grundlagen der Thermodynamik' [Einstein, 1903], Einstein builds the theory of another basis not used by Gibbs, namely the consideration of a single system in course of time (later called 'Zeit-Gesamtheit,' time assembly), and proves that this is equivalent to a certain virtual assembly of many systems, Gibbs' micro-canonical assembly. Finally, he shows that the canonical and micro-canonical distributions lead to the same physical consequences. (Born, 1949, pp. 164-165) Actually, the time ensemble had also not been invented by Einstein; it rather goes back to a paper of Ludwig Boltzmann (1871). However, Einstein presented matters in such a way that he must be considered, together with Gibbs, as the founder of the ensemble theory in statistical mechanics. 17 Though he did not obtain much recognition for his pioneering work, his efforts were not in vain. After all, in establishing the foundation of statistical thermodynamics he had had in mind only one goal — the proof of the molecular hypothesis — and he would achieve this goal very soon. 18 Some time in the first half of the year 1905 Einstein wrote again a letter to Conrad Habicht, in which he announced that he would soon send him copies of four different scientific papers: the first dealt with radiation (Einstein, 1905b); the second with methods to determine the real dimensions of atoms (Einstein, 1905f); the third with the irregular motion of particles suspended in fluids (Einstein, 1905c); and the fourth with the electrodynamics of moving bodies (Einstein, 1905d) . 19 In 1 Einstein's papers of 1902 and 1903 were hardly noticed. However, Boltzmann and J. Nabl cited them in their review of kinetic theory for the Encyklopddie der mathematischen Wissenschaften as demonstrating the fact that very general dynamical systems may behave in a thermodynamicaly manner (Boltzmann and Nabl, 1907, p. 549, footnote 145). As we have mentioned in footnote 84, in Einstein's mind Maxwell and Boltzmann had not completed this task. Also Gibbs, whose work he came to know only later, had not done so. In a review of Hendrik A. Lorentz' book, 'Les theories statistiques eh thermodynamique' (Lorentz, 1916), Einstein criticized Gibbs' approach in the following words: Everyone, who has studied mathematical theories, is familiar with the following painful experience: he verifies eagerly and laboriously each step of the deduction and at the end of his endeavours understands — nothing; he misses the guiding line of thought, which the author [of t h e text one reads] frequently suppresses, either because he cannot express it neatly or because he follows a kind of coquetry that used to be fashionable in earlier times, which appears to be quite ridiculous to the discerning [reader]. Against this evil only the unrestrained openness of the author can help; he should not be afraid of sharing with the reader even incomplete guiding ideas if they have assisted him in advancing his work. In theoretical physics there is hardly a field in which this requirement is more difficult to satisfy than statistical mechanics. Every expert will agree with me in the opinion that Gibbs has sinned heavily against this requirement in his pioneering book on the subject; many have read it, gone through its derivations and not understood it. (Einstein, 1916f, pp. 480-481) Conrad Habicht, a former fellow student of Einstein's at the E.T.H. in Zurich, continued to pursue his mathematical studies at the University of Berne. He maintained close contact with Einstein; especially, he joined Einstein and Maurice Solovine, a young Rumanian student of science and literature, in reading and discussing important books on philosophy and science. (The three of

Albert Einstein

and the Origin of Light-Quantum

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337

two of these papers Einstein succeeded in presenting his proof of the existence of molecules of a finite size. Especially, in the work entitled 'A New Determination of Molecular Dimensions' ('Eine neue Bestimmung der Molekiildimensionen,' Einstein, 1905f, 1906a), which he submitted as a doctoral thesis to the University of Zurich, he demonstrated that 'one can estimate the size of the molecules of the substance dissolved in a non-dissociated, dilute solution from the internal friction of the solution and the pure solvent and from the diffusion of the dissolved substance in the solvent, provided the volume of the molecule of the dissolved substance is large compared to the volume of a molecule of the solvent' (Einstein, 1906a, p. 289). 20 The other work dealing with the existence and size of molecules made use of the results of the foundations of thermodynamics (Einstein, 1905c). Einstein claimed that 'according to the kinetic theory of heat, particles which are suspended in fluids and can be seen under the microscope will have to perform — because of molecular heat motion — movements of such a magnitude, that they (the movements) may be easily observed with the help of a microscope' (Einstein, 1905c, p. 549). Einstein's procedure for arriving at this result was straightforward. From Eq. (4), which expressed the entropy of a mechanical system, he derived an equation for the 'osmotic pressure' of particles suspended in a liquid; this pressure was, of course, a consequence of the kinetic theory and it caused the particles to diffuse in the liquid. 21 Due to the irregularity of the heat motion the diffusion of the particles would be very irregular. Einstein assumed all suspended particles to move independently of each other; thus he found that their propagation, say in the rc-direction, was represented by a Gaussian distribution with the characteristic exponent, (—x2/4Dt), where x was the distance the particle traversed during the time interval t and D denoted the diffusion coefficient. Hence the average displacement of the particle in the ^-direction was given by the quantity, x2, or by (13) where D = ^-(871-fcP)- 1 . Wo

(13a)

them referred to their meetings jokingly as the sessions of the 'Olympia Academy.') In spring 1905 Habicht left Berne to accept a position as a high school teacher. Einstein's letter to Habicht was not dated, but must have been written after his departure and before the publication of Einstein's paper on the light-quanta in June 1905 (Einstein, 1905b). Einstein completed his thesis on 30 April 1905 (Einstein, 1905f). He submitted the paper based on it to Annalen der Physik several months later (Einstein, 1906a; it was received on 19 August 1905). Einstein was awarded his doctorate on 15 January 1906. The publication of the paper in the Annalen was delayed; it appeared in February 1906 with an addendum by the author signed January 1906. 21 Originally, osmotic pressure was defined as a property of salts dissolved in a solution of water. The molecules of the dissolved substance act in the liquid as do the molecules in a dilute gas; in particular, they exert a pressure depending on their number in a given volume and the temperature of the solution. The kinetic theory of matter now demanded the existence of osmotic pressure, even if the dissolved particles had larger size and became observable under the microscope.

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In Eq. (13a), R and NQ denote the gas constant and the number of molecules per gram-atom, respectively, k the coefficient of friction in the liquid and P the radius of the dissolved or suspended particles. Clearly, Eq. (13) provided a method of determining the number No from A^ and the knowledge of the radius P of the suspended particles, or determining the radius of the dissolved molecules from Ax and No, provided P is much larger than the molecules of the solvent. 22 On the basis of Einstein's theoretical work, Jean Perrin and his collaborators would carry out within a few years systematic experimental investigations of the molecular dimensions and the determination of the number iVo, finding for the latter values between 6 x 10 23 and 7 x 10 23 . (See Perrin in Langevin and de Broglie, 1912.) The first paper which Einstein, in his letter to Habicht mentioned above, promised to send very soon, bore the title 'On a Heuristic Point of View about the Creation and Conversion of Light' (' Uber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt,' Einstein, 1905b). He had signed the paper in Berne on 17 March 1905 and sent it immediately to Annalen der Physih, it was received the following day and published in the first issue of Volume 17, which was distributed on 9 June 1905. 'It is on radiation and energy of light,' he described its content, 'and it is very revolutionary, as you will see yourself ('sie ist sehr revolutionary Einstein to Habricht, 1905; quoted in Seelig, 1956, p. 74). What was the result that Einstein had obtained and why did he call it revolutionary? To understand this point we shall summarize his previous concern with the problem of radiation, especially heat radiation. Einstein had become interested in the problem of heat radiation quite early; thus he had read Planck's work 'shortly after [its] appearance' (Einstein, 1949, p. 47). The 'major question' that concerned him was: 'What general conclusions can be drawn from the radiation formula concerning the structure of radiation and even more concerning the electromagnetic foundation of physics?' (Einstein, 1949, p. 47). In connection with his work on the foundation of thermodynamics he had, of course, been most interested in Planck's application of Boltzmann's statistical definition of entropy to the radiation problem without actually being able to make real sense of 22 When Einstein postulated the irregular motion of suspended particles in liquids based on the molecular theory of heat, he was not certain whether it had already been observed or not. 'It

is possible,' he wrote, 'that the motions treated here are identical with the so-called "Brownian molecular motion;" the information available to me about the latter is, however, too imprecise, hence I could not come to a proper judgment about it' (Einstein, 1905c, p. 549). Several months later he knew that he had indeed given a theory of the phenomenon which had been reported first by the Scottish botanist Robert Brown in 1827 (Brown, 1828). Brown had observed the pollens of a plant suspended in water under a microscope and found that they executed an irregular motion. Later on it was discovered that all small particles suspended in liquids or gases performed such motions and t h a t the activity increased with decreasing size and growing temperature. The physicist Heinrich Friedrich Wilhelm Siedentopf of Jena, who had developed together with the chemist Richard Adolf Zsigmondy the ultramicroscope (an instrument to study extremely small objects), informed Einstein about the identity of his postulated motion of suspended particles and Brownian motion. In a second paper, entitled 'Zur Theorie der Brownschen Bewegung' ('On the Theory of Brownian Motion') Einstein generalized his previous treatment to include also a rotational type of motion of the suspended particles (Einstein, 1906b).

Albert Einstein

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it. 'It was,' he recalled later, 'as if the ground had been pulled out from under one, with no firm foundation to be seen anywhere, upon which one could have built' (Einstein, 1949, p. 45). In his endeavour to secure a new understanding he did not receive much help from the available scientific literature; in the years following Planck's publication no articles throwing new light on the problem had appeared, for instance, in Annalen der Physik, until 1904. Then, in the Boltzmann Festschrift of February 1904, two authors dealt with heat radiation: Heinrich Kayser from Bonn wrote on the determination of temperature of radiating gases (Kayser, 1904), and Max Abraham from Gottingen discussed certain consequences of radiation pressure (Abraham, 1904). Abraham had, in particular, derived the displacement law in the special form given to it by Max Planck (i.e., pv = (u3/c3)f(T/v)) and then, by integrating over all frequencies, the law of Stefan and Boltzmann. 23 Shortly after the publication of the Boltzmann Festschrift, Einstein, in his third paper on the molecular theory of heat, for the first time treated a problem connected with heat radiation, which we have mentioned above. The interesting point is that Einstein, like Abraham, dealt with Stefan-Boltzmann's law; but while Abraham had applied only electrodynamic and thermodynamic arguments, Einstein had introduced statistical arguments from kinetic theory. 24 The question then arose whether one could also obtain Planck's law for blackbody radiation from the same foundation. Of course, Einstein knew that Planck had derived his equation from a probability consideration similar to the one used by Boltzmann in kinetic theory; especially, he had given an expression for the distribution of energy among the resonators in a cavity filled with radiation. However, by applying his own statistical methods to the same problem Einstein succeeded in deriving not Planck's distribution but rather a different one. Indeed, when treating a cavity containing molecules, electrons and resonators (which he represented by electrons fixed at a given space point, and oscillating in one direction), he found the average kinetic energy of the resonator of frequency v as

TS = u„ = JLTt

(i4)

which deviated strongly from Planck's expression for the energy of the resonator. That is, according to kinetic theory a one-dimensional oscillator at temperature T had to possess an average kinetic energy of \{R/NQ)T, and the same amount of potential energy. With Eq. (14) the energy distribution in blackbody radiation would be described by the equation

p

» = —w0T>

23

(15)

T h e distribution law, pv = {v3/c3)f{T/v), with f(T/u) a function depending only on the quotient (T/v) may be obtained immediately from Max Thiesen's form of Wien's displacement law, Ex = T V ( A T ) . 24 M a x Abraham had tried to extend Willy Wien's derivation of the displacement law by removing a restriction; Wien, in his adiabatic processes, had discussed only the reflection of light from slowly moving walls.

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The Golden Age of Theoretical

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rather than by Planck's law. Hence there existed, as Einstein remarked explicitly, 'a difficulty pertaining to the theory of black radiation' (leine die Theorie der "schwarzen Strahlun^ betreffende Schwierigkeitf (Einstein, 1905b, p. 133), a difficulty which he wanted to analyze carefully and to overcome. To begin with, Einstein recognized easily that Planck's law for the distribution of energy among frequencies in blackbody radiation, that is, 8-Khv3

exp

(5)

passed over into Eq. (15) for large values of the quotient, T/v, i.e., for large wavelengths and large radiation densities. 'We thus arrive at the conclusion,' he noted, 'that the higher the energy density and the longer the wavelength of radiation are, the more applicable are the theoretical foundations employed by us; however, for short wavelengths and low radiation densities, these foundations fall completely' (Einstein, 1905b, p. 137). What were the theoretical foundations that had failed? Einstein was convinced that, in obtaining Eq. (15), he had applied mechanics and the thermodynamics correctly (like the expert whose duty it was to check thoroughly the physical basis of the inventions submitted to the Patent Office); he was also convinced that his statistical thermodynamics, which he had developed over the previous years, was the only one consistent with the theoretical foundations. Thus the question arose as to what had to be changed in order to arrive at Planck's law, a law described by two universal constants — the Boltzmann constant k and Planck's constant h, the latter determining the size of the finite energy elements. The constant k had already appeared in Einstein's theory of statistical thermodynamics as the constant R/NQ.25 But there was definitely no place at all for Planck's constant h. Even more, it seemed that the very introduction of a finite value for h — and consequently of finite elements in the statistical considerations — somehow contradicted the accepted principles of kinetic theory. Therefore, the question had to be answered whether or not one could give h a different physical meaning without violating the foundations of the theory. 26 Einstein knew that, to establish an interpretation of the constant h, he had to analyze the energy distribution of the heat radiation in the region where h did not drop out of Eq. (25a), but rather determined the structure of the radiation law: that is, for large values of the factor hu/kT in the exponential. This, however, was the region where Wien's law described the observations accurately. Now Willy Wien had derived his law under the special assumptions that the radiating parts in 25

Einstein proved the equality of R/No with k explicitly by taking the limit of large T/v in Planck's formula and comparing the result to Eq. (15). Then he calculated N^1, identifying it with the mass of the hydrogen atom, whose value he found to agree completely with Planck's result of December 1900 (Planck, 1900f, p. 244). 2 Einstein, in his review of Planck's Lectures on Heat Radiation, pointed out the fact that the author had not yet given a proper interpretation of the constant h by saying: 'The author points repeatedly to the necessity of introducing this universal constant h and emphasizes how important its physical significance (which is not treated in this book) is' (Einstein, 1906f, p. 766).

Albert Einstein

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a blackbody obeyed the Maxwellian velocity distribution and that the frequency of the emitted radiation depended only on the velocity of the emitter (Wien, 1896). Later, in 1900, when the experiments showed strong deviations from his law, Wien had argued that this had to be expected, for the statistical assumption used in 1896 was correct only for large values of the quotient u/T and did not apply to radiation of long wavelengths (Wien, 1900b). Wien had stated, at the same time, that the heat radiation satisfying his law could not be explained on the basis of the electrodynamic theory; rather, it exhibited a structure which was different in quality from that of the long-wavelength radiation, the radiation which obeyed the laws of classical electrodynamics. Einstein, in his analysis of the foundations of Planck's radiation formula in 1905, picked up Wien's idea of 1900. That is, he assumed that the heat radiation consisted of two parts: the long-wavelength radiation, described by the known laws of electrodynamics, and short-wavelength radiation, determined by laws still unknown. He wanted to discover the nature of these unknown laws. The type of radiation that interested Einstein was described formally by Wien's radiation law, which he wrote as p„ = au3 exp

It

' T

with two constants a and (3. (In Planck's notation the constants were a = 8irh/c3 and f3 = h/k, respectively.) In order to analyze the physical content of Wien's radiation law Einstein (following Planck's preferred method) studied the entropy of radiation associated with it. In particular, he considered the entropy S of radiation in a volume v, having the frequency v and energy Ev (where Ev = pvvdv is the energy in the frequency interval between v and v + dv), that is,

*'-%H5)-\

(16)

which he obtained from the definition of the entropy differential and the fact that the entropy of blackbody radiation takes on a maximum value. 27 Planck had derived a similar expression for the entropy of radiation, but Einstein went beyond Planck in deriving physical consequences from the volume dependence of S. If VQ and v denote the initial and final volumes, respectively, filled with heat radiation (obeying Wien's law), having the energy Ev, then the entropy difference, S — So, was given 27 Evidently, the total entropy of radiation of all frequencies in a volume v is given by an integral, v j 4>(pi,, v) dv, with 4>{p„,v) a function of the two variables pv and v, and this integral assumes a maximum value only if the function <j>(pw,v) is independent of the frequency. Then, for each frequency one has the relation: d<j>/dpv — T _ 1 . And one can obtain an expression for the inverse of the temperature, from which one calculates (by integration) the function 4> = 4>(P"iu) as:

pv

L

XOLV6)

Then the entropy of the radiation in the frequency interval v, v + dv will be v<j>(pu, v) dv.

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The Golden Age of Theoretical

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by the equation

S-So = f W - V

(17)

Einstein noted: This equation shows that the entropy of monochromatic radiation of sufficiently small density [such that it satisfies Wien's radiation law] varies with the volume according to the same rules as the entropy of a perfect gas or of a dilute solution. The equation just obtained [Eq. (17)] shall be interpreted in the following on the basis of the principle introduced by Mr. Boltzmann into physics, according to which the entropy of a system is a function of the probability of its state. (Einstein, 1905b, pp. 139-140) On first inspection these conclusions appear to be rather surprising, and one has to ask the question how Einstein arrived at them. The answer must be sought in the ideas and motivations guiding him at that time. Einstein had decided, at least since 1904, that one should apply the general molecular theory of heat or statistical mechanics to the problem of heat radiation; in particular, he had shown then that fluctuation considerations yielded reasonable results (Einstein, 1904). Encouraged by his success he investigated further consequences of the equations of statistical mechanics in radiation theory. Since he was most fascinated by the problem of fluctuations — as evidenced by his interest in such phenomena as the Brownian motion — he preferred to deal with equations related to it. Now the dependence of the entropy on the volume was indeed connected with fluctuations: if one interpreted Boltzmann's entropy-probability relation (S = (R/No)logW) as an equation for calculating the probability of a state from its entropy, then Eq. (17) allowed one to compare the probabilities of two situations, in which heat radiation (of low density and high frequency) of the same energy per frequency interval was concentrated in the volumes VQ and v, respectively. As a consequence Einstein's above conclusion can be perfectly understood. Einstein thought along these lines, as is evident from the discussion in his paper (Einstein, 1905b, §5 and 6). And he had no difficulty in recognizing the analogy of Eq. (17) to the equation expressing the volume dependence of the entropy of ideal gases or dilute solutions. In fact the latter (i.e., the volume dependence of the entropy) was well known at the time Einstein discussed heat radiation. For example, in the Boltzmann Festschrift, N. Schiller of Kharkov had presented a formula describing the entropy of gases in the process of diffusion (Schiller, 1904, p. 361, Eq. (20)). In it the proportionality of the entropy on the logarithm of the volume showed up explicitly. Incidentally, Einstein reviewed Schiller's paper in the Beibldtter zu den Annalen der Physik, and in this review he quoted Schiller's entropy formula (Einstein, 1905a, p. 238). After this outline of Einstein's motivation, we may return to the specific presentation of the arguments in his paper. He first derived the volume dependence of the entropy of a gas consisting of molecules according to the kinetic theory. According to Boltzmann, the difference between the initial and final entropy of a system which passes from a given initial state (entropy So) to a final state (entropy 5) is given

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by the equation S-S0

= -j^\nW,

(18)

where R and No denote the universal gas constant and the number of molecules per gram-atom, respectively, and W is the relative probability of the final state. 28 Then Einstein considered n molecules (of a given type) in a volume VQ, which move about in arbitrary directions and do not exert any actions upon each other except by elastic collisions (i.e., they constituted an ideal gas); he calculated the entropy difference of this state with another, in which the n molecules were concentrated in the volume v, a fraction of the initial volume VQ. He argued that, according to Boltzmann, the 'statistical probability' for the final state would be proportional to (v/v0)n; hence from Eq. (18) the entropy difference became 29

s

-*° = £'"(;;)"•

(19)

Einstein commented on Eq. (19) as follows: 'It should be noted that for the derivation of this equation — from which one can easily derive thermodynamically the Boyle-Gay-Lussac law [i.e., the equation of state for ideal gases] and identically the same law of osmotic pressure — no other assumption need be made about the law according to which the molecules move' (Einstein, 1905b, p. 142). Now the last step was quickly performed: all Einstein had to do was to write the right-hand side of Eq. (17) in a form which could be compared to the right-hand side of Eq. (19); evidently this was R ( v \ (N°/R)<.E"/W

s s

- » = So,nUJ

(178)

Einstein interpreted this equation as follows: 'If monochromatic radiation of frequency v and energy E is enclosed (by reflecting walls) within a volume VQ, the probability that at any arbitrary instant of time the total radiation energy is contained in a part v of volume v0 will be: W = (v/v0)(-N°/RKE»/*3u'>, (Einstein, 1905b, p. 143). Evidently the probability for the radiation was now described by an expression having exactly the same structure as that for the probability for n molecules, provided one took the factor (No/R)(Ev/[5v) in the exponent instead of n. Hence Einstein concluded finally: Monochromatic radiation of low density (within the range of validity of Wien's radiation formula) behaves, in a thermodynamic sense, as if it consisted of mutually independent radiation quanta of magnitude Rf3v/N0. (Einstein, 1905b, p. 143) To some extent the procedure outlined above seems to be straightforward and immediately evident. As we have pointed out, we may assume with good reason 28 N o t e t h a t Einstein in 1905 consistently used the ratio R/No instead of his earlier universal constant 2«. 29 T h e same Ansatz for the statistical probability was used by Marian von Smoluchowski in his paper for the Boltzmann Festschrift (Smoluchowski, 1904, p. 626).

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that Einstein had Eq. (19) (for the molecules in a gas) in mind for a long time. All he had to do was to derive the expression for the corresponding entropy of radiation satisfying Wien's law, Eq. (17), say with the help of Planck's papers in Annalen der Physik (Planck, 1901a,c), and to realize its formal relationship to Eq. (19). Still one has to ask the question: how did Einstein come to search for a relation between a dilute gas and Wien's radiation? One reason may perhaps be seen in the formal similarity between the energy distribution in Wien's radiation law and the energy distribution in a system consisting of independent gas molecules; in fact, Wien had obtained his law by referring explicitly to Maxwell's velocity (and energy) distribution. The physical argument in favour of his approach was, as we have pointed out above, that the molecules emitting radiation in a blackbody were described by Maxwell's distribution and, therefore, this distribution had to be reflected in the emitted radiation. For this purpose, of course, an equilibrium had to be assumed to exist between radiation and molecules. If that was true, Einstein concluded, then kinetic theory implied that heat radiation, on one hand, and the molecules, on the other, must lead to the same macroscopic effects. Especially, both must create an 'osmotic pressure,' say, on a mirror in a cavity. Now the osmotic pressure, p, created by n molecules of a dilute gas (or — in the original sense of the actual definition of the osmotic pressure — by n molecules dissolved in a liquid) in a volume v at temperature T was given by the equation pdv = TdS = ^f-n—.

(20a)

On the other hand, the radiation satisfying Wien's law yielded for p dv the result pdv==§LT—.

(20b)

pu v Evidently, the right-hand sides of Eqs. (20a) and (20b) had the same structure, provided the factors (RT/N)n and Ev/(3v were associated with each other. The consequence from this association agreed with the consequence derived from the expressions for probability. Therefore, the comparison of the osmotic pressure of a dilute gas with the pressure of radiation obeying Wien's law provided an immediate proof of the molecular structure of radiation. 30 The physical interpretation of the short-wavelength radiation obeying Wien's law also implied a physical interpretation of Planck's constant h. By substituting 30 Although Einstein did not give the procedure outlined above in his paper, he recalled later that considerations of the osmotic pressure had played an important role in his interpretation of Wien's radiation law (see Einstein, 1949, pp. 48-52). T h a t osmotic pressure considerations occupied Einstein in 1905 is also confirmed by their appearance in the paper on Brownian motion (Einstein, 1905c), which was composed at the same time as the paper on radiation. In any case, the procedure of arriving at the interpretation of Wien's radiation law via the consideration of the osmotic pressures can be obtained from equations given in the published paper (Einstein, 1905b, p. 142, footnote 1). (Note that in the equation for TdS a factor T is missing on the right-hand side, i.e., the expression R(n/No)(dv/v) must be replaced by RT(n/No)(dv/v).)

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the value of /3 (= h/k) and observing the equality of the constant k with the ratio R/No, one finds for the number of energy-quanta n the relation No E R pv

E hv

Thus the quantity {R/N)(iv indeed represented the size of energy quanta, in agreement with Planck's fundamental assumption of 1900. Einstein did not, however, write Eq. (21) in his paper, for he wanted to keep his distance from Planck's theory. 'At that time it appeared to me,' he explained a year later, 'that Planck's theory constituted in some respects a counterpart to my own work' (Einstein, 1906c, p. 199). It appeared to Einstein in 1905 that Planck, in formulating his theory of heat radiation, had moved away from the basis of kinetic theory and had employed an extra hypothesis to arrive at his radiation law; otherwise he would have obtained the radiation law given by Eq. (15). Indeed, it must have seemed as if the foundations of Einstein's and Planck's radiation theories were entirely different: Planck, on several occasions, had emphasized the fact that he had used a unified description of all wavelengths on the basis of Maxwell's electrodynamics, while Einstein had just arrived at the conclusion that radiation of short wavelengths possessed a structure that did not fit into the electrodynamic theory. In any case, Einstein demonstrated his independence from Planck by not using the latter's notation for the constants; he preferred to denote Boltzmann's constant k (which Planck had employed) by the ratio R/No of the gas constant and number of molecules per gram-atom and Planck's constant h by the quantity (R/No)0. Nevertheless he knew that he had given at least a partial answer to the question concerning the physical interpretation of Planck's constant h. For the radiation satisfying Wien's law he had proved the equality of h and (R/N0){3, and the product of (R/No)/3 and the frequency denoted the size of the independent energy-quanta of radiation. Einstein did not stop at this point but proceeded, as had become his custom in scientific work, to check certain consequences of the assumption of independent energy-quanta in radiation theory. To begin with, he expected that energy-quanta of radiation did not play a role in the phenomena of diffraction and propagation of light in vacuo; however, whenever radiation was created or transformed — i.e., a change of frequencies occurred — then the molecular structure of light might show up. 3 1 If monochromatic radiation (of sufficiently low density) behaves, as far as the volumedependence of its entropy is concerned, as a discontinuous medium consisting of energy quanta of magnitude R(3V/NQ [he remarked] then it is plausible to investigate whether the laws of the creation and transformation of light are so constituted as if light consisted of such quanta. (Einstein, 1905b, pp. 143-144)32 ^Although Einstein's and Planck's theories were different, both authors agreed on one point: the fact that propagation and diffraction of light were satisfactorily described by the electromagnetic theory. In 1905, Einstein had not yet developed the full corpuscular theory of the light-quantum including energy and directed momentum. " N o t e that for the purpose of describing these processes Planck invoked t h e concept of irreversibility, which offered him the possibility of entering into his statistical considerations.

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But what were the phenomena in which the quantum structure of radiation intervened? Einstein selected three examples from his knowledge of the scientific literature: first, the well-known phenomenon of fluorescence; second, the so-called photoelectric effect; third, the ionization of gases by ultraviolet light. In all three cases he demonstrated agreement of the observed phenomena and the consequences from his heuristic point of view concerning the nature of radiation. In the example of fluorescence — or photoluminescence, as Einstein called it — George Gabriel Strokes had found that the (visible) light emitted by fluorescent substances was excited mainly by ultraviolet radiation; hence he had established the rule, later called Stokes' rule, that the absorbed light had higher frequencies than the reemitted one. Einstein now explained this rule on the assumption that both the original and the newly produced light consist of energy quanta of magnitude (R/No)/3v, where v is the corresponding frequency. [He concluded:] Unless the photoluminescent substance can be regarded as a continuous source of energy [i.e., as violating the law of conservation of energy], the energy of the light-quantum that is produced cannot — according to the law of energy conservation — be greater than that of an initial lightquantum; hence the condition, (R/No)f}v2 < (R/N0)f3i>i, or u2 < v\, must hold. (Einstein, 1905b, p . 144)

With this argumentation, in which the name 'light-quantum' occurred for the first time, he provided an elegant derivation of Stokes' rule which could not be understood on the basis of the electromagnetic theory of radiation. 33 The next example which Einstein considered was the photoelectric effect. It had been discovered by Wilhelm Hallwachs, who in turn had been stimulated by previous experiments of Heinrich Hertz; he had shown, in particular, that some uncharged metallic surfaces acquired a positive charge if exposed to ultraviolet light (Hallwachs, 1888). 34 Joseph John Thomson and Philipp Lenard had then shown that the effect was caused by the fact that negatively charged corpuscles, the electrons, were emitted from the surfaces of metals (Thomson, 1899; Lenard, 1899). Lenard had continued to investigate the phenomenon and had presented detailed results in a long memoir published in Annalen der Physik (Lenard, 1902). 35 In this paper he had Einstein further concluded that his light-quantum hypothesis implied t h a t fluorescence radiation could be excited by incident light (of higher frequencies) having arbitrarily small intensity. He also added that exceptions from Stokes' rule might occur, if either the intensity of the exciting radiation was very large (so that one newly created quantum obtained its energy from several incident quanta) or if the radiation considered did not satisfy Wien's law. 34 Wilhelm Ludwig Franz Hallwachs was born on 9 July 1859 in Darmstadt. He studied physics at the Universities of Strasbourg and Berlin, obtaining his doctorate in 1883. He then became assistant to August Kundt in Strasbourg and Friedrich Kohlrausch in Wiirzburg, and later professor of physics at the Technische Hochschule of Dresden (1900-1922). He died in Dresden on 20 June 1922. Philipp Eduard Anton von Lenard was born in Pozsony (Pressburg), Hungary, on 7 June 1862. He studied physics successively at Budapest, Vienna, Berlin and Heidelberg and had among his teachers Robert Bunsen, Hermann von Helmholtz and Georg Hermann Quincke. After obtaining his doctorate from the University of Heidelberg in 1886 he became assistant to Quincke in Heidelberg and then t o Heinrich Hertz in Bonn; in Bonn he also received his Habilitation in 1892. In 1894 he was appointed Extraordinarius of physics at the University of Breslau, and one year later

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reported especially two important findings: first, in order to obtain electrons from a given metallic surface only the incident light of certain frequencies was effective; second, the velocity of the emitted electrons did not depend on the intensity of the incident radiation. He had argued, therefore, that the initial velocities of the emitted quanta [so did Lenard call the electrons] do not originate from light energy at all, but from the violent motions existing already before the illumination within the interior of the atoms; thus the resonance motions [of the electrons stimulated by the incident radiation] only play the role of a mechanism for the release [of electrons]. (Lenard, 1902, p. 170) Einstein interpreted Lenard's results differently; especially, he took the above statement as an indication of the fact that, 'The usual idea, that the energy of light is continuously distributed over the space through which it travels, meets with especially great difficulties when one tries to explain the photoelectric phenomena' (Einstein, 1905b, p. 145). He suggested the following solution of these difficulties. If the radiation falling on a metallic body possesses the structure of energy-quanta and one assumes that each energy-quantum gives its energy to a single electron, then the emitted electron receives the kinetic energy, EkiB=Ue=-^-l3u-P,

(22)

where P is the potential barrier (characteristic for the body under investigation) which the electron (of charge —e) has to overcome in order to leave the metallic surface. This kinetic energy had been measured by Lenard by subjecting the metallic surface to a positive potential II, which prevented the electrons from being emitted at all. The order of magnitude of his values agreed with Einstein's estimate from Eq. (22). Einstein further demanded: 'If the formula derived above is correct, then II — if drawn in Cartesian coordinates as a function of the frequency of the incident light — must be a straight line, the slope of which is independent of the nature of the substance studied' (Einstein, 1905b, p. 146). Evidently, the slope of the function is given by (R/N0)f3 or Planck's constant h. In 1905 Einstein could only state that: 'As far as I can see, our ideas are not in contradiction with the properties of the photoelectric action observed by Mr. Lenard' (Einstein, 1905b, p. 147). 36 A decade later Robert Andrews Millikan would report a he became Ordinarius (full professor) a t t h e Technische Hochschule in Aachen. In 1896 he moved to Heidelberg as Professor of Theoretical Physics. Prom 1898 t o 1907 he served as Professor of Physics at the University of Kiel; then he returned t o Heidelberg as Professor of Physics (1907 1930). Lenard died at Messelhausen on 20 May 1947. Philipp Lenard worked on theoretical mechanics, on the phenomena of phosphorescence and luminescence. From 1888 he investigated cathode rays; stimulated by Hertz, he invented the 'Lenard window,' which made it possible to isolate cathode rays from the source and t o observe them as free objects. For his pioneering investigations in the field of cathode rays he received t h e Nobel Prize in Physics in 1905. He also worked on the photoelectric effect and on the nature and origin of spectral lines. 36 Einstein drew attention in particular t o the fact that his hypothesis of light-quanta explained Lenard's observation t h a t the number of electrons was proportional to the intensity of the incident

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detailed verification of Einstein's formula, Eq. (22), and note especially: 'Planck's h has been photoelectrically determined with a precision of about .5 per cent and is found to have the value h = 6.57 x 1 0 - 2 7 [erg • sec]' (Millikan, 1916b, p. 388). The last phenomenon which Einstein considered in order to support his conception of a quantum-like structure of radiation was the ionization of gases by ultraviolet light. By assuming that the ionization of each molecule was due to the energy transfer from one energy-quantum of radiation, he found values for the ionization energy that were in agreement with the empirical data available at that time (Einstein, 1905b, Section 9). Einstein was very satisfied that all these phenomena fitted well into his new picture about the quantum structure of light. In the following years he would find further empirical facts which could be easily accounted for by the same assumption. These would confirm and develop his belief in the dual nature of radiation, an undulatory one described by the laws of classical electrodynamics and a corpuscular one described by the laws of molecular theory. He would also discover that his light-quantum theory and the blackbody radiation theory of Planck involving the quantum of action rested on the same foundation. Hence he would become convinced that both had to be combined into a new theory. This theory — the quantum theory — would have a different structure and be expressed by means of laws different from those of classical theories, i.e., of classical mechanics and electrodynamics (including special relativity theory), which were fully and completely established by the end of the first decade of the twentieth century. References M. Abraham, 1904, Boltzmann Festschrift, 85-93. L. Boltzmann, 1868, Site. ber. Akad. Wiss. (Wien) 58, 517-560. L. Boltzmann, 1871, Site. ber. Akad. Wiss. (Wien) 63, 712-732. L. Boltzmann, 1872, Site. ber. Akad. Wiss. (Wien) 66, 275-370. L. Boltzmann, 1877, Site. ber. Akad. Wiss. (Wien) 76, 373-435. L. Boltzmann, 1895c, Vorlesungen iiber Gastheorie I, Leipzig: J.A. Barth. L. Boltzmann and J. Nabl, 1907, Encypl. d. Math. Wiss. V/l, 493-557. M. Born, 1949, in Albert Einstein: Philosopher-Scientist (ed. P.A. Schilpp, New York: Tudor Publishing), 162-167. Robert Brown, 1828, Phil. Mag. (2) 4, 161-173. R.W. Clark, 1971, Einstein: The Life and Times, New York: World Publishing, paperback edition, New York: Avon Books, 1972. W.J. de Haas, 1916, Verh. d. Deutsch. Phys. Ges. (2) 18, 423-443. A. Einstein, 1901, Ann. d. Phys. (4) 4, 513-523. light. In addition, he considered the inverse of the photoelectric effect, that is, the creation of radiation when cathode rays fall on a metallic surface; in that case he derived from his lightquantum hypothesis the conclusion that the product of the frequency of the emitted radiation, u, and the factor R[3/N ( = h) was always smaller than the kinetic energy of the cathode rays (electrons), in agreement with the experimental results (Lenard, 1903a).

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Einstein, 1902a, Ann. d. Phys. (4) 8, 798-814. Einstein, 1902b, Ann. d. Phys. (4) 9, 417-433. Einstein, 1903, Ann. d. Phys. (4) 11, 170-187. Einstein, 1904, Ann. d. Phys. (4) 14, 354-362. Einstein, 1905a, Beiblatter Ann. d. Phys. 29, 237-238. Einstein, 1905b, Ann. d. Phys. (4) 17, 132-148. Einstein, 1905c, Ann. d. Phys. (4) 17, 549-560. Einstein, 1905d, Ann. d. Phys. (4) 17, 891-921. Einstein, 1905f, Eine neue Bestimmung der Molekuldimensionen, doctoral dissertation, University of Zurich, Berne: Wyss. A. Einstein, 1906a, Ann. d. Phys. (4) 19, 289-306. A. Einstein, 1906b, Ann. d. Phys. (4) 19, 371-381. A. Einstein, 1906c, Ann. d. Phys. (4) 20, 199-207. A. Einstein, 1906e, Beiblatter Ann. d. Phys. 30, 764-766. A. Einstein, 1906f, Ann. d. Phys. (4) 2 1 , 583-586. A. Einstein, 1916b, Verh. d. Deutsch. Phys. Ges. (2), 18, 173-177. A. Einstein, 1916f, Naturwiss. 4, 480-481. A. Einstein, 1949, Autobiographical Notes, in Albert Einstein: Philosopher-Scientist (Paul Arthur Schilpp, ed.), New York: Tudor, 1-95. A. Einstein, 1956, Autobiographische Skizze, in Helle Zeit-Dunkle Zeit, in memoriam Albert Einstein (C. Seelig, ed.), Europa Verlag, 9-17. A. Einstein and W.J. de Haas, 1915, Verh. d. Deutsch. Phys. Ges. (2) 17, 152-170. W. Hallwachs, 1888, Ann. d. Phys. (3) 3 3 , 301-312. B. Hoffmann, 1972, Albert Einstein: Creator and Rebel, New York: The Viking Press. H. Kayser, 1904, Boltzmann Festschrift, 38-42. L. Kollross, 1956, Erinnerungen, in Helle Zeit-Dunkle Zeit emorvum Albert Einstein (C. Seelig, ed.), Europa Verlag, 17-31. P.S. Laplace, 1806, Traite de mecanique celeste, Paris: reprinted in Oeuvres 4, Paris, 1845. P. Lenard, 1902, Ann. d. Phys. (4) 8, 149-198. P. Lenard, 1903a, Ann. d. Phys. (4) 12, 449-490. H.A. Lorentz, 1916, Les Theorie Statistiques en Thermodynamique, Leipzig-Berlin: B.G. Teubner. J.C. Maxwell, 1860, Phil Mag. (4) 19, 19-32. Jagdish Mehra, 1975a, Einstein and the Foundation of Statistical Mechanics, Physica 79A, 447-477. R.A. Millikan, 1916b, Phys. Rev. (2) 7, 355-388. H. Minkowski, 1907, Eneyl. d. Math. Wiss. VI/I, dated fall 1906, published 25 April 1907. N. Nernst, 1904, Boltzmann Festschrift, 904-915. J. Perrin, 1912, in P. Langevin and M. de Broglie, La Theorie du Rayonnement et les Quanta, Paris: Gauthier-Villars. M. Planck, 1901a, Ann. d. Phys. (4) 4, 553-563. M. Planck, 1901c, Ann. d. Phys. (4) 6, 818-831. M. Planck, 1904, Boltzmann Festschrift, Leipzig: J.A. Barth. M. Planck, 1905f, Verh. d. Deutsch. Phys. Ges. (2) 2, 237-245. N. Schiller, 1904, Boltzmann Festschrift, 350-366. C. Seelig, 1954, Albert Einstein: Eine dokumentarische Biography, Zurich: Europa Verlag. English translation (by M. Savill): Albert Einstein: A Documentary Biography, London: Staples Press.

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J.J. Thomson, 1899, Phil. Mag. (5) 48, 547-567. M. von Smoluchowski, 1904, Boltzmann Festschrift, 626-641. W. Wien, 1896, Ann. d. Phys. (3) 58, 662-669.

10 Niels Bohr and t h e Q u a n t u m Theory of t h e Atom* 1 In issue No. 151 (July 1913) of the renowned British journal Philosophical Magazine there appeared an article entitled 'On the Constitution of Atoms and Molecules;' it was communicated by Ernest Rutherford of Manchester and written by a sofar-unknown physicist, a Dr. phil. Niels Bohr of Copenhagen (Bohr, 1913b). The author announced that the article was the first of a series dealing with a theory of the constitution of atoms and molecules on the basis of Rutherford's atomic model and Planck's elementary quantum of action. At the close of the introduction to his paper Bohr remarked: 'In the present first part of the paper the mechanism of the binding of electrons by a positive nucleus is discussed in relation to Planck's theory. It will be shown that it is possible from the point of view taken to account in a simple way for the law of the line spectrum of hydrogen' (Bohr, 1913b). Who was this man who had solved this great problem of atomic theory? Niels Henrik David Bohr was born on 7 October 1885 in Copenhagen as the first son of Christian Bohr, Professor of Physiology at the University of Copenhagen, and his wife Ellen Adler, daughter of a Jewish banker. 2 Niels had an elder sister, Jenny, and a brother, Harald, who was two years younger. He received an Lecture delivered at the Niels Bohr Institute and NORDITA, Copenhagen, 15 September 1975, CERN (European Organization for Nuclear Research), Geneva, 25 September 1975, and at Universite libre de Bruxelles, 17 October 1977. This revised and enlarged version was published in The Historical Development of Quantum Theory (with Helmut Rechenberg, Springer-Verlag, New York, 1982, pp. 181-200). 1 T h e origin of Niels Bohr's theory of line spectra is one of the most frequently treated topics in the literature on the development of quantum theory. The literature includes the articles of Leon Rosenfeld (in Bohr, 1963) and John A. Heilborn and Thomas S. Kuhn (1969), and the monograph of Ulrich Hoyer (1974). An authoritative account of certain events was also given by Niels Bohr himself in his Rutherford Memorial Lecture of 1958 (Bohr, 1961a). Finally, there exists the record of interviews with Bohr himself and a Manchester colleague, the chemist George de Hevesy, in which the events of the second decade of the twentieth century leading up to Bohr's model of atomic structure have been discussed. 2 For the life of Niels Bohr we have consulted the biography by Ruth Moore (1966) and the articles of David Jens Adler, and Leon Rosenfeld and Erik Riidinger in the book edited by Stefan Rozental (Rozental, 1967, pp. 11-37 and 38-73), as well as the introductions to the early papers (preceding the ones on atomic constitution) by J. Rud Nielsen, contained in Niels Bohr, Collected Works, Vol. 1 (Bohr, 1972).

351

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excellent education, both at home — where learned professors, the colleagues of his father, frequently assembled — and at school.3 After attending Gammelholm School, where, despite some difficulties in composition, he passed with distinction, especially in mathematics and physics, Niels entered the University of Copenhagen to study mathematics, philosophy and physics. 4 He stood out as an unusually perceptive investigator: with his first research work, a precision measurement of the surface tension of water by the observation of a regularly vibrating jet, Niels Bohr won a gold medal of the Academy of Sciences and Letters in Copenhagen (Bohr, 1906).5 He then decided to take his master's and doctor's degrees in physics, while his brother Harald concentrated on mathematics. 6 Niels completed his thesis for the master's degree in July 1909 and continued to work on the same subject — the interpretation of the physical properties of metals on the basis of electron theory — for his doctoral dissertation, which he defended on 13 May 1911 (Bohr, 1911). In September 1911, supported by a fellowship from the Carlsberg Foundation, Bohr went to Cambridge to study under J.J. Thomson. 7 He entered Trinity College and attended the lectures of Thomson, Joseph Larmor and James Jeans; at the suggestion of Thomson, he also did experiments on positive rays in the Cavendish Laboratory. He anxiously waited for Thomson's comments on his doctoral thesis, a rough English translation of which he had brought along with him. 8 But Thomson 3 Besides other English customs, Christian Bohr also admired the game of soccer; thus he founded Akademisk Bold Klub, i.e., the University Soccer Club. Harald Bohr became one of the best soccer players in Denmark; he was even on the Danish team which won the silver medal at the Olympic Games in London in 1908. 4 Niels Bohr took courses in philosophy and logic from Harald H0ffding, a close friend of his father's, who taught the main system of philosophy from the sixteenth to the eighteenth century and aroused the interest of his students in the philosophy of Baruch Spinoza and Soren Kierkegaard. It was in the circle of philosophy students that Bohr learned his method of developing new ideas in discussion with others. Most of the physics courses were taught by Christian Christiansen, a physicist who belonged to the group of friends of Bohr's father. 5 I n this work Bohr also improved upon a theory of Lord Rayleigh. Later he published an English translation of his first work in Philosophical Transactions of the Royal Society of London (Bohr, 1909), with a shorter version of this paper appearing in Proceedings of the Royal Society (Bohr, 1910). 6 Harald Bohr was born on 22 April 1887 in Copenhagen. He obtained his master's degree in 1909, before Niels, and then left Copenhagen to continue his studies with Edmund Landau in Gottingen. He returned to Copenhagen in fall 1909 and defended his doctoral thesis on the theory of Dirichlet series in January 1910, after which he joined the faculty of the University of Copenhagen. Five years later he was appointed Professor of Mathematics at the College of Technology, and in 1930 he became head of the newly founded Institute of Mathematics at the University of Copenhagen. Harald often visited Gottingen to collaborate with Edmund Landau, hence the name 'Bohr' had a good reputation there even before Niels ever went there. Besides Gottingen, Harald Bohr had fruitful contacts with the English mathematicians Godfrey Harold Hardy and John Edensor Littlewood at Cambridge and Oxford, respectively. He died in Copenhagen on 22 January 1951. 7 T h e reason for this choice was the fact that Thomson was considered as one of the outstanding experts in electron theory and the conduction of electricity, which was the subject of Bohr's thesis. In his doctoral thesis Bohr had criticized some of Thomson's work. In particular, he had argued that in Thomson's calculation of the absorption of heat radiation by metals (Thomson, 1907a) the motion of the individual electrons had not been taken into account properly (Bohr, 1911; see English translation p. 358).

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was very difficult to approach for the somewhat shy Danish research student; he only suggested that the paper should be submitted to the Transactions of the Cambridge Philosophical Society.9 On the occasion of a visit to Lorrain Smith, a friend and colleague of his father's in Manchester, Bohr was introduced to Ernest Rutherford. Rutherford had just returned from the Solvay Conference in Brussels, and he impressed Bohr deeply. Bohr was happy to be invited to spend some time in Rutherford's laboratory in Manchester; he went there in the middle of March 1912 and enrolled immediately in an experimental course on radioactivity. 'When I came up there,' he recalled later, 'I can say that Rutherford was very nice and arranged that I could take part in courses by [Hans] Geiger. But it was so that just a few weeks later on I said to Rutherford that it would not work to go on making experiments and that I would better like to concentrate on the theoretical things' (Bohr, AHQP Interview, 11 January 1962, p. 11). Rutherford's laboratory in Manchester was a most active place at that time. The experiments of Hans Geiger and Ernest Marsden on testing Rutherford's scattering formula, and the related picture of atomic structure were still going on. Besides, a number of young and very good collaborators — including Charles Galton Darwin, Henry Gwyn Jeffreys Moseley and George de Hevesy — worked with Rutherford. Bohr felt very happy in the midst of these people. 'And Hevesy was most extraordinary,' he remarked many years later, 'he had such fine manners. He knew how to be helpful to a foreigner' (Bohr, AHQP Interview, 11 January 1962, pp. 11-12). Hevesy, who had just come to Manchester from Karlsruhe in order to learn experimental techniques in physics from Rutherford, was soon impressed by About his encounters with J.J. Thomson, Bohr reported to his brother Harald as follows: . . . Thomson has so far not been as easy to deal with as I thought the first day. He is an excellent man, incredibly clever and full of imagination (you should hear one of his elementary lectures) and extremely friendly; but he is so immensely busy with so many things, and he is so absorbed in his work that it is very difficult to get to talk to him. He has not yet had time to read my paper, and I do not know yet if he will accept my criticism. He has only talked t o me about it a few times for a couple of minutes, and only about a single point, namely about my criticism of his calculation of the absorption of heat rays. You may remember that I pointed out that, in his calculation of the absorption (contrary to the emission), he does not take into account the time taken by the collisions and, therfore, finds a value for the ratio between emission and absorption that is of the wrong order of magnitude for small periods of vibration. Thomson first said that he couldn't see that the duration of the collisions could have such a large influence upon the absorption. I tried to explain it t o him and gave him the next day a calculation of a very simple example... which showed it very clearly. Since then I have only talked with him about it for a moment, about a week ago, and I believe that he feels t h a t my calculation is correct; however, I am not sure but that he thinks that a mechancial model can be found which will explain the law of heat radiation on the basis of the ordinary laws of electromagnetism, something that is obviously impossible, as I have shown indirectly, and as it has, moreover, later been proved directly by McLaren... (Bohr to H. Bohr, 23 October 1911). Bohr's hope of having his doctoral thesis published in the Transactions of the Cambridge Philosophical Society did not materialize. After waiting for several months he received the request to cut it down to half of its size. At that time, in May 1912, he was occupied with other work, hence he did not do so.

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Bohr as well.10 Especially, Bohr concerned himself deeply with the structure of the atom. As Hevesy recalled many years later: 'When I first talked to him in 1912, he had quite clear[ly] the idea that the chemistry [i.e., the chemical properties of the atoms] resides in the outer electron arrangement, and mass and atomic charge in the nucleus It was a Sunday afternoon. I was at the house of Rutherford. Bohr was also present. I asked Rutherford: 'Alpha-particles clearly come from the nucleus, no doubt. But where do the beta-particles come from?' Rutherford answered, 'Ask Bohr.' Bohr was present and with no difficulty answered that electrons involved in transmutation processes come from the nucleus, and all other electrons come from the exterior of the atom' (Hevesy, AHQP Interview, 25 May 1962, p. 3). It was in the summer of 1912 that Niels Bohr finished the first draft of a paper on the constitution of atoms and molecules, in which he worked out his ideas. 11 In his memoir, Bohr started out from the 'Atom-model proposed by Professor Rutherford in order to explain the 'big scattering' of a-particles,' which he described as follows: 'The atoms consist of a positive charge concentrated in a point (in an extension very small compared with the dimensions of the atoms) surrounded by a system of electrons, which total charge is equal to that of the positive 'kern' [nucleus]; the kern is also assumed to be the seat of the mass of the atom' (Bohr, 1963, p. XXI). He immediately noticed that 'in such an atom there can be no equilibrium configuration, without motion of the electrons' and he set out to discuss the rotations of n electrons moving in a ring around the nucleus, finding that it had no 'stability in the ordinary mechanical sense,' because 'a ring, if only the strength of the central charge and the number of electrons in the ring are given, can rotate with an infinitely great number of different times of rotation, according to the assumed different radius of the ring; and there seems to be nothing (on account of the instability) to allow from mechanical considerations to discriminate between the different radii and times of vibration' (Bohr, 1963, pp. XXII-XXIII). At that point he had to make an assumption, which allowed him to fix the dimensions of the electron ring in the atom. In particular, he stated: 10

George Charles de Hevesy (or Georg von Hevesy) was born on 1 August 1885 at Budapest. He studied chemistry at the Technische Hochschule, Berlin, and at t h e University of Freiburg, where he obtained his doctorate in 1908. Then he went to E.T.H., Zurich, as assistant to Richard Lorenz (1908-1910), and to Karlsruhe as assistant to Fritz Haber. On Haber's suggestion he studied electron emission from liquid sodium alloys, and he was sent to Manchester in 1912. From Manchester, Hevesy returned t o the University of Budapest as lecturer (1913-1914) and, after his military service during World War I, as professor (1918-1920). In 1920 he joined Bohr's Institute of Theoretical Physics in Copenhagen as Associate; in 1926 he became Professor of Physical Chemistry at the University of Freiburg and later, after 1934, Professor at the University of Copenhagen. From 1943 to 1956 he worked at the Research Institute for Organic Chemistry in the University of Stockholm. Hevesy received the Nobel Prize in Chemistry for 1944 for his discovery of the use of radioactive traces in the study of chemical processes. He died in Freiburg on 5 July 1966. ^Bohr's manuscript, six carefully written and numbered sheets in Danish — which Bohr later entitled 'First draft of considerations contained in the paper 'On the Constitution of Atoms and Molecules' (written up to show these considerations to Prof. Rutherford) (June or July 1912)' — is reproduced in Bohr, 1963, pp. XXI-XXVIII. Bohr sent this memoir to Rutherford, accompanied by a letter, dated 6 July 1912.

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This hypothesis is: that there for any stable ring (any ring occurring in the natural atom) will be a definite ratio between the kinetic energy of an electron in the ring and the time of radiation. This hypothesis, for which there will be no attempt of a mechanical foundation (as it seems hopeless), is chosen as the only one which seems to offer a possibility of an explanation of the whole group of experimental results, which gather about and seems to confirm conceptions of the mechanisms of radiation as the ones proposed by Planck and Einstein. (Bohr, 1963, p. XXIII)12 With the above hypothesis Bohr evidently referred to the quantum hypothesis of Arnold Sommerfeld, which the latter had presented at the Solvay Conference of 1911. 13 Bohr now believed that he could add further strong support to this hypothesis from four empirical 'facts:' the periodic law of atomic volumes; Richard Whiddington's relation between the minimum velocity of cathode rays, which were able to excite characteristic X-rays, and the atomic weights of the substances (Whiddington, 1911); William Henry Bragg's conclusion that the stopping power of different metals for a-rays was proportional to the square root of their atomic weights (W.H. Bragg, 1905); and the heats of chemical combinations of compounds. As for the second of these 'facts,' a theoretical investigation seemed to be relevant, which Bohr was just about to complete. In this work he showed that the energy loss of charged particles, especially a-particles, in matter might be ascribed analogously to the dispersion of light and, by comparison with empirical data, he derived two important consequences for atomic theory: first, the hydrogen atom contains one electron, the helium atom two, and the lighter elements about ^A electrons, A being their atomic weight; second, the absorption of a-particles per unit area of equal weight of matter decreases for elements of increasing weight (Bohr, 1913a).14 Bohr dealt further with such considerations in his memoir on atomic structure. Especially, he calculated the stability of electrons in rings; he found that a ring Bohr had become interested in quantum theory in late 1911. Thus he wrote to his friend Carl Wilhelm Oseen on 1 December 1911: 'I am at the moment very enthusiastic about the quantum theory (I mean its experimental side), but I am still not sure this is not due to my ignorance. I can say the same, in a far higher degree, about my relation to the theory of magnetons. I very much look forward to trying to get all these things straight next term.' (This part of the letter is quoted in Heilborn and Kuhn, 1969, p. 230.) 1 Since Rutherford attended the Solvay Conference, he brought back copies of the reports to England. In his first paper on atomic constitution Bohr quoted the French version of the proceedings of the Solvay Conference (Langevin and de Broglie, 1912; see Bohr, 1913b, p . 2, footnote t ) . 14 Bohr's interest in this problem had been aroused by a paper of Charles Galton Darwin, who had calculated a curve fitting the d a t a on the absorption of a-rays rather well (Darwin, 1912). Darwin's paper appeared in June 1912, and Niels Bohr wrote about it to his brother: 'Things are not too bad at the moment; a few days ago I had a little idea about understanding the absorption of a-rays (it so happened, that a young mathematician here, C.G. Darwin (the grandson of the right Darwin) has just published a theory on this question and it seemed to me t h a t it was not on'y not quite right mathematically (this was however rather trifling) but quite unsatisfactory in its basic conception, and I have worked out a little theory about it, which, however modest, may perhaps throw some light over a few things concerning the structure of atoms' (Bohr to Harald Bohr, 12 June 1912; see Bohr, 1963, p. XVII). T h e two mathematical approximations of Darwin, which Bohr criticized, were: the neglect of the effect on the a-particles when they did not penetrate the atoms and the forces exerted on the atoms when a rapidly moving a-particle collided with an atom.

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containing up to seven electrons will be stable, but a ring with eight electrons possesses positive total energy, hence the eighth electron can leave the atom. He concluded: This, together with the fact that inner rings of electrons in Professor Rutherford's atommodel will have only very little influence (and always to the worse [i.e., towards an instability]) on the stability of outer rings, seems to offer a very strong indication of a possible explanation of the periodic law of the chemical properties of elements (the chemical properties is [sic] assumed to depend on the stability of the outermost ring, the 'valency electrons') by help of the atom-model in question. (Bohr, 1963, p. XXII)15 In the latter part of his memoir Bohr also calculated electron configurations in molecules; thus he proposed a model of the hydrogen molecule as consisting of two separated nuclei plus a ring, in which two electrons rotate around the axis connecting the nuclei (Bohr, 1963, pp. XXV-XXVII). It was in this part that Bohr introduced for the first time a 'special hypothesis,' i.e., E = K-v,

(1)

where E denotes the kinetic energy of an electron, v its frequency of rotation, and K a constant of the order of magnitude of Planck's constant. 16 During the second half of the year 1912 Niels Bohr made only little progress with his work on the constitution of atoms and molecules. In July he left Manchester and went back to Copenhagen, where he married Margarethe N0rlund on 1 August. 17 The young couple spent their honeymoon in England and Scotland and then returned home, where Niels had to give lectures at the University of Copenhagen. 18 But he also continued to work on atomic constitution. Thus he reported in a letter to Rutherford: 'I have made some progress with regard to the question of dispersion. The number of electrons in a Hydrogen- and a Helium-Atom calculated from the dispersion seems thus to work out nearer to respectively 1 and 2 if the forces acting on the electrons are assumed to vary inversely as the square of the distance, than if they, as in Drude's theory, are assumed to be of the elastic type' (Bohr to Rutherford, 4 November 1912, quoted in Bohr, 1963, p. XXXV). 19 On the other hand, Evidently, the above conclusions explained the periodic law of atomic volumes, i.e., the fact that the ratio of atomic weight over density is a periodic function of the position of the chemical elements in the periodic system (L. Meyer, 1870). 1 For the exact value of K, Heilbron and Kuhn — by using the numerical values given by Bohr in the memoir (Bohr, 1963, p. XXVII) and in his paper on the absorption of a-particles (Bohr, 1913a) — found the result 0.6h (Heilbron and Kuhn, 1969, pp. 250-251). 1 Margarethe N0rlund was the sister of the mathematician Niels Erik N0rlund, a good friend of Harald Bohr's. 18 Bohr was first appointed assistant to Martin Knudsen, the successor of his former teacher Christiansen, and he gave a series of lectures on the mechanical foundations of thermodynamics from 16 October to 18 December 1912. (See Bohr, 1963, p. XXXV.) Already in the letter accompanying the memoir to Rutherford, Bohr remarked that he had come across certain difficulties in explaining the results from dispersion: especially, the number of electrons, computed by Clive and Maude Cuthbertson (1910) were not integers but, e.g., 0.85 for hydrogen. Bohr drew attention to this difficulty also in his paper on the passage of a-particles through matter (Bohr, 1913a, p. 23).

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he became concerned again with the problem of stability. In order to devote full attention to the problem, he asked for a leave from his university duties and retired, in December 1912, with his wife to the countryside with the intention of writing a long and detailed paper on the constitution of atoms and molecules. At this time he also studied thoroughly the literature on that subject, especially the papers of John William Nicholson.20 A Christmas card to his brother Harald noted that 'one of us [i.e., Niels] would like to say that he thinks that Nicholson's theory is not incompatible with his own' (M. and N. Bohr, 23 December 1912, quoted in Bohr, 1972, p. 563), and about a month later he wrote a long letter concerning this matter to Rutherford, in which he also enclosed the final version of his a-particle paper. In this letter he remarked: 'The theory of Nicholson gives apparently results which are in striking disagreement with those I have obtained; and I therefore thought at first that the one or the other necessarily was altogether wrong. I have however now taken the following point of view' (Bohr to Rutherford, 31 January 1913, quoted in Bohr, 1963, p. XXXVII). Then he outlined the points of atomic theory in which he agreed with Nicholson and also those about which they disagreed. Both Nicholson and he, Bohr claimed, used Rutherford's model of atomic constitution, and in determining the dimensions and energy of atoms they both sought 'a basis in the relation between the energy and the frequency suggested by Planck's theory of radiation' (Bohr to Rutherford, ibid.). In contrast to Nicholson, however, Bohr assumed that these considerations just determined the permanent or natural state of atoms, i.e., the state of lowest energy, but Nicholson also treated atomic states of less stable character (i.e., the states of higher energy) in the same way. Thus Bohr, for the moment, just concentrated upon treating the atom in the ground state, even when he referred to the problem of spectral lines emitted by the atom. He took the point of view that the problem could not be fully discussed on the basis of the ordinary mechanics, but that one had to take into account Planck's theory of radiation as follows: If we thus assume that the systems considered [i.e., the atoms] are formed by a successive binding of the electrons by the nucleus until the whole system is neutral (compare the formation of a helium-atom from an a-particle), and further assume that the energy emitted as radiation by this binding is equal to Planck's constant multiplied by the frequency of rotation of the electron considered in its final orbit, we get results which seem to be in conformity with experiments. (Bohr to Hevesy, 7 February 1913, quoted in Bohr, 1963, pp. XXXII-XXXIII)21 These remarks show clearly that Bohr was not dealing yet with the calculation of the frequencies corresponding to the lines in the visible spectrum of an atom, again Bohr had met Nicholson earlier in Cambridge and discussed with him the problems of electron theory. About his meeting with Nicholson, Bohr had written to Oseen: 'I also had a discussion with Nicholson; he was extremely kind but I scarcely agree with him about much' (Bohr to Oseen, 1 December 1911; quoted in Heilbron and Kuhn, 1969, p. 258). 21 In a footnote to this sentence, Bohr remarked: 'The constant entering in the calculation [i.e., K in Eq. (112)] is not exactly equal to Planck's constant, but differs from it by a numerical factor, in conformity with what was to be expected from theoretical considerations' (Bohr to Hevesy, 7 February 1913).

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in striking contrast to Nicholson. However, only a month later he sent his new paper to Rutherford, whose 'first chapter is mainly dealing with the problem of emission of line-spectra, considered from the point of view sketched in my former letter [of 31 January 1913] to you' (Bohr to Rutherford, 6 March 1913). The new paper contained the full account of a complete theory of atomic constitution, including the concept of stationary states and the relation between the frequencies of the emitted spectral lines and the energy difference between stationary states. How had Bohr come to such far-reaching conclusions in such a short time? When Leon Rosenfled, one of Bohr's closest collaborators in later years, inquired about the motivation for the dramatic development, Bohr told him: 'As soon as I saw Balmer's formula the whole thing was immediately clear to me' (Bohr, 1963, p XXXIX). However, the actual process of arriving at the new results — going far beyond what Bohr had done previously — must have been much more involved. It is known that in those days of February 1913 Bohr had the opportunity of talking about spectroscopic questions with Hans Marius Hansen, who had worked earlier at Woldemar Voigt's Institute in Gottingen on the inverse Zeeman effect and knew spectroscopic data as well.22 From information obtained personally from Bohr, Rosenfeld later reconstructed the events as follows: Hansen asked Bohr how on his theory he would account for the spectral regularities. Bohr had not been interested so far in this aspect of the question, because he thought those spectra too complicated to give any clue to the structure of atomic systems. At this stage Hansen was able to contradict him by calling his attention to the great simplicity with which the spectral lines had been represented by Rydberg, who was more preoccupied to find formulae likely to express general laws than to achieve the greatest possible accuracy for single series. (Rosenfeld in Bohr, 1963, p. XL) By comparing the equation for the frequency of the Balmer lines, 23 i.e.,

—t-Q-i,),

(2)

and the formula for W, the total energy of the electron in a hydrogen atom, 7r 2 m e e 4 2tf 2

(3)

(m e and e are, respectively, the mass and charge of the electron, and K a constant). Equation (3) followed from mechanics and an application of Eq. (1). Bohr could have been brought to the idea that K might assume values that were integral multiples of Planck's constant. Then an interpretation of the right-hand side of Eq. (3) in 22 Hansen had spent the period from April 1911 to August 1912 in Gottingen. After his return to Copenhagen he wrote his doctoral thesis, which was accepted on 30 June 1913. (See Bohr, 1963, pp. XXXIX.) 23 According to Bohr's testimony, he found the Balmer formula for hydrogen in the book of Johannes Stark (1911, pp. 44-45). (See Bohr, AHQP Interview, also quoted in Hermann, 1969, p. 172, and Heilborn and Kuhn, 1969, p. 265.)

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terms of energy differences might indeed have been evident. 24 Still, on the way to arriving at this result, Bohr had to establish an intermediate step: that is, he had to assume that there was a discrete set of stable or stationary states in an atom, and that these states could be calculated in accordance with ordinary mechanics supplemented by certain quantum rules. Of course, indications of the existence of stationary states could be found in the work of Johannes Stark and also in his book on atomic dynamics (Stark, 1911) which Bohr consulted. However, it is more likely that he found stimulation in the papers of Nicholson; especially in the second paper on the constitution of the solar corona, Nicholson had drawn attention to the fact that the angular momentum of an electron ring might rise and fall only by integral multiples of h/2n, and that this change also occurred in connection with the process of emission of spectral lines (Nicholson, 1912b, p. 679). All Bohr had to do was to take over Nicholson's assumption concerning the discreteness of angular momentum and apply it systematically to determine all possible states of the hydrogen atom; this procedure would then yield Eq. (3), with K assuming the 25 values ^nh,n = 1,2,3 The relation between the energy differences of the stationary states, i.e., of states with different values of K, and the frequencies v of the emitted spectral lines then followed by assuming that the energy loss of the atom was radiated away in an energy-quantum hv.26 On 6 March 1913 Bohr sent the 'first chapter' of his work on the constitution of atoms and molecules to Rutherford and announced that further chapters would follow within a few weeks. In the following weeks he made a few additions and alterations in the first chapter, which became part one of a trilogy: part one, signed 5 April and containing mainly the theory of the one-electron atom and its line spectra, appeared in the July issue of Philosophical Magazine (Bohr, 1913b); part two, dealing with systems having only one nucleus, was sent off to Rutherford on 10 June and appeared in September (Bohr, 1913c); and part three, on systems of several nuclei (i.e., on molecules), was ready by August and appeared in the November issue of the same journal (Bohr, 1913e).27 The first paper opened with 24 Rosenfeld argued that Bohr had probably proceeded in this way in order t o arrive at his theory of the Balmer formula, (see Bohr, 1963, pp. XL-XLI.) 25 I t should be noted that Bohr, in his paper, explicitly quoted Nicholson's above-mentioned assumption (Bohr, 1913b, p. 15, footnote). He also knew that in an atom, besides the normal state, or the state of lowest energy, other states exist. (See Bohr to Rutherford, 31 January 1913.) 26 T h i s idea may be found already in Einstein's light-quantum paper of 1905 and in several papers of Stark; it appeared again in several papers of Thomson at that time (e.g., in Thomson, 1912, p p . 455-456). It is most likely that Bohr took it over from Thomson, whose works he studied carefully. 27 Concerning the dates and procedure of submission of Bohr's three papers on the constitution of atoms and molecules, we may refer to Rosenfeld's introduction in Bohr, 1963, pp. XLIII-XLVI. Rutherford critized originally the length of the first paper. 'I really think it desirable that you should abbreviate some of the discussions to bring it within more reasonable compass,' he wrote to Bohr and added, 'As you know, it is the custom in England to put things very shortly and tersely in contrast to the German method, where it appears to be a virtue to be as long-winded as possible' (Rutherford to Bohr, 25 March 1913). Bohr, however, went to Manchester to fight for every word of his manuscript and he succeeded in doing so.

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general considerations concerning the binding of electrons in an atom and also stated the principal assumptions used in the calculations, namely, the existence of stationary states and the frequency condition: '(1) That the dynamical equilibrium of the [atomic and molecular] systems in the stationary states can be discussed by help of the ordinary mechanics, while the passing of the systems between stationary states cannot be treated on that basis. (2) That the latter process followed by the emission of a homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by Planck's theory' (Bohr, 1913b, p. 7). Bohr calculated the energy of the electron in the electric field of a nucleus with charge Z\e\ by assuming the equality of the attractive force and the centrifugal force on the elliptical orbit with a major-axis 2a. Thus he obtained equations expressing the frequency of revolution UJ and the quantity a (which in the case of a circular orbit is the radius) in terms of the (negative) energy W and the parameters of the electron, that is, V2 \W\3/2 u> = — i I ' •K Ze ^me

, and

n

Ze2 2a=—y. \W\

,A. 4

In order to find the expression for the energy of the stationary states he used the quantum condition \W\=nh^ (n = 1,2,3,...), (5) which he motivated by saying: 'If we assume that the radiation emitted is homogeneous, the second assumption concerning the frequency of the radiation suggests itself, since the frequency of revolution of the electron at the beginning of the emission is 0' (Bohr, 1913b, p. 5). 28 With the help of the above equations, he finally obtained the energy states as W

" =

2n2mee'iZ2 al2

, „ „ . (n = 1,2,3,...).

... (6)

From this, by substituting the empirical values for the mass and charge of the electron, me and e, and for Planck's constant h, respectively, he obtained an ionization energy | Wi| of 13 eV for the hydrogen atom (Z — 1) and a diameter 2a\ (in the lowest state) of 1.1 x 1 0 - 8 cm. Then he applied Planck's theory to determine the frequency of the homogeneous radiation emitted by the hydrogen atom in the course of a transition from a state n\ to a state n^, arriving at the result

„_ \(Wn, - W . , ) - R (4j - 4 [ ) .

(7)

In Eq. (7) the constant R (= 2n2mee4/h3), which was called Rydberg's constant, assumed the value 3.1 x 10 15 (if the mass was given in grams, the charge in electrostatic 28 Evidently, Bohr hinted here at Planck's second quantum hypothesis of 1911, which he knew from Planck's report at the Solvay Conference.

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units, and Planck's constant in erg x s). 'The agreement between the theoretical and observed values,' Bohr noted, 'is inside the uncertainty due to experimental errors in the constants entering in the expression for the theoretical value' (Bohr, 1913b, p. 9). Equation (7) described, of course, the Balmer spectrum for hydrogen (for n2 = 2 and rii = 3,4,5,...) and the Paschen series of the same atom (for n2 = 3 and n\ = 4 , 5 , 6 , . . . ) , and predicted an ultraviolet series (for n2 = 1 and n\ = 2,3,4,...) which was found a year later by Theodore Lyman (1914). 29 In addition, Bohr noticed that on the basis of his theory, a series discovered some time earlier in the spectrum of the star £-Puppis (Pickering, 1896, 1897) and another found more recently by Alfred Fowler in a mixture of hydrogen and helium, both of which had been ascribed to hydrogen, were actually emitted by the ionized helium atom. 30 Bohr knew the importance of these results and wrote to Rutherford, already in his letter of 6 March 1913, that he should request Alfred Fowler to check his helium hypothesis. Rutherford, who was very interested in this experimental consequence, had Evan Jenkin Evans in his Manchester laboratory perform the experiments, which confirmed Bohr's conclusion (Evans, 1913). To Evans' letter to Nature, dated 11 August 1913, Fowler replied in another letter, dated 13 September, that 'Dr. Bohr's theory does not at present seem to me to give much evidence to helium, as the origin of lines in question' (Fowler, 1913a, p. 96). Fowler's main argument was that Bohr's frequency formula for ionized helium did not describe the data at all better than the old Balmer-Rydberg frequency formula with half-integral numbers. Upon this criticism Bohr himself answered immediately in a letter to Nature, dated 8 October (Bohr, 1913d). In it he refined his theory by introducing explicitly the motion of the atomic nucleus; thus the Rydberg constant in the case of hydrogen was actually 4 3 4 3 RH = 27rm e e //i • ( M H / ( M H + me)) rather than 27rm e e //i (which corresponded to the approximation of an infinitely heavy hydrogen nucleus, i.e., Mn = oo), and 9

Bohr also discussed the conditions under which the spectral lines show up (Bohr, 1913b, pp. 9 10). Thus he remarked, in particular, that in celestial bodies more lines of the Balmer series should show up from the following argument: for the higher states the diameter of the hydrogen atom increases rapidly, thus for « i = 12 it would be 1.6 x 1 0 - 6 cm, and for n\ = 33 it would be 1.2 x 10~ 5 cm. The latter dimension can only be achieved for low pressures. Under normal conditions, however, the yet higher Balmer lines might appear in the absorption spectrum. 3u I f he put R in the case of helium equal to 87T 2 m e e 4 //i 3 and took ni = 3, he obtained Fowler's series, and with n^ — 4 he got Pickering's series. Formerly these series had been described by the formula, V = R\—L—

-_J— [K/2)2 «/2)2j '

R being the Ryderg constant for hydrogen. Bohr wrote instead the equation 87r2mee4

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for helium he did not have to take four times the constant for hydrogen, but -n-He =

8ir 2 m e e 4 To h?

MHe ; M He + me

• -r-f

,

,„,

\p)

where Mn e is the mass of the helium nucleus. With it he described ten lines, which Fowler had attributed to three different 'helium' series, by one helium formula, predicating at the same time several hitherto unobserved lines. 31 Upon this development Fowler withdrew his objection and, in a further letter to Nature, wrote: 'I am glad to have elicited this interesting communication from Dr. Bohr, and I readily admit that the more exact form of his equation given above is in close accordance with the observations of the lines in question' (Fowler, 1913b, p. 232). While the spectra of one-electron atoms followed very easily from the new theory, much less could be said about other atoms. 32 Bohr investigated the theory of many-electron atoms, especially their stability, in part two (Bohr, 1913c). There he achieved a few qualitative results: he concluded, also motivated by the chemical properties of atoms and their organization in the periodic system of elements, that there could be only two, four or eight electrons in the inner rings surrounding the atomic nucleus. Moreover, he claimed that characteristic X-radiation arises when an electron, which is removed from an inner ring, returns to the ground state; the velocity of a cathode ray particle, should it be able to excite characteristic Xradiation of atoms with nuclear charge Z\e\, must exceed the value (2Tre2/h)Z (or 2.1 x 10SZ cm/s). This result agreed with Whiddington's earlier observations if one substituted for Z about half of the value of the atomic weights of substances. Finally, in part three, Bohr treated the structure of molecules and their stability, taking as examples the hydrogen molecule and a molecule consisting of two helium atoms (Bohr, 1913e). Bohr's theory of atomic structure was soon to be discussed publicly. Rutherford proposed that his former Danish disciple be invited to participate in the discussion on radiation, which was planned to be held at the 83rd Meeting of the British Association for the Advancement of Science in Birmingham. Bohr indeed went to Birmingham and attended the discussion on radiation theory on 12 September 1913. 33 It opened with a report by James Hopwood Jeans, who gave 'a masterly and concise' review of the problems starting from blackbody radiation and its theoretical analysis by Poincare, going over the most important quantum phenomena on to 'the recent work by Dr. Bohr, who has arrived at a convincing and brilliant explanation of the laws of spectral series.' (See the report on 'Physics at the British Association' 31 T h e s e lines (having wavelengths of 6560.3 A, 4859.5 A, and 4338.9 A, respectively) were subsequently found by Evans (1915), A. Fowler (1914) and Paschen (1916). Basically Bohr could only calculate the lines due to transitions of an electron from a large orbit. Then he found the formula v = F i ( n i ) — F2{n2) with limn->-oo F(n) = 27r 2 m e e 4 //i 3 ; that is, the same constant appeared in all spectra in agreement with the empirical d a t a (Bohr, 1913b, p . 12). Among the distinguished foreign scientists, who attended the Birmingham Meeting, were Madame Curie, Hendrik Lorentz, Ernest Pringsheim, Svante Arrhenius and Robert W. Wood. Lorentz and Pringsheim participated in the disucssion on radiation.

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in Nature 92, issue of 6 November 1913, especially p. 305.) Niels Bohr spoke after Jeans, Lorentz and Pringsheim. The report in Nature described his talk as follows: His [Bohr's] work had been referred to by previous speakers, and he gave a short explanar tion of his atom. His scheme of the hydrogen atom assumes several stationary states for the atom, and the passage from one state to another involves the yielding of one quantum.... Professor Lorentz intervened to ask how the Bohr atom was mechanically accounted for. Dr. Bohr acknowledged that this part of his theory was not complete, but the quantum theory being accepted, some sort of scheme of the kind suggested was necessary. (Nature, Volume 92, p. 306) After Bohr, other speakers, including Joseph Larmor, J.J Thomson and Oliver Lodge, addressed the Association on other problems of radiation and quantum theory. Altogether Bohr could be quite content with the reception of his theory, especially by Jeans, who referred to it as 'a most ingenious and suggestive... convincing explanation of the laws of spectral lines' (Jeans, 1914a, p. 379). The response from Thomson, on the other hand, was far less encouraging; he did not believe at all in the new quantum theory of the atom, especially since he had just presented a day earlier a new model of the atom, which — as he claimed — described many observed features (even the photoelectric effect) without the help of a quantum hypothesis (Thomson, 1913b); also, on another occasion at the Birmingham meeting he and Bohr had a little argument about the correct interpretation of a positive ion observed in experiments. 34 But his negative attitude, though it did not make Bohr happy, did not really hurt him. 'Thomson went out of physics,' he remarked later, 'and he never came into it really again, because he objected to the only positive way' (Bohr, AHQP Interview, 11 January 1962, p. 10). What counted most was that the scientific public got to know about his (Bohr's) model of atomic structure, which was the quantum theoretical completion of Rutherford's atom. And soon, strong support would be registered in favour of his theory. This support came from Henry Gwyn Jeffreys Moseley, a member of Rutherford's Manchester group. 35 The analysis of the radioactive emanations in the first decade of the twentieth century had revealed the nature of a- and /3-rays. This recognition, together with the investigation of the chemical properties of the members of the radioactive families — of the uranium as well as the thorium family — led to the so-called 'radioactive displacement law:' it stated that a daughter element arising from j5decay was located one place to the right of the parent in the periodic table (Fajans, 1913a,b; Soddy, 1913). Even before the displacement law was formulated, Antonius Thomson presented his new model of the atom in some detail also at the second Solvay Conference, which was held in Brussels from 27 to 31 October 1913. On this occasion Bohr's theory was not discussed at all. •"H.G.J. Moseley was born in Weymouth, England, on 23 November 1887. He was educated at Trinity College, Oxford, where he received his M.A. in 1910. Then he went to the University of Manchester as Demonstrator (1910-1912) and Lecturer (1911-1912). In late 1913 he joined the Clarendon Laboratory, Oxford, but after the outbreak of World War I he did military service with the Royal Engineers. Moseley was killed in action during the British Gallipoli Campaign at the Dardanelles on 10 August 1915.

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Johannes van den Broek had, in an article entitled 'Die Radioelemente, das periodische System und die Konstitution der Atome' ('The Radio-Elements, the Periodic System and the Constitution of Atoms'), organized the decay substances according to their weights and chemical properties in a scheme, which he related to the known periodic system of elements (van den Broek, 1913). Referring to the available models of atomic constitution, especially the models of Thomson and Rutherford, and to the empirically established hypothesis that the atomic weight of a substance was proportional to the number of what he called the 'intra-atomic charges,' i.e., the electrons, he had arranged the elements in the periodic system according to increasing atomic weight, with an average difference of two between the weights of two successive elements. 36 The main point of his organization was, however, that he attributed to each chemical element a number Z, starting from 1 for hydrogen to 118 for uranium, which he believed to be identical with the number of intra-atomic charges. In Manchester, where the experiments on the scattering of a-rays from matter had yielded reliable information about the nuclear charges of atoms, which, within experimental errors, should be identical to the ones assumed by van den Broek, his ideas were immediately accepted. 37 The number of intra-atomic charges, especially their binding, seemed to be connected with the emission of the characteristic X-rays, which Barkla and Sadler had discovered earlier in 1908 (see Thomson, 1912a). Hence Moseley, in summer 1913, began to investigate systematically this characteristic radiation of different emitting substances. 38 After a few months of Van den Broek had discussed the periodic system, especially the position of the radioactive elements in it, earlier (van den Broek, 1911). He had noticed already then t h a t the average difference in atomic weights of two successive elements in the periodic system was two. 37 T h u s C.G. Darwin recalled later: At this stage [i.e., after the establishment of Rutherford's atomic model in 1911] the whole Manchester laboratory believed in the undoubted existence of the atomic number, defined as the nuclear charge. The idea of what the number meant was perfectly precise, but the value of it could not yet be known precisely To those working at the time in Manchester t h e work on nuclear scattering made the idea quite convincing and it was fully accepted there, though in fact the proposal of atomic number was first published rather later elsewhere [i.e., by van den Broek]. (Darwin, 1955, pp. 6-7) Also Bohr in Copenhagen agreed more or less fully with van den Broek, as he wrote in a letter: 'When I returned home I found two papers in the Physikalische Zeitschrift. One was by [George de] Hevesy dealing with the radioactive properties of atoms; its results agreed totally with my ideas. The other was by van den Broek, who arrived in an empirical way at the same conceptions concerning the periodic system of elements as I did deductively with my theoretical conceptions' (Bohr to Oseen, 5 February 1913; German translation in Hoyer, 1974, p. 139). 38 C . G . Darwin, who assisted Moseley in the early stages of the work, reported later: He [Moseley] departed from the laboratory's exclusive devotion to radioactivity, and took up and developed some important points in the quite new science of X-ray spectroscopy. He then applied his technique to the discovery of the wavelengths of the characteristic X-rays, which had been studied by Barkla several years before. Working day and night by himself with a very characteristic excess of energy, and in spite of constantly pulling his apparatus t o pieces in order to improve it, he quickly got his main results. (Darwin, 1955, p. 8) Moseley used a method, in which the secondary characteristic radiation was reflected (according to the procedure of t h e Braggs) by a selected crystal; he observed the third-order pattern from a potassium ferro-cyanide crystal. This method had the advantage that the exposure time of the photoplates was comaparatively short, only about 5 min.

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trying different experimental methods, in which he employed both the ionization method and the photographic method to determine the wavelengths of the emitted X-rays, Moseley was able to photograph the principal lines of the X-ray spectra of most elements within a short time. In a letter, dated 16 November 1913, he reported to Niels Bohr about his results concerning the K-lines of elements from calcium to zinc. He described these results as 'extremely simple and largely what you would expect' (Moseley to Bohr, 16 November 1913). That is: each element provided two main lines, denoted by Ka and Kp, of which the a-lines were about five times more intense; further, the frequencies of the /3-lines were about 10% higher than those of the a-lines, their ratio having nearly, though not exactly, a constant value. Moseley put together his results in a frequency formula for the a lines,

va=VQ{z-i?(±-2-±y

(9)

which he also reported to Bohr. In Eq. (9), UQ denoted a constant (for all elements) which appeared in Rydberg's formulae. In his letter he gave a table containing the values derived for the atomic number Z. Especially, he drew attention to the finding that the correct sequence in the periodic table should be Fe-Co-Ni rather than Fe-Ni-Co (which appeared to follow from the higher atomic weight of cobalt), in agreement with the chemical properties. Thus he remarked in his published paper: We can confidently predict that in the few cases in which the order of the atomic weights A clashes with the chemical order of the periodic system, the chemical properties are governed by Z [the atomic number, which on account of the Rutherford-Bohr atomic model could be identified with the number of electrons in the atom]; while A is probably a complicated function of Z. (Moseley, 1913, p. 1031) Moseley continued to work on X-ray spectra in Oxford, where he had moved in late 1913, and completed his programme of measuring the frequencies of most elements. He published the results on the characteristic X-ray lines of another thirty pure substances, running from aluminum to gold (Moseley, 1914b). Because of experimental difficulties (especially, the disturbing influence of the continuous X-radiation) he was not able to observe the hardest, i.e., if-lines, for elements beyond silver; instead he had to register the softer L-lines. For the frequency of the L Q -lines, he obtained the equation 39 va = VQ(Z - b)2 (^

- i )

with

6-7.4.

(10)

Moseley's results — especially Eq. (9) — evidently spoke clearly in favour of the atomic model of Rutherford and Bohr. Hence Rutherford, in a letter of 6 December to Nature drew attention to them. In particular, he wrote: The original suggestion of van den Broek that the charge of the nucleus is equal to the atomic number and not to half the atomic weight seems to me very promising. This idea Moseley, like Bohr, denoted the atomic number by N. We have replaced it here by the later notation Z.

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has already been used by Bohr in his theory of the constitution of atoms. The strongest and most convincing evidence in support of this hypothesis will be found in a paper by Moseley in the Philosophical Magazine of this month. He there shows that the frequency of the X-radiations from a number of elements can be simply explained if the number of unit charges on the nucleus is equal to the atomic number. It would appear that the charge of the nucleus is the fundamental constant which determines the physical and chemical properties of the atom, while the atomic weight, although it approximately follows the order of the nuclear charge, is probably a complicated function of the latter depending on the detailed structure of the nucleus. (Rutherford, 1913, p. 423) T h u s Moseley h a d given t h e first detailed proof of t h e nuclear atom, confirming at t h e same t i m e Bohr's use of the q u a n t u m theory. 4 0 While Moseley's work on the characteristic X-ray spectra was motivated by the existence of t h e Rutherford-Bohr atomic model, another experimental investigation, which provided a strong evidence for one of B o h r ' s basic assumptions — the existence of stationary states in a t o m s a n d molecules a n d the frequency condition — was not connected with it. In spring 1912 Ernst J o h a n n Gehrcke a n d Rudolf Seeliger h a d found t h a t the light accompanying c a t h o d e rays moving in a t u b e filled with gas — for instance, nitrogen or mercury vapour — changed its colour towards longer wavelengths when t h e velocity of the electrons was reduced by a decelerating potential (Gehrcke a n d Seeliger, 1912). Their observation was clarified by the results, which J a m e s Franck a n d G u s t a v Hertz obtained two years later after a long series of experiments, stimulated by t h e earlier work of Emil W a r b u r g — Pranck's former teacher — on gaseous discharges. 4 1 Franck a n d Hertz h a d studied t h e con40 Moseley's work on the characteristic X-rays of elements, the importance of which was quickly recognized, stimulated a lively discussion on atomic models in England. Thus, in a letter dated

28 December 1913 and published in the issue of Nature of 1 January 1914, FVederick Alexander Lindemann stated that atomic theories including the quantum concept, but differing from Bohr's might explain the d a t a as well (Lindemann, 1914a). Bohr replied immediately that the dimensional arguments, used by Lindemann, did not suffice to calculate the spectrum (Bohr, 1914a). Moseley supported Bohr with a similar argument (Moseley, 1914a); but Lindemann reemphasized his point in a further letter, dated 25 January, and published in the issue of Nature of 5 February (Lindemann, 1914b). Again he claimed that Bohr's connection between atomic dimensions and the quantum of action was not as unique as he (Bohr) had assumed. Although Bohr did not reply publicly to this argument, he stated in a note — which has been preserved in the Bohr Archives in Copenhagen — t h a t Lindemann's example did not represent the physical situation properly (see Hoyer, 1974, pp. 202-203). Also John William Nicholson participated in the discussion; while he found Bohr's models of hydrogen and ionized helium attractive, he argued against the models of lithium, beryllium and boron — i.e., against coplanar rings assumed by Bohr in Part II (Bohr, 1913c) — on dynamical grounds (Nicholson, 1914a,b). Van den Broek, on the other hand, quoted facts in favour of Bohr's and his own ideas (van den Broek, 1914). 41 J a m e s Franck was born on 26 August 1882 at Hamburg. After attending Wilhelms-Gymnasium he left his hometown to study mainly chemistry at the University of Heidelberg (1901-1902) and physics at the University of Berlin. He received his doctorate under Emil Warburg in 1906, after which he became an assistant at the University of Frankfurt, and shortly afterwards an assistant to Heinrich Rubens at the University of Berlin (1906-1911). In 1911 he received his Habituation in physics and in 1916 he was appointed extraordinary (i.e., associate) professor in Berlin. Gustav Ludwig Hertz, a nephew of Heinrich Hertz, also come from Hamburg, where he was born on 22 July 1878. He studied physics at the Universities of Gottingen (1906-1907), Munich (19071908) and Berlin (1908-1911), where he received his doctorate under Ruben in 1911. He then served an assistant to Rubens (1911-1914). After his military duty in World War I (1914-1917),

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nection between the ionization potential — i.e., the potential difference which had to be applied to an electron in order to enable it to ionize atoms in a collision — of a given gas and quantum theory (Pranck and Hertz, 1911). Thus they obtained the values of the ionization potentials of several gases, such as helium, neon, hydrogen and oxygen, according to a method of Philipp Lenard (Pranck and Hertz, 1913a). Later they investigated specifically the collisions of electrons — which, after passing through a potential difference, possessed a definite, controlled kinetic energy — with atoms of inert gases (Franck and Hertz, 1913b,c). Continuing along these lines Franck and Hertz arrived at unexpected results, which Hertz presented at a meeting of the German Physical Society in Berlin on 24 April 1914. The report, which was published on 30 May in the Verhandlungen of the Society, closed with the following summary: 1. It will be shown that the electrons in mercury vapour undergo elastic collisions with the molecules until they obtain a critical velocity. 2. A method is described to measure this critical velocity up to an accuracy of 0:1 volts. This velocity is equivalent to the one obtained by electrons that have gone through a potential difference of 4.9 volts. 3. It will be shown that the energy of a 4.9 volt-beam [of electrons] equals the energy-quantum of the mercury lines [having wavelength] 253.6 fifi. 4. Reasons are given for the conclusion that a fraction of the collisions, in which the 4.9 volt-rays lose their energy, leads to ionization; hence 4.9 volts might represent the ionization potential of mercury vapour. Another part of the collision seems to stimulate light emission, and we suppose that it is connected with the emission of the line 253.6 /i/i. (Franck and Hertz, 1914a, p. 467) In the footnote appended to the last statement Franck and Hertz added: 'Meanwhile we have performed experiments testing the excitation of the 253.6 /i/x-line by 4.9 volt-rays, and have found a positive result. We shall report about it in the following issue of these Verhandlungen.' The experimental apparatus used by Franck and Hertz was very simple. Electrons, emitted from a heated platinum wire, were accelerated by a potential difference applied between the wire and a symmetrically placed cylindrical net (also of platinum) surrounding the wire (so that the wire formed the axis of the cylinder); outside the cylindrical net was fixed a cylinder of platinum foil, connected to a galvanometer, so that the current of the electrons falling on it could be measured; a tube of glass, which could be filled with any gas or vapour, covered the entire equipment. In the case of mercury vapour Franck and Hertz observed that the registered electron current increased with rising potential difference until a value of 4.9 V was reached, then it dropped suddenly; by further increasing the potential difference beyond 4.9 V the current again increased, but dropped at about 9.8 V; and being seriously wounded, he returned to Berlin and became Privatdozent there in 1917. In 1920 he joined the physics laboratory of the Philips Incandescent Lamp Factory in Eindhoven as a research physicist. In 1925 he was appointed Professor of Physics and Director of the Physics Institute at the University of Halle; three year later he moved to a similar position at t h e Technische Hochschule, Berlin. He resigned this position in 1935 and became director of the research laboratory of the Siemens Company in Berlin. Prom 1945 to 1954 he worked in the Soviet Union; then he was appointed Professor of Physics at the University of Leipzig (1954-1961). He died in Berlin on 30 October 1975.

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these sudden fall-offs occurred a t all integral multiples of t h e critical 4.9 V. Pranck a n d Hertz found a similar, t h o u g h less pronounced, behaviour if their a p p a r a t u s was filled with helium; there t h e critical potential difference was a b o u t 21 V. These phenomena, together with t h e occurence of t h e characteristic radiation of 2536 A units when t h e critical potential difference was applied t o t h e electrons in a mercury t u b e , was interpreted by Pranck and Hertz as follows: 1. The electrons are reflected by the mercury atoms without losing energy as long as their kinetic energy is below the amount hi/, where v is the frequency of the resonance line. 2. As soon as the kinetic energy of an electron reaches the value /ii/, this energy quantum will be transferred in one of the following collisions to the spectrum of frequency v [that arises from] within the atom. 3. The energy transferred is partly used for ionization, and partly it is emitted as light of frequency v. 4. The constant h calculated from these experiments is 6.59 x 10 - 2 7 erg • sec, with a possible error of 2 percent. (Pranck and Hertz, 1914b, p. 517) I n arriving a t t h e interpretation of their results Franck a n d Hertz referred explicitly to t h e ideas of Johannes Stark on t h e origin of discrete spectral lines. Stark h a d suggested several years earlier t h a t line spectra were created in connection with t h e process of ionization of a t o m s a n d molecules, a n d t h a t their frequency v was related to t h e ionization potential V t h r o u g h the quantum-theoretical equation fci/ = V ,

(11)

where h is P l a n c k ' s constant (Stark, 1908a, p 85; 1909c, p . 616). 4 2 Franck a n d Hertz stuck t o S t a r k ' s interpretation also in t h e following years, for they believed t o have shown conclusively t h a t t h e potential difference of 4.9 V led t o t h e ionization of mercury a t o m s . T h u s , in their review article in Physikalische Zeitschrift, after mentioning t h e empirical evidence for t h e ionization, t h e y stated: The applicability of quantum theory to ionization processes is essentially substantiated by these facts and [they added, that] it is not possible to make the assumption, as Bohr had done recently, that in helium the 20.5 volt-beams and in mercury the 4.9 volt-beams lead only to secondary ionization, such that the short-wavelength radiation [resulting from inelastic collisions] causes a photoelectric effect at the electrodes or at the impurities present in the gas. (Pranck and Hertz, 1916, p. 438) Bohr's different interpretation of t h e F r a n c k - H e r t z experiment h a d originated from a remark in his first paper on t h e constitution of a t o m s . After mentioning some 'instructive' calculations on the energy of ^-particles emitted from radioactive substances (Rutherford, 1912a, b), he h a d argued: These calculations strongly suggest that an electron of great velocity in passing through an atom and colliding with the electrons bound will lose energy in distinct finite quanta. As is immediately seen, this is very different from what we might expect if the result of the collisions was governed by the usual mechanical laws. The failure of the classical mechanics in such a problem might also be expected beforehand from the absence of 42

The ionization potential V, if multiplied by e, the charge of the electron, corresponds to the kinetic energy of the ionizing electron; it is 4.9 V for the mercury vapour.

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anything like equipartition of kinetic energy between free electrons and electrons bound in atoms. Prom the point of view of the 'mechanical' states we see, however, that the following assumption — which is in accord with the above analogy — might be able to account for the result of Rutherford's calculation and for the absence of equipartition of kinetic energy: two colliding electrons, bound or free, will, after the collision as well as before, be in mechanical states. Obviously, the introduction of such an assumption would not make any alteration necessary in the classical treatment of a collision between two free particles. But, considering a collision between a free and a bound electron, it would follow that the bound electron by the collision could not acquire a less amount of energy than the difference in energy corresponding to successive stationary states, and consequently that the free electron which collides with it could not lose a less amount. (Bohr, 1913 b, p. 19) According to this view t h e 4.9 V did not necessarily correspond t o the ionization potential of mercury. T h u s , in 1915, Bohr argued: Franck and Hertz assume that 4.9 volts corresponds to the energy necessary to remove an electron from the mercury atom, but it seems that their experiments may possibly be consistent with the assumption that this voltage corresponds only to the transition from the normal state to some other stationary state of the neutral atom. On the present theory we should expect that the value for the energy necessary to remove an electron from the mercury atom could be calculated from the limit of the single line series of Paschen, 1850, 1403, 1269 [Paschen, 1911]. For since mercury vapour absorbs light of wavelength 1850 [Stark, 1913a], the lines of this series as well as the line 2536 must correspond to a transition from the normal state of the atom to other stationary states of the neutral atom (see I, [i.e., Bohr, 1913b] p. 16). Such a calculation gives 10.5 volts for the ionization potential instead of 4.9 volts [McLennan and Henderson, 1915]. If the above considerations are correct it will be seen that Franck and Hertz's measurements give very strong support to the theory considered in this paper. If, on the other hand, the ionization potential of mercury should prove to be as low as assumed by Franck and Hertz, it would constitute a serious difficulty for the above interpretation of the Rydberg constant, at any rate for the mercury spectrum, since this spectrum contains lines of greater frequency than the line 2536. (Bohr, 1915b, pp. 410-411) A few years later Franck a n d Hertz finally agreed with B o h r ' s views a n d spoke, in a new review for Physikalische Zeitschrift, about the 'confirmation of Bohr's atomic theory by investigations of inelastic collisions between slow electrons and gas molecules' (Franck a n d Hertz, 1919, p . 132). 4 3 In 1925 they received t h e Nobel Prize in Physics for their discovery of t h e law governing the impact of a n electron u p o n a n a t o m , especially the verification of Bohr's hypotheses of the stationary s t a t e s and t h e frequency condition. T h o u g h it took some t i m e until B o h r ' s theory of atomic constitution was recognized by other scientists, Rutherford knew t h a t his former disciple h a d accomplished a n i m p o r t a n t piece of work. I n 1914, after Charles Galton Darwin was called t o war service, he invited Niels Bohr to succeed Darwin as Reader at the University of Manchester. Bohr accepted a n d joined Rutherford. At Manchester he continued ^ I n their new paper Franck and Hertz especially admitted that the ionization effect, which they had found to accompany the inelastic collisions of electrons with the mercury atoms at 4.9 V, had been proven to be a secondary photoelectric effect, as Bohr had assumed in 1915. (See van der Bijl, 1917.)

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t o extend his a t o m i c theory. J u s t two years later t h e University of C o p e n h a g e n invited him back t o take u p a new professorship of theoretical physics. References Bohr, N. (1906), Physics Prize Essay, submitted on 30 October 1906 to the Royal Danish Academy of Sciences and Letters (on the problem to investigate the vibrations of liquid jets, presented in February 1905), unpublished. Bohr, N. (1909), Phil. Trans. Roy. Soc. (London) A209, 281-317; reprinted Collected Works 1, pp. 29-80. Bohr, N. (1910), Proc. Roy. Soc. (London) A 8 4 , 395-403; reprinted in Collected Works 1, pp. 81-89. Bohr, N. (1911), Studied over Metallernes Elektrontheori, doctoral dissertation, University of Copenhagen: Thaning &: Appel; reprinted in Collected Works 1, pp. 167-290. English translation: Studies on the electron theory of metals, in Collected Works 1, pp. 291-395. Bohr, N. (1913a), Phil. Mag. (6) 25, 10-31. Bohr, N. (1913b), 'On the constitution of atoms and molecules' (Part I), Phil. Mag. (6) 26, 1-25, reprinted in Constitution of Atoms and Molecules (Bohr, 1963), pp. 1-25. Bohr, N. (1913c), Phil. Mag. (6) 26, 476-502. Bohr, N. (1913d), Nature 92, 231-232. Bohr, N. (1913e), Phil. Mag. (6) 26, 857-875. Bohr, N. (1914a), Nature 92, 553-554. Bohr, N. (1915b), Phil. Mag. (6) 30, 332-335. Bohr, N. (1963), On the Constitution of Atoms and Molecules: Papers of 1913 reprinted from the Philosophical Magazine with an Introduction by L. Rosenfeld, Copenhagen: Munskgaard, New York: W.A. Benjamin. Bohr, N. (1972), Collected Works, Volume 1: Early Work (J. Rud Nielsen, ed.), Amsterdam; North-Holland, New York: Elsevier. Bragg, W.H. (1905), Phil. Mag. (6) 10, 318-340. Cuthbertson, C , and M., Proc. Roy. Soc. (London) A 8 3 , 151-171. Darwin, C.G. (1912), Phil. Mag. (6) 23, 901-920. Darwin, C.G. (1955), The discovery of the atomic number (Niels Bohr 70th birthday Festschrift, ed. W. Pauli, London: Pergamon). Evans, E.J. (1913), Nature 92, 5. Fajans, K. (1913a,b), Phys. Zs. 14, 131-136; 14, 136-142). Franck, J., and Hertz, G. (1911), Verh. d. Deutsch. Phys. Ges. (2) 13, 967-971. Franck, J., and Hertz, G. (1913a,b,c), Verh. d. Deutsch. Phys. Ges. (2) 15, 34-44; 373-390; 613-620. Franck, J., and Hertz, G. (1914a,b), Verh. d. Demtsch. Phys. Ges. (2) 16, 457-467; 512-517. Franck, J., and Hertz, G. (1916), Phys. Zs. 17, 409-416. Franck, J., and Hertz, G. (1919), Phys. Z. 20, 132-143. Fowler, A. (1913a), Nature 92, 95-96. Fowler, A. (1913b), Nature 92, 232-233. Fowler, A. (1914), Proc. Roy. Soc. (London) A 9 0 , 426-430. Gehrcke, E., and Seeliger, R. (1912), Verh. d. Deutsch. Phys. Ges. (2) 14, 335-343. Heilbron, J.L., and Kuhn, T.S. (1969), The genesis of the Bohr atom, in Historical Studies in the Physical Sciences 1 (R. McCormmach, ed.), Princeton; Princeton University Press, pp. 211-290.

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Hoyer, U. (1974), Die Geschichte der Bohrschen Atomtheorie,Weinheim: Physik Verlag. Jeans, J. (1914a), Report on the 83rd Meeting of the British Association, Birmingham, 1913, London: John Murray. Langevin, P., and De Broglie, M. (Editors) (1912), La Theorie du Rayonnement et les Quanta-Rapports et Discussions de la Reunion tenue a Bruxelles, du 30 Octobre au 3 Novembre 1911, Paris: Gauthier-Villars. Lindemann, F.A. (1914a,b), Nature 92, 500-501; 92, 631. Lyman, T. (1914), Phys. Rev. (2) 3, 504-505. Meyer, L. (1870), Annalen der Chermie und Pharmacie, Supplement 7, 354-364. Moore, Ruth. (1966), Niels Bohr: The Man, His Science, and the World They Changed, New York: Alfred A. Knopf. Moseley, H.G.J. (1913), Phil. Mag. (6) 26, 1024-1034. Moseley, H.G.J. (1914a), Nature 92, 554. Moseley, H.G.J. (1914b), Phil. Mag. (6) 27, 703-713. Nicholson, J.W. (1912b), Monthly Notices Roy. Astr. Soc. (London) 72, 677-692. Nicholson, J.W. (1914a,b), Nature 92, 583-584; 92, 630. Paschen, F. (1916), Ann. d. Phys. (4) 50, 901-940. Pickering, E.C. (1896, 1897), Astrophysical Journal 4, 369-370 (1896); 5, 92-94 (1897). Rozental, S. (1967), Niels Bohr: His Life and Work as Seen by his Friends and Colleagues, Amsterdam: North-Holland. Rutherford, E. (1912a,b), Phil. Mag. (6) 24, 453-462; 893-894. Rutherford, E. (1913), Nature 92, 423. Soddy, F. (1913), Chemical News 107, 97-99. Stark, J. (1911), Prinzipien der Atomdynamik. II. Die elementare Strahlung, Leipzig: S. Hirzel Verlag. Stark, J. (1908a), Phys. Zs. 9, 85-94. Stark, J. (1909c), Phys. Zs. 10, 614-623. Thomson, J.J. (1912a), Phil. Mag. (6) 23, 449-457. Thomson, J.J. (1913b), Phil. Mag. (6) 26, 792-799. van den Broek, A. (1911), Phys. Zs. 12, 490-497. van den Broek, A. (1913), Phys. Zs. 14, 32-41. van den Broeck, A. (1914), Nature 9 3 , 241-243. Whiddington, R. (1911), Proc. Roy. Soc. (London) A 8 5 , 823-332.

11 Arnold Sommerfeld and A t o m s as Conditionally Periodic Systems*

Bohr's theory of t h e constitution of a t o m s and molecules received a mixed response on the Continent. His friends Carl Wilhelm Oseen from Uppsala and George de Hevesy from Vienna wrote enthusiastically a b o u t the progress brought a b o u t by it. Hevesy had even more encouraging news for Bohr, because he could report a b o u t a favourable opinion of Albert Einstein, w h o m he met a t t h e 85th Assembly of G e r m a n N a t u r a l Scientists (Naturforscherversammlung).1 He wrote: This afternoon I spoke with Einstein... then I asked him about his view on your theorie. He told me, it is a very interesting one, important one if it is right and so on and he had very similar ideas many years ago but had no pluck to develop it; I told him than that is established now with certainty that the Pickering-Fowler spectrum belongs to He. When he heard this he was extremely astonished and told me: 'Then the frequency of the light does not depend at all on the frequency of the electron' — (I understood him so??) And this is an enormous achievement [sic]. The theory of Bohr must be then wright. I can hardly tell you how pleased I have been and indeed hardly anything else could make me such a pleasure than this spontaneous judgement of Einstein.' (Hevesy to Bohr, 23 September 1913) However, other physicists reacted much less favourably to t h e theory. For example, when Bohr's theory (the first p a r t of Bohr's 1913 paper) was discussed shortly after its publication a t a joint physics colloquium of the University a n d the I n s t i t u t e of Technology (E.T.H.) at Zurich, M a x von Laue protested: 'This is all nonsense! Maxwell's equations are valid under all circumstances. A n electron in a circular orbit must emit radiation' (F. Tank to J a m m e r , 11 May 1964; quoted in J a m m e r , Lectures delivered at the Universite libre de Bruxelles, October 1977. Revised and enlarged version published in The Historical Development of Quantum Theory (with Helmut Rechenberg, Springer-Verlag New York, 1982, pp. 200-231). 1 The 85th Naturforscherversammlung took place in Vienna from 21 to 28 September 1913. Among the speakers were Einstein (who delivered a talk on the status of the gravitational problem on the morning of 23 September 1913), James Franck (on ionization by collisions), "Max von Laue and Walter Friedrich (on X-ray interferences), Johannes Stark (on electrical and optical changes of chemical atoms), Charles Glover Barkla (on characteristic X-rays), and Max Born (on application of the quantum theory to the law of Eotvos). (See Verh. d. Deutsch. Phys. Ges. (2) 15, No. 20, 30 October 1913, pp. 919-923.)

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1966, p. 86). 2 From Gottingen, Niels Bohr's brother Harald reported: 'People are very interested in your papers; but I have the impression that most of them — though not Hilbert — especially among the youngest, Born, Madelung, etc., do not believe in the objective correctness [of your ideas]; they find the assumptions 'bold' and 'full of fancy' (Harald Bohr to Niels Bohr, fall 1913). 3 Bohr had also sent the reprints of his three papers on the constitution of atoms and molecules to a number of selected scientists in order to learn their opinion about his work. Of these, especially Arthur Erich Haas, Heinrich Matthias Konen, Hantaro Nagaoka, and Arnold Sommerfeld congratulated him. 4 In any case, Bohr's work on atomic structure had immediately received some attention from the scientific community; moreover, by no means did the younger scientists respond more positively than older ones. Thus, for example, the sixty-seven-year-old President of the PhysikalischTechnische-Reichsanstalt, Emil Warburg, gave a favourable report on Bohr's theory to the German Physical Society in Berlin on 5 December 1913; he not only outlined the content of the theory but also applied it to explain the effects of magnetic and electric fields on the frequency of spectral lines (Warburg, 1913b).5 It might seem somewhat surprising that the one scientist in Germany — Johannes Stark — who had been most concerned with atomic and molecular structure in previous years and who had come close in many respects to the views entering Bohr's theory, stayed away from the discussion.6 But at that time he was preoccupied with his most important experimental discovery, that of the influence of a static electric field on the frequency of the emitted spectral lines. Since Zeeman had observed the influence of a static magnetic field on the frequency of spectral lines in 1896, the question had been asked whether or not a similar effect On the same occasion at the Zurich Colloquium Einstein was reported to have remarked: 'Very remarkable! There must be something behind it. I do not believe that the derivation of the absolute value of the Rydberg constant is purely fortuitous.' ('Sehr merkwiirdig, da mufi etwas dahinter sein; ich glaube nicht dafi die Rydbergkonstante durch Zufall in absoluten Werten ausgedriickt richtig herauskommt.' Tank to Jammer, 11 May 1964; quoted in Jammer, 1966, p. 86). The colloquium on Bohr's theory was attended by Franz Tank, then a student at the Swiss Federal Institute of Technology (E.T.H.) in Zurich. 3 According to Harald Bohr's letter (which was undated, but probably written in late October or early November 1913), the spectroscopist Carl Runge also doubted the correctness of Bohr's theory. In particular, he claimed that Bohr's correction to the Balmer formula due to the finite mass of the nucleus (Bohr, 1913e) did not describe the observed data correctly. Some of these responses have been quoted in Bohr, 1963, pp. LI-LIII, and in Hoyer, 1974, pp. 182-186. 5 Warburg's paper contained the first application of Bohr's theory to a new problem. Like Warburg, the senior physicist Wilhelm Wien at Wiirzburg also considered Bohr's theory with proper respect, especially after his student Heinrich Rau had interpreted his results dealing with the excitation of Helium lines by collisions with electrons in agreement with Bohr's theory (Rau, 1914). At about the same time as Warburg's paper was published (on 15 December), the essential results of Bohr's theory also became available through t h e detailed report which Paul Ewald gave in issue No. 25 of Physikalische Zeitschrift on the Birmingham Meeting of the British Association; he gave, especially, an account of James Jeans' review on the radiation theory including the report on Bohr's atomic model and the following discussion (Ewald, 1913, pp. 1298-1300). To some extent this fact may have prevented Stark from entering into a discussion concerning the priority of the ideas involved.

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existed due to the electric field.7 In 1901 Woldemar Voigt in Gottingen had tried to calculate the order of magnitude of the effect of an electric field on the spectral lines by assuming that the lines were emitted by bound electrons, with binding forces including nonharmonic terms — these ensured also that the molecular forces extended over a finite distance (Voigt, 1901a). In this way he described both the Kerr effect and the still undiscovered effect on the frequency of spectral lines; by suitably fitting the constants of the theory (especially the anharmonicity parameter of the molecular force) in case of the Kerr effect, he found that even for electrostatic fields of about 300 V/cm the effect on the frequency remained below 5 x 1 0 - 5 times the separation of the two sodium D-lines. 'By this,' Voigt concluded, 'the failure of the previous attempts to discover an electrical analogue to the Zeeman effect is fully accounted for' (Voigt, 1901a, p. 208). This result discouraged people for many years from searching for the electric effect. However, in 1913 Stark began to look for it, for he believed that Voigt's argument could not be upheld; since he pictured the atom as consisting of a collection of charged particles (i.e., positive and negative ions and electrons) and assumed that the spectral lines were caused by an ionization of the atoms, he concluded that electric fields must alter their charge distribution and, therefore, also influence considerably the emitted frequencies. In the fall of that year, finally, Stark succeeded in obtaining a positive result: on 20 November 1913 Heinrich Rubens presented Stark's paper, entitled 'Beobachtungen iiber den Effekt des elektrischen Feldes auf Spektrallinien' ('Observations on the Effect of the Electric Field on Spectral Lines'), at a meeting of the Prussian Academy (Stark, 1913b). At about the same time Stark sent a letter to Nature announcing his discovery (Stark, 1913c). The main results of his observations were: when a strong electrostatic field of up to 31,000 V/cm acted on radiating hydrogen canal rays (behind the cathode of a discharge tube), one found that the lines Ha and Hp, if viewed in a direction perpendicular to the electric field, were split into five polarized components; the electric vector of the middle components was 7 A strong hint concerning the existence of such an effect arose from the arguments of W. Voigt; he had established a theory connecting the Zeeman effect with the Faraday effect — i.e., the rotation of the plane of polarization of light under the influence of a magnetic field (Voigt, 1898). Now it was known that a static electric field also influenced the propagation of light by causing double refraction in crystals (the so-called Kerr effect), hence Voigt expected a definite influence of the electric field also on the frequency of emitted and absorbed discrete spectral lines (Voigt, 1899a, pp. 308-310, p. 318). 8 S t a r k expressed this motivation clearly in his first paper, in which he presented the discovery of the effect of electric fields on spectral lines (Stark, 1913b, pp. 933-934). In his Nobel lecture, delivered on 3 June 1920, Stark gave, however, a different reasoning. Thus, after referring to Voigt's theory, he stated: However, I was unable to accept the presupposition of the theory — namely, the assumption that the emission of a spectral line on the part of an atom was the work of only one single independently moving electron in the atom. In my view the structure of the whole atom was that of an individual, with all its parts interconnected, and the emission of a spectral line appeared to me t h e result of the coherence and cooperation of several electric quanta. (Stark, in Nobel Lectures in Physics, 1901-1921, p. 433) It would seem t h a t this point of view came in only later, i.e. after the discovery of the effect. (See Stark 1914c.)

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perpendicular to the direction of the static electric field, while the outer ones had an electric field vector in its direction; in the case of helium lines at wavelengths of 4472 A and 4026 A there appeared six components, respectively, three of which were polarized perpendicular and three parallel to the direction of the applied field. Stark further reported: 'It seems... that the electric separation of a spectral line (the distance between components), when measured in wavelengths, is in first approximation proportional to the first power of the field strength' (Stark, 1913b, p. 941). 9 A few weeks later, on 20 December 1913, he reported to the Gottingen Gesellschaft der Wissenschaften on an effect that was observed if one looked in the direction of the electric field, the so-called longitudinal effect ('Langseffekt,' Stark and Wendt, 1914). In the case of Ha- and .ff^-lines of hydrogen, Stark and Wendt found three unpolarized components, while in the case of helium the same number of components showed up at the same place as in the transverse effect, except that the components were unpolarized. While Stark was extending his researches and investigated, in collaboration with his students, the effect of the electric field on other spectral lines and on band spectra (Stark and Kirschbaum, 1914b), a note appeared in Physikalische Zeitschrift. Its author, Antonino LoSurdo of Florence, referred to his observation in summer 1913 that the lines emitted from hydrogen canal rays in thin tubes 'appear to be resolved in a remarkable manner,' and concluded by saying that he had seen Stark's note in Nature and that he had also 'observed the same effect [i.e., the longitudinal effect]' (LoSurdo, 1914, p. 122). 10 Stark continued to work on his effect and, by late 1914, arrived at complete separations of the first four lines of the Balmer series of hydrogen; while he found no further splitting of the if a -line, the Hp-line was decomposed into 13, iJ T into 15, and Hs into 17 components, both in transverse and longitudinal observations; in all cases the individual separation increased linearly with the increasing strength of the applied electric field (Stark, 1914d). The first to attempt a theoretical explanation of the Stark effect was Emil Warburg in Berlin in the paper quoted earlier (Warburg, 1913b). 11 He based his ap9

T h e proportionality of the separation of the components to the strength of the electric field was confirmed in further experiments by Stark and his assistant Heinrich Kirschbaum (1914a). 10 When, in the same issue of Physikalische Zeitschrift, LoSurdo's teacher Antonio Garbasso proposed to associate the new effect with the names of both Stark and LoSurdo (Garbasso, 1914, p. 123), Stark defended his priority vigorously in a letter, which appeared in the following issue (Stark, 1914a). LoSurdo published the details of his results in Atti della Reale Accademia dei Lincei, the first publication having been presented at the session of 21 December 1913 (LoSurdo, 1913). It is possible that LoSurdo indeed saw the effect first without knowing its origin. 11 Emil Warburg was born on 9 March 1846 at Altona. He studied chemistry and physics at the Universities of Heidelberg (with Bunsen, Helmholtz and Kirchhoff) and physics at the University of Berlin (with Magnus and Kundt). In 1867 he received his doctorate under Gustav Magnus, and in 1870 he became Privatdozent at the University of Berlin; two years later he was appointed extraordinary (i.e., associate) professor at the University of Strasbourg, where his friend August Kundt was t h e Ordinarius. They collaborated for many years and published important papers, e.g., on the friction and heat conduction in dilute gases and on the specific heat of mercury. In 1876 Warburg became Professor of Physics at the University of Freiburg; there he discovered and explained the phenomena of hysteresis in ferromagnetic materials. Moving to Berlin in 1895 as the

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proach on Bohr's atomic model, and considered the deformation of circular electron orbits by the electric field; especially, he claimed that the Bohr frequency condition had to be changed to hu-A = Wnx- Wn2 , (1) where A was an additional energy term denoting the work which the electrons had to do in order to restore their original orbit in the presence of the electric field.12 Thus he found that every line of the hydrogen atom would become smeared out symmetrically with respect to the original frequency u, the width being 2|Af |, with

^spsbra-

(2)

where ri\, me and e denote the integral number of the initial orbit, the mass and the charge of the electron, respectively, h Planck's constant and |E| the strength of the static electric field. Equation (2) yielded the right order of magnitude in the case of the separations of the Hp- and H 7 -lines. Because of the appearance of Planck's constant in the expression for the width, Warburg argued that it was 'probable that the electric effect [Stark effect] belonged to those phenomena which could not be explained on the basis of classical electrodynamics' (Warburg, 1913b, p. 1266). He also calculated, with the same model, the magnetic splitting of spectral lines, obtaining a result depending upon the quantum numbers of the stationary states — in contrast to observation. Bohr, to whom Warburg wrote about his results, replied that he had also meanwhile tried to obtain the Zeeman and Stark effects of hydrogen lines. 'In agreement with your considerations,' he wrote, 'I find that my theory allows me to compute the effects in question correctly. In contrast to your results, however, I arrived at the conclusion that the theory explains the line separations observed by Stark, and also the observations on the Zeeman effect, quantitatively' (Bohr to Warburg, 8 January 1914). 13 In a paper, which appeared in the March issue of Philosophical Magazine, Bohr presented these results (Bohr, 1914b). According to his treatment an external electric field also deformed the orbits of the electrons in free atoms. However, he claimed that only two stable motions remained, i.e., the electrons passed either parallel or antiparallel to the direction of the external electric field through the nucleus; thus the original frequencies were shifted by amounts of ±(3h/47r 2 m e e)|E|(n 2 — n2,), respectively.14 successor to August Kundt at the University, he became President of the Physikalisch-Technische Reichsanstalt in 1905. He retired in 1922 and died on 28 June 1931 at Gut Grunau near Bayreuth. During his Berlin years Warburg worked mainly on the phenomena of electric conduction, on the radiation problem and on photochemical reactions (proving the Einstein-Stark equivalence law). 12 Warburg calculated this work (A) in the approximation that the deformation was small (A -C Wni,Wn2) as A = ± o n i e | E | , where ani is the radius of the initial orbit, e the charge of the electron and | E | the strength of the electric field. 13 B o h r ' s attention had been drawn to the Stark effect by a letter from Ernest Rutherford (Rutherford to Bohr, 11 December 1913). Similar results were obtained by Garbasso and Ernst Gehrcke; while Garbasso's treatment was close to Bohr's (Garbasso, 1914), Gehrcke considered a kind of one-dimensional version of Bohr's atomic model (in which the electrons moved in a line: Gehrcke, 1914).

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With respect to the influence of a magnetic field on spectral lines, Bohr simply postulated that the (classical) Larmor precession was superimposed on the motion of the electron around the nucleus and this fact, although it did not alter the energy of the stationary states, changed the frequency condition to h(V±vL)

= Wni-Wnt,

(3)

where UL, denotes the Larmor frequency. Bohr's treatments of the Zeeman and Stark effects did not describe any finer details; in the case of the Stark effect only a part of the transverse effect could be explained (the lines polarized perpendicular to the external fields should not exist), and the complex Zeeman effect discovered by Thomas Preston did not follow either. (The classical Larmor precession had to be added to all electron orbits.) Still, Bohr was very optimistic in spring 1914 about the possibility of finally accounting quantitatively for all observed separations of spectral lines. Thus, for example, he claimed that the extremely small doublet splittings of the hydrogen lines had their origin in terms of the perturbation of the Coulomb force between the nucleus and the electrons (Bohr, 1914b). In searching for such terms he arrived later at the conclusion that relativistic corrections to the motion of the electrons might be the reason (Bohr, 1915a). However, he did not really get hold of this problem properly. There occurred a standstill in Bohr's work on atomic structure at Manchester. When, in spring 1916, he returned to Copenhagen, he found several publications of Arnold Sommerfeld's on his desk. Sommerfeld had achieved important progress by extending the theory to electron motions having several degrees of freedom. In the first decade of quantum theory physicists had dealt with systems of just one degree of freedom, especially the harmonic oscillator. The quantum of action entered into the description of these systems in the form of a condition, which Planck formulated as / /

dp dq = nh;

(4)

that is, the integral over the phase space (with p and q denoting the momentum and position variables, respectively) took on finite values. 15 While Planck had used Eq. (4) only for the harmonic oscillator, Paul Ehrenfest applied it also to the rigid rotator in a plane (Ehrenfest, 1913b). By studying the boundaries of the phasespace regions, he discovered that the angular momentum p assumed there the values h

•p="to>

f*\

(5)

^Planck used Eq. (4) in the derivation of his second quantum hypothesis (Planck, 1911a). Later, it frequently became the starting point of the discussion of systems other than the one-dimensional harmonic oscillator. Thus Peter Debye applied it in his Wolfskehl lecture of 1913 at the University of Gottingen; he was interested, in particular, in the anharmonic oscillator, hence he concluded that the areas in phase space were not bounded by ellipses (as in the case of Planck's oscillators) but by curves of different shapes (Debye, 1914, p. 29).

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with n = 0, 1, 2, etc. Ehrenfest, who favoured Planck's first q u a n t u m hypothesis, claimed t h a t in t h e motion of t h e quantum-theoretical r o t a t o r p<j, was always given by Eq. (5). T h e same condition also entered into Nicholson's a n d Bohr's theories of atomic constitution. Bohr, in particular, noticed t h a t his original condition quantizing t h e energy of an electron a r o u n d a nucleus was equivalent to J o h n William Nicholson's assumption t h a t the angular m o m e n t u m of the electron ring satisfied Eq. (5) (Bohr, 1913b, p . 15). 1 6 Although t h e r e had existed some concern previously with t h e t r e a t m e n t of systems of one degree of freedom in q u a n t u m theory, the serious discussion of this problem b e g a n only in 1 9 1 1 . 1 7 At t h e first Solvay Conference Henri Poincare raised, during t h e discussion of Planck's report, t h e question of whether t h e particular shape of t h e finite areas in t h e phase plane of a one-dimensional q u a n t u m system played any role. After Planck denied it, giving as a n example t h e one-dimensional rotator (in which case the energy values uniquely determined t h e s h a p e of t h e phase cells), Poincare asked further: What if there are more degrees of freedom? One may think of a resonator vibrating in all directions, which therefore possesses three degrees of freedom but is isotropic, i.e., it has the same period (of vibration) along all three axes. If one decomposes its motions in the direction of the axes, one will have to take for the motion parallel to the x-axis an energy ahu, with a an integer; for the [motions parallel to the] y-axis and z-axis, the energy will be /3hu and 7/11/ (/? and 7 being integers), respectively. Now, if one introduces new axes, then one will have different energies, say a'hv, fi'hv, •y'hi', in the direction of the new axes. However arbitrarily one chooses the new axes, a', /3', 7 ' must be integers; and that is impossible. (Poincare in Eucken, 1914, p. 99) Planck replied t h a t so far n o q u a n t u m hypothesis had been considered in t h e case of systems of several degrees of freedom, adding: 'Of course, I consider it by no means impossible to establish such a one' (Planck in Eucken, 1914, p . 99). O n the other hand, Einstein (in t h e discussion of Nernst's report at t h e Solvay Conference) emphasized t h a t certain difficulties might arise in dealing w i t h t h a t problem. 1 8 16 Bohr restated the quantum condition at the end of his first paper on the constitution of atoms and molecules as follows: ' In any molecular system consisting of positive nuclei and electrons in which the nuclei are at rest relative to each other and the electrons move in circular orbits, the angular momentum of every electron round the centre of its orbit will in the permanent state of the system be equal to h/2-n, where h is Planck's constant' (Bohr, 1913b, pp. 24-25). He added a footnote at the end of this sentence, which stated: 'In the considerations leading to the hypothesis we have assumed that the velocity of the electrons is small compared with the velocity of light' (loc. cit., p. 25). In part II Bohr discussed the extension of the hypothesis to relativistic motions; he claimed that also in these cases the angular momentum should still be constant, but different from the ratio of energy W over the frequency of revolution of the electrons (Bohr, 1913c, p. 479). l 7 For example, in his theory of specific heats of solids Einstein treated the elastic vibrations of a gram-atom as being caused by N three-dimensional oscillators, and endowed each of these with energy-quanta 3nhv, n = 0,1, 2 , . . . . (Einstein, 1906g). 18 Einstein gave thefollowing argument: 'If one modifies the statistical mechanical equation for the average energy E of a three-dimensional oscillator,

. _ ~

fE3exp(-E/kT)dE fE2exp(-E/kT)dE'

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In the years following the first Solvay Conference Planck concerned himself with answering Poincare's question concerning the structure of the phase space in quantum theory and the extension of the quantum hypothesis to systems of several degrees of freedom. He took this question especially seriously, for in his thinking a particular aspect played a role: the fact that the finite areas of phase space were connected with the probability for physical processes. If he, therefore, knew their structure, he could interpret the discontinuity occurring in the quantum-theoretical description as a discontinuity in the probability rather than as a discontinuity in energy or in the light-quantum. (See Planck, 1912a.) In his Wolfskehl lecture in April 1913 at the University of Gottingen he discussed a particular system of several degrees of freedom, the molecules of a monatomic gas (Planck, 1914b). These molecules had three translational degrees of freedom, and — according to the previous treatment of Otto Sackur and others (see Sackur, 1911) — a six-dimensional phase space consisting of finite cells of volume h3 (h being Planck's constant) was associated with each molecule. Planck claimed, however, that the situation was much more complex: In order that this quantity [i.e., f dxdydzdpxdpydpz, the finite volume in phase space] may possess a definite value, as required by quantum theory, the limits of the integral must be quite definite; and it is a task of fundamental importance for the theory to reveal precisely the significance of these limits. (Planck, 1914b, p. 7) For the moment he could give only preliminary indications as to how these boundaries might be found; he felt strongly that they depended on the interaction between the gas atoms. 19 More than two and a half years later, Planck finally found himself in a position to provide a definite answer, and he presented it in two contributions, entitled 'Die Quantentheorie fur Molekeln mit mehreren Freiheitsgraden' ('The Quantum Theory of Molecules with Several Degrees of Freedom'), at the meetings of the German Physical Society on 5 November and 3 December 1915 in Berlin (Planck, 1915b,c). In these papers Planck considered atomic systems of / degrees of freedom, whose phase space had to be divided, according to the known procedure, into finite cells of volume h.f, h being Planck's constant. Planck first proposed the following quantum condition: he required that the phase-space integral over all variables, pi,...,pf and
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un, should satisfy the condition dpi • • • dpf dqi • • • dqj = (nh)f

,

(6)

/ Jo

n being a positive integer ( = 0, 1, 2, etc.) (Planck, 1915b). Thus, for example, in the case of the spatial rotator (which possesses two degrees of freedom), he obtained the discrete energy states, n?h2/8n2A, where A denotes the moment of 20 inertia and n = 0, 1, 2 In his second contribution he tried to generalize Eq. (7); he especially assumed the existence of systems, in which the phase space did not just depend on the energy of the system under investigation (or on the number n denoting it), but also on other variables (Planck, 1915c). If these characteristic variables (including the energy) were denoted by g, g', g",..., and their discrete values determining the boundaries of phase cells by gn, g'q, g",..., the general quantum condition became

f9n f9q I"" dPl • • -dpf dqi--- dqf = (nhy(qhf (rhf Jo

• • •,

(6')

Jo Jo

with i + i' + i" + • • • = f. Evidently, in Eq. (6') several 'ordering numbers' {LOrdnungszahlen') n, q, r , . . . entered. Further, the / total degrees of freedom of the system could be adjusted in such a way to the characteristic variables that i were associated with g, i' with g', i" with g", etc.; and Planck called the degrees of freedom belonging to one particular variable 'coherent,' and those belonging to different variables 'incoherent' (Planck, 1915c, p. 441). As in the simpler quantum systems, in which only one ordering number n occurred, also in general cases the discrete values of the characteristic variables, gn, g'q, g'J,..., determining the boundaries of the phase-space regions, could be calculated. 21 Planck applied the new quantum conditions to the case of a mass point, which moves in space under the attraction of a central force whose strength increases linearly with the distance of the mass from the centre. He was not interested, however, in treating the Bohr atom; he did not believe in its basic assumption of discrete stationary states, but rather connected the origin of radiation with probability jumps at the boundaries of the quantum cells in phase space (Planck, 1915a,d). Only several months later, when he prepared Prom the discrete energy values u„, Planck also calculated the value which the oscillator assumes on the average in the phase-space region n, which is bounded by hyper-areas characterized by u = un and u = un+i, i.e., J

udpi • • • dpfdqi • • • dqf J0 " d-Pl • • • dpfdqi

•••dqf

(Planck evaluated the phase-space integral in the denominator to be ft/ times {(n + 1)^ — n?}.) Thus, in the case of the spatial rotator, un became equal to (h2/8ir2A)(n2 + n + |). 21 Planck proposed t h e following procedure: One takes the classical dependence, say, of g on the dynamical variables and considers a motion of the molecules only in the coherent degrees of freedom associated with g. It should be noted that the case of completely incoherent degrees of freedom denotes a nondegenerate system.

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a paper for Annalen der Physik summarizing the content of the two earlier notes, did he also give the results in the case of an electron moving in the Coulomb field of a positive nucleus, achieving basically the same results as did Sommerfeld in his theory of the relativistic hydrogen atom, with which he had become acquainted in the meantime (Planck, 1916a). The papers, which Planck presented to the German Physical Society in November and December 1915, did not contain the first extension of the quantum hypothesis to systems of several degrees of freedom. Two other publications on this subject had appeared earlier that year, both in journals which, because of war conditions, were not directly accessible in Germany. On 4 April 1915, Jun Ishiwara of the Physical Institute at Sendai University presented a paper to the Mathematical-Physical Society of Tokyo, entitled 'Die universelle Bedeutung des Wirkungsquantums' ('The Universal Significance of the Quantum of Action,' Ishiwara, 1915a). In this paper Ishiwara, who had spent some years in Europe previously and had acquainted himself with the progress in relativity and quantum theories, attempted to generalize the quantum hypothesis in such a way that it embraced Planck's hypothesis of finite elementary regions (cells) in phase space, the Nicholson-Bohr-Ehrenfest quantization of angular momentum, Eq. (5) as well as Sommerfeld's action-integral hypothesis of 1911. 22 In particular, he proposed the following new, fundamental hypothesis: Suppose an elementary object of matter, or a system of an enormously large number of elementary objects is in a state of stationary, periodic motion or in statistical equilibrium. Let its state be completely determined by the coordinates qi, q2, • • • ,Qf, and the corresponding momenta pi, p%,... ,pf. Then the motions in nature always occur in such a way that the decomposition of each state plane, qiPi, [i = 1 , . . . , / ] , is permitted into those regions of equal probability, the average value of which at a given point of the phase space,

1

/

iYl is equal to a universal

constant.

f pidqi h

^>

(7)

(Ishiwara, 1915a, p. 1 0 8 ) 2 3

22 J u n Ishiwara (or Ishihara) was born in Tokyo on 15 January 1881. He was educated at the University of Tokyo from where he graduated in theoretical physics in 1906. In 1911 he was appointed assistant professor in the faculty of science at Tohoku University, Sendai. Shortly afterwards he went abroad for further studies at the University of Munich (with Sommerfeld), at the E.T.H. in Zurich (with Einstein), and at the University of Leyden. He worked on relativity and electron theory, contributing articles to Annalen der Physik (Ishiwara, 1913) and to Jahrbuch der Radioaktivitdt und Elektronik (Ishiwara, 1912, 1914). After his return to Japan he concentrated on relativity and quantum theory. He was promoted to full professor at Tohoku University, but retired already in 1921 to devote himself to poetry. Ishiwara acted as Einstein's host during his visit to Japan in late 1922 and also translated several papers of Einstein's into Japanese. He died on 19 January 1947. (See the information provided by Pierre Speziali in Albert Einstein-Michele Besso: Correspondance 1903-1955, pp. 542-543.) 23 I t should be noted that Eq. (7) represented the first appearance in literature of what Sommerfeld later called the 'phase integral.' (Different statements, e.g., in Hund, 1967, pp. 33-34, cannot be substantiated.) Ishiwara motivated his quantum hypothesis on the basis of Sommerfeld's action integral hypothesis of 1911. (We have changed Ishiwara's notation slightly: he called qt the momentum and pi the position coordinates, and he referred to systems of j degrees of freedom.)

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Ishiwara immediately convinced himself that the new hypothesis yielded the previously well-established results of quantum theory, such as the expression for the chemical constants (Sackur, 1911), the Sommerfeld-Debye theory of the photoelectric effect (Debye and Sommerfeld, 1913) and Bohr's theory of atomic structure (Bohr, 1913b). However, in an important detail, he obtained a different result from Bohr's theory: if he assumed that the motion of an electron around the nucleus possessed two degrees of freedom, then the Balmer formula for the hydrogen spectrum followed quantitatively from Eq. (7) only if the nucleus of the hydrogen atom bore two positive charges and, therefore, the neutral atom contained two electrons. 24 Even earlier than Ishiwara, namely, in March 1915, John William Nicholson communicated a paper on 'The Quantum Theory of Radiation and Line Spectra' by his colleague William Wilson to Philosophical Magazine, which was published in the June issue (Wilson, 1915). Wilson, then working at King's College, London, wanted to establish the 'possibility of deducing the results of Planck and Bohr from a single form of quantum theory' (Wilson, 1915, p. 795). 25 He based his approach on two assumptions: first, that interchanges between dynamical systems (atoms) and the ether occur in a discontinuous manner; second, that between these discontinuous 24 Evidently Ishiwara's result arose from associating the electron with two degrees of freedom; hence the double of Bohr's value for the angular momentum states entered into the energy expressions. Ishiwara was satisfied with it, as he subscribed to some extent to Nicholson's opinion. (He had become familiar with Nicholson's theory of atomic constitution in London.) Ishiwara continued to discuss the theory of quantum systems in later papers; in one of them he changed his quantum hypothesis (7) to the following: the stationary states of atomic systems are given by the fact that their energy multiplied with the period of motion assumes integral multiples of Planck's constant (Ishiwara, 1915b, p. 318). 25 W . Wilson was born on 1 March 1875 in Goody Hills, near Mawbray, Cumberland. Prom 1893 to 1896 he studied geology, mathematics, physics and astronomy at the Royal College of Science, London; among his teachers were Arthur Riicker, William Watson, Norman Lockyer and Alfred Fowler. Then he taught in English schools and at the Berlitz School of Languages at various places in Rhineland. His savings allowed him to enroll in summer 1902 at the University of Leipzig to continue his studies in mathematics (under Carl Neumann, who gave him a thorough course in Hamiltonian mechanics) and physics. In 1906 he earned his doctorate summa cum laude, the physicist Otto Wiener being his supervisor. After returning to England, Wilson got a position as assistant lecturer in the Wheatstone Laboratory at King's College, London. He worked especially on the photoelectric effect and related phenomena and also contributed papers on theoretical problems, the first one on an application of quantum theory to explain the discharge of hot bodies (Wilson, 1913). Later he also wrote on relativity theory and again on quantum theory. In 1919 he was promoted to a readership at King's College, and two years later he was elected to the Hildred Carlile Chair of Physics at Bedford College, London. He retired in 1944 and died on 14 October 1965. Wilson was elected a Fellow of the Royal Society in 1923. In a letter, dated 7 June 1920, Wilson drew Sommerfeld's attention to the fact t h a t he had published the quantum conditions for systems having several degrees of freedom already in his first paper of June 1915. Sommerfeld admitted in his reply: 'You are right in t h a t I should have quoted your 1915 papers in my book [Atombau und Spektrallinien], as I did in my paper in the Annalen der Physik, Volume 51, 1916, p. 9' (Sommerfeld to Wilson, 14 July 1920). And he promised to cite Wilson in the second edition of Atombau und Spektrallinien. Sommerfeld further stated in his letter to Wilson that he had developed the extension of Bohr's theory — independently of Wilson — in winter 1914-1915, but that he had delayed the publication until Paschen had tested the fine-structure formula experimentally. He concluded by stating: 'The priority for the publication of the rule Jpdq — nh belongs to you without doubt' (Sommerfeld to Wilson, 14 July 1920).

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changes the systems are described by Hamiltonian mechanics. The steady motions of periodic systems, however, satisfy the equations Pidqi-riih,

1 = 1,2,3,...,

(8)

where the n* denote positive integers and the paths of integration extend over all values of the dynamical variables pi and qi that are assumed during the period v~ {i>i being the frequency and h Planck's constant). Using this hypothesis Wilson derived, in the case of N resonators, Planck's formula for the average energy of the resonator (entering into the fundamental equation of radiation theory). In the second part of his paper Wilson applied Eq. (8) in order to obtain (in the case of the circular motion of an electron around the nucleus) Bohr's formula for the kinetic energy of the electron. In addition, he found that the radiation emitted from any quantum system satisfied a generalization of Bohr's frequency conditions to systems having several degrees of freedom. In a second paper, communicated again by Nicholson in November 1915, Wilson further developed his views on the emission of spectral lines (Wilson, 1916). He also referred to Ishiwara's paper of April 1915 and stated the main points in which the two approaches differed: especially, that he, Wilson, preferred to assume Planck's original hypothesis of 1900, while Ishiwara tended towards Planck's second hypothesis of 1911; also Ishiwara extended the path of the integral (on the right-hand side of Eq. (7) over the full period of motion of the total system, while in Wilson's Eqs. (8) the integral for each degree of freedom extended over the particular period associated with that degree. 26 Then he applied his scheme to the case of an elliptical electron orbit in an atom by writing down two phase-space integrals of the type of Eqs. (8): one for the pair of dynamical variables, p^ (the angular momentum) and (j) (the angle), the other for the pair of radial variables, pr and r. He established a relation between their respective integral numbers, ni and ri2, and the eccentricity (e) of the elliptical orbit, (ni + n 2 ) V l - e 2 = n2 .

(9)

At this point Wilson stopped and did not draw any further conclusion, in contrast to Sommerfeld, who independently at about the same time had Eqs. (8) available. Arnold Sommerfeld, well-established Professor of Theoretical Physics at the University of Munich, had contributed to the quantum theory since 1911 by proposing a new quantum hypothesis involving the action integral. The application of this quantum hypothesis to aperiodic processes, such as the creation of photoelectrons or the motion of molecules in gases, seemed to be promising though it did not yield really conclusive results. Stimulated by Friedrich Paschen and Ernst Back's investigations of the Zeeman effects in strong external magnetic fields (Paschen and Back, 1912, 1913), Sommerfeld turned in 1913 to establish a theoretical description of the 2°Wilson explicitly criticized Planck's second hypothesis. (See Wilson, 1916, pp. 159-160.)

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anomalous Zeeman effects by generalizing Lorentz' original theory of the elastically bound electrons; he assumed in particular that electrons were isotropically bound in atoms (Sommerfeld, 1913). He continued to work on this subject, in competition with Woldemar Voigt of Gottingen (Voigt, 1913a,c,d), and submitted a second paper in March 1914 to the Gottingen Academy (Sommerfeld, 1914a). In the latter paper he simplified Voigt's classical theory considerably. Sommerfeld referred to a quantum-theoretical argument only at a minor point concerning the intensities of the Zeeman components. This did not mean that he was not interested in Bohr's quantum theory of atomic structure. Thus, when Bohr sent him a reprint of his paper in the July issue of Philosophical Magazine (Bohr, 1913b), he immediately wrote to him: Many thanks for sending me your highly interesting paper, which I read already in Philosophical Magazine. Since long I have thought of the problem of expressing the RydbergRitz constant in terms of Planck's constant h. I discussed it several years ago with Debye. Although I am still somewhat skeptical about atomic models in general, [I must say that] the calculation of that constant represents without question a great achiewement. By the way, the numerical agreement becomes even better with the new value of Planck's constant, h = 6.4 x 10 - 2 7 . (Sommerfeld to Bohr, 4 September 1913) At the end of his letter Sommerfeld asked Bohr whether he planned to calculate the Zeeman effect from his model. Bohr replied: 'I hope soon to publish a short note on the phenomena of magnetism and the Zeeman effect. I have been working for a long time with this problem, which seems promising on account of the close analogy between the hypothesis of the universal constancy of the angular momentum of the electron and the theory of magnetons' (Bohr to Sommerfeld, 23 October 1913). It might have been the skepticism against atomic models per se or, what was more likely, Sommerfeld's general attitude not to interfere with young people when they were on a promising track; in any case, Sommerfeld did not concern himself with the application of Bohr's atomic model to the problem of the Zeeman effect. However, he did use Bohr's model of molecules for the purpose of obtaining a generalization of Drude's dispersion formula (Sommerfeld, 1915d). And soon he became deeply involved in the problem of atomic constitution and in Bohr's theory; especially, he announced a course of lectures on 'Zeemaneffekt und Spektrallinien' ('Zeeman Effect and Spectral Lines') for the winter semester 1914-1915 (see Physikalische Zeitschrift 15, p. 860, 1914). It turned into a course on Bohr's theory, and Sommerfeld also talked about an extension of the latter, including a theory of elliptical orbits and the relativistic fine structure. 27 Still he waited with the publication of his results because of an important reason. At that time, i.e., between fall 1914 and fall 1915, Einstein worked on the final steps towards the theory of general 1 Sommerfeld mentioned this fact in his Autobiographische Skizze by recalling: 'My extension of Bohr's theory (elliptical orbits, fine structure) was already presented in winter 1914-1915 in my lecture course, but was first published in the beginning of 1916' (Sommerfeld, 1968, Volume IV, p. 678).

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relativity, and Sommerfeld was very interested in it. Since he was not certain whether the new theory of gravitation would also change his relativistic extension of Bohr's atomic model, he sent Einstein the manuscripts of two papers containing his results and asked for Einstein's opinion of them. Einstein replied to Sommerfeld in the beginning of December 1915: Herewith you get back both manuscripts, which I have looked through with interest. Planck is also working on a problem similar to yours (the quantization of phase space of molecular systems). He is concerned with spectroscopic questions as well. [Concerning the relativistic calculation of electron orbits he added:] General relativity will hardly be of any help to you, because its results coincide for these problems with those of special relativity theory. (Einstein to Sommerfeld, 9 December 1915) Sommerfeld was satisfied with this answer and he quickly went on to publish his papers on the extension of Bohr's atomic theory. The first paper, entitled 'Zur Theorie der Balmerschen Serie' ('On the Theory of the Balmer Series') and containing the nonrelativistic theory of the hydrogen atom, he presented already on 6 December 1915 to the Bavarian Academy (Sommerfeld, 1915b); he also made sure that the second paper was still included in the proceedings of the Academy for the year 1915, although he was not able to present it until the session of 8 January 1916 (Sommerfeld, 1915c). Sommerfeld opened his first paper by referring to the successes of Bohr's theory in explaining the hydrogen spectrum and the X-ray spectra of heavier elements, and wrote: Nevertheless I want to demonstrate that even the theory of the Balmer series exhibits in certain respects a gap, namely as soon as one admits non-circular (i.e., even in the case of hydrogen, eUiptical) orbits. [He added:] I shall fill this gap by deepening the quantum hypothesis, and I shall at the same time illuminate the special situation of the hydrogen spectrum. While the other elements exhibit a sequence of different series types (singlet, doublet and triplet series), hydrogen possesses (if one neglects the still little known multiple-line spectrum) only a single Balmer series. According to the concepts presented here, this fact can be explained as follows: in the Balmer series there coincides a sequence of series, i.e., each of its lines arises in several different ways, not only from circular orbits but also from elliptical orbits of given eccentricities. (Sommerfeld, 1915b, p. 425) This simplicity originated, according to Sommerfeld, in the simple constitution of hydrogen; for all other chemical atoms the different elliptical orbits led to different types of line spectra, such that in general each spectral line depended at least on On one hand, Sommerfeld took a great personal interest in Einstein's work; on the other, he was also charged with editing the third edition of a collection of papers on relativity theory, which had first appeared in 1913 under the title Das Relativitdtsprinzip (edited by Otto Blumenthal). For that purpose he wanted to include some papers on general relativity and he wrote Einstein a letter concerning the matter. In his reply, dated 28 November 1915, Einstein wrote about his recent progress in deriving the equations of the theory of gravitation, which yielded in particular a satisfactory explanation of the observed shift of the perihelion of the planet Mercury. We may mention in passing t h a t Sommerfeld had himself worked on electron theory early in the 1900s (see especially, Sommerfeld, 1904a,b). He had even discussed the case of electrons moving with speeds faster than light in vacuo (Sommerfeld, 1905).

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two sets of integral numbers, n, n' and m, m', and its frequency was described by the formula v = const. [4>{n, n!) — <j>(m, m')],

(10)

where (j>(n,n') and 4>(m,m') denoted the energy terms of the atom. 29 The task which Sommerfeld set out to accomplish in the first paper consisted, therefore, in deriving Eq. (10) from the assumption that electrons in atoms move on certain elliptical orbits. Sommerfeld did not refer in December 1915 to the earlier publications of Ishiwara and Wilson on the generalization of the quantum hypothesis to systems of several degrees of freedom. On one hand, these publications were not available to him; on the other, he had arrived independently — and earlier — at the same results. He rather referred to the much earlier work of Planck, especially to phase-space considerations (Planck, 1906), and to the extension of Planck's considerations to the anharmonic oscillator (Debye, 1914) and the rotator (Ehrenfest, 1913b). From such examples he concluded that in the quantum-theoretical phase space of momentum and position variables (p and q) certain cells of finite size existed, enclosed by boundaries — which in the case of the harmonic oscillator were concentric ellipses of increasing semiaxes — whose areas were given by integral multiples of Planck's constant. This condition, the 'quantum condition' (' Quanttn^ordexun^ as Sommerfeld called it, could be reexpressed as the 'phase integral' ('P/iaseranieyra/,' Sommerfeld, 1915b, p. 429) / /

dpdq = I pdq = nh,

(11)

i.e., as a line integral. Sommerfeld then showed the close relation of the phase integral to his earlier action-integral quantum condition and derived from it the positivity of the right-hand side of Eq. (11). Hence the 'quantum number' n must be a positive integer ('Quantenzahl,' Sommerfeld, 1915b, p. 430). 30 Sommerfeld extended the phase-integral approach to describe also the quantum-theoretical motion of systems having two degrees of freedom, especially to the elliptical orbits of an electron in the Coulomb field of a positively charged nucleus. An essential feature of the (classical) Kepler motion was the existence of a particular constant, the socalled 'area constant' ('Flachenkonstante1), which was identical with the angular momentum variable p^,. Now the quantum condition (11), when applied to the azimuthal motion of the electron, i.e., to the canonically conjugate pair of dynamical 29

I n the special case of hydrogen, the function <j>(n, n') should depend only on the sum of the two integers, n + n ' ; hence the energy term becomes proportional to (n + n')~2. 30 Sommerfeld found that the integral, J pdq, could be written as Jpdq = J pqdt = J const.q2dt in the case of a conservative system whose kinetic energy is proportional to q, the time derivative of the position variable q. Since the integrand of J pdq is proportional to the kinetic energy, the phase integral is indeed closely related to Sommerfeld's earlier action integral (provided the Lagrangian is replaced by the kinetic energy) and it assumes a positive value.

Arnold Sommerfeld

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variables p$ and <j> (where <j> is the azimuthal angle), yielded the equation /•2ir

/ Jo

1p^dj)

= 2npj, — nh.

(12)

Hence the angular momentum p^ had to assume values that were positive integral multiples of the quantity h/2ir.31 However, a new difficulty arose with the definition of the phase integral, a difficulty which did not exist in the one-dimensional motions considered thus far and which Sommerfeld realized immediately: 'For the Keplerian motion, however, we have two degrees of freedom, hence the concept of the phase integral ceases to be uniquely defined' (Sommerfeld, 1915b, p. 432). In order to obtain a hint concerning the removal of the above-mentioned ambiguity in the Kepler motion, Sommerfeld studied the expression for the energy of the electron in the elliptical orbit in the case of the hydrogen atom, i.e., Wn

h2

n*

'

K

'

with n denoting the quantum number of the angular momentum (see Eq. (12) and e n the eccentricity of the ellipse under consideration. From Eq. (13) there followed the frequencies v, 27r2mee4 fl-el l-eL\ { u ) which were emitted during the transitions of the electron (mass me and charge e) from an initial orbit with azimuthal quantum number m and eccentricity e m to a final orbit with quantum number n and eccentricity en. Evidently, the frequencies depended explicitly on the values of the eccentricities en and e m . From the fact that the hydrogen lines were sharp, Sommerfeld concluded: 'If we do not want to disallow from the very beginning that the electrons move on elliptical orbits besides circular ones, then inevitably the requirement arises that the eccentricities have also to be treated arithmetically by connecting them with certain integral numbers according to the quantum condition' (Sommerfeld, 1915b, p. 435). Such a quantum condition, however, seemed to be available immediately. One had just to introduce (in accordance with the motto: 'What is fine with <> / should also be all right with r,' as Sommerfeld used to say, see Jammer, 1966, p. 93, footnote 17) the phase-integral for the radial degree of freedom, i.e.,

/

prdr = n'h,

(15)

wherep r denotes the radial momentum variable (pr — me(dr/dt)) and r the distance of the electron from the nucleus (which changes in case the orbit has nonzero eccentricity). By inserting for pr the expression obtained from the equation of motion 1 Sommerfeld drew attention to the difference between his treatment and Bohr's. Bohr had established a relation between nh and the quantity 2Trmea2u>2, where a is the semi-major axis of the orbit and u the frequency of the revolution (Bohr, 1913b, pp. 3-4).

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(i.e., r = dr/dt = (e2/p^)€sm4>) into t h e left-hand side of Eq. (15), Sommerfeld derived a n equation expressing t h e eccentricity in t e r m s of t h e two q u a n t u m numbers n and n ' , i.e., 3 2

^•""OT-

(16)

Evidently, e now depended on b o t h q u a n t u m numbers separately. By combining Eqs. (13) and (16) Sommerfeld found a simple formula for the energy t e r m s of the hydrogen a t o m , namely, 27r 2 m e e 4

1

o n which he commented as follows: This result is surprising in the highest degree and [at the same time] of penetrating certainty ('schlagende Bestimmtheit'). Not only the additional allowed energy values have become discrete and are determined by integral numbers, but the previous denominator n 2 has just cancelled such that the result depends only on n + n'. Hence the energy is uniquely determined by the sum of the quanta of action, which we may distribute arbitrarily among the azimuthal and radial coordinates. It seems to me to be quite out of the question that such a far-reaching result should be ascribed to an algebraic accident. I rather see in it a convincing confirmation of the extension of the quantum hypothesis to the radial component, or of the separate application of this hypothesis to the two degrees of freedom of our problem. (Sommerfeld, 1915b, p. 439) I n any case, t h e above t r e a t m e n t also yielded Bohr's formula for t h e hydrogen spectrum.33 Altogether Sommerfeld believed confidently t h a t he was on t h e right track a n d t h a t his generalized q u a n t u m conditions, Eqs. (12) a n d (16), indeed allowed him to describe t h e hydrogen a t o m satisfactorily: t h a t is, t h e electrons in stationary states moved in ellipses, whose parameters — the semimajor axis a ( = h2(n + n ' ) 2 / 4 7 r 2 m e e 2 ) a n d eccentricity e — were completely determined by the q u a n t u m conditions. This situation justified the opinion, which he had already expressed earlier in 1915, ' . . . t h a t the laws governing t h e interior of the a t o m do not deviate so much from those of classical mechanics and electrodynamics, as one might think according t o Bohr's assumptions' (Sommerfeld, 1915d, p . 549). T h e q u a n t u m conditions did n o t restrict t h e validity of the classical laws but determined t h e initial values of t h e dynamical problem: while the q u a n t u m number n' = 0 signified, in agreement w i t h Eq. (16), circular orbits, the choice n = 0 denoted linearly degene r a t e orbits (in which the electron describes a linear p e n d u l u m line). Sommerfeld regarded the l a t t e r orbits as 'most problematical' ('hochst problematisch'), because ' . . . t h e electron will t h e n approach t h e nucleus infinitely closely and probably be 2

Equation (16) agreed, of course, with Wilson's Eq. (9) upon identifying Wilson's quantum numbers n\ and ni with Sommerfeld's n' and n, respectively. 3 Also the mass correction could be incorporated in the same way as in Bohr's theory; i.e., in considering the finite ratio of the mass of the electron over the mass of the nucleus, me in Eq. (13') had to be replaced by the reduced mass meM/(M + roe), M being the mass of the nucleus.

Arnold Sommerfeld

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repelled by it' (Sommerfeld, 1915b, p. 446). Excluding such degenerate orbits, he found that to any line in the hydrogen spectra several transitions may contribute: for example, in the case of the Balmer Ha-lme, with initial state (m + m') = 3 and final state (n + n') = 2, six transition from one stable orbit to the other were possible; in the case of the i^-line, with initial and final states denoted by the quantum number sums (m + m') = 4 and (n + n') = 2, respectively, eight transitions were possible, etc. 'One may, however, doubt,' Sommerfeld added, 'whether each of these transitions is possible, or whether perhaps only those transitions are admitted that are associated with a loss of quanta' (Sommerfeld, 1915b, p. 447). He expected, in particular, that the Stark effect of the lines would throw some light on the question of allowed transitions, and argued: 'According to our interpretation, in each Balmer line an entire sequence of frequencies having different origins coincides. The [external] electric field will influence the different elliptical orbits in a different manner; hence it will separate the originally coinciding frequencies. In the case of hydrogen the influence will be stronger than for other elements, where such a coincidence [of the field-free lines] cannot be expected' (Sommerfeld, 1915b, p. 449). His prediction agreed with the known data on the Stark effect (Stark, 1914d). For atoms having a more complex structure than the hydrogen atom — and the one-electron ions, He + , L i + + , etc. — more spectral series should show up, because their frequencies depend on four quantum numbers, n, n' and m, m' due Eq. (10). These frequencies could be written as v = const.[/„/ (n) - / m / (m)],

(10')

where fn'{n) — <j>{n,n') and / m ' ( m ) = <j>(m,m'). Thus different series should arise because of different values of the parameters n' and m'. Sommerfeld suggested especially the following relations: the spectroscopic 'principal' series arose from the combinations fs{n) — fp(m), with n = 1 and m — 2, 3, 4, etc.; the 'first subordinate' series from combinations fp(n) — fd(n), with n = 2 and m = 3, 4, 5, etc.; and the 'second subordinate' series from combinations fp(m) - fs(n), with n = 2 and m = 3, 4, 5, etc. (The fp, fd and fs, denote functions with special values for the quantum numbers n' and m'.) At that point Sommerfeld was aware of the possibility that the structure of orbits might be even more complicated than considered so far. He noted: 'At any rate, when one considers the structure of atoms and the shape of electron orbits, the restriction [of the motion] to the plane — which so far has been extended from the hydrogen atom to other elements — cannot be upheld ultimately' (Sommerfeld, 1915b, p. 453). In order to describe spatial orbits in quantum theory he proposed to introduce a third quantum number, n", which determines the phase integral involving the coordinate z perpendicular to the original plane. Sommerfeld concluded his first paper on the structure of the hydrogen atom by answering the question which Poincare had posed in 1911: how far did the quantum conditions, Eqs. (12) and (15), depend on the coordinate system? For this purpose he started from the relation of the phase integral to his older action integral, that is,

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Physics

Tdt = Y, •*

Pidqi = h(n + n' + • • • ) .

U7)

i=l J

where T denotes the kinetic energy of the system under investigation. Since in a periodic system the left-hand side (= 2rT, where T is the average kinetic energy) is a constant of motion, the sum of quantum numbers, n + n' -\ , cannot depend on the coordinate system. In the case of the hydrogen atom, which he described by two quantum numbers, n and n', Sommerfeld argued that both quantum conditions, Eqs. (12) and (15), did not depend on the coordinate system: on one hand, the angular momentum p
T h e important point in the treatment of the hydrogen atom, as Sommerfeld noticed, was that the angular coordinate <j> represented a cyclic coordinate; i.e., tj> did not occur in the Hamiltonian function describing the hydrogen atom (Sommerfeld, 1915b, pp. 454-455). 35 T h e doublet structure of the hydrogen lines had been discovered by Albert A. Michelson with the help of interferometric methods (Michelson, 1891). In the following decades several experimentalists had measured the frequency differences. 36 T h u s Paschen wrote to Sommerfeld on 7 February 1915: I am enclosing a calculation of the Pickering series of hydrogen and note that Fowler — using a Geissler tube filled with hydrogen and helium — obtained at high voltages the series which have nearly the same values as Pickering's. On the basis of Bohr's theory Fowler thinks t h a t they are possibly helium lines. Because of the deviations of Pickering's wavelengths [from Fowler's], however, one cannot exclude the possibility t h a t Pickering's stellar lines are not identical with Fowler's and represent hydrogen lines. T h e question is still open, and it is treated here by one of my students who, at present, is [on military duty] at the front. I wish to inform you meanwhile that also according to our own [experimental] researches, Fowler's lines are probably helium lines, but we leave open the possibility that Pickering's lines are different and belong to hydrogen. We shall publish the proofs only after all the experiments have been completed. Unfortunately this cannot be done now because of the war.

Arnold Sommerfeld

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of the year Paschen reported his first results on the separation of hydrogen doublets: 'My measurements yield: A A ^ = 0.09 A' (Paschen to Sommerfeld, 12 December 1915). At the same time he discussed possible experimental errors, such as the ones caused by Doppler effect and Stark effect. He also did not exclude the possibility, mentioned earlier by Bohr (1914b, p. 521), that the duplicity of the hydrogen lines might come from a Stark effect within the atom. Only detailed calculation, thought Paschen, would decide this question. Several weeks later, after having received copies of Sommerfeld's reports to the Bavarian Academy, he wrote again: 'The results of your theory, which you have reported, are very attractive. I believe your notion [concerning the origin of doublet structure] is more probable than mine' (Paschen to Sommerfeld, 27 December 1915). Paschen carried out further experiments, and he finally concluded: 'My measurements are now finished, and they agree everywhere most beautifully with your fine structures' (Paschen to Sommerfeld, 25 May 1916). The experimentalist Paschen was completely satisfied, for now the experimental and theoretical results agreed within an error of 1 0 - 3 A units. 37 Sommerfeld had based his theory of fine structure, to which Paschen referred above, on the assumption that the doublets observed in the spectrum of hydrogen arose from 'the finitely different, discrete eccentricities of our quantized ellipses' (Sommerfeld, 1915c, p. 461). The calculation of this effect was straightforward. Sommerfeld started simply from the equation of motion of a relativistic electron (with mass me/^l — v2/c2, where me is the rest mass of the electron, v its velocity, and c the velocity of light in vacuo) in a Coulomb field of a stationary nucleus of charge Z\e\, and obtained the total energy W, expressed as an expansion in rising powers of a quantity b2, as

7

4 7J

87*

(18)

where W0(= — (m e e 4 Z 2 /2p^)(l-e 2 )) is the nonrelativistic energy, and the quantities 6 2 (= Z2e4/p2j>c2) and 7 2 ( = 1 — b2) become, in the nonrelativistic limit (for small velocities of the electron), equal to zero and unity, respectively. In applying the quantum conditions to the relativistic Kepler motion, Sommerfeld had to take into account the fact that the orbit in this case was not stationary (or closed), but that

The main difficulty in the experimental investigation consisted in establishing reliable absolute value for the observed frequencies. 37 Paschen published a full account of his results on the spectra of hydrogen and ionized helium in a paper, entitled 'Bohrs Heliumlinien' ('Bohr's Helium Lines') and submitted it in late June 1916 to Annalen der Physik. The final remarks in this paper were: 'That this work has advanced beyond the stage of a not-understood-contradiction with Bohr's theory, is due to the helpful encouragement of my colleague Sommerfeld, whose untiring efforts enabled us to recognize in the incomplete experimental d a t a the wonderful laws of his beautiful theory. For this I wish to express my grateful thanks to Mr. Sommerfeld (Paschen, 1916, p. 940).

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its perihelion advanced after each revolution by an angle (27r/7) — 27r.38 Hence he replaced the quantum conditions (12) and (15) by

I

2

*h

o

and

f

2TT

Prf,# = p
(12')

r2ir/-y

= nh-t2 ( ' -l)=rih. /o Vvl Jo As a consequence, the eccentricity e was given not by Eq. (16) but by Prdr

(15')

Thus the energy W, Eq. (18), finally depended on both quantum numbers n and n' separately (due to the factor 7 2 in front of n in the denominator of the expression for the eccentricity). Sommerfeld rewrote it as an expansion, w

=

_2n2meeiZ2 h2{n + n')

aZ2 +

(1 2

+

n_

{n + n') V 4 n .

"

(19)

where a represented a small dimensionless parameter, 27re2 he

(19a)

which assumed the value 0.7 x 1 0 - 3 (see Sommerfeld, 1916b, p. 51, Eq. (12a)). It should be noted that for obtaining Eq. (19) Sommerfeld did not use the quantum condition (12') but rather the condition (12). 39 The reason for his procedure was that in the other case the factor multiplying the term linear in a became such Sommerfeld knew about the perihelion shift from the corresponding astronomical problem. According t o special relativity the perihelion shift, e.g., of the planet Mercury, would be 7" of arc per century, while in general relativity Einstein obtained 43" per century. In the case of the hydrogen atom the force of attraction is the Coulomb force; the gravitational forces may be neglected in comparison with it, hence general relativity does not give any sensible correction to the perihelion shift in the electron orbit in the hydrogen atom. (See also Sommerfeld, 1915c, pp. 467-468.) 39 Sommerfeld motivated this step in an addendum to his fine-structure paper, dated 10 February 1916, with the following words: The new Ansatz may also be understood perfectly well as a quantum Ansatz from the point of view of a coordinate system corotating with the rotation of the perihelion, by the introduction of which [i.e., of the new quantum Ansatz] the problem of the distribution of quanta in the relativistic case is reduced to the same problem in the non-relativistic case, as described in our paper I [Sommerfeld, 1915b]; thus any arbitrariness or uncertainty occurring in the Ansatz has been removed. (Sommerfeld, 1915c, p. 500) By combining Eq. (12) with Eq. (15') one found the result, 1 — e 2 = 7 2 n 2 / ( n ' +*fri)2, rather than Eq. (16'), and this — when substituted in Eq. (18) — yielded Eq. (19). (Note that Sommerfeld, in his paper for the Bavarian Academy, denned a different parameter a, namely, (7r 2 e 2 //ic) 2 , hence his expression for W, the relativistic hydrogen energy (Sommerfeld, 1916b, p. 56, Eq. 22) looks very different from our Eq. (19).)

Arnold Sommerfeld

and Atoms as Conditionally

Periodic Systems

393

that it disagreed with the spectroscopic data. In his final paper for Annalen der Physik (Sommerfeld, 1916b), Sommerfeld excluded still another possibility. Since the quantity j 2 had to be positive, it seemed to follow that the angular momentum Ptf, assumed a minimum value, i.e., Ze2/c. Hence, instead of Eq. (12), one might expect the quantum condition

2n(P4>-^-]

= nh

(12")

for the azimuthal degree of freedom. Such a condition also followed from Max Planck's quantum theory of systems having several degrees of freedom (Planck, 1916a, §11). Karl Schwarzschild commented on it in a letter to Sommerfeld: 'Since the lower limit of p^ is not zero, but e2Z/c, it seems to me that the discrete values of pj, must be put equal to e2Z/c + {h/2ir)n (n = 1,2,3,...). This does not change the doublets, but spoils the Balmer formula' (Schwarzschild to Sommerfeld, 21 March 1916). Sommerfeld definitely wanted to keep the Balmer formula. The relativistic correction only had to determine the fine structure of each Balmer line. In that connection he added an historical remark: The quantity a, which will play an important role in all the following formulae, is also of some historical interest. Einstein remarked once [Einstein, 1909a] that the quantity e 2 /c has the same dimension and order of magnitude as h, and thought of deriving h from this quantity. Later Jeans [in his report to the Birmingham Meeting of the British Association] thought that the accurate form of the numerical factor was given by the equation h/2ir = (47re)2/c. Now we see that this numerical factor, namely our a above, becomes the fundamental quantity for describing the fine structure of spectral lines. (Sommerfeld, 1916b, pp. 51-52) The consequences of the relativistic energy formula (19) were quite evident. An energy term having (n + n') equal to 2 occurred as a doublet, due to the two possibilities of distributing 2 among n and n' (excluding n = 0, of course); a series term with quantum number 3 occurred as a triplet, etc. This multiplicity of terms would result in a rather large number of fine-structure components of the Balmer lines. Sommerfeld, however, reduced the number of possible transitions by additional conditions, stating that the quantum numbers of the initial state, m and m', always had to be larger than or equal to the corresponding ones, n and n', of the final state, that is, 40 m>n and m!>n'. (20) Thus he obtained the following fine structure in the case of Balmer lines. The constant term (with n + n' = 2) possesses a doublet structure and causes a frequency separation AI/H = 2/K2meeAac/l&h2 (or 0.31 c m - 1 -c). Each doublet of the Balmer series is accompanied by satellites; i.e., each i/ Q -component by one, each Hpcomponent by two, each i? 7 -component by three satellites, etc. 'Because of the large 40

Sommerfeld derived the relations (20) from the positivity of the phase integrals (Sommerfeld, 1915b, p. 448).

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The Golden Age of Theoretical

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uncertainty of the if-lines and the small distances of these additional components, there is little prospect of observing them,' wrote Sommerfeld (1915c, p. 480). 41 Similarly the so-called Paschen series of hydrogen, whose constant term possesses the quantum number n + n' = 3 , consists essentially of triplet lines with satellites, the triplets being separated by frequency differences (8/81)AVH and (&/21)AvH, respectively. In the case of the corresponding series of ionized helium, the latter differences had to be multiplied by a factor of 16; Paschen's data on these helium 'triplets' indeed agreed perfectly with Sommerfeld's theoretical prediction. Also in the X-ray spectra of many-electron atoms the fine structure had to show up; the separation of the components there became very large due to the fact that the effective charge entering, say, the if-spectra was (Z — 1) from Moseley's observations. 42 Sommerfeld's quantization method of periodic systems with several degrees of freedom immediately obtained an important application. His former student, Paul Sophus Epstein, then interned as an enemy alien in Munich but in a position to work on theoretical problems, attacked with its help the calculation of the Stark effect of spectral lines. 43 His interest in this problem had been aroused a couple of years earlier, when Stark had come to Munich to talk about his experimental discovery. Now, in early 1916, Epstein succeeded in integrating the equations of motion of an electron, moving in the Coulomb field of a nucleus of charge Z\e\ and an external homogeneous field of strength |E|, by using parabolic coordinates, £ and rj, in the plane defined by the direction of the electric field (Epstein chose the x-direction) and the radius vector joining the nucleus (at the origin) and the electron. 44 As a third variable of the problem, Epstein chose ij), the angle between the above-mentioned (moving) plane and a fixed plane also going through the origin. In these variables, the equations of motion of the electron were separable; in particular, there followed two equations, one involving only the coordinate £ and the canonically conjugate momentum p$, and another involving 77 and pv, which could be integrated to yield the equations

Pt = y/K(0 41

and

Pr,

= y/Mnj,

(21)

I n the case of Ha the satellites are separated from the components by the differences (8/81) AVH and (8/27) A I / J J , respectively. 42 Sommerfeld similarly discussed the L-spectra, where the effective charge is Z — I, where I is of the order of 8 (Sommerfeld, 1915c, pp. 496-497). 43 P . S . Epstein was born in Warsaw on 20 March 1883. He studied at the University of Moscow from 1901 to 1906, where Petr Nikolaevich Lebedev was his teacher. Then he became assistant (1906-1909), docent (1909-1910) and assistant professor (1910-1911) at the Agricultural Institute in Moscow. In 1911 he went to the University of Munich to continue his studies in theoretical physics under Sommerfeld. He was particularly interested in relativity theory and published his first paper on this subject in Annalen der Physik (Epstein, 1911). In 1914 he obtained his doctorate with a thesis on an optical problem: the diffraction of light from a parabolic cylinder. After the outbreak of World War I he stayed on in Munich and worked for his Habilitation, which he received in 1919 at the University of Zurich. Two years later, Epstein went to the United States and became Professor of Physics at the California Institute of Technology. He retained this position until his retirement in 1953. He died on 9 February 1966 in Pasadena. 4 T h e parabolic coordinates are related to Cartesian coordinates x and y in this plane by the equations x = | ( f 2 — n2) and y = £r).

Arnold Sommerfeld and Atoms as Conditionally Periodic Systems

395

/i(£) and f2(v) being functions of the third order in £ 2 and T/2, respectively. Also the Hamiltonian function could be written as a sum of three separate terms, i.e., H — s/m{ ;<*!> + Jo

f y/hffidt + / " y/W)dri,

(22)

Jo

involving the same functions /i(£) and ^2(77) and a constant a connected with the angular rotation described by the angle V>-45 By quantizing the three degrees of freedom according to the phase-integral method with quantum numbers n i , ri2 and 713 for the £-, 77-, and V-phase integrals, Epstein obtained for the energy W the expression 27r 2 m e e 4 Z 2 3/i 2 |E| , , ,00, = ~T2i 7-^-, ^ ~ Q 2 711 ("i ~ n 2 ) ( " i + "2 + na) • (23 h2(ni + n 2 + n3)2 8Tr2meZ\e\ On the right-hand side he neglected terms of higher order in the electric field strength |E|. He identified the first term on the right-hand side of Eq. (23) with the energy of the atom in the absence of an external electric field.46 The effect of the electric field on the spectral lines, therefore, yielded a separation of the frequencies, according to the formula W

Ai/ =

3/ilEI i\ I K m i - "i 2 )(mi +m2 + m 3 ) - (n x - n 2 )(ni +n2+

2

n3)},

(24)

where the sets of triple numbers, (mi, 7712,7713) and (711,712,713), refer to the initial and final states of the one-electron atom, respectively. By substituting positive integral values for these triple numbers into Eq. (24) Epstein found that the Stark components of a hydrogen line lay symmetrically on both sides of the original frequency. He then used Sommerfeld's 'selection rule' ('Auswahlprinzip,' i.e., ni < m~i, 7i2 < 7712, and 713 < 7713), and arrived at a complete description of Stark's data. Finally, from the data he concluded the empirical rule that 'even-numbered differences 7713 — 713 give rise to p-polarization [i.e., polarization in the direction of the external field], while odd-numbered to the s-polarization [i.e., polarization perpendicular to the electric field]' (Epstein, 1916a, p. 148, footnote 3). Epstein submitted a short note on his results on 26 March 1916 to Physikalische Zeitschrift, where it appeared in the issue of 15 April (Epstein, 1916a). He knew that he had to hurry because at the same time Karl Schwarzschild was working on the same problem. Hence he wanted to establish his priority, even before the detailed publication of the theoretical derivation was ready; it appeared finally in the issue of 45

p^,, the momentum variable corresponding to 1/), is given by i/ma. The fact that three quantum numbers replace the two quantum numbers of the one-electron atom in the field-free case does not change its energy. Evidently, Epstein's sum, ni + ni + 713, must be equal to Sommerfeld's n + n't in that case. 46

396

The Golden Age of Theoretical

Physics

25 July of Annalen der Physik (Epstein, 1916c). 47 The astrophysicist Schwarzschild had become interested in the problem two years earlier and had discussed the Stark effect from the point of view of classical theory (Schwarzschild, 1914a). 48 In it he obtained the correct order of magnitude of the separation of the components. 49 On 30 March 1916 be presented his paper, entitled 'Zur Quantenhypothese,' at a session of the Prussian Academy; it was published on 11 May, the day he died. In it, Schwarzschild again studied the motion of an electron in an atom under the influence of an electric field, but in contrast to his previous paper he now used quantum theory explicitly, especially Planck's and Sommerfeld's quantum treatment of systems having several degrees of freedom.50 For the 'conditionally periodic' Epstein later recalled his competition with Schwarzschild on the quantum theory of the Stark effect as follows: I started to work [on the theory of the Stark effect] Now after some time again I saw Sommerfeld. It was already in the middle of the war. I had no key to the Institute anymore, being an alien, but Sommerfeld was interested in my access t o literature, because I wrote an encyclopedia article [for the Encyklopddie der mathematischen Wissenschaften — Epstein had already written an article in 1915 on special diffraction problems in optics]. He was the editor of this volume [V]. So at one of the reunions, he told me, 'I wrote Schwarzschild, that he should work on this article.' Now I was a little crestfallen, because I regarded this as a stab in the back, since he knew that I was writing it. And Schwarzschild was a mathematician of unbelievable energy; he could do everything in a twinkling. I of course couldn't reproach him, but I decided, 'Now I have no prospects unless Schwarzschild should go to heaven.' And the next day when I was going to bed, I had the idea of the limit. You see, I knew already how the electron moves, and I knew how to do it. I got up at 5 o'clock the next morning and by 10 I had the formula. And then the same morning I brought it to Sommerfeld. And what do you know, the same afternoon he got a letter from Schwarzschild, and Schwarzschild had the wrong formula. It was the same order of magnitude, but didn't agree, on t h e positions of the lines. Sommerfeld wrote Schwarzschild, 'This morning Epstein brought me the formula of the Stark effect, and this afternoon we got your letter. But Epstein's formula agrees with the observation.' When Schwarzschild got [his result], he immediately announced in the Berlin Academy that he would speak about it. And he did, before he wrote the letter to us, so he reported it wrong in the Academy. By that time however I had already sent my announcement that came out just one day before he delivered that lecture in the Academy. But in the lecture he corrected it, and in the proof he removed all the discrepancies, and it came out correct also. Of course the two final papers came out much later. So I had the priority by one day. (Epstein, A H Q P Interview, 25 May 1962, p. 11). 48 K a r l Schwarzschild was born on 11 October 1873 at Frankfurt-am-Main. He studied astronomy at the Universities of Strasbourg (1891-1893) and Munich (after 1893), obtaining his doctorate under Hugo von Seeliger in 1896. Three years later he became Privatdozent in Munich, and in 1902 Professor of Astronomy at the University of Gottingen. There he became interested in problems of astrophysics, especially the radiation emitted from the sun, and in electron theory. He continued both of these interests after his appointment as Director of the Potsdam Observatory in 1909. During the last year of his life he published two important papers on Einstein's theory of gravitation. Schwarzschild died on 11 May 1916 in Berlin. (See Schwarzschild's obituary notice by Sommerfeld, 1916a). 49

Schwarzschild found, in particular, the solutions of the classical equations of motion and expanded them into a Fourier sum; he identified the frequencies which occurred there with the frequencies of emission. Thus he arrived at the result that each spectral line of the free atom would split into a symmetrical triplet under the influence of a homogeneous electric field. 50 Schwarzschild explained the fact that the earlier classical calculation yielded the correct order of magnitude for the Stark effect without any reference to the quantum hypothesis in the following

Arnold Sommerfeld

and Atoms as Conditionally

Periodic Systems

397

systems ('bedingt periodisch,' Schwarzschild 1916, p. 548) of / degrees of freedom the integration of the equations of motion led in general to solutions for the position coordinates, which depended only on / angle variables Wi (which increased linearly in the course of time, i.e., Wi = 2-mJit + const.) and / constants a*.51 The latter also represented the momenta, canonically conjugate to Wi, and were called action variables. The / pairs of action and angle variables thus described the system fully, and Sommerfeld's phase-integral conditions could be applied, yielding discrete values for the action variables, ai = rii—+Ci

(j = l , . . . , / ) ,

(25)

with the constant c* possibly depending on the degree of freedom. In order to obtain the solution of the equation of motion of a quantum-theoretical multiply periodic system one had, therefore, first to introduce action-angle variables by a (classical) canonical transformation of the original variables, say with the help of the Hamilton-Jacobi differential equation. The energy function of the system could then be expressed by action variables alone, and through the quantum conditions the discrete energy values followed immediately. Schwarzschild used his method to calculate the Stark effect in one-electron atoms and the motion of free body rotating around its centre of gravity. 52 While Epstein gave the first satisfactory solution of the Stark effect on the basis of the Bohr-Sommerfeld theory of atomic structure, Peter Debye, another former student and collaborator of Sommerfeld's, dealt with the influence of external homogeneous magnetic fields on the spectral lines. 53 In order to obtain his results on the Zeeman effect, which he first presented to the Gottingen Academy on 3 way: the frequencies of the electrons in stationary orbits are of the order of magnitude of the limiting frequencies of the hydrogen series; now there existed a relation between these limiting frequencies and the quantum of action. Through that backdoor Planck's constant sneaked into Schwarzschild's classical result (Schwarzschild, 1914a, p. 24). 5 ^SchwarzschiId's approach to the atom as a multiply periodic system went back to Carl Gustav Jacob Jacobi (see Jacobi, 1866). Jacobi had solved the Kepler problem (of attraction of one body by the gravitational field of a second heavier body) by using the partial differential equation for the action function (the so-called Hamilton-Jacobi equation); especially, he had shown that two separate differential equations arose, each involving only one set of dynamical variables. Later his method of the separation of variables in the differential equation for the action function was extended by Paul Gustav Stackel in his Habituation thesis dealing with, as t h e astronomers said, 'conditionally periodic' systems of any finite degree of freedom (Stackel, 1891). As suitable variables for the separation were the action-angle variables; they had been employed, for example, by the French astronomer Charles-Eugene Delaunay in his theory of the lunar motion (Delaunay, 1860, 1867). Henri Poincare (1892, 1893, 1899) and Carl Wilhelm Ludwig Charlier (1902, 1907) used them extensively in their books on celestial mechanics. 52 I n his calculation of the Stark effect Schwarzschild assumed that the atomic system exhibited a certain degree of degeneracy: he claimed, in particular, that the electrons in the presence of electric fields also described closed orbits; hence the motion was degenerate and he had to introduce special quantum conditions (similar to the ones Planck had used in late 1915) which took into account this degeneracy. In a further paper, submitted in August 1916, Epstein pointed out the essential equivalence of Schwarzschild's treatment to his own (Epstein, 1916e). 53 Debye had already concerned himself earlier with Bohr's theory of atomic constitution and had used Bohr's model of the hydrogen molecule (Debye, 1915a).

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June 1916 (Debye, 1916a) and later, in September, also submitted to Physikalische Zeitschrift (Debye, 1916b), Debye used Schwarzschild's method of integrating conditionally periodic systems. In particular, he solved the Hamilton-Jacobi differential equation for the one-electron atom in the presence of a static magnetic field of strength |H| in spherical coordinates, (the radius) r and (the angles) 9 and <j>, that is, 2 2

d£ ^ejf^\ dt

±{9S\ + ^2
2

sin 2

+ ~(27r^)2r2sin20+^=O,

(26)

where VL was given by the equation

and, was, therefore, equal to the classical Larmor frequency. In the approximation that the term quadratic in VL, could be neglected, this equation yielded a solution for the action function S, in which S appeared as the sum of three integrals, R, 6 and $, plus a term proportional to the time; the integral R depended only on the radial variable r, the integral 0 only on the angle 6, and the integral $ only on the azimuthal angle . These integrals had now to satisfy Sommerfeld's quantum conditions; i.e., if one extended the integrals R, 0 and $ over a full period of the variables involved — the path of integration in R ranged from a minimum value r min to a maximum value r m a x and back to r m j n , the integration path in 0 ranged from 9 = 9Q to 9 — -K — 0Q with 9Q a constant angle, and the integration path in $ ranged from <j> = 0 to <j> = 2-K — then each of them had to be equal to an integral multiple of Planck's constant h. Denoting their associated quantum numbers by mi, vn.2 and 7713, Debye found for the action integral the result

S = f{r,0,4>) +

2n2mee4Z2 h2

1 (mi

+m2

+ TO3)2

' L + m^hvL t,

(27)

where f(r,9,<j>) is a function of the position coordinates alone. Now the terms in the square bracket represent the energy of the system; hence the frequencies of the one-electron atom in the static homogeneous magnetic field became 27r 2 m e e 4 Z 2

h3

1 (mi + m 2 + m 3 ) 2

1 + (m 3 - n 3 )j/ L , (n\ + n 2 + n 3 ) 2

(28)

where the quantum numbers mi,7712,7713 define the initial and 711,712,713, the final state of the atom. Evidently, the first term had to be interpreted as the frequency of the free atom, and the second term as the shift due to the magnetic field. Prom the occurrence of the Larmor frequency VL in the result, Debye concluded that the normal Zeeman effect 'also occurs here with the right magnitude as a consequence

Arnold Sommerfeld and Atoms as Conditionally Periodic Systems

399

of the general quantum hypothesis' and that the 'usual triplet arises if one admits for 7713 and 713 only the smallest integral numbers' (Debye, 1916b, p. 511). Debye also used Schwarzschild's approach to derive Sommerfeld's results concerning the relativistic fine structure (Debye, 1916c). At the same time, Sommerfeld independently concerned himself with the Hamilton-Jacobi differential equation for the one-electron inhomogeneous magnetic and electric fields and for the relativistic one-electron atom (Sommerfeld, 1916d). Like Debye, he now postulated the quantum conditions for the action function S; i.e., he required that the equations

/

^•dqi

= nih

(i = l , . . . , / )

(29)

be satisfied for multiply periodic systems of / degrees of freedom. These were equivalent to the integrals, f Pidqi, for the partial differential quotients, dS/dqi, did represent the momentum variables pi. Sommerfeld called the integrals, J(dS/dqi)dqi, with paths extending over a full period of the system, the 'moduli of periodicity' of S, and formulated the quantum condition as: 'The moduli of periodicity of the Jacobi action function are integral multiples of Planck's quantum of action' (Sommerfeld, 1916d, p. 499). The solution of a quantum-theoretical multiply periodic system thus implied two steps: first, one had to introduce variables in which the action function consisted of a sum of terms, each depending only on one pair of canonically conjugate variables — in these variables, then, the problem was separable; second, one had to evaluate each term in the action function, especially to calculate their moduli of periodicity. The evaluation of the angular phase integrals presented no difficulty; however, the radial phase integral possessed the general structure

J^dr = f\lA+ 27 + ^dr>

(3°)

where the quantities A, B and C were independent of the radial variable. 54 Sommerfeld now carried out the integration, which went along a closed path from r = rmin to r = r m a x and back (which was indicated by the closed circle) in the complex r-plane. There the integrand, the square-root function, possessed a cut on the real axis from rm\n to r m a x and two poles, one at r = 0 and the other at r = 00. Then 54

The radial phase integrals of the one-electron atom (in the presence of a magnetic field) and of the relativistic one-electron atom did have the structure given in Eq. (30). In the case of the Stark effect the position coordinates were the parabolic ones, f and 7/; then the phase integrals, f p$d£ and f pvdn, could both be transformed to f

I

A+

2~B

J \l ^

C

+ +Dxdx

^

f

I

A+

2B

= J \l ir

+

C~

1

dx+ D

f

xi 2 J

y/A + 2B/A + C/x2

dx,

where A, B, C and D are quantities independent of x. The evaluation of the second integral can be reduced to that of the first: if f \JA + B/x + C/x2 dx becomes a function J(x), then the second integral will be (D/2)(dJ(x)/dx).

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The Golden Age of Theoretical

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he replaced the original contour of integration by an equivalent one, i.e., two circles around the singularities at r = 0 and r = oo and applied the residue formula, obtaining 55 /

V^ + 2 7 + ^

= -™ (v^ - 3=) •

(31)

By adding this method of complex integration to Schwarzschild's extension of the Hamilton-Jacobi theory of conditionally periodic systems, Sommerfeld arrived at a powerful tool for dealing with atomic problems. 56 The Sommerfeld-Schwarzschild theory would dominate the calculations of atomic structure in years to come. In spite of the fruitfulness of the theory of multiply periodic systems in dealing with the problems of atomic physics, a difficulty remained: the method worked only for those problems in whose dynamical description the variables associated with each degree of freedom could be separated. However, not all Hamiltonians occurring in atomic physics possessed the property of separability. Hence Einstein, in a contribution entitled iZum Quantensatz von Sommerfeld und Epstein' ('On the Quantum Law of Sommerfeld and Epstein') and presented at the meeting of the German Physical Society in Berlin on 11 May 1917, proposed a more general method, which made it possible to treat nonseparable problems as well (Einstein, 1917b). In it he started from the observation that for an arbitrary system of / degrees of freedom only the sum, Y^iPi <%> where i runs over all the degrees of freedom, was an invariant quantity and might, therefore, be assigned the definite value of an integral multiple of Planck's constant. Still Einstein claimed that it was possible to derive several quantum conditions similar to the ones used by Sommerfeld, Schwarzschild, Debye and Epstein. For this purpose he considered integrals of the sum YliPi ^9«i extended over a closed path in the space of the position variable qt. Einstein found that for any problem of atomic theory there existed a finite number of closed paths for which the integral §Y^iPid