Invariance: 100 Years of Relativity

e

E B * A i u ? f F -0

(1_0>)-I/»

k

7

(V, v, W) (X, Y, Z)

(Xu y„ z,)

1 Electric displacement and magnetic intensity vectors in the original are here represented for the sake of agreement with present common practice, by current symbols for electric intensity and magnetic induction vectors, respectively. This includes Maxwell's displacement current density. 0 In units of the speed of light in oacuo (i.e., in units such that c = 1).

[ 9 ] "I have likewise not hesitated to publish these few partial results, even if at this very moment the entire theory may appear to be in danger by the discovery of cathode rays." 3 1. Lorentz Transformation The "fundamental formulas" {of the MaxwellLorentz theory), using rationalized Gaussian units and choosing the units of length and of time so that the speed of light equals unity, are (see Table I! 4 : i=(dE/dt)+pu= B=vxA,

VxB,

(1)

E = - ( r ) A / d < ) --V4>,

(2)

dB/d<=- V xE,

(dp/dt) + V-p\i = 0,

(3) •

[1.1]

(4)

V - E = P,

(•">)

d/dt+V-A = 0,

(6)

•*=-p.

(7)

DA=-pu,

82 Lorentz and Poincare Invariance f = „[E+(uxB)].

[1.2]

(8)

[10] "These equations admit of a remarkable transformation discovered by Lorentz, which is of interest because it explains why no experiment is capable of making known to us the absolute motion of the universe." This transformation has the form f see Table 11 x' =

yl(x-0t),

t' = z' = lz,

u' = ly,

:

MI

yl(t-fix), [1.3]

(9)

where 1 = 1(13) is, to begin with, an arbitrary constant. 5 The associated transformation of our Eqs. ( l ) - ( 8 ) can be deduced step by step, beginning with Eqs. (7). We determine first the transformation of the charge density p. This is accomplished by combining the kinematic law of transformation of volumes with the following physical assumption 6 : The charge of an electron is an invariant of our transformation.

(10)

The law of transformation of volumes can be obtained as follows7: "Let us consider a sphere carried along with the electron in its uniform translational motion, and let ( x - u 0 2 = r2

^&^0

be the equation of this moving sphere, whose volume is 4irrV3. The transformation will change it into an ellipsoid whose equation is easily found. In fact, one easily deduces from Eqs. (9) x=

yl-'(x'+(3t'),

t=

yl-l(t'+0x'),

: = l-

[1.3 h's]

(9')

Thus the equation of the ellipsoid becomes 7s[/i'+(/3-«i)i'-|3«1j:']s

+ t [x/-7«,a'4-/?A')]2 = /V.

Chap. 2. Special Relativity ... This ellipsoid moves uniformly; {i.e., its equation) reduces to

^ ( l - f e ) ^ £

83

for t'=0 -it

(zj'-ypUixSy^l-r2,

i-i

and it has the volume (iirr3/^)li/y(l

— ffui)."

From the ratio of volumes thus obtained and our assumption (10) it follows that the transformed density p satisfies the equation p' = 7 i - ' ( l - / 9 u i ) p .

[1.4]

(11)

Since by Eqs. (9) the transformation equations for the particle velocity vector u are given by the following "rule for addition of velocities," «I'=(MI-/3)/(1-/3M1),

w/ = u y / T (l-/3ui) (7 = 2,3),

(12)

[1.4 6i's]

(13)

we have also p'u1' =

yl-3{u1-0)p,

p'uj = l-lpUj

( j = 2,3).

Comparing these results with those of Lorentz's paper [loc. int., Eqs. (7) and (8)], it is seen that there is agreement for the transformation of pu, but not of p. In addition, the equation of conservation of charge, Eq. (4), cannot hold for the primed quantities of Lorentz since they do hold in the present case. In fact, consider the Jacobian D of (t+\p, x+Xpu) with respect to (t, x). We find that dD/a\\x^~Di=(.dP/dt)

+ V-pia.

(14)

On the other hand, since (t' + \'p, x' + X'p'u') is for \'= l*\ connected with the corresponding unprimed quantities by the Lorentz transformation (9), it follows that D' = D* Hence ZV = l~*Dh and our result follows from Eq. (4) and the definition (14) of Di. Now, it is found from (9) that [notation in (7)] a ' = Z- 2 Q

(!•">)

Invariance Hence it can be deduced from Eqs. (11) and (13) that the .solution of Eqs. (7) in the primed quantities is3 *'-7*-l(*-/M,)F

A/ = l-iA,

( j = 2,3).

[1.7]

(16)

[11] "These formulas differ considerably from those of Lorentz, but the difference bears in the last analysis only on the definitions." Applying the relations (2) to the primed quantities, we obtain the transformation equations for the field vectors, Ex' = l->E\, E,/ = yl-*-{Ey-pB,), E/ = yl-*{E,+0Bv),

B/ = l--Bx, By'^yl-^Bi+fiE,), B,' =

yl-*(B,-0Ev), [1.9]

(17)

which are identical with those of Lorentz. The proof of the validity of the remaining equations [1.1] in the primed quantities [i.e., the proof of their covariance under Lorentz transformations) now follows from the covariance results already obtained. In fact, Eq. (6) can be deduced from Eqs. (7) and (4); and Eqs. (1), (3), and (5) can be deduced from Eqs. (7), (2), and (6). It remains to discover the law of transformation of the force density f of Eq. (8). This is accomplished by assuming the covariance of Eq. (8). Use of Eqs. (11), (13), (17), and (8) then yields the result fx' = yt*(h-fri-f), f/ = l~6fi

(J = 2,3).

[1.11]

(18)

{For instance,

X[(l-0ui)£i+«s(3i-|8ff*)-ui(fls-/3£1)] =

l-iyP{El+ihB3-u3B.i-l3\i-E),

then using Eq. (S).J

Chap. 2. Special Relativity ...

85

By combining Eqs. (IS) and (11) we obtain the transformation equations for the force- F per unit charge:

F/ = i-V-lF;

U = 2,3).

[1.11 W«]

(19)

The.se equations differ significantly from those found by Lorcntz [loc. tit., Eq. (10)]. [12] "Before going further it is of importance to seek the cause of this significant divergence. It evidently derives from the fact that the formulas for u/ are not the same, whereas the formulas for the electric and magnetic fields are the same." Considering now the stability of the electron, it is seen that on the assumption of an exclusively electromagnetic origin of its inertia, the condition of its equilibrium requires the vanishing of f in its interior, and in view of Eqs. (18), this is indeed a covariant condition. Unfortunately, it is not an admissible condition; for instance when u = 0, we must have E = 0 , and hence p = V • E = 0. So we must look for additional noneleclromagnetic forces that insure the electron's equilibrium and find conditions that they must satisfy "in order that the equilibrium of electrons is not disturbed by the transformation." 2. Principle of Least Action The principle of least action for Lorentz's equations is formulated here in a form that differs slightly from that presented by Lorentz.1" We start with the action integral 7 = /di/cfr[(EJ+B2)/2-A-i] (dr=d 3 z),

[2.1]

(20)

where the quantities to be varied are taken to be connected by the relations V-E=p,

B = VxA,

i=(dE/dt)+pu. [2.2]

(21)

./ is to be minimized subject to the following additional conditions:

86 Lorentz and Poincare Invariance The state of the system is fixed at the integration limits t = U and t = ti. "All our functions vanish at infinity." u

(22) (23)

Varying at first only A, we find after integrating by parts, and using the second relation in (21) and condition (23), 0 = &/= JcMr[B • (V x 5A) - i • 5A] = -Jdldr(i-

(24)

VxB)-5A,

which implies the field equation (1). When the latter equation is substituted in Eq. (20), we obtain, again by an integration by parts and use of the second of Eqs. (21), the alternative expression J = fdtdTU&-B*)

[2.4]

(25)

:

. • =fr«-==3 "lj' f U h ^

Frrv - —

5 i; v, ^-

-»

-< J>~- ( t ' ~ — - -

/C

{which serves as the starting point for the relativists discussion in the next section). Returning to the expression (20), we now vary the other quantities. We apply Eq. (1) [see Eq. (24) ] and allow for the first of Eqs. (21) with the aid of the Lagrange multiplier . Our variation assumes then the form &/ = /
<;

[2.7]

(26)

When 5p=0 and 5(pu) = 0 , the vanishing of this variation yields the second of Eqs. (2). If we now allow also for this equation, then Eq. (26) reduces to &/=/<&M>5p-A-$(pu)],

[2.9]

(27)

and this expression must lead to our remaining equation, Eq. (8). Indeed, if | = |(xo, t) denotes the displacement of the volume element dr from a

Chap. 2. Special Relativity ...

87

given initial position Xo, and f is as denned in Table I, then we have by a generalization of D'Alembert's principle, 8J=-Jdtdii-St,

[2.10]

(28)

where SJ is given in Eq. (27). In order to transform SJ, Eq. (27), into a form involving only the variations <5$;, we apply the principle of conservation of charge, in the sense that the charge inside a given element of volume is unchanged during the variation in the position or shape of the volume, whether the variation be actual or virtual. It follows that in addition to the usual "equation of continuity" (i.e., equation of conservation of charge) (dp/dt) + (pUi).i = 0 [_f.i=(df/dxi);

repeated-index summation convention]

(29)

we have the similar equation for the virtual motion of the charges (noting that xi = d%/dt since x = x<>+£), (3p/a«) + O(dfc/
(30)

where t is the parameter for the virtual changes (as t is for the actual changes), and we distinguish by the respective symbols d and d local variation (i.e., keeping x constant) and convective variation (i.e., keeping io constant). Hence (dp/dt)8€=SP=-(PS^),i, (31) since, of course, the 5 | in Eq. (28) are of convective character, so that, S|=(d§/de)&.

(32)

Again, using the familiar connection between local and convective derivatives, and Eq. (32), we have Sui = Se(dUi/de) = 8eZ(dui/de)

-u
Invariance

= Se£(d2^/dtde) =

]

(d/dt)5Zi-Ui,Mj

= (a/ao*fc+ (*&) ,/Ui-Ui.&i. Hence, adding the zero represented by Eq. (30), and using Eq. (31), we find12 8(pUi)

=Ui8p+pdUi

= -( P w,s$ ; ),y+(a/aO(^0 + (^iUi).i.

[2. IS]

(33)

When we substitute the results (31) and (33) into Eq. (27) and perform the appropriate integrations by parts with due account of conditions (22) and (23), we obtain 5./ = jdtrlrp&.j-

A {jUi+ (dAj/dt) +/I,.,«,-]5?;-

= -fdldrpi-i~

(dAj/dQ +

lAij-AjJuiytj

= -/ri^rp[E4-(u xB)]• 5i Eq. (S) now follows at once upon substituting this result in Eq. (28). 3. Lorentz Transformation and the Principle of Least Action [13] "Let us see if the principle of least action gives us the reason for the success of the Lorentz transformation." We consider first the behavior of expression (25) under the transformations (9). We find that the Jacobian of x', y', z', t' with respect to x, y, z, t is I*, so that dt'dr' = l*didr.

(34)

On the other hand, Eqs. (17) imply the identity J4(E'2-B,2)=E2-B2.

(35)

Consequently, if we take U = — °° and U = » , so that the limits of integration do not change

Chap. 2. Special Relativity

...

89

under our transformations, then 00

J' = ffffdt'dr'i

( E " - B ' 2 ) =J.

(35')

— 00

We investigate next the transformation properties of Eq. (28). [14] "For this, it is at first necessary to compare 5|' and 5§. Let us consider an electron whose initial coordinates are x<,, yn, zr>; its coordinates at the instant t will be z. = *i,.+$.

(1 = 1,2,3).

If one considers the corresponding electron after the Lorentz transformation, it will have the coordinates £i'= 71(^-00,

x/ = lx,

( . / = 1 , 2),

where J!.-' = xoi+S.-'

(i=l,2,3);

(30)

but it will only attain these coordinates at the instant t' =

yl{l-flx/)."

It is therefore necessary to consider the variations &Si when we also vary /; so that, using the same notation as before, we have Sx = S£, + uSl, and correspondingly, Sx' = Si'+u'St'. Combining the latter equation with the transformation equations (9) for (5x, St), we obtain upon setting 81 = 0, and using Eqs. (12) (recalling that l-,8* = 7 - s ) , *& = yl~l (1 - j8«i) «&',

% = l~l («*/ - PyuMi') •

When these equations and Eq. (34) are applied to the integral in Eq. (28), the result is an integral

90

Lorentz and Poincare In.vaiia.nce

of the same form in the primed quantities, provided /,' are connected with /,• by Eqs. (18). Thus "the principle of least action leads to the same result as the analysis in Sec. 1." Referring to Eqs. (17), [15] "we see that E2—B2 is not changed by the Lorentz transformation, except for a constant factor; the same is not true of the expression E 2 +B 2 which appears in the energy." 4. The Lorentz Group The Lorentz transformations form a group. In fact, if we combine the transformation (9) with the transformation arl"

= 7T(i1'-/3'«,)l

x/' = l%;

r = 7'Z'(J'-j8V), the resultant transformation is xi = 7 I (xi — /3 t), I" =

r.j =1 i.j,

y"l"(t-I3"x),

where 0"=(/3+/3')/( 1+0/3'), T"

l" = W. 2

= 7 7 '(l+/3/3') = (l-/3" )- 1 ' 2 -

If we set 1=1, and take —/3 = e infinitesimal, then by Kqs. (9) applied to x/ — x.i = 5x,, L' — t = 8t, we have Sx.i = Sx = el,

Sxj = 0,

Sl' = (X

This is an "infinitesimal generating transformation" of the group, call it 7\. In Lie's notation we can write Ti = l(d/dx)+£(d/dt).

(37)

If we take « = 0 and set l=l+ol, we find that •8xi = Xi5l, M = t5l. We have thus another infinitesimal transformation of the group, T0, where T0 =

Xi(d/dXi)+t{d/dt).

(38)

Chap. 2. Special Relativity

...

91

Because the Maxwell-Lorentz equations are invariant under spatial rotations we also have the infinitesimal transformations T^tia/dxA+Xjid/dt)

(.7 = 2,3).

(39)

In addition, the Lie commutators, [I7,-, Tj~]= TiTj-TjTi, reduce to £Tit Til^Xiid/dxA-Xiid/dXi),

(40)

which are easily verified to correspond to rotations in the Xi, x, plane. "We are thus led to envisage a continuous group, which we shall call the Lorentz group." It is generated by the seven infinitesimal transformations represented in Eqs. (37)-(40). Every transformation of this group can be decomposed into a transformation of the form Xi' = lXi,

t' = lt

and a linear transformation that leaves invariant the quadratic form x2—£2. It can also be represented by a transformation of the form (9) preceded and followed by a suitable rotation. [16] "But for our purpose we need only consider a portion of the transformations of this group; we have to suppose that I is a function of e, and it is a question of choosing this function in such a way that this group portion, which I shall call P, forms also a group. Let us turn the (coordinate) system by 180° around the y axis: we rediscover a transformation that must also belong to P. But this amounts to changing the sign of z, x', z, and z'; one finds thus [in our present notation, but writing here t for -0) x' = yl{x-d), z' = lz,

y' = ly,

t' = yl(t-ex)

[8.2]

(41)

Hence I does not change when t is replaced by —«. On the other hand, if P is a group, then the transformation inverse to (9), which is

92

Lorentz and Poincare Invariance x' =

yl-l(x-d), L' =

yl-i(L-fx)

mast also belong to / 3 ; it must therefore be identical to (2) jour (41) J; i.e., l = l~K One muKt thus have / = 1." 1

o 9

i I

i i

l^S^^^M

:„,

A mere outline of Poincar6's paper—as suggested some time ago p i . M. Schwartz, Amer. J. Phys. 33, 170 (1965)]—would hardly serve the needs of historical research relating to special relativity. This is not contradicted, in my opinion, by a recent paper on the subject [C. Cuvaj, Amer. J. Phys. 36, 1102 (1968)]. The extent of the richness of information provided by the Rendiconti paper on Poincar6's role in the history of relativity is made evident in a historic study now in preparation. 1 After the manuscript of the present paper was completed my attention was called to a literal translation of Pts. I and III of Poincard's paper in C. W. Kilmister, Special Theory of Relativity (Pcrgamon, New York, 1970). This now makes these two parts more readily physically accessible, but the utility of the present and projected prdcis remains; in particular, Pt. II (which forms 40% of the original paper), dealing with the frequently quoted "Poincar6 stresses," is certainly of considerable interest. * Poincar6 uses here the curious term rayons magnilocathodiques. * Equations (l)-(6) are of course not all independent, and their arrangement in [1.1] may appear strange at this point. Considerations developed in Sees. 2 and 3 provide at least partial clarification. 6 The significance of the transformation parameter 0 is obvious from the context. Poincar6 refers to both I and /3 as "two arbitrary constants," but the intension is obvious. * Although this assumption is not set off in the original exposition, Poincarc's recognition of its primary significance is implicit in his discussion. 7 Because of its historical interest, Poincard's roundabout method of deriving this kinematic result is reproduced verbatim.

s;;\ ;\4f-\ ill • •The original text has X'-PX (and later, £>i'=-*"'£i) which is an obvious (and harmless) misprint. That D' = D can be seen, for instance, if we denote t by xo, ixa/dt by «„ (<* = 0, 1, 2, 3), and write A', for x.+ Xp«0; we have then, letting L stand for the matrix of a Lorentz transformation (9), and using the repeated-index summation convention: OX.' /dxt' -L„(dXyl det || dX.'/dx,'

OxfiL-ht,

l l - D ' - detL-detX"'•£>=/).

Chap. 2. Special Relativity

...

It is a little curious that PoineanS does not give here the much simpler and direct proof of the LorertU covariance of Eq. (4), which in the above notation, and writing J* for pua, is as follows: J J = l~,LaiJ')t I'OJa'/dxJ 9

= LaidJs/dXyL'1^

= SypdJj/dx-, -

dJt/dX).

In the notation of footnote 8, l^QAJ = -l-,L„tJ> = l^LatO.-ip, i.e., 0(-'l«' — I'^LatAj) =0. An argument for discarding possible additive functions of the A„ that satisfy the wave equation is therefore required to complete the proof that .4„' = i~'L««Ag. The same type of argument also enters in other similar deductions that follow. 10 Poincare has reference probably to Lorentz's variational treatment in his paper "Contributions to the Theory of Electrons," Proc. Acad. Amsterdam 5, 608 (1003). 11 Understood here of course is sufficient rapidity of tending to zero at spatial infinity, so that surface integrals obtained from integration by parts tend to zero. 12 This derivation of Eq. (33) is a more direct version of that given in the original text.

93

94

Lorentz and Poincare Invariance

Poincare's Rendiconti Paper On Relativity. Part II H. M. SCHWARTZ Department of Physics University of Arkansas FayeUeviUe, Arkansas 7S701 (Received 14 January 1972)

This is a continuation of the modernized presentation of Poincari's Rendiconti paper (begun in the November 1971 issue of this Journal). It covers Sees. 5-8 of that paper, dealing vrith its central theme, as indicated by its title, "On the dynamics of the electron," the subject being of interest to both the historian of the classical theory of the electron and the historian of relativity.

The purpose and scope of the present modernized rendition of Poincare's Rendiconti paper is described in the introductory remarks to Pt. I.1 A few additional words are needed to introduce Pt. II. Although the contents of Pt. II (Sec 5-8 of the original work) deal with the structure and dynamics of "electrons," and are, therefore, in the first place, of interest to the historian of the development of the theory of charged particles at the turn of the century, there is also a great deal here which ia relevant to the early history of relativity theory. Because of the latter fact, the listing of some of the translated portions with the aid of bracketed numbers, begun in Pt. I, is continued. In general, all numberings in Pt. II (except for the references) are a continuation of those in Pt. I, so that in referring to Pt. I, no explicit mention of "Pt. I " is needed. Although a number of new symbols arise in Pt. II, it seemed unnecessary to extend Table I of Pt. I. Wherever at all feasible, the original notation is retained, and the few nontrivial changes that are introduced are explained in the references. However, no reference is made (nor was it made in Pt. I) to the replacement of Poincare's antiquated notation for different types of differentiation.

Chap. 2. Special Relativity ...

95

Because Pt. II contains material which is somewhat intricate and not now too familiar, and explanations and references in the original text are not always ample, the number of explanatoryfootnotes to Pt. II is rather large. Helpful consultation on a number of points with my colleague, Lieber, is gratefully acknowledged. ON THE DYNAMICS OF THE ELECTRON S. Langevin's Waves The inhomogeneous wave equations (7) are known to have the retarded-potential solutions ^=(4x)-1/(Pl/r)dr1, A=(4x)- 1 /(piUi/r)rfr 1

(dn =
[5.2] (42)

where

r>=(x-x1)*.

(43)

Langevin has shown that Eqs. (42) admit of a "particularly elegant form" when applied to the field "produced by a single electron." 2 Let Xo = Xo(0 describe the motion of a given "molecule of the electron," 3 and set Xi=x„{li) = Xo+| for given fixed x and t. Then 4 dxu = dxoi+ (dti/dxoj) dx0j+ (<3£./%) dth

(44)

and by (43), dli = — dr=n-dxi,

n=(x—x x )/r.

(45)

Since d%/dtx = xi(tl) =Ui, we find from (44) that (Sij—uunfjdxij^

(8is+dti/dxoj)dxoj,

(46)

from which it follows that Jidn-Jdro,

[5.3]

(47)

96 Lorentz and Poincare Invariance where J\, Jo are the determinants of the respective matrices in the left and right sides of (46). An immediate expansion gives5 Ji = l+u,

-Ui-n.

(48)

On the other hand, if we denote by x the position vectors of the "molecules of the electron" in the neighborhood of the chosen molecule, all taken at the time h associated with this molecule, then the corresponding element of electric charge is6 dqi=pidr

(dT=d?x).

(49)

When we apply Eq. (44) to determine the dxit the last term is absent since now dtx = 0, and instead of (47) we have the relation df =Jodro, where J0 involves x—Xo. In substituting this equation into Eq. (49) we may set J0=JoCombining Eqs. (49), (47), (48), and (42) we obtain then the desired result: *=(47r)-iJ[rfgi/r(l+co)],

A=(4x)-'/[u 1 cZ( 7 l /r(l+co)].

(50)

"If we deal with a single electron, our integrals reduce to a single element, provided only sufficiently distant points x are considered for which r and oi have sensibly the same values at every point of the electron." Referring to Eqs. (2), we sec that the expressions for the electric and magnetic fields contain terms that involve the velocity but not the acceleration of the electron, as well as terms that involve the acceleration, and in fact, linearly. Langevin calls the former terms collectively "velocity wave," and the latter terms, "acceleration wave." We consider first the velocity wave. By a Lorentz transformation we can bring the electron instantaneously to rest at the instant k. In this reference frame, S', Eqs. (50) reduce to7

Chap. 2. Special Relativity

A'=0,

4>'=e/4xr'

( / = | x ' - n ' |),

...

97

(50')

and therefore, E'=(e/4rr")(x/-xl/).

B' = 0,

Transforming back to the original reference frame it is then found that 8 B = pxE, E= 7 P(e/4irr / 3 )[x-:K 1 - •P(«-«i):.

[5.4]

(51)

Turning to the acceleration wave, it will now be shown that the fields satisfy the following three conditions: E l = IB

. ; »S£r^!SM

I

t'.y^. £•$?

E-B=0,

E - n = B - n = 0.

(52)

This result can likewise be derived by the application of a Lorentz transformation, in this instance, to an electron which oscillates in such a way "that the displacements and the velocities are infinitely small, but the accelerations are finite."9 For such an electron the relations (52) follow from the corresponding results in "Hertz's celebrated paper Die Krdfte elektrischer Schwingu?ige?i nach der Maxwell''schen Theorie,10 in the case of a very distant point." In fact, the. general validity (i.e., for finite electron velocity) of the first of Eqs. (52) is an immediate consequence of Eq. (35). Similarly, the second of Eqs. (52) follows from the relation P E ' . B ' = E - B , that is, deducible from Eqs. (17). To prove the Lorentz covariance of the last of Eqs. (52), we note that the first two of these equations imply that E=Bxn,

B = n x E , [5.C]

(53)

where n is defined in (45), and that by Eqs. (9) and (43), rx' = rv'=lru, so that

11

x'-xi'=yl(rx-pr), r,' = lrt

[5.7]

(54)

Invariance E-r / = 7 Z- 1 E-r,

B'-r' = yl-lB-r.

(55)

Another derivation of the Lorentz covariance of the relations (52) is based on "simple considerations of homogeneity."12 In fact, (j>, Ai are functions of Xi—xu and Ui = dxa/dt\, homogeneous of order —1 with respect to xit t, xu, k and their differentials. Hence the derivatives of , 'A ,• with respect to Xi, t (and consequently also the two fields E and B) are homogeneous of degree — 2 with respect to the same quantities, when we recall, moreover, that the expression t— ii = r = | x—Xi [ is homogeneous {i.e., of degree 1} with respect to these quantities. Now these derivatives or these fields depend on the x—Xi, the velocities dxi/dti, and accelerations d?x/dtf; they consist of a term independent of the accelerations (velocity wave) and a term linear with respect to the accelerations (acceleration wave). But dxi/dti is homogeneous of degree 0 and d?X\/dtf is homogeneous of degree — 1; from which it follows that the velocity wave is homogeneous of degree —2 with respect to x — xn and the acceleration wave is homogeneous of degree — 1. Therefore, at a very distant point the acceleration wave predominates and can therefore be identified with the total wave. Furthermore, the homogeneity law shows that the acceleration wave is similar to itself at a distant point and at an arbitrary point. It is therefore similar at an arbitrary point to the total wave at a distant point. But at a distant point the perturbation {i.e., the electromagnetic field] can propagate itself only in plane waves, so that the two fields must be equal, mutually perpendicular, and perpendicular to the direction of propagation. 6. Contraction of Electrons [17] Let us consider a uniformly and rectilinearly moving electron. We have seen that with the aid of a Lorentz transformation

Chap. 2. Special Relativity ... 99 one can reduce the study of the field produced by such an electron to the case of a stationary electron; the Lorentz transformation replaces thus a moving real electron by a motionless ideal electron.13 We wish now to calculate the "electromagnetic masses of the electron." It is necessary, therefore, to determine the "total energy due to the motion of the electron" as well as the corresponding electromagnetic momentum. We cannot employ Eqs. (51) because they apply only to distant field points when it is permissible to treat the electron effectively as a point structure. But "the energy is located principally in the parts of the ether that are nearest to the electron." [18] Concerning this subject many'hypotheses can be made. According to that of Abraham the electrons are spherical and nondeformable. Then upon applying the Lorentz transformation since the real electron is spherical, the ideal electron will become an ellipsoid— 7 - V 2 +2/ / 2 +2 / 2 = Z2r!

(r is the radius of real electron) .u (56)

Lorentz assumed, on the contrary, that it is the ideal electron which is spherical and that "electrons in motion become deformed." Thus it is the real electron which becomes an ellipsoid, whose semiaxes are r/ly, r/l, r/l, where r is the radius of the ideal electron. Now the field produced by the ideal electron, i.e., the field in the rest-frame S' of the electron, satisfies the equations16 B'=0,

E' = - V V ,

(57)

and by (17), we have therefore for the "real field": Bi=0,

Bz = - pE,,

B3=pE, [6.1] (58)

Ei-PES,

Ej-yPE/

( i = 2,3)

100 Lorentz and Poincare Invariance It follows that if we write"

C = i/(52J+B32)dr,

(59)

then, in the first place, C'=0,

C=jPB,

(60)

B' = y-H-lB.

(61)

and since17 dr = 7?dr, A' = yl~lA,

Moreover, by the spherical symmetry in the charge distribution of the Lorentz ideal electron, (62)

B' = 2A'.

Thus, the "total energy," E, the "action per unit time," 18 L, and the x component of the electromagnetic momentum, D, are given by the following equations fusing (58) in the last equation]: E=

A+B+C=(Z+p)lyA',

L=A+B-C=2h-lA', D = f(ExB)4h-=>20B=4PylA'.

(63)

"But one ought to have certain relations" between E, L, and D, namely,19 dD/dp=0-1(dE/d(3),

E=L-/3(dL/d0),

(64)

"from which"20 D = -dL/d0,

E=l+0D.

[6.2]

(65)

[19] The second of equations (2) {Eqs. [6.2]} is always satisfied; but the first one is satisfied only if21 Z = ( 1 - ^ ) 1 " = T-" 3 ,

(66)

Chap. 2. Special Relativity ...

101

i.e., if the volume of the ideal electron is equal to that of the real electron, or also, if the electron's volume is a constant; this is Langevin's hypothesis.22 "This is in contradiction with the result of Sec. 4 and with the result obtained by Lorentz in a different way. It is this contradiction that needs to be explained." To begin with, let us note that whatever the chosen hypothesis, we have [since by (60) and (61), B-C=(l-l?)B = l7-1B'l

L = h-\A'+B'), or since C =0, L = ly~lL'.

[6.3]

(67)

This result can be related to Eq. (35') by noting that by (9), t'= ly~1t—ffx', and hence dt'= ly-'dt (dx'=0). Let us consider now a general hypothesis, which embraces those of Abraham, Lorentz, and Langevin as special cases: the semiaxes of the real electron are r, Or, Br, and hence those of the ideal electron are ylr, I6r, Wr.

(68)

The three special models under consideration are then given by the following determinations: . r = const.,

0= 1

(Abraham)

1 = 1,

yr = const.,

d=y

(Lorentz)

Z = T -i/3 ;

yir- const.,

e=y

(Langevin).

(69)

Whether the electron's charge is supposed to be distributed on its surface as on a conductor, or uniformly throughout its volume, we have23 E' = A'+B'=
(70)

102

Lorentz and Poincare Invariance

The function

L=

(71)

and noting that for Abraham's model we have23 L = ( a / r ) [ ( ^ - l ) / / 3 ] l n [ ( l - / 3 ) / ( l + / 3 ) ] (a const). Taking into account the first set of Eqs. (69), it follows then that

1

)=(^)ln[(l+/3)/(l-i9)].

[6.5]

(72)

Now, all the electron-structure models (69) involve a constraint represented by a relation between r and 8 of the form r = bdm

(73)

(b const).

In fact, corresponding to the three cases in (69), we have, respectively, m arbitrary,

m = — 1,

m—-

(74)

By Eqs. (71) and (73), 9(0/y) (75) by2^ The condition of equilibrium for the electron "when one does not assume the intervention of any other forces except those of constraint" is given by the equation L=

[6.6]

dL/dd = 0,

(76)

which combined with Eq. (75), yields for the logarithmic derivative of

(77)

On the other hand, by expansion in powers of 0, we obtain from (72) that (d/du) \n
whnnw = l.

(77')

Chap. 2. Special Relativity

...

103

Comparing with (77), wo see [bearing in mind that the argument of

IXmf, (see PL I, Table I ] , (78)

and hence the principle of least action can be written (see Pt. I, Table I ) : SJ = fXS&rdt

= SXdrSZJt = I (dD/dt) 5$xdL

(79)

f

. ••£*

On the other hand, J = fLdt, and by our assumption that r = r(8), L can be considered as a function of 6 and j3. Hence, &/ = JI(5L/5/3)50+ (5L/50)30]di.

**jji

Since 8P=d(8$x)/dt, parts,

'^J,

(80)

we find by integration by

- fDSffdt = / (dD/dt)

sytt.

Comparison of Eqs. (79) and (80) show that Hi*

-D = dL/d0,

dL/d6=Q,

and since dL/dff = (dL/dfi) + (dL/dO) (dB/dp),

(80')

104 Lorentz and Poincare Invariance we see that Eqs. (76) and the first of Eqs. (65) are indeed equivalent. [20] The conclusion is that if the electron is- subject to a constraint between its three axes, and if no other forces intervene except the forces of constraint, then the shape assumed by this electron when it is in uniform motion, cannot be such that the corresponding ideal electron is spherical, except when the constraint is that the volume be constant, in agreement with Langevin's hypothesis. What additional forces, then, are needed "to account for Lorentz's law, or more generally, for every law different from that of Langevin?" [21] The simplest hypothesis and the first that we should examine, is that these supplementary forces admit of a special potential depending on the three axes of the ellipsoid, and hence on 8 and r. Denoting this potential by F(d, r), and retaining the symbol L for our previous Lagrangian function, the new expression for our action integral is J=

S[L+F{d,r)-]dt,

and the equilibrium conditions are represented by the equations (d/dd) (L+F) =0,

(d/dr) (L+^F) =0. [6.8]

(81)

If we assume the connection (73), then only the first of Eqs. (81) needs to be retained, with L taking on the expression (75) so that dL/dd = ( - m r V + T ' V ) / ^ . By (72) and (77'),
«/(!)=-(2/3)a.

(82)

Therefore, for d = y, dF/d6 = -dL/dd=(m+i)X a/bd™*3, and hence (the integration constant being immaterial),

Chap. 2. Special Relativity ... F = -(m+i)a/(m+2)b8m+\

105 (83)

For the Lorentz model [second of Eqs. (74) ] this is F=a/360.

(84)

On the other hand,25 if r, 0 are treated as independent variables, then by Eqs. (71) and (81), we find for the case where d = y and relation (73) holds, that dF/dr = a/bWm+\

dF/dd = § (a/60™+3). [6.9]

(85)

One can satisfy these equations by an expression of the form F = Ar°80,

[6.10]

(86)

where A, a, /S are constants. For the case 0 = 7 and (73) one finds then in a few simple steps, a = 3s,

/3 = 2s,

s= -(m+2)/(3m+2),

A=a/ab^1.

[6.11]

(87)

We now note, that since the volume of the ellipsoid is proportional to r3©2, it follows from Eqs. (86) and (87) that F is proportional to the volume raised to the power s. But for the Lorentz model, s = l. [22] One recovers thus Lorentz's hypothesis provided one adds a supplementary potential proportional to the volume of the electron. 7. Quasistationary Motion It remains to be seen if this hypothesis on the contraction of electrons accounts for the impossibility of manifesting absolute motion, and I shall begin with studying the quasistationary motion of an isolated electron, or one subjected only to the action of other distant electrons.

106 Lorentz and Poincare Invariance By quaaistationory motion of an electron one understands "motion involving sufficiently small variations of the velocity so that the magnetic and electric energies due to the electron's motion differ little from what they would be in the case of uniform motion." More precisely, and confining ourselves for the moment to the case of an isolated electron, we can define this motion in terms of our Lagrangian function L. When the electron motion is uniform, i.e., when the velocity, u, of its center of mass is constant, then L depends only on the m and on the shape parameters r and 8. In the case of general nonuniform motion, L will depend not only on these five variables but also on their time derivatives of all orders. But for quasistationary motion "the partial derivatives of L with respect to the successive derivatives of w,-, r, 6 are negligible compared with the partial derivatives of L with respect to the quantities u,-, r, 8 themselves." "The equations of such motion can be written": (d/d8)(L+F)

= (d/dr)(L+F)=0,

(88)

[7.1] (d/dt) (dL/din) = -

ft

(89)

ffidr.

Here F has the same significance as in Sec. 6, and the force f is that produced by all the other electrons of our system. Since L depends on the Ui only through u= | u |, therefore, retaining the symbol D for the magnitude of the electromagnetic momentum, we have,26 dL/dm = (dL/du) (m/u) =-D(ui/u),

(90)

and hence, d dL D . - - — = -Ui-D at dUi u

M,dDui . — U+u, vr du u

[7.2J

(91)

where we use the dot to indicate differentiation with respect to t.

Chap. 2. Special Relativity ...

107

In particular, when Ui = u8n, then: - (d/dt) (dL/dUi) = (dD/du)u

(i=l)

= (D/u)ui

(i = 2,3), (92)

which, recalling (89), explains why Abraham calls dD/du and D/u the longitudinal and the transverse masses, respectively. In the case of Lorentz's model we have27 D = - dL/du = - dL/du.

(93)

On the other hand, upon substituting in L and 0 their functions of u as deduced from (88), we find that 23 L = A (1 —u2)1/-, and proper choice of units,23 we can arrange to the constant A = 1. Therefore, if we sot L = /is=(l-u2)"2

for r Eqs. by a have

(94)

and M=u(du/dt)

su-ii,

we find by Eqs. (91) and (S9) that our equation of quasistationary motion is of the form / i - ' u + / r t l / u = F,

[7.5]

(95)

where30 F = /Wr'.

(95')

In proving Eq. (95) we make use of the following relations implied by Eqs. (93) and (94): dD/du = A-*,

(1/V-) (dD/du) - {D/u') = hr*.

We now prove the theorem that Eq. (95) is covariant under Lorentz transformations. First we note that Eqs. (12) can be written: W =1^-/3,

ixu/=Uj/y

( i = 2,3),

(96)

where M=l-/3Ul,

(96')

108 Lorentz and Poincare Invariance and that, therefore,

Using also the relation31 dt'=ytidt, we find that dui/dtf^ui/r'^, dm' dt'

DM-_(j8//*)«jtti] 7

y

( i = 2,3).

By somewhat lengthy but straightforward algebra, we then obtain the relations: 'I

"*. M' =

-/SA^/TV+M/TV,

A'- 1 (dui'/(ft')+A , -H«i'iV = Ctti+A-*(« l -^)A/]/MA [7.6] (97) h'-l{du//dt')

+h'~3u/M'=

(ui+hrHtjAD/ph* [7.7]

Again, since Eq. (95) implies that F-u = M/A3, Eqs. (19) and (11) with Z = l, and (96') yield: Flf-(Fl-0h-*M)/v,

F/ = Fj/yv

(i-2,3). [7.9]

(98)

Combining Eqs. (97), (95), and (98), we sec the truth of our theorem. We now sharpen this theorem by showing that among the members of the class of models under consideration, that of Lorentz is the only one that satisfies the theorem. First we observe that if one supposes that L = A then one finds immediately that one must have 1 = 1. In fact, with this assumption Eqs. (97), (98), and the primed version of (95) subsist with the right-hand sides multiplied, respectively, by l~l, l~2, and l~l. If then Eq. (95) is to be identical with its primed version (i.e., if it is to be Lorentz covariant) then 1=1 is obviously a necessary condition.

Chap. 2. Special Relativity ...

109

In general, we may of course in proving our uniqueness theorem restrict ourselves to the special case w,=0 ( i = 2, 3) and hence deal with the equations of motion (92). Introducing the notation, dD/du^f(u), D/u=g(u), (99) we have then [recalling Eqs. (89) and (95'), and noting that u i = u ] : f{u)ux = Fu

g{u)HLi = F}

(.7=2,3),

(100)

and for the transformed equations we have: /(«') du'/dt' = F1' = F1(1- pUl) /ZV = Fx/P, g(u')du//dt

= F/ = Fi/Pylx

( i = 2,3).

(101)

Therefore, /(«)iii = P/(«')d«i'/(«', g(u)uj = PyM(u')du//dt'.

[7.11]

(102)

But, by Eqs. (96) and (96'), dui'/dt' = U1/7V 3 ,

• =

.."•3

|

l

\

du//dt'

= Uj/y2ii2,

Hence we deduce from Eqs. (102) the identity

\ :

yVo(U)/g(u)=f(U)/f(u),

: : •

(103)

where 17»(u-jS)/(l-jSu). • - ' • ' • " • • :

^

'

By letting g(u)/f(u)=n(u), the identity (103) in u and /S, becomes Q([/)^(M)/TV-

But for u = 0 we have U = —/3 and /x = l- Hence, n(-/3)=Q(0)/72 = O ( 0 ) ( l - ^ ) .

(104)

In variance Since by Eq. (99), of

^

J

(")

wdD/au we deduce from the differential equation (104), setting —0 = u, and £2(0) =m~l, that D (u) = A [u (1 —u1) 1 / 2 >; A const, and hence from Eqs. (99) and (96) that3*

g(u) =

{A/u)[u/{l-v?-)wy,

(>) = -*?. (

?lWJ

u 0

~

V

w-/3\(l-w2)1/:!(l-/32)1/V '

But by Eqs. (101) and (100), g(vf) Hence,

=g(u)yn/P.

(u- p) '"-1 ( i - /32) -™'2 = - u-"-1 ( i - p2)

-i/n-\

This identity clearly implies that m = \, 1 = 1. [23] Thus, Lorentz's hypothesis is the only one which is compatible with the impossibility of manifesting absolute motion; if one admits this impossibility one must admit that electrons when in motion contract so as to become ellipsoids of revolution whose two axes remain constant; one must then admit, as we have shown in the preceding section, the existence of a supplementary potential proportional to the volume of the electron. [24] Lorentz's analysis is thus fully confirmed, but we can better account for the true reason of the matter under consideration; this reason must be sought among the considerations of Sec. 4. The transformations which do not change the equations of motion must form a group, and this cannot take place except when 1 = 1. Since we must not be able to recognize if an electron is at rest or in a state of absolute motion, it is necessary that when it is in motion, it undergoes a

Chap. 2. Special Relativity

...

Ill

deformation which must be precisely that which is imposed on it by the corresponding transformation of the group. 8. Arbitrary Motion The extension of the preceding results to the general case of arbitrary motion of an electron is easily accomplished with the aid of the action principle developed in Sec. 3. It suffices to add to the action / given by Eq. (25), "a term representing the supplementary potential F of Sec. 6," of the form

the J2 extending over all the electrons of the system under consideration. Since the covariance of J under Lorentz transformations has been proved in Sec. 3, that of the total action, J+Jy will be established, when this covariance is shown ito hold for Ji. This is accomplished by noting first that inasmuch as for each electron, F = COOT,

"bio being a coefficient special to the electron and r its volume," we can write:

where the integral is extended over all of space and the function co0 "vanishes outside of the electrons, and inside each electron it is equal to the coefficient appropriate to that electron."33 Thus, J i = fuqdnU. Now, by Eq. (34) for 1=1, dr'dl' = drill. Hence, the equality of ./i and the Loretnz-tnvnsformed action J-l will be assured provided u0' =o>o. But this is so— [25] because if a point belongs to an electron, the corresponding point after the Lorentz transformation still belongs to the same electron."

112 Lorentz and Poincare Jnvariance This proof that J'+Ji'=J+Ji provides also an answer to the question posed at the end of Sec. 1. [26] If the inertia of electrons is exclusively of electromagnetic origin, if they are subject only to forces of electromagnetic origin or to forces that generate the supplementary potential F, then no experiment can disclose absolute motion. What are these forces that generate the potential F1 They can be evidently identified with a pressure acting in the interior of the electron; it is as if each electron were hollow and subject to a constant internal pressure (independent of volume); the work of such a pressure would evidently be proportional to the change in volume. Now, by Eqs. (86) and (87) applied to Lorentz's model (m = l ) , F = Ar3ffl, where A = a/364,

(105)

so that "our pressure is equal to A, up to a constant coefficient, which is moreover negative." Let us now evaluate the "experimental mass of the electron" (i.e., its rest mass). By Eqs. (71), (82), and (73),

L=

a(l-u1)l'I/b,

so that for w « l , L=(a/6)[1-(«V2)1 and conseqeuently "the mass whether longitudinal or transverse, will be a/6," Since a is a "numerical constant," it follows from Eq. (105) that "the pressure which is generated by our supplementary potential is proportional to the fourth power of the experimental mass of the electron."

Chap. 2. Special Relativity

...

113

"Since the Newtonian attraction is proportional to this experimental mass, one is tempted to conclude that there exists some relationship between the causes which generate gravitation and this supplementary potential." 1

H. M. Schwartz, Amer. J. Phys. 39, 1287 (1971). P. Langevin, J. Phys. 4, 165 (1905). Poincare" refers to this paper at the end of this section: "I shall content myself with referring for further details to the article of Mr. Langevin in the Journal de Physique (the year 1905)." He does not mention earlier derivations relating to Eqs. (50), but references to Lienard, Wiechert, and Schwartzschild are contained in Langevin's paper. 3 A more literal translation of Poincare^s moUcule in the present context would be "particle," but "particle of the electron" sounds no less strange to us. What is meant here, of course, is an infinitesimal element of charge at the point under consideration. * We employ the repeated-indices summation convention. Free Roman indices range over 1, 2, 3. 5 det || iij+a-ibj || =««t(«n+a<60 (Jft+ajW (Su+at&s) = l+a,b,-. * Poincare' uses the symbol p instead of q, and later [in Eq. (50') ] he introduces the symbol m for the electron charge, although the symbol e for that quantity was already then used by most investigators in electrodynamics (including Lorentz and Langevin). It should be noted that "electron" meant then an elementary charged particle of either sign. 7 See Ref. 6. * Equations [5.4] in the original text are written for the special case g = Q3, 0, 0), when they follow from Eqs. (17) and (9): The first set, namely, B t = 0, B j ^ T p B j . j is deducible directly from the transformation (17) and its inverse, while the second set is obtained by using these equations together with Eqs. (17) and (9). The general form given in (51) can be proved easily by using the general transformation formulas corresponding to Eqs. (17) and (9). ' The electron's motion may be thought of as represented by the equation x =a sinwi with aw1 finite and a infinitesimal. 10 H. Hertz, Ann. Phys. Chem. 36, 1 (1888). II By Eqs. (17) and (54), I

E' • r1 = l-*ZExyl {rx - 0r) + 7 (Ev - pB.) lry +y (E, +0By) Zr,] = 7l-'(ET-0[rS„-(BXr)I]), and the first of Eqs. (55) follows from (53), remembering that r =rn. The second of Eqs. (55) is obtained in similar fashion.

114

0

Lorentz and Poincare In variance

?

11 The contents of the present section deal not with new results, but with methods of proving old results. The preceding method serves to illustrate the application of Lorentz transformations. The following method shows the flavor of Poincarg's mathematical style. It is therefore presented verbatim (except for change in mathematical notation). " The italics are not in the original. Note also that the statement [17] must have reference to the center of mass of the electron, unless the electron is assumed to be in some sense absolutely rigid. 14 We arrive by means of our Lorentz transformation to the equation following Eq. (9'), which reduces to Eq. (56) when ui=/3, Uj = u s =0. » Cf. the last remark in Ref. 13. 11 Our retention in (59) of PoincarS's symbols A, B need cause no confusion with the magnitudes of the vectors A, B (seePt. I, Table I). " T h e Jacobian 3(i / )/3(x) for the transformation (9) when t = const is -yf. 11 That is, the field Lagrangian. Poincar6's symbol for this quantity, H, is even in conflict with his own symbol for A, (seePt. I, Table I). a Unfortunately, Poincar6 gives no indication as to how he arrived at these relations, nor any references. To present here any discussion of these developments would be inconsistent with the avowed purpose of the present work, as stated in the introduction to Part I. Perhaps the most useful single reference in this connection is M. Abraham, Theorie der EUktrizil&t (B. G. Teubner, 1905), Chap. 3, Vol. II, Eqs. (64) correspond to Eqs. (Ilia) and (Illb) in the latter book. One must note that Abraham's and Poincart's definitions of the Lagrangian function differ in sign, and bear in mind the differences in the underlying

9.

ft/ )3

\.

* i

basic assumptions. One must also remember that Poincare"s discussion in the present and in the following two sections presupposes (as does also Abraham's discussion) that the electron's mass is purely electromagnetic. " The converse is also true, so that Eqs. (64) and (65) are actually equivalent. In fact, by the two equations in (64), dE/dp = -0d*L/dp =0dD/d0, i.e., dD/dB = -PL/dp so that D=-0, the constant vanishes and we get the first of Eqs. (65). Then the second of Eqs. (65) is equivalent to the first of Eqs. (64). It follows then conversely that Eqs. (65) imply the first of Eqs. (64), and that the latter equation together with the first of Eqs. (65) imply the second of Eqs. (64). 11 The second of Eqs. (65) is identically satisfied by virtue of the relations (63), but applying the latter to the first of Eqs. (65), we get the equation:

Chap. 2. Special Relativity

...

115

O = D+dL/d0 A'[l(^y+3dy-l/d0)

+37-W]

=A'(lf)-y+3y-W/df}), or, «fl/I--U#/3(l-/S»). Hence 1= (1— /31)1", the multiplicative integration constant being unity, since obviously we must have 1 = 1 when 0 = 0 . a See Ref. 17. U M . Abraham, Gottinger Nachrichten 1, 20 (1902), p. 37. See also Ref. 19. " Bearing in mind also that a purely electromagnetic origin of the electron's inertia is being assumed. * The purpose of this treatment is to obtain F explicitly in terms of r and 9 so as to arrive at the result cited in [22]. The result (84) is in fact obtained from the general expression (86) [and (87)]] when the conditions for the Lorentz model are introduced. Note also that the result (83) is consistent with Langevin's model involving no supplementary potential [see third equation in (74) ]. M See Ref. 19. If we denote the electromagnetic momentum by P, then D = | P |, while in Sec. 6, D = Px. The relation, D = —dL/du, introduced in (90), is seen to be equivalent to the first equation in (80'), when we note that u can be identified there with Q3, 0, 0). 27 See equation below (80') and the second equation in (80'). " B y Eq. (73), since 8 = y and m=— 1 for Lorentz's model. " A s is shown in Sec. 8, A =a/b — rest mass of the electron. M That we can use this notation and also apply (as we do presently) the transformation Eq. (19) without contradiction with the meaning of the symbol F given in Table I, follows from assumption (10). " B y (9) for 1 = 1, dt' = y(dt-0dx) =ydt(l-pu,). 51 Noting the identity: (1 - / } « ) ' - ( u - 0 ) J - (1 -«») (1 -P). u

The symbol u0 is thus now used as a 6-type function. This appears to involve the tacit assumption of the absolute invariance of the coefficient wo, similar to the invariance stated in (10) for electric charge. 14

116

Lorentz and Poincare In variance

On the Electrodynamics of Moving Bodies By

A.

EINSTEIN

Translated from "Zur Electrodynamik bewcgter KSrper," Amuilen derPhysik, 17, 891(1905).

IT IS known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case. Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the "light medium," suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjcture (the purport of which will hereafter be called the "Principle of Relativity") to the status of apostulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies. The introduction of a "luminiferous ether" will prove to be superfluous inasmuch as the view here to be developed will not require an "ab-

Chap. 2. Special Relativity

solutely stationary space" provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place. The theory to be developed is based—like all electrodynamics— on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters. I. Kinematical Part § 1, Definition of Simultaneity Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good. In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the "stationary system." If a material point is at rest relatively to this system of co-ordinates, its position can be denned relatively thereto by the employment of rigid standards of measurement and the methods of Euclidean geometry, and can be expressed in Cartesian co-ordinates. If we wish to describe the motion of a material point, we give the values of its co-ordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear as to what we understand by "time." We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, "That train arrives here at 7 o'clock," I mean something like this: "The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events."* It might appear possible to overcome all the difficulties attending the definition of "time" by substituting "the position of the small hand of my watch" for "time." And in fact such a definition is satisfactory when we are concerned with defining a time exclusi* We shall not here discuss the inexactitude which lurks in the concept of simultaneity of two events at approximately the same place, which can only be removed by an abstraction.

...

117

118 Lorentz and Poincare In variance vely for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or—what comes to the same thing—to evaluate the times of events occurring at places remote from the watch. We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this coordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought. If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an "A time" and a "B time." We have not defined a common "time" for A and B. The latter time can now be defined in establishing by definition that the "time" required by light to travel from A to B equals the "time" it requires to travel from B to A. Let a ray of light start at the "A time" tA from A towards B, let it at the "B time" tB be reflected at B in the direction of A, and arrive again at A at the "A time" t'A. In accordance with definition the two clocks synchronize if ta—tA

= tA — tjj-

We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid: 1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B. 2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other. Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary

Chap. 2. Special Relativity ... clocks located at different places, and have evidently obtained a definition of "simultaneous," or "synchronous," and of "time." The "time" of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock. In agreement with experience we further assume the quantity 2AB — = c, ?A-tA to be a universal constant—the velocity of light in empty space. An essential point is that we have defined time by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it "the time of the stationary system."

§ 2. On the Relativity of Lengths and Times The following reflexions are based on the principle of relativity and on the principle of the constancy of the velocity of light. These two principles we define as follows: 1. The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion relative to each other. 2. Any ray of light moves in the "stationary" system of coordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Here , . light path velocity = — r^ ; time interval where time interval is to be taken in the sense of the definition in §1Let there be given a stationary rigid rod; and let its length be / as measured by a measuring-rod which is also stationary. We now imagine the axis of the rod lying along the axis of x of the stationary system of co-ordinates, and that a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod, and imagine its length to be ascertained by the following two operations:

119

120 Lorentz and Poincaxe Invariance (a) The observer moves together with the given measuring-rod and the rod to be measured, and measures the length of the rod directly by superposing the measuring-rod, in just the same way as if all three were at rest. (b) By means of stationary clocks set up in the stationary system and synchronizing in accordance with § 1, the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring-rod already employed, which in this case is at rest, is also a length which may be designated "the length of the rod." In accordance with the principle of relativity the length to be discovered by the operation (a)—we will call it "the length of the rod in the moving system"—must be equal to the length / of the stationary rod. The length to be discovered by the operation (b) we will call "the length of the (moving) rod in the stationary system." This we shall determine on the basis of our two principles, and we shall find that it differs from /. Current kinematics tacitly assumes that the lengths determined by these two operations are precisely equal, or in other words, that a moving rigid body at the epoch t may in geometrical respects be perfectly represented by the somebody at rest in a definite position. We imagine further that at the two ends A and B of the rod, clocks are placed which synchronize with the clocks of the stationary system, that is to say that their indications correspond at any instant to the "time of the stationary system" at the places where they happen to be. These clocks are therefore "synchronous in the stationary system." We imagine further that witli each clock there is a moving observer, and that these observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks. Let a ray of light depart from A at the time* tA, let it be reflected at B at the time tB, and reach A again at the time t'A. Taking into consideration the principle of the constancy of the velocity of light we find that te-tA —

and c—v

tA-tB=

—— c+v

where rAB denotes the length of the moving rod—measured in the stationary system. Observers moving with the moving rod would * "Time" here denotes "time of the stationary system" and also "position of hands of the moving clock situated at the place under discussion."

Chap. 2. Special Relativity ... thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous. So we see that we cannot attach any absolute signification to the concept of simultaneity, but that two events which, viewed from a system of co-ordinates, are simultaneous, can no longer be looked upon as simultaneous events when envisaged from a system which is in motion relatively to that system.

§ 3. Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatively to the Former Let us in "stationary" space take two systems of co-ordinates, i.e. two systems, each of three rigid material lines, perpendicular to one another, and issuing from a point. Let the axes of X of the two systems coincide, and their axes of Y and Z respectively be parallel. Let each system be provided with a rigid measuring-rod and a number of clocks, and let the two measuring-rods, and likewise all the clocks of the two systems, be in all respects alike. Now to the origin of one of the two systems (k~) let a constant velocity v be imparted in the direction of increasing x of the other stationary system(K), and let this velocity be communicated to the axes of the co-ordinates, the relevant measuring-rod, and the clocks. To any time t of the stationary system K there then will correspond a definite position of the axes of the moving system, and from reasons of symmetry we are entitled to assume that the motion of k may be such that the axes of the moving system are at the time t (this "t" always denotes a time of the stationary system) parallel to the axes of the stationary system. We now imagine space to be measured from the stationary system K by means of the stationary measuring-rod, and also from the moving system k by means of the measuring-rod moving with it; and that we thus obtain the co-ordinates x, y, z, and f, t],'Crespectively. Further, let the time t of thestationary system be determined by means of the clocks at rest in the stationary system, for all points thereof at which there are clocks, by means of light signals in the manner indicated in § 1; similarly let the time x of the moving system be determined for all points of the moving system at which there are clocks at rest relatively to that system by applying the method, given in § 1, of light signals between the points at which the latter clocks are located.

121

122 Lorentz and Poincare Invariance To any system of values x, y, z, t ,which completely defines the place and time of an event in the stationary system, there belongs a system of values | , r\, £, x, determining that event relatively to the system k, and our task is now to find the system of equations connecting these quantities. In the first place it is clear that the equations must be linear on account of the properties of homogeneity which we attribute to space and time. If we place x' = x—vt, it is clear that a point at rest in the system k must have a system of values x', y, z, independent of time. We first determine x as a function of x', y, z, and t. To do this we have to express in equations that x is nothing else than the summary 01 the data of clocks at rest in system k, which have been synchronized according to the rule given in § 1. From the origin of system k let a ray be emitted at the time x0 along the X-axis to x', and at the time TI be reflected thence to the origin of the co-ordinates, arriving there at the time T 2 ; we then must have -|(T 0 + T 2 ) = xu or, by inserting the arguments of the function x and applying the principle of the constancy of the velocity of light in the stationary system: i [ T ( 0 , 0,0, 0 + r (o, 0, 0, / + —

2[

\

c—v

+ -±-\

] = x(V,

c+vj J

\

0,0, / + —

V

c — v)

Hence, if x' be chosen infinitesimally small, 1/1 2 \c — v \c — v

1 \ dx dt c + vj

(—+4-) c + vj

or

dx

v

8x -+• dx'

1 dx c — v dt '

dx _

It is to be noted that instead of the origin of the co-ordinates we might have chosen any other point for the point of origin of the ray, and the equation just obtained is therefore valid for all values of x', y, z. An analogous consideration—applied to the axes of if and Z— it being borne in mind that light is always propagated along these axes, when viewed from the stationary system, with the velocity y/{c2—v2), gives us

S-* £-* Since x is a linear function, it follows from these equations that

Chap. 2. Special Relativity ... Where a is a function 4>{v) at present unknown, and where for brevity it is assumed that at the origin of k, t = 0, when x = 0. With the help of this result we easily determine the quantities I, t], C by expressing in equations that light (as required by the principle of the constancy of the velocity of light, in combination with the principle of relativity) is also propagated with velocity c when measured in the moving system. For a ray of light emitted at the time x = 0 in the direction of the increasing £ ex

l = flC

('-;r^b4

or

But the ray moves relatively to the origin of A:, when measured in the stationary system, with the velocity c—v, so that x c — v - t. If we insert this value of t in the equation for | , we obtain I = a -5—- x . C*

—V1

In an analogous manner we find, by considering rays moving along the two other axes, that

n = cx =

ac[t-^x>)

where

„ I

2,

=t,

x' = 0.

and

£= a

Thus 7] = a

—y 2

Substituting for x' its value, we obtain = <]>(v)p(t-vxlc2),

I =

z

V( — v )

V(c — v ) T

c —=r c

4,(v)P(x-vt),

t) =
V(l-«7c 2 ) '

123

124

Lorentz and Poincare Invariance

and (j> is an as yet unknown function of v. If no assumption whatever be made as to the initial position of the moving system and as to the zero point oft, an additive constant is to be placed on the right side of each of these equations. We now have to prove that any ray of light, measured in the moving system, is propagated with the velocity c, if, as we have assumed, this is the case in the stationary system; for we have not as yet furnished the proof that the principle of the constancy of the velocity of light is compatible with the principle of relativity. At the time t = x = 0, when the origin of the co-ordinates is common to the two systems, let a spherical wave be emitted therefrom, and be propagated with the velocity c in system K. If (x, y, z) be a point just attained by this wave, then x2+y2+z2

=

C 2,2

Transforming this equation with the aid of our equations of transformation we obtain after a simple calculation S 2 +»? 2 +C 2 = C 2 T 2 .

The wave under consideration is therefore no less a spherical wave with velocity of propagation c when viewed in the moving system. This shows that our two fundamental principles are compatible. In the equations of transformation which have been developed there enters an unknown function of v, which we will now determine. For this purpose we introduce a third system of co-ordinates K', which relatively to the system A: is in a state of parallel translatory motion parallel to the axis of S, such that the origin of co-ordinates of system A: moves with velocity —v on the axis of S. At the time t = 0 let all three origins coincide, and when / = x = y = z — 0 let the time t' of the system £ ' be zero. We call the co-ordinates, measured in the system K', x', y', z', and by a twofold application of our equations of transformation we obtain f = )i?(-u)(r + r!/c 2 ) x' = 0 ( - v) /3( - v ) (I + vx) y = <j)(-v)r) Z'= <$>(-v)t,

= ( - v)x, = (j)(v)(p{-v)y, =
Chap. 2. Special Relativity

...

Since the relations between x',y', z' and x, y, z do not contain the time t, the systems K and K' are at rest with respect to one another, and it is clear that the transformation from K to K' must be the identical transformation. Thus * W ( - t ; ) - 1. We now inquire into the signification of <j>(v). We give our attention to that part of the axis of Y of system k which lies between | = 0, t] = 0, C = 0 and f = 0, rj = /, C = 0. This part of the axis of H is a rod moving perpendicularly to its axis with velocity v relatively to system K. Its ends possess in K the co-ordinates

and

xi = vt,

yi = - 7 7 - - ,

x2 — vt,

y2 = 0,


zi --= 0

z 2 = 0.

The length of the rod measured in AT is therefore //(w); and this gives us the meaning of the function (j>(v). From reasons of symmetry it is now evident that the length of a given rod moving perpendicularly to its axis, measured in the stationary system, must depend only on the velocity and not on the direction and the sense of the motion. The length of the moving rod measured in the stationary system does not change, therefore, if v and — v are interchanged. Hence follows that l/(j)(v) = lj{ —v), or (j)(v) = 4>(-v). It follows from this relation and the one previously found that (v) = 1, so that the transformation equations which have been found become T = P(t-vx/c2),

I = 0(x-vt), n = y, C = z, where j8= l/V(l-f2/c2).

§ 4. Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks We envisage a rigid sphere* of radius R, at rest relatively to the moving system k, and with its centre at the origin of co-ordinates of k. The equation of the surface of this sphere moving relatively * That is, a body possessing spherical form when examined at rest.

125

126 Lorentz and Poincare Invariance to the system K with velocity v is

The equation of this surface expressed in x, y, z at the time t = 0 is 2

2

(V(l-« /c ))

:

+ / + Z2 = R2.

A rigid body which, measured in a state of rest, has the form of a sphere, therefore has in a state of motion—viewed from the stationary system—the form of an ellipsoid of revolution with the axes

RV(l-v2lc2),

R, R.

Thus, whereas the Y and Z dimensions of the sphere (and therefore of every rigid body of no matter what form) do not appear modified by the motion, the ^dimension appears shortened in the ratio 1 : ^/{\ —v2jc2), i.e. the greater the value of v, the greater the shortening. For v = c all moving objects—viewed from the "stationary" system—shrivel up into plane figures. For velocities greater than that of light our deliberations become meaningless; we shall, however, find in what follows, that the velocity of light in our theory plays the part, physically, of an infinitely great velocity. It is clear that the same results hold good of bodies at rest in the "stationary" system, viewed from a system in uniform motion. Further, we imagine one of the clocks which are qualified to mark the time t when at rest relatively to the stationary system, and the time r when at rest relatively to the moving system, to be located at the origin of the co-ordinates of k, and so adjusted that it marks the time x. What is the rate of this clock, when viewed from the stationary system? Between the quantities x, t, and T, which refer to the position of the clock, we have, evidently, x = vt and

Therefore, x = tVV-v2/c2)

= /-(l-VU-^/c2))/

whence it follows that the time marked by the clock (viewed in the stationary system) is slow by 1 — VG —v2/c2) seconds per second, or—neglecting magnitudes of fourth and higher order—by \v2\c2. From this there ensues the following peculiar consequence. If at the points A and B of K there are stationary clocks which, viewed

Chap. 2. Special Relativity

in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by yfv 2 /c 2 (up to magnitudes of fourth and higher order), t being the time occupied in the journey from A to B. It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide. If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds, then by the clock which has remained at rest the travelled clock on its arrival at A will be jtv2/c2 second slow. Thence we conclude that a balance-clock* at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions.

§ 5. The Composition of Velocities In the system k moving along the axis of X of the system K with velocity v, let a point move in accordance with the equations £ = W(t,

T) = WVX,

C = 0,

where w( and wv denote constants. Required: the motion of the point relatively to the system K. If with the help of the equations of transformation developed in § 3 we introduce the quantities x, y, z, / into the equations of motion of the point, we obtain x

= T~r

FT ''

1 + VW(/C2

V(l-« 2 /c 2 )

,

1 + VW(/C2

z = 0. Thus the law of the parallelogram of velocities is valid accord* Not a pendulum-clock, which is physically a system to which the Earth belongs. This case had to be excluded.

...

127

128

Lorentz and Poincare Invaiian.ee

ing to our theory only to a first approximation. We set

"•-(£)•+(£)•• M>2 = wf+W 2 ,,

a = t a n - 1 wy/wx, a is then to be looked upon as the angle between the velocities v and w. After a simple calculation we obtain _ V[(v2 + w2 + 2vw cos a) — (vw sin a/c2)2] 1+OTCosfl/e 2 It is worthy of remark that v and w enter into the expression for the resultant velocity in a symmetrical manner. IT w also has the direction of the axis of X (axis of S), we get y -

v

+w 2 1 + vw/c

It follows from this equation that from a composition of two velocities which are less than c, there always results a velocity less than c. For if we set v = c—x, w = c — X, % and A being positive and less than c, then 2c — x — X V = c^r r-—57- < c. 2c — « — X+KX/C

It follows, further, that the velocity of light c cannot be altered by composition with a velocity less than that of light. For this case we obtain

„

c+w

V =

j - = c. l + w/c We might also have obtained the formula for V, for the case when v and w have the same direction, by compounding two transformations in accordance with § 3. If in addition to the systems K and k figuring in § 3 we introduce still another system of co-ordinates k! moving parallel to k, its origin moving on the axis of X with the velocity w, we obtain equations between the quantities x,y, z, rand the corresponding quantities of &', which differ from the equations found in § 3 only in that the place of "v" is taken by the quantity v+w 1 + vw/c2 from which we see that such parallel transformations form a group, as they must do.

Chap. 2. Special Relativity

We have now deduced the requisite laws of the theory of kinematics corresponding to our two principles, and we proceed to show their application to electrodynamics.

II. Electrodynamical Part § 6. Transformation of the Maxwell-Hertz Equations for Empty Space. On the Nature of the Electromotive Forces Occurring in a Magnetic Field During Motion Let the Maxwell-Hertz equations for empty space hold good for the stationary system K, so that we have 1 dX _ dN c dt ~ dy

dM dz '

1 dL c dt

dY dz

dZ ~dy~'

1 dY _ dL c dt ~ dz

dN dx '

1 dM c dt

dZ

dX

~dx

1 dZ _ dM c dt ~ dx

dL dy '

1 dN dX c dt ~ dy

W dY dx

where (X, Y, Z) denotes the vector of the electric force, and (L, M, N) that of the magnetic force. If we apply to these equations the transformation developed in § 3, by referring the electromagnetic processes to the system of coordinates there introduced, moving with the velocity v, we obtain the equations

7£-*{'K"}-£{'K")}^{'K*)}-!{'K")}-Swhere 0 = I/V(I-»»/c*).

...

129

130

Lorentz and Poincaxe Invariance

Now the principle of relativity requires that if the MaxwellHertz equations for empty space hold good in system K, they also hold good in system k; that is to say that the vectors of the electric and the magnetic force—(A", Y', Z') and (L', M', N')—of the moving system k, which are defined by their ponderomotive effects on electric or magnetic charges respectively, satisfy the following equations: l_ dX^ _ dlT__dAr_ drj c 0T dx dt] dtC ' 0

dY'

dZ'

~W

drj

1 dM' c dx

dZ'

dX'

0f

0C

dL' dZ' dM' \_dZ^__dJ^__dlJ_

\_ dN'

c

c

dX' dr]

dY' 0£

1 dY' dY' c dx ox dx

01/ 01/

dN' dN'

~ 0C dt

01 dt

0|

drj '

1 dU c dx

dx ~

Evidently the two systems of equations found for system k must express exactly the same thing, since both systems of equations are equivalent to the Maxwell-Hertz equations for system K. Since, further, the equations of the two systems agree, with the exception of the symbols for the vectors, it follows that the functions occurring in the systems of equations at corresponding places must agree, with the exception of a factor y>(v), which is common for all functions of the one system of equations, and is independent of 5, rj, £ and x but may depend upon v. Thus we have the relations X' = f(v)X, L' = y)(v)L, r

= y,(v)p(Y-^N\,

Z' = y(v)(l(z+-M\,

M' = W(v)p(M + j Z \ , N'=

y>(v)p(N- — Y\.

If we now form the reciprocal of this system of equations, firstly by solving the equations just obtained, and secondly by applying the equations to the inverse transformation (from k to K), which is characterized by the velocity —v, it follows, when we consider that the two systems of equations thus obtained must be identical, that ip(v)f(— v) = 1. Further, from reasons of symmetry* y(v) = y)(—v), and therefore

* If, for example, X=Y=Z = L = M = Q, and N ^ 0, then from reasons of symmetry it is clear that when v changes sign without changing its numerical value, Y' must also change sign without changing its numerical value.

Chap. 2. Special Relativity ... V(«) = U and our equations assume the form X' = X,

U = L,

z' = p(z+jM\,

N' =

P(N--Y\.

As to the interpretation of these equations we make the following remarks: Let a point charge of electricity have the magnitude "one" when measured in the stationary system A", i.e. let it when at rest in the stationary system exert a force of one dyne upon an equal quantity of electricity at a distance of one cm. By the principle of relativity this electric charge is also of the magnitude "one" when measured in the moving system. If this quantity of electricity is at rest relatively to the stationary system, then by definition the vector (X, Y, Z) is equal to the force acting upon it. If the quantity of electricity is at rest relatively to the moving sysem (at least at the relevant instant), then the force acting upon it, measured in the moving system, is equal to the vector (X', Y', Z'). Consequently the first three equations above allow themselves to be clothed in words in the two following ways: 1. If a unit electric point charge is in motion in an electromagnetic field, there acts upon it, in addition to the electric force, an "electromotive force" which, if we neglect the terms multiplied by the second and higher powers ofv/c, is equal to the vector-product of the velocity of the charge and the magnetic force, divided by the velocity of light. (Old manner of expression.) 2. If a unit electric point charge is in motion in an electromagnetic field, the force acting upon it is equal to the electric force which is present at the locality of the charge, and which we ascertain by transformation of the field to a system of co-ordinates at rest relatively to the electrical charge. (New manner of expression.) An analogous situation holds with "magnetomotive forces." We see that electromotive force plays in the developed theory merely the part of an auxiliary concept, which owes its introduction to the circumstance that electric and magnetic forces do not exist independently of the state of motion of the system of co-ordinates. Furthermore it is clear that the asymmetry mentioned in the introduction as arising when we consider the currents produced by the relative motion of a magnet and a conductor, now disappears. Moreover, questions as to the "seat" of electrodynamic electromotive forces (unipolar machines) now have no point.

131

132 Lorentz and Poincaxe Invariance § 7. Theory of Doppler's Principle and of Aberration In the system K, very far from the origin of co-ordinates, let there be a source of electrodynamic waves, which in a part of space containing the origin of co-ordinates may be represented to a sufficient degree of approximation by the equations X = X0 sin 0, Y = Y0 sin 0, Z = Z 0 sin 0 ,

L - L0 sin
where 0 = colt

(lx + my + nz)\.

Here (X0, Y0, Z0) and (L 0 , M 0 , A^) are the vectors defining the amplitude of the wave-train, and /, m, n the direction-cosines of the wave-normals. We wish to know the constitution of these waves, when they are examined by an observer at rest in the moving system k. Applying the equations of transformation found in § 6 for electric and magnetic forces, and those found in § 3 for the co-ordinates and the time, we obtain directly X' = X0 sin 0', Y' = P(Yo-vN0/c) sin 0', Z' = p(Z0 + vMo/c) sin 0',

L' = Lo sin 0', M' = P(M0-hvZ0/c) sin 0', N' = P(N0-vYo/c) sin 0'.

0' = a)'{r-i-(/'| + »i'ij + n'0} where w' =

"

co§(l-lv/c), I-vjc 1-lv/c'

•_

m

' =

n

]9(l-/»/c)*

From the equation for to' it follows that if an observer is moving with velocity v relatively to an infinitely distant source of light of frequency v, in such a way that the connecting line "source—observer" makes the angle

Chap. 2. Special Relativity ... given by the equation V

=

V

1—cos <j>.v/c

This is Doppler's principle for any velocities whatever. When (f> = 0 the equation assumes the perspicuous form v

=

v

1/ -i

1- •

y i+vfc We see that, in contrast with the customary view, when v = — c, If we call the angle between the wave-normal (direction of the ray) in the moving system and the connecting line "source—observer" (p', the equation for /' assumes the form cos 6 —vie cos

= -|jt, the equation becomes simply cos <j>' = —v/c. We still have to find the amplitude of the waves, as it appears in the moving system. If we call the amplitude of the electric or magnetic force A or A' respectively, accordingly as it is measured in the stationary system or in the moving system, we obtain A

_

(l-cos^.v/cf

~A

l-i> 2 /c 2

which equation, if = 0, simplifies into A'2

„

l-vfc l+v/c

'

It follows from these results that to an observer approaching a source of light with the velocity c, this source of light must appear of infinite intensity.

§ 8. Transformation of the Energy of Light Rays. Theory of the Pressure of Radiation Exerted on Perfect Reflectors Since A2/Sn equals the energy of light per unit of volume, we have to regard /4'2/&r, by the principle of relativity, as the energy of light in the moving system. Thus A'2/A2 would be the ratio of

133

134 Lorentz and Poincare Invariance the "measured in motion" to the "measured at rest" energy of a given light complex, if the volume of a light complex were the same, whether measured in K or in k. But this is not the case. If /, m, n are the direction-cosines of the wave-normal of the light in the stationary system, no energy passes through the surface elements of a spherical surface moving with the velocity of light: (x-lct)2

+ (y-mct)2

+ (z-nct)2

= R2.

We may therefore say that this surface permanently encloses the same light complex. We inquire as to the quantity of energy enclosed by this surface, viewed in system k, that is, as to the energy of the light complex relatively to the system k. The spherical surface—viewed in the moving system—is an ellipsoidal surface, the equation for which, at the time x = 0, is 2

W-ipev/cy+to-mPhlcy+G-nPSvlc? = -R .

If S is the volume of the sphere, and S' that of this ellipsoid, then by a simple calculation

V(l-a2/c2)

£_ S

1 — cos <j>. v/c '

Thus, if we call the light energy enclosed by this surface E when it is measured in the stationary system, and E' when measured in the moving system, we obtain

— ~E~

A 2S

' ' - l~cos

A2S

$-vlc

V(l-v2/c2)

'

and this formula, when <j> = 0, simplifies into E ~ \

l+v/c'

It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law. Now let the co-ordinate plane 5 = 0 be a perfectly reflecting surface, at which the plane waves considered in § 7 are reflected. We seek for the pressure of light exerted on the reflecting surface, and for the direction, frequency, and intensity of the light after reflexion. Let the incidental light be defined by the quantities A, cos (p, v

Chap. 2. Special Relativity ... (referred to system K). Viewed from k the corresponding quantities are 1—cos (p.vjc A' = A V(l-«2/c2) ' cos <£' =

cos
, V

1—cos S.vlc =

j)

•—

.

V(l-v2/c*) For the reflected light, referring the process to system k, we obtain A" = A'. COS (j)" = —COS (j)'

v" = v'. Finally, by transforming back to the stationary system K, we obtain for the reflected light A1" _ j " l + c o s <j>"-v/c _ V(i~v2/c2) ,,, _

cos cp" + v/c _ I + c o s "v/c

V

= V —-rr. V(l-v2/c2)„ , '

1 — 2'cos (j).v/c + v2fc2 l-v2/c2 '

(l + v2/c2)cos (j)-2v/c 1 - 2 cos 4>.vjc + v2/c2' 1— 2 cos
= v-

The energy (measured in the stationary system) which is incident upon unit area of the mirror in unit time is evidently A2(c cos

A2 (cos 4> — v/c)2 in l-v2/c2

In agreement with experiment and with other theories, we obtain to a first approximation 2- ——cos2 (j). on

135

136 Lorentz and Poincare Invariance All problems in the optics of moving bodies can be solved by the method here employed. What is essential is, that the electric and magnetic force of the light which is influenced by a moving body, be transformed into a system of co-ordinates at rest relatively to the body. By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies. § 9. Transformation of the Maxwell-Hertz Equations when Convection-Currents are Taken into Account We start from the equations dX ) c -dT+U*\ dY

]_

dN dM -dy—dz-'

1 dL c dt

1

dL

dN

1 dM dZ c dt ~ dx

]

dM

dL

1 dN c dt

=

c

j_

dZ

c

-¥t- * \--dx—dj'

+u e

dY "dz

dX dy

dZ •

dy'

dX ' dz'' dY dx '

where dX q =

dY

-dx-+-dJ

+

dZ

1F

denotes 4n times the density of electricity, and (ux, uy, uz) the velocity-vector of the charge. If we imagine the electric charges to be invariably coupled to small rigid bodies (ions, electrons), these equations are the electromagnetic basis of the Lorentzian electrodynamics and optics of moving bodies. Let these equations be valid in the system K, and transform them, with the assistance of the equations of transformation given in §§ 3 and 6, to the system k. We then obtain the equations

i f ex'

;i

7\-dr+u*\

1 f dZ'

,}

dN' dr)

dM' dC

1 dL' c dr

dL' dC

dN' 01

1 dM' c dr

dM'

dL'

dS,

dr)

1 dN' c dr

~ dC

dZ' dr]

dZ'

dX'

dY'

~

dt ~'dC~ dX' drj

dY' d$

Chap. 2. Special Relativity ... where ux — v l-uxv/c2 u

'

P(l-uxv/c2)'

"

Ul " c ~ P(l-uxv/c2)

'

and ,

dX'

BY'

dZ'

...

. „,

Since—as follows from the theorem of addition of velocities (§ 5)— the vector (wf, un, wc) 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 kinematical principles, the electrodynamic foundation of Lorentz's theory of the electrodynamics of moving bodies is in agreement with the principle of relativity. In addition I may briefly remark that the following important law may easily be deduced from the developed equations: If an electrically charged body is in motion anywhere in space without altering its charge when regarded from a system of co-ordinates moving with the body, its charge also remains—when regarded from the "stationary" system K—constant.

§ 10. Dynamics of the Slowly Accelerated Electron Let there be in motion in an electromagnetic field an electrically charged particle of charge e (in the sequel called an "electron"), for the law of motion of which we assume as follows: If the electron is at rest at a given epoch, the motion of the electron ensues in the next instant of time according to the equations d2x eX, dt2 m£y2=eY,

dt

d2z = eZ dt2

137

138

Lorentz and Poincare Invariance

where x, y, z denote the co-ordinates of the electron, and w the mass of the electron, as long as its motion is slow. Now, secondly, let the velocity of the electron at a given epoch be v. We seek the law of motion of the electron in the immediately ensuing instants of time. Without affecting the general character of our considerations, we may and will assume that the electron, at the moment when we give it our attention, is at the origin of the co-ordinates, and moves with the velocity v along the axis of X of the system K. It is then clear that at the given moment (/ = 0) the electron is at rest relatively to a system of co-ordinates k which is in parallel motion with constant velocity v along the axis of X. From the above assumption, in combination with the principle of relativity, it is clear that in the immediately ensuing time (for small values off) the electron, viewed from the system k, moves in accordance with the equations

-

^

=

- '

in which the symbols £, r\, C, r, X', Y', Z' refer to the system k. If, further, we decide that when t = x — y — z — 0 then T = £ = V = C = 0, the transformation equations of §§ 3 and 6 hold good, so that we have I = P(x-vt), r] = y, Z = z, x = P(t-vx/c2) X' = X, Y' = P(Y-vN/c), Z' = p(Z+vM/c). With the help of these equations we transform the above equations of motion from system k to system K, and obtain (Px ~d~F cPy ~dP cP-z

=

=

-

e

x

1

•Mr-7»)

(A)

•syK").

Taking the ordinary point of view we now inquire as to the "longitudinal" and the "transverse" mass of the moving electron.

Chap. 2. Special Relativity

We write the equations (A) in the form cPx mB3 -—-2 = eX = eX', ' dt m 2

P Sr =

s

P(Y-7N)=eY'<

-*•£=*(*+»

= eZ',

and remark firstly that eX', eY', eZ' are the components of the ponderomotive force acting upon the electron, and are so indeed as viewed in a system moving at the moment with the electron, with the same velocity as the electron. (This force might be measured, for example, by a spring balance at rest in the last-mentioned system.) Now if we call this force simply "the force acting upon the electron,"* and maintain the equation—mass X acceleration = force—and if we also decide that the accelerations are to be measured in the stationary system K, we derive from the above equations 111

Longitudinal mass = Transverse mass

2

{Vl-v /c2)3 m \-v2lc2 '

With a different definition of force and acceleration we should naturally obtain other values for the masses. This shows us that in comparing different theories of the motion of the electron we must proceed very cautiously. We remark 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 (in our sense of the word) by the addition of an electric charge, no matter how small. We will now determine the kinetic energy of the electron. If an electron moves from rest at the origin of co-ordinates of the system K along the axis of X under the action of an electrostatic force X, it is clear that the energy withdrawn from the electrostatic field has the value

eXdx. As the electron is to be slowly accelerated,

and consequently may not give off any energy in the form of radiation, the energy withdrawn from the electrostatic field must be put down as equal to the energy of motion W of the electron. Bearing * The definition of force here given is not advantageous, as was first shown by M. Planck. It is more to the point to define force in such a way that the laws of momentum and energy assume the simplest form.

...

139

140 Lorentz and Poincare Invariance in mind that during the whole process of motion which we are considering, the first of the equations (A) applies, we therefore obtain W = \ eXdx = m V B3v dv = mc21, * ,-ll. J Jo }Vl-^2/c2 J Thus, when v = c, W becomes infinite. Velocities greater than that of light have—as in our previous results—no possibility of existence. This expression for the kinetic energy must also, by virtue of the argument stated above, apply to ponderable masses as well. We will now enumerate the properties of the motion of the electron which result from the system of equations (A), and are accessible to experiment. 1. From the second equation of the system (A) it follows that an electric force Y and a magnetic force N have an equally strong deflective action on an electron moving with the velocity v, when Y = Nvjc. Thus we see that it is possible by our theory to determine the velocity of the electron from the ratio of the magnetic power of deflexion Am to the electric power of deflexion Af, for any velocity, by applying the law

This relationship may be tested experimentally, since the velocity of the electron can be directly measured, e.g. by means of rapidly oscillating electric and magnetic fields. 2. From the deduction for the kinetic energy of the electron it follows that between the potential difference, P, traversed and the acquired velocity v of the electron there must be the relationship

3. We calculate the radius of curvature R of the path of the electron when a magnetic force N is present (as the only deflective force), acting perpendicularly to the velocity of the electron. From the second of the equations (A) we obtain

_fl_2 dt

t--±l

~ R

m c

AM/7\-*\ 2 ]j \

c J

or mc2 v/c 1 ~7~ ' V ( I - « 2 / c 2 ) ' If ' These three relationships are a complete expression for the laws according to which, by the theory here advanced, the electron must move.

Chap. 2. Special Relativity ... In conclusion I wish to say that in working at the problem here ...dealt with 1 have had the loyal assistance of ray friend and colleague M. Besso, and that I am indebted to him for several valuable suggestions.

141

142

Lorentz and Poincare Invariance

Concepts which have proved useful for ordering things easily assume so great an authority over us, that we forget their terrestrial and accept them as unalterable facts. "conceptual

necessities,"

"a priori

scientific progress is frequently errors.

origin

They then become labeled as

situations," etc.

The road of

blocked for long periods by such

It is therefore not just an idle game to exercise our ability to

analyze familiar concepts, and to demonstrate which their justification

and usefulness

depend,

the conditions

on

and the way in

which these developed, little by little, from the data of experience. In this way they are deprived of their excessive authority." Einstein (Phys. Zeitschr. 17,101 (1916))

Einstein and Ehrenfest sharing their love of music. From a drawing by Maryke Kamerlingh Onnes.

Chap. 2. Special Relativity ...

143

The Principle of Relativity and the Fundamental Equations of Mechanics* by Max Planck, (presented in the meeting on 23. March 1906.) The "principle of relativity", recently introduced by H. A. LORENTZ 1 and, in a more general form by A. EINSTEIN 2 , states that of two reference frames (x, y, z, t) and (x', y1, z',t'), related by: x' =

,

1

(x - Vt),

y' = y,

z' = z,

t' =

. \ ,

(t - Kx)

m

(c velocity of light in vacuum) both can be used for the basic equations of mechanics and electrodynamics with same justification. Neither frame can therefore be considered to be "at rest" (or preferred). If this statement proves true in general, it implies such a great simplification of all problems of the electrodynamics of moving bodies, that the question of its validity deserves to be placed at the center of all theoretical research in this area. Apparently, the most recent measurements by W. KAUFMANN 3 have solved this problem in a negative sense, so that any further investigation seems to be unnecessary. However, in view of the not-so-simple theory behind these experiments, I do not think one can entirely exclude the possibility that further developments of the principle of relativity prove to be compatible with the observations. Neither do I wish to give decisive weight to the consideration that, according to the principle of relativity, a moving electron would be subject to a special deformation work. After all, one can consider this work to be generally part of the kinetic energy of the electron. It is true that an electrodynamic explanation of inertia remains an open question; on th ; other hand, an advantage arises in that one need not ascribe a spherical shape or any other form to an electron to attain a certain dependency of inertia on velocity. However that may be: a physical idea of the simplicity and generality as that contained in the principle of relativity deserves to be tested in more ways than one, and to be proven wrong if faulty. No better way of testing it can be found than 'Translated by T. Kleinschmidt and W. Kern, Univ. Heidelberg and Univ. of Massachusetts Dartmouth. 'H.A. Lorentz, Versl. Kon. Akad. v. Wet. Amsterdam 1904, S. 809. 2 A . Einstein, Ann. d. Phys. (4) 17, 891, 1905. 3 W . Kaufmann, Sitzungsber. d. preu. Akad. d. Wiss. 1905, S.949; Ann. d. Phys. (4) 19,487, 1906.

144 Lorentz and Poincare Invariance searching for the consequences that it leads to. In this point of view, the following investigation might be of some use. It treats the problem of determining the form of the fundamental equation of motion of mechanics that have to replace the usual Newtonian equations for the motion of a free point particle mx = Fx,

my - Fy,

m'z = Fz

(2)

if the principle of relativity is taken into account. According to this principle the simple equations (2) only hold for a point at rest (x = 0, y — 0, z — 0). For a point with a finite velocity V = y/x2 + y2 + P

(3)

they need to be generalized. It is true that for arbitrary values of v one could simply define quantities Fx,Fy,Fz as products of mass and acceleration and label them components of the force, which causes the movement, as is the custom in many treatments of mechanics. However, the force defined in this manner would not have an independent physical meaning. In particular, it would lose its simple relationship with the potential energy. This is because the principle of relativity demands that in a "primed" reference frame defined by equations (1), the equations mx' = Fx,

my'=Fy,

mz'= F'z

be valid as well. But then the relations between Fx,Fy,Fz and FX,F'FZ would become quite complicated as can be seen by deriving them from the relations between x and x1 resulting from equation (1). A simple physical interpretation of these quantities becomes impossible. In order to become familiar with the general relation existing between acceleration and moving force, it is advisable to start with a special case in which one knows the relationship between the components of the moving force in both reference frames. Such a case is an electromagnetic field in vacuum acting on a point particle with mass m and charge e. In that case, we have the relations 1

E

'- = OT (a + 7*') •

"'' = W^

("' - 7 £ 0

for the electric and magnetic fields in both reference frames (1). Imagine a point particle at the origin of the "unprimed" reference frame (x, y, z,t) with velocity components x, y, z in that frame. We ask for its equations of motion. This question has a unique answer when we imagine the point particle to be at the origin of a second reference frame which moves with constant velocity components J

A . Einstein, 1. c , S. 909.

Chap. 2. Special Relativity ...

145

x,y,z with respect to the original frame. Let the x-axis of this frame be along the velocity direction v of the point particle with magnitude given by (3). Then the particle is at rest in the new frame and the equations of motion in the new frame are valid in the simple form (2), where the force, causing the motion is given by the product of electric charge e and electric field strength. Now let us transform the equations of motion to a second frame in which the x-axis points again in the direction of v, but which is at rest in the (x,y,z,t) frame. On one side, equations (1) serve to transform the components of the acceleration, on the other side equations (4) serve for the force components, by simply replacing i; with V everywhere. Finally, a simple rotation brings this frame into the (x,y, z,t) frame. After performing all these elementary calculations one obtains the equations of motion in the form TTIX

GX

6

eEx - -j{iEx

+ yEy + zEz) + -{yH2 - zHy),...

(5)

These three equations can be verified by considering that, according to the principle of relativity, the equations have to remain correct when replacing unprimed quantities by primed ones everywhere, while maintaining the constants c, e and m. This is indeed confirmed in general because of the relations (1) and (4), for an arbitrary value of V. To simplify the equations of motion as much as possible, we multiply them with x,y and z, respectively, and add them. The result is e(xEx + yEy + zEz) =

m(xx + yy+ z'z)

(l-£)§

Substituting this in (5) yields mx

dt \ y i _ i ; 2 / c 2

= Fx,etc.

(6)

where we have used eEx + -(yHz-zHy)

= FXl...

These equations contain the solution of the problem posed above. They represent the generalization of the Newtonian equations of motion (2) demanded by the principle of relativity. Comparing them with the Lagrangian equations of motion: d

fdL

*lai)=F--

W

where L denotes the Lagrangian, we obtain L = —mc \ 11

+ const.

(8)

146 Lorentz and Poincare Invaiiance Multiplying equations (7) with xdt, ydt and zdt, respectively, and adding them yields the energy theorem dL .dL .dL d ( x— + y— + z— - L ) = Fxdx + Fydy + Fzdz. ox oy oz From this relation the expression of the energy H of the point particle follows: .dL .dL .dL TT H = Z — + y— + z— - L Ox Oy oz

mc2 . + const. L _ v2

The equations of motion (7) can also be represented in the form of Hamilton's Principle:

I {SL + A)dt = 0, to

where the time t and the initial and the final position remain unvaried. The quantity A denotes the virtual work: A = Fx5x + FySy +

Fjz.

Finally we write down Hamilton's canonical equations of motion. For this purpose we introduce momentum coordinates px, py, pz, where dL Px = dx

mi

v^T

Regarding the energy H as a function of px, py, pz, and using the abbreviation 2 Pl+ PI+PI = P2 w e obtain y H = mc2\

/

I -\

v2 r-rr + const m^c1

and the Hamiltonian equations of motion become:

dt

*'

dx__dH_ dt dpx'

dt

"'

dy__dH_ dt dpy'

dt

dz _ dH dt dpz

All these relations are valid in the (x, y, z, i) frame used here as well as in any other frame (x', y', z', t') connected with it by equations (1).

Chap. 2. Special Relativity

...

SPACE AND T I M E BY H. MINKOWSKI

T

H E views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. I First of all I should like to show how it might be possible, setting out from the accepted mechanics of the present day, along a purely mathematical line of thought, to arrive at changed ideas of space and time. The equations of Newton's mechanics exhibit a two-fold invariance. Their form remains unaltered, firstly, if we subject the underlying system of spatial co-ordinates to any arbitrary change of position; secondly, if we change its state of motion, namely, by imparting to it any uniform translatory motion; furthermore, the zero point of time is given no part to play. W e are accustomed to look upon the axioms of geometry as finished with, when we feel ripe for the axioms of mechanics, and for that reason the two invariances are probably rarely mentioned in the same breath. E a c h of them by itself signifies, for the differential equations of mechanics, a certain group of transformations. The existence of the first group is looked upon as a fundamental characteristic of Bpace. The second group is preferably treated with disdain, so that we with untroubled minds may overcome the difficulty of never being able to decide, from physical phenomena, whether space, which is supposed to be stationary, may not be after all in a state of uniform translation. Thus the two groups, side by side, lead their lives entirely apart. Their Utterly heterogeneous character may have discouraged any attempt to compound them. B u t it is precisely when they are compounded that the complete group, as a whole, gives us to think. W e will try to visualize the state of things by the graphic method. L e t x, y, z be rectangular co-ordinates for space, and let t denote time. The objects of our perception invariably include places and times in combination. Nobody has ever noticed a place except at a time, or a time except at a place. B u t I still respect the dogma that both space and

147

148

Lorentz and Poincare Invariance

time have independent significance. A point of space at a point of time, that is, a system of values x, y, z, t, I will call a world-point. The multiplicity of all thinkable x, y, z, t systems of values we will christen the world. With this most valiant piece of chalk I might project upon the blackboard four world-axes. Since merely one chalky axis, as it is, consists of molecules all a-thrill, and moreover is taking part in the earth's travels in the universe, it already affords us ample scope for abstraction; the somewhat greater abstraction associated with the number four is for the mathematician no infliction. Not to leave a yawning void anywhere, we will imagine that everywhere and everywhen there is something perceptible. To avoid saying " matter " or " electricity " I will use for this something the word " substance." We fix our attention on the substantial point which is at the worldpoint x, y, z, t, and imagine that we are able to recognize this substantial point at any other time. Let the variations dx, dy, dz of the space co-ordinates of this substantial point correspond to a time element dt. Then we obtain, as an image, so to speak, of the everlasting career of the substantial point, a curve in the world, a world-line, the points of which can be referred unequivocally to the parameter t from - oo to + GO . The whole universe is seen to resolve itself into similar world-lines, and I would fain anticipate myself by saying that in my opinion physical laws might find their most perfect expression as reciprocal relations between these world-lines. The concepts, space and time, cause the x, y, ^-manifold t = 0 and its two sides t > 0 and t < 0 to fall asunder. If, for simplicity, we retain the same zero point of space and time, the first-mentioned group signifies in mechanics that we may subject the axes of x, y, z at t = 0 to any rotation we choose about the origin, corresponding to the homogeneous linear transformations of the expression x2 + y* + z\ But the second group means that we may—also without changing the expression of the laws of mechanics—replace x, y, z,tbj x - at, y - fit, z - yt, t with any constant values of a, /9, 0. Now what has the requirement of orthogonality in space to do with this perfect freedom of the time axis in an upward direction ?

Chap. 2. Special Relativity

...

To establish the connexion, let us take a positive parameter c, and consider the graphical representation of cH* - x* - y2 - z* = 1. It consists of two surfaces separated by t = 0, on the analogy of a hyperboloid of two sheets. We consider the sheet in the region t>0, and now take those homogeneous linear transformations of x, y, z, t into four new variables x, y', z, t', for which the expression for this sheet in the new variables is of the same form. It is evident that the rotations of space about the origin pertain to these transformations. Thus we gain full comprehension of the rest of the transformations simply by taking into consideration one among them, such that y and z remain unchanged. We draw (Fig. 1) the section of this sheet by the plane of the axes of x and t—the upper branch of the hyperbola cH2 - x% — 1, with its asymptotes. From the origin O we draw any radius

FIG.

1.

149

150

Lorentz and Poincare Invariance

vector OA' of this branch of the hyperbola; draw the tangent to the hyperbola at A' to cut the asymptote on the right at B ' ; complete the parallelogram OA'B'C; and finally, for subsequent use, produce B'C to cut the axis of x at D'. Now if we take OC' and OA' as axes of oblique co-ordinates x', t', with the measures OC = 1, OA' = 1/c, then that branch of the hyperbola again acquires the expression c3?* - x* = 1, t'> 0, and the transition from x, y, z, t to x, y, z, t' is one of the transformations in question. With these transformations we now associate the arbitrary displacements of the zero point of space and time, and thereby constitute a group of transformations, which is also, evidently, dependent on the parameter c This group I denote by Gc. If we now allow c to increase to infinity, and 1/c therefore to converge towards zero, we see from the figure that the branch of the hyperbola bends more and more towards the axis of x, the angle of the asymptotes becomes more and more obtuse, and that in the limit this special transformation changes into one in which the axis of f may have any upward direction whatever, while x approaches more and more exactly to x. In view of this it is clear that group Gc in the limit when c = co , that is the group Gm, becomes no other than that complete group which is appropriate to Newtonian mechanics. This being so, and since Gc is mathematically more intelligible than G^, it looks as though the thought might have struck some mathematician, fancy-free, that after all, as a matter of fact, natural phenomena do not possess an invariance with the group G B , but rather with a group Gc, c being finite and determinate, but in ordinary units of measure, extremely great. Such a premonition would have been an extraordinary triumph for pure mathematics. Well, mathematics, though it now can display only staircase-wit, has the satisfaction of being wise after the event, and is able, thanks to its happy antecedents, with its senses sharpened by an unhampered outlook to far horizons, to grasp forthwith the far-reaching consequences of such a metamorphosis of our concept of nature. I will state at once what is the value of c with which we shall finally be dealing. It is the velocity of the propagation of light in empty space. To avoid speaking either of space or of emptiness, we may define this magnitude in another way, as the ratio of the electromagnetic to the electrostatic unit of electricity.

Chap. 2. Special Relativity

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The existence of the invariance of natural laws for the relevant group Ge would have to be taken, then, in this way:— From the totality of natural phenomena it is possible, by successively enhanced approximations, to derive more and more exactly a system of reference x, y, z, t, space and time, by means of which these phenomena then present themselves in agreement with definite laws. But when this is done, this system of reference is by no means unequivocally determined by the phenomena. It is still possible to make any change in the system of reference that is in conformity with the transformations of the group Gc, and leave the expression of the laws of nature unaltered. For example, in correspondence with the figure described above, we may also designate time t', but then must of necessity, in connexion therewith, define space by the manifold of the three parameters x, y, z, in which case physical laws would be expressed in exactly the same way by meanB of x, y, z, t' as by means of x, y, z, t. We should then have in the world no longer space, but an infinite number of spaces, analogously as there are in three-dimensional space an infinite number of planes. Three-dimensional geometry becomes a chapter in four-dimensional physics. Now you know why I said at the outset that space and time are to fade away into shadows, and only a world in itself will subsist. II The question now is, what are the circumstances which force this changed conception of space and time upon us ? Does it actually never contradict experience ? And finally, is it advantageous for describing phenomena ? Before going into these questions, I must make an important remark. If we have in any way individualized space and time, we have, as a world-line corresponding to a stationary substantial point, a straight line parallel to the axis of t; corresponding to a substantial point in uniform motion, a straight line at an angle to the axis of t; to a substantial point in varying motion, a world-line in some form of a curve. If at any world-point x, y, z, t we take the world-line passing through that point, and find it parallel to any radius vector OA' of the above-mentioned hyperboloidal sheet, we can introduce OA' as a new axis of time, and with the new concepts of space and time thus given, the substance at the

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world-poinfc concerned appears as at rest. We will now introduce this fundamental axiom :— The substance at any world-point may always, with the appropriate determination of space and time, be looked upon as at rest. The axiom signifies that at any world-point the expression c2dt2 - dx* - dy2 - dz1 always has a positive value, or, what comes to the same thing, that any velocity v always proves less than c. Accordingly c would stand as the upper limit for all substantial velocities, and that is precisely what would reveal the deeper significance of the magnitude c. In this second form the first impression made by the axiom is not altogether pleasing. But we must bear in mind that a modified form of mechanics, in which the square root of this quadratic differential expression appears, will now make its way, so that caBes with a velocity greater than that of light will henceforward play only some such part as that of figures with imaginary co-ordinates in geometry. Now the impulse and true motive for assuming the group Gc came from the fact that the differential equation for the propagation of light in empty space possesses that group Gc. * On. the other hand, the concept of rigid bodies has meaning only in mechanics satisfying the group G^,. If we have a theory of optics with Gc, and if on the other hand there were rigid bodies, it is easy to see that one and the same direction of t would be distinguished by the two hyperboloidal sheets appropriate to Gc and G,,,, and this would have the further consequence, that we should be able, by employing suitable rigid optical instruments in the laboratory, to perceive some alteration in the phenomena when the orientation with respect to the direction of the earth's motion is changed. But all efforts directed towards this goal, in particular the famous interference experiment of Michelson, have had a negative result. To explain this failure, H. A. Lorentz set up an hypothesis, the success of which lies in this very invariance in optics for the group Gc According to Lorentz any moving body must have undergone a contraction in the direction of its motion, and in fact with a velocity v, a contraction in the ratio

1: VI " «7c2.

* An application of this fact in its essentials has already been given by W. Voigt, Gdttinger Naohrichten, 1887, p. 41.

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This hypothesis sounds extremely fantastical, for the contraction is not to be looked upon as a consequence of resistances in the ether, or anything of that kind, but simply as a gift from above,—as an accompanying circumstance of the circumstance of motion. I will no w show by our figure that the Lorentzian hypothesis is completely equivalent to the new conception of space and time, which, indeed, makes the hypothesis much more intelligible. If for simplicity we disregard y and z, and imagine a world of one spatial dimension, then a parallel band, upright like the axis of t, and another inclining to the axis of t (see Fig. 1) represent, respectively, the career of a body at rest or in uniform motion, preserving in each case a constant spatial extent. If OA' is parallel to the second band, we can introduce t' as the time, and x as the space co-ordinate, and then the second body appears at rest, the first in uniform motion. W e now assume that the first body, envisaged as at rest, has the length I, that is, the cross section P P of the first band on the axis of x is equal to I . OC, where OC denotes the unit of measure on the axis of x; and on the other hand, that the second body, envisaged as at rest, has the same length Z, which then means that the cross section Q'Q' of the second band, measured parallel to the axis of x, is equal to I. O C . W e now have in these two bodies images of two equal Lorentzian electrons, one at rest and one in uniform motion. But if we retain the original co-ordinates x, t, we must give as the extent of the second electron the cross section of its appropriate band parallel to the axis of x. Now since Q'Q' = I. OC, it is evident that QQ = I. OD'. If dx/dt for the second band is equal to v, an easy calculation gives OD' = O C V 1 "

W , 2

therefore also P P : QQ = 1 : Jl - v /c2. But this is the meaning of Lorentz's hypothesis of the contraction of electrons in motion. If on the other hand we envisage the second electron as at rest, and therefore adopt the system of reference x' t', the length of the first must be denoted by the cross section P ' P ' of its band parallel to O C , and we should find the first electron in comparison with the second to be contracted in exactly the same proportion; for in the figure P ' P ' : Q'Q' = OD : O C = O D ' : OC = QQ : P P . Lorentz called the t' combination of x and t the local time of the electron in uniform motion, and applied a physical

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construction of this concept, for the better understanding of the hypothesis of contraction. But the credit of first recognizing clearly that the time of the one electron is just as good as that of the other, that is to say, that t and t' are to be treated identically, belongs to A. Einstein.* Thus time, as a concept unequivocally determined by phenomena, was first deposed from its high seat. Neither Einstein nor Lorentz made any attack on the concept of space, perhaps because in the above-mentioned special transformation, where the plane of x, t' coincides with the plane of x, t, an interpretation is possible by saying that the a;-axis of space maintains its position. One may expect to find a corresponding violation of the concept of space appraised as another act of audacity on the part of the higher mathematics. Nevertheless, this further step is indispensable for the true understanding of the group Grc, and when it has been taken, the word relativitypostulate for the requirement of an invariance with the group Gc seems to me very feeble. Since the postulate comes to mean that only the four-dimensional world in space and time is given by phenomena^ but that the projection in space and in time may still be undertaken with a certain degree of freedom, I prefer to call it the postulate of the absolute world (or briefly, the world-postulate).

ni The world-postulate permits identical treatment ofkfche four co-ordinates x, y, z, t. By this means, as I shall now show, the forms in which the laws of physics are displayed gain in intelligibility. In particular the idea of acceleration acquires a clear-cut character. I will use a geometrical manner of expression, which suggests itself at once if we tacitly disregard z in the triplex x, y, z. I take any world-point 0 as the zero-point of spacetime. The cone c2t2 - x2 - y2 - z2 = 0 with apex 0 (Fig. 2) consists of two parts, one with values t < 0, the other with values t > 0. The former, the front cone of O, consists, let us say, of all the world-points which " send light to O," the latter, the back cone of O, of all the world-points which " receive light from O." The territory bounded by the front cone alone, we may call " before " 0, that which is bounded by • A. Einstein, Ann. d. Phys., 17, 1905, p. 891; Jahib. d. Radioaktivit&t und Elektronik, 4, 1907, p. 411.

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the back cone alone, " after" O. The hyperboloidal sheet already discussed F = c't1 - x1 - y2 - z* = 1, t > 0 lies after 0. The territory between the cones is filled by the one-sheeted hyperboloidal figures - F = x* + y2 + z* - cH* = ¥ for all constant positive values of k. We are specially interested in the hyperbolas with O as centre, lying on the latter figures. The single branches of these hyperbolas may be called briefly the internal hyperbolas with centre O. One of these branches, regarded as a world-line, would represent a motion which, for t = - co and t = + co, rises asymptotically to the velocity of light, c. If we now, on the analogy of vectors in space, call a directed length in the manifold of x, y, z, t a vector, we have to distinguish between the time-like vectors with directions from 0 to the sheet + F = 1, t > 0, and the space-like vectors

Fia. 2.

with directions from O to - F = 1. The time axis may run parallel to any vector of the former kind. Any world-point between the front and back cones of O can be arranged by means of the system of reference so as to be simultaneous with 0, but also just as well so as to be earlier than 0 or later than 0. Any world-point within the front cone of 0 is necessarily always before O ; any world-point within the back cone of 0 necessarily always after O. Corresponding to passing to the limit, c = oo , there would be a complete flattening out of the wedge-shaped segment between the cones into the plane manifold t = 0. In the figures this segment is intentionally drawn with different widths. We divide up any vector we choose, e.g. that from 0 to x, y, z, t, into the four components x, y, z, t. If the directions

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of two vectors are, respectively, that of a radius vector OE from 0 to one of the surfaces + F = 1, and that of a tangent RS at the point R of the same surface, the vectors are said to be normal to one another. Thus the condition that the vectors with components x, y, z, t and xx, ylt zv tx may be normal to each other is cHtx

- ZiCj - yyx

- zzx — 0 .

For the measurement of vectors in different directions the units of measure are to be fixed by assigning to a space-like vector from 0 to - F = 1 always the magnitude 1, and to a time-like vector from 0 to + F = 1, t > 0 always the magnitude 1/c. If we imagine at a world-point P (x, y, z, t) the worldline of a substantial point running through that point, the magnitude corresponding to the time-like vector dx, dy, dz, dt laid off along the line is therefore dr = -y/^dt*

- dx1 - dif - dz'.

The integral $dr = r of this amount, taken along the worldline from any fixed starting-point P 0 to the variable endpoint P, we call the proper time of the substantial point at P. On the world-line we regard x, y, z, t—the components of the vector OP—as functions of the proper time T ; denote their first differential coefficients with respect to T by x, y, z, i ; their second differential coefficients with respect to T by £. y> z> ^; and give names to the appropriate vectors, calling the derivative of the vector OP with respect to T the velocity vector at P, and the derivative of this velocity vector with respect to r the acceleration vector at P. Hence, since cH2 - x2 - y* - z* = c 2 ,

we have cHt - xx - yy - zz = 0, i.e. the velocity vector is the time-like vector of unit magnitude in the direction of the world-line at P, and the acceleration vector at P is normal to the velocity vector at P, and is therefore in any case a space-like vector. Now, as is readily seen, there is a definite hyperbola which has three infinitely proximate points in common with the world-line at P, and whose asymptotes are generators of

Chap. 2. Special Relativity

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a " front cone " and a " back cone " (Fig. 3). Let this hyperbola be called the hyperbola of curvature at P . If M is the centre of this hyperbola, we here have to do with an internal hyperbola with centre M. L e t p be the magnitude of the vector M P ; then we recognize the acceleration vector at P as the vector in the direction M P of magnitude If x, y, z, l are all zero, the hyperbola of curvature reduces to the straight line touching the world-line in P , and we must put p = oo. IV To s r i o w that the assumption of group Gc for the laws of physics never leads to a contradiction, it is unavoidable to undertake a revision of the whole of physics on the basis of this assumption. This revision has to some extent already been successfully carried out for questions of thermodynamics and heat radiation,* for electromagnetic processes, and finally, with the retention of the concept of mass, for mechanics.f F o r this last branch of physics it is of prime importance to raise the question—When a force with the components X, Y, Z parallel to the axes of space acts at a world-point P (x, y, z, t), where the velocity vector is x, y, z, £,.what must we take this force to be when the system of reference is in any way changed ? Now there exist certain approved statements as to the ponderomotive force in the electromagnetic field .in the cases where the group G c is undoubtedly admissible. These statements lead up to the simple rule:—When the system of reference is changed, the force in question transforms into a force in the new space co-ordinates in such a way that the appropriate vector with the components £X, FIG. 3.

tY, tZ, tT, where T = i (%X + %Y +

-Z)

is the rate at which work is done by the force at the world* M. Planck, " Zur Dynamik bewegter Systems," Berliner Berichte, 1907, p. 542; also in Ann. d. Phys., 26, 1908, p. 1. + H. Minkowski, " Die Grundgleichungan fiir die elektromagnetiBchen Vorgange in bewegten Korpern," Gottinger Nachrichfcen, 1908, p. 58.

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point divided by c, remains unchanged. This vector is always normal to the velocity vector at P . A force vector of this kind, corresponding to a force at P, is to be called a " motive force vector " at P . I shall now describe the world-line of a substantial point with constant mechanical mass m, passing through P. L e t t h e velocity vector at P , multiplied by m, be called the " m o m e n t u m vector" at P, and the acceleration vector at P , multiplied by m, be called the " force vector " of the motion at P . With these definitions, the law of motion of a point of mass with given motive force vector runs thus :— * The Force Vector of Motion is Equal to the Motive Force Vector. This assertion comprises four equations for the components corresponding to the four axes, and since.both vectors mentioned are a priori normal to the velocity vector, the fourth equation may be looked upon as a consequence of the other three. In accordance with the above signification of T, the fourth equation undoubtedly represents the law of energy. Therefore the component of the momentum vector along the axis of t, multiplied by c, is to be defined as the kinetic energy of the point mass. The expression for this is mc2-,- = mc 2 /«yi - v^lc2 i.e.-, after removal of the additive constant mc2, the expression ^mv2 of Newtonian mechanics down to magnitudes of the order 1/c1. It comes out very clearly in this way, how the energy depends on the system of reference. But as the axis of t may be laid in the direction of any time-like vector, the law of energy, framed for all possible systems of reference, already contains, on the other hand, the whole system of the equations of motion. At the limiting transition which we have discussed, to c = oo, this fact retains its importance for the axiomatic structure of Newtonian mechanics as well, and has already been appreciated in this sense by I. E. Schutz.*/ W e can determine the ratio of the units of length and time beforehand in such a way that the natural limit of velocity becomes c = 1. If wa then introduce, further, * H . Minkowski, loc. cit., p 107. Cf. also M. Planck, Verhandlungen der physikalischen Gesellschaft, i, 1906, p. 136. *tl. R. Scafitz, " D a s Prinzip der absolution Erhaltung der Energie," Gottinger Nachr., 1897, p. 110.

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*/ - 1 t = s in place of t, the quadratic differential expression dr* = - dx2 - dya - dz* - ds* thus becomes perfectly symmetrical in x, y, z, s ; and this symmetry is communicated to any law which does not contradict the world-postulate. Thus the essence of this postulate may be clothed mathematically in a very pregnant manner in the mystic formula 3 . 105 km = *J - 1 sees.

The advantages afforded by the world-postulate will perhaps be most strikingly exemplified by indicating the effects proceeding from a point charge in any kind of motion according to the Maxwell-Lorentz theory. Let us imagine the world-line of such a point electron with the charge e, and introduce upon it the proper* time T from any initial point. In order to find the field caused by the electron at any world-point Pj, we construct the front cone belonging to Pi (Fig. 4). The cone evidently meets the world-line, since the directions of the line are everywhere those of time-like vectors, at the single point P. We draw the tangent to the world-line at P, and construct through P x the normal P,Q to this tangent. Fia. 4. Let the length of PiQ be r. Then, by the definition of a front cone, the length of PQ must be rjc. Now the vector in the direction PQ of magnitude e/r represents by its components along the axes of x, y, z, the vector potential multiplied by c, and by the component along the axis of t, the scalar potential of the field excited by e at the world-point P. Herein lie the elementary laws formulated by A. Lienard and E. Wiechert.* Then in the description of the field produced by the electron we see that the separation of the field into electric * A. Lienard, " Champ <51eotrique et magnetique produit par une charge concentres en un point et anim£e d'un mouvement quelconque," L'Eclairage Electrique, 16, 1898, pp. 5, 53, 106; B. Wiechert, " Elektrodynamisohe Elementargesetze," Arch. N^erl. (2), 5,1900, p. 549.

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160 Lorentz and Poincare Invariance

and magnetic force is a relative one with regard to the underlying time axis ; the most perspicuous way of describing the two forces together is on a certain analogy with the wrench in mechanics, though the analogy is not complete. I will now describe the ponderomotive action of a moving point charge on another moving point charge. Let us imagine the world-line of a second point electron of the charge' eu passing through the world-point ¥1. We define P, Q, r as before, then construct (Fig. 4) the centre M of the hyperbola of curvature at P, and finally the normal MN from M to a straight line imagined through P parallel to Q P r With P as starting-point we now determine a system of reference as follows:—The axis of t in the direction PQ, the axis of x in direction QPj, the axis of y in direction MN, whereby finally the direction of the axis of z is also defined as normal to the axes of t, x, y. Let the acceleration vector at P be x, y, z, t, the velocity vector at P x be xu yv zx, ix. The motive force vector exerted at P x by the first moving electron e on the second moving electron ex now takes the form

- ee^k - £ ) * , where the components Six, Siy, Stz, Sit of'the vector Si satisfy the three relations cStt - Six = p ,

Sty = - j - , Stz = 0,

and where, fourthly, this vector Si is normal to the velocity vector at P^ and through this circumstance alone stands in dependence on the latter velocity vector. When we compare this statement with previous formulations * of the same elementary law of the ponderomotive action of moving point charges on one another, we are compelled to admit that it is only in four dimensions that the relations here taken under consideration reveal their inner being in full simplicity, and that on a three dimensional space forced upon us a priori they cast only a very complicated projection. In mechanics as reformed in accordance with the worldpostulate, the disturbing lack of harmony between Newtonian •K. Schwarzwald, G5ttinger Nachr., 1903, p. 182; H. A. Lorentz, Enaykl. d. math. Wissensoh., V, Art. 14, p. 199.

Chap. 2. Special Relativity

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mechanics and modern electrodynamics disappears of its own accord. Before concluding I will just touch upon the attitude of Newton's law of attraction toward this postulate. I shall assume that when two points of mass m, ml describe their world-lines, a motive force vector is exerted by m on m^, of exactly the same form as that just given in the case of electrons, except that + mmx must now take the place of - eev W e now specially consider the case where the acceleration vector of m is constantly zero. L e t us then introduce t in such a way that m is to be taken as at rest, and let only mx move under the motive force vector which proceeds from m. If we now modify this given vector in the first place by adding the factor i ~ l = * / ! - v^c1, which, to the order of 1/c2, is equal to 1, it will be seen f that for the positions xv yv zv of m1 and their variations in time, we should arrive exactly at Kepler's laws again, except that the proper times TX of mx would take the place of the times ^. F r o m this simple remark it may then be seen that the proposed law of attraction combined with the new meghanics is no less well adapted to explain astronomical observations than the Newtonian law of attraction combined with Newtonian mechanics. T h e fundamental equations for electromagnetic processes in ponderable bodies also fit in completely with the worldpostulate. As I shall show elsewhere, it is not even by any means necessary to abandon the derivation of these fundamental equations from ideas of the electronic theory, as taught by Lorentz, in order to adapt them to the worldpostulate. The validity without exception of the world-postulate, I like to think, 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. In the development of its mathematical consequences there will be ample suggestions for experimental verifications of the postulate, which will suffice to conciliate even those to whom the abandonment of old-established views is unsympathetic or painful, by the idea of a pre-established harmony "between pure mathematics and physics. + H. Minkowski, loc. oit., p. 110.

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162 Lomntz and Poincare Invariance

The Theory of Relativity and Science* W. Pauli

The special theory of relativity was linked up with the mathematical group concept, as it had already come to light in the mechanics of Galileo and Newton, now sofirmlyestablished on an empirical basic. In this system of mechanics all states of motion of the observer, or, expressed mathematically, all coordinate systems which arise from each other by a uniform motion of translation without rotation, are equally privileged. Since the state of rest of a mass does not require any particular cause for its maintenance, the same assumption had to be made in classical mechanics for the state of uniform motion, since the latter arises from the state of rest by one of the transformations contained in the group of mechanics. This formulation of the law of inertia of classical mechanics is of course not the original one, but takes account of the later development of the group concept in the mathematics of the 19th century. The development of electrodynamics during the same period culminated in the partial differential equations of Maxwell and K A. Lorentz. It was evident that these did not admit the group of classical mechanics, since in particular the fact that the velocity of light in vacuo is independent of the state of motion of the light-sources is contained in them as a -consequence.

* Helvetica Physica Acta, Supplement IV, pp. 282-286 (1956).

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163

Would it now be necessary to abandon as only approximately valid the property whereby the laws of nature admit a group, or is the group of mechanics perhaps only approximately valid, and should it be replaced by a more general group, valid for both mechanical and electromagnetic processes? The decision was in favour of the second alternative. This postulate could be arrived at by two paths. Either one could investigate by pure mathematics what is the most general group of transformations under which the equations of Maxwell and Lorentz which were well known at this time, preserve their form. This path was followed by the mathematician, H. Poincare. Or one could determine, by critical analysis, those physical assumptions which had led to the particular group of the mechanics of Galileo and Newton. This was the path followed by Einstein. He showed that, from the general standpoint of the equivalence of all coordinate systems moving with constant velocity with respect to each other, the invariance of simultaneity of spatially separated events, in the sense in which it is assumed in classical mechanics, involves the special additional supposition of the possibility of infinitely great signal velocities. If this supposition is dropped and replaced by the assumption of a finite maximal signal velocity, time is also transformed, and the group, mathematically speaking, leaves invariant an indefinite quadratic form in four dimensions, three of space and one of time. The electrodynamics of Maxwell and Lorentz did in fact turn out to be invariant under the group of transformations determined by Einstein on the basis of these general considerations, if the maximal signal velocity was identified with the velocity of propagation of light in vacuo. Both Einstein and Poincare took their stand on the preparatory work of H. A. Lorentz, who had already come quite close to this result, without however quite reaching it. In the agreement between the results of the methods followed independently of each other by Einstein and Poincare I discern a deeper significance of a harmony between the mathematical method and analysis by means of conceptual experiments (Gedankenexperimente), which rests on general features of physical experience.

164 Lorentz and Poincare Invariance

Einstein's first paper on relativity H. M. Schwartz Department of Physics, University of Arkansas, Fayetteville, Arkansas 72701 (Received 4 April 1975; revised 24 May 1976) Because of its exceptional significance in the history of great ideas in science, Einstein's first paper on relativity, especially its first part, deserves a more careful translation into English than presently exists. A new and annotated translation of this first part is presented here, together with a brief discussion of certain aspects of Einstein's paper.

A. INTRODUCTION With the recent perceptible awakening of an interest among physicists in the history of their subject, the older of the two pillars of current terrestrial physics, the special theory of relativity, has become the subject of intensive historical studies.1 Yet rather remarkably, seven decades after its publication, Einstein's trailblazing paper,2 "Zur Elektrodynamik bewegter Korper" ("On the Electrodynamics of Moving Bodies"), does not yet have a fully satisfactory translation into English. The only English translation available until recently, contained in the widely known collection of original papers on relativity,3 falls short as a completely reliable historical document, not so much because it is marred by a few outright mistranslations— which are in any case easily spotted—but because of its failure in important instances to convey properly the intent of a passage or of the nuances in its original presentation.4 A new translation has indeed appeared recently,5 but unfortunately its newness is limited essentially only to correcting the few flagrant mistranslations.6 For these reasons, and since Einstein's first paper on relativity represents one of the most remarkable intellectual achievements,7 a thoroughgoing English translation of this work would be of obvious interest. In the meantime, an attempt in this direction is presented in Sec. B of the present paper. It covers only the first of the two parts of Einstein's paper. It is this part which is of greatest interest historically, and the one where changes in the older translation are especially in order. A few observations concerning the second part of the paper are contained in Sec. C.

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The references to Sees. B and C include comments on the text of the paper, and remarks concerning Refs. 3 and 5. The translations of the original footnotes to the first part of the paper are labeled with lower case roman letters (a, b, and c) and are located at the end of the present paper as the first three references. Except for one formula which is labeled as (A), none of the formulas are numbered in the original. Thus all the numbering of formulas by Arabic numerals is supplied in the translation. This is, however, the only essential deviation from the original. Even the punctuation of the original paper has been followed as closely as feasible, and the lengthy sentences of the original text have been retained almost throughout without dissection, so as to preserve as far as possible a sense of the flow of thought in the original writing.8 In addition to the comments on the original text contained in the footnotes, a few observations of a more systematic character are presented in the concluding section. A more complete discussion relating to this subject is deferred for presentation in a future paper dealing with certain questions in the history of special relativity (referred to in Ref. 1).

B. TRANSLATION OF THE INTRODUCTION AND THE FIRST PART OF EINSTEIN'S PAPER That Maxwell's electrodynamics—as presently conceived—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena, is well known. Consider, for instance, the electrodynamic interaction of a magnet and a conductor. The observed phenomenon depends here only on the relative motion of the conductor and the magnet, whereas according to the usual conception, the two cases in which either one or the other of the two bodies is in motion must be strictly differentiated. For, if.the magnet is moving and the conductor is stationary, there arises in the vicinity of the magnet an electric field of a certain energy, which generates a current in those places where parts of the conductor are situated. However, if the magnet is stationary and the conductor is moving, then no electric field arises in the vicinity of the magnet, whereas in the conductor there appears an electromotive force to which in itself there corresponds no energy, but which—assuming equality of the relative motion in the two considered cases—gives rise to electric currents of the same magnitude and the same course as those produced by the electric forces in the first case. Examples of a similar kind, as well as the unsuccessful attempts to verify that the earth moves relative to the "light medium," l) lead to the conjecture that, not only in mechanics but in electrodynamics as well, no properties of

166 Lorentz and Poincare Invariance phenomena10 attach to the idea of absolute rest, but rather that the same electrodynamic and optical laws hold in all coordinate systems in which the equations of mechanics are valid, as has already been proved for first-order quantities.1' We shall raise this conjecture (whose content will be called in the sequel the "Principle of Relativity") to the status of a postulate, and in addition we shall introduce another postulate that is only seemingly inconsistent with the former, namely, that light in empty space always propagates with a definite velocity V which is independent of the state of motion of the emitting body. These two postulates suffice for arriving at a simple and consistent electrodynamics of moving bodies on the basis of Maxwell's theory for stationary bodies. The introduction of a "luminiferous ether" will prove to be superfluous, inasmuch as according to the conception that will be developed, there will be no need for the introduction of an "absolutely stationary space" endowed with special properties, or for the assignment of a velocity vector to any point of empty space at which electromagnetic processes occur. Like any other electrodynamic theory, the theory to be developed is based on the kinematics of the rigid body, since the assertions of every theory concern relations between rigid bodies (coordinate systems), clocks, and electromagnetic processes.12 Not taking account sufficiently of this circumstance is the source of the difficulties with which the electrodynamics of moving bodies has to contend with at present.

r

I. Kinematics part /. Definition of simultaneity

vj

•

«

*

Consider a coordinate system in which Newton's equations of mechanics are valid. In order to distinguish this system from other coordinate systems to be introduced later, and for the sake of clarity of presentation, it will be called the "stationary system." If a material point is at rest relative to this coordinate system, its position with respect to it can be determined with the aid of rigid measuring rods by the employment of the methods of Euclidean geometry, and can be expressed in terms of Cartesian coordinates. If we wish to describe the motion of a material point, we give the values of its coordinates as functions of time. It is well to bear here in mind that such a mathematical description has physical sense only when it has been first established clearly what is to be understood here by the word "time." We must note that all our judgments in which time plays a role are always judgments of simultaneous occurrences. When I say, for example: "That train arrives here at 7 o'clock," it means something as follows: "The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events."2-13

Chap. 2. Special Relativity ...

167

It might appear that it would be possible to surmount all the difficulties connected with the definition of "time" by substituting "the position of the small hand of my watch" for "time." Such a definition does in fact suffice when it is a question of defining the time exclusively at the place where the watch happens to be located, but the definition no longer suffices when it is a matter of connecting in time a series of events occurring at different places, or—what amounts to the same thing—to establish the time of events that occur at places remote from the watch. We could, of course, content ourselves with determining the time of every event by having an observer, situated together with a clock at the origin of coordinates, associate the position of the clock's hand with the arrival through empty space of a light signal produced at the event. But, as we know from experience, this association entails the drawback of not being independent of the standpoint of the observer supplied with the clock. We arrive at a much more practical determination by the following consideration. If a clock is situated at the point A of space, then an observer at A can determine the times of the events in the immediate vicinity of A by noting the positions of the hands of the clock which are simultaneous with these events. If there is also a clock at the point B of space—we wish to add, "a clock of precisely the same construction as that located at A"—then the times of events in the immediate vicinity of B can be likewise determined by an observer situated at B. But without further stipulation it is impossible to compare the times of an event at A and of an event at B\ we have so far defined only an "/1-time" and a "fl-time," but have not defined a for -/1-and-S common "time." The latter time can, however, be defined if we establish by definition that the "time" required by light to travel from A to B is equal to the "time" which it requires to travel from B to A. Thus let a light ray depart from A towards B at the "/J-time" tA, be reflected at B towards A at the "fi-time" tB, and return to A at the ",4-time" tA'. The two clocks run in synchrony by definition, when tB~tA

= tA' -tB.

(1)

We assume that this definition of synchronism admits of no contradictions and can be applied for any number of points; that the following conditions are, thus, generally satisfied: 1. When the clock at B is synchronous with the clock at A, then the clock at A is synchronous with the clock at B. 2. When the clock at A is synchronous both with the clock at B and with the clock at C, then also the clocks at B and C are mutually synchronous. We have thus established with the aid of certain (imagined) physical procedures14 what is to be understood by synchronously running clocks at rest at different places, and

168 Lorentz and Poincare Invariance have thereby evidently acquired a definition of "simultaneous" and of "time." The "time" of an event is the indication, simultaneous with the event, of a stationary clock located at the place of the event, and which, for all time determinations, runs in synchrony with a particular stationary clock. We also stipulate, in agreement with experience,! 5 that the quantity 2AB/(tA'- tA) = V (2) is a universal constant (the velocity of light in empty space). The essential thing is that we have defined time by means of clocks at rest in the stationary system; we call the time so defined, because of this association with the stationary system, "the time of the stationary system." 2. On the relativity of lengths and times

I'

|rp>'/.--'^-,R.--"-

The following considerations are based on the principle of relativity and on the principle of the constancy of the velocity of light. We define these two principles as follows: 1. The laws in accordance with which the states of physical systems vary are not dependent on whether these changes of state are referred to one or to the other of two coordinate systems that are in uniform translational relative motion. 2. Every light ray moves in the "stationary" coordinate system with the fixed velocity V, independently of whether this ray is emitted by a stationary or a moving body. Here velocity = (light path)/(time interval),

!•.; 7 /

where time interval is to be taken in the sense of the definition hi Sec. 1. Let there be given a stationary rigid rod, and let its length be / as measured by a measuring rod which is also stationary. We now imagine the axis of the rod placed along the .Y-axis of the stationary coordinate system, and then a uniform parallel translation (of velocity v) imparted to it along the .Y-axis in the direction of increasing x. We now inquire as to the length of the mooing rod, which we suppose to be established by the following two operations: (a) The observer moves together with the aforementioned measuring rod and the rod to be measured, and measures the length of the rod directly, by laying off the measuring rod along it, in the same way as when the rod to be measured, the observer, and the measuring rod are at rest. (b) By means of stationary clocks set up in the stationary system and nchronized according to Sec. 1, the observer ascertains the points in the stationary system where the ends

Chap. 2. Special Relativity ...

169

of the rod are situated at a specified time t. The distance between these two points, measured with the measuring rod previously employed, which is in this case stationary, is likewise a length, which can be denoted as the "length of the rod." According to the principle of relativity, the length obtained with the operation (a), which we shall call "the length of the rod in the moving system," must be equal to the length / of the stationary rod. The length obtained with the operation (b), which we shall call "the length of the (moving) rod in the stationary system," will be determined by us on the basis of our two principles, and we shall find that it differs from /. The kinematics in general use involves the tacit assumption that the lengths determined by our two operations are exactly equal, or, in other words, that at a given instant of time t, a moving rigid body is completely replaceable in geometrical respects by the same body, when it is at rest in a definite position. We imagine further that at the two ends (A and B) of the rod16 there are placed clocks which synchronize with the clocks of the stationary system, i.e., whose readings correspond at any given instant to the "time of the stationary system" at the places where they happen to be; these clocks are thus "synchronous in the stationary system." We also image that each clock is accompanied by an observer, and that these observers apply to both clocks the criterion established in Sec. 1 for the synchronization of two clocks. Suppose that, at the timeb tA, a light ray departs from A, is reflected at B at the time tB, and returns to A at time tA'. Taking account of the principle of the constancy of the velocity of light, we find h-tA= rAB/( V-v) and t/ -tB = rAB/{V+v). where rAB denotes the length of the moving rod—as measured in the stationary system. Observers accompanying the moving rod would thus find that the two clocks are not synchronous, whereas observers in the stationary system would declare them to be synchronous. We thus see that we cannot attach any absolute significance to the concept of simultaneity, but that two events, which when viewed from a given system of coordinates are simultaneous, can no longer be considered as simultaneous events when viewed from a system moving relative to the given system. 3. Theory of coordinate and time transformations from the stationary system to one having relative to it a uniform translational motion Let there be given in the "stationary" space17 two coordinate systems, i.e., two systems consisting each of three

In variance mutually perpendicular rigid material lines issuing from a point. Let the A"-axes of the two systems coincide, and the Y- and Z-axes be respectively parallel. Let each system be provided with a rigid measuring rod and a number of clocks, and let the two measuring rods, as well as all the clocks of both systems, be exactly alike. Let there be imparted now to the origin of one of the two systems (k) a (constant) velocity o in the direction of increasing x of the other, stationary system (A"), which velocity is also communicated to the coordinate axes, the measuring rod in question, and the clocks. To each time t of the stationary system K there corresponds then a definite position of the axes of the moving system, and from symmetry considerations we are justified in assuming that the motion of K can be such that the axes of the moving system at time t (by "r" is always represented a time of the stationary system) are parallel to the axes of the stationary system. We suppose now that space18 is measured in the stationary system K with the stationary measuring rod and in the moving system k with the comoving measuring rod, and that the respective coordinates x.y.z and £,rj,f are thus established. Further, let the time t of the stationary system be determined at all its points where stationary clocks are located, with the aid of these clocks, by the method of light signals presented in Sec. 1; likewise let the time r of the moving system be determined at all its points where clocks arc located and are at rest relative to it, by applying the method of light signals, described in Sec. 1, between the points at which the latter clocks arc located. To every set of values x.y.z.t which determine completely the place and time of an event in the stationary system there corresponds a set of values £,77,f,r which determine that event with respect to the system k. and the problem that must now be solved is finding the system of equations that connect these quantities. To begin with, it is clear that the equations must be linear owing to the homogeneity properties which we attribute to space and time.19 If we set x' = x — ot, then it is clear that to a point at rest in the system k there corresponds a fixed set of values x'.y.z independent of time. We determine first r as a function of x'. y, z, and t. To this end we must express in terms of equations that r is just a representation of the aggregate20 of readings of the clocks at rest in k, which have been synchronized according to the rule given in Sec. 1. Suppose that at the time r 0 a light ray is sent out from the origin of the system k along the A"-axis towards x' and is there reflected at time T\ back to the origin of coordinates, which it reaches at time ri; we must then have ( r o + r 2 ) / 2 = r,

Chap. 2. Special Relativity ...

171

or, inserting the arguments of the function T, and applying the principle of the constancy of the velocity of light in the stationary system,21

I [ , ( 0 , 0 , 0 . ) + r(0,0,0,r

+

^

=

+

-^)_

4<°'°-' + 7rJ-

(3)

From this it follows by choosing x' infinitely small that I I 1 2\V-v

1 \ ch_ _ jh_ V + oJdt dx'

_J dr V-vdt

or dr

^

v

:

+

dr

^ 3 ^ ^ 7 = o-

(4)

It is to be noted that instead of the origin of coordinates we could have chosen any other point as the starting point of the light ray, so that the equation just obtained holds for all values of x'.y.z-22 An analogous consideration—applied to the H- and Z-axes—yields the following equations, when it is noted that light is always propagated along these axes with the velocity (V2 — v2)'/2 when viewed from the stationary system: - = 0, dy

^ = 0. d:

(5)

From these equations it follows, since r is a linear function, that r = a[t-ox'/{V--v-)\. (6) where a is for the moment an unknown function <j}(u), and it is assumed for the sake of brevity that, at the origin of k. t = 0 when r = 0. Using this result, it is easy to determine the quantities £,7j,f by expressing by means of equations that (as required by the principle of the constancy of the velocity of light together with the principle of relativity) light propagates with the velocity V also when measured in the moving system. For a light ray emitted at time r = 0 in the direction of increasing |, we have = = Vr

or Z=

aV[t-vx'/{V^-o-)\.

But, as measured in the stationary system, the ray moves with velocity V — u relative to the origin of k, so that we have

172 Lorentz and Poincaxe Invariance x'/(V -u) = t. Inserting this value of t in the equation for £, we find £=

aV2x'/(V2-v2).

Analogously, we find, by considering light rays moving along the other two axes. 7,= VT = aV[t - ux'/iV2 - u2)], where yj(\n

_ „2)l/2 =

X>=Q.

ti

so that i?= [aV/{Vi - 02)i/2]y,

f = [aV/(V2 -

v2Y'2]z.

Substituting for x' its value [i.e., x - vt], we obtain T = 4>(v)f3{t - vx/V1), i=4>{v)p{x-vt), 7, = (o)y, f = « ( o ) z . (7) where 0=[l-(«;/K)2]-'/2 and <> / is as yet an unknown function of v. If we make no assumption about the initial position of the moving system and about the zero point of T, one has to add a constant to the right-hand side of each of these equations. We must now prove that every light ray propagates with the velocity V, as measured in the system k, if this is the case, as.we have assumed, in the stationary system; for we have not yet proved that the principle of the constancy of the velocity of light is consistent with the principle of relativity. At the time I = r = 0 let a spherical wave be emitted from the common origin of the two coordinate systems, propagating in the system K with the speed V. Thus if {x.y.z) is a point reached by this wave at the time t, then x2+y2

+ z2=

V2f-.

We transform this equation with the aid of our transformation equations and obtain, after a simple calculation, f- + rp + r2 = KV.

^Sf^

The wave under consideration is therefore also as viewed in the moving system, a spherical wave with velocity of propagation V. This shows that our two fundamental principles are mutually compatible. In the transformation equations that have been developed there enters still an unknown function <\> o( u. which wc will now determine.

Chap. 2. Special Relativity ...

173

To this end we introduce yet a third coordinate system K', which is in such a state of parallel-translational motion relative to the system k. parallel to the H-axis, that its origin moves along the H-axis with the velocity —u.2J Let all the three origins of coordinates coincide at the time t = 0 and let the time t' of the system K' vanish when / = x = y = ; = 0. We denote by x'.y'.z' the coordinates measured in the system K\ and obtain by a twofold application of our transformation equations /' = 0 ( - r ) t f ( - r ) ( r + £t-/K2) = (-L')tf(-<')(sC + VT) = 0(1')0(-O).t. / = (/>(-'')>? = (l>(v)(-L-)y. :' = (/>(-<•')£•= (t0'/>(-(-')r.

Since the relations between x'.y',:'. and x.y.: do not contain the time /, the systems K and K' are mutually at rest, and it is clear that the transformation from K to K' must be the identity transformation. Therefore, 4>(u)(~o) = I.

We now investigate the significance of (<;). We direct our attention to that part of the W-axis:4 of the system k which lies between £ = 0,»; = 0, f = 0 and £ = 0,77 = /, f = 0. This part of the //-axis is a rod that moves perpendicularly to its axis with the velocity u relative to the system K, and whose ends possess in K the coordinates25 x\ = ut,

_y, = l/
j,.= 0

and xi = vt,

yi = 0 ,

zi = 0.

Hence, the length of the rod as measured in K is !/(v), and the meaning of the function (j>(v) is established. But from symmetry considerations it is clear that, as measured in the stationary system, the length of a given rod moving perpendicularly to its axis cannot depend on the direction and sense of the motion but only on its speed. The length of the moving rod, as measured in the stationary system, does not change therefore upon interchanging v and — v. From this follows that or 4>{v) = (-«).

(8)

From this relation and the one obtained previously it follows that (j>(u) = 1, so that our transformation equations reduce to r = 0[t -

S = 0ix-ot).

{v/V-)x], V

= y,

r=--.

(9)

174

Lorentz and Poincare Invariance where 0=[\-(u/V)i\-W.

(10)

4. Physical meaning of the derived equations as concerns mooing rigid bodies and moving clocks We consider a rigid sphere^ of radius R, which is at rest relative to the moving system k. and whose center lies at the origin of coordinates of k. The equation of the surface of this sphere, which moves with velocity o relative to the system K, is £2 + „2 + f2 =

R2

When expressed in x.y.z, the equation of this surface is at time / = 0 (1 - v2/V2)~Kx2

'• '4 / . • - ? "

+ y2 + z2 = R2.

A rigid body, which is of spherical shape when surveyed in a state of rest, has therefore when in a state of motion—as viewed from the stationary system—the shape of an ellipsoid of revolution with the axes

A

R[I -

><"~D

*S"-

{p/V)*yn,R.R.

Thus, while the Y- and Z-dimensions of the sphere (and hence also of every rigid body of arbitrary shape) do not appear to be changed by the motion, the A"-dimension appears to be shortened in the ratio 1:(1 — {u/V)2]]/2, and hence the more so the greater the value of v. For o= Kail moving objects—viewed from the "stationary" system— shrink into plane-like structures. For superlight speeds our considerations become senseless; we shall find, moreover, in the following discussion that the velocity of light plays in our theory the role of an infinitely large velocity. It is clear that the same results hold for bodies at rest in the "stationary" system, as viewed from a system in a state of uniform motion. We imagine further one of the clocks which are capable of furnishing the time / when at rest in the stationary system and the time T when at rest in the moving system, placed at the origin of coordinates of k, and set so as to mark the time r. How fast does this clock run when viewed from the stationary system? We obviously have the following equations between the quantities x, t, and T, which refer to the position of this clock: r = [\ - {»/V)2]-V\t

-

vx/V2)

and x = at. Hence,

r = r[i - {u/vy-yr- = t - \\ - [i - (o/^]'Z2!*,

Chap. 2. Special Relativity ...

175

from which it follows that the indication of the clock (viewed in the stationary system) is retarded by 1 — [1 — {o/V)1] '/2 sec per second, or by u2/2 V2 sec per second—to fourth and higher order. From this arises the following peculiar consequence. If at the points A and B of K there are stationary clocks that are synchronous as viewed in the stationary system, and if the clock at A is moved with the velocity o along the line joining A and B, then upon the arrival of this clock at B the two clocks no longer synchronize; instead, the clock that has moved from A to B lags behind the one which has stayed at B by tvl/2V2 sec (to fourth and higher order), when t is the time of transit of the clock from A to B. It is seen immediately that this result also remains true when the clock moves from A to B along an arbitrary polygonal line,26 and so also when the points A and B coincide. If one assumes that the result proved for a polygonal line26 holds likewise for a continuously curved line, one obtains the proposition: If there are two synchronous clocks at A, and one of these is moved with uniform speed along a closed curve until it returns back-to A after an interval of t sec, then upon its arrival the latter clock lags by tv2/2V2 sec behind the clock that has not been moved. From this one concludes that a balance clock27 at the Earth's equator must go more slowly by a very small amount than a precisely similar clock located at one of the Earth's poles, and otherwise subject to identical conditions. 5. Addition theorem for celocities In the system k moving with the velocity u along the .Y-axis of the system K. let a point move according to the equations £ = WCT, n = w,r, f = 0, where Wj and wn denote constants. It is desired to find the motion of the point relative to the system K. If one substitutes in the equations of motion of the point the quantities x,y,:,t given in the transformation equations developed in Sec. 3, one obtains 1 -{v/V)lV/2\ \1 + CwJV -JV J) ' • ' V 1 + vwcJV2 w„t. z = 0. 22

Thus the law of the parallelogram of forces holds in our theory only to first approximation. We set

and [correcting an obvious misprint]

176

Lorentz and Poincare Invariance a = arctan(vv,,/Wj); « is then to be looked upon as the angle between the velocities v and w. After a simple calculation it is found that 28 [u2 + w2 + 2cw cosa — (vw sina/ V)2]*/-2 (ID 1 + uw cosa/ V2. It is noteworthy that o and w enter in the expression for the resultant velocity in a symmetrical way. If w has also the direction of the ,V-axis (H-axis), we obtain U=

U =-

v+ w + vw/V2

lt follows from this equation that from the superposition of two velocities that are smaller than V there always results a velocity smaller than V. For if we set v = V — K, W = V — X, where K and X are positive and smaller than V, then V(2V - K - X) U =< V. 2VX + K\/V It follows further that the velocity of light V cannot be changed when it is combined with a "subluminal velocity." 29 In this case we obtain £/ = •

V+ w = V. + w/V

(12)

In the case that v and w have the same direction, we could have also obtained the formula for U by the composition of two transformations according to Sec. 3. If in addition to the systems K and k appearing in Sec. 3 we introduce a third coordinate system k' moving parallel to k, whose origin moves with the velocity w along the H-axis, we obtain equations between the quantities x.y.z.t and the corresponding quantities of k', which differ from those obtained in Sec. 3 only by the replacement o f ' V by the quantity v+w 1 + vw/V2' we sec from this that—as must be the case—such parallel transformations form a group. We have now derived the required laws of kinematics which correspond to our two principles, and proceed to show their application to electrodynamics.

C. REMARKS CONCERNING THE SECOND PART OF EINSTEIN'S PAPER The second part of Einstein's paper deals with'applications of relativistic kinematics to optics (Sees. 7 and 8) and to electron dynamics (Sec. 10), after the establishment of the Lorentz invariance of the Maxwell equations (Sec. 6) and of the Maxwell-Lorentz equations (Sec. 9), 3 0 respectively.

Chap. 2. Special Relativity ...

177

In Sec. 6, in addition to deriving the relativistic covariance of Maxwell's equations for the electromagnetic field in a vacuum, Einstein points out the consequent relative nature of the concepts of electric and magnetic "force." Section 7, entitled "Theory of Doppler's Principle and of Aberration," treats not only these subjects, but also the Lorentz transformation of the "amplitude of the electric or magnetic force" associated with a monochromatic electromagnetic wave. Denoting this amplitude "as measured respectively in the stationary and moving system" by A and A', he finds (first formula on p. 57 of POR 31 ) 32 : A'I A = j3[l - (u/T)cos<£]. This formula takes on added interest by the result obtained in Sec. 8 (see pages 57-58 of POR) for the ratio E'/E of the Lorentz-transformed and the original values of the energy of the associated light pulse, namely,33 £'/£ = j3[l - {v/V) cos]. In Einstein's words, referring to this formula: "It is noteworthy that the energy and the frequency of a light complex [i.e., a light pulse] vary with the state of motion of the observer according to the same law." But, surprisingly, Einstein does not refer to the associated quantum law for light, enunciated by him earlier in 1905.34 The last section of Einstein's paper contains the first attempt to formulate relativistic mechanics, and as such it is, of course, of great historical interest. This attempt was, however, not wholly successful, owing to a partly infelicitous appraoch, which led to the following equations of motion for a charged particle, in an external electromagnetic field (in POR, p. 62, the letter m is substituted for Einstein's M):

dr-

M/33 dt-

dc- upV

' M

$\

v1 )•

(A)

VI

Einstein was fully aware of the questionable elements in his approach35; moreover, luckily, by a restricted use of the first, and correct equation of the set (A), he obtained the correct relativistic expression for the kinetic energy of a particle (last equation on p. 63 of POR), which enabled him to obtain in 1905 his first enunciation of the equivalence of mass and energy.36 The paper concludes with a remarkable acknowledgment to Einstein's "friend and colleague, M. Besso." 37 D. DISCUSSION When Einstein's first paper on relativity is read with due attention to its content and its style, and proper account is

178 Lorentz and Poincare Invariance

Gr\l>**

taken of what is known about its author in the decade ending in 1905, and of the state of basic physics during that period, there arise a number of questions that are of considerable interest to the historian and to the philosopher of science. Most of these questions have already been the subject of penetrating investigations,1 and further discussion will be presented in the last reference in Ref. 1. The present discussion will be limited largely to examining the idea expressed in Ref. 7. That the creation of the special theory of relativity in Einstein's sense was in some respects even a more remarkable achievement than the creation of the general theory may be questioned by many—Einstein's worldwide fame was certainly associated with the latter theory. What is extraordinarily unique about Einstein's formulation of special relativity is the constructive introduction of radically new ideas concerning time and space. After those ideas have been sufficiently assimilated, it is difficult to appreciate fully the intellectual feat involved in that step. It was unquestionably one of rare physico-philosophic intuition and intellectual daring. That Einstein himself must have been aware (if perhaps only dimly) of the magnitude of the contribution contained in the first part of his paper is shown by the unusual pains taken there in explaining the relativity of simultaneity and the ensuing novel conceptions of time and space, as contrasted with the rapid development of the second part, dealing with applications to the electrodynamics of moving bodies—the avowed object of the paper. At the same time, the exposition in Part 1 gives also the impression of being in large measure a spontaneous recording of the flow of exciting ideas. Along with great lucidity, there exist, therefore, not surprisingly a few minor scarcely detectable logical flaws. The most apparent and perhaps the strangest lapse—and one which, unfortunately, has been imitated in many textbooks—js contained in the phrasing of the principle of relativity in Law 1, Sec. 2. The laws of physics are stated there to be independent of whether they are referred to "one or the other of two coordinate systems that are in uniform translational relative motion." Without specifying explicitly that one of the coordinate systems is inertial, the statement lacks, of course, in precision. Less apparent and more innocuous are the logical questions which are left open, concerning the relationship between the definition of simultaneity in terms of Eq. (1) and postulate 2 of Sec. 2. These questions touch the discussion in Sec. 3 leading to the crucial transformation equations (9) and (10). That discussion, incidentally, clearly illustrates the spontaneous-outpouring character of the exposition, and at the same time represents convincing evidence (if any were needed) that in 1905 Einstein was unaware that the result

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179

(9) was discovered by Lorentz a year earlier. Moreover, Einstein himself never reverted to the arguments emploved in Sec. 3.31* As an indication of how these arguments can be sharpened—if we pause to examine them—with a simultaneous clarification of the above-mentioned logical questions, let us consider Einstein's discussion through Eqs. (1) and (3)-(5), culminating in the result (6) or, equivalently, the first of Eqs. (7). At this point Einstein uses an alternative expression of his second postulate, namely: 2'. The speed of light in vacuum has the same value in every inertial frame. What can be shown is that both postulate 2' and the result (9) can now be obtained very simply by using an essentially self-evident relationship, namely, that the speed of relative motion of two inertial frames is the same as measured in each of these frames (with the use, of course, of identical standards of length and duration).39 Applying this relationship, we have (by understandable reasoning involving the spirit of Einstein's first postulate) t = H-») 0'2{T + (v/V'i)z], where V is the speed of light as measured in the frame k, and /?' is the corresponding value of (10). Eliminating T from this equation and the equation in (7), and identifying (—u) with 4>{v) = 4> by Einstein's isotropy-of-space argument [discussion leading to Eq. (8)], we find H = a[x - (V2/v) (\ - 1/0 W 2 ) ' ] . a = (V'2/V2)(32.

<13)

Since £ must be proportional to x-vt, it follows that 02/T-=l. This equation will violate the principle of relativity unless ff' = /3, i.e., V = V, which proves postulate 2'. At the same time, a in Eq. (13) assumes the value j3, and Eq. (13) reduces to the second of Eqs. (9). ACKNOWLEDGMENTS The author wishes to express his indebtedness for helpful suggestions to the referees, and to Professor Lothar Schafer of the Chemistry Department of the University of Arkansas. The author also thanks Dr. Otto Nathan, trustee of the estate of Albert Einstein, for his kind permission to publish in this Journal the translation of the first part of Einstein's paper, Ref. 2. 'The imprecision which inheres in the idea of the simultaneity of two events at (approximately) the same place and which must be likewise bridged

180

Lorentz and Poincare Invaxiance over by an abstraction will not be discussed here. "Time" signifies here "'time of the stationary system," as well as "the position of the hands of the moving clock situated at the place under consideration." '•Thai is. a body which is of spherical shape when examined at rest. 'The first such study was published in this Journal: G. Holton. Am. J. Phys. 28, 627 (I960). This paper is reprinted (with omission of two paragraphs of the introductory remarks) in G. I lolton. Thematic Origins of Scientific Thought: Kepler to Einstein (Harvard U. P., Cambridge, MA, 1973). Further papers on the subject by the author are also included in this book. Up-to-date references on the history of special relativity will be included in a forthcoming study. "Lorentz. Poincare. Einstein, and the Theory of Relativity." : A. Einstein, Ann. Phys. (Leipzig) 17, 891 (1905). 'A. Einstein el al.. The Principle of Reiatirily (Dover. New York; reprinting of a translation by VV. Pcrrett and C i. B. Jeffery, first published in 1923), pp. 37-65. This will hereafter be referred to as POR. 4 This is attested to by the observations contained in a number of references of the present paper. 5 C. W. Kilmister, Special Theory of Reiatirily (Pcrgamon, New York, 1970), pp. 187-218. This will hereafter be referred toasSTOR. "These consist of a number of utter mistranslations of isolated words, and of a misleading amplification of a passage, which has been corrected by C. Scribner, Jr., Am. J. Phys. 31. 398 (1963). Another mistranslation, reproduced in STOR, is pointed out in Ref. 23. Moreover, in addition to not reproducing the original notation, as is also true of POR, and reproducing a misprint only found in POR, this translation includes, apparently inadvertently, two footnotes not found in the original paper, Ref. 2, without reproducing all the additional footnotes contained in POR. 'It can be argued that in certain respects this achievement surpasses even Einstein's creation of the theory of general relativity. "Only in one instance, namely in the line above Eq. (7), is a minor explanatory item (enclosed in brackets) added to facilitate the reading of the text. A few changes in mathematical notation have been introduced for typographical convenience. This is the literal translation of Einstein's "Lichtmedium," for which "luminiferous ether" is the more common designation, which he. in fact, employs later in the paragraph. "That is to say, no observable physical properties. ' 'This statement reflects Einstein's ignorance in 1905 of Lorentz's relevant work of 1904. 12 The original "die Aussagen eincr jeden Theorie" is translated in POR as "the assertions of any such theory." but the addition of the word "such" to the original phrase is unwarranted. Every theory, i.e., every fundamental theory, was then apparently to Einstein, as to most of his contemporaries, reducible essentially to a consideration of electromagnetic processes (with the possible exception of.gravitation). This might constitute also an explanation for Einstein's referring earlier explicitly only to "electrodynamic and optical laws" in his first pronouncement of the "principle of relativity." '•The original in footnote a reads "Die Ungenauigkeit, welche in dem Begriffe dcr Gleichzeitigkeit zweier Ereignisse an (annahernd) demselben Orte steckt und gleichfalls durch eine Abstraction iiberbruckt wcrden muss, soil hicr nicht erortert werden" (italics supplied). This is not rendered quite accurately by the following translation in POR: "We shall not here discuss the inexactitude which lurks in the concept b

f\'" *

o

•

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of the simultaneity of two events at approximately the same place, which can only be removed by an abstraction." It is true that the word "gleichfalls" (likewise> appears to be redundant in Einstein's statement, as it stands, but it may well be a slip of the pen, reflecting the author's awareness of the many other tacit assumptions that necessarily enter in the framing of a fundamental theory of physics, and that are basically in the nature of abstractions. l4 The rendition in POR of the original "gewisser (gedachter) physikalischer Erfahrungen" as "of certain imaginary physical experiments" is hardly accurate. On the other hand, the literal translation of "Erfahrung," namely, "experience," is also hardly appropriate in the present context. 15 The original phrase is, "Wir setzen noch der Erfahrung gemass fest." This could also be translated—as is done in POR—"In agreement with experience we further assume," but the logical status of this statement connected with relation (2) is not quite the same as that of the related assumption announced in the introduction and reiterated in Sec. 2. See the pertinent discussion in Sec. D. ,6 That is, of the mooing rod. 'That is to say, in the physical space associated with the inertial coordinate system referred to in the first paragraph of Sec. 1. and designated there as the "stationary system." l8 There appears to be some logical conflict in the present use of the word "space" and the previous one, which is commented upon in Ref. 17. Apparently, this is an instance of Einstein's terminology being still steeped in the older mode of thinking in kinematics, while he was in the process of laying the foundations for a revolutionary change in that very mode of thinking. "This is a laconic statement in need of amplification. 20 The more direct translation in POR of the word "Inbegriff" as "summary" does not appear to adequately convey the idea intended in this sentence. Inbergriff'is, in fact, not an easily translatable word. 21 Actually, it is only the special case associated with Eq. (1) that is being used here (cf. Sec. D). 22 This observation is superfluous when account is taken of the linearity of the transformation equations, which property is, in fact, utilized presently in arriving at Eq. (6). ^That is to say, moving with this velocity relative to the coordinate frame k. This is explicit in the original phraseology, and is indeed what is required to arrive at the next set of equations. But this part of the sentence ' ("dass stch desscn Koordinatenursprung mit dcr Geschwindigkcit — v auf der H-Achse bewege") is mistranslated in POR as "such that the origin of coordinates of system k moves with velocity — v on the axis of X." This mistranslation is reproduced in STOR, but with the symbol X replaced by the original symbol S. 24 The letter H (used also earlier) stands for the capital of rj. 23 The order of the following two coordinate sets should be reversed for agreement with that employed earlier. 26 This has not been proved. The clock experiences an acceleration (in fact, an infinite one) at the corners of the polygon, and therefore, strictly, an additional assumption is required here to the effect that the rate of a clock is not affected by acceleration. !7 The following footnote is included in POR: "Not a pendulum clock, which is physically a system to which the Earth belongs. This had to be excluded." This footnote is not contained in the original paper, but is

182

Lorentz and Poincare Invariance reproduced in STOR. In POR the last term under the square root in Eq. (11) has an obvious misprint, not contained in the original paper, nor in the German edition of POR, but reproduced in STOR. 29 Since the relation (12) is actually valid for any value of w, this statement must be taken to imply Einstein's taking here for granted the "upperlimit" property of the speed of light. J0 Einstein does not yet use in this paper the designation "Maxwell-Lorentz equations." •"Einstein's symbol Kis replaced in POR (see Ref. 3) bye. 32 Einstein outlines a proof of this important relationship in his subsequent comprehensive paper on relativity, Jahrb. Radioakt. Elektron. 4, 409-462 (1907), p. 431. This essay will form the subject of sequel papers to the present one. "In the derivation of this result there occurs the curious coincidence that the same (somewhat unusual) method.for finding the ratio of the volumes attributed by observers in two coordinate frames having uniform relative motion, to a sphere moving with constant velocity, is also found in Poincare's Rendiconti paper on relativity. See, e.g., H. M. Schwartz, Am.J.Phys.39, 1289 (1971), p. 1290: or STOR. p. 151. An outline of the proof by this method of the first equation on p. 58 of POR is given in H. M. Schwartz, Introduction to Special Relativity (McGraw-Hill, New York, 1968), p. 415 (answer to problem 2-5). A proof that is based more obviously on relativistic kinematics is also indicated there. M A. Einstein, Ann. Phys. (Leipzig) 17, 132 (1905). "Referring to the "longitudinal" and "transverse" masses corresponding to Eqs. (A), he remarks (Ref. 2, p. 919): "Naturally, we shall obtain other values for the masses with other definitions of force and of acceleration: from this it is seen that one must proceed with utmost caution when comparing different theories of the motion of the electron." A footnote is contained in POR (p. 63) pointing out a more suitable definition of force, later introduced by Planck and generally accepted. This footnote is, of course, not in the original paper, but is included in STOR. " A . Einstein, Ann. Phys. (Leipzig) 18, 639 (1905); POR, pp. 69-71. "This acknowledgment is remarkable, in the first place, because until fairly recently the identity of this important friend of Einstein's was not generally known. Now there is a book about him and his correspondence with Einstein, namely, P. Spezialli, Albert Einslein-Michele Besso: Correspondence 1905-1955 (Hermann, Paris, 1972). This book contains translations of the German letters into French, notes, and an illuminating biographical sketch of Besso by Spezialli. The acknowledgment is also intriguing in the questions it raises as to the nature of the assistance which Einstein received from Besso. The translation in POR, reproduced in STOR, namely, "and I am indebted to him for several valuable suggestions," is not an altogether precise rendition of the concluding phrase "und dass ich demselben manche vcrtvolle Anregung verdanke," which strictly means "and that I am obliged to him for some valuable stimulation." 3, Sec, e.g., his derivations of the Lorentz transformations in the reference in Ref. 32, and his popular exposition. Relativity (Crown, New York, 1961), Appendix I. •"This result may be taken to follow from a straightforward application of the principle of sufficient reason. Einstein uses it without explanation in the second of Refs. 38 (first paragraph on p. 117). 2s

Chap. 2. Special Relativity

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On the Origins of the Special Theory of Relativity* GERALD HOLTON

Department of Physics, Harvard University (Received May 9, 1960) Einstein's early work on relativity theory is found to be related to his other work at that time (e.g., in subject matter and style). In addition to this element of internal continuity one finds also—as a key to a new evaluation of the significance of Einstein's contribution—an external continuity with the classic, Newtonian tradition governing restrictions on permissible hypotheses. On the other hand, Einstein's work is shown to have been, in important respects, more independent of other contemporary work in this field than has recently been proposed. These continuities and discontinuities are set forth to make the point that philosophical studies of scientific work should proceed on historically valid ground. Some guiding principles are indicated for dealing with conflicting source materials for such studies.

W

H E N I received the persuasive invitation to speak today on a problem of theory construction and of the logic of discovery, I noted particularly the request to bring out the historical-sociological aspects. This directive was a pleasant surprise, for I recalled t h a t Hans Reichenbach had flatly declared himself for the opposite view when he said " T h e philosopher of science is not much interested in the thought processes which lead to scientific discoveries- • •, that is, he is not interested in the context of discovery, but in the context of justification". 1 If, therefore, I shall make some remarks on the origins of Einstein's special theory of relativity, I will be disobeying the Reichenbachian dictum. However, I draw further strength for this resolution from Einstein, who himself declared for the value of the historical treatment of the rise of key theories in science. In fact, it is appropriate to say at the very outset to an audience consisting primarily of philosophers of science t h a t sound historical investigations have lately perhaps been overlooked as important bases of sound philosophical discussions. Some examples come to mini immediately. The crux of the Copernican revr ition was initially not, as is maintained in son ; philosophical works, a pragmatic search for the smallest num* Presented at the Symposium -Theory Construction in Logical and Historical Perspective on December 27, 1959, organized by Section L (History and Philosophy of Science) of the AAAS, the American Philosophical Association, and the Philosophy of Science Association. Based on work-inprogress, supported in part by a grant of the National Science Foundation. ' P . A. Schilpp, editor, Albert Einstein: PhilosopherScientist (Library of Living Philosophers, 1949), p. 292.

183

184

Lorentz and Poincaxe Invariance

^

•

•

\

\

I

•

>!b

m HW. ? . < i , ,V>

•w »

1

ber of components with which to build a world system, nor was it the establishment of the possibility of relativism in the choice of coordinate systems. Rather, as historians of science have shown, it was a return to an earlier, even an Aristotelian austerity concerning the type of motion judged to be suitable for the construction of the world system, mixed with a commitment to a neo-Platonic epistemology t h a t looked for the warrant of reality in a new direction. T h e importance of Kepler is not t h a t he was a mystic, an obsessed searcher for empirical rules, or a master of the intuitive, "personal" way to scientific knowledge; on the contrary, it can be shown that he was the first of the modern mathematical physicists, the first to look with some success for one dynamical explanation of all celestial and terrestial motions. Galileo, we have had to relearn only recently, was not the patron saint of laboratory experimentation, as philosophers of science have at times maintained. Concerning the abuse to which Newton's work has had to lend itself, the less said the better. Einstein's work has not been immune from this fate. I am suggesting t h a t in this case, as in the others, we build our philosophical analyses of science on real ground instead of dubious models, that we examine what physics was like in Einstein's time, what he did and said, how he came to do and say these things, and how he changed his mind—not once, but often. I urge this not as an easy program—for it is not that—and even not just because it is in principle better to do justice to the work of a man on his own terms rather than to use his work for a purpose which may have been inherently foreign to him. 1 urge this, rather, because I believe that a future source of strength of scholarship in the philosophy of science lies in philosophical analysis of historical cases. I speak of Einstein's work because his case is both typical and special. T h e rise of relativity theory shares many features with the rise of other important scientific theories in our time, and in addition it is of course very much more: To find another work that illuminates as richly the relationship between physics, mathematics, and epistemology, or between experiment and theory,

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or one with the same range of scientific, philosophical and general intellectual implications, one would have to go back to Newton's Principia. The theory of relativity was a key development, both in physical science itself and also in modern philosophy of science. The reason for its dual significances is that Einstein's work provided not only a new principle of physics, but, as A. N. Whitehead said, "a principle, a procedure, and an explanation." Accordingly, the commentaries on the historical origins of the theory of relativity have tended to fall into two classes, each having distinguished proponents: the one views it as a mutant, a sharp break with respect to the work of the immediate predecessors of Einstein ; the other regards it as an elaboration of then current work, e.g., by Lorentz and Poincare. To my mind, the Einsteinian innovation is understood best by superposition of both views, by seeing the discontinuity of methodological orientation within an historically continuous scientific development.2 Before we come to discuss this, and if we take seriously my point of view, we should first be ready to investigate a number of real problems of the historical or even "historicalsociological" kind: What are the sources for a study of the origins of the special theory of relativity (RT) and what is their probable reliability? What was the state of science around 1905, what were the contributions which prepared the field for the RT, and what did Einstein know about them ? What were the steps by which Einstein reached the conclusions he published in 1905 ? To what extent was this work a member of a continuous chain having as its immediate predecessors Lorentz and Poincare? What was the role of experiment in the genesis of the RT, and what the role of the existence of contradictory hypotheses? What part played epistemological analysis in Einstein's thought? What was the early reception of the RT among scientists? I n particular, what was Einstein's relation with Mach, Lorentz, and Planck? What may we say about the style of Einstein's work and his personal orientations? What, if anything, in the 2 G. Holton, IX Congreso International de Historia de las Ciencias, Guiones de las Communtcacioncs (BarceionaMadrid, 1959), Vol. II, p. 41.

186

Lorentz and Poincare

G*,

>t

••O.... -

Invaiiance

origins and content of the RT is typical of other theories with great impact on science ? And even, what methodological principles for the study of the history of science emerge from this study? We would find that the existing literature is not always of help in studying such questions. The literature on the RT is of course vast. LeCat3 listed over 3400 scientific papers in the field up to 1922, with an approximately exponential growth giving a sevenfold increase in seven years. Biographically or philosophically oriented analyses are also fairly numerous (for example, by Schlick, Reichenbach, Frank, Meyerson, Cassirer, Whitehead, Wenzel, Grunbaum, Polanyi, Margenau, Lenzen, Bridgman, and Northrop.) It may be remarked there has so far been no full-scale historical study (although one is now in progress). A number of valuable essays exist in this direction (for example, by Born, Dugas, Kuznetsov, von Laue, Pauli, Straneo, and Whittaker); these are generally concerned with the chronological development of physics, and typically constitute a portion of a longer work having a purpose different from that of a primarily historicalphilosophical study. For the latter, the best source is at present indeed Einstein's own set of papers. CONTINUITY IN EINSTEIN'S "WORK To these papers we must turn to discover, for example, the elements of continuity linking Einstein's first publication on the RT with his other work at the time and with the older tradition itself. After the paper of 1905/ Einstein returned to the exposition of the RT several times, and each restatement is of interest. For instance, in his book Uber die spezielle und die

allgemeine

Relativitatstheorie* he emphasized in his introduction that "the author has made the greatest effort to present the main ideas- • -on the whole in the sequence and in such context as they in fact arose." It is not surprising that the sequence given there is not in accord with the sequence of » Maurice LeCat, Bibliographic de la Rclativili (Bruxelles, 1924). 4 A. Einstein, Ann. Physik 17, 891 (1905). ! A. Einstein, (Braunschweig, 1916).

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steps in the 1905 paper itself, but the historian of science finds an interesting problem in the fact that neither of these is in accord with other autobiographical or biographical accounts. When one studies the relativity papers in the larger contextual setting of Einstein's other scientific papers, particularly those on the quantum theory of light and on Brownian motion which also were written and published in 1905, one notices two crucial points. While the three epochal papers of 1905—sent to the Annalen der Physik at intervals of less than eight weeks— seem to be in entirely different fields, closer study shows that they arose in fact from the same general problem, namely, the fluctuations in the pressure of radiation. In 1905, as Einstein later wrote to von Laue,6 he had already known that Maxwell's theory leads to the wrong prediction of the motion of a delicately suspended mirror "in a Planckian radiation cavity." This connects on the one hand with the consideration of Brownian motion as well as to the quantum structure of radiation, and on the other hand with Einstein's more general reconsideration of "the electromagnetic foundations of physics" itself.7 One also finds that the style of the three papers is essentially the same, and reveals what is typical of Einstein's work at that time. Each begins with the statement of formal asymmetries or other incongruities of a predominantly esthetic nature (rather than, for example, a puzzle posed by unexplained experimental facts), then proposes a principle—preferably one of the generality of, say, the second law of thermodynamics, to cite Einstein's repeated analog}'—which removes the asymmetries as one of the deduced consequences, and at the end produces one or more experimentally verifiable predictions. Specifically, Einstein's first paper on the quantum theory of light opens in a typical manner: "There exists a radical formal difference between the theoretical representations which physicists have constructed for themselves concerning gases and other ponderable bodies on the one hand, and 'Letter of January 17, 1952 (unpublished). See also

Max Born in Finfzig Jahrc Relativitdtsthearie, edited by A.7Mercier and M. Kervaire (Bern, 19SS), pp. 248-249. See footnote reference 1, p. 47.

Poincare Invariance Maxwell's theory of electromagnetic processes in so-called empty space on the other hand." 8 The significant starting point is a formalistic difference between theoretical representations in two fields of physics which, to most physicists, were so widely separated that no such comparison would have invited itself and therefore no such discrepancy would be noted. The discrepancy Einstein points out is between the discontinuous or discrete character of particles and of their energy on one hand, and the continuous nature of functions referring to electromagnetic events and of the energy per unit area in an expanding wave front on the other hand. The discussion of the photoelectric effect, for which this paper is mostly remembered, occurs toward the end, in a little over two pages out of the total sixteen. The prescription for obtaining an experimental verification of his point of view is given in a single, typically succinct Einsteinian sentence (straightline relation with constant slope between frequency of light and stopping potential for all electrode materials). Inhissecond paper published in 1905,9 Einstein points out in the second paragraph that the range of application of classical thermodynamics may be discontinuous even in volumes large enough to be microscopically observable. He ends with the equation giving Avogadro's number in terms of observables in the study of particle motion, and with the one-sentence exhortation: "May some investigator soon succeed in deciding the question which has been raised here, and which is important for the theory of heat!" Significantly, Einstein reported the following year10 that only after the publication of this paper was his attention drawn to the experimental identification, as long ago as 1888, of Brownian motion with the effect whose existence he had deduced as a necessity from the kinetic-molecular theory. In his autobiographical notes he repeats that he did the work of 1905 "without knowing that observations concerning Brownian motion were already 8

A. Einstein, Ann. Physik 17, 132 (1905). • A. Einstein, Ann. Physik 17, 549 (1905). 10 A. Einstein, Ann. Physik 19, 371 (1906).

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long familiar". 11

The third paper of 190512 is, of course, Einstein's first paper on the RT. He begins again by drawing attention to a formal asymmetry, i.e., in the description of currents generated during relative motion between magnets and conductors. The paper does not invoke explicitly any of the several well-known experimental difficulties— and the Michelson and Michelson-Morley experiments are not even mentioned when the opportunity arises to show in what manner the RT accounts for them. At the end, Einstein briefly mentions here, too, specific predictions of possible experiments (giving the equation "according to which the electron must move in conformity with the theory presented here").13 RETURN TO A CLASSIC RESTRICTION ON HYPOTHESES

The recognition of these common elements in the three papers prepares us for the essential realization that the fundamental postulates appearing in each of the three papers are heuristic. The heuristic nature of the postulate of relativity was from the beginning apparent to Einstein (as he asserted in 1907 and later) because of the restriction of the RT to translational motions and to gravitation-free space." The study of the three papers together reveals also the extent to which Einstein's RT represents an attempt to restrict hypotheses to the most general kind and the smallest number possible—a goal on which Einstein often insisted.15 In the 19Q5 paper on RT, he makes, in addition to the "See footnote reference 1, p. 47. See also L. Infeld, Albert Einstein (New York, 1950), p. 97-98. 12 A. Einstein, Ann. Physik 17, 891 (1905). " See footnote reference 12, p. 921. " On a few occasions, although not in the original paper, Einstein made this point [e.g., Ann. Physik 23, 206 (1907)3: "The relativity principle [is to be regarded]- • • solely as a heuristic principle, which, considered by itself, contains only assertions about rigid bodies, clocks, and light signals." " Cf. A. Einstein, "The Problem of Space, Ether, and the Field in Physics," Ideas and Opinions by Albert Einstein, translated and revised by Sonja Bargmann (New York, 1954), p. 282: "The theory of relativity is a fine example of the fundamental character of the modern development of theoretical science. The initial hypotheses become steadily more abstract and remote from experience. On the other hand, it gets nearer to the grand aim of all science, which is to cover the greatest possible number of empirical facts by logical deductions from the smallest possible number of hypotheses or axioms."

In variance two "conjectures" raised to "postulates" (i.e., of relativity and of the constancy of light velocity) only four other hypotheses: one of the isotropy and homogeneity of space, the others concerning three logical properties of the definition of synchronization of watches. In contrast, H. A. Lorentz's great paper which appeared a year before Einstein's publication 18 and typified the best work in physics of its time—a paper which Lorentz declared to be based on "fundamental assumptions" rather than on "special hypotheses"—contained in fact eleven ad hoc hypotheses : restriction to small ratios of velocities v to light velocity c; postulation a priori of the transformation equations (rather than their derivation from other postulates); assumption of a stationary ether; assumption that the stationary electron is round; t h a t its charge is uniformly distributed ; t h a t all mass is electromagnetic; t h a t the moving electron changes one of its dimensions precisely in the ratio of (1—n 2 /c 2 )* to 1; that forces between uncharged particles and between a charged and uncharged particle have the same transformation properties as electrostatic forces in the electrostatic system; t h a t all charges in atoms are in a certain number of separate "electrons"; that each of these is acted on only by others in the same a t o m ; and t h a t atoms in motion as a whole deform as electrons themselves do. It is for these reasons t h a t Einstein later maintained t h a t the RT grew out of the Maxwell-Lorentz theory of electrodynamics "as an amazingly simple summary and generalization of hypotheses which previously have been independent of one another- • •," 17 If one has studied the development of scientific theories, one notes here a familiar t h e m e : the socalled scientific "revolution" turns out to be at bottom an effort to return to a classical purity. This is not only a key to a new evaluation of Einstein's contribution, but indicates a fairly general characteristic of great scientific "revolutions." In" H. A. Lorentz, Proc. Acad. Sci. Amsterdam 6, 809 (1904). This paper, originally presented as part of the proceedings of the meeting of April 23, 1904, was first published in June, 1904 in the Dutch language edition of the Proceedings [12, 986-1009 (1904)]. 17 See footnote reference 5, p. 28. See also A. Einstein. Scientia 15, 338 (1914).

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deed, while it is usually stressed that Einstein challenged Newtonian physics in fundamental ways, the equally correct but neglected point is the number of methodological correspondences with earlier classics, for example, with the Principia. Here a listing of some main parallels between the two works must suffice: the early postulation of general principles which in themselves do not spring directly from experience; the limitation to a few basic hypotheses18; the exceptional attention to epistemological rules in the body of a scientific work; the philosophical eclecticism of the author; his ability to dispense with mechanistic models in a science which in each case was dominated at the time by such models19; the small number of specific experimental predictions ; and the fact that the most gripping effect of the work is its exhibition of a new point of view. The central problem, moreover, is the same in both works: the nature of space and time, and what follows from it for physics. Here, the basic attitudes have in both cases more in common than appears at first reading. That Newton's absolute space and absolute time were not meaningful concepts in the sense of laboratory operations, was, of course, not the original discovery of Mach; rather, it was freely acknowledged by Newton himself. But Einstein was also quite explicit that in replacing absolute Newtonian space and time with an infinite ensemble of rigid meter sticks and ideal clocks he was not proposing a laboratory-operational definition. He stated it could be realized only to some degree, "not even with arbitrary approximation," and that the fundamental role of the whole conception, both on factual and on logical grounds "can be attacked with a certain right."20 Thus the RT

•

/

" Wolfgang Pauli, in Theory of Relativity [(B. G. Teubner, Leipzig, 1921 and Pergaraon Press, New York, 1958), p. 5], unwittingly draws forceful attention to this particular point when summarizing his analysis of the RT in the following words: "The postulate of relativity implies that a uniform motion of the center of mass of the universe relative to a closed system will be without influence on the phenomena in such a system." Note the correspondence with the main hypothesis in the last edition of the Principia. " Cf. Max von Laue, Naturwissenschaften 43, 1 (1956). "•"Les Prix Nobel en 1921-1922 (Stockholm, 1923), p. 2. See also A. Einstein, Naturwissenschaften 6, 692 (1918).

192 Lorentz and Poincaxe Invariance merely shifted the locus of space time from the sensorium of Newton's God to the sensorium of Einstein's abstract Gedankenexperimentsr—as it were, the final secularization of physics. In his tribute on the occasion of the 200th anniversary of Newton's death, Einstein wrote: "I must emphasize that Newton himself was better aware of the weakness inherent in his intellectual edifice that the generation of learned scientists which followed him. This fact has always aroused my deep admiration • • • ."21 He then immediately draws attention to the fact that "Newton's endeavors to represent his system as necessarily conditioned by experience and to introduce the smallest number of concepts not directly referable to empirical objects is everywhere evident." He recalls that Newton regarded the law of gravitational interaction as a heuristic device, "not supposed to be a final explanation, but a rule derived by induction from experience." When the essay ends with Einstein clearly associating himself with a view of causality which he characterizes as "Newtonian," he could well have widened the context of that remark.

Chapter 3

Inquiries Regarding the Constancy of the Speed of Light 3

3

W. Ritz (1908), R. C. Tolman (1910), J. Kunz (1910), D. F. Comstock (1910), W. Pauli (1921).

The Postulate of the Constancy of the Speed of Light. Ritz's and Related Theories 1 W . Pauli

It will be shown in the next section that the constancy of the speed of light, together with the principle of relativity, 2 leads to a new concept of time. For this reason W. Ritz 3 and, independently, Tolman, 4 Kunz 5 and Comstock 6 have raised the following question: Could one not avoid such radical deductions and still preserving agreement with experiment, by refusing the constancy of the speed of light and accepting only the first postulate? It is clear then that one would have to give up not only the idea of the existence of an aether but also Maxwell's equations for the vacuum, so that the whole of electrodynamics would have to be re-created. Only Ritz has succeeded in accomplishing this in a systematic theory. He accepts the equations curl E + — — = 0, div H = 0, c at so that the field intensities can be derived, just as in the conventional electrodynamics, from a scalar and vector potential IdA E = —grad (f> — , ~c~di' However, the equations ation s

•h

pdVpi

H == curl A.

(l/c)pvdVp.

A(p,t)--

r

pp']t'=t-(r/c) '

• / •

[rpp']t'=t-(r/c)

of the conventional electrodynamics are now replaced by

J

[rpp'\t'=t-(r/[c+vr])

J

[rpp'lt'=t-(r/[c+vr])

This corresponds to the assumption that it is only the speed of a light wave

194

Chap. 3. Inquiries Regarding the Constancy of the Speed ...

195

relative to its source, and analogously the speed of an electromagnetic disturbance relative to the electron, which is equal to c. We shall call all theories which are based on this assumption "emission theories". Since the relativity principle is automatically satisfied by all such theories, they can all explain Michelson's interferometer experiment. It remains to be investigated whether they are also consistent with the results of other optical experiments. 7 References 1. An excerpt translated from W. Pauli, Collected Scientific Papers by Wolfgang Pauli (Edited by R. Kronig and V. F. Weisskopf, 1964, Interscience Publisher, New York.) pp. 11-12. 2. That is, the first postulate. 3. W. Ritz, 'Recherches critiques sur Pelectrodynamique generale', Ann. Chim. Phys., 13 (1908) 145 [Coll. Works, p. 317]; 'Sur les theories electromagnetiques de Maxwell-Lorentz', Arch. Sci. Phys. Nat., 16 (1908) 260 [Coll. Works, P. 447]; 'Du role de Tether en physique', Riv. Sci., Bologna, 3 (1908) 260 [Coll. Works, p.477]; see also P. Ehrenfest, 'Zur Frage nach der Entbehrlichkeit des Lichtathers', Phys. Z., 13 (1912) 317; 'Zur Krise der lichtather-hypothese', lecture delivered in Leyden, 1912 (berlin, 1913). 4. R. C. Tolman, Phys. Rev., 30 (1910) 291 and 31 (1910) 26. 5. J. Kunz, Amer. J. Sci., 30 (1910) 1313. 6. D. F. Comstock, Phys. Rev., 30 (1910) 267. 7. For further discussions, see W. Pauli, The Theory of Relativity (translated by G. Field, Pergmon Press, London, 1958) pp. 6-9.

196

Lorentz and Poincare Invariance

Critical Researches on General Electrodynamics Walter Ritz Translated (1980) by Robert S. Fritzius 1 from Recherches critiques sur I'Electrodynamique Generale, Annales de Chimie et de Physique, 13, 145, 1908.

Introduction Electric and electrodynamic phenomena have acquired in the course of these last years more and more importance. They include Optics, the laws of radiation and the innumerable molecular phenomena associated with the presence of charged centers, ions and electrons. Finally, with the notion of electromagnetic mass, mechanics itself seems obliged to become a chapter of general electrodynamics. In the form given to it by H. A. Lorentz, Maxwell's theory would thus become the turning point towards a new conception of nature, where the laws of electrodynamics, considered as primary, would The Introduction and First Part are available in hard-copy from R.S. Fritzius, 305 Hillside Drive, Starkville, MS 39759. [email protected]. For a short biography of Ritz, see appendix C in this volume.

Chap. 3. Inquiries Regarding the Constancy of the Speed ...

197

contain the laws of motion as special cases and would play the fundamental role in the physical theories which, until now, have belonged to mechanics. Under these circumstances, it is plainly desirable to have a rigorous criticism of the foundations of this theory, to give it the degree of clarity and precision that mechanics itself reached only recently after much controversy. It is in order to ask which hypotheses are essential and can be deduced from observations, which others are logically useless or can be discarded without experience ceasing to be adequately represented, and finally, which are those which can be, and should be (Oeuvres 318) rejected; a question which is asked principally in regard to absolute motion. In the first part of his Lessons on Electricity and Optics2 Poincare devoted some classic pages to the criticism of the more or less distinctive theories of Maxwell himself and of Hertz. Therefore I will concern myself only with the form that the theory took in the hands of Lorentz, a form that presents well known advantages. Some of his results can easily be extended to the other theories. Here again, I only have to recall or to complete the ideas put forward by Poincare and more importantly by Lorentz who was well aware of the different aspects under which his theory could be interpreted. In general, I set aside the phenomena of molecular order, dependent on the corpuscular theory of electricity. That fruitful concept is evidently independent in large part of ideas that we can develop about the mode of action of electric charges on one another via the ether medium, which are more specifically the object of electrodynamic equations. The result of these researches has not been favorable to the existing theories. The discussions about the difficulties that they raise show that the difficulties have a common origin intimately linked to the concept of ether, which is the basis of all these theories. We will see specifically that: 1° From a strictly logical point of view, the electric and magnetic forces, which in appearance play in the theory a role so fundamental are notions that we can eliminate entirely; they only contain in reality the relations of space and time. We thus return to the old elementary actions with the sole difference that they are no longer instantaneous. 2° The theory permits an infinite number of solutions, each satisfying all the conditions, but incompatible with experience and even leading for example to perpetual motion. To remove these solutions we must admit by hypothesis formulae for retarded potentials. These formulae introduce irreversibility in electrodynamics (Oeuvres 319) whereas the general equations permit reversibility. I show that contrary to accepted ideas that they can't be deduced from a proper specialization of the initial state. They constitute H. POINCARE, Electricite et Optique: La lumiere et las theories 2nd edition, Paris 1901

electrodynamiques;

198

Lorentz and Poincare Invariance

a new hypothesis, making useless the partial differential equations. To clarify this hypothesis it is necessary to distinguish the elementary actions; it is to renounce Maxwell's fundamental idea of rejecting them. 3° The notion of localization of energy in the ether is vague and allows many simple solutions. 4° The Impossibility described by Maxwell to reduce gravitation to the same notions. That the negative energy involved would correspond to an unstable system shows that these ideas do not have general applicability to the forces of nature. 5° Action and reaction are not equal, and this inequality, in the manner in which it is deduced from the introduction of absolute velocities, is contrary to experience. 6° Kaufmann's experiments on the electric and magnetic deviability of 0 rays of radium don't prove that the mass of electrons is entirely of electromagnetic origin and dependent on their absolute velocity, because on the first hand, nothing obliges us to believe, as in Lorentz's theory, that the forces are linear functions of velocity, (this may be true at small velocities), and that, on the other hand, one of Trouton and Noble's experiments shows that the expression of electromagnetic momentum as a function of velocity from which Abraham has deduced the one of electromagnetic mass is certainly inexact. 7° The theory of Maxwell and of Lorentz starts from a system of absolute coordinates, that is to say, independent of all motions of matter. In order to be in agreement with experimental results, which have always, in optics and electrodynamics, as well as in mechanics, confirmed the principle of relative motions, we are obliged, then, to eliminate this absolute system by hypotheses of little credibility, thus eliminating the notion of solid bodies, and the concept of the invariability of ponderable masses. It will be necessary also, to change the principles of kinematics, to consider the rule of the velocity parallelogram just as a first approximation, valid at small speeds, (Oeuvres 320) and to make time and simultaneity completely relative notions. It would be regrettable, for the economy of our thinking if we had to live with all the complications listed above. I think, that instead of kinematics, it will be the ether hypothesis, and with it, the representation of phenomena by partial differential equations, that must be abandoned. The necessity to explain why bodies do not meet any resistance from the ether as they pass through it, and the fact that they do not modify its state, and many other considerations, have created a simple physical space out of Fresnel's mechanical ether, perfectly penetrable by matter, a system of absolute coordinates. The ether is now only a mathematical abstraction and its elimination would only be the final phase of a long evolution.

Chap. 3. Inquiries Regarding the Constancy of the Speed ...

199

This conclusion, as I set it forth, is not at all involved with a return to actions at a distance. Nevertheless, it indeed collides head-on with many currently accepted ideas, and I am the first to admit that a hypothesis which has rendered such great services to science can't be condemned for the sole reason that it presently raises some seemingly inextricable difficulties. We should always hope for future solutions of these difficulties, or accept the idea that they are inherently a part of things, and independent of our models. This is, fortunately, not the case. This is what I have sought to demonstrate in the Second Part of this work, but the theory which I will present does not pretend to be a satisfactory and definite solution to a problem so difficult. Its primary purpose is to show how large is the unknown part which, in spite of recent advances, still exists in this domain, and in what measure, it's much smaller than we would be tempted to believe. Experimental evidence may be considered as confirmation of Maxwell and Lorentz's theory, even when we adopt, as I will do, the remarkable ideas of this latter savant on the atomic constitution of electricity, the nature of conduction current and of dielectrics, in a nutshell, the theory of electrons. These researches will show that it is not necessary to introduce absolute motion and thus to upset kinematics and the notion of time; relative velocities alone will suffice. There will be no use of notions subject to criticism such as polarization, the electric vector the magnetic (Oeuvres 321) force, etc., but only the notions of time, space, and electric charges, these latter only playing, like the masses in mechanics, the role of coefficients conveniently chosen and invariable for a given ion or electron. In a certain sense it is a mechanical theory of electricity. But I have not believed it advisable to bring in the more or less complicated latent mechanisms which play such an important role in Maxwell's theory. Those hypotheses are unnecessary, and, one must say, barely satisfactory. It suffices, indeed, to recall that ponderable bodies must pass through these complicated mechanisms without disturbing them and without feeling sensible action, even when their speed reaches that of celestial bodies. Impenetrability, in particular, doesn't exist in the mechanical theories, and this is the one point which isn't always sufficiently placed in evidence. Experience has shown that actions are not instantaneous; also it hasn't revealed any trace of a medium which could exist in materially empty space. I therefore felt I could restrict myself to give to the law of propagation of these actions, a very simple kinematic interpretation borrowed from the emanative theory of light and satisfying the principle of relativity of motion. Fictitious particles are constantly emitted in all directions by electric charges. They keep on moving indefinitely in straight lines with constant speed, even through material bodies. The action under gone by a charge depends uniquely on the disposition, velocity, etc., of these particles

200

Lorentz and Poincare Invariance

in its immediate surroundings. The particles are therefore simply a concrete representation of kinematic and geometric data. These hypotheses are sufficient for the purely critical objective that I suggest to reach here. They permit study in detail of the law of elementary actions between electrons in motion and show in particular, that this law, almost entirely unknown for great speeds, requires, even at small speeds, an indeterminate parameter K, which is not without analogy with the one that Helmholtz has introduced in his theory. I need to specify the temporary scope of these hypotheses. Indeed, when the particles (or, if we like, the actions or energies) emitted by an electrified body reach another electric charge and modify its motion, the principle (Oeuvres 322) of action and reaction demands that they undergo on their part, a deviation, or a change, and it is very remarkable that Fizeau's experiment on the entrainment of waves, like certain other facts of optics, is not compatible with the hypothesis admitted here, and demands such a reaction. It's the opposite that happens in the ether hypothesis, as Poincare so presented it. Hertz's theory, which satisfies the principle of action and reaction, is incompatible with Fizeau's experiment. Lorentz's theory, which doesn't satisfy it, explains the experiment perfectly. But Poincare has shown that in giving a momentum to the radiant energy, everything falls into place; therefore this hypothesis is natural if this energy is projected instead of being propagated. It is precisely this that permits safeguarding this principle in the new model that I propose. We can even foresee the possibility of obtaining, by these principles, the electrodynamic terms that depend on velocity and acceleration, using only the consideration of propagation 3 , a problem that Gauss posed in his well know letter to W. Weber, and that Maxwell's theory didn't solve because it introduces to these terms a special quantity, the vector potential. I will return to these questions.later. The remarks which precede are sufficient to explain why I didn't address optics in this criticism. In many respects, the new theory will therefore bring the reader back to some older classical ideas, which seemed destined to be forgotten. The interpretation of certain experiments will necessarily be modified. In particular, perhaps part of or the whole of mass will be electromagnetic in origin, but it will be constant and won't depend on an absolute velocity. It is the forces, and not the mass, that changes. Kaufmann's experiments also permit this new viewpoint. The new formulae are applicable to gravitation; they permit abolishment in large part the most apparent divergence which exists at this time between calculation and experiment regarding the perihelon motion of Mercury. 3

The word "projection" may have been intended here. Trans.

Chap. 3. Inquiries Regarding the Constancy of the Speed ...

201

The theory of electrons has constituted a first partial return from Maxwell's ideas to others much older, and for those who consider as indispensable a new evolution in (Oeuvres 323) the same sense, Lorentz's hypotheses, which have been so fruitful, will maintain their importance, and the mathematical form that he gave them will continue in many cases to be the most elegant and the most practical. FIRST PART 1. - RECALL OF LORENTZ'S THEORY 4 We know that Maxwell did not work on hypotheses on the nature of electric currents. Lorentz claims that all conduction current results from motion of electric particles, which are subjected to a kind of friction in conductors and to elastic forces in dielectrics. Many experiments have, in the last years, confirmed this hypothesis. This concept has permitted Lorentz to consider only the dielectric ether in his fundamental equations. Renouncing a purely mechanical explanation and the impenetrability of matter, Lorentz considers the ether as immobile and even present in the interior of ions and electrons. The ions and electrons modify it physically, and this modification, which is difficult to picture in a concrete fashion, is characterized by two vectors; the electric or dielectric displacement vector E, whose components are Ex, Ey, Ez, and the magnetic vector H (Hx, Hy, Hz). The electric charges are fixed to the ions which are considered undeformable. Where the electric density, p is measured in electrostatic units at xyz, at time t, the system of coordinates being connected to the stable ether, and t; the velocity of the electric substance in (x ,y, z, t) and c, the speed of light; we have, for these constraints, the system of equations ,TS

,TT

(I) v '

l

9

E

„

V

curlH = -——\-4np— H j. c 8t c

(TT\

III) y

IT?

~

1 9 H

curlE = '

— c dt

{III)

divE = 4TI7>

(IV)

divH = 0

4

H.A. LORENTZ, Archives need., vol. XXV, 1892; Versuch einer Theorie der elektrischen und optischen Erscheninungen in bewegten Korpern, Leiden, 1895; Elektronentheorie: Enzyklopadie der math, Wissenschaften. Bd. V, Heft 1, Leipzig, 1904-POINCARE, Electricite et Optique, Chap, III, p. 422.

202

Lorentz

and Poincare

Invaxiance

(V)

-£ + divpv = 0

The field therefore created in the ether by other charges present, exerts on the element of charge pdr the mechanical force represented in magnitude and direction by the vector Fpdr where (VI)

F = E+-(V

xH)

c

In this theory there is no magnetism. The magnetization comes from Amperian particle currents. By means of certain hypotheses, to which we will return, this system of equations integrates itself by the introduction of retarded potentials. As a matter of fact we show that all solutions of the system (I) to (V), where we assume the data p, vx,vy, vz can be expressed in the form 1 d A

(VII)

E=-div--^ c dt

(VIII)

H = curlA

the functions $ (the scalar potential) and Ax, Ay, Az (components of the vector potential) being continuous with their first derivatives in all space, from zero to infinity, and satisfying the equations 1 d2$ — -

(IX)

1 d2Ax ,2 dt2 1 d2Ay

(X)

,

A

=

A A

LS/ix A

{ ?T5i*

A

4,,, _ 4ivpVx ~ c _ AirpVy

and ldd> divA

(xi)

= -cTt

Lorentz satisfies these conditions by setting (XII)

U(x, (x-x>y

+

l\2 (y-y>y+(Z-z')

Chap. 3. Inquiries Regarding the Constancy of the Speed ... 203 Ax=t±fff^dT' (XIII)

Ay^Uff^dr' I A 2 = ...

These expressions have the form of newtonian potentials, with the difference that instead of taking the value of p at point x'y'z' at time t, we have to take it at the previous time t' = £ — r/c the time r/c being necessary because of the propagation. This is what we will point out along with Lorentz by the notation \p'] = p{x',y',z?,t-^), [p'v'x] = p(x\ y', z', t - r-)Vx(x', y\ z',

t-'-).

The field is therefore completely determined, and in introducing expressions (XII) and (XIII) into formulae (VII) and (VIII) (Oeuvres 326) and hence into (VI), we obtain for Fx d[p'}

<*">

'-JJH-KV, v* d [p'v'x] | vy d [p'v'y] | c2 dx r c2 dx r vx d [p'v'x] vy d [p'v'x] c2 dx r c2 dy r

and similar expressions for Fy,Fz. d d dt ' = m

Id [p'v'x] c2dt vz d [p'v'z] c2 dx r vz d \p'v'x}, c2 dz r

In introducing the total derivative d dx-

+ Vx

d d *d-y + Vzdz~>

+ v

and setting

(XV)

L(x,y,z,t,vx,vy,vz)

= f f f M { 1 - v*b>'*] + vvty\ +

v

*[v'*\]dTi

Schwarschild5 has found for Fx the remarkable form

This is the form of Lagrange's equations. The expressions (XIV) and (XVI) give the force undergone by an electric point of unit charge, expressed by s

Gottinger

Nachr., Math.-Phys.

Klasse, 1903, p. 126.

204

Lorentz and Poincare Invariance

means of elementary actions analogous to the ones considered in the old electrodynamics, without the notion of non-instantaneous transmission that we find again in Gauss and C. Neumann. A charge e\ sensibly point-like, exercises, under very general considerations, on another similar charge e', a force

,

N

f-'(-§HM*:)>

(* >

{-'(-t+£i£),

y//

where A_

i

f

i - ^ K ] + ^ K ] + ^K3

r 2 = [x(t) - x'(t - T-)f + [y(t) - y'(t - r-)f + [,(*) - z'(t - ^ ) ] 2 This expression reduces itself, to a first approximation, to the law of distance squared. We can therefore name it the generalized Newton's law. This explicit expession will be given later on. In these formulae the notion of field doesn't come into play. It is very remarkable that Clausius, like Weber, in seeking to account for the electrodynamic actions by means of actions at a distance, depending on positions, velocities and accelerations of the electric points, has derived the same formulae (XV) and (XVI) with the only difference that the actions are instantaneous, so that we have to take the values of p' and v' at time t and not at time t — - . c

This very remarkable result, due to Schwarzschild, shows that Lorentz's theory is very close to the old theories. The first form that Lorentz gave to his theory was less abstract in the sense that, following the path that Maxwell laid out, he started from Lagrange's equations, introducing two kinds of variables, the first ones determining the positions of the electrified particles, the others the state of the ether. We attribute to this latter one kinetic energy, without determining its internal motions. It suffices that they exist. Hamilton's principle permits, likewise, restricting the variations under certain conditions, to obtain the fundamental equations (I) to (VI) by considering the electrical energy i - j{Ex2 on J

+ Ey2 +

E2)dr

as potential energy, and the magnetic energy l-j(Hx2

+ Hy2 + Hz2)dr

Chap. 3. Inquiries Regarding the Constancy of the Speed ...

205

as kinetic energy.*' This deduction is fairly complicated and can be done in various forms. 7 These two aspects of Lorentz's theory are clearly distinct. The second one is similar to Larmor's theory, 8 which, in leading to the same formulae offers precise hypotheses on the motions of ether in an electromagnetic field as borrowed from Lord Kelvin's conceptions on gyrostatic ether. Ether is incompressible and moves itself in the direction of magnetic force lines. We know that Maxwell and Hertz explain the mechanical forces that a substances experiences in an electromagnetic field by the pressure that ether is supposed to exert on the substance, and conversely: the action of one is equal and opposite to the reaction of the other in all respects. These pressures, as Helmholtz has shown,9 tend to set the ether (supposedly incompressible) in motion; they are, by unit volume, proportional to the time derivative of Poynting's radiant vector S: (XVIII)

S = — £ x H Aire

Lorentz considers ether as immobile; he is therefore inclined to Maxwell's theory of pressures and, by that, the equality of action and reaction, the non-compensated force being characterized by the vector ^jf.10 Let's observe, in concluding this brief exposition, that if we admit unknown motions in ether, the solution of the equations is determinable only for quantities close to the order of the speed of the ether divided by that of light. 2 - CRITICISM OF NOTIONS OF ELECTRIC AND MAGNETIC FIELDS We know that the introduction. of the notion of force into mechanics was the subject of much criticism. This notion calls for muscular sense, whereas the ideas cf space and time are primarily of tactile and visual origin: and the irreducible psychological duality introduced at the very base of this fundamental science leaves a certain discomfort in the mind, justifiably so, for it seems very obvious that the notion eliminates itself in each particular case. Whether we measure the forces by masses and accelerations, or by elastic deformations, whether we oppose their effects with those of gravity, 6

LORENTZ, Proc. Amsterdam Acad., 1903, p. 608; Elektronen-theorie, p. 165 to 170. LARMOR, Aether and Matter, Cambridge,' 1903, Chap. IV. SCHWARZSCHILD, Gottinger Nachr. 1903, p. 125. 8 LARMOR, Proc. Roy. Soc, Vol. LIV, 1893, p. 438; Aether and Matter. 9 Gesammelte Abhandlungen, Vol. Ill, p. 526. 10 POINCARE Electricite et Optique, p. 448. 7

206

Lorentz and Poincare Invariance

etc., what we really observe and measure is always a displacement, or the absence of a displacement. Again, in this latter case, we only end up defining the difference of two forces. In the equations of mechanics, as applied to any particular example, there remain only the relations of space and time, with certain coefficients properly chosen and invariable which are the masses or other physical constants. With regard to pure logic, it is therefore with good reason that many experts have rejected the introduction of the notion of forces in the fundamental expressions as being useless. Modern electrodynamics is entirely based on the notion of electric and magnetic forces. If this were absolutely necessary it would be regrettable. But it isn't so. These notions eliminate themselves in the equations. They are logically useless. In the final analysis the theory only expresses the existence of certain relations of time and space, as it is in the case of mechanics. It will therefore be preferable to express these relations directly. We thus come back to the classical elementary actions. In fact, what are the exact definitions of the vectors for E and H fields? I say that these vectors are defined by the theory itself. Thus, without knowing the significance of these symbols we can at once, by means of certain hypotheses that (Oeuvres 330) we will examine in the next section, integrate the fundamental equations by the method of retarded potentials, and we will be led to expressions (XIV) and (XVI). The equations of motion for a material point of charge e, of mass m and of coordinates x, y, z will be m

"

/

m

^

W

= eFx(x1,yi,zi\

vxl,vyi,v2l,t)

If we desire to take into account the action of the electron on itself, or of liaisons, d'Alembert's principle has to be applied, and we have, in extending the integration over the whole of the electron, and designating by Sx\,6yi,Szi the virtual displacements compatible with the liaisons, and by Hi, pi the densities of the substance and of electricity,

(2)

J{{j*i-^r

-

PiF*(*i,-)]6*i

After having replaced Fx, • • • with the value (XIV) or (XVI) (the terms solely relative to the electron will play a special role), we will have in (1) and (2) only relations of space and time, even when fi — 0, that is to say, when mass is entirely of electromagnetic origin. Now, I say that Lorentz's equations don't effectively express anything more than (1) and (2). That is to say, that the field never plays a role in

Chap. 3. Inquiries Regarding the Constancy of the Speed ...

207

pure ether. In fact, we can only determine the field's magnitude and direction by positioning a body and observing the mechanical forces that it feels or rather its motions and those of the ions in its near vicinity, motions which are indicated by luminous, thermal, chemical, etc., phenomena. Therefore, we only know F, and that, only at such points of £1,2/1,21, where there is electrified matter, and we deduce E and H by reasoning (which is not always so simple when we have to consider absolute motion). This is to say that, in all cases, to know the formula (Oeuvres 331) that gives F as the result of elementary actions exerted by an element of charge on another element of charge, and that this second representation, with regard to the facts, is exactly equivalent to the first, which is based on the field and its partial differential equations which only play a purely mathematical role. We can, if we please, get along without the notion of electric and magnetic fields. It is important to specify the sense of this affirmation. In the theory of light, for example, everything thus represented with regard to Lorentz's theory can be derived from elementary actions between ions in the luminous source, those in the dielectrics or conductors which constitute the optical apparatus, and finally those in the retina or photographic plate which receives the impression. Thus, we are accustomed, for example, to describe the phenomena of diffraction that we observe in the case of a slit used with a screen by considering with Fresnel that the points of ether situated in the slit as so many centers of disturbance. This does not conform to the equation for retarded potentials though. Electric charges are the only points of origin for waves. Lorentz's theory, or the law of elementary actions, will explain it as the combined action of ions in the source and in the screen; besides it is easy to show, using Huygen's principle as Kirchoff has it, the equivalence of the two methods with regard to results. It wouldn't be allowed any longer to say that the field is a purely mathematical intermediate which we could do without, if it were possible to perceive its existence in a region of ether without placing any matter in the region. That is what would have happened, for example, if ether, under the influence of a field, were to be susceptible to modification or were to move itself more or less as Hertz wants it to, and as Lord Kelvin demands. 11 Interference experiments would have put this speed into evidence. These ideas were generally very widespread; but we know that the (Oeuvres 332) experiment, conducted several times, 12 gave only negative results, as do all the experi11 Lord Kelvin, Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light, London, 1904, p. 159: "It is absolutely certain that there is a definite dynamical theory for waves of light, to be enriched, not abolished, by electromagnetic theory." 12 5eein particular O. LODGE, Phil. Trans., Vol. CLXXXIV, 1893; HENDERSON and HENRY, Phil. Mag., 5th Series, Vol. XLIV, 1897, p. 20.

208

Lorentz and Poincare Invariance

ments designed to prove the existence of ether. The hypothesis of all these motions, on the other hand, has not lent itself to any plausible mechanical explanation of electrical actions in their effects. Lorentz was therefore led to exclude it in the recent statements of his theory; and that is what gives the go-ahead for us to eliminate the notions of force and field in this theory without touching any fact of reality or any possible experiment according to it. Lorentz has already indicated 13 this point of view: "We therefore see, in the new way I'm going to present it, Maxwell's theory draws nearer to the older ideas. We can even, after we have established the simple formulae that describe the motions of the particles, leave out the reasoning that spawned them and consider looking at these formulae as expressing a fundamental law comparable with those of Weber and Clausius." The actions, however, are no longer instantaneous; and we have seen, in light of this important restriction, that there is even an identity with Clausius' law. We can easily see that the notion of field introduces the notion of absolute motion as soon as the velocities come into play, either in the field expressions or in the expression for its action on bodies. It isn't likely anymore that it depends only on coordinates and accelerations. Sections 3-8 are omitted. 9. - ABSOLUTE MOTION In placing the ether hypothesis at the base of electrodynamics and optics we of necessitiy introduce, at least for the propagation of light and electric actions, a system of coordinates, independent of ordinary matter. We should therefore expect, and we have indeed long expected, an influence of absolute motion with respect to the assumed ether. -We know that the experiments have always been negative. Lorentz's theory gives this result when we consider first order terms. But the experiments (Oeuvres 361) of Michelson and Morley, Trouton and Noble, and Lord Rayleigh, which should have shown second order effects, have, contrary to the theory, also given negative results. Lorentz and FitzGerald have therefore assumed that all bodies experience a contraction of the relation (1 — v2/c2)~ll2 in the direction of their velocity v. We thus account for the observed negative results. 14 To explain this contraction, Lorentz calls to mind that, according to his theory, for a system of 13

Arch, neerl., Vol. XXV, 1892, p. 433. M. Planck has shown that if we assume that the density of the ether at the earth's surface is less than 50,000 times greater than that in the interplanetary medium, without any appreciable change in its properties resulting, we may possibly reconcile the aberration theory with the hypothesis that the ether is entrained in the earth's motion (see 14

Chap. 3. Inquiries Regaxding the Constancy of the Speed ...

209

electric charges S, at rest, in equilibrium, this same system, when assumed to be antimated by a uniform translational motion v, will still be in equilibrium if we modify its dimensions by the indicated ratio. Therefore, if the molecular actions obey the law of electrostatic actions, and if we can exclude the molecular motion, the molecules of a solid body should of necessity take the position of equilibrium, and the accepted contraction will take place. It is evident that this hypothesis confuses our notions of solids. The invariability of certain bodies, when we transfer them from one place to another, when we change their direction or their speed, gives us the experimental definition of distance and of other geometric scales. The bodies that we use, necessarily participating in the motion of the earth, will always have an infinity of movements and rotations which change their dimensions and since we don't have any way to precisely determine the absolute motion which comes into play here these deformations remain absolutely unknown. How do you physically define the true length of a body? Does the assertion of the reality of this contraction have any sense? It results, from the researches of Einstein, to which we return later on, that the answer is negative. The question of stability brings forth a second objection. (Oeuvres 362) A system of electric charges, subject only to electrostatic forces, is never in stable equilibrium. This is evident when the sole restriction imposed is the conservation of electricity. In changing all the dimensions of the system by the ratio of 1 to 1 + e, the charges of elements of corresponding volume being equal, we will have performed a deformation compatible with the conditions of the system. The energy will fall to (jr^) t h of its original value. The equilibrium, therefore, is not stable. The sphere, for example, is for a deformable electron a shape of unstable equilibrium, and its the same if we suppose, with Bucherer and Langevin, that its volume is invariable15 a fortiori, whereas in Lorentz's hypothesis this restriction doesn't exist. To obtain a solid body we therefore need to add forces of a very different character from those of electrostatic forces, or of liaisons other than incompressibility, or finally of whirling motions giving a dynamic equilibrium. But in all of these cases, Lorentz's explanation no longer applies so this explanation doesn't seem acceptable. Poincare has finally objected to the hypothesis as being incomplete. New experiments could bring new terms into evidence and we would require new hypotheses if, as expected, results are negative. The question of the complete LORENTZ, Enzyklop. math. Wiss., vol 5, art. 133, p. 104). This would be a very strange property of the ether. ls This is what Ehrenfest pointed out (Physik. Zeitschr., vol. 7, 1906, p. 302). For gravitation the equilibrium would be stable; but in changing the attractions into repulsions the energy changes sign and the equilibrium becomes unstable.

210

Lorentz and Poincare Invariance

elimination of absolute motion was therefore posed and was addressed by Lorentz, 1 6 Poincare, 1 7 and Einstein 1 8 . It is no longer permissible to overlook the difference between "local time" and "true time", which was an essential point when we were content to explain the negative results observed up until now and a few others analogous. For us to render a full account, lets consider two points A and B which move with a constant absolute velocity v in the direction A B . A luminous wave, starting from A at instant t, will arrive (Oeuvres 363) at B at instant t'. It will have to travel the distance AB + (t — t')v with the speed c we have then t

,_

t =

AB + (f - t)v c

C — V

The duration of transmission will depend on v, and its changes include a first order term -ABv/c, the correction (including higher order terms) being precisely the one we have to apply to the true time to get the local time. Lorentz 1 9 showed t h a t for terrestrial phenomena, this correction is without influence. In particular, to determine the speed of light directly we are obliged to make it cover a closed path which brings it back to its starting point; thus eliminating the first order terms. So, in the example considered, if the wave emitted at A is reflected at B, it will arrive at A after a time

c— v

c+ v

c

c*

But it will be otherwise for astronomical Phenomena. In the determination of the speed of light by the occultations of satellites, we don't use a closed path, thus, the perturbation, that the hypothesis of a translation of the solar system with respect to the ether would cause in the observed retardation, would be of a first order and of observable magnitude. Indeed, the delay of an occultation can reach * (where d is the diameter of the terrestrial orbit), or about 1000 seconds. An absolute speed of the solar system (having no connection with the system's motion with respect to the closer fixed stars) equal to 30 km per second in the plane of the ecliptic would bring about a correction of about 1 0 0 0 3 ^ , 3 = 0.1 second for the maximum observed delay, a correction which would change sign according 16

Amsterdam Proceedings, 1903-1904, p. 809. Comptes rendus, vol.. CXL, 1905, p. 1504. ™Ann. der Physik, vol. XVII, 1905, p. 891. 19 Versuche einer Theorie, etc., p. 82ff.

17

Chap. 3. Inquiries Regarding the Constancy of the Speed ...

211

to the relative position of the Earth and the satellite in relation to the direction of translation. The systematic differences for a (Oeuvres 364) long system of observations can then reach 0.2 second, a quantity which is on the order of those which we observe in astronomy. 20 If, therefore, we do not wish to admit that the speed of light depends on that of the bodies emitting it and purely relative, like all speeds (and the ether concept, alone, prevents drawing out of the relativity principle this so natural consequence), we will have to modify the definition of time. Lorentz has enunciated the hypotheses which would allow giving to the equations for an entrained system, as well as for the axes of coordinates, in uniform motion of translation v parallel to the x axis, the same form as for the case of rest. He admits that all the masses are functions of velocity, thus abandoning the principle of the conservation of mass. 21 We must also, as we formerly revealed, suppress.the notion of solid bodies and introduce a new definition of time. It will be the variable ,

/

V2

VX

which will play the role of time in the equations. Time, thus defined doesn't satisfy almost any of the axioms assigned to the notion of time in the ordinary sense. Two events, simultaneous for observer A, but taking place at different points, are not simultaneous anymore for a second observer B [who is] in motion with respect to the first. Simultaniety becomes a relative notion. Two equal times for observer A will not be so for B. The parallelogram rule for velocities is only approximated. Thus for v and v' being the speeds of two bodies which are moving in opposite directions in relation to a primary system of coordinates, the relative speed of the first body with respect to the second, i.e., the velocity that an observer traveling along with the second body would observe, is not v — v', but v — v' cr 20

Supposing that the laws of gravitation are modified by motion as the laws of electrodynamics are, the corrections would be only of the second order and would not cancel the first order term. 21 The word mass, in the theories of Lorentz, Poincare, and Einstein, has no precise meaning anymore. The number representing it depends on the motion of the system of coordinates, the motion being absoutely arbitrary. But the force depends also on this motion and it is not the two members of the equation d2x/dt2 = X,... but indeed their relation alone that remains invariable when we change this motion.

212

Lorentz and Poincare Invariance

it remains steadfastly inferior to the speed of light. For two (3 rays emitted in opposite directions by "a grain of radium, each with a speed of 250,000 km per second, the relative speed will not be 500,000 km per second, but 294,000 km per second. The words "speed", "time", etc., have therefore acquired a significance very different from that which they normally have, and now only have a relative sense. The ether in this new Kinematics will not play any role because it no longer furnishes a system of absolute coordinates. But this concept will oblige us to replace the simple axioms of the conservation of mass, the invariability of solids, the parallelogram of velocities, etc., axioms which we should abandon only as a last resort, with complicated relationships presenting considerable difficulties to the imagination (similar to those of curved space in three dimensions), and which we generally can't treat rigorously, except by analytical considerations. We must add that this theory was presented by Lorentz with all reservations. Einstein (loc. cit.) has presented the same results in a different form. He admits, a priori, for the speed of light, a law which by its nature involves a large amount of arbitrariness; the comparison with that which we will adopt in the Second Part of this work will demonstrate this sufficiently. It leads, with the principle of relativity, to a definition of the simultaneity of two events at two different points, of which he makes a relative notion, and more generally [leads] to the new kinematics which has just been discussed. The simultaneity included in the (Oeuvres 366) definition of the length of a body in motion in relation to the fixed marks of a standard measure (since it will be a matter of pointing simultaneously to the two extremities of the body, otherwise the body would move during the interval) this body will appear to be of a different length to an observer at rest, according to its more or less large speed (although its true length remains invariable). We therefore avoid the contractions admitted by Lorentz, or rather we see that their reality is only a question of definition. Einstein verifies that Lorentz's equations are thus made independent of absolute motion, and that the law admitted by him for the propagation of light is in accordance with the equations. These then, in the measure that they express this law of propagation, become superfluous. Moreover, the reasoning does not demonstrate at all, as some authors have beleived, that these transformations are the only group which leaves Lorentz's equations invariable. This problem has rather to do with Poincare's methods. 17 Bucherer 22 was led, by considerations on the relativity of motions to abandon the notion of ether. Lorentz's equations should always be applied Physik. Zeitschr., vol. 7, 1906, p. 553.

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213

by assuming that the system of coordinates is at rest with respect to the point P whose motion we study. Bucherer only considers the case of uniform motions; the action on an electron propelled with a relative velocity u = v'—v in relation to P, according to formula (13) (where we set v' = u, v — 0), will be given by the formula *x

ee'cospx pi ^i+

u2-3up2 2c2 ; '

t

v ~ -

For two closed currents, we verify easily (see the Second Part) that only the terms proportional to vv' play a role, with the accelerations excluded, which we haven't taken into consideration, and terms in v2, v'2: it is moreover what should happen if the intensity of the force is proportional to the product of the intensities of the currents. The action of two elements of current on each other will from then on be %% dsds 2—cos(p, x)[—cos(ds, ds') + 3cos(p, ds)cos(p, ds')] The hypotheses of Ampere are verified. The action is parallel to the line of junction p of the elements; but the parenthesis should be —2cos(ds, ds') + 3cos(p, ds'). Bucherer's hypothesis is therefore irreconcilable with the laws of closed currents. This is because the notion of field ceases to be applicable there. We are therefore brought back to the complicated hypotheses which were set forth. We must say, in closing, that these complications occur not only at great speeds, but also in Fizeau's experiment on the entrainment of waves, for example. In effect, according to the principle of relativity, an observer carried along in the translational movement of a transparent body will find, for the speed of propagation of waves in this environment, the same value as if he were at rest (supposing the period to be the same, or the dispersion negligible). We would conclude that, in ordinary kinematics, the waves are totally carried along by the substance. There is nothing to it. The term velocity has a new meaning, and in reality, Lorentz's proof continues to be applicable. We come back to Fresnel's coefficient. 10 - SUMMARY AND CONCLUSIONS We know that ether was, at first, only one of the numerous fluids of physics, but with the new experiments having proved to Fresnel that light waves are transversal, we had to create a body analogous to elastic solids.

214

Lorentz and Poincare Invariance

But then, how can the other bodies move through it without experiencing any appreciable resistance? The question was all the more difficult with the problem of aberration obliging us to admit that the ether does not participate in the earth's translational motion, so that all bodies are constantly traversed by a current of ether of 30 km per second with zero effects despite the rigidity of ether. We must add that the elasticity of this body is quite singular, seeing that its resistance to compression would be zero, which wouldn't happen for a finite solid. We can, it is true, appeal to Lord Kelvin's rotational elasticity, by being (Oeuvres 368) careful not to take into account the perturbation that would be brought to this ingenious mechanism by the brutal passage of an animated body with a speed of 30 km per second. The difficulties increased when, the identity of light and electrical oscillations having been shown, we had to extend this system of explanations to electromagnetism as a whole. Poincare 23 has exposed some of the strangeness to which we are led. Besides, experiments have refused to accord to ether that primordial property of bodies; movement. Fizeau's experiments (interpreted by Lorentz), and those of Lodge, and others, have concurred in their negative results. Ether is not entrained by the movement of matter, nor by that of charged or magnetized bodies, nor by currents, etc. The hypothesis of such motions, itself, does not permit obtaining a mechanical explanation of electrodynamics. We resigned ourselves to accept the absolute rest of ether. The hypothesis of a complete compenetration permitted the avoidance of the difficulty relative to the movement of bodies through ether. This latter has become what Drude calls a "Physical Space", the seat of electric and magnetic energy, and polarizations. It furnishes a system of coordinates independent of all matter, and to which Maxwell's and Lorentz's equations must be referred. This is already too much abstraction. It still is not enough. Indeed, according to these views, ether could also be the seat, of phenomena independent of matter, and thus manifest its existence. Nothing of the kind; and, in order to explain it, we needed a new hypothesis, discarding all waves that don't diverge from a material element of volume. The role of ether is again reduced. We have seen that, from now on, we can completely set aside the notion of field and the consideration of what is passing in the ether, and to be content with elementary actions of charges on each other (exactly as in the older theories of Gauss, Weber, Riemann and Clausius, but with a finite time of transmission). We thus express the same facts, but by including the hypothesis on the divergence of waves and the consequential (Oeuvres 369) irreversibility, which the equations of partial derivatives are powerless 23

Electricite et Optique, 2nd ed.; And proposed by Larmor's theory., p. 577ff.

Chap. 3. Inquiries Regarding the Constancy of the Speed ...

215

to express, the ether has become a system of absolute coordinates, a mathematical abstraction. The equations of partial derivatives are an intermediate mental construct which, however, aren't sufficient in themselves. Finally, this phantom of ether itself did not stand up under the scrutiny of experiment. It seems well accepted t h a t absolute motion cannot be put into evidence. We have seen t o which hypotheses, disrupting all the principles of physics, we must have recourse to, to render account of this result. T h e only conclusion which, from then on, seems possible to me, is t h a t ether doesn't exist, or more exactly, that we should renounce the use of this representation that the motion of light is a relative motion like all the others, that only relative velocities play a role in the laws of nature; and finally that we should renounce the use of partial differential equations and the notion of field, in the measure that this notion introduces absolute motion. As I already stated in the Introduction, this overly negative conclusion needs two complimentary remarks; a simple representation for the new mode of light motion; and the demonstration t h a t a theory satisfying these principles is possible. The habit t h a t we have to "substantiate", if I dare to express myself so, a habit which we owe to the old caloric, magnetic, etc., fluids and the new energy fluid, makes indispensable, indeed, the introduction of a representation which makes us realize what happens t o light and electrical forces when, having left a body, they don't act on still another. A theory which wouldn't admit such a representation would be considered by many as introducing actions at a distance, simply retarded. Moreover, as Poincare noted (Science et Hypothesis, p. 199), 2 4 and this is one of the reasons t h a t we can judge in favor about the existence of ether, mechanics would have it t h a t the state of a system depends only on the immediately preceeding states. It wouldn't be so anymore if we cancelled all intermediates. Actually, we thus save only a convention, which perhaps doesn't have any extreme usefulness. We have seen t h a t we can't arbitrarily give the initial s t a t e of ether, which must satisfy (Oeuvres 370) the formulae of retarded potentials. It is to say t h a t the consideration of the system during a finite period is not avoided effectively. On the other hand, the pressure exerted by light on a mirror, even in a vacuum, for example, is contrary to the equality of action and reaction when applied to the material alone. We will therefore have to "substantialize" the radiant energy t o save the principle and t h a t of the conservation of energy whenever there is a body in which the radiation doesn't meet any material obstacle in certain directions, and for which the energy cannot, consequently, ever be fully recovered. These principles will then become conventions, in part at least, but for a greater advantage to the economy of our thought. 24

This is probably page 119. Trans.

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Chapter 4

Extended Relativity and its 4-Dimensional Symmetry 4

"H. Reichenbach (1928), A. Griinbaum (1963), W. F. Edwards (1963) L. Hsu (1997), J. P. Hsu (1997), D. A. Schneble (1997).

The Philosophy of Space and Time Hans Reichenbach Simultaneity x After we had specified the unit of time, which is the first metrical coordinative definition of time, we were led to the problem of uniformity, which is the second metrical coordinative definition of time and deals with the congruence of successive time intervals. There is however a second type of time comparison that concerns parallel time intervals occurring at different points in space rather than consecutive time intervals occurring at the same point in space. The comparison of such time intervals leads to the problem of simultaneity and hence to the third metrical coordinative definition of time. Although it had been known for some time that uniformity is a matter of definition— Mach,1 for instance, asserted the definitional character of the uniformity of time emphatically—the definitional character of simultaneity was recognized first by Einstein and has since become famous as the relativity of time. Einstein immediately applied his solution of the problem of simultaneity to theoretical physics and for this reason the epistemological character of his discovery has never been clearly distinguished from the physical results. Therefore, we shall not follow the road taken by Einstein, which is closely connected with the principle of the constancy of the velocity of light, but begin with the epistemological problem. To see this problem clearly, we must start with a distinction which originated with the work of Einstein. We shall distinguish between the simultaneity at the same place and the simultaneity of spatially separated events. Only the latter contains the actual problem of simultaneity; the first is strictly speaking not a simultaneity of time points, but an identity. Such a concurrence of events at the same place and at the same time is called a coincidence. In a strict coincidence there is actually no comparison of space or time since position and time are identical for both events. Practically speaking, such an i E. Mach, The Science of Mechanics, The Open Court Publishing Co., Chicago and London, 1919, p. 223.

218

Chap. 4. Extended Relativity ...

219

identity never occurs since we could no longer distinguish the two events. But an approximate coincidence can be realized, in the example of two colliding spheres or two intersecting light rays. Simultaneity plays no essential role even in the case of a roughly approximated coincidence, because a time comparison of distant events shows such slight differences in the determination of the time of neighboring events that they can be ignored. We can therefore treat the problem of the comparison of neighboring events similarly to the problem of coincidence and restrict our investigation to the comparison of distant events.^This investigation will lead us to the result that the simultaneity of distant events is based on a coordinative definition. We shall demonstrate this result by showing that a comparison of time has the characteristic properties of a coordinative definition. We therefore maintain: First, it is impossible to ascertain whether two distant clocks are set "correctly" in their indication of time; second, they can be set arbitrarily and yet no contradiction will arise. Following the first line of thought, we may ask how one can determine the simultaneity of distant events. We shall consider events as distant, if the distance between them is large compared with the dimensions of the human body. The perceptual judgment of simultaneity is thus not sufficient under these circumstances. We may hear for instance the sound of thunder and notice at the same time that the hands of our watch point to 8:50. The determination of simultaneity which we make here is a comparison of neighboring events; we compare the moment when the watch indicates 8:50 with the moment when the sound of thunder reaches our ear and not with the instant of its occurrence. If we want to derive from this time determination the actual time at which the thunder occurred, we must have additional physical facts. We must know the distance which the sound has traveled and the velocity of sound, before we can calculate backwards from 8:50 to the time at which the thunder took place. But are there no other means? It is well known that we can avoid using the velocity of sound in the given example, if we observe the lightning rather than the thunder. Let us say that the lightning was i For a rigorous treatment of the comparison of neighboring events, see A., § 8.

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Lorentz and Poincare Invariance

observed at 8h 49m 50s; we may then consider this time as the time at which both lightening and thunder occurred. Is this statement true? Obviously, in this type of time determination the situation is changed quantitatively but not in principle. The light of the lightning also requires a certain amount of time to reach the eye, and our judgment therefore concerns again the moment at which the light reaches our eye and not the moment when the lightning actually occurred. Only because this time difference is extremely small can we ignore it for practical purposes. It can easily be seen that the time comparison of distant events is possible only because a signal sent from one place to another is a causal chain. This process leads to a coincidence, i.e., a comparison of neighboring events, and from the time measurement thus obtained we can determine the time of the distant event only with the help of an inference. What assumptions are contained in this inference? This inference requires besides the knowledge of the distance also the knowledge of the velocity of the signal. How can this velocity be measured? In principle, there exists only one method, which we shall schematize as follows. The signal leaves a point P i at the time ti and reaches a point Pz at the time U. Its velocity is then given by the quotient of the time interval t2—ti and the distance P2 — P\. Therefore, two time measurements are required which have to be made at different places. We can think of them as given by two clocks located at P\ and P2. If the indication of the time interval t2—ti is to be meaningful, however, the two clocks must have been synchronized previously, i.e., it must have been determined whether their hands occupied the same positions at the same time. In order to measure a velocity, therefore, the simultaneity of distant events must already be known. Is this statement correct? Did not Fizeau measure the velocity of light differently? Fizeau indeed used an arrangement which did not require the simultaneity of distant events. We can schematize his measuring arrangement as follows. In Fig. 16 a light ray is sent from

Fig. 16. Round trip of a light signal.

Chap. 4. Extended Relativity

...

221

A at the time t\ = 12:00; it is reflected at the point B, which is at a distance I from A, and finally returns to A at h = 12:06. It has required 6 minutes to travel twice the distance I, and its velocity is thus given by the ratio of these two numbers. In this arrangement, time is measured only at A, only one clock is used and the simultaneity of distant events does not affect the problem. Of course, in the actual experiment the time interval was much smaller than 6 minutes, even though I was several kilometers long, but Fizeau measured it by an ingenious device involving a rotating gear. We have simply chosen larger numbers to clarify the illustration. On closer examination, we notice that this measurement contains a certain untested assumption, namely, that the velocity of light is the same in both directions along /. For instance, if it were less, in the direction AB than in the direction BA, the velocity of light as calculated by Fizeau would correspond to neither of the two velocities, but represent an average of the two. How can we prove this assumption by Fizeau? It seems that it can be proved only if the time t% is known at which the light ray reaches B. This means, however, that we are again employing two clocks and a comparison of distant events. Our assertion that the measurement of any velocity in one direction presupposes a knowledge of simultaneity is therefore correct. Thus we are faced with a circular argument. To determine the simultaneity of distant events we need to know a velocity, and to measure a velocity we require knowledge of the simultaneity of distant events. The occurrence of this circularity proves that simultaneity is not a matter of knowledge, but of a coordinative definition, since the logical circle shows that a knowledge of simultaneity is impossible in principle. We also notice that the second characteristic of a coordinative definition, namely its arbitrariness, is satisfied. It is arbitrary which time we ascribe to the arrival of the light ray at B. If we assume it to be 12:03, the velocity of light becomes equal in both directions. If we assume it to be 12:02, the light ray requires 2 minutes in one direction and 4 minutes in the other; this assumption is equally compatible with Fizeau's measurements. It does not make sense, therefore, to call the time 12:02 false or improbable, since we are here concerned not with an

222

Lorentz and Poincare Invariance

empirical statement but with a definition. This definition determines at once the velocity of light and simultaneity, and such a determination can therefore never lead to contradictions. If we wish to determine by velocity measurements which events are simultaneous, we shall always obtain that simultaneity which has already been introduced by definition. It is this consideration that teaches us how to understand the definition of simultaneity given by Einstein h=h+\[tz-h) (1) which defines the time of arrival of the light ray at B as the mid-point between the time that the light was sent from A and the time that it returned to A. This definition is essential for the special theory of relativity, but it is not epistemologically necessary. Einstein's definition, too, is just one possible definition. If we were to follow an arbitrary rule restricted only to the form (2) ' 2 = ' i + £('3-'i) 0<*<1 (3) it would likewise be adequate and could not be called false. If the special theory of relativity prefers the first definition, i.e., sets e equal to \, it does so on the ground that this definition leads to simpler relations. It is clear that we are dealing here merely with descriptive simplicity, the nature of which will be explained in § 27. The arbitrariness is restricted only by condition (3) which specifies that h must lie between ti and h; otherwise the signal would arrive at B at a time earlier than its departure from A. The epistemological significance of this restriction will be discussed in detail in § 22. These considerations have shown that simultaneity is a matter of a coordinative definition. Simultaneity also has the peculiar dual character which we can most easily observe in the definition of the unit of length. What we mean by a unit of length can be defined conceptually: a unit of length is a distance with which other distances are compared. Which distance serves as a unit for actual measurements can ultimately be given only by reference to some actual distance. The same is true of simultaneity. We can give a conceptual definition of "simultaneity": two events at distant places are simultaneous if the time scales at the respective places indicate the same time value for these events. What time points of parallel time scales do receive the same time value can ultimately be determined

Chap. 4. Extended Relativity

...

223

only by reference to actual events. This reference is essentially of the form:'' These particular events are to be called simultaneous.'' We say with regard to the measuring rod as well as with regard to simultaneity that only "ultimately" the reference is to be conceived in this form, because we know that by means of the interposition of conceptual relations the reference may be rather remote. We may recall here the example of the determination of a unit of length by reference to a color, mentioned in § 4, where the reference is not directly to a spatial distance. Correspondingly, we find that the reference in the definition of simultaneity is commonly not in terms of the occurrence of arbitrary events, but in terms of light, i.e., a physical process, the properties of which are utilized in the definition of simultaneity. In this fashion we are able to replace a direct reference by a description of operations which can easily be repeated, since it is commonly understood what is meant by " l i g h t " and by these operations. The definition of simultaneity through the use of light signals, for instance Einstein's definition, cannot be compared to the definition of the meter by means of the Parisian standard meter, but is to be compared to the definition of the meter by means of the earth's circumference. In this definition the physical phenomenon the earth's circumference corresponds to the physical phenomenon light in the definition of simultaneity, and the rule "count off 40 million times" corresponds to the rule "send a light signal from A to B and back and set the time of arrival at B equal to the average of the two time values at A." Such a rule does not change the nature of the coordinative definition, since what is meant by "light" and " t h e circumference of the e a r t h " can ultimately be determined only through a direct reference. The conceptual definition which we related to the coordinative definition of simultaneity may appear empty; it is tautological to define simultaneity as the equality of time values on parallel time scales. But the situation is no different for any other conceptual definition. All conceptual definitions are tautological in this sense, since they deal exclusively with analytic relations. A concept is coordinated to a combination of certain other concepts and derives its meaning only from these other concepts. The conceptual definition of the unit of length is also a tautology in this sense. Yet the desire for a different conceptual definition of simultaneity has a certain justification. We

224

Lorentz and Poincare Invariance

mean more when we speak of simultaneity; we are searching for a rule that restricts the determination of the parallel time scales in a special fashion. An answer to this question can only be given by the causal theory of time which we shall develop in § 21 and § 22. We anticipate, however, that this investigation will not eliminate the relativity of simultaneity but only justify the restriction of arbitrariness given in (3).

References Hans Reichenbach, Philosophie der Raum-Zeit-Lehre (Berlin, Walter de Gruyter Company, 1928); The Philosophy of Space and Time (Translated by Maria Reichenbach and John Freund, Dover Publications, 1958) pp.123-129. For a short biography of Hans Reichenbach, see appendix D in this volume.

Reprint with permission. Copyright ©1958 by Maria Reichenbach. All right reserved.

Chap. 4. Extended Relativity ...

>m ^^%^

v

-^>V*^

Hans Reichenbach (1891-1953)

225

226

Lorentz

and Poincare

Invariance

Special Relativity in Anisotropic Space W. F . EDWARDS

Utah State University, Logan, Utah (Received 9 January 1963) An investigation of the propagation of light, assuming absolute spacial distances, the veracity of the principle of causality, and that the average speed of light measured over a closed path is a constant G, shows that the speed of light may have values between e / 2 and is a function of direction. G e n e r a t e d Lorentz transformations are found which are shown to be equivalent to those conventionally adopted. T h a t an absolute simultaneity cannot be established electromagnetically is shown to follow. The arbitrary nature of indirect observables such as time dilation is discussed.

I. INTRODUCTION A

' I HE apparent fact that one cannot unam-*- biguously measure the "one-way" velocity of light between two points allows certain liberties in synchronizing clocks and leads to interesting differences in the formalism of the special theory of relativity. The measurement of the "average" speed of light emitted from point 0, traveling through a distance d to A and then back to 0 can easily be accomplished by means of distance measurements and time measurements using only one clock. This "two-way" speed is, of course, 6. On the other hand, to measure the one-way speed between two distant points the distance traveled by the pulse and the time lapse between its emission at 0 and its arrival at A, must be established. The time measurement involves two clocks which must be synchronized. This synchronization can not be accomplished by sending a light signal from 0 to A for that method would presume knowledge of the quantity whose value is desired. On the other hand, the synchronization might be affected by setting two clocks while at point 0 and then by carrying one to A. It appears that if the effect of the motion upon the translated clock is known, then this method might be feasible. In the past, this method was discarded for lack of this knowledge or for other reasons to be mentioned.

Chap. 4. Extended

Relativity

...

Two schools of thought have arisen in the absence of a one-way speed of light measurement. The attitude of the first group, and one that might have been held by Einstein, 1 is that no observable difference would result if the speed of light really were anisotropic (i.e., had a value other than 6 in some direction).21^ The second holds that relativity theory contains an important assumption that has not and possibly cannot be tested and that, because of this, the foundations of relativity are uncertain. 4 Some suggest that one must "take on faith" Einstein's simultaneity ("We establish by definition that the 'time' required by light to travel from A to B equals the 'time' it requires to travel from B to A.").1 Even recently, serious suggestions have been made of experiments futilely designed to measure the one-way velocity of light.4

•^j)i

rt,M

Concerning the necessity of taking on faith the isotropy of space, may I preliminarily remark that if the adoption of one definition of simultaneity (or, which is equivalent, the one-way speed of light) could in any way lead to different observable results than those predicted by the adoption of another, then a test of the isotropy of space would exist. If, however, no experiment can be devised to make this test, then surely it makes no difference what one assumes about the simultaneity of clocks so long as the two-way speed of light equals S. This study investigates the consequences in the special theory of relativity of adopting definitions of simultaneity other than that used by Einstein. The following facts are found: (1) Certain restrictions are placed upon clock synchronization within a single coordinate system ' A . Einstein, Ann. Physik 17, 891 (1905). 2 H. Reichenbach, The Philosophy of Space and Time (Dover Publications, Inc., New York, 1958), p. 124. 3 A. Grunbaum, "Logical and Philosophical Foundations of the Special Theory of Relativity," in Philosophy of Science, edited by A. Danto and S. Morgenbesser (Meridian Books, New York, 1960), Sec. 2. 4 M. Ruderfer, Proc. I R E 48, 1661 (1960); also see, P. M. Rapier, Proc. I R E 49, 1322 (1961).

227

In variance which have apparently been overlooked. (2) There are an infinite number of correct Lorentz transformations corresponding with the infinity of possible clock synchronizations. (3) These new transformations give the same results when applied to observable effects. (4) An example illustrates the fact that one cannot discover the actual properties of light propagation or establish an absolute simultaneity by moving a master clock about, for clocks would change in such a way as to frustrate any attempts. Finally, (5) using electromagnetic means alone, the measurement of the one-way speed of light or the establishment of an absolute simultaneity cannot be accomplished. II. DEFINITIONS

To obtain Lorentz transformations when other than isotropic space is assumed, we must first inquire as to the restrictions which limit the various possible choices of clock synchronization. Let us imagine that at each point within the coordinate system resides a clock at rest. The clocks are synchronized using light signals. It is assumed that an absolute measure of spatial distance using "rigid" rods exists. Any problems entailed by this assumption are neglected. The time at any point in space is simply the reading of the clock at that point. The time lapse between events at any two points is defined as the difference between the readings of the clocks at those two points. The speed of propagation of any causal signal producing events at two points is clearly the distance between the points divided by the elapsed time. The two-way speed of a light pulse traveling from point 0 to point A and by reflection, back to 0, is an observable quantity that does not depend upon the definition of simultaneity for only the clock at 0 need be used. Using the definition of time lapse, the following is seen to be an identity: toAO — toA~{~tAOt

(1)

Chap. 4. Extended

Relativity

...

229

where the subscripts indicate the path of the light signal and the first and last subscripts identify the clocks that are read when the signal reaches those points. Using the definition of speed and letting S z.

/ y\—y^y

IX

y

J/

X FlG. 1. The "circulation" path of a light signal emanating from the origin and traversing path OABCO by reflection from mirrors. It is assumed that an experiment would show that the time taken in traversing such a path would give the value 6 for the average speed. This thought experiment is used to find the velocity of light as a function of direction.

indicate the two-way speed of light, we find that 2x/Q

= (X/COA) +

(X/CAO)

or (1/C0A)

+ (1/CAO)=2/G.

(2)

The choices of C0A and cAo are restricted in such a w a y that the sense of "cause" is preserved. In other words, a light signal starting at 0 cannot reach A before it leaves 0. Since t0A must be positive for such signally connected events, so must Co A be positive; and since this also holds for CAO, we see from Eq. (2) that neither can be smaller than 6/2. Therefore, the speed of light in any direction indicated by the direction cosines a, /?, and y has the restriction e/2
(3)

It should be clear that the choice of synchronization of clock A with respect to clock 0 determines the value of COA, or vice versa, and cAo follows from Eq. (2). Thus, we can use the ideas of clock synchronization or one-way speed of light interchangeably.

230 Lorentz and Poincare Invariance Further limitations on the arbitrariness of clock synchronization are found by imagining an experiment where a light signal from 0 reflects from mirrors appropriately placed so that the path is OABCO, as in Fig. 1. One can easily imagine light traversing each of the straight paths in such a way that the two-way speed for each path is equal to 6 but the "circulation" speed, COABCO, is equal to °°. This would occur if

COA — CAB = CCO= °°

and

COC = CCB = CBA — CAO

= 6/2, so that conditions (2) and (3) are met.

&\y&

but a value for the speed of light other than e would be measured using only one clock. Although an explicit measurement of this circulation speed has not been made, I anticipate the result and assume the value of Q for the remainder of this paper. III. DEPENDENCE OF SPEED UPON DIRECTION

To find the speed of light as a function of direction, we begin by writing the identity t0ABC0 = t0A~\~tA.BJrtBC~\-tc0,

(4)

from which we obtain, by assuming the value Q for the circulation speed,

x+y+z+ixt+yt+z^

e

x

y

z

- =—+—+— Cx

Cy

CZ

(*J+y*+z*)» c(-

-P,

(5)

-7)

where cx, cy, and cz are the speeds in the directions of the positive x, y, and z axes, respectively [i.e., Ci = c(l,0,0), etc.], and c( — a, —/3, —7) is the speed in the direction of the negative radius vector from the origin. For the later derivations, it is convenient to alter the form of Eq. (5). Using the definitions of the angle cosines and referring to Eq. (2) to establish that lA(a,/3,7) = ( 2 / e ) - [ l A ( - a , - 0 , - 7 ) ] , Eq. (5) becomes

Chap. 4. Extended

Relativity

...

231

e c(a,j8,7)

= a(l cj

)+j8(l \

)+7(l c„/ \

)•

(6)

cz

R(«^,*)

FIG. 2. The relationship between the speed of light and the corresponding speed variable.

We now introduce variables": X=

the

following

"speed

l-e/cx,

K=l-e/c„ z=i-e/c„ i?=l-e/c(a,/3,7).

(7)

Note that as the speed of light in a particular direction varies from 6/2 to «> the corresponding speed variable varies from —1 to + 1 , as shown in Fig. 2. The restriction indicated by Eq. (3) therefore becomes -1
(8)

The speed in the opposite direction may be found simply by reversing the sign on the speed variable as can be seen by combining Eqs. (7) and (2). Equation (6), giving the speed of light as a function of direction, now becomes R(0t,P,y)=aX

+ l3Y+yZ.

(9)

This result shows that, in general, the speed of light in any direction is determined when the speed in the direction of the x, y, and z axes are given. Furthermore, any clock synchronization leading to light speeds which are functions of

Invariance

direction and which do not agree with Eq. (9) contradicts the principle of causality earlier mentioned. 6 IV. TYPES OF SPACE We now investigate the type of "space" one obtains if clocks along the z axis are synchronized so as to make c z =oo. Letting Z = l , Eq. (9) becomes R = aX+pY+y.

(10)

The arbitrariness that generally exists in X and Y is not found in this special case as is shown by the following argument: Let ,3 = 0 so that we are restricted to the xz plane. Where 8 is the polar angle, we have 7 = cos0, a = sin0, and R = Xsind+cos8.

(11)

We now find that X is limited to a single value by requiring that R lie within the range —1 to 1. The extreme values of R are obtained by taking the 8 derivative of Eq. (11), setting it equal to zero and solving for X, which results in X = tan8m,

(12)

where 8m is the value of 9 corresponding with an extreme value of R. Substituting this back in Eq. (11) we obtain 2?w = l/cos0„.

(13)

To keep Rm within the allowable range, the only values of 8m that may be admitted are 0 or x. In either case, we find from Eq. (12) that X = 0 or C i = 6 . By symmetry we see also that F = 0 or c„ = S. Using the above values for X and F, Eq. (10) becomes 6 Reichenbach used such a contradictory synchronization, though he apparently wished to retain the principle of causality. (See p. 162, Ref. 2.) If the value of his parameter "a" is taken so that the speed of light in the positive x direction is J e then, for instance, the value of the speed of light of a signal emanating from the negative x axis and traveling parallel to the y axis to the point equidistant from the two axes is —3.16, a negative value!

Chap. 4. Extended Relativity

R = y=cosO,

...

233

(14)

giving

c(9) = e / ( l - c o s 0 ) . The conclusions are that in this space where cz— oo we must have cz = cy = Q and the speed of light in any direction is a function of the polar angle only. If the speed of light in one direction is infinite, then the speed in every other direction is determined. With this definition of simultaneity, one would picture a wavefront of light emitted by a point source as a paraboloid of revolution as in Fig. 3. It is interesting that Huyghen's construction appears to be as applicable in this case as in the case of spherical fronts emanating from point sources. V. LORENTZ TRANSFORMATIONS IN ANISOTROPIC SPACE

For the work that follows, the relationship between time measurements in isotropic and anisotropic space is needed. Let the coordinate variables without subscripts indicate general space, while those with subscript i refer to isotropic space. The origins of the coordinate systems representing these spaces are coincident, the axes are aligned and the systems are at rest

FIG. 3. The form of a wavefront emanating from a point source situated at the origin if cI = cy=
with respect to one another. The relationship between the readings of clocks at a position

Invariance indicated by radius vector r, whose direction cosines are a, /?, and y is <(r)=<,-(r)-r{(l/e)-[l/c(a,a,7)]},

(15)

or t=*U-(r/e)R.

(16)

Using Eq. (16) the relationship between instantaneous velocities is dx

dti

dt

dt

V=-=V—

r \e

c(a,$,y)J r

c2(«-/3,7)

VrVc(a,p,y)

"I"1 .

(17)

J

To derive the generalized Lorentz transformations, Einstein's second postulate is modified to read: the two-way speed of light in a vacuum as measured in two coordinate systems moving with constant relative velocity is the same constant regardless of any assumptions concerning the oneway speed. The equation of the wavefront emanating from a point source of light situated at the origin is r = c(a,p,7)t. (18) Using the fact that a = x/r, p = y/r, and y = z/r, and applying Eq. (7), Eq. (9) takes the form r/c(a,0,y)=(l/e)lr-

(xX+yY+zZ)l,

(19)

so that Eq. (18) becomes r = xX+yY+zZ+et.

(20)

This new form for the equation of the wavefront may be used to find the Lorentz transformations between any two systems. For simplicity we examine the case where symmetry about the x axis exists; and cv = cz=Q, so that F = Z = 0. Thus, Eq. (20) may be written r = xX+Qt.

(21)

Chap. 4. Extended Relativity ...

235

The prime system S' is moving with constant velocity v relative to and as measured in the unprimed system. This motion is in the direction of the positive x axis. A different simultaneity is allowed for in the 6" system as indicated by and X' which may be different from X. r' = x'X' + et'.

(22)

Writing Eq. (21) in Cartesian coordinates and squaring both sides leads to the form x2(l-X2)+y2+z2-QH2-2eXxt

= 0.

(23)

A similar equation must hold in the primed system. As in standard derivations of the Lorentz transformations, we choose the following equations with A, B, and D to be determined. x'=D(x-vt),

(24a)

y'=y,

(24b)

z' = z,

(24c)

f=Al+Bx.

(24d)

Substituting these in the primed version of Eq. (23), and setting the left-hand side of the resulting equation equal to the left-hand side of Eq. (23), collecting terms and setting the coefficients of the coordinate variables equal to zero, we obtain the following set of equations e2B2+2eX'BD-(l-X'i)D2+l-X2 &A B + QX'AD 2 2

i

= 0,

eX'vBD +v(l-X")Di~QX 1

i

2

Q A -2eX'vAD-v (l-X' )D -e

(25a)

= 0, (25b) = 0.

(25c)

The solution to these equations follows:

A = r,li+(v/e)(X'+xn B = (71/e)l(v/e)(x*-i)+x-x'l, D =v

(26a) (26b) (26c)

where v={ll

+ (vX/e)J-v2/&)-K

(27)

Combining Eqs. (26) with Eqs. (24) we have

the general Lorentz transformations in a space where clocks along the y and z axes are set so as to make cv = c,= 6 in both systems and c% and cx> are undetermined. x' = v(x-vt),

(28a)

y-y,

(28b)

z' = z,

(28c)

t'=v{[i+(v/e){x+x')-]t +t(v/e)(X*-i)+x-x'l(x/e)}.

(28d)

Let us now examine a few limiting cases of the above transformations. First, letting X' = X = 0 we obtain the traditional Lorentz transformations for isotropic space as is easily verified. The equations in spaces where cx = cx> = °° , or X = X' = \ are interesting. In this case Eqs. (27) and (28) give x'=[l/(l + 2iy'e)i](x-0.

(29a)

y' = y,

(29b)

z' = z,

(29c)

*'=(l+2»/e)**.

(29d)

The fact that the transformation of time in these two systems does not involve a spatial coordinate, is particularly interesting. If we set v = 0, then Eq. (28d) reduces to a form of the time transformation, t' = t+ (X+X') (x/ e) [similar to Eq. (16)], which suggests that one can obtain the generalized transformation equations simply by changing the time and velocity terms in the Lorentz transformations for isotropic space according to Eqs. (16) and (17). This, in turn, implies that the only real difference in the forms of the Lorentz transformations is in how one wishes to synchronize clocks. This is indeed the case. VI. THE IDENTICAL NATURE OF THE TRANSFORMATIONS

Using the subscript i again to indicate a quantity in isotropic space, Eqs. (28) are now shown to reduce to the familiar transformations in all cases.

Chap. 4. Extended Relativity ...

237

Using Eq. (17) we find the velocity of the S' system as measured in the 5 system (which assumes cy = cz = (5) to be v=Vi/ii-(viX/e)2.

(30)

From Eq. (16) the transformations of the time coordinates are clearly t' =

ti'-x'X/Q,

t = U-xX/e.

(31)

When these equations are substituted in the transformation Eqs. (28) the familiar Lorentz transformation equations result: 3c' = [ l / ( l - o i 1 , / e i ) * ] ( * - o i 0 .

(32a)

y'=y,

(32b)

z' = z,

(32c)

*'=[i/(i-».7e*)*]('—We*)-

(32d)

It should therefore be clear that, as far as special relativity is concerned, the same experimental results are obtained by assuming isotropic space regardless of what function of direction the speed of light "really" is. If one were able to establish absolute simultaneity using some signal having an infinite two-way speed, then presumably one particular form of Eq. (9) would give the actual manner in which light propagates. But, as we have seen, within relativity, this form would give the same results as would that which simply presumes that space is isotropic, or, for that matter, as would any other presumption concerning the speed of light, even if space is not in actuality so constituted. The equation one uses or the manner in which one sets his clocks is simply a matter of convenience. VII. ON THE MOTION OF CLOCKS

As mentioned in the Introduction, one method that comes immediately to mind, for the absolute synchronization of clocks in a coordinate system, is to set two clocks at the same point in space and then to carry one to all other points in the coordinate system to synchronize the rest of the clocks. The question that is raised is the

Invariance

effect of the motion upon the transported clock. In Reichenbach's excellent book on space and time, this clock transport method is discussed.6 He concludes that the method must fail if the time of the master clock cannot be shown to be independent of path and velocity. I feel, however, that this very motion effect could conceivably be the means to establish absolute simultaneity. If, for instance, it could be shown that, for a particular type of motion, the effect would cause the transported clock (after being synchronized at M) to differ with a clock at point A (that was set by 0 using a light signal after arriving at that point) by an amount that was a function of the true nature of light velocities as predicted by the theory of relativity and a function of the type of motion, then the observed value would determine the absolute simultaneity, at least relative to the whole experimental basis of relativity. The type of motion that is adopted in discussing this question, is as follows.7 The moving clock M, at rest in the S' system is given a constant velocity such that it passes point 0 and later point A. Clocks 0 and A are at rest in the S system. All clocks are on the x axis. In principle, when the clock is at the origin, it can be synchronized with the clock at rest at that point with a vanishingly small error. Later, the clock passes point A where clock M is read. The velocity of M is determined by clock 0 and A, the latter having been set using light signals. As in Sec. V, the types of space are limited to those symmetric about the x axis, with F = Z = 0 in all coordinate systems considered. We inquire as to the difference At' in the reading of the S' clock at point A and at point 0, the distance between being Ax. Note that At' is a directly observable quantity since it involves only one clock. If At' does not remain invariant under 8

H. Reichenbach, Ref. 2, Sec. 20. One may wish to consider the case where clock M is initially a t rest at point O, accelerates in a negligible time to velocity v, then travels with this velocity till it arrives "at A where it accelerates, again in a negligible time, to rest. The result of general relativity that the time change of clock M during these accelerations approaches zero as the time of acceleration approaches zero reduces this problem to the one considered in the text. 7

Chap. 4. Extended Relativity ...

239

changes in the nature of space, given by X and X', for a constant velocity between the two points as determined by the clocks in the 5 system, then a method would exist to establish absolute simultaneity. Using Eq. (28d) to find the time lapse At' in terms of At and Ax, we have At' = v{£l + (v/Q)(X+X')lAt + t(v/e)(X*-l)+X-X'-](Ax/e)).

(33)

Since Ax' = 0, we have from Eq. (28a) Ax = vAt. Substituting this in Eq. (33) and using Eq. (27) to eliminate -q results in At' = i(l + (vX/e)i-(v2/e*)yAt

= At/r,.

(34)

To see whether or not At' changes when under a different clock synchronization, we transform At and v/Q according to Eqs. (16) and (17). Generalizing Eq. (16) to give the relationship between time measured at different positions under two arbitrary clock synchronizations and limiting the positions to the x axis gives At = Ata+(Ax/e)(Xa-X),

(35)

where subscript "a" is used to indicate quantities in the new system with clock synchronization relating to the actual universe. The corresponding form of the velocity transformation is v = va/Zl+va/e(Xa-X')J

(36)

Substituting for At and v in Eq. (34) and using the fact that Ax in Eq. (35) equals v,Ata, we obtain At' = {11+ (vaXa/Q)J-

(va*/&)))Atat

(37)

which, by comparison with the form of Eq. (34), clearly yields the result At' = Ata'.

(38)

We conclude that if one chose a clock synchronization different from nature's intrinsic one, no observable difference would result. Thus a clock that is moved to all positions in a coordinate system to set the other clocks does not establish an absolute synchronization.

240 Lorentz and Poincare Invariance VIII. THE RELATIVITY OF SIMULTANEITY

5^> ™FV-V

To mention the fact that two spatially noncopunctal events simultaneous in one system are not simultaneous in a system moving with constant relative velocity is now standard in textbooks. Now, however, it is easy to show that with a proper setting of clocks in the S' system, two events simultaneous in one system can always be made simultaneous in a second regardless of the value of the relative velocity, and the setting of clocks in the first system. This fact, which was previously discussed by Grunbaum, 8 can easily be proved in the case where Y=Z = 0. If simultaneity exists in the S system so At = 0, Eq. (33) becomes

At' = vt(v/e)(X*-l)+X-X'l(Ax/e).

(39)

Setting S' clocks so that

^ & \ ^

T

' -

X'=(v/e)(X*-l)+X

(40)

clearly gives the result, At' = Q and the events are simultaneous in the second system. By a simple analysis of Eq. (40) it can be shown that X' is between — 1 and + 1 for any allowed values of X and V. An interesting result of choosing X' so that events simultaneous in the 5 system are also simultaneous in the S' system, is that the transformation of the time difference At' between any two events separated by Ax and At in the 5 system is not a function of the spacial separation Ax, but is given simply by At'= At/r)

(41)

as is easily verified. This is the same as Eq. (34) but the former equation applied only to clocks at rest in the two systems so that Ax was equal to zero. We thus can have the same time dilation regardless of the spacial separation of the two events in the 5 system. Figure (4) illustrates the effect of assuming different values of the velocity of light upon the concepts of past, present, and future. Assume 8

A. Grunbaum, Ref. 3, p. 410.

Chap. 4. Extended Relativity ... two events, one occurring at the origin and the other at any point in hyperspace. If the second is above the x axis it is definitionally future (definitional present is abbreviated d.p.) but not necessarily causally future because there is some question as to whether or not a signal having velocity less than that of light could connect the two event points. If isotropic space is assumed, then the causally nondetermined region is as occurs in the figures between the causal future and causal past. A light cone gives the separation between these regions. Figures 4(b) and 4(c) show the situation under different definitions of simultaneity. Note that an event just above the positive x axis is causally and definitionally future when c x = °°. One can see how the concepts of past, present, and future are altered. In Fig. 4(a), lines representing the definitional present, corresponding to those in 4(b) and 4(c) but transformed to the new time system ta, are shown. IX. DIRECT AND INDIRECT OBSERVABLES

Since one form of the Lorentz transformation is always equivalent to all other forms obtained by changing the definition of simultaneity, any direct observable (such as two readings on a single clock) could not be changed by either a physical or a mathematical thought alteration of space time. But such quantities as time lapse involving two clocks separated in distance with respect to the observer, are not directly observable and involve the definition of simultaneity which is, to some extent, arbitrary. Even the time dilation formulas or the concepts, past, present, and future do not involve direct observables. Some of the mathematical quantities are directly observable and others are not. Nevertheless, the application of the mathematical quantities to physical experiments result in quantities which are directly observable and these quantities are invariant under any change of clock reading or the manner in which light "really" travels. Such quantities as the standing-

241

In variance et

ct

ct

FIG. 4. Showing light cones in hyperspace under three definitions of simultaneity. In the figures d.p. means definitional present. The figures are discussed in the text.

wave pattern in a waveguide, the diffraction pattern in a slit experiment, or the detector frequency in the observation of radiation emitted by a moving source are completely independent of how fast light really travels in one direction or how one sets his clocks. One might well argue about the reality of time dilation, or the relativity of simultaneity or other indirect observables, but in the final analysis, the argument is academic unless a two-way signal velocity greater than Q is discovered outside of electromagnetism. As far as any measurement is concerned, one can adopt any view he wishes consistent with the fact that the circulation speed of light is <3 (if, indeed, it is). For most problems the most convenient assumption to make is still that of isotropic space. ACKNOWLEDGMENTS

I wish to thank William E. Dibble for many interesting discussions and Akeley Miller for critically reading the manuscript.

Chap. 4. Extended Relativity ...

243

Four-dimensional symmetry of taiji relativity and coordinate transformations based on a weaker postulate for the speed of light. - I LEONARDO HSUC 1 ), JONG-PING HSU( 2 ) and

DOMINIK A. SCHNEBLE( 2 )

C) Physics Department, University of California at Berkeley - Berkeley, CA 91*720, USA (2) Physics Department, University of Massachusetts Dartmouth North Dartmouth, MA 0271*7, USA (ricevuto il 4 Aprile 1995; approvato il 2 Aprile 1996)

Summary. — The power of the four-dimensional symmetry of taiji relativity is demonstrated through an analysis of Edwards' transformation based on the weaker postulate that only the two-way .speed of light is a universal constant. His transformation, involving Reichenbach's time, is shown to be inconsistent with Lorentz-group properties and the relativistic energy-momentum expression for particles. However, using the symmetry of taiji relativity as a guiding principle, we can obtain a new four-dimensional transformation which does not have these difficulties. We show that Reichenbach's concept of time is compatible with the four-dimensional symmetry of physical laws. No known experiments can distinguish physical laws in this four-dimensional formalism («extended relativity") from those in special relativity. PACS 03.30 - Special relativity. PACS 11.30.Cp - Lorentz and Poincare invariance.

Only at the beginning of the twentieth century, after the creation of the space-time four-dimensional symmetry of special relativity, was it recognized that the concept of symmetry played an important role in physics [1]. Subsequently, all basic symmetries, such as space-time symmetry, right-left symmetry, time-reversal symmetry etc., have been taken for granted. In 1957, however, when right-left symmetry was found to be violated in weak interactions, it was a shock to the physics community. Nowadays, many symmetries, such as CPT symmetry and isotropy of space in relation to the speed of light, are still under experimental investigation. In view of the ultraviolet divergence difficulties in quantum field theory, one may wonder whether the four-dimensional symmetry still holds at extremely high energies. Recently, Devlin has informed us that this symmetry has been tested and confirmed by measuring the decay lifetime of A;s° in flight at several hundred GeV [2]. The four-dimensional symmetry is one of the most thoroughly tested symmetry

244 Lorentz and Poincare Invariance principles in physics. In the beginning of his book Invariance Principles and Elementary Particles [3], Sakurai quoted from C. N. Yang's Nobel Lecture: «Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.» This quotation summarizes the essence of symmetry in physics, which will be illustrated below by a non-trivial analysis of different viewpoints of the physical world to show how four-dimensional symmetry is critical to any theory. In this paper, we would like to demonstrate the importance of using four-dimensional symmetry as a guiding principle for discussing physical laws from different viewpoints of time and space. We first analyze Edwards' original attempt in 1963 to formulate a relativity theory based on a weaker postulate for the speed of light [4], namely that «the two-way speed of light in a vacuum as measured in two coordinate systems moving with constant relative velocity is the same constant regardless of any assumptions concerning the one-way speed» [5]. He derived space and time transformations involving infinitely many possible physical times which can be physically realized through Reichenbach's convention of clock synchronization and which include relativistic time as a special case [4]. Edward's transformations were shown to be consistent with many experiments related to the propagation of light. Nevertheless, we show that they do not form the Lorentz group so that, in general, physical laws are not invariant under his transformation. Moreover, it also leads to an incorrect expression for the relativistic energy-momentum of a particle. In view of these results, one may conclude that assuming the universality of the 2-way speed of light and Reichenbach's concept of time is wrong. However, that is not the case. We show that, using symmetry, Reichenbach's general convention of time can be accommodated in a new four-dimensional formalism which is consistent with the relativistic energy-momentum of a particle and the Lorentz group. Such a four-dimensional formalism may be termed «extended relativity» which includes special relativity as a special case. These results are physically and pedagogically interesting for the following reasons: first, as early as 1910 several physicists, including Ritz, Tolman, Kunz and Comstock, raised the question of whether it was possible to construct a relativity theory without postulating the constancy of the (one-way) speed of light [6]. They all failed because the importance and the general structure of the fundamental four-dimensional symmetry were not recognized [5,7]. Now, we are able to answer their question affirmatively. Second, people are usually puzzled by discussions of Reichenbach's convention of time and the impossibility of the unambiguous measurement or test of the one-way speed of light in the literature [4,5]. They ask: Can Reichenbach's convention of time be consistent with the Lorentz-group properties of transformation between two inertial frames? Are the ideas of Reichenbach and Edwards viable? Our results suggest that the key to answering all these questions is four-dimensional symmetry. The physical theory of taiji relativity based solely on the first postulate of relativity, i.e. the principle of relativity for physical laws, has been discussed recently [8]. Such a theory has a four-dimensional coordinate transformation between any two inertial frames, F(w, x, y, z) and F'(w', x', y', z'). The absence of a second postulate in taiji relativity makes it

Chap. 4. Extended Relativity ...

245

impossible to factor the inherent evolution variable w, called «taiji-time», into a well-defined velocity and time as in special relativity. As a result, time and speed of light are undefined in taiji relativity. Nevertheless, the four-dimensional taiji relativity is consistent with all known experiments [7,8]. We show that Edwards' weaker postulate for the speed of light is a further restriction of taiji relativity, just like the second postulate of special relativity. Thus, extended relativity discussed in this paper gives a more restricted view of the four-dimensional physical world than that of taiji relativity. It also shows the power of the first postulate of relativity from the vantage point of four-dimensional symmetry. 2. - Edwards' transformation with Reichenbach's time Let us consider two inertial frames F and F', where F' is moving along the x-axis. Suppose there are two identical clocks, clock 1 located at the origin of the F frame and clock 2 at point x on the x-axis. A light signal starts from the origin (event 1) at time ti, it reaches clock 2 (event 2) at time i2 and returns to the origin (event 3) at time t s . Reichenbach's concept of time can be realized by synchronizing clock 2 to read t2 by the relation [4] (2.1)

*2 = t1 + e[«3-* 1 ],

where e is restricted by 0 < £ < 1, so that causality is preserved, e.g., t2 cannot be earlier than i : . The same synchronization can be done to clocks in the F' frame: (2.2)

*2=«i+e'[*s'-*i].

The special case £ = £' = 1/2 corresponds to the conventional synchronization of special relativity. One can easily verify that the two-way speed c2w of light along the x-axis in F is given by (2.3)

c2w =

2Lj\t%-tx\,

which is independent of t2 and Reichenbach's parameter s. Similarly, in the F' frame, we also have a constant two-way speed of light, (2.4)

c2'w, = 2 L 7 [ i 3 ' - i i ' ] .

Following Edwards, we shall assume these two-way speeds of light to be the same, (2.5)

c 2 ' w = c2w = c .

The synchronization of clocks in F and F' according to (2.1)-(2.5) defines the Reichenbach times which include relativistic time as a special case [4]. In special relativity, since an Einstein clock at x is synchronized according to tE = = t0 + x/c (where t0 is the time of the clock at the origin of F) and the corresponding Reichenbach clock at x reads tR = t0 + 2ex/c2w = t, which follows from (2.1), (2.4) and (2.5), tE and t are related by (2.6)

tE = t-(2E-l)x/c

= t-qx/c,

Similarly, in the F' frame, we have the relation (2.7)

ti=t'

-q'x'/c,

q'=2e'-l.

q=

2e-l.

246 Lorentz and Poincaie Invariance For simplicity and without loss of generality, we set q = 0 in (2.6), so that physics in the F frame is identical to that in special relativity. We concentrate on physical implications of (2.7) in the F' frame to see whether Edwards' transformation contradicts experiments. Using (2.6) with q = 0, (2.7) and the Lorentz transformation involving Einstein time tE and ££> o n e c a n obtain the coordinate transformation between inertial frames F and F' [4],

\p = V/c,

y = l/(l-02)1/2,

where V is the speed of F' as seen from F. This type of transformation with Reichenbach's time was first derived and discussed by Edwards [4]. It is important to note that A) Space-time (cf , x', y', z') is no longer a four-vector under the transformation (2.8). B) As V—»0, F and F' become the same inertial frame, but (2.8) does not reduce to the identity transformation (2.9)

t' = {t- q'x),

x'=x,

y'=y,

z'=z.

This shows that Edwards' transformations do not form the Lorentz group, except for the special case q' = 0. O The «invariant four-dimensional interval» under Edwards' transformation is (2.10)

c2t'2 + 2q'x'ct'-x'2(l-q'2)-y'2-z'2

= c2t2-x2-y2-z2

,

q =0.

We may remark that if q * 0, then c2t2 - x2 - y2 - z2 will be replaced by c2t2 + + 2qxct - x2(l — q2) - y2 - z2 At can be shown that this quadratic form holds also for infinitesimal intervals.

3. - Difficulties of Edwards' transformation The transformation (2.8) was shown to be consistent with many experiments related to the propagation of light. Moreover, since Edwards' transformation can be obtained by a «change of time variables", (2.7), in the Lorentz transformation, one might think that it is equivalent to the Lorentz transformation [4]. However, we show that this is not the case because Edwards' transformation (2.8) does not have the Lorentz-group properties in general due to the property of t'. Thus, Edwards' transformation (2.8), as it stands, violates four-dimensional symmetry. As a result, one can show explicitly (see (3.4) below) that the time t' leads to an incorrect expression for the relativistic energy-momentum of a particle, which contradicts experiments in general, except for the special case q'=0. (The correct transformation takes the form (4.3) rather than (2.8), as we shall see later.)

Chap. 4. Extended Relativity ...

247

To wit, let us consider Edwards' transformation (2.8), where we have chosen a frame F in which the Reichenbach time equals the Einstein time by setting q = 0. We stress that this frame can be chosen arbitrarily. (If one wishes, one can set q' = 0 instead of q = 0. The following arguments still hold with F and F' interchanged.) For a free particle, we have the actions S and S' in F and F', respectively: ds 2 = c2dt2 - dx2 - dy2 - dz 2 ,

S= ~jmeda=JLdt, (3.1) L= -mc2(l-i;2/c2)1/2,

inF(q = 0),

S' = - Jracds' = \h' dt' , (3.2)

ds' 2 = c2dt'2 + 2 g ' d x ' c d f - d z ' 2 ( l - g' 2 ) - dy'2 L ' ( E ) = -mc2[(l

- q'vi/cf

- v'2/c2)1/2

,

dz'2,

in F ' ( l > 9' > - 1 ) .

Note that S = S' because ds = ds' which can be verified by taking differentials of (2.8). The constant c in (3.2) is the universal 2-way speed of light measured in F'. The Lagrangians L and L' lead to the following momenta in F and F', respectively: (3.3)

px(E) = dL/dvx = mvx/(l-v2/c2)1/2

(3.4)

px(E) = (mcq'-mq'v±+mvx)/[(l-q'vx/c)2-v'2/c2]1/2,

= px,

etc. in F; etc. i n * " ,

according to Edwards' formalism. We see clearly that the momentum px (E) in F' is in contradiction to experimental results, except for the special case q' = 0. Similarly, one can show that the result for the energy, defined by p'(E)-v' - L', in F' is also incorrect. Before dismissing the expression (3.4) as incorrect, one should see whether (3.4) is the same as the result of changing time variable of the relativistic momentum px (SR) of special relativity in the F' frame, (3.5)

px (SR) = mux / ( l - u'2lc2)1'2

,

ux = dx'/dt& ,

Under a change of time variable (2.7), we have dx'/dt' ie. (3.6)

etc. in F' .

= (dx'/dti)(dtz/dt'),

etc.,

u'=v'/{l-q'vi/c).

And from (3.5) and (3.6), we have (3.7)

p,'(SR) = m ^ / [ ( l - q'vx/cf

- v'2/c2?'2

*

px{E),

which differs from the momentum px (E) in (3.4). This example shows that the lack of four-dimensional symmetry in Edwards' formalism based on transformation (2.8) with 1 > q' > - 1 makes it untenable. Of course, there are many other difficulties related to the lack of symmetry, such as the non-invariance of the Maxwell equations, the Klein-Gordon equation, laws in quantum electrodynamics, etc. under transformation (2.8). In the next section, we show that if one uses four-dimensional symmetry of taiji relativity as a guiding principle, all these difficulties can be resolved.

248 Lorentz and Poincare Invariance 4. - Extended relativity—a four-dimensional theory with Reichenbach's time The method for the construction of a four-dimensional symmetry framework without necessarily having the usual relativistic time has been discussed before [5,7]. The logically simplest case is the four-dimensional symmetry of taiji relativity which is based solely on the first postulate of relativity [8,7]. It can be applied to guide the construction of the four-dimensional framework for the present case with Reichenbach's time. For simplicity and without loss of generality, we choose q = 0 in synchronizing clocks in the F frame, so that we have the usual relativistic time t = tE. In the F' frame, clocks are synchronized to read Reichenbach's time V. An event is, as usual, denoted by (ct, x, y, z) in F. However, following taiji relativity [8], the same event must be denoted by (b't',x',y',z') in F' within the four-dimensional symmetry framework. We stress that it is necessary to introduce the function V so that (b' t', x', y', z') = (w' ,r') transforms like a four-vector and the laws of physics can display four-dimensional symmetry. Otherwise, laws of physics cannot be invariant under the transformation from F to F'. With synchronization of clocks in F and F' as discussed previously, the times t and t' are related as in (2.8): (4.1)

t'=y[(l-Pq')t-(P-q')z/c],

q'=2e'-l.

This may be considered as an assumption. Indeed, from the viewpoint of taiji relativity [8], this is the second postulate which is the same as assuming the universality of the two-way speed of light over a closed path in any inertial frame, as discussed in sect. 2. As usual, we start with the invariance of the fourdimensional interval s 2 , (4.2)

s2 = b'2t'2 - x'2 -y'2-z,2

= c2t2 - x2 - y2 - z2,

to derive the four-dimensional transformation. We may remark that the relation (4.2) or [c2t2 — x2 - y2 - z2]/s2 = 1, s2 > 0, is nothing but the law of motion (i.e. the energy-momentum relation, p2 - p2 = m2c2), for a free particle with a mass m > 0. (This equivalence can be seen more easily by using the infinitesimal form ds 2 in both F and F' frames. See also sect. 6 below.) By the usual method for deriving the Lorentz transformation or the taiji transformation [8], we can derive the extended four-dimensional transformation involving Reichenbach's time, \ w' = b' V = y(ct (43)

\fi = Vlc,

fix),

x' = y(x-fict),

y'=y,

z'-

z\

2 1 2

y = l/(l-/3 ) / ,

where t' is given by (4.1). This extended transformation completely determines the function b': (4.4)

b' = (ct - fix)/[t(l -pq')-(fi-

q')x/c],

i.e. if x and t are known, we can always calculate the value of b' in (4.3) and (4.4). Note that the fourth dimension in F' is now b't' = w' rather than t' or ct', where w' may be called «lightime». Although the function V and the time t' separately have complicated transformation properties, we stress that b' and t' are separately well defined, in sharp contrast to b' and t' in taiji relativity,

Chap. 4. Extended Relativity ... 249 and that the product b' t' = w' transforms as the fourth component of the coordinate four-vector. The property of V is completely determined by (4.4) or the transformation (4.3). From now on we use w' = b' t' to display four-dimensional symmetry of physical laws. The extended transformation of velocities can be derived from (4.3): ' c' =d(b't')/dt' (4.5)

| vx =

=

[dt/dt']y(c-0vx),

(dt/dt')y(vt-0c),

Vy = (dt/dt ')vv,

vz' = (dt/dt ')vz,

where dt/dt' may be obtained from differentiation of (4.1), (4.6)

l/[dt/dt']

= dt'/dt =

y{[l-Pq']-[p-q'][dx/dt]/c}.

We stress that the definition c' = d(b't')/dt' is quite natural because [d(b't')f - dr' 2 = 0 is to be interpreted as the law of light propagation, c' 2 d£' 2 - dr' 2 = 0. Note that the transformation of the ratios v'/c' is precisely the same as those in special relativity (4.7)

vi/c'=(vx/c-p)/(l-vxp/c);

v^/c' = (vy/c)/[y(l - 0vx/c)].

Let us consider the property of c' in F'. Its value depends on its direction of propagation. Suppose a light signal propagates along an angle 8 as measured in F and 6' as measured in F', we have (4.8)

vx = c cos 6,

vx = c' cos 9' .

From the inverse transformation of (4.5), we have c = (dt'/dt)[y(c' (4.9)

—fivx)},i.e.

c' = c/{(di'/dt)[y(l+i3cos0')]} = c'(0' ).

Evidently, the average speed of light over any closed path in F is a constant c. Now we prove that the average speed of a light signal over any closed path is also c, though the speed of light is no longer isotropic in F'. Suppose a light signal travels along the vectors r[, where i = 1, 2, ..., N, which form a closed path on the (x', ?/')-plane in F'. The total distance Lt'ot and the total time 7Yot are given by N

(4.10)

LU = 2 r> ,

N

and 7"^ = 2 r{ /c'(01);

i=i

N>1.

i = i

The average speed c'w of the light signal over this closed path is N

(4.11)

ca'v = Lt'ot /7Yot = L ^ /{Lit /c + (Q 7c) 2 r[ cos B[ } = c, i=

I

where we have used (4.12) vi = c'(0!)cos0!, (4.13)

c ' ( 0 O = c/{(d«7d«) i [y(l + /3cos0O]},

(4.14)

(dt'/dt)i = (l + q'cosBi)/[y(l N

(4.15)

2n'cos0-=O. i = 1

+ bcosB^]},

i=

l,2,...,N,

250 Lorentz and Poincare Invariance Equation (4.15) is a property of a closed path in F'. Thus, the average speed of light over an arbitrary closed path is a universal constant c in extended relativity. We note that a closed path for a light signal as observed by F observers is in general not a closed path as observed by F' observers. Suppose a signal starting from 0 (0'), i.e. the origin of F (F'), travels to a point A (A') on the x-axis and is reflected back to B = 0 (B' * 0') as observed in F (F'). Evidently, the two-way speed of this signal in F is c. One may ask: What is the average speed of this light signal as measured in F't From Reichenbach's time (4.1) and extended transformation (4.3), we have the space and time coordinates of these events in F', (4.16)

t'(O') = x'(O')

= 0,

(4.17)

t'(A') = y{[l - Pq'WA)

(4.18)

t'(B') = y{[l-

- [0 + q' }x(A)/c) , x'(A')

pq']t(B) - [p + q']x(B)/c}

, x'(B')

= y(x(A) - pct{A)), = y(x(B) - pct(B)).

Since t(0) = x(0) = x(B) = 0, x(A) = ct(A) = L and t(B) = '2L/c, eqs. (4.16)-(4.18) lead to the «average speed» ca'v(nc): (4.19)

ca'v(nc) = = {x'(A')+

\x'(B')-x'(A')\}/{t'(A')+

\t'(B')

-t'(A')\}

= c/[l -

0q'],

for such a non-closed path in F'. The result (4.19) can also be derived from (4.5) directly because this average speed of light can be considered as two events which satisfy Ax = x(B) — x(0) = 0. As far as constant velocities are concerned, Ax = 0 is equivalent to dx = 0 or vx = 0. Thus using the expression for c' for a non-closed path in F', we have (4.20)

c'(v i = 0) = c / ( l - / 8 7 ' ) I

where we have used the relation (4.6) with the condition dx/dt = 0, (4.21)

(dt'/dt) = y(l-Pq'),

vx = 0.

4. - Basic postulates of extended relativity The physical foundation of the extended transformation (4.3) is based on two postulates: the invariance of the form of physical laws and Reichenbach's time (4.1). This latter may also be regarded as a convention of time. As we have discussed previously, the relation (4.1) is directly related to the clock synchronization. To be specific, we have used the relation between Reichenbach's and Einstein's times, together with the Lorentz transformation, to obtain (4.1). One may ask: Is it possible to derive the extended transformation (4.3) without using Einstein's time and the Lorentz transformation as a crutch? The answer is yes. Instead of postulating (4.1), one can use its equivalent postulate: namely, that the 2-way speed of light over a closed path in any frame is a universal constant and independent of the motion of the light sources. This postulate was first made by Edwards for his formalism [4]. Let us now derive the relation (4.1) based on the two basic postulates of extended relativity:

Chap. 4. Extended Relativity ...

251

1. The principle of relativity for physical laws: The form of a physical law must be invariant under coordinate transformations. (In other words, physical laws must display four-dimensional symmetry.) 2. The 2-way speed of light over a closed path is a universal constant and is independent of the motion of sources. To derive (4.1) from these two postulates, we first observe that the first postulate leads to (4.2) and (4.3) but not to (4.1). Since the F' frame moves along the x-axis, its time t' can only be a linear function of t and x, (5.1)

t'=Pt

+ Qx,

where unknown coefficients P and Q are to be determined. From (4.5) and (5.1), we have (5.2)

c'( + ) = (dt/dt')lcy(l-fi),

(5.3)

c'(-)

(5.4)

(dt/df),

(5.5)

(dt/dt')2 =

dx/dt=+c,

= (dt/dt')2cy(l+P),

dx/dt=-c,

= P + Qc, P-Qc.

Postulate 2 implies that (5.6)

L'/c'( + )+L'/c'(.-)

= 2L'/cim = 2L'/c.

It follows from (5.2)-(5.6) that (5.7)

P + /3cQ = l / y .

Without loss of generality, we may express Q in terms of another parameter q' such that (5.1) is more closely related to the form (4.1) in Edwards' transformation, (5.8)

Q=-y(P

+ q')/c.

The relations (5.1), (5.7) and (5.8) lead to (5.9)

t'=y[(l-Pq')t-(P

+

q')x/c],

which is precisely the basic relation of Reichenbach's time (4.1). 6. - Physical implications of extended relativity Although extended relativity involves a class of different concepts of time, realized by Reichenbach's procedures of clock synchronization with the universal 2-way speed of light, all physical laws have the four-dimensional forms which are identical to those in special relativity. Let us first demonstrate that extended relativity leads to a correct expression for momentum, in contrast to Edwards' formalism. The invariant action for a free particle

252 Lorentz and Poincare Invariance in F' is (61)

\

s

' = - j™cds' = JL'dt' 2

2

2

| ds' = c' dt'

2

, 2

c = ciw = c2w, 2

- dx' -dy' -

dz' ,

c' = d(b't')/dt'

.

Note that c in (6.1) is the two-way speed of light which is a universal constant. The Lagrangian L' in the F' frame takes the form (6.2)

L'= -mcc'{\

-

v'2/c'2)1'2,

which leads to the momentum p', (6.3)

p' =

{mcv'/c')/(l-v'2/c'2)^2.

The «energy» p0' is defined as the zeroth component of the momentum by (6.4)

p0' = (p'-v' -L')/c'

=

These form the momentum 4-vector four-dimensional invariant relation

mc/{l-v'2/c'2)1'2. p'M = (pi,p')

which

satisfies

the

p 0 ' 2 - p ' 2 = ra2c2.

(6.5)

By Noether's theorem, this is the conserved energy-momentum in extended relativity. We see clearly that the momentum p' and the energy p0' in F' are consistent with those seen in high-energy experiments. As a matter of fact, the-momentum p'x given by (6.3) is precisely the same as the momentum px' (SR) in (3.5) for the F' frame in special relativity because of the relations (3.6) and (2.7) and the invariant property that ds 2 in F' for special relativity is the same as ds' 2 given by (6.1) (i.e. ds2 = c2dt'i - dx'2 -dy'2dz'2 = c'2dt'2 - da;'2 - dy'2 - dz'2): (6.6)

- q'v'x/c)[c2 - u'2]1'2}

p'(SR) = mo'c/{(l

= p' .

This example shows that for Reichenbach's time to be viable it must be embedded in the four-dimensional framework. Of course, the Klein-Gordon equation with the form (^-fc2)9P = 0

(6.7)

is invariant under the extended transformation (4.3) as well because the differential operators 3/l = d/dx'1,

(6.8) 1

and

3; = 3/3a;'^,

1

are 4-vectors, where x ' = (ct, r), x' * - ( w \ r') = (&' V, r'). 7. - Remarks and conclusions We would like to make some remarks as follows: 1) It appears that a particular concept of time, such as Reichenbach's time, cannot be judged to be correct or not on its own merits. Only when it is combined with

Chap. 4. Extended Relativity

... 253

another postulate which specifies a symmetry framework, can one make a proper judgement as to its correctness [5,7]. Although we do not have the usual relativistic time dilation in transformation (4.3) of extended relativity, explicit calculations of the lifetime of a particle decay in flight show that the theory is consistent with experiments. (See appendix B.) 2) If we use the inverse transformation of (4.3) to express ct and x in the function b' given by (4.4), we would obtain (7.1)

b' = e-q'x'/t'

.

This turns out to be consistent with the function V obtained from comparing the four-dimensional interval (4.2) and (2.10) (or from (2.7) and (7.2) below). 3) Although Edwards' transformation (2.8) can be obtained by a simple «change of time variables" (2.7) in the Lorentz transformation, this does not imply that the two transformations are physically equivalent. We have shown in (3.2) that Edwards' transformation leads to an incorrect momentum because of its lack of four-dimensional symmetry, as shown in eqs. (3.3)-(3.7). In this sense, the principle of relativity for physical laws has not been really incorporated in his formalism. The main reason is that (ct', x', y', z') is not a four-vector. In contrast, the present formalism of extended relativity based on the new transformation (4.3) between coordinate four-vectors (w, x, y, z) and (w', x', y', z'), where w = ct and w' = b' t', is explicitly consistent with this principle. If one compares the space-lightime transformation (4.3) with the Lorentz transformation, one sees that (7.2)

cti = b't',

where t^ and t' are, respectively, Einstein's time and Reichenbach's time. However, extended relativity is not a trivial change of time variables. If extended relativity is just a change of time variables given by (7.2), we should be able to obtain all results in extended relativity from the corresponding result in special relativity by the relation (7.2). To wit, the momentum (6.3) indeed can be obtained from usual relativistic momentum (3.5) by a simple change of variables (7.2). However, this is not always the case. For example, a plane wave in F' is described by an invariant function (7.3)

exp[i((o'tE-k'-r')]

in special relativity. By a simple change of time variable (7.2), one obtains (7.4)

exp[i(a)'b't'/c-k'-r')]

which leads to an incorrect wave 4-vector (
v'X'=c',

in extended relativity. This implies that the wave 4-vector in F' is (7.6)

(fco',t') = ( a > 7 c ' , * ' ) .

We note that if quantum mechanics is formulated on the basis of extended relativity, the 4-momentum of a photon should be proportional to the extended-wave 4-vector (7.6) rather than (a> '/c, k'). We may remark that from (7.3) one can obtain the correct

254 Lorentz and Poincare Invariance invariant phase, involving (7.6) and (w', x', y', z'), by changing both quantity related to time (e.g., the frequency). In doing this, essentially know the correct wave 4-vector in extended relativity. Therefore, time variable in special relativity appears not to be a useful approach extended relativity.

time and the one must first changing the for obtaining

4) After the transformation of time variables in (2.7), we are not left with the same theory (special relativity) because, strictly speaking, special relativity is a theory which possesses the four-dimensional symmetry with a universal constant for the one-way speed of light and the Einstein time t& has the usual time dilatation property. However, after the transformation of time in (2.7) (which actually violates and destroys the second postulate of special relativity), one has extended relativity which possesses the four-dimensional symmetry with a universal constant only for the two-way speed of light (i.e. Edwards' weaker postulate for the speed of light) and the Reichenbach time t' given by (5.9) does not have t^he usual time dilatation property in general. But we show in appendix B that the experimental results of lifetime dilatation of unstable-particles decay in flight can be understood in extended relativity in terms of the dilatation of their decay lengthi 5) Although c' differs in different directions in F', if one compares speeds of various physical objects in a given direction in F', the speed of light is still the maximum speed in the Universe. This can be seen from the expressions for extended relativistic momentum and energy in. (6.3) and (6.4) or the extended transformation of velocities (4.5). Furthermore, even though the velocities c' and v' measured by using Reichenbach's time depend on the parameters q', as shown in (4.5) and (4.6), the experimental results (such as the Michelson-Morley experiment, the conservation laws of momentum p' and «energy» p 0 ', the dilatation of lifetimes, etc.) turn out to be independent of q'. For example, the momentum p' in (6.3) depends on the ratio v'/c' = (dr'/dt')/(dw')/dt' = dr'/dw' which is independent of time t' or q'. Thus, the restriction for q' in (3.2) is really not necessary from the experimental viewpoint. The basic reason for these properties is that, within the four-dimensional symmetry framework, the inherent evolution variable in the laws of physics is the lightime w' rather than Reichenbach's time t', as indicated by taiji relativity [8]. (Cf. appendix B.) From the discussions in 3) and 4), we conclude that extended relativity and special relativity are two logically different theories—one has a universal one-way speed of light and the other has a universal two-way speed of light (which includes the former as a special case). Nevertheless, they both have four-dimensional symmetry, so that physical laws have the same forms in all inertial frames. We have examined all known experiments. Strangely enough, these experiments cannot distinguish extended relativity from special relativity. For any covariant law in special relativity, there is a corresponding law of the same form in extended relativity. This suggests that extended relativity is equivalent to special relativity, at least as far as known physics is concerned. We believe this indicates that physical properties such as our concept of time (e.g., Einstein's relativistic time or Reichenbach's time, etc.) and the corresponding speed of light in the four-dimensional symmetry framework are human conventions rather than the inherent nature of the physical world. This conclusion was already indicated by the results of fourdimensional symmetry of taiji relativity [8].

Chap. 4. Extended Relativity ... 255

This work was supported in part by The Jing Shin Research Fund of the UMass Dartmouth Foundation. One of us (JP) would like to thank Yuan-Zhong Zhang for correspondences.

APPENDIX A

Group properties of the space-lightime transformation If an object is at rest in F, i.e. v = (0, 0, 0), the extended velocity transformations (4.5) lead to (A.1)

c' = c / ( l - # 7 ' ) s c ' ( 0 ) ,

vl=-0c/(l-Pq')*vHO),

^ = vi = 0.

This implies that the speed of the F frame, measured from F', is different from the speed of F' measured from F. When one thinks about it, there is no reason, and in particular, no experimental evidence (obtained independently of the second postulate of special relativity) that the speed of F measured from F' should be - V. One might claim that common sense dictates that the two speeds be equal and opposite. However, as is evidenced by special relativity and quantum mechanics, common sense is often a poor guide in modern physics. Consider the ratio v'x (0)/c' (0) in (A.l). This ratio is a constant, independent of dtjdt', and satisfies (A.2)

P's-vi(0)/e'(.0)=+p.

Thus, using the ratio of velocities, {$' and /}, one restores the symmetry of velocities between F and F'. The inverse transformation of (4.3) can be written as \ct = y{b'f

+P'x')

\x = y'(x' +P'b't'),

=

Y'Q't'+P'x'), y = y',

z = z';

y' = y.

Let us consider also another frame F" moving with a constant velocity Vt = = (V-!, 0, 0), as measured from F and VI = (V{, 0, 0), as measured in F'. Note that the ratio V{/c' is related to Vx by (4.7)[7]: (A.4)

Vl/c'

^(Vjc-frlil-pVjc).

From (A.1), (A.2) and (A.4), we see that instead of V, one should use the ratio V/c to characterize the relative motion between inertial frames in extended relativity, as this ratio will always be constant and, more importantly, is independent of the parameter q'. The four-dimensional transformation between F and F" is given by (A.5)

bKt" = Yi(ct-Pix),

x" = yi(x-plCt),

y" = y,

z" = z,

where (A.6)

t" = y i [ ( l - piq")ct

- (0J -

q")x]/c.

From (4.3) and (A.5), we can obtain the transformation between F' and F", (A.7)

b"t" = y[(b't'-p[x'),

x'^r'Ax'-pib't'),

y' = y',

z" = z' ,

256

Lorentz and Poincare Invariance

where (A.8)

fc

= (Pi-P)/a-0iP)

= Vl/c',

(A.9)

y; = y 1 y ( l - / 3 l / 3 ) = l / ( l - ; 3 i 2 ) 1 / 2 ,

(A.10)

t" =

px = Vx/c, Yl

= 1/(1 - /J?)1/*,

y'l[(l-!3'1q")b't'-(P'1-q")x'yc.

This result (A.7), together with other properties such as the existence of an inverse transformation and associativity, demonstrates that the set of extended fourdimensional transformations forms the Lorentz group. This Lorentz-group property is the core of the four-dimensional symmetry and is crucial for extended relativity to be consistent with experiments.

APPENDIX B

Decay rate and «lifetime dilatation» Fundamental wave equations such as (6.6) with (6.7) show that four-dimensional symmetry dictates the evolution parameter to be the lightime w rather than t [8]. We must use the lightime w i n a general inertial frame as the evolution variable for a state 0 ( S ) (w) in the Schroedinger representation [8,9]: (B.l)

itid
=H{SHw)0{S)(w),

H™ = H^ + H{S),

because the evolution of a physical system is usually assumed to be described by a «Hamiltonian operator» H = P0 for which the corresponding evolution parameter is the lightime w. Both H = P0 and d/dw transform as the fourth component of a 4-vector. Naturally then, the transition probability within extended relativity is defined directly in terms of the lightime w. The usual covariant formalism of perturbation theory can be applied to quantum field theory based on extended relativity [8]. For example, let us consider the decay rate I\ 1 —> 2 + 3 + ... + N) for a physical process 1 —» 2 + 3 + ... + N. In a general frame, it is given by [8,9] (B.2)

c|(/|SK)| 2 d 3 x 2 d 3 p 2 d3xNd3pN r ( 1 ^ 2 + 3 + ... +A0 = lim f w (2Po2ftf ' " (2pQNhf

which has the dimension of inverse time. The decay lifetime r is given by r = l / r ( l —» -> 2 + 3 + ... + N). To illustrate the calculation of a decay rate, let us consider a simple example, ie. the muon decay n" ( p ^ - ^ e " (p2) + v(1(p3) + v e (p 4 ) with the usual V - A coupling. The decay S-matrix in momentum space is, as usual, given by [9] J ( / | S | i } « (P0lP02P03P04) -1 <5 4 (Pl-P2-P3-P4)M a c> [M3C = (G/V2)[v ( i (p 3 )y"(i-)'5)Mp 1 )][e(P2)yA(i-y5)v e (p4)].

Chap. 4. Extended Relativity ... 257 The muon lifetime r can now be calculated and the result is (B.4)

l / r = r ( 1 ^ 2 + 3 + ... +N) <=c a

J _ f d ^ d ^ ^ POI J

P
4 ( p i

P03

_

p 2

_

p 3

_

p 4 ) E | M s c | 2

P04

s

Pta

Everything to the right of l / p 0 1 in (B.4) is invariant under the extended transformation so that the decay lifetime r is indeed dilated: (B.5)

T<x(pf + m 1 2 ) 1 / 2 = poi-

One should keep in mind, however, that (B.2) is really for the decay length, as shown in ref. [8]. We use the universal two-way speed of light c to convert it into lifetime. I t is the decay length which is really measured in the laboratory.

REFERENCES [1] YANG C. N., Phys. Today, 33 (1980) 42. [2] GROSSMAN N., HELLER K., JAMES C. et al, Phys. Rev. Lett, 59 (1987) 18.

[3] SAKURAI J. J., Invariance Principle and Elementary Particles (Princeton University Press, Princeton) 1964, p. v. [4] EDWARDS W. F., Am. J. Phys., 31 (1963) 482; REICHENBACH H., The Philosophy of Space and Time (Dover, New York, N.Y.) 1958. Edwards' transformation of space and time (2.8) was rederived and discussed by several authors. See, for example, WINNIE J. A., Philos. Sci., 37 (1970) 81, 223; YUAN-ZHONG ZHANG, Experimental Foundations of Special Relativity (Science Publishers, Peking) 1979, pp. 14-21; Gen. Relativ. Gravit, 27 (1995) 475. [5] For a detailed discussion of the universal constant of the one-way speed of light and its implications, see Hsu J. P., Nuovo Cimento B, 74 (1983) 67; 88 (1985) 140; 89 (1985) 30; Phys. Lett. A, 97 (1983) 137; Hsu J. P. and WHAN C, Phys. Rev. A, 38 (1988) 2248, Appendix. [6] PAULI W., Theory of Relativity (Pergamon, London) 1958, pp. 5-9 and references therein. [7] Hsu L. and Hsu J. P., Nuovo Cimento B, 111 (1996) 1283; (it is the preceding article in this issue); Experimental tests of a new Lorentz-invariant dynamics based solely on the first postulate of relativity. - II, UMassD preprint (1995). [8] Hsu J. P. and Hsu L., Phys. Lett. A, 196 (1994) 1. [9] See, for example, BJORKEN J. D. and DRELL S. D., Relativistic Quantum Mechanics (McGraw-Hill, New York, N.Y.) 1964, pp. 261-268 and 285-286; SAKURAI J. J., Advanced Quantum Mechanics (Addison-Wesley, Reading, Mass.) 1967, pp. 171-172 and 181-188.

258 Lorentz and Poincare Invariance

Four-dimensional symmetry of taiji relativity and coordinate transformations based on a weaker postulate for the speed of light. - II JONG PING HSU (*) and LEONARDO HSU (2)

C1) Physics Department, University of Massachusetts Dartmouth North Dartmouth, MA 02747, USA (2) Physics Department, University of California at Berkeley - Berkeley, CA 94720, USA (ricevuto il 15 Maggio 1996; approvato il 9 Luglio 1996)

To Prof. Ta-You Wu for his wonderful and tireless teaching of physics and his ninetieth birthday

Summary. — Extended relativity is a theory of four-dimensional symmetry with Reichenbach's time, in which only the 2-way speed of light is a universal constant. It includes special relativity as a special case. The theory is shown to be consistent with experiments such as Fizeau's experiment, aberration of light and precision Doppler shifts. The formulations of classical and quantum electrodynamics are discussed. They are shown to be dependent on the four-dimensional symmetry rather than on the usual constant one-way speed of light. The four-dimensional symmetry also dictates a new coordinate transformation, called the "Wu transformation", for constant-linear-acceleration frames. PACS 03.30 - Special relativity. PACS 11.30.Cp - Lorentz and Poincare invariance.

We continue to demonstrate that the four-dimensional symmetry is necessary and essential [1] for discussing physical laws from Reichenbach's viewpoint of time [2] or Edwards' weaker postulate for the speed of light [3]. Edwards attempted in 1963 to formulate a relativity theory based on a weaker postulate that the 2-way speed of light in a vacuum is a universal constant. He derived space and time transformations which involve Reichenbach's time but which do not form a four-dimensional Lorentz group in general. As a result, it leads to an incorrect expression for the relativistic energy-momentum of a particle in the Lagrangian formalism of mechanics and electrodynamics, as showrt in paper I [1]. Furthermore, it appears to be impossible to obtain invariant forms of Maxwell equations and the Dirac equation if Reichenbach's time is used as an evolution variable, as we shall see later. The reason is that Reichenbach's time does not transform covariantly as the zeroth component of the coordinate 4-vector in general. Therefore, the lack of four-dimensional symmetry makes Edwards' original transformations untenable. Recently, we have formulated and discussed taiji relativity [4] based solely on the

Chap. 4. Extended Relativity ...

259

first postulate of relativity, i.e. the invariance of physical laws. A four-dimensional transformation between any two inertial frames, F(w, x, y, z) and F'(w', x', y', z'), is derived. Since taiji relativity does not make any assumption regarding the speed of light, the zeroth components w and w' cannot be factored into a well-defined speed of light and time. In fact, the speed of light and usual time (measured in seconds) are unspecified and undefined in the theory. However, the theory of taiji relativity possesses the four-dimensional symmetry which is shown to be the only essential ingredient for the theory to be consistent with previous experiments. This sheds light on the difficulty encountered by Edwards' transformations. In paper I, we show that, guided by the four-dimensional symmetry of taiji relativity, Reichenbach's general convention of time (or, equivalently, the universal 2-way speed of light) can be used as the "second postulate" for the construction of a new four-dimensional formalism of coordinate transformation which is termed extended relativity. The second postulate is necessary to, factorize, say, w' into a well-defined velocity function b' (called "ligh") and Reichenbach's time V in the F' frame, i.e. w' = d ' f which is called "lightime". (See eqs. (2.1)-(2.3) in sect. 2.) It turns out that the lightime w', rather than Reichenbach's time, plays the role of evolution variable in physical laws and makes extended relativity consistent with established energy-momentum of a particle, the Lorentz group, etc. Furthermore, the covariant lightime embedded in the four-dimensional symmetry is also crucial for the formulation of a covariant quantum electrodynamics (QED) based on extended relativity, as we shall see in sect. 6. 2. - Extended relativity—A theory with universal 2-way speed of light Reichenbach's synchronization procedure amounts to imposing a second postulate [1] upon taiji relativity [4], so that we have a well-defined "extended" time, which includes Einstein's time as a special case. For simplicity and without loss of generality, we choose q = 0 in the synchronization of clocks in the F frame, so that we have Einstein's time t = t E ; while in the F' frame, clocks are synchronized to read Reichenbach's time t'. An event is, as usual, denoted by (ct, x, y, z) in F. Following taiji relativity [4], the same event must be denoted by (b' t', x', y', z') in F' which is moving along the + x axis with a constant velocity ft = V/c, as measured in F. We stress that it is necessary to introduce the "ligh function" V so that- {b'f, x', y', z') = (w',r') transforms like a four vector and laws of physics can display four-dimensional symmetry. It was shown by Edwards on the basis of the universal 2-way speed of light that times t and V were related by [3] (2.1)

t' = Y[(l-l3q')t-(p-q')x/c],

q' = 2e'-l,

P = V/c,

which was the basic property of Reichenbach's time. Assuming relation (2.1) is effectively the same as assuming the universality of the 2-way speed of light over a closed path in any inertial frame within four-dimensional framework. The extended four-dimensional coordinate transformation was derived [1], w' = b't' = y(ct-fix), (2.2)

0 = V/c,

x' = y(x- j3ct),

y' = y ,

2

y = l/(l-0 )V*,

_b' = (ct- fix)/[t(l - [3q') - (fi - q') x/c] = c - q' x' jf ,

z' = z\

260 Lorentz and Poincare Invariance where t' is given by (2.1) and b' is determined by (2.1) and (2.2). Although V and V in (2.1) and (2.2) separately have complicated non-covariant transformation property, their product w' = b't' has a simple transformation property and is covariant. The extended transformation of the velocities r and acceleration a can be derived from (2.2): we obtain (2.3)

e' = d(b't')/dt'

=

v^=[dt/dt']y(c-fivx),

v; = (dt/dt')y(vx-fie);

vy' = (dt/dt')

vy ,

v'z = (dt/dt')

vt

and a^ = dc'/dt' = [d2t/dt'2] 2

2

2

2

ax' = (d t/dt' )y(vx-pc)

(2.4)

ay' = (d t/dt' )

where vx =dx'/dt', (2g)

\l/[dt/dt'l 2

\d t/dt'

2

+

vy + [dt/dt' 2

az =

y(c - pvx) - [dt/dt'fyfiax

,

[dt/dt'fyax, fay,

2

(d t/dt' )vz+[dt/dt'faz;

vx = dx/dt,

etc. and dt/dt'

= dt'/dt = = (dt/dt')

may be obtained from (2.1),

y{[l-l3q']-[p-q']vx/c},

Z,

Z=[(P-q')ax/c]/[(l-l3q')-(P-q')vx/c].

We stress that the definition c' = d(b't')/dt' is quite natural because when ds 2 = 0, 2 ie. [d(b' t')f - dr' = 0 is the law of light propagation, c ,2dt '2 - dr' 2 = 0. Note that we still' have the four-dimensional law for the propagation of light in the F' frame, although the speed of light is not isotropic and Reichenbach's time is not covariant. Also, the transformations of the velocity ratios v ' / c ' turn out to be precisely the same as those in special relativity, (2.6)

v'/c1 = (vy/c)/[y(l

vl/c'=(vx/c-P)/(l-vxp/c);

-0vx/c)],

etc.

This is crucial for the extended coordinate transformations (2.2) to form the Lorentz group. For the ratios of accelerations, we have J ^/a,;

= [Z{vx - c/3) + (dt/dt')

yax]/[Z(vx

- c0) - (dt/dt')

\a;/a^=[Zvy+(dt/dt')ay]/[Z(vx~cfi)-(dt/dt')yliax],

y0ax], etc.,

which are more complicated than those in special relativity because q' ^ 0 in Z given by (2.5). The momentum p* = (p0, px, pv, pz) and wave vector k1" = (k0, kx, ky, kz) transform like a coordinate four-vector (2.2). For example, we have (2.8)

fc0'

= y(k0- Pkx),

kx = y(kx -0ko),

ky = ky ,

kz = kz.

For a light wave, k= (kcosd, ksind, 0) and k' = (k' cos8', k'sin9', leads to the formula for the aberration of light, (2.9)

cos0'=(cos0-/S)/(l-/3cos0),

(2.10)

sin0' = s i n 0 / [ y ( l - 0 c o s 0 ) ] ;

0), eq. (2.8)

Chap. 4. Extended Relativity ... 261 where we have used k = | k | = k0 for a light wave. These results are the same as those in special relativity because they both have the four-dimensional symmetry. 3. - Some experimental implications of extended relativity In the Fizeau experiment, the observed drag coefficient can be explained by the addition law (2.6) for velocity ratios with vx /c = 1 /n: (3.1)

v'/c'=(l/n-p)/a-P/n),

[3 = V/c.

The speed of light relative to the medium (at rest in the F frame) is vx = c/n in F, where n is the refractive index of the medium. Now, if the medium is moving with speed V = fie parallel to the direction of light, the ratio vx /c' observed by a person at rest in F' is given by (3.1) as (3.2)

vx' /c' = 1 /nL = (1 /n - 0) / ( l - 0/n) s i jn - fi(l - 1 jn2),

where nL is the "effective refractive index" of the moving medium with velocity + V. Assuming each tube in the Fizeau experiment has length L' and the speed of water is ±V= ±[ic, the optical path difference AL' of the two beams of light is (3.3)

A L ' = 2L'TC1 -2L'nL

(3.4)

\/n'±

= AL'n2{V/c){\

-

l/n2),

=\/n±(V/c){l-l/n2)

as expected, since the optical path is just the distance in vacuum equivalent to the actual path length traveled by each beam [7]. The formula for the aberration of light can be obtained from the inverse transformations of (2.9) and (2.10): (3.5)

tg0 = sin0'/[y(cos0' + )3)].

This shows the deviation of light when transforming to a new reference frame. Note that the invariant law for the propagation of light c' 2 d*' 2 -Hr' 2 = 0

(3.6)

does not refer to any specific source and, hence, it holds for light emitted from any source. From a microscopic viewpoint, the state of motion of a macroscopic source of light is actually irrelevant because photons are emitted from atoms in violent motion which may not even be uniform. 4. - Doppler shifts of frequency and atomic energy levels The extended transformation (u)' /c', k') is given by (2.8): (4.1)

(o'/c' = y(cu/c -f}kx),

fcj

of wave four-vectors k= (a>/c,k) = y(kx -

fiw/c),

ky = ky , fc; = kz.

This implies that the Doppler wavelength shift in extended relativity is (4.2)

and k' =

l / l ' = y(l/2-/3/A) = ( l / A ) [ ( l - / 3 ) / ( l + / 3 ) ] 1 / 2 ,

262 Lorentz and Poincare Invariance where kr = 2n/k, ky = kz = 0, &J = 2 J T / A ' . This is the same as that in special relativity. However, the consistency of the Doppler frequency shift in (4.1) with the result of laser experiments is more subtle because c' is not a constant in extended relativity. Since experiments which measure the frequency shift involve the absorption of photons by atoms, we must first re-examine the nature of atomic levels from the point of view of extended relativity. In extended relativity, Dirac's Hamiltonian Hv for a hydrogen atom is hctidip/dw = HDy, [HD= - a-Pc - fine2 - e2/(inr),

P=-ifcV,

which is given in the Dirac equation (6.6) below. Using the usual method, it can be shown that (4.3) leads to atomic energy levels (44)

\E„ = m c 2 / { l + ai/[n -h,+ \ho=j +1/2,

(h2 -

a^l2fy/*,

ae = e2/(4jtch).

Thus when an electron jumps from a state nl to another state re2, it will emit or absorb an energy quantum chk0: (4.5)

En2 - Enl = chk0 = hw,

inF,

(4.6)

E^ - E^ = chko = cfiu> ' /c',

in F',

where c' and c in (4.6) do not cancel in F', except for the special case q' = 0. If two photons with "energy" ((o0/c) ch and (a>'0/c') ch are emitted from two hydrogen atoms at rest in F and F', respectively, then by. the equivalence of F and F' frames, (4.7)

(aj0/c)cti=(a)'0/c')ch.

Here, the ratio a>'0/c' is isotropic. However, if (CD/C) ch and (a>'/c') ch are energies of the same photons measured from F and F', respectively, then they are related by (4.1) (4.8)

w ' / c ' = y(a>/c)( 1 -/?cos 9),

kx = kcosd = {CD/C) cos8 .

Evidently, only when c'—c do we have the usual relation for the Doppler frequency shift. In general, we can only talk about an "energy shift" or a "A;0 shift" in extended relativity because the energy of a quantum particle is not always proportional to the corresponding frequency in all frames. In other words, the energy [1] of a particle or A;0 is the zeroth component of a 4-vector, but the frequency (defined on the basis of Reichenbach's time) is in general not covariant. Experimentally, one never measures the frequency directly unless constant one-way speed of light is presupposed (so that the frequency becomes well-defined in all frames); instead, what one really measured is the shift of atomic levels in interaction with radiation or a laser that is measured in a general frame. Thus, (4.4)-(4.8) are consistent with precision measuring experiments of Doppler shifts. Since Reichenbach's time is used in F', the speed c' of a photon and its frequency co' in F' is anisotropic in general; however, the ratio cu'/c' is isotropic. Four-dimensional symmetry dictates the mixture of two waves in F' in terms of (w1, r') and (CD' /c', k')

Chap. 4. Extended Relativity ...

263

rather than (£', r') and (a)', k'), so their superposition is in general given by (4.9)

Af, sini^w'-k[-r')

+ B0 sin(k^w'

-k^-r'),

w' =

b't'.

Only in the F frame, where c is, by definition, isotropic and constant, does one have the usual expression A0 sin(a> l t-k 1 -r) + B0 sin(a>2^ ~ k2-r).

5. - Classical electrodynamics in extended relativity Although the one-way speed of light in extended relativity is not a universal constant, one still has the universal 2-way speed of light c. Thus, the usual action for a free particle, - jmc ds, is an invariant in extended relativity. If a charged particle with mass m and charge e is moving in an electromagnetic field in a general frame, the invariant action is assumed to be [1] (5.1)

S = - J weds - (e/c) J A „ dx" - (l/4c) JF^F*" dsrdw ,

(5.2)

F„, = d,Av - dvAM ,

(5.3)

ds = (g/lvdx'xdxv)1^

(5.4)

e/c= -1.6021891 •10- 20 (4^) 1 / 2 (g. C m) 1 / 2 ,

= dt(Cz-v2)1/2,

C = dw/dt,

v = dr/dt,

where gMV = ( 1, - 1 , - 1 , - 1 ) , x* = (w, x, y, z), e/c is in Heaviside-Lorentz units and Af, is the same as the usual electromagnetic vector potential in special relativity. Note that C = dw/dt is in general not a constant in extended relativity. For a charged particle moving in an electromagnetic potential field A^, the invariant action is given by S cp , (5.5)

Scp=JLcpdt,

Lcp =

rac(l--y2/C)1/2-(e/c)A^da;7dt,

which consists of the first two terms in (5.1), dx^/di = (C, v) and A^ = (A0, -A). The canonical momentum of a charged particle is now given by

(5.6)

\P = 3Lm/d\ = p + eA/c, p/ ' [ p =(mcv/C)/(l-?;2/C2)1/2.

Note that we have used the universal 2-way speed of light c to make p having the usual dimension of mass times velocity. In contrast, there is no universal speed of light in taiji relativity, so that the covariant momentum must have the dimension of mass [4]. Following taiji relativity, we define the covariant Hamiltonian H as H= [(5L/3v)-v - L](c/C) = cp0 + eA 0 , p0 = m c / ( l - ' y 2 / C 2 ) 1 / 2 .

264 Lorentz and Poincare Invariance Note that the factor c/C in definition (5.7) is necessary for the Hamiltonian H/c to transform as the zeroth component of the momentum 4-vector, H/c = P0. Otherwise, the Hamiltonian will not be meaningful. From (5.6), (5.8) and H/c = P0 we have (5.8)

(P 0 - eAo/cf - (P - eA/cf

= m2c2.

The Lagrange equation of motion for a charged particle in the electromagnetic field has the usual form dp"ds = (e/c) F"v dxv/ds ,

(5.9)

where p* = (p°, p) and x^ = (w, - r ) . Making the substitutions P-*-ihV relativistic Klein-Gordon equation (5.10)

[(ihd/dw-

eA0/cf

and P0^>ihd/dw,

- (-ihV - eA/cf

we obtain the extended


For a continuous charge distribution in space, the second term in (5.1) should be replaced by -fA^J^dPrdw. In this case, the variation of (5.1) leads to the invariant Maxwell equations in a general frame

[3xF>"' + d/lF"xdvFi<'=0.,

d^d/dx",

xx = x(w,r),

Thus, Maxwell's equations have the invariant form, even if the one-way speed of light is not universal. This is consistent with the result in taiji relativity. 6. - Quantum electrodynamics based on extended relativity We demonstrate that an extended quantum electrodynamics (QED) can be formulated because extended relativity has the four-dimensional symmetry with the lightime w' as the evolution parameter in the F' frame. This would be impossible if one attempts to use Reichenbach's time t' in Edwards' transformation [3] as the evolution parameter because V does not transform as the zeroth component of a 4-vector. In extended QED, the invariant action Sq involves rp and Au is assumed to be (6.1)

Sq=JLd4x,

L = ylyf'(ihdll +

eAfl/c-mc]ip-{l/4c)FflvFiJV,

where e < 0 and d4£ = dwd 3 r. For quantization of the fermion fields, for example, the "canonical momentum" jib conjugate to i/>b is defined as usual to be (6.2)

jzb = dLy,/9(d0ipb),

Lv =

y)[yf'itid/t-mc]yj,

and the Hamiltonian density for a free electron is H - itd0ip - Lv. For free-photon (Au) and electron (ip) fields, we have (6.3)

A„{w, r) = (1/F 1 / 2 ) 2 (ft/Zpo)1 / 2 . •L4(p, a) eM(a) expl-ip-x/h]

+Ar(p,

a) e^a)

exp[ip-x/h]],

Chap. 4. Extended Relativity ... (6.4)

265

ip(w, r) = (1/y 1 / 2 ) 2 (m/po)1/*. p;s

'

•[b(p, s)u(p, s) exp[-ip-x/h]

+ dHp, s) v(p, s) exp[ip-x/h]],

p-x = p>1x",

where f [ A ( p , a ) , A t ( p ' > a ' ) ] = (5pp.<5aa.) (6.5)

+

[{6(p, s), 6 T (p', s')] = <5PP<5SS.,

[d(p, s), d T (p', s') =^PP-<5SS. ;

and all other commutators vanish. The Dirac equation in extended relativity can be derived from (6.1). We obtain (6.6)

ih dip/dw = [a- (ihV + eA/c) - fimc + eA0 /c] \p .

In view of the equations of motion (5.10) and (6.6), we must use the lightime w in a general frame as the evolution variable for a state <S)(w) in the Schroedinger representation: (6.7)

34><S>(W) ih

= H®Hw) &S)(w),

H(S) = H™ + HlS),

dw because the evolution of a physical system is assumed to be described by a Hamiltonian operator which has the same transformation property as that of w. The usual covariant formalism of perturbation theory [5] can also be applied to quantum field theory based on extended relativity. To illustrate this point, let us briefly consider the interaction representation and the S-matrix based on extended relativity. The transformations of the state vector <3E>(w) and operator 0 from the Schroedinger representation to the interaction representation are (6.8)

O(w) = 4>(I)(w) = exp [iH^w/h]

0>(S)(w),

(6.9)

0(w) = 0 ( I ) (w) = exp [iH&S) w/h] 0<S) exp [ - z#0(S) w/h]

Since 0 ( S ) and 0(w) are the same for w = 0, we have (6.10) (6.11)

ih—— dw

= Hl(w) 3>(w),

HY = exp[itf0(S)w//i] H±S) exp [ - itf0(S) w / ^ '

O(w)=exp[iH^S)w/h]O(0)exp[-iH^)wh].

The [/-matrix can be defined in terms of the lightime w: O(w) = U(w, w0) O(w;0), U(wQ, w0) = 1. It follows from (6.10) and (6.11) that dU(w, wa) ih — ° = Hj(w) U(w, w0). dw If a physical system is in the initial state O; at lightime w0, the probability of finding it in the final state <£>{ at a later lightime w is (6.12)

(6.13)

|(4>f | U(w, w0) *,) | 2 = | Un(w, w0) | 2 .

266 Lorentz and Poincare Invaiian.ee Evidently, the average transition probability per unit lightime for 4>f—>
(6.14)

As usual, we can express the S-matrix in terms of the ^/-matrix, ie. S = £/(°°, - °°) and obtain the following form: (6.15)

S=l-(i/h)

j Hl(w)dw+(-i/hf

j Hj(w)dw

— 00

j

Hl(w')dw'+....

— 00

—00

For w-dependent operators, one can introduce a w-product W (corresponding to the usual chronological product), so that one can write (6.15) in an exponential form: (6.16)

S = W exp | Hl(x")dsr

(6.17)

•ii/h) J

Htix^dwd^r

= HJ(iv).

For simplicity, one may set h = c = 1, where c is the 2-way speed of light. (These are the "natural units" in extended relativity.) For the QED Lagrangian (6.1), we can derive Feynman rules (based on extended relativity). Let us summarize the Feynman rules for QED with the Lagrangian L in (6.1) with a gauge-fixing term O^A") 2 /(2a'): (6.18)

LQED = L - 0 „ A " ) 2 / ( 2 a ' ) ,

h = c=\,

e < 0.

The covariant photon and electron propagators are (6.19)

- i[g,v - (1 - a ' ) k„ kv /{kf + ie),

(6.20)

i/(yflp"-rn

+ ie).

The vertex factor is (6.21)

ley".

There is a factor e^ for each external photon line and a factor u(p, s) for each absorbed electron, a factor - 1 for each closed fermion loop, etc. These rules are identical to those in the conventional theory, if the natural unit is used. Thus, if one calculates scattering cross-sections and decay rates (with respect to the lightime w) of a physical process in a general frame, one will get the same result as that in special relativity [5]. For example, let us consider the decay rate T(l —>2 + 3 + ... + N) for a physical process l - * 2 + 3 + ...+iV. It is given by (6.22)

r ( l - > 2 + 3 + ...+iV) = lira f

\(f\S\i)\2 w

d3x2d3p2 (27thf

'"

d3xNd3PN (2jthf

which has the dimension of inverse length. The decay length D is given by D = l/r(l—>2 + 3 + ... + N). Thus in extended relativity, we have the dilatation of the decay length for a particle decay in flight. Since we have the universal 2-way speed of

Chap. 4. Extended Relativity ...

267

light c, we can use it to define the lifetime r of a particle by (6.23)

x = D/c.

We stress that such a lifetime has little to do with Reichenbach's time in general. The result (6.23) is also consistent with experiments because the decay length dilatation is what one actually measured in the high-energy laboratory. This is not surprising because we use the same four-dimensional symmetric Lagrangian as that in special relativity. 7. - Limiting four-dimensional symmetry and accelerated Wu transformation One may wonder whether the power of four-dimensional symmetry is strong enough to say something about constant-linear-acceleration (CLA) frames. The attempt to generalize the coordinate transformation for inertial frames to that for CLA frames through a symmetry consideration is very natural because the transformation for a CLA frame must reduce to that for an inertial frame in the limit of zero acceleration. So far, no satisfactory transformation for such non-inertial frames has been obtained in the literature, even though one has general relativity and the correspondence principle [6]. All those accelerated transformations discussed previously are not based on a symmetry principle and do not naturally reduce to the four-dimensional transformation for an intertial frame when the acceleration approaches zero. By a stroke of luck, we have found a transformation for CLA frames which does reduce to the correct four-dimensional transformation in the limit of zero acceleration and reduces to the Galilean accelerated transformation when the velocity is small. For simplicity of notation and calculations in an accelerated frame, let us denote a CLA frame by F(w, x, y, z) and an inertial frame by Fi(wi, x r , ylt z{). Suppose a CLA frame F(w, x, y, z) is moving with a constant acceleration a, so that its velocity is /3 = aw + p0, along the +x axis. We find that the accelerated transformation between Fi and F is given by wl = yp(x + 1/ayl) (7.1)

P0/ay0, Vi = y ,

xr = y(x + 1/ay f,)-l/ay 0, P = aw + p0,

1

y = 1/(1-ft') /*,

zx = z ; y o =l/U-0 2 o) 1/2 ,

which will be called the "Wu transformation". If one wishes, one may define Wi = ct\, where t\ is the usual Einstein time, in (7.1) for easy comparson with special relativity. (But this definition is not necessary for deriving experimental results.) The inverse Wu transformation of (7.1) is wi + Po/ayo (7.2)

a(*i + l / a y 0 ) 2

_ 0o a

x = [(x, + l / a y 0 ) - (w, +/S 0 /ay 0 ) 2 ] 1 / 2 y = yu

1/ayl,

z = zx.

One can verify that (7.1) and (7.2) reduce to four-dimensional transformations of the form (2.2) in the limit of zero acceleration a. (See appendix.) We may remark that the

268

Lorentz and Poincare Invariance

coordinate transformation between two CLA frames can be derived on the basis of (7.1) or (7.2). From the viewpoint of limiting four-dimensional symmetry, the CLA transformation must be expressed in terms of the Cartesian coordinates rather than other coordinates, just like the Lorentz transformation. Furthermore, the coordinates of CLA frames should play the same role and have a similar physical meaning as those of inertial frames. This appears to be different from the usual viewpoint that coordinates for accelerated frames have no physical meaning. The Wu transformation (7.1), based on the four-dimensional symmetry, differs from that obtained by Moller [6] based on the approximate principle of equivalence in general relativity because they give different spatial measurements by meter sticks or the Bohr radius of hydrogen atoms. We believe that such a difference should be tested by, say, measuring a Doppler shift of wavelength emitted from a source with a constant linear acceleration. We may remark that the constant acceleration a in (7.1) can be shown to be related to constant change of "energy" (or "moving mass") per unit length measured in an inertial frame. This differs from the usual definition of acceleration in (2.4). It is interesting to note that such a constant acceleration a dictated by the limiting four-dimensional symmetry is precisely what has been actually realized in linear accelerators in laboratories. Physical implications of the Wu transformation and their experimental tests will be discussed in a separate paper. 8. - Remarks and discussions In the formulation of QED in sect. 6, the electron is, as usual, assumed to be a point particle. However, if the physical electron is really a fuzzy point (in the sense of fuzzy set theory with a bell-shape membership function having a width L0) rather than a geometric point, then there will be a departure from the four-dimensional symmetry at short distances or large momentum [7]. A fuzzy-point model of a particle has been interpretated as follows: a particle by itself is a structureless-point particle, but it can simultaneously exist at different places with a different probabilities. As a result, the position uncertainty of such a quantum particle has a minimum width Acc~L 0 . The Coulomb potential will be modified when r
Chap. 4. Extended Relativity ...

269

dimensional physical laws. There are three universal and fundamental constants in QED based on extended relativity. Special relativity is a special case (q-q' = 0 in (2.2)). c) Common relativity: it is based on two postulates. The additional second postulate is a common time t' = t for all observers [8]. The speed of light is, roughly speaking, relative. Lightimes w and w' are evolution variables in four-dimensional laws. There are only two universal and fundamental constants in QED based on common relativity, precisely the same as those in taiji relativity. Common relativity has the unique advantage for dealing with many-particle systems where canonical evolution of the system is essential and for obtaining covariant thermodynamics and invariant Planck's law of black-body radiations [9]. One may ask: how can one realize the evolution variable w' in the extended coordinate transformation (2.2) by physical means? Since the invariant phase of an electromagnetic wave in the F' frame is given byfc0'w' - k' l r', wherefc0'= | k' |, we can define the lightime w' in terms of fc0\ J u s t as the length can be defined by the wavelength A' or |k' |. We note that the "clocks", which show lightime in this theory, are the same as those in taiji relativity [4] because they have exactly the same four-dimensional transformation property. However, the taiji-time w' in F' cannot be factored into two well-defined b' and t' because of the absence of a second postulate while the lightime w' in extended relativity and common relativity can be factored into two well-defined functions b' and t', as shown in (2.1), (2.2) and ref. [8]. Our discussions show that it is extremely important to be aware of what quantities are actually measured in the experiments and what effects the assumption of a universal speed of light may have had on the interpretation of the results. For example, we have seen in paper I that the lifetime dilatation of unstable particle decay in flight has little to do with the property of Reichenbach's time with a general parameter q or q', because the lifetime r is basically defined as the decay length divided by the universal 2-way speed of light c. The basic reason is that the four-dimensional symmetry dictates that the decay rates in, say, QED based on extended relativity can only be defined in terms of the covariant lightime w or w' which has the dimension of length. The constant 2-way speed of light in extended relativity is in general not the maximum speed of physical objects in the universe. Rather, it is the one-way speed of light in a given direction, that is the maximum speed of any object in that direction, as shown in (2.3). This holds for any inertial frame. It is worthwhile to note that this property of light, bt ng the "maximum speed" of all physical objects in any given direction, is a logical consequence of the first postulate of relativity, as shown in taiji relativity [4]. We have examined a number of experimental tests of special relativity and the formulations of classical electrodynamics and QED. All of them are consistent with extended relativity. These discussions can be generalized to other field theories such as unified electroweak theory and quantum chromodynamics. As we have seen, only the four-dimensional symmetry of physical laws is absolutely essential for explanations of experimental results and for the formulation of classical electrodynamics and QED; the universality of the one-way speed of light is irrelevant. In this connection, we stress that according to the theory of taiji relativity [4], the universality of the one-way or the two-way speed of light is a convention rather than an inherent part of the physical world [10].

270

Lorentz and Poincare Invariance

Note added in proofs Suppose one writes dwj = y(Tdw+ Uftdx), d«i = y(Vdx + Wfidw), dyl = dy, dz} = dz; y = 1/(1- f}'2)1/2, where T, U, V and W are four unknown functions of x and w. The new Wu transformation (7.1) for a constant-linear-acceleration (CLA) frame can be derived from the postulate of the limiting four-dimensional symmetry of taiji relativity and the initial condition that a CLA transformation reduces to the spatial identity r : = r when the taiji-time w = 0 and the initial "velocity" >30 = 0. This initial condition holds also for the Lorentz transformation. Thus, once the principle (or the first postulate) of relativity is rigorously stated to include the limiting cases, the concept of acceleration is determined in the physical theory based on "extended four-dimensional framework". Within the present conceptual framework, the taiji-time w in the Wu transormation (7.1) or (A.2) is a primary concept and has the dimension of length. The motion of physical objects, including light signals, is a derived concept and described by dimensionless "taiji-velocities" dr/dw. The taiji-time w can be realized by computerized "Leonardo clocks" [4]: We could program any Leonardo clock in a CLA frame F to obtain a reading w, from the nearest clock in an inertial frame Ft and, based on its F1 frame position Xi and given parameters a and /30, compute the taiji-time w it should display, iv = (wx + (l0/ay0)/[a(xi + l/ay0)] -l30/a. (See (7.2).) In the limit of zero acceleration w shown on a Leonardo clock will automatically reduce to the taiji-time in the four-dimensional transformation, w = z/oC^'i + Pox\)- It will not reduce to relativistic time, unless the second postulate of universal constant for the speed of light (w = ct, W\ = ct{) is made in this limit [41.

This paper is dedicated to Prof. TA-YOU W U for his wonderful and tireless teaching of physics and his ninetieth birthday. The work was supported in part by The Jing Shin Research Fund of the UMass Dartmouth and by a grant from the Potz Science Fund.

APPENDIX

Limiting four-dimensional symmetry and constant-linear-acceleration frames For simplicity, let us denote a CLA frame by F(w, x,y,z) and an inertial frame by F\(wi, Xi, yi, zi). Suppose a CLA frame F(w,x,y,z) is moving with a constant acceleration a, so that its velocity is (A.1)

/3 = aw + p0,

along the + x axis. Guided by the limiting four-dimensional symmetry, we find that the linearly accelerated transformation between Fl and F should be Wi = yfi(x + (A.2)

l/ay20)-p0/aya,

xl = y{x + l/ayl)-l/ay0,

P = aw + I30,

Vi = y, 2 1 2

y = l/(l-/3 ) / ,

Zi = z ,

y 0 = 1/(1 - / ^ ^ .

This is a generalization of the accelerated transformation obtained by Wu and Lee [7] based on a kinematic approach to satisfy limiting four-dimensional symmetry. It will be

Chap. 4. Extended Relativity ...

271

called the "Wu transformation". When /?0 approaches zero, the accelerated transformation (A.2) can reduce to the well-known transformation obtained in ref. [7], provided one uses a new time T :fi = aw + f}0 = tgh (at). Furthermore, one can verify that the Wu transformation (A.2) indeed reduces to four-dimensional transformations in the limit of zero acceleration a—•(): W\ = y 0 ( w + /*0x), Xi = y0(x + fi0w), yi = y, z\ = z; where y 0 = 1/(1 - Pi)1'2. With the definitions wx = cti and w = ct, the "extented relativistic time" t in the CLA frame F is completely determined by (A.2). In other words, if the time tx and the position x\ of an event as observed in the inertial frame F\ is known, then the corresponding time t in the CLA frame can be calculated, provided the constants c, a and /30 are given. Such a time t in a CLA frame can be physically realized by computerized "Leonardo clocks" [4]: evidently, the time t in the CLA frame is not the relativistic time in general and, hence, the constant c by itself is not physically meaningful. Only when the acceleration a vanishes, the time t in (A.2) for the frame F reduces to the relativistic time and c becomes physically meaningful. We may remark that the definitions W\ = ctx and w = ct are not necessary for deriving observable results because we may directly use w as evolution variables. When p0—>0, the inverse of the Wu transformation (A.2) leads to f w — c i : / ( l + ax{) = 0 ^ ( 1 — axi), \x=

2

2

ctx = wl,

( l / a ) [ l + 2axl + a (xf - c tf)]^

2

~ l/a~xI

- c2at?/2 .

Thus, c2a is related to a constant acceleration g in Newtonian mechanics by the relation a = g/c2

(A.4)

when velocities are small. In this sense, the Wu transformation (A.2) is a four-dimensional generalization of the Galilean transformation for accelerated frames in classical mechanics. From (A.1) we obtain (A.5)

ds 2 = c 2 At? - drf =firoodw2 - d r 2 ,

9oo

= y<(yo 2 + ax)2,

where x in a CLA frame F is restricted to the region x > xs = ~\/{ay\) which may be pictured as a "wall singularity" at x,. We may remark that finite Wu transformation (A.2) implies that the space-time of the CLA frame F(W, x, y, z) is flat, i.e. the Riemann curvature tensor vanishes, i?';ftm = 0, which can also be directly calculated by using the metric tensors in (A.5). The velocity of a fixed point xr in Fj as measured by F-observers using evolution variable [4] w is dx/dw with xx fixed. From (A.2), we find (A.6)

(dx/dw)Xi = Asroo)1/2;

x > - l/(a r 2 0 ),

1 > £2.

We see that only in the approximationfiroo— 1 do we have (dx/dw)Xl~ /3 and (A.7)

(d 2 x/dw 2 )x 1 * s a = constant.

We note that the Wu transformation (A.2) holds for general W\ and w. In the limit of zero acceleration, it reduces to the four-dimensional taiji transformation [4]. If one wishes, one may define (A.8)

wI = ctI and w = bt,

b = {ctY - # r i ) / [ * i ( l —/3g') - (£ - q')

xx/c\,

272

Lorentz and Poincare Invaxiance

where ti and t are, respectively, Einstein's time and "extended" Reichenbach's time (and b is the corresponding "ligh function"), then the limit of zero acceleration of (A.2) is the extended transformation (2.2) (where the inertial frame F' corresponds to the CLA frame F of (A.2) in the limit of zero acceleration). One can formulate, say, classical electrodynamics in a CLA frame. According to taiji relativity, physical results in the CLA frame F should be independent of the definition in (A.8).

REFERENCES [1] Hsu L., Hsu J. P. and SCHNEBLE D., NUOVO Cimento B, 111 (1996) 1299. This is referred as paper I in the text. [2] REICHENBACH H., The Philosophy of Space and Time (Dover, New York) 1958. [3] EDWARDS W.F., Am. J. Phys., 31 (1963) 482. [4] Hsu J. P. and Hsu L., Phys. Lett. A, 196 (1994) 1; 217 (1996) 359; Hsu L. and Hsu J. P., Nuovo Cimento B, 111 (1996) 1283. [5] See, for example, BJORKEN J. D. and DRELL S. D., Relativistic Quantum Mechanics (McGraw-Hill, New York) 1964, pp. 261-268 and pp. 285-286; SAKURAI J. J., Advanced Quantum Mechanics (Addison-Wesley, Reading, Mass.), 1967, pp. 171-172 and pp.181-188; WEINBERG S., The Quantum Theory of Fields (Cambridge University Press, New York) 1995, pp. 134-147. [6] M0LLER C , Danske Vid, Sel. Mat.-Fyz., xx, No. 19 (1943); FOCK V., The Theory of Space Time and Gravitation (Pergamon, New York) 1958, pp. 206-211; Wu T. Y. and LEE Y. C, Int. J. Theor. Phys., 5 (1972) 307; TA-YOU "WU, Theoretical Physics, Vol. 4, Theory of Relativity (Lian Jing Publishing Co., Taipei) 1978, pp. 172-175. [7] Hsu J. P., Nuovo Cimento B, 80 (1984) 183; 88 (1985) 140; Hsu J. P. and PEI S. Y., Phys. Rev. A, 37 (1988) 1406. [8] For a detailed discussion of common time in four-dimensional framework and its implications, see Hsu J. P., Nuovo Cimento B, 74 (1983) 67; 88 (1985) 140; 89 (1985) 30; Phys. Lett. A, 97 (1983) 137; Hsu J. P. and WHAN C, Phys. Rev. A, 38 (1988) 2248, appendix. [9] Hsu J. P., Nuovo Cimento B, 93 (1986) 178. [10] In other words, all physical results in taiji relativity or extended relativity can be derived by simply using the quantities (w, x, y, z) and (w', x', y', z') without ever mention time t or V (measured in seconds) and speeds of light or other physical objects.

Chapter 5

The Splendid Union of Special Relativity and Quantum Mechanics 5

5

P. A. M. Dirac (1927-1928), S. Tomonaga (1946), J. Schwinger (1948), R. Feynman (1949), F. Dyson (1949).

The Quantum Theory of the Emission and Absorption of Radiation

By P. A. M.

St. John's College, Cambridge, and Institute for Theoretical Physics, Copenhagen.

DIRAC,

(Communicated by N. Bohr, For. Mem. R.S.—Received February 2, 1927.) § 1. Introduction and Summary. The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed sufficiently to form a fairly complete theory of dynamics. One can treat mathematically the problem of any dynamical system composed of a number of particles with instantaneous forces acting between them, provided it is describable by a Hamiltonian function, and one can interpret the mathematics physically by a quite definite general method. On the other hand, hardly anything has been done up to the present on quantum electrodynamics. The questions of the correct treatment of a system in which the forces are propagated with the velocity of light instead of instantaneously, of the production of an electromagnetic field by a moving electron, and of the reaction of this field on the electron have not yet been touched. In addition, there is a serious difficulty in making the theory satisfy all the requirements of the restricted

274

Chap. 5. The Union of Special Relativity and Quantum ...

275

principle of relativity, since a Hamiltonian, function can no longer be used. This relativity question is, of course, connected with the previous ones, and it will be impossible to answer any one question completely without at the same time answering them all. However, it appears to be possible to build up a fairly satisfactory theory of the emission of radiation and of the reaction of the radiation field on the emitting system on the basis of a kinematics and dynamics which are not strictly relativistic. This is the main object of the present paper. The theory is non-relativistic only on account of the time being counted throughout as a o-number, instead of being treated symmetrically with the space co-ordinates. The relativity variation of mass with velocity is taken into account without difficulty. The underlying ideas of the theory are very simple. Consider an atom interacting with a field of radiation, which we may suppose for definiteness to be confined in an enclosure so as to have only a discrete set of degrees of freedom. Resolving the radiation into its Fourier components, we can consider the energy and phase of each of the components to be dynamical variables describing the radiation field. Thus if E r is the energy of a component labelled r and 0r is the corresponding phase (defined as the time since the wave was in a standard phase), we can suppose each E r and 0r to form a pair of canonically conjugate variables. In the absence of any interaction between the field and the atom, the whole system of field plus atom will be describable by the Hamiltonian H = ZrE, + H 0

(1)

equal to the total energy, H 0 being the Hamiltonian for the atom alone, since the variables E r , 6r obviously satisfy their canonical equations of motion •j? _

3H _

i _ 3H

_

When there is interaction between the field and the atom, it could be taken into account on the classical theory by the addition of an interaction term to the Hamiltonian (1), which would be a function of the variables of the atom and of the variables E„ 0r that describe the field. This interaction term would give the effect of the radiation on the atom, and also the reaction of the atom on the radiation field. In order that an analogous method may be used on the quantum theory, it is necessary to assume that the variables E r , 0f are q-numbers satisfying the standard quantum conditions 0,Er — E r 0 r = ih, etc., where h is (2TT) -1 times the usual Planck's constant, like the other dynamical variables of the problem. This assumption immediately gives light-quantum properties to

276

Lorentz and Poincare Invaxiance

the radiation.* For if vr is the frequency of the component r, 2-nvTQr is an angle variable, so that its canonical conjugate ET/2TCV,. can only assume a discrete set of values differing by multiples of h, which means that B r can change only by integral multiples of the quantum (2-KJI) vr. If we now add an interaction term (taken over from the clasical theory) to the Hamiltonian (1), the problem can be solved according to the rules of quantum mechanics, and we would expect to obtain the correct results for the action of the radiation and the atom on one another. I t will be shown that we actually get the correct laws for the emission and absorption of radiation, and the correct values for Einstein's A's and B's. In the author'3 previous theory,f where the energies and phases of the components of radiation were c-numbers, only the B's could be obtained, and the reaction of the atom on the radiation could not be taken into account. I t will also be shown that the Hamiltonian which describes the interaction of the atom and the electromagnetic waves can be made identical with the Hamiltonian for the problem of the interaction of the atom with an assembly of particles moving with the velocity of light and satisfying the Einstein-Bose statistics, by a suitable choice of the interaction energy for the particles. The number of particles having any specified direction of motion and energy, which can be used as a dynamical variable in the Hamiltonian for the particles, is equal to the number of quanta of energy in the corresponding wave in the Hamiltonian for the waves. There is thus a complete harmony between the wave and light-quantum descriptions of the interaction. We shall actually build up the theory from the light-quantum point of view, and show that the Hamiltonian transforms naturally into a form which resembles that for the waves. The mathematical development of the theory has been made possible by the author's general transformation theory of the quantum matrices.J Owing to the fact that we count the time as a c-number, we are allowed to use the notion of the value of any dynamical variable at any instant of time. This value is * Similar assumptions have been used by Born and Jordan [' Z. f. Physik,' vol. 34, p. 886 (1925)] for the purpose of taking over the classical formula for the emission of radiation by a dipole into the quantum theory, and by Born, Heisenberg and Jordan [' Z. f. Physik,' vol. 35, p. 606 (1925)] for calculating the energy fluctuations in a field of black-body radiation. t ' Roy. Soc. Proc.,' A, vol. 112, p. 661, § 5 (1926). This is quoted later by, loc. cit., I. % ' Roy. Soc. Proc.,' A, vol. 113, p. 621 (1927). This is quoted later by loc. cit., II. An essentially equivalent theory has been obtained independently by Jordan [' Z. f. Physik,' vol. 40, p. 809 (1927)]. See also, F. London, ' Z. f. Physik,' vol. 40, p. 193 (1926).

Chap. 5. The Union of Special Relativity

and Quantum ...

277

a q-number, capable of being represented by a generalised " matrix " according to many different matrix schemes, some of which may have continuous ranges of rows and columns, and may require the matrix elements to involve certain kinds of infinities (of the type given by the 8 functions*). A matrix scheme can be found in which any desired set of constants of integration of the dynamical system that commute are represented by diagonal matrices, or in which a set of variables that commute are represented by matrices that are diagonal at a specified time.f The values of the diagonal elements of a diagonal matrix representing any q-number are the characteristic values of that q-number. A Cartesian co-ordinate or momentum will in general have all characteristic values from — oo to + oo , while an action variable has only a discrete set of characteristic values. (We shall make it a rule to use unprimed letters to denote the dynamical variables or q-numbers, and the same letters primed or multiply primed to denote their characteristic values. Transformation functions or eigenfunctions are functions of the characteristic values and not of the q-numbers themselves, so they should always be written in terms of primed variables.) If/(£, V)) is any function of the canonical variables \k, 7jt, the matrix representing/at any time t in the matrix scheme in which the E,k at time t are diagonal matrices may be written down without any trouble, since the matrices representing the £,k and r)fc themselves at time t are known, namely,

u m = ^'8(ST)>

1 •(2)

vj* W) = -ih 8tex'-Si")...S ( ^ ' - ^ - i " ) 8' &'-!•/) S ( k + i ' - W ) - J Thus if the Hamiltonian H is given as a function of the \k and 7)*, we can at once write down the matrix H(^' £"). We can then obtain the transformation function, (£'/ a ') sa7> which transforms to a matrix scheme (a) in which the Hamiltonian is a diagonal matrix, as (£7 a ') must satisfy the integral equation H ({•'!•") at" (57a') = W («') . (r/a'),

(3)

of which the characteristic values W(oc') are the energy levels. This equation is just Schrodinger's wave equation for the eigenfunctions (£'/a')> "which becomes an ordinary differential equation when H is a simple algebraic function of the * Loc. cit. n , § 2. •)• One can have a matrix scheme in which a Bet of variables that commute are at all times represented by diagonal matrices if one will sacrifice the condition that the matrices most satisfy the equations of motion. The transformation function from snch a scheme to one in which the equations of motion are satisfied will involve the time explicitly. See p. 628 in loc. cit., II.

278 Lorentz and Poincare Invariance 5* and?]j. on account of the special equations (2) for the matrices representing 2,k and t\k. Equation (3) may be written in the more general form H (5T) &,' ( W = ih 3 (57«')/3«,

(3')

in which it can be applied to systems for which the Hamiltonian involves the time explicitly. One may have a dynamical system specified by a Hamiltonian H which cannot be expressed as an algebraic function of any set of canonical variables, but which can all the same be represented by a matrix H(i;'£"). Such a problem can still be solved by the present method, since one can still use equation (3) to obtain the energy levels and eigenfunctions. We shall find that the Hamiltonian which describes the interaction of a light-quantum and an atomic system is of this more general type, so that the interaction can be treated mathematically, although one cannot talk about an interaction potential energy in the usual sense. It should be observed that there is a difference between a light-wave and the de Broglie or Schrodinger wave associated with the light-quanta. Firstly, the light-wave is always real, while the de Broglie wave associated with a lightquantum moving in a definite direction must be taken to involve an imaginary exponential. A more important difference is that their intensities are to be interpreted in different ways. The number of light-quanta per unit volume associated with a monochromatic light-wave equals the energy per unit volume of the wave divided by the energy (2TCA)V of a single light-quantum. On the other hand a monochromatic de Broglie wave of amplitude a (multiplied into the imaginary exponential factor) must be interpreted as representing o2 lightquanta per unit volume for all frequencies. This is a special case of the general rule for interpreting the matrix analysis,* according to which, if (£'/*') or <|v (!;*') is the eigenfunction in the variables Z,k of the state a' of an atomic system (or simple particle), | (jv (£*')|2 is the probability of each ?,k having the value £jfc', [or | <|v(£*/) | 2 dE,x' &£,%'... is the probability of each \k lying between the values \k and Ek + d£,k', when the %k have continuous ranges of characteristic values] on the assumption that all phases of the system are equally probable. The wave whose intensity is to be interpreted in the first of these two ways appears in the theory only when one is dealing with an assembly of the associated particles satisfying the Einstein-Bose statistics. There is thus no such wave associated with electrons. * Loc. cit., H, §§ 6, 7.

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and Quantum ...

279

§ 2. The Perturbation of an Assembly of Independent Systems. We shall now consider the transitions produced in an atomic system by an arbitrary perturbation. The method we shall adopt will be that previously given by the author,! which leads in a simple way to equations which determine the probability of the system being in any stationary state of the unperturbed system at any time.J This, of course, gives immediately the probable number of systems in that state at that time for an assembly of the systems that are independent of one another and are all perturbed in the same way. The object of the present section is to show that the equations for the rates of change of these probable numbers can be put in the Hamiltonian form in a simple manner, which will enable further developments in the theory to be made. Let H 0 be the Hamiltonian for the unperturbed system and V the perturbing energy, which can be an arbitrary function of the dynamical variables and may or may not involve the time explicitly, so that the Hamiltonian for the perturbed system is H = H 0 + V. The eigenfunctions for the perturbed system must satisfy the wave equation

ih mdt = (H0 + V) 4>, where (H 0 + V) is an operator. If <]> = E ^ ^ r is the solution of this equation that satisfies the proper initial conditions, where the tp/s are the eigenfunctions for the unperturbed system, each associated with one stationary state labelled by the suffix r, and the ar's are functions of the time only, then I aT 12 is the probability of the system being in the state r at any time. The ar's must be normalised initially, and will then always remain normalised. The theory will apply directly to an assembly of N similar independent systems if we multiply each of these a r 's by N* so as to make S r | ar | 2 = N. We shall now have that | ar | 2 is the probable number of systems in the state r. The equation that determines the rate of change of the aT's is§ ihdr = E s V„a„ where the V„'s are the elements of the matrix representing V. imaginary equation is - ihd* = X,Y„*a* = -Lji*Y„.

(4) The conjugate (4')

t Loc.tit.I. % The theory has recently been extended by Born [' Z. f. Physik,' vol. 40, p. 167 (1926)] so as to take into account the adiabatic changes in the stationary states that may be produced by the perturbation as well as the transitions. This extension is not used in the present paper. § Loc. tit., I, equation (25).

280 Lorentz and Poincare Invariance If we regard aT and ih at* as canonical conjugates, equations (4) and (4') take the Hamiltonian form with, the Hamiltonian function F x = Z„a,.*Vrja„ namely,

£& = 1 JUi. <&

iA 3 a r * '

a


*-


3Fl

8a r

We can transform to the canonical variables N„ <j>T by the contact transformation ar = N,**-**^*, a,* = N r V* /A . This transformation makes the new variables N r and j>T real, N r being equal to aji* = \arl2, the probable number of systems in the state r, and ^r/A being the phase of the eigenfunction that represents them. The Hamiltonian F x now becomes Fx = S ^ W N . V * - " " ' * , and the equations that determine the rate at which transitions occur have the canonical form

A more convenient way of putting the transition equations in the Hamiltonian form may be obtained with the help of the quantities 6r = o r e- iW ' i/A J

b* == a* e i W ^ ,

W r being the energy of the state r. We have | br | 2 equal to | ar | 2 , the probable number of systems in the state r. For br we find

with the help of (4). If we put V„ = v„e' (Wr_w,) " A , so that vT, is a constant when V does not involve the time explicitly, this reduces to ih br = W A + Z.tvA = S.H.A,

(5)

where HT, = W r S„ + v„, which is a matrix element of the total Hamiltonian H = H 0 + V with the time factor e'( w '- w ')(/» removed, so that H „ is a constant when H does not involve the time explicitly. Equation (5) is of the same form as equation (4), and may be put in the Hamiltonian form in the same way. I t should be noticed that equation (5) is obtained directly if one writes down the Schrodinger equation in a set of variables that specify the stationary states of the unperturbed system. If these variables are i;ft, and if H(5'£") denotes VOL. cxiv.—A.

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281

a matrix element of the total Hamiltonian H in the (5) scheme, this Schrodinger equation would be ih 3+ (l')ldt = S r H (l-T) + m,

(6)

like equation (3'). This differs from the previous equation (5) only in the notation, a single suffix r being there used to denote a stationary state instead of a set of numerical values £*' for the variables E,k, and bT being used instead of cp (£'). Equation (6), and therefore also equation (5), can still be used when the Hamiltonian is of the more general type which cannot be expressed as an algebraic function of a set of canonial variables, but can still be represented by a matrix H ( £ T ) or H„We now take bT and ih br* to be canonically conjugate variables instead of o r and ih ar*. The equation (5) and its conjugate imaginary equation will now take the Hamiltonian form with the Hamiltonian function F = 2WH^..

(7)

Proceeding as before, we make the contact transformation br = Nr* «-*'*

b* = Nr* eis'lh,

(8)

to the new canonical variables N„ 0r, where N r is, as before, the probable number of systems in the state r, and 6r is a new phase. The Hamiltonian F will now become •F=S„H r ,N r *N^e i ^-«->/*, and the equations for the rates of change of N r and 6r will take the canonical form N - _ ^

e -3F

The Hamiltonian may be written F = S,W,N r + 2 rs «„ N r i N,* J » " W*.

(9)

The first term ~LrWjST is the total proper energy of the assembly, and the second may be regarded as the additional energy due to the perturbation. If the perturbation is zero, the phases 9r would increase linearly with the time, while the previous phases T would in this case be constants. § 3. The Perturbation of an Assembly satisfying the Einstein-Bose Statistics. According to the preceding section we can describe the effect of a perturbation on an assembly of independent systems by means of canonical variables and Hamiltonian equations of motion. The development of the theory which

282

Lorentz and Poincare Invariance

naturally suggests itself is to make these canonical variables q-numbers satisfying the usual quantum conditions instead of c-numbers, so that their Hamiltonian equations of motion become true quantum equations. The Hamiltonian function will now provide a Schrb'dinger wave equation, which must be solved and interpreted in the usual manner. The interpretation will give not merely the probable number of systems in any state, but the probability of any given distribution of the systems among the various states, this probability being, in fact, equal to the square of the modulus of the normalised solution of the wave equation that satisfies the appropriate initial conditions. We could, of course, calculate directly from elementary considerations the probability of any given distribution when the systems are independent, as we know the probability of each system being in any particular state. We shall find that the probability calculated directly in this way does not agree with that obtained from the wave equation except in the special case when there is only one system in the assembly. In the general case it will be shown that the wave equation leads to the correct value for the probability of any given distribution when the systems obey the Einstein-Bose statistics instead of being independent. We assume the variables br, ihbr* of § 2 to be canonical q-numbers satisfying the quantum conditions br. ih b* — ih br* .br = ih or and

bfi* — b*br = 1, bjb, — b,br = 0, bjb* - b*br = 0

b*b* — b*br * = 0, (« ^ r).

The transformation equations (8) must now be written in the quantum form br = (Nr + 1)* e-'8'"1 = e-*/»N,* b* = N r V'' A = «*/* (Nr + 1)*,

(10)

in order that the N„ 0, may also be canonical variables. These equations show that the N r can have only integral characteristic values not less than zero,j" which provides us with a justification for the assumption that the variables are q-numbers in the way we have chosen. The numbers of systems in the different states are now ordinary quantum numbers. f See § 8 of the author's paper ' Boy. Soc. Proc.,' A, vol. I l l , p. 281 (1926). What are there called the c-number values that a q-number can take are here given the more preoise name of the characteristic values of that q-number.

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283

The Hamiltonian (7) now becomes

= SJEJtf,* (N. + 1 - *„)**<*-** in which the H,, axe still c-numbers. sponding to (9)

(11)

We may write this F in the form corre-

F = E , W A + Xr,vrFr* (N, + 1 - 8J

e**-«/*

(11')

in which it is again composed of a proper energy term S^WyN, and an interaction energy term. The wave equation written in terms of the variables N r isf ih | + (W, N 2 ', N / ...) = F+ (N/, N,', N , ' ...),

(12)

where F is an operator, each 0r occurring in F being interpreted to mean ih 3/3N/. If we apply the operator e±i0'lh to any function / ( N / , N 2 ', ... N / , ...) of the variables N x ', N 2 ', ... the result is e * * / * / ^ ' , N 2 ', ... N / f ... ) = e ^ / ^ 7 ( N 1 ' ) N 2 ', ... N / ... ) = /(N x ', N 2 ', ... N / T 1, ... ). If we use this rule in equation (12) and use the expression (11) for F we obtain^ tt!N,',N,'...) = Sr4H„N/* ( N /

+

i _ S„)i<j, ( N l ', N 2 ' . . . N / - 1 , . . . N / + 1, ...). (13)

We see from the right-hand side of this equation that in the matrix representing F, the term in F involving e1 (Sr ~ , , ) / h will contribute only t o those matrix elements that refer to transitions in which N r decreases by unity and N, increases by unity, i.e., to matrix elements of the type F ^ ' , N 2 ' . . . N / . . . N , ' ; N x ', N 2 ' ... N / - 1 ... N / + 1 ...). If we find a solution ^ ( N / , N 2 ' ...) of equation (13) that is normalised [i.e., one for which Sir,',N,'... I "K^i', N 2 ' . . . ) | 2 = 1] and that satisfies the proper initial conditions, then | <\> (Nj', N 2 ' ...) I2 will be the probability of that distribution in which N x ' systems are in state 1, N 2 ' in state 2, ... at any time. Consider first the case when there is only one system in the assembly. The probability of its being in the state q is determined by the eigenfunction j- We axe supposing for definiteness that the label r of the stationary states takes the values 1, 2, 3, .... % When a = r, f (N/, N 2 '... Nr' - 1... N,' + 1) is to be taken to mean f (Nj'N,'... N / . . . ) .

284 Lorentz and Poincare Invariance ^(Nj', N 2 ', ...) in wkich all the N"s are put equal to zero except N,', which is put equal to unity. This eigenfunction -we shall denote by <\i {q}. When it is substituted in the left-hand side of (13), all the terms in the summation on the right-hand side vanish except those for -which r = q, and we are left with

which is the same equation as (5) with <\i {q} playing the part of bq. This establishes the fact that the present theory is equivalent to that of the preceding section when there is only one system in the assembly. Now take the general case of an arbitrary number of systems in the assembly, and assume that they obey the Einstein-Bose statistical mechanics. This requires that, in the ordinary treatment of the problem, only those eigenfunctions that are symmetrical between all the systems must be taken into account, these eigenfunctions being by themselves sufficient to give a complete quantum solution of the problem.f We shall now obtain the equation for the rate of change of one of these symmetrical eigenfunctions, and show that it is identical with equation (13). If we label each system with a number n, then the Hamiltonian for the assembly will be H A = £„H (n), where H (n) is the H of § 2 (equal to H 0 + V) expressed in terms of the variables of the nth system. A stationary state of the assembly is denned by the numbers rlt r2... rn ... which are the labels of the stationary states in which the separate systems lie. The Schrodinger equation for the assembly in a set of variables that specify the stationary states will be of the form (6) [with H A instead of H], and we can write it in the notation of equation (5) thus :— ihb(r^a

...) = S,„,,...H A (r x r 2 ... ; s^

...)b(SjS2...),

(14)

where H A (rjr 2 ... ; SjS^ ...) is the general matrix element of H A [with the time factor removed]. This matrix element vanishes when more than one sn differs from the corresponding r n ; equals H r . , , when sro differs from rn and every other sn equals r„ ; and equals S n H,. r , when every s„ equals r„. Substituting these values in (14), we obtain ihb^rz ...) =S m S f a l t ,JH r i J > .6(f 1 r,... r»_i««r» + i...) + 2^Hrwrw6(r1r,...). (15) We must now restrict b ( r ^ ...) to be a symmetrical function of the variables r,. r» ... in order to obtain the Einstein-Bose statistics. This is permissible since if b (r^,, ...) is symmetrical at any time, then equation (15) shows that f Loc. eft., I, § 3.

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285

b(r1r2 ...) is also symmetrical at that time, so that b ( r ^ ...) will remain symmetrical. Let N r denote the number of systems in the state r. Then a stationary state of the assembly describable by a symmetrical eigenfunction may be specified by the numbers N1} N 2 ... N r ... just as well as by the numbers rv r2 ... r„ .... and we shall be able to transform equation (15) to the variables N"1; N 2 .... We cannot actually take the new eigenfunction b (Nx, N 2 ...) equal to the previous one 6 ( r ^ ...), but must take one to be a numerical multiple of the other in order that each may be correctly normalised with respect to its respective variables. We must have, in fact, ST* r,... | & far, . . . ) | 2 = 1 =S f f a l r,...|6(N 1 , N 2 . . . ) | 2 , and hence we must take | b(Nj, N 2 ...) | 2 equal to the sum of | b [r1rz ...) | 2 for all values of the numbers rlt r 2 ... such that there are N t of them equal to 1, N 2 equal to 2, etc. There are N \[NX ! N 2 ! ... terms in this sum, where N = 2rN r is the total number of systems, and they are all equal, since &(rjr 2 ...) is a symmetrical function of its variables rx, r 2 .... Hence we must have 6(N 1 ,N 2 ...) = ( N ! / N 1 ! N 2 ! . . . ) H ( r 1 r 2 . . . ) . If we make this substitution in equation (15), the left-hand side will become ih (N, ! N 2 ! ... /N !)* b (Nx, N 2 ...). The term BTmSrb {rxrz ... r m _ x smrm+1 ...) in the first summation on the right-hand side will become [ N 1 ! N 2 ! . . . ( N r - l ) ! . . . ( N , + l ) ! . . . / N ! ] * H r s 6 ( N 1 , N 2 . . . N r - l . . . N . + l...), (16) where we have written r for rm and s for sm. This term must be summed for all values of s except r, and must then be summed for r taking each of the values rv r 2 .... Thus each term (16) gets repeated by the summation process until it occurs a total of N r times, so that it contributes N r [N x ! N 2 ! . . . (Nr - 1)!... ( N . + l ) ! .../N!]* H„& ( N ^ N , . . . N r - 1 ...N. + 1 . . . ) = Nr*(N3 + l ) i ( N 1 ! N 2 ! . . . / N ! ) i H r ^ ( N 1 , N 2 . . . N r - l . . . N J + l . . . ) to the right-hand side of (15). Finally, the term I ^ H ^ ^ (rv r 2 ...) becomes L f N f H f f .6(f 1 r,...)=S f N f H f r .(N 1 !Br a l.../NI)*6(N 1 ,N a ...). Hence equation (15) becomes, with the removal of the factor (Nx ! N 2 I.../N !)*, ihb (N 1 ( N 2 ...) = S . 2 U N , * ( N . + l p H ^ { S v N 2 . . . N , - l . . . N. + 1 . . . ) + S,N r H n 4(N 1> N 2 ...),

(17)

286

Lorentz and Poincare Invariance

which, is identical with (13) [except for the fact that in (17) the primes have been omitted from the N's, which is permissible when we do not require to refer to the N's as q-numbers]. We have thus established that the Hamiltonian (11) describes the effect of a perturbation on an assembly satisfying the EinsteinBose statistics.

§ 4. The Reaction of the Assembly on the Perturbing System. Up to the present we have considered only perturbations that can be represented by a perturbing energy V added to the Hamiltonian of the perturbed system, V being a function only of the dynamical variables of that system and perhaps of the time. The theory may readily be extended to the case when the perturbation consists of interaction with a perturbing dynamical system, the reaction of the perturbed system on the perturbing system being taken into account. (The distinction between the perturbing system and the perturbed system is, of course, not real, but it will be kept up for convenience.) We now consider a perturbing system, described, say, by the canonical variables Jk, oik, the J's being its first integrals when it is alone, interacting with an assembly of perturbed systems with no mutual interaction, that satisfy the Einstein-Bose statistics. The total Hamiltonian will be of the form H T = H p (J) + E B H(n), where H p is the Hamiltonian of the perturbing system (a function of the J's only) a n d H (n) is equal to the proper energy H 0 (n) plus the perturbation energy V(w) of the nth system of the assembly. H (n) is a function only of the variables of the nth system of the assembly and of the J's and w's, and does not involve the time explicitly. The Schrodinger equation corresponding to equation (14) is now ih'b ( J ' , r 1 r 2 . . . ) = E J , , E v v . . H T ( J ' , r 1 r 2 . . . ; J ' , V l . . . ) 6(J",

Va

..,),

in which the eigenfunction 6 involves the additional variables Jk'. The matrix element H T (J', rxr2 ...; J", sts2 ...) is now always a constant. As before, it vanishes when more than one s„ differs from the corresponding r n . When sm differs from rm and every other sn equals r„, it reduces to H (J'rm; J"s m ), which is the (J'r m ; 3"sm) matrix element (with the time factor removed) of H = H 0 -f- V, the proper energy plus the perturbation energy of a single system of the assembly; while when every sn equals r„, it has the value H P (J') SJ/J/, + 2 „ H (J'r„; J*r n ). If, as before, we restrict the eigenfunctions

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287

to be symmetrical in the variables ra, r 2 ..., we can again transform to the variables N 1; N 2 ..., which will lead, as before, to the result * 6 (J', N x ', N , ' ...) = H P ( J ' ) b(J', Wv N , ' . . . ) + S J „ 2 r . A ' i ( N / + l - S „ ) i H ( J ' r ; J" S )&(J",N 1 ',N 2 '...N/-1...N/ + 1...) (18) This is the Schrodinger equation corresponding to the Hamiltonian function F = H P (J) + S r .,H„N r *(N i + l-8„)*e < «-*>'\

(19)

in which H„ is now a function of the J's and w's, being such that when represented by a matrix in the (J) scheme its (J' J") element is H (J'r; J"s). (It should be noticed that H„ still commutes with the N's and 0's.) Thus the interaction of a perturbing system and an assembly satisfying the Einstein-Bose statistics can be described by a Hamiltonian of the form (19). "We can put it in the form corresponding to (11') by observing that the matrix element H (J'r; J"s) is composed of the sum of two parts, a part that comes from the proper energy H 0 , which equals W r when J / = J*' and s = r and vanishes otherwise, and a part that comes from the interaction energy V, which may be denoted by v (J'r ; J"s). Thus we shall have

H„ = Wr S„ + v„, where vr, is that function of the J's and w>'s which is represented by the matrix whose (J' J") element is v (JV ; J's), and so (19) becomes F = H P (J) + S r W f N r + Sr..i;„Nr* (N, + 1 - S J ^ * - * ' * .

(20)

The Hamiltonian is thus the sum of the proper energy of the perturbing system H P (J), the proper energy of the perturbed systems S f W ^ , and the perturbation energy 2,, ,»„!*,* (N. + 1 - SJ* jV'-W. § 5. Theory of Transitions in a System from One State to Others of the Same Energy. Before applying the results of the preceding sections to light-quanta, we shall consider the solution of the problem presented by a Hamiltonian of the type (19). The essential feature of the problem is that it refers to a dynamical system which can, under the influence of a perturbation energy which does not involve the time explicitly, make transitions from one state to others of the same energy. The problem of collisions between an atomic system and an electron, which has been treated by Born,* is a special case of this type. Born's method is to find a periodic solution of the wave equation which consists, in BO far as it involves the co-ordinates of the colliding electron, of plane waves, * Barn,' Z. f. Phyedk,' vol 38, p. 803 (1926).

288 Lorentz and Poincare Invariance representing the incident electron, approaching the atomic system, which are scattered or diffracted in all directions. The square of the amplitude of the •waves scattered in any direction with any frequency is then assumed by Born to be the probability of the electron being scattered in that direction with the corresponding energy. This method does not appear to be capable of extension in any simple manner to the general problem of systems that make transitions from one state to others of the same energy. Also there is at present no very direct and certain way of interpreting a periodic solution of a wave equation to apply to a non-periodic physical phenomenon such as a collision. (The more definite method that will now be given shows that Bom's assumption is not quite right, it being necessary to multiply the square of the amplitude by a certain factor.) An alternative method of solving a collision problem is to find a non-periodic solution of the wave equation which consists initially simply of plane waves moving over the whole of space in the necessary direction with the necessary frequency to represent the incident electron. In course of time waves moving in other directions must appear in order that the wave equation may remain satisfied. The probability of the electron being scattered in any direction with any energy will then be determined by the rate of growth of the corresponding harmonic component of these waves. The way the mathematics is to be interpreted is by this method quite definite, being the same as that of the beginning of §2. We shall apply this method to the general problem of a system which makes transitions from one state to others of the same energy under the action of a perturbation. Let H 0 be the Hamiltonian of the unperturbed system and V the perturbing energy, which must not involve the time explicitly. If we take the case of a continuous range of stationary states, specified by the first integrals, afc say, of the unperturbed motion, then, following the method of § 2, we obtain ih d (a') = f V (a'a*) do." . a (a'),

(21)

corresponding to equation (4). The probability of the system being in a state for which each <xfc lies between xk' and oc^'-f^a*' at a n v t u i i e *s I a (<*') l ^ i ' • ^ • • • when a (a') is properly normalised and satisfies the proper initial conditions. If initially the system is in the state a0, we must take the initial value of a (a') to be of the form a 0 . S(a' — a 0 ). We shall keep a0 arbitrary, as it would be inconvenient to normalise a (a') in the present case. For a first approximation

Chap. 5. The Union of Special Relativity

and Quantum . ..

289

we may substitute for a (a") in the right-hand side of (21) its initial value. This gives ih d(cc') = a°V(a'a°) = a M * ' * V I W M ~ W M I ' / * » where v (x'aP) is a constant and W (a') is the energy of the state a.'. Hence „

<>i[W<«')-W(aW/A_ 1

a a M = «• 8 V - «°> + A M ;

L W ( 0

_

W

^-

TO 0

For values of the xk such that W (a') differs appreciably from W (a ), a (a') is a periodic function of the time whose amplitude is small when the perturbing energy V is small, so that the eigenfunctions corresponding to these stationary states are not excited to any appreciable extent. On the other hand, for values of the ock such that W (a') = W (a 0 ). and <xfc' j± xk° for some k, a (a') increases uniformly with respect to the time, so that the probability of the system being in the state «' at any time increases proportionally with the square of the time. Physically, the probability of the system being in a state with exactly the same proper energy as the initial proper energy W (<x°) is of no importance, being infinitesimal. We are interested only in the integral of the probability through a small range of proper energy values about the initial proper energy, which, as we shall find, increases linearly with the time, in agreement with the ordinary probability laws. We transform from the variables aj, a 2 ... ix, to a set of variables that are arbitrary independent functions of the oc's such that one of them is the proper energy W, say, the variables W, y 1; y 2 , ... y„_i. The probability at any time of the system lying in a stationary state for which each y* lies between yk and Tfc' + ^Yfe' i s n o w (apart from the normalising factor) equal to

dYl' . dy2' ... dfS

f | a(a') ] » ^ K ' ^ / - O

^

(23)

For a time that is large compared with the periods of the system we shall find that practically the whole of the integral in (23) is contributed by values of W' very close to W° = W(oc°). Put a ( a ' ) = a ( W ' ) T 0 and dfa', « 2 ' . . . : 0 / 3 ( W ' , y / ... y u _i') = J ( W , y')Then for the integral in (23) we find, with the help of (22) (provided yA' 9^ -fk0, for some k) f | a (W, y') |2 J (W, y') dW'

= I *° |»J I v (W, y'; W°, y°) |2 J (W, y')

fe

^ V y i

~ ^

= 21a012 f |v(W',y'; W°,y°)l2 J ( W / , y ' ) [ l - c o s ( W ' - W ° ) i / A ] / ( W ' - W 0 ) 2 . d W ' = 2\a°\*tlh.

f | v (W° + hxjt, y'; W°, y°) | 2 J (W°+ Jix/t, y') (1 - cos x)jxl. dx,

290 Lorentz and Poincare Invariance if one makes the substitution (W'—W°)t/A = x. For large values of t this reduces to 21 o° | 2 tfh. 1 v (W°, Y ' ; W°, Y°) | 2 J (W°, y') f

(1-cos x)l& . dx

J —»

= 2u | a01« * / i . | «(W°, T ' ; W°, Y°) | 2 J (W°, y'). The probability per unit time of a transition to a state for which each yk lies between yk' and yk + ^Y*' ^ thus (apart from the normalising factor) 2* | a° |»/& . |«(W°, Y ' ; W°, Y°) | 2 J (W°, T ') d T l ' • d Y l ' . . . <*Y«_I', (24) which is proportional to the square of the matrix element associated with that transition of the perturbing energy. To apply this result to a simple collision problem, we take the a's to be the components of momentum px, pv, p, of the colliding electron and the Y' S to be 9 and , the angles which determine its direction of motion. If, taking the relativity change of mass with velocity into account, we let P denote the resultant momentum, equal to (px2 + J>y2 + Pt2)*> and E the energy, equal to (m2ci-{-'Pic?)i, of the electron, m being its rest-mass, we find for the Jacobian 3 (E, 6, )

c2

Thus the J (W°, Y') of the expression (24) has the value J (W°, Y') = E ' P sin B'/c2,

(25)

where E' and P' refer to that value for the energy of the scattered electron which makes the total energy equal the initial energy W° (i.e., to that value required by the conservation of energy). We must now interpret the initial value of a (a'), namely, a0 8 (a' — a 0 ), which we did not normalise. According to § 2 the wave function in terms of the variables a.k is b (a')= a (a') e - f w '' A , so that its initial value is o° 8 (a' - a0) e-iWt'h = a0 8{px' - px°) 8 «

- py°) 8 (pz' - p°) e"™""*.

If we use the transformation function* (x'/p') =

(2n1i)-sliei^"':c'lh,

and the transformation rule * (x') = |(x'/j>') i> (p') dp,' dpv' dp,', we obtain for the initial wave function in the co-ordinates x, y, z the value o°

(2nh)-s,2ei'!-"''-°x'lhe-iW'tlh.

* The symbol x is used for brevity to denote x, y, z.

Chap. 5. The Union of Special Relativity

and Quantum ... 291

This corresponds to an initial distribution of | a 0 1 2 (2-Kh)~z electrons per unit volume. Since their velocity is P V / E 0 , the number per unit time striking a unit surface at right-angles to their direction of motion is \a°\2P°c2/(27rA)3E°. Dividing this into the expression (24) we obtain, with the help of (25), i-K*(2Tzh)*^\v(P'; p°)\^0smV

dVdfi.

(26)

This is the effective area that must be hit by an electron in order that it shall be scattered in the solid angle sin 0' dQ' d' with the energy E'. This result differs by the factor (27cA)2/2mE' . P'/P° from Born's.* The necessity for the factor P'/P° in (26) could have been predicted from the principle of detailed balancing, as the factor | v (p'; p°) | 2 is symmetrical between the direct and reverse processes.^ § 6. Application to Light-Quanta. We shall now apply the theory of § 4 to the case when the systems of the assembly are light-quanta, the theory being applicable to this case since lightquanta obey the Einstein-Bose statistics and have no mutual interaction. A light-quantum is in a stationary state when it is moving with constant momentum in a straight line. Thus a stationary state r is fixed by the three components of momentum of the light-quantum and a variable that specifies its state of polarisation. We shall work on the assumption that there are a finite number of these stationary states, lying very close to one another, as it would be inconvenient to use continuous ranges. The interaction of the light-quanta with an atomic system will be described by a Hamiltonian of the form (20), in which H P ( J ) is the Hamiltonian for the atomic system alone, and the coefficients vn are for the present unknown. We shall show that this form for the Hamiltonian, with the vT, arbitrary, leads to Einstein's laws for the emission and absorption of radiation. The light-quantum has the peculiarity that it apparently ceases to exist when it is in one of its stationary states, namely, the zero state, in which its momentum, and therefore also its energy, are zero. When a light-quantum is absorbed it can be considered to jump into this zero state, and when one is emitted it can be considered to jump from the zero state to one in which it is * In a more recent paper (' Nachr. Gesell. d. Wias.,' Gottingen, p. 146 (1926)) Born has obtained a result in agreement with that of the present paper for non-relativity mechanics, by rising an interpretation of the analysis based on the conservation theorems. I am indebted to Prof. N. Bohr for seeing an advance copy of this work. t See Klein and Rosseland, ' Z. f. Physik,' vol. 4, p. 46, equation (4) (1921).

292 Lorentz and Poincare Invariance physically in evidence, so that it appears to have been created. Since there is no limit to the number of light-quanta that may be created in this way, we must suppose that there are an infinite number of light-quanta in the zero state, so that the N 0 of the Hamiltonian (20) is infinite. We must now have 80, the variable canonically conjugate to N 0 , a constant, since 0O = 3F/3N 0 = W 0 + terms involving N„-* or (N0 + 1)-* and W0 is zero. In order that the Hamiltonian (20) may remain finite it is necessary for the coefficients i^, v^ to be infinitely small. We shall suppose that they are infinitely small in such a way as to make v^$£ and v<^£ finite, in order that the transition probability coefficients may be finite. Thus we put UH) (N0 + 1)* e-w°lh = v„ VorNjeW = v*, where vr and vr* are finite and conjugate imaginaries. We may consider the vr and v* to be functions only of the J's and w's of the atomic system, since their factors (N0 + 1)* e~"VA and N 0 V° / A are practically constants, the rate of change of N 0 being very small compared with N 0 . The Hamiltonian (20) now becomes F = H P (J) + S r W r N f + S^oKNVV*/" + vr*(NT + l ) » e - * ^ + S r „ o S ^ o t>„Nr* (N. + 1 - S„)le< «'-°»\

(27)

The probability of a transition in which a light-quantum in the state r is absorbed is proportional to the square of the modulus of that matrix element of the Hamiltonian which refers to this transition. This matrix element must come from the term ivNrVr'"> in the Hamiltonian, and must therefore be proportional to N/* where N / is the number of light-quanta in state r before the process. The probability of the absorption process is thus proportional to N,'. In the same way the probability of a light-quantum in state r being emitted is proportional to (N/ + 1), and the probability of a light-quantum in state r being scattered into state s is proportional to N / (N/ -j-1). Eadiative processes of the more general type considered by Einstein and Ehrenfest,| in which more than one light-quantum take part simultaneously, are not allowed on the present theory. To establish a connection between the number of light-quanta per stationary state and the intensity of the radiation, we consider an enclosure of finite volume, A say, containing the radiation. The number of stationary states for light-quanta of a given type of polarisation whose frequency lies in the t * Z. f. Physik,' vol. 19, p. 301 (1923).

Chap. 5. The Union of Special Relativity and Quantum ...

293

range vr to vr -\- dvr and whose direction, of. motion lies in the solid angle dar about the direction of motion for state r will now be A V ^ V ^ G V / C 3 . The energy of the light-quanta in these stationary states is thus N / . 2TtAvr. Av^dv^o^/c 3 . This must equal Ac""1ITivri
(28)

so that N / is proportional to I, and"(N/ -f- 1) is proportional to I r + (27tA)vr3/c2. We thus obtain that the probability of an absorption process is proportional to I„ the incident intensity per unit frequency range, and that of an emission process is proportional to I r + (2TcA)vr3/c2, which are just Einstein's laws.* I n the same way the probability of a process in which a light-quantum is scattered from a state r to a state s is proportional to I r [ I , + (2izh)vT3/c2], which is Pauli's law for the scattering of radiation by an electron, f §7. The Probability Coefficients for Emission and Absorption. We shall now consider the interaction of an atom and radiation from the wave point of view. We resolve the radiation into its Fourier components,, and suppose that their number is very large but finite. Let each component be labelled by a suffix r, and suppose there are ar components associated with the radiation of a definite type of polarisation per unit solid angle per unit frequency range about the component r. Each component r can be described by a vector potential kT chosen so as to make the scalar potential zero. The perturbation term to be added to the Hamiltonian will now be, according to the classical theory with neglect of relativity mechanics, c - 1 S r «f X,, where X, is the component of the total polarisation of the atom in the direction of KT, which is the direction of the electric vector of the component r. We can, as explained in § 1, suppose the field to be described by the canonical variables N„ 0r, of which N r is the number of quanta of energy of the component r, and 9 r is its canonically conjugate phase, equal to 2-jzhv,. times the 8 r of §1. We shall now have «r = a r cos 8, /h, where aT is the amplitude of JC„ which can be connected with N r as follows:—The flow of energy per unit area per unit time for the component r is \KC~Xar2vr2. Hence the intensity * The ratio of stimulated to spontaneous emission in the present theory is just twice its value in Einstein's. This is because in the present theory either polarised component of the incident radiation can stimulate only radiation polarised in the same way, while in Einstein's the two polarised components are treated together. This remark applies also to the scattering process. t Pauli, \Z. f. Physik,' vol. 18, p. 272 (1923).

294 Lorentz and Poincaie Invariance per unit frequency range of the radiation in the neighbourhood of the component r is I r = ^7tc -1 a r 2 v r 2 a r . Comparing this with equation (28), we obtain aT = 2 (AVf/cff^'N,*, and hence Kr = 2 (Avr/<wr)* Nr* cos 8r/A. The Hamiltonian for the whole system of atom plus radiation would now be, according to the classical theory, F = H P (J) + S r (27tAvr) N r + 2c- 1 S r (Avr/<w,)1 2 W cos 8,/A, (29) where Hp (J) is the Hamiltonian for the atom alone. On the quantum theory we must make the variables N r and 0r canonical q-numbers like the variables J*, wk that describe the atom. We must now replace the Nr- cos 9r/A in (29) by the real q-number \ {Nr* «*'* + e-*/* N/"} = i {Nr* e*'» + (Nr + 1 ) * «-*•'»} so that the Hamiltonian (29) becomes F = H P (J) + 2 r (27civr) N r +A* c-3 S, (v,/
x

' 0\ tta(«,) = 8.v + t>(«'«°H i[W(a')-W(a°)]/A'

Chap. 5. The Union of Special Relativity and Quantum ...

295

corresponding to (22). If, as before, we transform to the variables W, y x , Y2 ••• Yu-i> we obtain (when y' ?± y°) a (WY) = t»(W', y ' ; W°, Y°) [l-«'<*'-•"•> «*J/(W'-W°). The probability of the system being in a state for which each yk equals ytr *s S W ' | a (W y')| 2 . If the stationary states lie close together and if the time t is not too great, we can replace this sum. by the integral (AW) - 1 | a (WY) I2 dW', where AW is the separation between the energy levels. Evaluating this integral as before, we obtain for the probability per unit time of a transition to a state for which each yj. = yfc' 2-K/h AW . 1 v (W°, y'"; W°, Y°) |2.

(32)

In applying this result we can take the Y'S to be any set of variables that are independent of the total proper energy W and that together with W define a stationary state. We now return to the problem defined by the Hamiltonian (30) and consider an absorption process in which the atom jumps from the state J° to the state J' with the absorption of a light-quantum from state r. We take the variables Y' to be the variables J ' of the atom together with variables that define the direction of motion and state of polarisation of the absorbed quantum, but not its' energy. The matrix element v (W°, y ' ; W°, y°) is now Wc-** (vr/cjr)1/2 X, (J°J')N r °, where X, (J°J') is the ordinary (J°J') matrix element of X,- Hence from (32) the probability per unit time of the absorption process is

To obtain the probability for the process when the light-quantum comes from any direction in a solid angle da, we must multiply this expression by the number of possible directions for the light-quantum in the solid angle da, which is da ar AW/27c/i. This gives dc*±\±r

(JoJ') | 2 Nr° = da — L - 2 | X, (J°J') | * I r

with the help of (28). Hence the probability coefficient for the absorption process is l/2irA2cvra. |X r (J°J')| 2 , in agreement with the usual value for Einstein's absorption coefficient in the matrix mechanics. The agreement for the emission coefficients may be verified in the same manner.

296 Lorentz and Poincare Invariance The present theory, since it gives a proper account of spontaneous emission, must presumably give the effect of radiation reaction on the emitting system, and enable one to calculate the natural breadths of spectral lines, if one can overcome the mathematical difficulties involved in the general solution of the wave problem corresponding to the Hamiltonian (30). Also the theory enables one to understand how it comes about that there is no violation of the law of the conservation of energy when, say, a photo-electron is emitted from an atom under the action of extremely weak incident radiation. The energy of interaction of the atom and the radiation is a q-number that does not commute with the first integrals of the motion of the atom alone or with the intensity of the radiation. Thus one cannot specify this energy by a c-number at the same time that one specifies the stationary state of the atom and the intensity of the radiation by c-numbers. In particular, one cannot say that the interaction energy tends to zero as the intensity of the incident radiation tends to zero. There is thus always an unspecifiable amount of interaction energy which can supply the energy for the photo-electron. I would like to express my thanks to Prof. Niels Bohr for his interest in this work and for much friendly discussion about it. Summary. The problem is treated of an assembly of similar systems satisfying the Einstein-Bose statistical mechanics, which interact with another different system, a Hamiltonian function being obtained to describe the motion. The theory is applied to the interaction of an assembly of light-quanta with an ordinary atom, and it is shown that it gives Einstein's laws for the emission and absorption of radiation. The interaction of an atom with electromagnetic waves is then considered, and it is shown that if one takes the energies and phases of the waves to be q-numbers satisfying the proper quantum conditions instead of c-numbers, the Hamiltonian function takes the same form as in the light-quantum treatment. The theory leads to the correct expressions for Einstein's A's and B's.

Chap. 5. The Union of Special Relativity and Quantum ...

297

The Quantum Theory of the Electron By P. A. M. DIRAC St. John's College, Cambridge (Communicated by R. H. Fowler, F.R.S.—Received January 2, 1928)

THE new quantum mechanics, when applied to the problem of the structure of the atom with point-charge electrons, does not give results in agreement with experiment. The discrepancies consist of "duplexity" phenomena, the observed number of stationary states for an electron in an atom being twice the number given by the theory. To meet the difficulty, Goudsmit und Uhlenbeck have introduced the idea of an electron with a spin angular momentum of half a quantum and a magnetic moment of one Bohr magneton. This model for the electron has been fitted into the new mechanics by Pauli*, and Darwin t , working with an equivalent theory, has shown that it gives results in agreement with experiment for hydrogen-like spectra to the first order of accuracy. The question remains as to why Nature should have chosen this particular model for the electron instead of being satisfied with the point-charge. One would like to find some incompleteness in the previous methods of applying quantum mechanics to the point-charge electron such that, when removed, the whole of the duplexity phenomena follow without arbitrary assumptions. In the present paper it is shown that this is the case, the incompleteness of the previous theories lying in their disagreement with relativity, or, alternatively, with the general transformation theory of quantum mechanics. It appears that the simplest Hamiltonian for a point-charge electron satisfying the requirements of both relativity and the general transformation theory leads to an explanation of all duplexity phenomena without further assumption. All the same there is a great deal of truth in the spinning electron mo[* Proc. Roy. Soc. (A), 117, 610 (1928).] * Pauli, Z.f. Physik, vol. 43, p. 601 (1927). t Darwin, Roy. Soc. Proc, A, vol. 116, p. 227 (1927).

298

Lorentz and Poincare Invariance

del, at least as a first approximation. The most important failure of the model seems to be that the magnitude of the resultant orbital angular momentum of an electron moving in an orbit in a central field of force is not a constant, as the model leads one to expect.

§ 1. Previous Relativity Treatments The relativity Hamiltonian according to the classical theory for a point electron moving in an arbitrary electro-magnetic field with scalar potential A0 and vector potential A is F=

E+£^+,p+^A)+mV>

where p is the momentum vector. It has been suggested by Gordon* that the operator of the wave equation of the quantum theory should be obtained from this F by the same procedure as in non-relativity theory, namely, by putting W = ih

dt'

pr—— ih

d

dxr

r=

1,2,3,

in it. This gives the wave equation Fip = -[ih

cdt

+Zr

-ih

dxr

Ar\ +m2c2

w

= 0, (1)

the wave function ip being a function of x\, x2, x3, t. This gives rise to two difficulties. The first is in connection with the physical interpretation of tp. * Gordon, Z.f. Physik, vol. 40, p. 117 (1926).

Chap. 5. The Union of Special Relativity

and Quantum ...

299

Gordon, and also independently Klein,1" from considerations of the conservation theorems, make the assumption that if rpm, ipn are two solutions

and Imn = —~2— | —iKVm grad yn-yn

gradipm) + 2—Amy)my)„I

are to be interpreted as the charge and current associated with the transition m-—n. This appears to be satisfactory so far as emission and absorption of radiation are concerned, but is not so general as the interpretation of the nonrelativity quantum mechanics, which has been developedt sufficiently to enable one to answer the question: What is the probability of any dynamical variable at any specified time having a value lying between any specified limits, when the system is represented by a given wave function i/>„? The Gordon-Klein interpretation can answer such questions if they refer to the position of the electron (by the use of £>„„), but not if they refer to its momentum, or angular momentum or any other dynamical variable. We should expect the interpretation of the relativity theory to be just as general as that of the non-relativity theory. The general interpretation of non-relativity quantum mechanics is based on the transformation theory, and is made possible by the wave equation being of the form (H- W)W = 0,

(2)

i.e., being linear in W or dfdt, so that the wave function at any time determines the wave function at any later time. The wave equation of the relativity theory must also be linear in W if the general interpretation is to be possible. t Klein, Z.f. Physik, vol. 41, p. 407 (1927). % Jordan, Z. /. Physik, vol. 40, p. 809 (1927); Dirac, Roy. Soc. Proc, A, vol. 113, p. 621 (1927).

300

Lorentz and Poincare Invariance

The second difficulty in Gordon's interpretation arises from the fact that if one takes the conjugate imaginary of equation (1), one gets W e \2 / e V = 0, -( + —A0\ + I - P + —A) +m2c2 which is the same as one would get if one put — e for e. The wave equation (1) thus refers equally well to an electron with charge e as to one with charge —e. If one considers for definiteness the limiting case of large quantum numbers one would find that some of the solutions of the wave equation are wave packets moving in the way a particle of charge — e would move on the classical theory, while others are wave packets moving in the way a particle of charge e would move classically. For this second class of solutions Wh&s a negative value. One gets over the difficulty on the classical theory by arbitrarily excluding those solutions that have a negative W. One cannot do this on the quantum theory, since in general a perturbation will cause transitions from states with W positive to states with Wnegative. Such a transition would appear experimentally as the electron suddenly changing its charge from - e to e, a phenomenon which has not been observed. The true relativity wave equation should thus be such that its solutions split up into two non-combining sets, referring respectively to the charge — e and the charge e. In the present paper we shall be concerned only with the removal of the first of these two difficulties. The resulting theory is therefore still only an approximation, but it appears to be good enough to account for all the duplexity phenomena without arbitrary assumptions.

§ 2. The Hamiltonian for No Field Our problem is to obtain a wave equation of the form (2) which shall be invariant under a Lorentz transformation and shall be equivalent to (1) in the limit of large quantum numbers. We shall consider first the case of no field, when equation (1) reduces to

Chap. 5. The Union of Special Relativity

(-p2o+V2+m*c2)W

and Quantum . ..

= 0

301

(3)

if one puts W

.,

8

The symmetry between po and pi, p2, p3 required by relativity shows that, since the Hamiltonian we want is linear in /70, it must also be linear in pu p2 and pz. Our wave equation is therefore of the form (po + <xiPi + <x2P2 + ct3p2+P)y> = 0, (4) where for the present all that is known about the dynamical variables or operators <x1; <x2, <X3, /3 is that they are independent of po, PuP2,P3, i-e-, that they commute with t, xx, x2, x3. Since we are considering the case of a particle moving in empty space, so that all points in space are equivalent, we should expect the Hamiltonian not to involve /, xi, x2, x3. This means that ai, a 2 , a 3 , /3 are independent of t, x\, x2, x3, i.e., that they commute with/? 0 , Pi, Pi-, p3. We are therefore obliged to have other dynamical variables besides the co-ordinates and momenta of the electron, in order that ai, a 2 , a 3 , /? may be functions of them. The wave function rp must then involve more variables than merely xx, x2, x3, t. Equation (4) leads to 0 = (-/'o + ai/7i + a2!p2 + a3/'3 + j8)(/'o + ai/7i-l-aj|P2 + a3/73 + /5>= [-^+2'af'7?+i:(a 1 a 2 + a2a1)/71/?2 + i32+Z'(a1/3 + /5a1)/71]i/;,

(5)

where the U refers to cyclic permutation of the suffixes 1, 2, 3. This agrees with (3) if oc2r = 1,

a r a, + a,a r = 0

P2 = m2c2,

(r ^ s)

r, s = 1, 2, 3.

ccrp+p
If we put/3 = a 4 wc, these conditions become ccl = 1

«./t«.v+o!.va.fl = 0

(/J^D)

(JL,

v — 1, 2, 3, 4.

(6)

We can suppose the a^'s to be expressed as matrices in some

302

Lorentz and Poincaxe Invariance

matrix scheme, the matrix elements of a^ being, say, a (£'£"). The wave function ip must now be a function of C as well as x-j, x2, x3, t. The result of a^ multiplied into ip will be a function (cn^ip) of x%, x2, x3, t, C defined by (oc^vO (x, t, C) = i V a ^ C C ) y(x, ^ £')•

We must now find four matrices a^ to satisfy the conditions (6). We make use of the matrices ot =

0 1

1 0,

0 i

<*i

-i 0.

0- 3

=

which Pauli introduced* to describe the three components of spin angular momentum. These matrices have just the properties o\ = 1

GTOS

+
(r 9± s),

(7)

that we require for our a's. We cannot, however, just take the cr's to be three of our a's, because then it would not be possible to find the fourth. We must extend the a's in a diagonal manner to bring in two more rows and columns, so that we can introduce three more matrices QI, Q2, QS of the same form as ax, a2, a 3, but referring to different rows and columns, thus:

Ol =


<

0 1 0 0 1 0 0 0 0 0 0 1 ' 0 0 1 0 1 0 0 -1 i 0 0 0 0

* Pauli, loc. cit.

0 0 0 0 1 0 0 -1

(Jo =

<

0 —i / 0 0 0 0 0

0 0 0 0 0 —i i 0

Chap. 5. The Union of Special Relativity

Qi

0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0

1 0 £3 = < 0 0

Qo = <

0 0 0 0 , 1 0 > 0 0 - 1 0 0 -1

0 0 i 0

and Quantum ...

0 —i 0 0 0 0 i 0

303

0 —/

0 0

-

The g's are obtained from the o's by interchanging the second and third rows, and the second and third columns. We now have, in addition to equations (7) 1

and also

QrQs+QsQr = 0 QrO, =

(r 7* s).

(T)

fftQr.

If we now take ai = giffi,

a2 = gicr2,

a 3 = gicr3,

a4 = g 3 .

all the conditions (6) are satisfied, e.g., ,2 _

QiaiQiffi

= Qi°i =

aia 2 = Q1O1Q1O2 = Q1O1P1 = —

l

Qx^^x

-a2ai.

The following equations are to be noted for later reference Q1Q2 = 1Q3 =

-Q2Q11

(8)

CTiO'2 = /(T3 = —OiCx J

together with the equations obtained by cyclic permutation of the suffixes. The wave equation (4) now takes the form [po+Qi(o, v) + Q3mc]y = 0,

(9)

where o denotes the vector (ou o2, cr3). ((o,p) = £orpr, orpr being a matrix product).

304

Lorentz and Poincare Invariance

§ 3. Proof of Invariance under a Lorentz Transformation Multiply equation (9) by Q3 on the left-hand side. It becomes, with the help of (8), [QaPo+iQ2(0iPi + O2P2 + 03P3) + mc]y = 0. Putting Po = ipi, we have

Qz = 7i, Q&r = yr, [zTy^+mcfy = 0,

r = 1, 2, 3, /z = 1, 2, 3, 4.

(10) (11)

The p transform under a Lorentz transformation according to the law P'M =

Evd/jypv,

where the coefficients a are c-numbers satisfying 2^/JlfiyClfjx =

Ore»

-"T<2UT^IT

=

O/iar •

The wave equation therefore transforms into [iZy^+mcty = 0,

(12)

where y'M = Era^y,. Now the y , like the a , satisfy yl = 1,

y^y,+yryM = 0,

(^ ^ v).

These relations can be summed up in the single equation yMyv+yvyM = 2d/iV. We have y't/y'v+y'vy'^ = Erxa^civxiyryx+yxyr) — 2Urxaf^avxdrx Thus the y^ satisfy the same relations as the y ^ Thus we can put, analogously to (10) V'i = Q3,

7r =

QiOr

where the £"s and a's are easily verified to satisfy the relations

Chap. 5. The Union of Special Relativity and Quantum ...

305

corresponding to (7), (7') and (8), if g'2 and g'x are defined by £ 2 = - W 2 7 3 > Qi =

-iQ&v

We shall now show that, by a canonical transformation, the g"s and cr"s may be brought into the form of the g's and cr's. From the equation g'32 — 1, it follows that the only possible characteristic values for g'3 are ± 1. If one applies to g'3 a canonical transformation with the transformation function g[, the result is

eieiCeO -1 = - gaeKei) - 1 = - eS • Since characteristic values are not changed by a canonical transformation, g'3 must have the same characteristic values as — g'3. Hence the characteristic values of g3 are + 1 twice and — 1 twice. The same argument applies to each of the other @"s, and to each of the or"s. Since g'3 and o'3 commute, they can be brought simultaneously to the diagonal form by a canonical transformation. They will then have for their diagonal elements each + 1 twice and — 1 twice. Thus, by suitably rearranging the rows and columns, they can be brought into the form Qz and g3 respectively. (The possibility g3 = ±o'3 is excluded by the existence of matrices that commute with one but not with the other.) Any matrix containing four rows and columns can be expressed as c+£rcror+Zrc'rQr+Zrscrsgrcs

(13)

where the sixteen coefficients c, cr, c'r, crs are c-numbers. By expressing o[ in this way, we see, from the fact that it commutes with Qz = & a r ) d anticommutes* with o3 — a3, that it must be of the form o{ = c1ai + c2cr2 + C3i93cri + C32(?3cr2, i.e., of the form 0

cii2

0

0

floi

0

0

0

0 0

0 ai3

a3i 0

10 0

* We say that a anticommutes with b when ab — — ba.

306 Lorentz and Poincare Invariance

The condition a'x = 1 shows that aX2a2X = 1, a3iai3 = 1. If we now apply the canonical transformation: first row to be multiplied by (a2i/a12)112 and third row to be multiplied by (ai3/a2i)112, and first and third columns to be divided by the same expressions, a'x will be brought into the form of o^ and the diagonal matrices a'3 and £3 will not be changed. If we now express g'x in the form (13) and use the conditions that it commutes with a'x = ax and a'3 = a3 and anticommutes with Gz = £?3> w e s e e th a t ^ must be of the form pi = C1Q1 + C2Q2.

The condition g'x2 = 1 shows that c'x2 + c'22 = 1 , or c[ = cos 8, c'2 = sin 6. Hence g'x is of the form

Pi

*

=

'0 0 0 0 •< ie I e 0 0 eie

e~ie 0 0 0

0 e~ie 0 0

r

If we now apply the canonical transformation: first and second rows to be multiplied by e'e and first and second columns to be divided by the same expression, p^ will be brought into the form QX, and ax, az, QZ will not be altered. Q'2, and a'2 must now be of the form g2 and a2, on account of the relations ig'2 = Q'3Q'X, io'2 = a a

'z 'x-

Thus by a succession of canonical transformations, which can be combined to form a single canonical transformation, the p/'s and a "s can be brought into the form of the p's and tr's. The new wave equation (12) can in this way be brought back into the form of the original wave equation (11) or (9); so that the results that follow from this original wave equation must be independent of the frame of reference used.

Chap. 5. The Union of Special Relativity

and Quantum ...

307

§ 4. The Hamiltonian for an Arbitrary Field To obtain the Hamiltonian for field with scalar potential A0 and usual procedure of substituting for p in the Hamiltonian for thus obtain

an electron in an electromagnetic vector potential A, we adopt the p0 + e/c-Ao for po and p + e/c-Ao no field. From equation (9) we

Po + — Ao + Qiia, p + — A\+Q3mc ip = 0.

(14)

This wave equation appears to be sufficient to account for all the duplexity phenomena. On account of the matrices o and a containing four rows and columns, it will have four times as many solutions as the non-relativity wave equation, and twice as many as the previous relativity wave equation (1). Since half the solutions must be rejected as referring to the charge + e o n the electron, the correct number will be left to account for duplexity phenomena. The proof given in the preceding section of invariance under a Lorentz transformation applies equally well to the more general wave equation (14). We can obtain a rough idea of how (14) differs from the previous relativity wave equation (1) by multiplying it up analogously to (5). This gives, if we write e' for e/c, 0 = [ — (po + e'A0)+Qiip, X[(po+e'A0)+QX(a, 2

= [-(po+e'A0)

p + e'A)+g 3 mc] v + e'A) + Q3mc]ip

+ (a,V + e'Af + m2c2

+ Ql{(o,V + e'A)(po+e'A0)-(po

+ e'Ao)(a,V + e'A)})y. (15)

We now use the general formula, that if B and C are any two vectors that commute with a

(a, B) (a, C) = ZolBxd+EipKJiBiCz + o&iBiPi) = (B, Q + iZo^Cz-BiC) = (B, C) + /(o, BXC).

(16)

308

Lorentz and Poincare Invariance

Taking B = C = p+e'A, we find (a,p + eA')2 = (p+ e 'A) 2 + z2a3 [(pi+e'Ai) (p2 + e'A2)-(p2 + e'A2) (px + e'AJ] = ($ + e'A? + he'(o, curl A). Thus (15) becomes 0

-(p0+e'Ao)2+(-p + e'Xf + m2c2 + e'h(a, curl A) 1 6A" — z'e'/zpWa, grad ^40+c a? V = [ - ( p 0 + ^ o ) 2 + ( p + e'A)2 + m2c2 + rt(o, H.) + ie'hei(a, E)]-ip,

where E and H are the electric and magnetic vectors of the field. This differs from (1) by the two extra terms eh . __ ieh — (o, H) + —

,

_.

e i (o,E)

in F. These two terms, when divided by the factor 2m, can be regarded as the additional potential energy of the electron due to its new degree of freedom. The electron will therefore behave as though it has a magnetic moment eh/2mc. a and an electric moment iehjlmc. QXa. This magnetic moment is just that assumed in the spinning electron model. The electric moment, being a pure imaginary, we should not expect to appear in the model. It is doubtful whether the electric moment has any physical meaning, since the Hamiltonian in (14) that we started from is real, and the imaginary part only appeared when we multiplied it up in an artificial way in order to make it resemble the Hamiltonian of previous theories.

§ 5. The Angular Momentum Integrals for Motion in a Central Field We shall consider in greater detail the motion of an electron in a central field of force. We put A = 0 and e'A0 = V(r), an arbitrary function of the radius r, so that the Hamiltonian in (14) becomes

Chap. 5. The Union of Special Relativity

F = p0+V+Qi(a,

and Quantum ...

309

p) + g3mc.

We shall determine the periodic solutions of the wave equation Fip = 0, which means that po is to be counted as a parameter instead of an operator; it is, in fact, just 1/c times the energy level. We shall first find the angular momentum integrals of the motion. The orbital angular momentum m is defined by m = xxp, and satisfies the following "Vertauschungs " relations myXi — xifni = 0, mipi-pimx = 0, mXm = ihm,

m\X2 — x2mx = ihxz 1 m1p2-p2ml = ihpz i, 2 mrmi — mim. = 0, J

(17)

together with similar relations obtained by permuting the suffixes. Also m commutes with r, and with/? r , the momentum canonically conjugate to r. We have rnxF-Fni! = ei{mi(o, p ) - ( o , p)/ni} = Pi(o, /wip-pmi) = ihQ1(o2p3-o3p2), and so mF—Fm = i/iQi o X p .

(18)

Thus m is not a constant of the motion. We have further GXF-FGX

= ei{ffi(a, p ) - ( a , p ) ^ } =

=

Gi(ffao-offi,p)

liQxfapz-OzPa),

with the help of (8), and so oF—Fc = — 2zgi a X p . Hence (m+^ha)F-F(m

+ j;ha) = 0.

Thus m+-|7za ( = M say) is a constant of the motion. We can interpret this result by saying that the electron has a spin angular

310

Lorentz and Poincare In variance

momentum of \hc, which added to the orbital angular momentum m, gives the total angular momentum M, which is a constant of the motion. The Vertauschungs relations (17) all hold when M's are written for the m's. In particular MXM = ihM

and M 2 M 3 = M3M2.

Mz will be an action variable of the system. Since the characteristic values of w 3 must be integral multiples of h in order that the wave function may be single-valued, the characteristic values of Mz must be half odd integral multiples of h. If we put

M2 = (; 2 -i)/z 2 ,

(19)

j will be another quantum number, and the characteristic values of Mz will extend from {j—\)h to {—j+\)h*. Thus j takes integral values . One easily verifies from (18) that m2 does not commute with F, and is thus not a constant of the motion. This makes a difference between the present theory and the previous spinning electron theory, in which m2 is constant, and defines the azimuthal quantum number k by a relation similar to (19). We shall find that our j plays the same part as the k of the previous theory.

§ 6. The Energy Levels for Motion in a Central Field We shall now obtain the wave equation as a differential equation in r, with the variables that specify the orientation of the whole system removed. We can do this by the use only of elementary non-commutative algebra in the following way. In formula (16) take B = C = m. This gives (a, m)2 = m2 4- /(a, m X m) (20) 2 2 = (m + ihaf-h(a, m)-|/z a -/i(a, m) 2 = M -2A(a,m)-f/z 2 . * See Roy. Soc. Proc, A, vol. I l l , p. 281 (1926).

Chap. 5. The Union of Special Relativity and Quantum ...

Hence

311

{(a, m) + h}2 = M2 + \h2 = j2h2.

Up to the present we have defined j only through j 2 , so that we could now, if we liked, take jh equal to (a, m) + /z. This would not be convenient since we want/ to be a constant of the motion while (o, m)+/2 is not, although its square is. We have, in fact, by another application of (16), (a, m)(a, p) = /(a, mXp), hence (m, p) = 0, and similarly (a, p)(a, m) = /(a, pXm), so that (a, m)(a, p) + (o, p) (a, m) = iEcx(mipi-mzp>l+p
Qzjh-h.

Hence (a, x) (a, p) = rpr+iQ3jh. Introduce the quantity e defined by

(22)

312

Lorentz and Poincare Invariance

re = gi(o, x).

(23)

Since r commutes with 01 and with (a, x), it must commute with e. We thus have r2s2 — [gi(a, x)]2 = (c, x)2 = x2 = f1 or e2 = 1 .

Since there is symmetry between x and p so far as angular momentum is concerned, QX (a, x), like Qi (o, p), must commute with M andy. Hence e commutes with M and/. Further, s must commute with pr, since we have (a, x) (x, p ) - ( x , p) (a, x) = ih(o, x), which gives re(rpr + ih) — (rpr + ih)rs = ihrs, which reduces to epr-pre

= 0.

From (22) and (23) we now have regi(o, p) = rpr + ig3jh or ei(°>P) = epr + iep3Jh/r. Thus F - p0+ V+ epr + ieQ3Jh/r+Q3mc.

(24)

Equation (23) shows that e anticommutes with g3. We can therefore by a canonical transformation (involving perhaps the x's and p's as well as the cr's and g's) bring e into the form of the g2 of § 2 without changing g3, and without changing any of the other variables occurring on the right-hand side of (24), since these other variables all commute with e. ieg3 will now be of the form 1Q2Q3 — = —QX, so that the wave equation takes the form Ftp = [po+V+ Q2pr- Qijh/r+ Q3mc]y) = 0.

Chap. 5. The Union of Special Relativity

and Quantum ...

313

If we write this equation out in full, calling the components of ip referring to the first and third rows (or columns) of the matrices y)a and ipp respectively, we get (Fy)* = (po+V)ya-h—y)(}

ipp+mcrp* = 0,

(Fy)p = (po+V)ipf}+h-z-ipa

y)*-mcipf} = 0.

The second and fourth components give just a repetition of these two equations. We shall now eliminate ipa. If we write hB for po+ V+mc, the first equation becomes

{w+7)n = Bvwhich gives on differentiating a2 drz

j d r dr _B_ ~ h

j

„ d

dB

dr

dr

Jh ip a

— (po + V — mc) ipa-\ (po+Vf-mW h2

j •WP +

T

+

1

1 dV dV\ I d

j \

-Bh^)[c-r+^^

This reduces to 82 xpis + Br2

(po+Vf-mW

7(7+1)

1 dV(d

A

2

/z

(25) The values of the parameter po for which this equation has a solution finite at r = 0 and r = °° are 1/c times the energy levels of the system. To compare this equation with those of previous theories, we put xpp = r%, so that

314

Lorentz and Poincar^ Invariance

a 22 Z + 2 a

ar

r

%+

dr

(po+Vf-mW h2 1_ dV_ ~ dr Bh

j(j+iy 7+1 dr

X

= 0.

(26)

If one neglects the last term, which is small on account of B being large, this equation becomes the same as the ordinary Schroedinger equation for the system, with relativity correction included. Since j has, from its definition, both positive and negative integral characteristic values, our equation will give twice as many energy levels when the last term is not neglected. We shall now compare the last term of (26), which is of the same order of magnitude as the relativity correction, with the spin correction given by Darwin and Pauli. To do this we must eliminate the d%jdr term by a further transformation of the wave function. We put % = B-^Xi, which gives a2

2 a „

(po+Vy-mW 2

r2

h

+

Bh 7 ar

2" Bh dr2

jXj+l)

+

4 B2h2 \ dr

%i

Xi = 0.

(27)

The correction is now, to the first order of accuracy, 1 [j dV Bh\r dr

1 d2V 2 dr2 V

where Bh = 2mc (provided p0 is positive). For the hydrogen atom we must put V — e2/cr. The first order correction now becomes

Chap. 5. The Union of Special Relativity

and Quantum ...

315

If we write —j fory+ 1 in (27), we do not alter the terms representing the unperturbed system, so

< 28 '>

OT>

will give a second possible correction for the same unperturbed term. In the theory of Pauli and Darwin, the corresponding correcting term is

- 2 ^

(

°'

m)

when the Thomas factor \ is included. We must remember that in the Pauli-Darwin theory, the resultant orbital angular momentum k plays the part of our j . We must define k by m2 =

k(k+l)h2

instead of by the exact analogue of (19), in order that it may have integral characteristic values, like/. We have from (20) (a, m)2 = k(k+l)h2-h(c, or

m)

{(a,m) + |/z} 2 = (k + iW,

hence (a, m) = kh

or

-{k+\)h.

The correction thus becomes

which agrees with (28) and (28')- The present theory will thus, in the first approximation, lead to the same energy levels as those obtained by Darwin, which are in agreement with experiment.

316

Lorentz and Poincare

Invaiiance

The Radiation Theories of Tomonaga, Schwinger, and Feynman F. J.

DYSON

Institute for Advanced Study, Princeton, New Jersey

(Received October 6, 1948) A unified development of the subject of quantum electrodynamics is outlined, embodying the main features both of the Tomonaga-Schwinger and of the Feynman radiation theory. The theory is carried to a point further than that reached by these authors, in the discussion of higher order radiative reactions and vacuum polarization phenomena. However, the theory of these higher order processes is a program rather than a definitive theory, since no general proof of the convergence of these effects is attempted. The chief results obtained are (a) a demonstration of the equivalence of the Feynman and Schwinger theories, and (b) a considerable simplification of the procedure involved in applying the Schwinger theory to particular problems, the simplification being the greater the more complicated the problem. I. INTRODUCTION

A

S a result of the recent and independent discoveries of Tomonaga, 1 Schwinger,* and Feynman,* the subject of quantum electrodynamics has made two very notable advances. On the one hand, both the foundations and the applications of the theory have been simplified by being presented in a completely relativistic way; on the other, the divergence difficulties have been at least partially overcome. In the reports so far published, emphasis has naturally been placed on the second of these advances; the magnitude of the first has been somewhat obscured by the fact that the new methods have been applied to problems which were beyond the range of the older theories, so that the simplicity of the methods was hidden by the complexity of the problems. Furthermore, the theory of Feynman differs so profoundly in its formulation from that of Tomonaga and Schwinger, and so little of it has been published, that its particular advantages have not hitherto been available to users of the other formulations. The advantages of the Feynman theory are simplicity 1 Sin-iriro Tomonaga, Prog. Theoret. Phys. 1, 27 (1946); Koba, Tari, and Tomonaga, Prog. Theoret. Phys. 2, 101 198 (1947); S. Kanesawa and S. Tomonaga, Prog. Theoret. Phys. 3, 1, 101 (1948); S, Tomonaga, Phys. Rev. 74, 224 (1948). "Julian Schwinger, Phys. Rev. 73, 416 (1948); Phys. Rev. 74, 1439 (1948). Several papers, giving a complete exposition of the theory, are in course of publication. •R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948); Phys. Rev. 74, 939, 1430 (1948); J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945). These articles describe early stages in the development of Feynman's theory, little of which is yet published.

and ease of application, while those of TomonagaSchwinger are generality and theoretical completeness. The present paper aims to show how the Schwinger theory can be applied to specific problems in such a way as to incorporate the ideas of Feynman. T o make the paper reasonably self-contained it is necessary to outline the foundations of the theory, following the method of Tomonaga; but this paper is not intended as a substitute for the complete account of the theory shortly to be published by Schwinger. Here the emphasis will be on the application of the theory, and the major theoretical problems of gaugeinvariance and of the divergencies will not be considered in detail. The main results of the paper will be general formulas from which the radiative reactions on the motions of electrons can be calculated, treating the radiation interaction as a small perturbation, to any desired order of approximation. These formulas will be expressed in Schwinger's notation, but are in substance identical with results given previously by Feynman. The contribution of the present paper is thus intended to be twofold: first, to simplify the Schwinger theory for the benefit of those using it for calculations, and second, to demonstrate the equivalence of the various theories within their common domain of applicability.* * After this paper was written, the author was shown a letter, published in Progress of Theoretical Physics 3, 20^ (1948) by Z. Koba and G. Takeda. The letter is dated May 22, 1948, and briefly describes a method of treatment of radiative problems, similar to die method of this paper.

Chap. 5. The Union of Special H. OUTLINE OF THEORETICAL FOUNDATIONS

Relativistic quantum mechanics is a special case 01 non-relativistic quantum mechanics, and it is convenient to use the usual non-relativistic terminology in order to make clear the relation between the mathematical theory and the results of physical measurements. In quantum electrodynamics the dynamical variables are the electromagnetic potentials Af{t) and the spinor electron-positron field i W r ) ; each component of each field at each point r of space is a separate variable. Each dynamical variable is, in the Schrtidinger representation of quantum mechanics, a time-independent operator operating on the state vector <S of the system. The nature of $ (wave function or abstract vector) need not be specified; its essential property is that, given the $ of a system at a particular time, the results of all measurements made on the system at that time are statistically determined. The variation of <J> with time is given by the Schrodinger equation tA[d/3<>

-1/

H{r)dr

*,

(1)

where H(i) is the operator representing the total energy-density of the system at the point r. The general solution of (1) is :

l>(i) = exp

Z-Wlf

H{T)dr * 0 ,

(2)

with $o any constant state vector. Now in a relativistic system, the most general kind of measurement is not the simultaneous measurement of field quantities at different points of space. It is also possible to measure independently field quantities at different points of space a t different times, provided that the points of space-time a t which the measurements are made lie outside each other's light cones, so that the measurements do not interfere with each other. Thus the most comprehensive general type of measurement is a measurement of field quantities at each point r of space at a time t(t), Results of the application of the method to a calculation of the second-order radiative correction to the KleinNishina formula are stated. All the papers of Professor Tomonaga and his associates which have yet been published were completed before the end of 1946. The isolation of these Japanese workers has undoubtedly constituted a serious loss to theoretical physics.

Relativity

and Quantum

...

317

the locus of the points (r, t(j)) in space-time forming a 3-dimensional surface <x which is spacelike (i.e., every pair of points on it is separated by a space-like interval). Such a measurement will be called "an observation of the system on the results of measurements of A are the same in the state $ as in the state B4>. Therefore, the results of measurement of the field quantities at r' in the state (i(r')) are the same as if the state of the system were *(
-vmj t(r)H(T)dT\*o,

(3)

which differs from (i(r')) only by a unitary factor commuting with these field quantities. The important fact is that the state vector $(cr) depends only on a and not on r'. The conclusion reached is that observations of a system on a give results which are completely determined by attributing to the system the state vector (cr) given by (3). The Tomonaga-Schwinger form of the Schrodinger equation -is a differential form of (3). Suppose the surface 0, which we denote by d$/da(t) and call the functional derivative of * with respect to a at the point r. From (3) it follows that tAc[d*/3<7(r)] = H(r)*, (4) and (3) is, in fact, the general solution of (4). The whole meaning of an equation such as (4) depends on the physical meaning which is attached to the statement "a system has a constant state vector $o." In the present context, this statement means "results of measurements of

318

Lorentz

and Poincare

Invariance

field quantities at any given point of space are independent of time." This statement is plainly non-relativistic, and so (4) is, in spite of appearances, a non-relativistic equation. The simplest way to introduce a new state vector * which shall be a relativistic invariant is to require that the statement "a system has a constant state vector * " shall mean "a system consists of photons, electrons, and positrons, traveling freely through space without interaction or external disturbance." For this purpose, let fl-(r)=ffo(r)+Hl(r), (5) where Ho is the energy-density of the free electromagnetic and electron fields, and Hi is that of their interaction with each other and with any external disturbing forces that may be present. A system with constant * is, then, one whose Hi is identically zero; by (3) such a system corresponds to a $ of the form $(
-[''A]JV)#»(r)
(6)

It is therefore consistent to write generally

*(«)=rw*w,

(7)

thus defining the new state vector * of any system in terms of the old *. The differential equation satisfied by * is obtained from (4), (5), (6), and (7) in the form

;fc[a*/a
(8)

Now if g(r) is any time-independent field operator, the operator

where Hi{x<>) is the time-dependent form of the energy-density of interaction of the two fields with each other and with external forces. The left side of (9) represents the degree of departure of the system from a system of freely traveling particles and is a relativistic invariant; Hi(x0) is also an invariant, and thus is avoided one of the most unsatisfactory features of the old theories, in which the invariant Hi was added to the non-invariant Ho. Equation (9) is the starting point of the Tomonaga-Schwinger theory. HI. INTRODUCTION OF PERTURBATION THEORY

Equation (9) can be solved explicitly. For this purpose it is convenient to introduce a oneparameter family of space-like surfaces filling the whole of space-time, so that one and only one member
I

Hi(x)dx

is denoted the integral of Hi(x) over the 4dimensional volume between the surfaces
f

Hi{x)dx

are denoted integrals over the whole volume to the past of no and to the future of
Hi(x)dx\

g(xo) = (r(
(9)

and the general solution of (9) is

*(
(12)

Chap. 5. The Union of Special Relativity Expanding the product (10) in ascending powers of Hi gives a series HdxddXi+i-i/hcY

dxA

Hl(xl)Hl(.x,)dx1+---.

(13)

Further, U is by (10) obviously unitary, and

iy-i= f? = l + (i/hc) j

H&Jdx^V/hc)*

and Quantum

...

small perturbation as was done in the last section. Instead, H* alone is treated as a perturbation, the aim being to eliminate H' but to leave H' in its original place in the equation of motion of the system. Operators S(
Hl(x2)H1(xl)dxi+•

• •. (14)

It is not difficult to verify that U is a function of
(17)

Suppose now a new type of state vector fl(o-) to be introduced by the substitution ¥(
dxA

319

(18)

By (9), (15), (17), and (18) the equation of motion for fl(o-) is *ac[dG/3
(19)

The elimination of the radiation interaction is hereby achieved; only the question, "How is the new state vector fl(cr) to be interpreted?," remains. It is clear from (19) that a system with a constant J2 is a system of electrons, positrons, and photons, moving under the influence of their mutual interactions, but in the absence of external fields. In a system where two or more particles are actually present, their interactions alone will, in general, cause real transitions and scattering processes to occur. For such a system, it is rather "unphysical" to represent a state of motion including the effects of the interactions by a constant state vector; hence, for such a system the new representation has no simple interpretation. However, the most important systems are those in which only one particle is actually present, and its interaction rv. ELIMINATION OF THE RADIATION with the vacuum fields gives rise only to virtual INTERACTION processes. In this case the particle, including the In most of the problem of electrodynamics, the effects of all its interactions with the vacuum, energy-density Hi(x<>) divides into two parts— appears to move as a free particle in the absence of external fields, and it is eminently reasonable #,(*„) =-ff<(*o)+ff'(*o), (15) to represent such a state of motion by a constant ir , (*o) = -Cl/c]j,(*o)4 1 ,(*o), (16) state vector. Therefore, it may be said that the operator, the first part being the energy of interaction of HT(xo) = (5(
320

Lorentz

and Poincare

Invariance

single physical particle deviates, in the external field, from the motion represented by a constant state-vector, i.e., from the motion of an observed "free" particle. If the system whose state vector is constantly 12 undergoes no real transitions with the passage of time, then the state vector 12 is called "steady." More precisely, 12 is steady if, and only if, it satisfies the equation

mass, and should have been used instead of ilo(r) in the definition (6) of T(o-). Consequently, the second bracket should have been used instead of Hi(t) in Eq. (8). The definition of S(
5(oc)n = n.

The value of 5m can be adjusted so as to cancel out the self-energy effects in S( =o) (this is only a formal adjustment since the value is actually infinite), and then Eq. (21) will be valid for one-electron states. For the photon self-energy no such adjustment is needed since, as proved by Schwinger, the photon self-energy turns out to be identically zero. The foregoing discussion of the self-energy problem is intentionally only a sketch, but it will be found to be sufficient for practical applications of the theory. A fuller discussion of the theoretical assumptions underlying this treatment of the problem will be given by Schwinger in his forthcoming papers. Moreover, it must be realized that the theory as a whole cannot be put into a finally satisfactory form so long as divergencies occur in it, however skilfully these divergencies are circumvented; therefore, the present treatment should be regarded as justified by its success in applications rather than by its theoretical derivation. The important results of the present paper up to this point are Eq. (19) and the interpretation of the state vector 12. The state vector * of a system can be interpreted as a wave function giving the probability amplitude of finding any particular set of occupation numbers for the various possible states of free electrons, positrons, and photons. The state vector 12 of a system with a given t o n a given surface <J is, crudely speaking, the ^ which the system would have had in the infinite past if it had arrived at the given ¥ on a under the influence of the interaction H'(xo) alone. The definition of 12 being unsymmetrical between past and future, a new type of state vector 12' can be defined by reversing the direction of time in the definition of 12. Thus the 12' of a system with a given * on a given a is the *

(21)

As a general rule, one-particle states are steady and many-particle states unsteady. There are, however, two important qualifications to this rule. First, the interaction (20) itself will almost always cause transitions from steady to unsteady states. For example, if the initial state consists of one electron in the field of a proton, IIT will have matrix elements for transitions of the electron to a new state with emission of a photon, and such transitions are important in practice. Therefore, although the interpretation of the theory is simpler for steady states, it is not possible to exclude unsteady states from consideration. Second, if a one-particle state as hitherto defined is to be steady, the definition of 5(
SmcWMW) + (H,(r)-ta^'(i)j3f(r)).

The first bracket on the right here represents the energy-density of the free electromagnetic and electron fields with the observed electron rest-

H'(x0) = H'(.ro) +HS(xa) = ff'(x0) -8MC^(I,)*W.

' Here Schwinger's notation ^ = ^-*j3 is used.

(22)

Chap. 5. The Union of Special which the system would reach in the infinite future if it continued to move under the influence of H'ixo) alone. More simply, ft' can be defined by the equation Q'(
(23)

Since S(°o) is a unitary operator independent of
The Schwinger theory works directly from Eqs. (19) and (20), the aim being to calculate the matrix elements of the "effective external potential energy" HT between states specified by their state vectors ft. The states considered in practice always have ft of some very simple kind, for example, ft representing systems in which one or two free-particle states have occupation number one and the remaining free-particle states have occupation number zero. By analogy with (13), S{<m) is given by S(
Relativity

and Quantum

...

321

satisfactory agreement with experimental results. In this paper the development of the Schwinger theory will be carried no further; in principle the radiative corrections to the equations of motion of electrons could be calculated to any desired order of approximation from formula (25). In the Feynman theory the basic principle is to preserve symmetry between past and future. Therefore, the matrix elements of the operator HT are evaluated in a "mixed representation;" the matrix elements are calculated between an initial state specified by its state vector Qi and a final state specified by its state vector ft/. The matrix element of HT between two such states in the Schwinger representation is ft2*ifrft1 = Q2'*5(«>)ii'rfti,

(26)

and therefore the operator which replaces HT in the mixed representation is 5,(i,)=5(»)ffrW

= S(«)(SW)-'ff'W5W.

(27)

Going back to the original product definition of S(o) analogous to (10), it is clear that S(=°) X(5(cr)) _I is simply the operator obtained from S(o) by interchanging past and future. Thus, R(c) = 5 ( « ) (5(
xj H'(Xl)dXi + (-i/hcyf dx, dxA

H'(xl)H'(x,)dxt

+ • • •,

(24)

and (S(
x j

dxA

dx„xLH'(xn),

~"

dx,---

[-..,

[#'(*,),

[ f f & f O . i f ' W ] ] - ] ] - (25)

The repeated commutators in this formula are characteristic of the Schwinger theory, and their evaluation gives rise to long and rather difficult analysis. Using the first three terms of the series, Schwinger was able to calculate the second-order radiative corrections to the equations of motion of an electron in an external field, and obtained

.X

f

H'(xt)H'(xl)dxi+---.

(28)

The physical meaning of a mixed representation of this type is not at all recondite. In fact, a mixed representation is normally used to describe such a process as bremsstrahlung of an electron in the field of a nucleus when the Born approximation is not valid; the process of bremsstrahlung is a radiative transition of the electron from a state described by a Coulomb wave function, with a plane ingoing and a spherical outgoing wave, to a state described by a Coulomb wave function with a spherical ingoing and a plane outgoing wave. The initial and final states here belong to different orthogonal systems of wave functions, and so the transition matrix elements are calculated in a mixed representation. In the Feynman theory the situation is

322

Lorentz

and Poincare

Invariance

analogous; only the roles of the radiation interaction and the external (or Coulomb) field are interchanged; the radiation interaction is used instead of the Coulomb field to modify the state vectors (wave functions) of the initial and final states, and the external field instead of the radiation interaction causes transitions between these state vectors. In the Feynman theory there is an additional simplification. For if matrix elements are being calculated between two states, either of which is steady (and this includes all cases so far considered), the mixed representation reduces to an ordinary representation. This occurs, for example, in treating a one-particle problem such as the radiative correction to the equations of motion of an electron in an external field; the operator HF(XO), although in general it is not even Hermitian, can in this case be considered as an effective external potential energy acting on the particle, in the ordinary sense of the words. This section will be concluded with the derivation of the fundamental formula (31) of the Feynman theory, which is the analog of formula (25) of the Schwinger theory. If FJxi),

•••,

F„(x„)

are any operators defined, respectively, at the points Xi, • • •, .v„ of space-time, then PiFtixJ,

•••,

/%(.v„))

(29)

will denote the product of these operators, taken in the order, reading from right to left, in which the surfaces
H'(x.)). Since the integrand is a symmetrical function of the points .v,, • • •, .v„ the value of the integral is just n! times the integral obtained by restricting the integration to sets of points .Vi, • • •, x, for which cr(x,) occurs after cr(x,+i) for each i.

The restricted integral can then be further divided into ( » + l ) parts, t h e / t h part being the integral over those sets of points with the property that o-(xo) lies between
/„ = n ! E

«ff(i„-l)

dxy

dx„

x r <&,_,••• r dxixn'ixd--H'{x^)H'{x,)H'(x^

• • •#'(*„).

(30)

Now if the series (24) and (28) are substituted into (27), sums of integrals appear which are precisely of the form (30). Hence finally ^(*o) = X:(-v7"0»[l/»!]/, = £ ( - i / / i c ) " [ l / » ! ] f dxr-»-0

•/_„,

XP(H'(Xo),

H'(Xl),

f

dx,

J

H'(*.)).

(3D

By this formula the notation IIr(xa) is justified, for this operator now appears as a function of the point Xo alone and not of the surface u. The further development of the Feynman theory is mainly concerned with the calculation of matrix elements of (31) between various initial and final states. As a special case of (31) obtained by replacing 77' by the unit matrix in (27),

S(«) = £; (-(7/ic)"[l/«!] f dxr-- f dx„ XP(U'(xd,

•••, W'CO).

(32)

VI. CALCULATION OF MATRIX ELEMENTS

In this section the application of the foregoing theory to a general class of problems will be explained. The ultimate aim is to obtain a set of rules by which the matrix element of the operator (31) between two given states may be written down in a form suitable for numerical evaluation, immediately and automatically. The fact that such a set of rules exists is the basis of the Feynman radiation theory; the derivation in this section of the same rules from what is

Chap. 5. The Union of Special fundamentally the Tomonaga-Schwinger theory constitutes the proof of equivalence of the two theories. To avoid excessive complication, the type of matrix element considered will be restricted in two ways. First, it will be assumed that the external potential energy is

-ff'(zo) = - [lAQ/„(*„).4/(*o).

(33>

that is to say, the interaction energy of the electron-positron field with electromagnetic potentials A/ixo) which are given numerical functions of space and time. Second, matrix elements will be considered only for transitions from a state A, in which just one electron and no positron or photon is present, to another state B of the same character. These restrictions are not essential to the theory, and are introduced only for convenience, in order to illustrate clearly the principles involved. The electron-positron field operator may be written ^.W = E«»aWa„, (34) where the <£„«(:<:) are spinor wave functions of free electrons and positrons, and the a„ are annihilation operators of electrons and creation operators of positrons. Similarly, the adjoint operator


(35)

where d„ are annihilation operators of positrons and creation operators of electrons. The electromagnetic field operator is Af(x) = Z (A.„(x)b,+A.S(x)6.),

(36)

where b, and 6, are photon annihilation and creation operators, respectively. The chargecurrent 4-vector of the electron field is j,(x)=ieo(>(x)y^(x);

(37)

strictly speaking, this expression ought to be antisymmetrized to the form7 j,(x) = i*a:|iP„(*)iM*) - * « ( * ) * . ( * ) ) (7,).a.

(38)

but it will be seen later that this is not necessary in the present theory. Consider the product P occurring in the n'th 'See Wolfgang Pauli, Rev. Mod. Phys. 13, 203 (1941), Eq. (96), p. 224.

Relativity

and Quantum

...

323

integral of (31); let it be denoted by P„. From (16), (22), (33), and (37) it is seen that P„ is a sum of products of ( « + l ) operators ^„, (n + 1) operators i£„, and not more than n operators A„, multiplied by various numerical factors. By Q, may be denoted a typical product of factors ^„, $„, and At, not summed over the indices such as a and M, so that P „ is a sum of terms such as (?„. Then Q„ will be of the form (indices omitted) Q» =tf(*.-o)>K*.-o)!?(*.-i)>K*.-i)-• •$. Clearly, one of the annihilation operators must annihilate the electron in state A, one of the creation operators must create the electron in state B, and the remaining operators must be divisible into pairs, the members of each pair respectively creating and annihilating the same particle. Creation and annihilation operators referring to different particles always commute or anticommute (the former if at least one is a photon operator, the latter if both are electronpositron operators). Therefore, if the two single operators and the various pairs of operators in the product all refer to different particles, the order of factors in the product can be altered so as to bring together the two single operators and the two members of each pair, without changing the value of the product except for a change of sign if the permutation made in the order of the electron and positron operators is odd. In the case when some of the single operators and pairs of operators refer to the same particle, it is not hard to verify that tiie same change in order of factors can be made, provided it is remembered that the division of the operators into pairs is no longer unique, and the change of order is to

324

Lorentz

a n d Poincare

Invariance

be made for each possible division into pairs and the results added together. It follows from the above considerations that the matrix element of Q„ for the transition A —>B is a sum of contributions, each contribution arising from a specific way of dividing the factors of Q„ into two single factors and pairs. A typical contribution of this kind will be denoted by M. The two factors of a pair must involve a creation and an annihilation operator for the same particle, and so must be either one $ and one ^ or two A ; the two single factors must be one $ and one \j/. The term M is thus specified by fixing an integer k, and a permutation ro, ru • • •, r„ of the integers 0 , 1 , • • •, n, and a division ( W i ) , (W2), • • •, (Sk,th) of the integers j u •••,]„ into pairs; clearly m = 2h has to be an even number; the term M is obtained by choosing for single factors
XP(A(x.J,A(.*n))--

-P{A{x.h),A(x:»)),

(40)

a factor e being inserted which takes the value ± 1 according to whether the permutation of ^ and yp factors between (39) and (40) is even or odd. Then in (40) each product of two associated factors (but not the two single factors) is to be independently replaced by the sum of its matrix elements for processes involving the successive creation and annihilation of the same particle. Given a bilinear operator such as A,{x)A,(y), the sum of its matrix elements for processes involving the successive creation and annihilation of the same particle is just what is usually called the "vacuum expectation value" of the operator, and has been calculated by Schwinger. This quantity is, in fact (note that Heaviside units are being used) (A>(x)A,{y))a =

not be given here, because it turns out that the vacuum expectation value of P{A„{x)rA,{y)) takes an even simpler form. Namely, {P{AAx),A,{y)))o

(41)

where DF is the type of D function introduced by Feynman. Dp(x) is an even function of x, with the integral expansion 5?(.t) = - [ i 7 2 i r 2 ] f

exp[ia.r 2 ]da,

(42)

•'0

where x2 denotes the square of the invariant length of the 4-vector x. In a similar way it follows from Schwinger's results that (P(t.(x)My)))>

= b(x,y)Sr,.(x-y),

(43)

where Srt.(x) = - (.y,(.d/dx.) +
(44)

reciprocal Compton wave-length of the ri(x,y) is —1 or + 1 according as a{x) or later than
Af(x) = — [ i / 2 x : ] I

exp[ta.r-' —«oV4a]
Substituting from (41) and (44) into (40), the matrix element M takes the form (still omitting the indices of the factors $, <j/, and A of Q„) Af = til

(h(xi,Xr,)Sr(x,-Xn))

Xmh>'cVr(x.i-x,j))P($(xk)Mx'K)).

(46)

i

The single factors ^{xt) and ${xrk) are conveniently left in the form of operators, since the matrix elements of these operators for effecting the transition A—>B depend on the wave functions of the electron in the states A and B. Moreover, the order of the factors $(xt) and
\hctiv.{Dm+iD\{x-y),

where Dm and D are Schwinger's invariant D functions. The definitions of these functions will

= \hc!>t,DF{x-y),

(iSF(x,-xr,))U(i^DF(x,i-xh)) 1

Xt{xk)f(xn),

(47)

with

«'=«rii(*.-.*'0.

(48)

Chap. 5. The Union of Special Relativity

and Quantum ...

325

Now the product in (48) is ( — 1)', where p is the ing section it will be shown how this solution-innumber of occasions in the expression (40) on principle can be reduced to a much simpler and which the ii of a P bracket occurs to the left of more practical procedure. the ty. Referring back to the definition of e after VII. GRAPHICAL REPRESENTATION OF Eq. (40), it follows that t' takes the value + 1 or MATRIX ELEMENTS — 1 according to whether the permutation of $ Let an integer n and a product P„ occurring and