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into (9) we find consistency for any postulated function
(4) and the length of the rod 1 appeared as x
2
-1
12
* L. Jánossy, Theory and Practice of the Evaluation of Measurements. Clarendon Press, Oxford, 1965.
where the monotonous function 9? expresses the kind of distortion of the scales. We note that in terms of the distinguished scale the product of two lengths gives an area — namely S = kh, and the areas so obtained are themselves additive. In practice there seems to be no point in introducing non-additive scales for quantities if there is a possibility of introducing alsó additive representations. It must be emphasized, however, that it is not trivial that for certain quantities additive measures can be introduced. Whether or not such measures can be introduced in a particular case is a question which can be decided experimentally, as was shown e.g. in the case of the measures of electric charges in 101. 106. So as to see the role of the distinguished representation more clearly we give the following example. Let us consider a straight rod. Under practical conditions we may consider a rod to be straight if its contour can be made to coincide with a free string under tension — or if a beam of light can be made to move along it. Neither of the two definitions claim to be more than practical definitions with the help of which it can be decided with a limited accuracy whether or not a rod appears to be practically straight. We may mark on such a straight rod a number of consecutive points $ , SPi,. . F u r t h e r m o r e we may match a number of rods so that the rod x „ l > k = 0, 1, 2 n fits exactly between the points % and ty,. We can now ascertain whether or not the rod x when turnéd round still fits between the points $ and Once this has been found to be the case, we may order the rods according to their length thus we may establish in a qualitative manner e.g. that 0
k
k
kl
t
t*
A
< t
t A
< .• •
We can now ascribe arbitrary measures to the lengths of the rods; we may put •R( «) = ki > r
r
where the r should be positive numbers obeying kl
r
kíll
< ...
kJt
(15)
i.e. we choose the Iarger the measure the longer the rod. 107. A distinguished representation may be looked for in which the measures of the lengths are additive, i.e. the measures obey the relations r» + r
lm
=r
km
for k < l < m.
(16)
Such a representation is obtained if we put kl — l ~
r
r
l>k
k
r
(17)
where r = r^, r = r are the measures of the lengths of rods fitting in between the points ty and ty, respectively and ty . The measures obtained by (17) are additive; they give aconsistent system of measures if the k — 1, 2 , . . . can be chosen in such a manner that the r obtained by (17) obey (15). 108. Whether or not it is possible to obtain a distinguished representation with measures r obeying (15) and (16) depends on the physical properties of the rods. The everyday experience shows that using solid rods, additive measures can be obtained indeed inside the margin of error of measurement. We may also reverse the argument and state that we define as ideál solid rods the ones with the help of which an additive scale of lengths can be obtained. Constructing a scale of measures with real rods we may find small inconsistencies. Such inconsistencies can be made use of to determine deformations which real rods suffer when moved about and these deformations give the deviations in behaviours of the real rods from ideál solid rods. 109. Further we note that if we obtain additive measures r , for the lengths of rods then these measures can also be replaced by t
k
ok
0
k
r k í
kl
kl
k
r'ki = ar , k
a>0.
The r' are the lengths expressed in a new scale. We may also introduce a new scale using a factor a < 0. Doing so we reverse our convention and ascribe smaller measures to larger rods. Such a convention is unusual but nevertheless is internally consistent. kl
B. SYSTEMS OF SPACE COORDINATES 1. DETERMINATION OF COORDINATE VECTORS
110. We may now analyse the method of how to construct three-dimensional systems of coordinates. Consider for this purpose a set of fixed points . .., ^S„. The distances between the pairs of points can be measured with the help of rods and thus we can obtain (additive) measures K(*ki) = r„
k,l = 0, 1, 2 , . . . N
of the distances between various pairs ?$ and % of points. k
We expect the measures thus obtained to obey the relations +r
ri k
lm
r
=
k m
(18)
,
where the equality sign stands if the points ty , ty, and ty lie along a straight line. If the relations (18) are found to hold for the measures of distances taken for any group of three points then this can be taken as a qualitative check of consistency of our method. The above result supports in particular the assumption that our measuring rods behave like solids. 111. For to obtain measures of the coordinates of the points ty we may introduce relative position vectors x so that certain vector quantities k
m
k
kI
= *« define (in a representation K) the position of % relative to % . From experience we can take it that the position of a point relative to another is given by a three component quantity, therefore we suppose t
OT/,2' kl,3
kl — kl,l>
T
k
r
r
thus r i = 1, 2, 3 are the components of r . We may try to look for such a representation K in which the relative position vectors are additive, i.e. such that kIi
w
r« + f/m = r*m-
(19)
The relation (19) stands for three relations, i.e. for one relation for each of the three components of the vectors. The relations (19) are automatically satisfied if we suppose r « = i-/ -
r
k = 0, 1,. .., n
/,
fc
(20)
where r
/
=
r
k
0/»
T
0k •
=
T
We can take the r as the coordinate vectors of the points ?$ k = 0, 1, 2 , . . . the point ty with coordinate vector r = 0 being taken as the origin of K. 112. So as to obtain a statement which can be checked experimentally we make an assumption as to the connection between the position vectors r and the measures r . The distances r are supposed to be measured with solid rods, therefore we suppose the law of Pythagoras to be valid, thus fc
k
0
kl
0
kt
r£, =
kl
for
rf,
k, 1 = 0, 1 , . . .,
where we have written short 3 kl
T
=
X kl,i • I=L r
n
(21)
Relation (21) is valid in an orthogonal system of reference. In terms of skew coordinates we can suppose in the place of (21) rf, = T Gr kl
(22)
kl
where G is a symmetric and positive definite mátrix. In particular supposing G = 1, (22) reduces to (21). The following considerations will be carried through in representations where we do not specify whether they are orthogonal or skew. In this way we can considerably simplify the actual calculations and the results can be applied in orthogonal representations if this is desirable. For the moment we take thus (22) as a hypothetical connection between coordinate measures r and measures r of distances. The precise meaning of "orthogonal" respectively "skew" representations will be elucidated further below. 113. So as to check (22) experimentally we can write in place of (22) making use of (20) also w
kl
(r.-rOGfo-r*)^,.
(23)
In the case of N + 1 points ty , ^ . . . , ty the relations (23) give N(N + + l)/2 equations for the 3N components of the vectors r k = l,2,...,N. For sufficiently large values of N the system (23) becomes overdetermined and from the purely mathematical point of view (23) need not admit of solutions. If nevertheless in a particular case the overdetermined system (23) does admit of solutions, then this circumstance cannot be taken to be accidental. If in a given case relations (23) do not lead to contradiction this circumstance reflects on the properties of the measuring rods used. We may conclude (generalizing the result of 108) that the measuring rods can be taken to behave like ideál solids — if the measures of distances r obtained with them can be expressed in terms of quadratic expressions of the form (23). 0
u
N
k
kl
2. EXPLICIT DETERMINATION OF COORDINATE MEASURES
114. Explicit solution of (23) can be obtained in the following manner. Consider four points % (24) which four points should not lie in one pláne. (How this can be ascertained for four given points will be discussed further below.) We may consider a coordinate system such that ^ fixes its origin and the axes of the system lie in the direction $ ^P/t k = 1, 2, 3. Furthermore choosing the units along the axis suitably we can take the coordinate vectors of these points to be given by 0
0
r = 0,0,0, 0
r
1
=
1,0,0,
r = 0, 1, 0, 2
r = 0, 0, 1. 3
(25)
Writing down (23) for k = 0, / = 1, 2, 3 we find G„ = rf,
/ = 1, 2, 3
(26)
where we have written r, in place of r . Making use of (26) we obtain from (23) for k, l = 1, 2, 3 0/
G = j{ri
+ r?-rl).
u
(27)
Thus (26) and (27) give the elements of G if the coordinates of the four points (24) are to be given by (25). It will be necessary to suppose further below that det G # 0.
(28)
Whether or not the above relation stands, depends on the numerical values of the distances r in terms of which we express the elements of G. We have to require therefore that the points (24) should have distances in terms of which (28) is fulfilled. The latter requirement is equivalent to the requirement that the points (24) should not lie in one pláne. Thus we have to start our procedure with four points (24) with mutual distances satisfying (28). 115. Considering five points M
*o. %,
%
$3,
m>3.
m
(29)
Writing down (23) for the mutual distances we obtain relation 4 - 2(Gr ) + r G r = r | . m
fc
m
m
(30)
m
Supposing for the moment that r Gr = r ,
(31)
2
m
m
we can write in place of (30) G r = DC"> m
where D
( m )
is a vector with components ^ " Y ^ + r i - r í J
* = 1,2,3,
m>4
(32)
thus r m
=
G
-i
D ( m )
The relation (33) gives explicitly the coordinate vector r in terms of measured distances only.
( 3 3 )
m
of a point ty„
3. QUESTION OF CONSISTENCY
116. Introducing (33) into (23) we find that the relations (23) are indeed satisfied for the coordinate vectors of the five points (29) provided (31) is satisfied. Introducing (33) into (31) we find a relation D( >G- D > = r . m
1
(m
(34)
2
m
The latter relation gives a quadratic equation for Solving the above equation, we can express r in terms of the remaining nine distances between the five points (29). [Since (34) is a quadratic equation, we obtain in generál two solutions for r . ] If the measured value of r coincides with one of the solutions of (34), then (25) and (33) give a consistent set of coordinate vectors for the points (29). Relation (34) gives thus a test of consistency; it shows whether or not the distances between the five points (29) can be expressed in a form (23). 117. Adding a sixth point to the five points (29) we can determine the coordinate vectors r„ of *)3„ as m
m
m
r„ =
G-W>
where the elements of D are obtained from an expression of the form (32). The coordinate vectors thus obtained can be taken to be consistent provided we have apart from (34) also (n)
r Gr„ = r\ n
and
r„Gr = i - (r* + r%- r * J . m
Increasing the number of points we can construct expressions which give the values of the coordinate vectors provided a consistent set of coordinate vectors exist at all. In case of N > 4 points, the coordinate vectors obtained have to fulfil
conditions. 4. VARIOUS REPRESENTATIONS
118. It remains to investigate how far our procedure is affected by the arbitrary choice (25) of the coordinate vectors of the four standard points. We note first that equation (33) is only meaningful if (28) is fulfilled, i.e. if G exists. We have thus to choose the points S$ so that the determinant of the mátrix G should not vanish. It can be taken that the latter condition is fulfilled if we choose the four standard points (24) so as not to lie in one pláne. - 1
k
det G = 0
(35)
as the definition for points to lie in one pláne. If det G ^ 0 we find it to b positive definite. 119. If we succeed in determining coordinate vectors r k = 0 , 1 , 2 , . . . , N to N + 1 points such that these vectors satisfy all the relations (23) then we can take linear transforms fc
r' = Sr + s k
k = 0, 1, 2,...
k
N
(36)
alsó as consistent measures of the coordinate vectors. Indeed from (36) it follows that r - r, = S-V/c' - *í) = W - O S " .
(37)
1
fc
Introducing thus (37) into (23) we find (f * ~ ''i)G'(r - r',) = rli
(38)
k
with G' = S - G S - . 1
(39)
1
Thus we see that provided the coordinate vectors r satisfy the overdetermined systems (23), then the transformed coordinate vectors r' satisfy the relations (38). 120. When constructing coordinate measures for the coordinate vectors t of the points ty we assumed in 114 (25) particular values for the measures of the coordinate vectors r of the points ?$ k = 0, 1, 2, 3. If we were to assume in place of (25) that in somé representations K' the coordinate vectors are given by k
k
k
k
k
rí =
r ,
r'
kr2
ktS
k
k = 0,1,2,3
(40)
where the r' are chosen arbitrarily, then we can regard the r' as transforms of the originál r^, the transformation having the form kJ
k
t = Sr + s k
fc
k = 1, 2, 3
with $ik — k,i ~ 'o,t r
r
— ó.i r
k,l=
1,2,3.
(41)
Applying the transformation (41) to any of the coordinate measures of ty in the representation K we obtain its representation in K'. The representation K' could alsó be obtained directly by assuming the values (40) in place of the values (25) for the coordinate measures of the points ty k = 0, 1, 2, 3. The only restriction to be imposed on (40) is that the coordinate measures have to be chosen so that k
k
dct|r^-.ri|5É0 the latter condition is necessary for to make det S # 0 and to make the transformation (41) a reversible one. From the above considerations it follows that providéd consistent coordinate vectors r can be introduced, supposing distances to be given by (23) then consistent coordinate vectors can alsó be found if G is replaced by G' as defined by (39). 121. For given matrices G and G' we can determine S so as to satisfy (39). We may write e.g. S = G'- G . k
1 / 2
1 / 2
We see that if it is possible to construct consistent measures óf coordinate vectors for one assumed value of the mátrix G then it is possible to find consistent representation for any other choice G' in place óf G. We conclude therefore, that we can decidé by measurement whether or not it is possible to represent a set of measured distances r by a positive definite quadraticform of the difference of coordinate vectors. No information can be obtained about the elements of the mátrix G occurring in the quadratic form. In particular we can obtain orthogonal coordinates in which G' = 1 if we put r' = G ' !. u
1 2
Thus the coordinate vector of the point ty„ in the orthogonal representation is obtained as r
'
m
=
G
-l/2 (m)_
(42)
D
122. So as to formulate our results in a more generál form we denote @ the metric tensor. We say that ® is represented in a system of reference K by K(®) = G, where G is always a symmetric positive definite mátrix. The representations G = K(&)
and
G' = K'(®)
of the metric tensor are connected by a relation G' = S ^ G S -
1
provided the coordinate measures relatíve to K and K' are connected by r' = Sr + s. In particular we can obtain an orthogonal representation as described in 121. C. PROBLEMS CONNECTED WITH COORDINATE REPRESENTATIONS 1. REMARK O N " N O N - E U C L I D E A N " GEOMETRY
123. The above statements can also be formulated in another way. If the measured distances r between the points of a set can be expressed by a quadratic form (23), then one might conclude that the space in which the points are situated is "Euclidean". Or if no consistent coordinate measures can be obtained one might conclude that the space involved is "nonEuclidean". We do not think, however, that such a conclusion has any meaning. The fact that the overdetermined system (23) possesses solutions r , k = 0, 1, 2 , . . . , n seems to us to reflect upon the method of measurement of the distances r and in particular upon the physical properties of the measuring rods used. Roughly speaking one may conclude from the consistency of the measures that the measuring rods made use of are behaving like rigid bodies, i.e. if the measuring rods are turnéd or shifted they do not change their length. Of course the procedure described provides only necessary conditions for the rods to behave like rigid rods. 124. A further aspect of the question is as follows. As relation (23) is a (generalized) form of the law of Pythagoras we come therefore to conclude that the latter law can be tested experimentally. This statement appears at first sight paradox as the law of Pythagoras is usually proved with the help of the axioms of geometry. In fact no paradox is involved. The axioms of geometry simply reflect the properties of ideál solids. The experimentál test described above is a test to the effect that our measuring rods behave like ideál solids. u
k
kl
2. COORDINATE TRANSFORMATIONS A N D DEFORMATIONS
125. nr>
x
m
Let us consider a system of points % with coordinate vectors = 1. 2 , . .., n. In a particular representation K we have m
K(tJ
= r . m
We obtain the coordinate vectors in another representation K' with the help of a linear transformation; we may write K\i ) m
= t' = Sr + s. m
(43)
m
A linear transformation of the coordinate vectors may alsó be interpreted in a different manner, by writing in place of (43)
C = Tr
m
+ t
(44)
where detT # 0 and t is a constant vector.
5*
Fig. 14. Scheme of a deformation
We can regard the r* as the coordinate vectors of points ty* m = 1, 2 , . . . , / ! in the representation K. Thus we may suppose In the above sense the transformation (44) produces from a system of points fy *Js , . . . , $ „ another system • • •> The transformation (44) describes thus a deformation of the configuration of a set of points: the deformation being expressed in measures relative to K (see Fig. 14). 126. Let us consider the deformation (44) in measures of coordinate vectors of two systems of reference K and K'. Thus suppose lt
2
K(x) = r,
K\x) = r-
and r and r' are connected by (43). Transforming both sides from (44) according to (43) we find Sr* + s = STr + St + s, m
(45)
expressing r from (43) we have m
r = S - ^ - S - V
(46)
m
Inserting (46) into (45) we find C = Tr'
m
+ t\
T' = S T S -
(47) (48a)
1
t' = (1 - T')s + St.
(48b)
Relation (47) gives the connection between r*' = *'(r*)
and
r' = K'(t ) m
m
i.e. it gives the connection between the coordinate vectors of $ in the representation K'. 127. In a more generál notation we may also write
mj=r ,
m
and ^3*
(49)
m
where % stands for the deformation which shifts the points S$ into points ty*. The representations of % relatíve to K and K' can be written m
K(%) = T, t K'(%) «= T', t'. The relations between the representations T, t and T', t' of £ are given by (48a, b). The points ty m = 1, 2 , . . . n can also be taken to be the points constituting somé physical system m
a = %,
Applying the deformation operation 2 to the points of & we obtain another system q* = ..., «p;, and thus we may also write D* =
£(&)
where £}* is a deformed and displaced version of |Q the operator £ giving the deformation.
3. ORTHOGONAL TRANSFORMATIONS a. DEFINITIONS
128. In particular we may be interested in deformations, which we shall denote by £) which leave the distances i , between pairs of points ty , of JQ unchanged. Writing O in place of T we find from (44) k
k
r* = Or* + t,
(50)
k
and therefore
*ti =
thus
Or , w
r£ = r Ő G O r . 2
w
(51)
w
Provided O satisfies the relation ŐGO = G
(52)
we find from (51) r
r*i = ki
k,l=\,2,
...
Thus the deformation mátrix O produces deformations which leave the measures r unchanged. We denote such matrices orthogonal matrices and transformations (50) orthogonal transformations. In particular in an orthogonal representation K with G = 1 (52) reduces kl
0
to
ŐO = 1. 129. Equation (52) gives the definition of the representation of an ortho gonal mátrix in one particular system of reference. We may write O = X(£>). From (48a) and (39) we find 1
O' = S O S ,
1
1
G' = S G S ,
(53)
where O' = K'(£>)
and
G' = K'{®).
Thus introducing (53) into (52) we find Ő G O ' = G'. We see therefore that reference in which we 130. Relation (52) However, since G is a
the definition (52) is independent of the system of represent © and £). gives nine equations for the nine elements of O. symmetric mátrix only six out of the nine equations
are independent and the matrices obeying (52) form a set depending on three independent parameters. We may denote this by writing O in place of O where p stands for the parameters. We may also write £) for the mátrix giving a particular deformation and p
p
K(£) ) = O
K(ti) = p.
or
p
v
Thus we write p for the parameters defining a particular orthogonal de formation mátrix and write p for its representation in a system K. b. GROUP CHARACTE R OF ORTHOGONAL MATRICE S
131. The orthogonal matrices O , O , . . . form the socalled ortho gonal group. Indeed, taking the determinant of both sides of (52) we find since det G > 0 det O = ± 1 . (54) p
q
We see thus that there exist orthogonal matrices with determinant + 1 and others with determinant —1. From (54) it follows that any orthogonal mátrix O possesses an inverse O p ; thus multiplying (52) from the left with Op and from the right by Op we find l
p
Őp^GOp = G, 1
thus Op" is also an orthogonal mátrix. Furthermore if O orthogonal matrices then 1
ŐpGOp = Ö „ G O , = G
p
and O
q
are (55)
thus the product O, = O O p
q
is also an orthogonal mátrix. Indeed, multiplying (52) from the left by O , and from the right by O remembering O = O Ő we find q
r
q
p
Ő G O = G. r
r
The unit mátrix obeys also (52) and matrices are always associative, thus we see that the matrices obeying (52) fulfil the postulates of a group and thus they form indeed a group. The matrices with det O = + 1 form a subgroup which subgroup may be called the proper orthogonal group. Since the O form a group thus their transforms to another system of reference, i.e. the matrices p
O . = SOpS" , p
1
(56)
form alsó a group. The relation (56) defines alsó the transformation of the representation of the parameters p from one system of reference to another. 132. In particular we may consider coordinate transformations S taken with orthogonal matrices S = O . We write O in place of O, to signify that 0 ( , ) does not represent an operator but is the mátrix of a coordinate transformation. In case of one orthogonal coordinate transformation we may write in place of (56) (<0
tq)
O;. = OWOPO'o- .
(57)
1
Because of the group character of the transformations the product on the right hand side of (57) defines an orthogonal mátrix O - which is an element of the representation of O in K'. Thus an orthogonal transformation of the orthogonal group changes only the parameters of the deformation operators. It is convenient to define the parameters of the coordinate transformations such that we have p
OW = O , ;
(58)
1
using the above definition, we have r;; = 0 « O J R W =
r
w
i.e. the representation of x in K will become equal to the representation of r*, in K' if the coordinate transformation K -* K' is taken with the same paraméter as the operator producing the deformation 0, -* Q*. It can be seen easily that using the definition (58) the coordinate transformation O corresponds to a deformation according to O of the system of reference. 133. The question may be raised as to what are the common features of the various representations O , O -, OJ-, . . . of a deformation mátrix ? From (56) we see that a change of representation of Dj, leaves the eigenvalues of the mátrix O unchanged. The eigenvalues of an orthogonal mátrix O with real elements can be shown to have the form u
m
q
p
p
p
p
e'V-'M, thus they depend on one real paraméter q> only. As can be seen easily an orthogonal deformation r* = O r„ + t p
can always be regarded as a shift and a turning round through an angle
= m = $ (r, t) solutions of the equations (0)
t
i=L,2.
The law of conservation of energy can be written similarly as Xjo + K£> = Kf> +
(6)
with Kp=^mpw?*
i,k=\,2.
231. Applying the Lorentz principle to the collision we expect that the Lorentz deform of the elastic collision described by (4) and (6) should deseribe another collision which alsó obeys the laws of conservation of energy and momentum. We investigate how the expressions (3) have to be modified so as to make (4) and (6) Lorentz invariant.
Considering a Lorentz deformation with a deformation mátrix A we can calculate the velocities the particles take up in the deformed version. We find applying the results of 207 y
w<*>* =
i,k= 1,2
(7)
where we write $ for the addition according to Einstein's addition formula in 207 eq. (9). So as to simplify notation we shall use the upper index k = 1, 2 for the states of the particles before and after the collision and denote by upper indices k = 3,4 the quantities referring to the state of the particles before and after the collision in the deformed configuration. Thus we use the notation w
f>*
=
w
mj*)* = mj*+"),
f+2>,
p<*>* = pf >, +2
i,Jfc=l,2.
(8)
Considering in particular collisions so that the velocities w^ and v are all parallel we can write in place of (7) using the notation (8) fc)
VWT>
1+
—T-
From (9) it follows also that (k)
vw
where 2rt*> = — = J
/ = 1,2 k= 1,2,3,4.
r
w«
2
232. The law of conservation of momentum for the deformed system can be written p(3) + p f = p(4) pW (11) +
Introducing (9) and (10) into (5) using the notation (8) we find _ (fc+2) (fc+2) ~i m =
Pi
w
t
i
w< k)
+
.
v
_
±!_ p+2y (fc+2) • B
5
OT
( ]
UA>
Introducing (12) into (11) and writing = pf^mf^ we find after multiplication with y / l ' — tP/c a relation of the following form: 2
(/#> + vnífioéP
+ (f/P + 0 4
' = (/>í + wf^VS0 + (PÍ + vmf>)<£?
1
2>
2)
(13) with W
( 1 4
)
The law of conservation of momentum which is expressed for the originál pair of particles by relation (4) appears in the form (13) for the transformed pair. The law (4) is Lorentz invariant provided (13) is necessarily fulfilled if (4) is fulfilled. We investigate the circumstances under which (4) and (13) follow from each other. The terms of (13) not containing v reduce to (4) provided the tx\ have all the same values. We can suppose k)
«f*) = 1
i = 1, 2
k = 1, 2
(15)
and find from (14) and (15)
mf^mfif^
">>
Í
=
h
l
r
íl-%-
(16)
^=1,2,3,4
where m and m are constants. The terms of (13) containing v cancel provided (16) holds and alsó the relation mP + mf> = ni? + n%> (17) is satisfied. As the result of a short calculation we find that it follows from (4), (16) and (17) that x
2
mi > + m > = < > + m > . 3
3
4
2
2
233. Summarizing our considerations we see that in the case of a linear elastic collision it follows from the relation pf>+rf> = p f +
p > ( 2 2
(18)
together with the relation m + M = f ) + mf)
m
m
w
(19)
that the corresponding relations referring to the Lorentz deformed configurations are alsó valid; i.e. it follows from (18) and (19) that PÍ 3 , + P ( 2 3 ) = PÍ 4 ) + P ( 2 4 )
and where the transformation from the originál system of particles with k = 1, 2 to the deformed system with k = 3, 4 has to be taken in accord with (16). In particular we conclude that the momentum of a partiele with rest mass m moving with a velocity w has to be taken as P=
^
r.
•
(20)
The latter formula can be interpreted by writing p = m(w)w,
m
m(w>) =
. (21) yj 1 - W /C 234. We see that the law of conservation of momentum (18) can then and only then be formulated in a Lorentz invariant manner, if we add to it the relation (19). The relations (18) and (19) together appear Lorentz invariant if the transformation of mass and momentum is taken to be in accord with (21). The classical law of conservation of energy expressed by (6) is not Lorentz invariant; we see immediately from our formalism that in generál 2
2
even if (6) is satisfied in the originál configuration. The classical law of conservation of energy can, however, be replaced by the invariant relation (19). The latter relation for small values of the velocities reduces in a good approximation to the classical energy expression. Indeed, multiplying (19) by c we can write 2
+
with E\ = k)
=
+
(22)
' = = = mfi + — m-wf? + terms of higher order. y 1 - w^ jc 2
k)
2
2
Thus we can take where E = rriiC is the rest energy of a partiele and E\ — E can be taken as the kinetic energy. We see thus that the relativistic energy of a partiele can be defined by k)
2
t
t
E
=
^
L
=
(23)
where m is the rest mass and w the velocity of the partiele — and we find that the relativistic energy of particles is exactly conserved in elastic collisions. 235. We have obtained relations (20) and (23) as sufficient conditions for the invariant formulation of the conservation laws. Regarding, however, the independent parameters involved in the consideration, one can see easily that for given values m,{0) these conditions are alsó necessary. Thus supposing that the momentum vector has a direction parallel to w energy and momentum have necessarily to be assumed to have the dependence upon w as given in (20) and (23). (In the energy expressions there remains of course always the possibility of adding arbitrary constant values to the expressions £,-.) 3. INELASTIC COLLISIONS
236. The conservation laws can be formulated in an invariant manner only if both the laws of conservation of energy and of momentum are taken together. That the energy and momentum relations are indeed connected can be seen from the following example. Consider an inelastic collision of the following type. Two particles of equal rest masses mtö) = m (0) = m move towards each other with velocities and wl = - w. 2
0
1)
Suppose the particles to collide and stick together after having touched each other and to come to rest in the end. We suppose thus wf> = - w « = w
= w<> = 0 2
and we have m vf
><1> W 2
0
= p>= 0 2
2
and PÍ + P 1)
( 1) 2
= PÍ + P . 2)
( 2) 2
Transforming the collision with a velocity v = w we find wf> =
2w ,2 '
wf> = 0;
w^ = w 4)
4) 2
= w,
the momenta of the transformed system are found to be „(3) _
_
2
W
W
2
PÍ = P
„(3) _ o
0 C
4)
2
/n w 0
4) 2
thus we find , / 1 - w /c 2
2
thus for w > 0 the momentum relation in the transformed system is not fulfilled. The reason for this discrepancy is that the energy relation is not fulfilled in the originál version of collision, indeed we have IT?) = Ef = -j^£==,
E? = E? = m c . 2
0
/1-H' /C ' 2
X
2
B. EQUIVALENCE OF MASS AND ENERGY 237. The inelastic collision can, however, be treated in a Lorentz invariant manner if we suppose with Einstein the principle of the equivalence of mass and energy. According to this principle a system which has a totál energy E possesses a mass m = Ele .
(25)
2
If the inelastically colliding particles stick together, then their kinetic energy is transformed into somé other type of energy e.g. the particles may heat up. We have thus 22Í > + E? = £ f > + £ > + Q = - r ^ — , 2
1
2
y/ 1 -
Vf /C 2
(26)
2
where Q is the energy of the heat produced in the collision. With the help of (25) and (26) we find thus for the effective mass 2m of the complex of the two particles which have stuck together
0
2m = 0
Ej» + E?> C
5 2
2m
0
= — —
^/l -w /c 2
, 2
and thus we nave in the place of (24) P! + P ( 3)
3) 2
= Pl + P , 4)
4)
2
where p(4)
=
p(4) _
m
„
and
Thus the conservation of momentum is fulfilled in the transformed colüsion if we take into consideration the increase in mass caused by the heating up of the particles. 1. REMARK O N THE MECHANISM OF INCREASE OF MASS WITH ENERGY
238. The increase of mass with energy if a macroscopic body is supplied with heat can be understood immediately. The heat increases the average thermal velocity of the atoms of the body and thus the increase of mass of the body is caused by the changes of the masses of the molecules due to the increased thermal motion. Since the collisions between individual atoms can be taken to be elastic ones, an inelastic collision between two macroscopic bodies resulting in heating up, can be taken alternatively as a system of elastic colüsions between the atoms of the colliding bodies and thus it becomes evident that the collision as a whole will be in accord with the relativistic laws valid for elastic collisions. The problem is somewhat more involved, if the macroscopic collision produces not simply heat, but also elastic or inelastic deformations. The energy stored in these deformations corresponds also to change of mass in accord with (25); the mechanism of this increase will be dealt with further below.
C. DISTANT COLLISIONS 239. Two particles acting upon each other at a distance influence mutually their motion. The problem of the conservation laws in this case is somewhat more complicated.
SchematicalJy a collision at a distance can be pictured by supposing that the changes of momentum of the colliding particles occur suddenly. We may thus suppose that (as shown in Fig. 21) the partiele 1 is deflected in the point A at a. time t and it changes there its momentum from a value p W t pC2). At the same instant the partiele 2 changes its momentum p into p at a point B at somé distance from A. If we have 2
0
2
1 )
2 )
PÍ +P 2 =Pf + P , 1>
(
1)
)
2)
2
then the collision can be regarded as an elastic one.
Fig. 21. Scheme of a distant collision
Considering the Lorentz deformed version of the collision we have alsó
However, transforming the motion of the particles we find that in the transformed system the changes of momenta do not take place simultaneously. The momentum of the one partiele changes at a point A* at somé time '*> while the change of momentum of the second partiele takes place at a point B* at a time t = t% and in generál 1
=
If, say, t% < t% then in the interval t%
< t < t*
the momentum is not conserved since the momentum of the first partiele has already changed while that of the second partiele has still remained unchanged. If we consider instead of a sudden collision a continuous interaction at a distance, then it is to be expected that even if the momentum balance is restored after the interaction has ceased nevertheless the sum of momenta of the two particles does not remain constant in the course of the interaction. Or, even if it were constant in a particular configuration, then it will not be constant in other Lorentz deformed configurations.
The energy and momentum of the whole system can, however, be taken to be conserved if we consider that the interaction between distant particles is always transmitted by a radiation field. As will be seen in more detail, the totál energy and momentum consisting of that of the particles and fields remains strictly constant in the course of the collision. 1. EXPERIMENTÁL EVIDENCE
240. The pre-relativistic form of the conservation laws of energy and momentum expressed by relations (4) and (6) were obtained and confirmed observing particles with small velocities. Observing particles moving with velocities small as compared with that of light, these laws cannot be distinguished from the relativistically invariant laws expressed by the relations (18), (19) and (22). Observing collisions between fast moving elementary particles the discrepancies between the two sets of laws become very pronounced and such collisions do not obey the laws expressed by (4) and (6) but rather the relativistic laws expressed by (18), (19) and (22). While at least qualitatively the relativistic laws of collisions were confirmed by actual experiments, it would be useful to analyse the observational facts more critically than it was done so far, so as to obtain a quantitative confirmation of the relativistic laws of collisions. We note that in somé of the relativistic collisions the effects of radiation mentioned in 239 are strongly felt; an important example is the collision of fast electrons with atomic nuclei emitting an appreciable amount of bremsstrahlung in the course of the collision. D. MECHANICAL LAWS IN TERMS OF FOUR-VECTORS AND TENSORS 241. The considerations about elastic collisions can be simplified mathematically making use of four-vectors. The concept of four-vectors and tensors is well known, nevertheless we give in the appendix a short account of vector and tensor analysis, so as to point out certain particular features of the subject which are relevant to the conception developed here. Here we note that the energy and momentum of a partiele can be expressed by a (covariant) four-vector n = ,-e. P
(27)
The deformed form of the energy-momentum vector can be written n* = Á - n , 1
(28)
where A is a Lorentz mátrix. (It is immaterial whether we take (27) and (28) in an orthogonal representation or not.) In particular considering a partiele at rest relative to a system of reference K we can write for its energy momentum vector in this representation q
I I = 0, 0, 0, - m c .
(29)
2
0
Applying a deformation A we find for the deformed system T
my, - wc
II* =
(30)
2
with m=mB = (>
Thus we see that supposing the energy and momentum of a partiele to form a four-vector, we obtain the same expressions for energy and momentum which we derived from the Lorentz principle by direct calculation in 232. 242. Supposing energy and momentum to form a four-vector, we automatically ensure that the formalism thus obtained is in accord with the conservation laws. Indeed, if we consider the elastic collision between two particles, we may write instead of (18) and (19) 233 n < > + n< > = n i > + n < 1
1
2
2
(3i)
2 ) 2
with TT(*) _
n
(*)
'=1,2
_ pW
Relation (31) contains both the laws of conservation of energy and that of momentum. The relation (31) expresses automatically a Lorentz invariant law — indeed applying the operator A ~ to both sides of (31) we obtain n f + n > = n< > + n >, 3
4
2
4
2
where H
(*+2>
=
njk)* = x^injfc)
= \,i.
i,k
243. The derivation of the energy and momentum expressions as it was obtained in 233 is more circumstantial than that given here with the help of four-vectors. However, the derivation of 233 differs — in principle — from that obtained with the help of four-vectors. Indeed, the derivation of 233 shows that the energy and momentum expressions m\ 0
E=
mc 0
2
are the only possible relativistic generalizations of the non-relativistic laws. The considerations in terms of four-vectors give merely one possible generalization of the non-relativistic law and it is not obvious from the fourvector formalism whether or not other generalizations are possible? This remark is important because it is a fact that the measures of most physical quantities can be expressed in terms of four-vectors or four-tensors — nevertheless there is no reason why all physical quantities should necessarily be expressed by vectors and tensors. This is particularly important in connection with certain problems of gravitation. 1. NEWTON'S LAWS
244.
Newton's law, which can be written F=
f
(32)
is not an invariant one. Indeed, even if we take p to be the space part of a four-vector, the derivative of such a vector into time is not invariant. Nevertheless (32) can be taken to be valid in any system of reference. Making use of the transformation properties of p we can obtain the measure of the force F in various representations. However, the measures thus obtained are not Lorentz invariant in the sense that they do not form parts of fourvectors.
245.
(33)
Introducing
into (32) we find
JT^Tj?
or we can write
F = F
1
with
F, =
W
(i-
+
(34)
F2,
F =
m,y
x
2
2
m,y
2
where F and F respectively y and v are the components of F respectively y parallel and perpendicular to v, thus x
x
2
2
F = y(yF)/v\ x
F = F - F, a
(35)
y = v - v
l5
(36)
2
and similarly = y(yy)/v , 2
V l
2
finally m (1 v*lc ?' '
m (1 i; /c ) '
Q
m
'
=
2
Q
2
m
=
'
2
2
1/2
( 3 ? )
We see thus that a mass is accelerated by a given force to a different extent if the force acts parallel or if it acts perpendicular to the velocity of motion. 246. Relations which can be expressed in terms of tensors are obtained if we consider the mechanics of continua; a partiele e.g. can be regarded as a small cloud of mass. Supposing thus a mass distribution representing a partiele, we can sup pose that inside an element of volume ÖV v/e find őp momentum and öE energy. The force acting upon the matter inside ő V can be written as (38)
SV = főV
where f is the density of force. Intersecting the cloud of mass into elements of volume 5V we can take those elements to move together with the cloud — and thus because of the Lorentz contraction we can take 2
2
ŐV = ŐV fl-v /c
(39)
0y
where v is the velocity of flow of matter contained in bV. Rewriting (32) we find with the help of (38) and (39) I =
'
.
— —— • 2
l.T»/
2
Jl-v lc dt
The latter relation can be taken as the space part of a fourvector relation. Thus the fourforce density dU.
Í41)
forms a fourvector. 2. THE E NE RGY-MOME NTUM TE NSOR
247. The relation (41) gives the density of force which is needed to máin tain the state of motion of a matériái system with a momentum density p(x). Ifthe matériái system possesses internál stress, then the expression (40) must be extended. One finds that the state of matériái moving and alsó possessing internál stresses can be expressed by an energy momentum
tensor of the form a
-c^x
-q
cru)
where p is the density of momentum, q the density of the flow of energy, u the density of energy and a the internál stress tensor. Writing DivT=/ (43) we obtain an invariant relation between the tensor T describing the state of the matériái and the vector / describing the outer forces which are needed to maintain the state described by T. Indeed, if we have a state which (in a particular representation) is described by a tensor T with a = 0, then (43) reduces exactly to (41). In configuration where a # 0 we can take div a to represent the internál force density acting upon the volume element of the médium. In a closed system we have DivT = 0; separating space and time components we can also write őp div a +
~
=
0
dt
div q +
du dt
= 0
the above relations can be taken as the continuity relations for the flow of momentum and of energy and thus a can also be taken as to represent the density of flow of momentum. In such cases where T is symmetric such that 2
p = q/c , 2
the density of flow of energy is (apart from the factor l/c ) numerically equal to the density of momentum. The latter relation is describing the inertia of energy.
CHAPTER VIII
THE ELECTROMAGNETIC FIELD
248. For the understanding of a number of relativistic phenomena it is necessary to make use of Maxwell's theory of the electromagnetic field. We give here a short presentation of the well-known theory, partly to facilitate understanding, but alsó because we prefer a formulation, which deviates a little from the usual formulations and is adopted to somé extent to our own points of view. A. MAXWELL'S EQUATIONS 249.
Maxwell's equations are usually written as rotE = - ± B c
(a)
rotH =
— D + 4ni c
(b)
divD =
4nq
(c)
divB =
0.
(d)
(1)
The connection between E, D and H, B can be written as D = E + 4;tP
B = H + 4;rM
where P and M are the electric and magnetic polarizations. The current density i and charge density q are connected by the continuity equation div i + — q = 0 c
(2)
and we may add the expression for the force density acting upon a current of density i accompanied by a charge density q f = pE + i x B.
1. ANOTHER FORMULATION
250. We prefer in the following to write Maxwell's equations in a way, which though equivalent with (la-d) differs a little in form.* We shall thus write rot E = - — B c
(a)
rotB=
— É + 4m c
divE=
4nQ
(c)
divB=
0
(d)
efr
ef[
(b)
(3)
where we take i
= i + rot M + — P c
eff
(4)
Geff=e-divP. It can be seen that (3a-d) together with (4) are mathematically equivalent to (la-d) and it follows also that provided (2) stands, we have also divi
e f f
+ — éerf = 0,
c
(5)
and we may also write eff = eeffE + Í fXB.
f
ef
Relations (3a-d) give a set of differential equations for E and B only. The current and charge densities i and Q can be taken as the currents and charges flowing through matter together with the currents and charges flowing inside the atoms; f is the force density including the forces acting upon polarized atoms. Writing down the system of Maxwell's equations in the form of (3a-d) we think that they express the purely electromagnetic properties of the field and the role of matter is introduced through {4) which gives the currents i and charges g which depend on the state of matter. 251. P and M together with g and i characterize certain features of the state of matter; they determine the interactions between matter and the electromagnetic field. As outer fields cause polarization, move charges and efí
C{{
eff
eff
eff
* A more detailed analysis of Maxwell's equations I have given in Acta Phys. Hung., 20, 59, 1966, and Acta Phys. Hung., 20, 67, 1966.
change current densities, the change of the quantities P, M, Q and i are affected by the field acting upon matter. The exact mode of the change of these quantities depends not only upon the field, but alsó upon the particular properties and of the state of the matter on which the field acts. In older treatises it was usual to consider the simple assumption to the effect that P = KE and M = x'B (6) i =
a r e
s=l+
4UK
= 1+
\i
Anx'p
thus 1
where e is the dielectric constant and fi the magnetic permeability. The assumption to the effect that matter can be characterized by the distributions of a dielectric constant and a magnetic permeability and electric conductivity is a very primitive assumption on the physical nature of matter. We have to drop the simple assumption (6) and consider instead P, M, i and Q as characteristics of the state of matter the change of which depend partly upon the fields acting on matter but alsó on the physical state in which the matériái under consideration is found. We shall deal further below with a particular process for which it is important to replace (6) by more complicated relations. 252. From Maxwell's equations one can derive two relations which can be written as follows div^* —«= -Ei c
e f f
(7)
with
ff = -J-(ExB) « =
J - ( E + B ). 2
2
We can take c§ to be the density of the flow of energy, while u to be the density of electromagnetic energy.
Similarly we can form a tensor T =
— ( E o E + BoB) + lw
(8)*
which obeys the relation divT + - á = - f . (9) c Relations (7) and (9) give thus the distribution and flow of energy and momentum in an electromagnetic field. Indeed, the left hand expression of (7) gives the continuity equation of the flow of energy; if the right hand expression is equal to zero, then it describes a state of affairs where the energy flow deposits energy at a rate div § and the deposited energy appears in the form of electromagnetic energy of density u. The right hand side of (7) gives the rate with which energy is transformed from electromagnetic energy into other forms of energy. Similarly relation (9) gives the distribution of the flow of momentum which is described by Maxwell's tensor T; §jc can be taken as the density of electromagnetic momentum. The right hand side of (9) gives the rate of momentum transformed into non-electromagnetic form. I have given a detailed discussion of the relations (7) and (9) elsewhere.** eff
B. SOLUTIONS OF MAXWELL'S EQUATIONS 253. With the help of equations (3a) and (3b) the distributions E(r, t), B(r, /) can be calculated for any value of / provided the currents and charge densities i ff(r, t) and Q (r, t) are known and provided we impose an initial e
efí
* The expression (8) giving Maxwell's tensor differs from the förm it is usually given. We prefer the form (8) for two reasons: 1) The tensor (8) contains only E and B and is symmetric; in the usual form the tensor contains also D and H and is unsymmetric. The tensor in the form (8) expresses the purely electromagnetic momentum densities, while if D and H are also made use of, then we obtain a tensor which contains also part of the elastic energy and momentum which is produced in the raattér upon which the field acts. 2) The sign of T as given in (8) is the opposite to the sign usually used. We prefer this choice of the sign because T as defined by (8) obeys the continuity relation (9), i.e. the components of T can be taken to represent the flow of momentum density. In regions, where f = 0, (9) expresses the conservation of electromagnetic momentum. For regions with f # 0 (9) expresses the fact that the electromagnetic momentum increases or decreases but the increase or decrease is balanced by the change of mechanical momentum; the latter change appears in the form of a force acting upon matter. ** L. Jánossy: Acta Phys. Hung., 20, 6 7 - 7 9 , 1966. e[I
ef[
condition of the form E(r, 0) = E (r)
B(r, 0) = B (r)
0
0
(10)
upon the field strength. Relations (3c) and (3d) play the role of supplementary conditions. It can be seen easily that the latter conditions if fulfilled at t = 0 remain fulfilled at any other time, provided the currents and charges satisfy (5). Thus (5) and (3d) impose merely a restriction on the initial conditions. 254. Maxwell's equations can alsó be expressed in terms of potentials. Writing E = - g r a d $ - — Á,
B = rotA,
(11)
equations (3a-d) can be transformed into the so-called wave equations, namely V A 2
V
5- Á = - 4ni*eff
(a)
e
= -
\ &
(b)
4itQ
tfí
div A + — $ = 0. c
(12)
(c)
Equations (12a-c) are equivalent to equations (3b-c) in the following sense. We may prescribe an initial condition for the potentials in the following form: A(r, 0) = A (r) Á(r, 0) = Á (r) #(r, 0) = (r). 0
0
0
Because of (12c) we have é(r, 0) =
0
(14)
and therefore (13) must not include an arbitrary initial condition for The system (12a) and (12b) together with the initial conditions (13), (14) determine A, í> uniquely for any value of t # 0. Relation (12c) plays the role of a supplementary condition. It can be seen that the latter condition will be fulfilled automatically for any value of t if it is fulfilled for one value of í, and provided the current and charge densities satisfy the continuity relation (5). The solution of (12) thus obtained and introduced into (11) provides a set of field strengths E(r, t), B(r, r) which for themselves satisfy (3a-d). The
latter field strengths satisfy initial conditions of the form (10), namely E(r, 0) = - grad
0
0
B(r, 0) = rot A (r) = B (r). 0
0
The initial conditions (15) fulfil automatically (3c) and (3d) for t = 0 as it must be required for proper initial conditions. We see thus that solutions of (12a-c) provide with the help of (11) solutions of (3a-d). 1. G A U G E TRANSFORMATION
255. So as to see also the inverse of this statement we note that the equations (15) can be reversed. Indeed, relations (15) are satisfied if we put e.g. 4> (r) = 0 0
Á„(r) = - cE (r) 0
(15a)
1 C rotB (r') dx 4TJ |r-r'| 0
From (15a) it follows also div A„(r) = 0
and therefore
#o( ) r
=
0-
Giving therefore the functions E (r) and B (r) [in accord with (3c-d)] we can always construct functions A (r), A (r),
0
0
A' = A + grad V
0
0
0
X
C"
V = 0
(16)
for all values of r and t. Indeed replacing in (11) A and tf> by A' and the values of E and B will not be affected. The transformation (16) is the so-called gauge transformation — and we see that the solutions of Maxwell's equations can be expressed by potentials differing in gauge. 256. Because of the ambiguity of gauge it is often suggested that the potentials A and d> do not represent real physical quantities but are only convenient mathematical expressions with the help of which the solutions of Maxwell's equations can be obtained. We think, however, this not to be the case. So as to show why we think that A and $ have their good physical meaning, we investigate in a little more detail the set of wave equations. 2. RETARDED POTENTIALS
257. A set of particular solutions of the wave equations (12a-c) in terms of the current and charge densities can be given as follows
j-^^rfV,
A«(r,0»
(a) (17)
(b) with
/' = / - ] r - r' | /c.
(c)
j
The integrals (17) give indeed solutions of (12a-c) as can be shown inserting them into (12a-c). The latter integrals give, however, only particular solutions of (12), as the solutions (17) fix already A and
(0)
(0)
(0)
V A 2
( 0 )
- 4 A ' = O, ( 0
1 ..
y2$(o)
$(0) o, =
divA<°> + - í » = 0. c Equations (18) admit of non-trivial solutions.
(18)
It can be seen easily that we can always construct solutions of (12) of the form
A(R, 0 = A « ( R , t) + A<°\r, t) 1 cP(r, t) = &"Kt, f) + <£ (r, 0 (0)
j
so that A(r, /) and <£(r, /) provide the solutions of Maxwell's equations (3) with an arbitrary initial condition (10). We note that A and
(0
W
(0)
7
3. A D V A N C E D
259.
The wave equations (12a-c) admit also solutions of the form A<*»(r,0= *W(r,0 = J
with
POTENTIALS
\j~~^dh',
(a)
^
(b)
^T-r'7
t" = t+ | r - r ' |/c.
(19)
(c)
The latter expressions are called advanced potentials. The kinematic interpretation of (19a-c) is that the charges and currents at times í" > t determine the potentials at the time t in a point P with the coordinate vector R. Such an "action from the future" has, of course, no physical meaning; somé authors are inclined to reject the physical significance of both potentials not only because of the uncertain gauge, but also because of the possibility of expressing fields in terms of advanced potentials.
260. In our opinion the retarded potentials have a good physical meaning while the advanced potentials have no such meaning*. The advanced potentials can be eliminated from the theory in a simple fashion. Let us write A { i } resp. A {i} for retarded and advanced potentials derived from a current density i according to (17) respectively (19). Writing A{i} for an arbitrary solution of the wave equation corresponding to a current density i, we find considering the linearity of the equations w
(a)
A{i} = A "{i'} + A {i"} (
(ű)
i' + i* = i,
(20a)
and similarly for the scalar potentials tp{ } e
= d>«{ '} + * « { g * } e
e
' + e" = Q-
(20b)
We can thus split the potentials which correspond to source densities i, q in retarded and advanced parts corresponding to source densities i', q' respectively i", q". B y splitting i, q suitably we can achieve that (20a) and (20b) give solutions of (12) satisfying arbitrarily given initial conditions. Superficially it may thus appear as if it was necessary to make use of both retarded and advanced potentials so as to obtain solutions of Maxwell's equations with arbitrary initial conditions. 261. We note, however, that an arbitrary solution of the wave equations can also be written A{i} = AW{I} + (AWfc} - AWOx}).
(21a)
Indeed if we put i
1 =
i'-i
(21b)
i" = i - i ' ,
then (21ab) become equivalent to (20a). Furthermore we may write A « { l } - A W { i } = A«»{i }. 1
1
1
A {'i} thus defined is a solution of the homogeneous wave equation (18). Thus writing in place of (21a) (0)
A{i} = A('>{i} + A ^ } ,
(22a)
and similarly for the scalar potential (r)
<*•{(?} = * {«?} + *«»{ei},
(22 b)
we see that the field described by A and 4> can be split either according to (21a, b) into a retarded and advanced contribution, or according to (22) into a retarded and a homogeneous part. In our opinion the splitting up according to (22) reflects upon the physical contents of the potentials. * L. Jánossy: Acta Phys. Hung., 20, 5 9 - 6 6 , 1966.
4. W A N D E R I N G WAVES
262. It can be concluded therefore that any soiution of Maxwell's equations can be represented as a superposition of a retarded solution, i.e. the solution which is obtained as the solution arising from the retarded actions of currents and charges, and of a homogeneous solution, i.e. an electromagnetic field without sources. The question arises whether in reaüty fields corresponding to homogeneous solutions exist? Such solutions, if they are realized in nature, have to be regarded as "wandering waves" which are traversing the universe but did not arise originally from moving charges. It is a question of experiment to decidé, whether or not such wandering waves occur in nature. 263. We remark that whenever we meet electromagnetic waves we immediately search for their sources, i.e. we suppose immediately that they are of the retarded kind. When astronomers observed electromagnetic waves coming from the universe, they immediately made theories to their origin. It seems not unlikely that wandering waves do not exist at all and that waves occurring in nature are all of the retarded type. The latter hypothesis involves that Maxwell's equations are giving only necessary conditions for the motion of electromagnetic fields — and that in nature only particular solutions of these equations, i.e. those which are represented by retarded potentials do really occur. As an example of an "advanced" solution of Maxwell's equation we mention the solution corresponding to a spherical wave contracting with the velocity c and finally disappearing in one point where a suitable charge (waiting for the wave) absorbs it. It is quite clear that such solutions of Maxwell's equations are only mathematical possibilities and that such solutions do not correspond to real phenomena. In generál a process described by advanced potentials only (like the example of the contracting spherical wave) can be regarded as the time reversals of processes which can be described by retarded potentials only. If we suppose that electromagnetic processes occurring in nature can always be expressed in terms of retarded potentials, then we have to conclude that electromagnetic processes are irreversible. Indeed, the time reversal of any real process leads to one, which would have to be expressed in terms of advanced potentials — and therefore the reversed process is excluded by our hypothesis. 264. As an argument in favour of the occurrence of advanced solutions it was put forward that retarded potentials correspond to the emission of waves, while advanced potentials represent the process of absorption. We do not want to discuss the problems of quantum electrodynamics here. However, we note that as far as we are inside the limits of validity of classical theory, there is certainly no need to suppose the existence of advanced fields for to understand the process of absorption.
The totál exchange of energy and momentum of the electromagnetic field and matter was given in 252. The latter equations show that an electromagnetic field may transfer energy and momentum to matter and thus lose corresponding amounts of field energy and momentum. This process follows from Maxwell's equations; to understand its mechanism we note that an atom while it is interacting with radiation is excited and emits secondary radiation. If energy is transferred to the atom in the process, then the secondary radiation is emitted with such phase that it extinguishes by interference part of the incoming waves and thus reduces the energy contents of incoming radiation. Thus processes of absorption can be unambiguously accounted for in terms of the retarded fields only. We note, further, that it is quite irrelevant to the question of advanced potentials that for certain technical calculations (for the reason of mathematical simplicity) engineers sometimes make use of advanced solutions.
C. MAXWELL'S EQUATIONS IN TERMS OF FOUR-TENSORS 265. In the sections below we use four-vector notations in a form which is explained more precisely in Appendixes I and II. Maxwell's equations can also be expressed in terms of four-tensors. Consider a Lorentz system of reference K in the terms of which Maxwell's equations have the form as given in 250-254. The relation given there can be re-written. Denote ¥ = A, - c
(23)
Ieff = >eff, - cq
(24)
the four-potential and tfí
the four-current density. In accord with (4) we can also write I
e f f
=I +
Divn,
= i, -
CQ
(25)
where I
is the conductive part of the four-current and I I is an antisymmetric mátrix representing the polarization, so that Ilik = Mi i7
j4
= ~cP-,
i, k, l = cyclic permutation of 1, 2, 3 i = 1, 2, 3.
266.
Using the above notation the continuity relation (5) can be written Divl
=0.
e f f
(26)
The wave equations (12) can be written D i v ¥ = 0,
K
}
(where we write L for the Laplace operator in four-dimensions, see Appendix I, 449). The field strength can be obtained from the four-potential as F = Rot *F.
(28)
Comparing (28) with (11) we find that Fik = ' B t — Fa c
U k, / = cyclic permutation of 1, 2, 3 i = l, 2, 3.
= E,
(28a)
Maxwell's equations in the form (3a-d) can thus be written Div F =
47rl
e f f
Div F = 0,
(29)
with ~ 1 (*) F = y((e-F))
(30)
(see Appendix II 468). 267. The energy and momentum can be expressed by a four-tensor, i.e. (> 1 ~ ~ T = - — ( F - F + F-F). 2
(31)
Separating (31) into space and time components one fmds with the help of (7a, b) and (8) ik
T
= ik> T
T
i4
= r, 4
c§
= -
T
h
u
= c-u.
(31a)
Relations (7) and (8) can thus be written (2)
DivT + ^
e f f
= 0,
(32)
where ^ef =fef , f
f
-cEi
e f f
thus L^eff is the density of the ponderomotoric four-force.
(32a)
268. The relations (23)-(30) give Maxwell's equations — including the expressions of the conservations of energy and momentum — in terms of four-vectors and tensors. We note, however, that this formulation in itself does not give any new physical result. Indeed, if Maxwell's equations as given by three-vectors and tensors are valid in the measures of a system K, then the four-vectors and tensors relations as given in 265—267 are automatically valid — as they represent the same mathematical relations just expressed with a different notation. Furthermore, submitting the four-vector and tensor relations to an arbitrary reversible coordinate transformation x' = f(x) then according to the results of Appendix I the relations (23)-(32) retain their form provided we define suitably the operations Div, Rot, L. The latter result is not surprising. It shows merely that Maxwell's equations express objective laws which can thus be expressed in terms of arbitrary measures of coordinates, vectors, etc. The simple form in which Maxwell's equations can be expressed using four-vectors and tensors proves that the four-potentials, four-currents, etc, can be taken as a kind of distinguished measures which are particularly suitable for to describe an electromagnetic field. 1. RETARDED FOUR-POTENTIAL
269. A new result — which cannot be regarded merely as change of notation — is obtained if we consider the retarded potential solutions of Maxwell's equations. Relations (17) can also be written as
Y(x) = J
Ieff(x
^
+X)
/ R 3
t
(a) (33)
T= - Rjc
(b) I
X = R, T R = | R |. (c) The expressions (33) contain explicitly the space and time parts of coordinate vectors and therefore it is not obvious that the relations (33) are invariant. We show presently how the invariance of (33) can be proved with respect to reversible linear transformations. We consider a linear coordinate transformation x' = Sx + s.
(34)
According to Appendix I we have for any covariant vector field A'(x') = S-^AÍx). Thus applying S to both sides of (33) remembering that T and I covariant fields, we have _ 1
eff
are
S - I ( x + X) = i; (x' + SX) 1
eff
fr
and therefore Y'(»') = J
l
°"( '+ x
S X )
d
*
R
( 3 5
)
with T=-R/c.
(36)
We note that the coordinate transformation (34) does not affect the variables of integration and therefore in the integrál on the right of (35) is to be integrated just as (33) into the components of R. We introduce X' = SX
(37)
as new variables of integration. We note that (37) expresses the relations between the three components of R and those of R'; the fourth components T of X and T of X' have to be expressed in terms of R respectively R'. We define X' so that X'gX' = 0
(38)
with g = S" rs- . 1
(39)
1
It is convenient to separate space and time components; we write thus for the sake of simplicity U B
Aj a)
_ [G V ~ (v - C
g
One finds alsó using known algebraicai relations det S - = det U(a - (TJ- A)B)
(40a)
det g = - det G(C + VG V).
(40b)
1
1
2
_1
We obtain from (39) and (40) G = UU-c BOB,
(a)
V = AU - c aB,
(b)
2
2
2
2
C = c a - A, 2
2
(c)
and also det S = c/c'
c'=V-detg.
with
(d)
270. Separating the space and time components of (37) we may write with the help of (40) R = UR'
+ AT',
(a) (b)
T = — Rjc = B R ' + aT';
(42)
eliminating T from the first relation with the help of the second we find AR R +
v - ^ ] r .
ac
Differentiating the above relation into R ' and taking the determinant of the differentiated expressions of both sides we find as the result of a simple calculation ARI 1 , l dR R +1 —det
1
ac } R —
= det S~7a;
(43)
ŐR'
expressing R in terms of R ' and 7" with the help of (42) and (41) we find AR 1 (44) R + = — (- C T' + V R ' ) . 2
ac
ac
However it follows from (38) 2
C T'
= VR'-
C\/R'GR'
with V O V G = G +
- - ^ -
(45)
where we take 7" to be the solution of (38) so that T
<
0.
Thus inserting from (44) and (45) into (43) we find 1 ~R
-det
dR dR
1
(46)
271.
Introducing (47) into (35) we obtain Iefffr' + X') d R'. R' 3
(48)
We see thus that in the system K' the four-potential can be expressed by the same expression as in K. The interesting feature of this result is the manner in which the retardation is introduced in the system K', Considering the action of sources from a point P' to a point P we note that the action which arrives in P at a time t' starts at the time + t' where \T'\ is the time of flight from P' to P; that this is indeed the case follows from the fact that 7" is obtained as the suitable solution of (38). Furthermore according to 163 the expression (47) for R' gives just such a value that 2R'\c' = tp . + t'p. P
P
i.e. 2R'/c' gives the time of the return flight of a signal of light between /' and P'. Thus the expression (48) has to be taken supposing that the measures of three dimensional distances R' in K' are given by the times of return flights of signals covering the distance. The latter result shows the consistency of our considerations. 2. THE MOTION OF LIGHT SIGNALS I N TERMS OF MAXWELL'S EQUATIONS
272. Maxwell's equations written down in terms of four-tensor contain implicitly the propagation tensor g as the expressions for the vector operators contain g. We introduced g as a tensor describing the orbits of signals of light; we supposed in a purely phenomenological manner that the orbit of a signal of light obeys Xgx = 0. (49) So as to show the consistency of the theory, we have to show that (49) follows Maxwell's equations expressed in terms of an arbitrary system of reference. 273. So as to show this to be the case we consider as a first step the propagation of an electromagnetic pláne wave. Maxwell's equation can be
written V¥ = 0,
D i v Y = 0.
(50)
The system (50) permits pláne wave solutions of the form * = !(«),
(51)
(remembering that x(p) is a contravariant four-vector) where a is a zerovector obeying o - a = 0,
(52)
and f(w) is a four-function satisfying o-f'(u) = 0
(53)
for any value of u. Consider a four-point the orbit of which is given by x(p); this point will rest on a pláne of constant phase, provided «x(/>) = constant, thus oi0») = 0.
(54)
Relation (49) can be fulfilled e.g. by putting x(p) = g - a = a g - , 1
(55)
1
as can be seen from (52). From (54) it follows using again (48) x(p)gxQ>) = 0.
(56)
We see thus that the planes of constant phase of the waves contain points moving according to (56). 274. That signals of light — which can be taken as somé kind of wave packets — move also according to (56) is a statement going further than the result of the preceding paragraph. The latter statement can be seen to be correct considering the retarded solution of the wave equation. Indeed a signal is produced if somé atoms in the vicinity of a point r around a time t oscillate for a short period. The oscillation can be described by a source density
0
0
Ieff( , 0 r
differing from zero if r ~ r , t ~ í . 0
0
In an orthogonal representation the potentials in a point r at a time t are given by
r
-r
t ' = t - ~ — -
The integrand on the right-hand side vanishes unless t' ~
r ~ r, 1
t
0
0
thus we obtain potentials noticeably different from zero in four-points with four-coordinates r, t obeying t
~
t. 0
c We see thus that the disturbance in x gives rise to a spherical wave expanding isotropically with a velocity c. Such spherical waves can be used as signals of light — just as it was needed for the determination of coordinate measures. Changing from the orthogonal representation to an arbitrary straight representation, we find that the expansion of the spherical wave in the new representation can be described by a quadratic expression of the form (56). Thus the retarded solution of Maxwell's equations in an arbitrary representation lead to propagation of light in accord with (56). 0
D. MAXWELL'S EQUATIONS AND THE LORENTZ PRINCIPLE 275. In the preceding section we have shown that Maxwell's equations can be expressed in a consistent form which form is independent of the system of reference. Physically new statements are obtained if we apply the Lorentz principle to Maxwell's equation. Let us consider an electromagnetic field In a straight representation we may write K(%) = F(x), where F may contain the potentials, the field strength and sources of the field in the representation K. According to the Lorentz principle we expect that provided % is a field moving in accord with Maxwell's equations then £ ®)=r f l
should alsó be such a field. 276. The latter statement can be seen to be correct considering Maxwell's equations expressed by four-tensors. To show this consider e.g. the tensor
F(x) expressing the field strength of j$. We have F*(x*) = M ^ x j M
1
with
(57) X* = M X + (JL q
and M is a Lorentz mátrix obeying q
M„gM = g.
(58)
q
The transformation (57) has the form of a coordinate transformation: therefore if F(x) obeys Maxwell's equation in K, then the deformed field quantities F*{x*) obey Maxwell's equations in a system of reference K' the measures of which are obtained as ( 5 ,
x' = M " x + vl . with
M
( < , ) _
'= M . q
Since the transformation is a Lorentz transformation we have
m
= A"(A) = g,
therefore the Operators Grad, Div, etc. are identical in AT and K'; thus from the fact that the representation of the field % in K' obeys Maxwell's equa tions, it follows that the representation of the deformed field obeys these equations in K. 277. In place of (57) one can also write F*(x) = M ^ M
1
(59)
with i = M ^ í x [x).
(60)
Relation (59) is identical with (57) but the former expresses more clearly the fact that F*(x) just like the undeformed field F(x) is the representation relative to the same system of reference K. Explicitly written we have for the field strengths of a deformed field E*(x) = E,(x) + 2?JE (x) 1
(vx B fx)))
2
(61) B*(x) = B iíx) +
j
Ő[B (X) + 2
(vxE(x))j
where we denote with suffix 1 respectively 2 the component of a vector parallel, respectively, perpendicular to v. Thus e.g. E = v(vE)/u
2
1
E2 = E — E^.
Splitting into components parallel and perpendicular to v we can alsó write in the place of (61) E?(x) = E i ® ,
BJ(x) = B j ®
and E*(x) = I ? ( E ( X ) -
|(vxB (x))
2
2
(62)
B?(x) = i?JB (x) + i ( v x E (x)) 2
2
The inverse of transformation (60) split into space and time components can be written x = r, t AND
r = B(i - ví) + r 1 = B(t - vr/c ) t
(63)
2
2
where r and r are the components of r parallel respectively perpendicular to v. x
2
1. T H E FIELD OF A POINT CHARGE
278. Consider the field of a point charge e at rest in the point r = 0; we find for the field strengths B(r) = 0
for any value of t.
The transformed field is found with the help of (61) and (63) E*(r,
0=
Be(r - ví)
Be B*(r,0= — ( v x r ) cs
(64)
with s = B^ 2
2
- \t) + r 2
2
So as to see the significance of (64) we note that for velocities M c w e have B ~ 1 and s ~ r. In the latter approximation we find that the moving charge carries away the electric field as if the field together with the charge förmed a solid. A magnetic field is induced which is circling round the charge. The above result is verified experimentally; it amounts to the BiotSavart law giving the magnetic field of convection currents. We see that — from the experimentál point of view — the constant c appearing in the
latter relation has to be identified with the critical velocity c' rather than with the velocity c of propagation of electromagnetic waves. 279. So as to see the relativistic effects we consider for a fixed value of t the electric field in a pláne Ti =
\t,
i.e. in a pláne perpendicular to v which contains the charge. We have r i
- \t = 0,
.s = | r | = r 2
and therefore Et(r,t) =
Be -^-,
i.e. the field strength is increased by a factor B relative to that of the charge at rest. The longitudinal field with r = 0 2
and
s — Br
gives an electric field
Thus the field strength is reduced by a factor l/B in the longitudinal direction. In the extrémé relativistic case when Bf>\ the field carried by the charge is concentrated into a small region in the vicinity of the pláne perpendicular to v and moving with the charge. The field thus moves with a velocity v ~ c, it is nearly transversal and the electric and magnetic field strengths are nearly equal. Thus the field carried by the fast moving charge resembles strongly an electromagnetic pláne wave accompanying the charge. The latter result can be compared with experiments; it is verified indirectly by observing collisions between fast moving charged particles.* 280. The measure of charge e* of the moving partiele may be defined by Gauss theorem, i.e. 2
e* = -^-J*E*(r,/>ÍS,
(65)
where the integration is to be taken at a fixed time t over a closed surface containing the moving particle. Inserting the field strengths as obtained from (64) into (65) we find e* = e. Thus maintaining the definition (65) we conclude that the measure of the charge does not change if the partiele is set to move. * Sec for details e.g. L. Jánossy: Cosmic Rays. Clarendon Press, Oxford 1950. 2nd ed.
CHAPTER IX
RELATIVISTIC EFFECTS OF THE ELECTROMAGNETIC FIELD A. EFFECTS OF THE FIRST ORDER
281. Developing into powers of v/c we can classify effects for velocities v < c as of the first order, of the second order or of higher orders. The essentially relativistic effects are of the second (or higher) orders. Somé effects of the first order have, however, played a great role historically and it was not always fully realized that the latter effects can all be accounted for without making use of relativistic concepts. Such misunderstandings appeared in connection with the Doppler effect (not including the perpendicular Doppler effect) the effect of aberration of light and in particular in connection with Fizeau's result giving the velocity of light in a moving médium. It was claimed sometimes incorrectly that "Fizeau's experiment gives an experimentál proof that the relativistic law of addition of velocities is the correct law, the classical addition law being only approximately correct". The latter statement was already criticized in chapter VI208. Discussing the mechanism of first order effects we shall show in particular in this chapter that the results of Fizeau can be understood in terms of the classical theory of propagation of light in refracting média and it is not necessary to make use in this treatment of relativistic concepts. 1. EFFECTIVE FIELD STRENGTHS
282. The claim that the first order effect can be explained without using relativistic concepts has to be made a little more precise. One finds that the first order effects can be correctly interpreted supposing (1) that a closed physical system when made to move with a constant velocity v does not deform noticeably. This assumption implies that for the purpose of the first order approximation we can neglect the second order effects like the length contraction and the slowing down of the rates of clocks. We suppose e.g. when giving the theory of aberration that the telescope moving together with the Earth behaves like a rigid body and does not suffer deformations while the orbital velocity of the Earth changes. (2) We suppose that an electromagnetic field of strengths E and B acts upon a physical system Q moving with a velocity v as if Q was at rest and
placed into a field with field strength E
eff
= E + -(vxB), c
B
eff
= B - -(vxE). c
The expressions (1) describe first order effects only; they can be obtained considering the Biot-Savart law and the law of induction. From the latter it follows that an electric charge moving with a velocity v possesses a magnetic field B (r) = e
while a moving magnetic pole m produces an electric field of strength E (r) = - •m v x r m
Supposing the principle of action and reaction to hold (at least up to terms of the order of v/c) from the above two relations (1) can easily be obtained. The interpretation of first order effects as the aberration and Fizeau's result can be obtained in a much more elegant way using the Lorentz transformation then with the help of detailed classical considerations. We give the latter considerations only so as to show that these effects can indeed be understood without the use of relativistic concepts. The fact that these effects can also be interpreted making use of Lorentz transformations proves the consistency of the theory. 2. THE FIELD OF DIPOLES
283.
The field of an electric dipole situated in r = 0 can be written E(r) = - grad
,
B = 0,
(2a)
where p is the dipole moment. Similarly we find for the field of a magnetic dipole Bír) = - g r a d - ^ , r
E = 0,
(2b)
where m is the magnetic dipole moment. The field of an electric or a magnetic dipole moving with a velocity v can be obtained as the Lorentz deform of the fields (2a) and (2b).
The moving dipoles of both kinds possess both electric and magnetic fields. The electric field of a moving electric dipole is found to be — apart from terms of the order of v jc — to be that of the originál dipole moving together with the dipole. The magnetic field has a complicated structure. The main (non-relativistic) features of the magnetic field of a moving electric dipole can be obtained, if we take in place of a dipole a thin strip, which is polarized perpendicular to its surface (see Fig. 22) and which is moving parallel to its surface. We see thus that the electrically polarized strip, when set to move, shows a magnetic field corresponding to a magnetic polarization given by (2b). 2
2
+ + ++++ +
Fig. 22. Scheme of a moving strip polarized perpendicularly to the velocity
The strip behaves as if the electric dipoles when set to move would develop magnetic dipole moment dm = v x í/p/c. The above result has to be taken with care; the electrically polarized strip behaves as if it would contain magnetic dipoles of moment dm =
MdS.
However, the magnetic fields of the single electric dipoles when set to move differ from those of magnetic dipoles of moment dm. The superposition of the fields of the moving electric dipoles is, however, equal to that of the superposition of the equivalent magnetic dipoles. The strip can be taken to consist of electric dipoles of cross-section dS with an electric dipole moment dp =
TdS.
The Poisson charge representing the polarization consists of surface charges of density + P on the surfaces of the strip. The moving strip carries then convection currents of opposite signs on its surfaces; the surface densities of the currents are + | v x P \jc and they are currents which produce a magnetic field corresponding to a magnetic polarization M = vxP/c.
(2c)
Similarly, a magnetized strip, when set to move, produces an electric field and we find P = _ v x M/c for the apparent electric polarization of the magnetized strip.
(2d)
B. TRANSFORMATION PROPERTIES OF FOUR-CURRENTS 284. For the understanding of somé electromagnetic phenomena it is necessary to consider higher order effects also. Consider a uniform charge distribution with constant density Q . The current charge density distribution can be written 0
I = 0, 0, 0, - CQ . 0
0
A Lorentz deformed version of the distribution is obtained; applying the operator A to I„ we find T
I* = í, -
CQ,
with í
=
v c
Q =
o
Bg . 0
285. The current in a conductor consists of moving charges of one sign and a system of stationary charges of the opposite sign. The two charge distributions compensate each other statically. We may thus write for the current density in a conductor I = Ii + I , 2
with I = 0, CQ,
II = í, ~ CQ,
2
thus 1 = 1,0.
(3)
The deformed current density corresponding to (3) can be written I* = i*, (vi)^ with i* = Bi
1
+
i, 2
(4)
ii, i being the components of i parallel respectively perpendicular to v. It is interesting that the deform of the neutral four-current with density (3) gives a four-current (4) which possesses an excess charge with density 2
Q* =
-(VÍ)B/C.
(5)
The usual interpretation of (5) is that the density Q* occurs because "simultaneity depends on the system of reference". We do not think the above statement to make sense, but we do think that (5) has a very simple physical content.
1. THE ELECTRIC FIELD OF A MOVING CURRENT
286. So as to see how the excess charge Q* comes about consider a rectangular current as shown in Fig. 23. Suppose charges of density Q to move around the conductor with a constant velocity w and suppose that the conductor contains charges with density — g which are at rest. The current density is thus of the form (3) and the system has a magnetic field only.
Fig. 23. Scheme of a closed current moving with a velocity v
Consider the deformed version of the rectangular current which moves as a whole with a velocity v in the direction AB. In the latter configuration the charges moving in the circuit will have velocities (see (7) in 205) Wy
=
v+ w
, and
vw
c
v—w .
vw
c
2
2
in the sections AB and CD and therefore they have velocities
W
1
= w —v = 1
vw
1+ — cT 2
and w|lW
2
=w - v= 2
vw 1*
relative to the conductor. Supposing vw > 0 we find W
y
<
W
2
thus the particles flowing faster in the section C -» D than in the section A -+ B. Since the particles are thus moving with a variable velocity through the conductor their density changes accordingly, the actual number of particles to be found per unit volume being inversely proportional to the velocity of progress. (This effect can be compared with the density of traffic along a road. In sections where the traffic is slowed down by somé obstacle the density of vehicles increases.) The densities of the moving charges in the sections AB and CD are thus equal to ei =
-^r-e =
^ - ( e + WO
1
c
2
where we have written gw/c = i. Neglecting thus terms in v /c we see that the arms AB resp. CD of the circuit show excess charge densities + Aq with 2
2
Aq =
Q*
= iv/c.
We conclude that the electromotoric force which drives the charges evenly around the circuit, when the circuit is at rest, produces a nonuniform motion if the circuit is moving with a velocity v. The unevenly moving charges produce in the deformed systems excess charges through effects which can be compared with a "traffic jam". These excess charges give rise to an electric field. 287. The current shown in Fig. 23 acts at large distances like a magnetic dipole m the direction of which is perpendicular to the pláne of the current. If the circuit is set to move with a velocity v in the direction A -> B, then the excess charges ±Aq forming along AB and CD produce a field which at greater distances appears as that of an electric dipole of moment it = — v x m/c; the latter is in accord with solutions (2d) 283 relating to the transformation properties of electric and magnetic dipoles. The above derivation is making use of the addition formuláé of velocities and thus the electric field of a moving closed current appears as a second order effect. The electric dipole field of a moving magnet can be taken to be a first order effect in accord with 283. This comparison shows that the distinction between first and second order effects is not a sharp one.
C. FURTHER EFFECTS OF THE FIRST ORDER 1. DOPPLER EFFECT A N D ABERRATION
288. The above effects can be interpreted up to first order terms with the help of relations (1). However, as the higher order effects are alsó of somé interest we give here the exact calculations which include Lorentz deformations also. It can be seen easily that the relation obtained concerning the Doppler effect and aberration can be obtained in a first order ápproximation without making use of the Lorentz transformation but using only relations (1); this gives an effective field strength acting upon a moving system O. Let us consider a source of monochromatic light in a point A at rest and at a large distance from the origin of the system of coordinates K. The waves arriving from A produce in the vicinity of the origin oí K a field which can be described in a good ápproximation as a pláne wave. The waves satisfy Maxwell's equations, thus they can be expressed in terms of a four-potential •F(x) = «P cos 2K 0
+ 4>)
(<XX
(6)
where *F , a are constant four-vectors and <j> is a constant. Inserting (6) into Maxwell's equations we find that 0
a-a = 0
' F - a = 0. o
(7)
Writing a = x, — co
co > 0.
(8)
From (7) and (8) it follows also that co = ex,
(9)
we can take K to be the wave vector. We shall also write k thus k is the unit vector pointing into the direction of propagation of the waves. From (6) it follows F(x) = F sin 2n (ctx + 0
F = - 2rox x W . 0
0
<j))
289. Taking the Lorentz deformed configurations of the field considered in the previous paragraph, we obtain the field of a source A* moving with a velocity v relative to K. Using the formuláé of 277 we find F*(x) = F J sin 27c(aA_^ +
(10)
with F*, = A L . F . A ^ .
(11)
aA _ = A_ a = o*
(12)
We may write T
and in place of (10) F*(x) =
FJ
sin 2JI
( O * X + <j>).
From (11) and (12) we see that o and F change in the course of deformation like a four-vector and a four-tensor. So asto obtain the wave length frequency and direction of propagation of the deformed wave we introduce the quantities 0
a* = x*, —co*, co* = ex*
(13)
and also x* = —
with k * ' = 1.
Inserting (13) into (12) we find ,<x = Bx* + Bvco*/c co = B(co* - vx*)
(a) (b)
2
(14)
where x* and x* are the components of x* parallel resp. perpendicular to v. 2. FREQUENCIES OF THE DOPPLER EFFECT
290.
Taking the square of both sides of (14a) and remembering that x*x* = x*v = 0
we find 5 |cos#*- - | c 2
+sin #*
where we have written x* \/x*v — cos •&*,
2
thus d* is the angle between x* and v. Remembering the connection between B and vjc (15) can also be written x=x*5|l--^cos#*j .
(16)
From (9) and (13) we have further co* = cox*/x. Thus we may write in place of (16)
T
•
(17)
COS0*
1
c
Starting from the inverse expression of (14) we obtain in place of (17) similarly co
J
—
(18)
.
1 H — cos 0 c
3. EFFECT OF ABERRATION
291.
Multiplying (17) and (18) we find further l/B = 2
|i
+
-!icos#j j l - - % o s 0 *
and thus cos#* =
cos •&
v H
— • V 1 -\— cos V
(19)
c
The latter expression can also be written sin &*
sin 1 + — cos# c
Neglecting terms of second order in vjc we can also write
= # _ &* „ 1 i # c
S
n
(20)
where 0 is the angle of aberration. We come back to the discussion of this effect in 293. Equation (17) gives the frequency distribution of the Doppler effect, the latter expression is identical with that obtained from the purely phenomenological consideration of 38. 4. INTENSITIES IN THE DOPPLER EFFECT
292. So as to obtain the inténsity distribution of the radiation emitted by a moving source we calculate the energy density of the deformed wave. The field strength of the pláne wave arising from the source at rest can be written E =
B =
nA,
x E,
k
(21)
where TC is the unit vector in the direction of polarization, thus TTk
= 0
Jt = 2
k
2
= 1.
A is the amplitude of the wave. The inténsity of the wave can be characterized by the density of energy, thus u = -L(E
ö7t
2
+ B )=^. An
(22)
2
The energy density of the deformed wave is given by u* = ~(E** on
+ B**) = ^ .
(23)
47t
Introducing the expressions for E* and B* from (61) in 277 we obtain with the help of (23) as the result of a short calculation ^*/yt = 5|l
cos & =
+ -cosí c
\k/v.
Starting instead of (21) from relations B* = k * x E*
E* = n*A*
cos &* = k * yjv
and expressing E and B in (22) in terms of E* and B* we find also A*
1
A
5Jl-yCos#*
The relation gives the inténsity of the deformed wave as the function of direction of incidence.
In particular we find for the forward and backward directions c
A* ~A~
1
for
é = 0,
for
= n.
B\l + In the extrémé relativistic case if v 4B
2
c then we have for •» = 0
1 (24) for S* = 0 48* From (24) we see that in the extrémé relativistic case, i.e. for v ~ c, B > 1 the intensity of the deformed radiation will be very small in all directions except inside a narrow cone the axis of which points into the direction of v. ~A
5. OBSERVATION OF THE EFFECT OF ABERRATION OF STAR LIGHT
293. The considerations of the previous paragraph give the change of a radiation field from which takes place if the source is adiabatically accelerated and is made to change from © to @*. The observations of fixed stars show that their apparent positions change while the orbital velocity of the Earth changes. In the latter observation not the source @ is subjected to deformation, but the instrument Q of observation. The instrument Q suffers continuous deformations while the orbital velocity of the Earth changes and we take it that at two times, say, t = 0 and t = t the instrument has configurations Q respectively 0
D í = S (Oo), n
where S„ is the Lorentz transformation describing the deformation caused by the change of the orbital velocity of the Earth. Observing the radiation % we observe the effect of % upon £}„ and compare it with the effect of % upon £}*. However, we register the results of both observations relative to systems of reference K respectively K', relative to which the instrument at the time of observation is at rest. We compare thus the effect of F =Kfö) upon öo = *(Go) with the effect of F = * ' ( 8 ) upon e'o* = K'(G3). Since K and K' are the rest systems of Q respectively of Q* we have 0
Öo* =
Qo •
Thus we observe F respectively F' with instruments which appear to have the same configuration; thus from the point of view of the observer it appears as if not the instrument had changed from Ci -* ö * but as if the radiation field had changed from 0
3r-£-«,©) = S*. We may therefore apply the formuláé obtained in 288 — referring to change of radiation field if the source is made to move — also to the case, where the sources have remained stationary, but the instrument has been made to move. In particular eq. (20) 291 can thus be taken to give the apparent change of position of a star while the velocity of the Earth changes. 294. It is important to note that there exists a physical difference between the processes where the source is made to move and that when the instrument instead is made to move. In the former process there appears a particular type of radiation emitted while the source is accelerated. The radiation field F* reaches the observer only after the radiation emitted in the transient period has passed away. If the instrument Cl is accelerated instead, then the deformations take place practically instantaneously and no transient phase is to be expected. The above considerations can also be formulated as follows. If an instru ment of observation is placed into the radiation field of, say, a star then it indicates the direction of the Pointing vector § =
4n (E x B ). c
If the instrument is set to move with a velocity v then it will react upon the field as if its field strength had changed to values E and B as given by (1) and it will indicate the direction not of § but of eff
eff
471 e?eff =
(E ff x B e
eff
);
c is (neglecting higher order terms) equal to
the angle between § and § the angle of aberration. Let us consider e.g. a narrow metallic tűbe through which we view a star. The tűbe acts as a wave guide and the star light can pass through it only ifit is pointing into the direction of §. If the direction of the tűbe is changed then the heat losses caused by currents — produced by the electric field strength in the walls of the tűbe — extinguish the beam. If the tűbe is moved with a velocity v then E instead of E is responsible for the losses and we have to adjust the tűbe into the direction of § so as to allow the wave to pass. t í {
eff
e { (
Instead of the metál tűbe we can consider any other instrument, e.g., a telescope or — as was done by B radley — a telescope fiiled with water. Any such instrument will indicate the direction of § . The above consideration, though elementary, seems nevertheless to be worthwhile — as we find in the literature misconceptions regarding the effect of aberration. e{{
6. PROPAGATION OF LIGHT IN A RE FRACTING M É D I U M
295. Let us consider a source of light A and a receptor in a point B at the distance /. Suppose A starts to emit light at a time t = 0; the front of the emission will reach B at a time t = Ijc. This can be seen e.g. by taking
S Fig. 24. Penetration of waves through a moving médium
the retarded potential *F(x) produced by the source A in the point B; we have T(r , 0 = 0 B
for
/<— c
where r is the coordinate vector of B and / = | r — r |. 296. If we place a diffracting slab S between A and B (see Fig. 24) then we know from experience that the front reaching B will arrive at somé time t > t, the delay being caused by the transit of the electro magnetic waves through S. So as to understand more clearly the process we note that the retarded action of A arrives in B exactly at the time / = l/c as it can be seen from the explicit expression for 'F(x). The observed delay is caused by the secon dary processes arising in S. Indeed, when the radiation emitted by A falls on S, then the atoms of S start to oscillate and to emit secondary radiation. Thus the primary wave emerging out of S will be accompanied by the secondary radiation it has excited in S and — as we show presently — the secondary radiation extin guishes by interference the primary radiation for a time At = ti-t. Thus the radiation emitted by A is felt in B only at a time t = t + A t when B
B
A
x
x
the interference between primary and secondary waves has ceased to extinguish the field. 297. So as to see the above process in more detail suppose A to be far to the left and thus we can take that a pláne wave falls upon S. The propagation of the pláne wave in S can be described by
V A-X'A
= - 4m,eff
2
V 4>-\cp=-4no „ c
(25)
2
e
div A
S
= 0. c Supposing the S to be an uncharged dielectric we can take + —
•eff
= r o t M + —P
(26)
and supposing the field strength not to be too large P = xE
M = x'B
(27)
x' = -
If the dielectric is homogeneous we can suppose grad / = grad x = 0 (inside S) and thus div E = x div P = 0
and
Q
=
E(T
0.
We may thus write in place of (25) o. V A
1 ••
4rcx •
5- A =
.
,
„
E - 4tí7' rot B
c
(a)
c
div
A =
0.
(b)
Expressing E and B in terms of the potentials (see 254 (11)) we find E=
A
c rotB= - V 2 A ; we have in place of (28a) (1 - 4 T T / ) V A 2
-
1 + 47CX
(28)
or
V A--1-Á = 0
(29)
2
with V=clJen
e = 1 + 4UK
=
1—
—. T
4nx
We find thus that the planes of constant phase inside 5 are propagated with a velocity V < c. It must be emphasized that the radiation in S consists of the superimposed effect of the incident pláne wave and the radiation emitted by the atoms of S. The sources of the latter radiation can be taken as the inner atomic currents i given explicitly by (26). The front of the compound wave penetrates with a velocity V < c into S. There is thus a region into which the primary wave (travelling with the velocity c) has penetrated — but into which the compound wave (travelling with the velocity V) has not penetrated. Inside this region the secondary waves extinguish fully the primary wave as can be seen from solving (29). The latter equation gives the behaviour of the totál radiation — i.e. that of the primary wave superimposed with secondary radiation. Constructing the solution of Maxwell's equation for the region to the right of S, we find a delayed pláne wave. The front of the latter wave starts when the compound wave arrives at the right-hand surface of S. efr
a. DISPERSION
298. The above consideration is mathematically exact but it does not give a full account of the phenomena. The velocity of propagation of a front in a diffracting médium is experimentally found to be V = c/n(co) where n(co) is the geometrical refractive index for radiation of frequency co. The considerations obtained above are only in agreement with experiment as long as n(co)
~
Jhe,
The latter relation is fulfilled for very low frequencies co only. 299. The physical reason for the discrepancy is the incorrect assumption about the connection (27) between field and polarization. Consider e.g. the electric polarization: if we switch on a field E it cannot polarize atoms instantaneously, but it takes somé time until it overcomes the inertia of the atomic electrons and thus polarizes the individual atoms.
Considering in a first approximation the atomic electrons to be bound by harmonic forces, we have to replace the first relation (27) by P/co
+ P = x E
2 0
(30)
where co is the frequency corresponding to the harmonic force binding the electrons. Relation (30) was first given by Sommerfeld. A similar expression could be given for the connection of M and B; for the sake of simplicity — as we want to give here only a qualitative description of the phenomena — we consider a non-magnetic médium with ti = 1 or x — 0. 300. Using thus the relation (30) in place of (27) we can write (supposing M = 0) 0
and we have
V A--^A = - — P .
(31)
2
c
c
In place of (30) we can also write 1 •
1
- - A = - ( P / ( ü c x
+P)
and thus differentiate (31) into t 1
v (PK + p)
(IV)
dwy,
..
2 (PR + P) =
2
c
where
..
p,
2
c
(ív) d«p P =
The above relation is a differential equation of the fourth order for P. Pláne wave solution can be obtained with p = P , cos (Kr — co t) (
co |
co
a>l)
c
2
2
c
2
2
t
2
co
K =
l [
2
í
CO | 2
2
alj
« 1
47CXÜ)
1
2
+ 4tcx
r
co
2
and thus we find for the velocity of propagation of phase planes
We see that the effect of dispersion at least in a qualitative form is obtained from Maxwell's equation if we consider also the inertia of the atomic electrons. As the result of a further simple calculation one finds that in the interval / -* t between the arrival of the primary wave and the compound wave the interference does not completely extinguish the field. Thus we expect a phenomenon which was sometimes called "Vorláufer", i.e. a wave which appears before the main front arrives. Although the Vorláufer certainly exists, we have good reasons to believe that the latter phenomenon cannot be observed experimentally — we cannot go into details here. x
7. THE EXPERIMENT OF FIZEAU
301. According to Maxwell's theory we can obtain the phase velocity V(co) = c/n(co) of an electromagnetic wave passing through a médium. Consider thus a slab S through which a wave of frequency co is propagated with a velocity V(co). According to the Lorentz principle we may conclude that using a slab S* moving with a velocity v relative to S we expect that a wave with frequency co* will be propagated with a velocity V*(co*) = V(
v+ V(co) vV(co) '
neglecting terms of high order in v/c we find V*(co*) = V(co) + and co
1 v\ln\co))
The velocity of propagation of a wave of frequency co in a médium moving with a velocity v is therefore 1 n (5)
V*(co) = V(co) + v 1
2
(32)
co = co I 1 -
Relation (32) follows from the assumption that both the electromagnetic fields and their interactions with the atoms of 5 obeyed the Lorentz principle. As there exists somé misunderstanding about the significance of (32) (see chapt. VI 208) we show that (32) can be obtained from Maxwell's theory also. Furthermore we point out that (32) describes an effect of the order of v/c and therefore this effect is not an essentially relativistic effect. 302. The atoms of S* are moving with a velocity v, therefore the atoms behave as if the field acting upon them had components E B
= E + - ( v x B) c
e r f
e f f
(33)
= B - - ( v x E) c
Taking the constants of polarization x and x' we have thus for the induced polarization P = xE M = Z'B . (34) 0
eff
0
eff
We denote with the suffixes " 0 " that P and M refer to the polarization of the moving atoms. From the result of 283 we find that the atoms moving with a velocity v and showing polarization according to (34) are equivalent to matter at rest with polarization 0
0
P = P + -(v x M ) c 0
0
(35)
M= M --(vxP ) 0
0
With the help of (33), (34) and (35) we find thus p = xE +
X
+
M = x'B — *
X
c
(v x B) - \
c
v x (v x E)
* (v x E) — ~ v x (v x B).
+
c
We may express E and B in terms of the vector potential A and thus we can express i
eff
= rotM + —P, c
also in terms of A and its derivatives. Using this expression the wave equation V A - -^Á = -4m 2
e f f
(36)
appears as an equation containing A only. Supposing A to represent a pláne wave, i.e. A = A cos (Kr - cot) (37) 0
we obtain from (36) and (37) an amplitude relation. Supposing A to be parallel to v we find in particular V - 2 \ \ - \ \ v V - V = 0, 2
2
(38)
where V = co/K is the velocity of propagation of the wave in the moving médium and V = c/n 0
n = E/i; 2
neglecting terms of the order of v /c we obtain from (38) 2
v =
c
-
n
2
+
l-l)r
(39)
in agreement with the result obtained further above. We see thus that the velocity of propagation of planes of constant phase of an electromagnetic wave in a moving dielectric can be calculated from Maxwell's equations. Doing so one has to take into consideration that the incident wave acts upon moving atoms — and that the secondary radiation which interferes with the primary wave is emitted by moving atoms. The above considerations could be easily extended by taking into consideration the effects of inertia of the electrons as it was due in 299. As the result of such a calculation the effect of dispersion enters into the result — and we are led in place of (37) to the equation (32) in 257. We see thus that relation (39) giving the velocity of propagation of electromagnetic waves in a moving médium can be obtained from purely electromagnetic considerations without using explicitly the Lorentz principle.
D. EFFECTS OF THE SECOND ORDER 1. ACTION OF A CHARGE U P O N ITSELF
303. The force with which a moving electric charge acts upon itself can be worked out with the help of Maxwell's equation. Let us consider a charge e at rest at t = 0; the charge distribution may be given by g(r) so that |g(r) cPt = e and Q(T) = 0 r> a thus the charge is contained in a sphere with radius a. The electric field E(r) can be worked out and the force upon itself can be taken as F = Je(r)E(r)rf r.
(40)
3
If the charge is at rest the latter vanishes as can be seen from symmetry. 304. We calculate the field of the partiele if it is accelerated so that its velocity is zero at t = 0. We can thus write for the velocity and displacement of the partiele • 2
1
\ — vt
vr • 2 If we take the partiele to move like a rigid body, then the charge and current distribution at the time t can be taken as i(r,
0
=
víg
Q(T,Í) = Q
a =
|r
- y T í
s
(r-yví
The vector potential can thus be written according to eq. (17) in 257
A(r,
.•A|,(, + »-L,|,-i)1
0= -
R
c where we have put t' = t
R c
Differentiating into t and neglecting terms of the order of l/c or smaller, we find 3
-ÍÁ(r,0)=-^
R) similarly we have e
|
r
R_^
+
4>(r,0) = |
l V j R
2
/ c
2
L J-d V 3
^
and neglecting again higher order terms _1_.
r
Ő |r + R - vT^ 7 c | 2
d 3 R
e
— grad 4>(r, 0) = —
dr
(41)
R
Developing the integrand in powers of R we can take 1 grad Q\r + R - j-vR lc \ 2
2
= grad Q(T + R) -
\ 2
1 R? . d q(t + R) + smaller terms. 2 c 2
(42)
2
őtf
Introducing (42) into (41) and integrating by parts twice we find neglect ing the small terms grad#(r,0) = grad * (r) +
+V
0
^
The field of the accelerated partiele can thus be written
E(r) = E ( r ) - - ^ J 0
l
-
^
2
2R
]
^
L
3 d
R
R
where E (r) = grad <í>(r) 0
0
is the field of the partiele at rest. 305. Integrating over the whole of the accelerated partiele considering (40) we find s
3
F<> = Je(r)E(:r)d r =
U
with
Uf/(l^f)Üͱ^rtA.
(43,
U is a mátrix the components of which have the dimensions of energy. 306. As can be seen easily the orders of magnitudes of the elements of U are comparable with
where U is the electrostatic energy of the charge. The equation of motion of the charge can be written 0
wv = F
+
(ou,)
(44)
where F is the outsidé force and F* the self force with which the charge acts upon itself. Introducing (43) into (44) we find also (out)
J)
ml
+ —5-
v = F<° . u,)
(45)
Thus we see that the outside force accelerates the charged partiele less than the corresponding neutral one. The partiele behaves thus as if its mass had increased as the result of the charge. Since U is a tensor, the excess mass of the charged partiele appears to depend on direction. If we take as an order of magnitude relation U~
1U
0
we find that the effective mass of the charged partiele is about m
efí
x m-\
f- • c
Writing m
e{{
— m + ám, taking a uniformly charged sphere one finds
and thus
- 1 ) so the apparent increase in the mass caused by the electromagnetic reaction is 2/3 of that corresponding to the relativistic increase in mass expected from the Lorentz principle.
2. MASS DEFECT
307. Considering a pair of opposite point charges +e and — e at a distance R from each other, one finds
Am = — — 5 - (1 - cos #). Rc* 2
We see thus that the attraction between the charges causes a decrease in the apparent mass. Indeed accelerating a pair of particles with opposite charges, we find that the field of each of the particles acts upon the other exerting a force in the direction of the outside force. Thus the inner forces increase the acceleration and the system as a whole can be accelerated more easily than the system of the unconnected particles. The latter effect is connected with the mass defect observed in a system of particles bound together. 308. From the considerations of 237 one is led to expect that a closed system, e.g. a partiele, has a mass E where E is its totál energy. If E is changed in any way the mass is expected also to change — e.g. if we charge a neutral system and supply an electrostatic energy to the system the mass is expected to change by Am = 0
.
(46)
The calculation of the force which an electrically charged system acts upon itself gives an apparent mass change which has the same order of magnitude as expected from (46) but differs from this value to somé extent. It is believed that real particles behave in accord with the relativistic formuláé. The reason of the diserepancy is the following. Charging a system electrically we produce a stress which has to balance the Coulomb forces with which the charge acts upon itself. The stress is caused by the interaction of the atoms; the latter interaction may be expected also to be of retarded nature and thus the interatomic forces give rise also to a self force. If the system as a whole obeys the Lorentz principle, then it is to be expected that the reacting forces caused by stress and the electromagnetic self force produce together a mass increase which corresponds to the relativistic mass increase.
E. RELATIVISTIC MECHANICS OF A CONTINUUM 309. In a closed physical system we expect momentum and also energy to be conserved and therefore we have DivT
= 0;
( m )
(47)
the above relation gives four equations for the ten components of T , it gives therefore only a necessary condition for the motion of the médium to which T refers. We have to add to (47) the equations of motion of the médium, so as to be able to determine its motion from initial conditions. 310. In a configuration where (m)
(m)
DivT "° = F # 0 (
(48)
we have a system in which energy and momentum are not conserved Such a case may arise, if the system under consideration is under the influence of an outer force. The most important cases are, where a closed physical system is under the influence of an electromagnetic field. We can write for the energy momentum tensor of the latter and, as was shown in 252, Div T gives the rate of energy and momentum transferred by the electromagnetic field upon matter. In a mechanical system which is under the influence of an electromagnetic field we can suppose that, on account of the interaction, the totál energy and momentum are conserved. Thus we may suppose (el)
Div(T
+ T< >) = 0;
(m)
e,
(49)
the latter equation describes the circumstance that the density of force with which the electromagnetic field acts upon the matériái system provides exactly the force density — F which according to (48) is necessary to balance the force density + F which is produced by the mechanical stresses inside the system. 1. INTERPRETATION OF THE T R O U T O N - N O B L E EXPERIMENT
311. The field of a charged condenser can be worked out and in terms of the field strengths the ponderomotoric force equals F
(el)
= Div T , (el)
the moment of force acting upon the condenser can be expressed with the help of the density M
(el)
=
x
x
F
(el).
the totál moment of force can be expressed by the antisymmetric tensor of third order m = $M fdx *, 7 = 1 , 2 , 3 . kl
k
Considering a charged condenser at rest, we find /?#/> = 0
*, 7 = 1 , 2 , 3 .
Transforming the four-tensor M we obtain the tensor M * describing the state of the deformed system. In generál the space components of the latter will not vanish, even if those of the former did vanish, i.e. we have ( e l )
(el)
m{f* # 0. Thus the electromagnetic forces acting between the plates of the condenser produce a non-vanishing moment of force if the condenser is in a state of translational motion. The system consisting of two charged condenser plates is, however, not a closed system. To maintain the system, there is need for a mechanical support keeping the plates apart from each other and internál forces which counterbalance the Coulomb repulsion of the charge upon the plates. The totál system including electromagnetic forces and mechanical stresses can be described by an energy momentum tensor T = T + T — and since the latter system is in equilibrium we have (el)
Div T = 0 thus
( m )
M = 0.
From the above relation it follows that mu^rrtiP
+ m'tf = 0,
and also mf, = mif* + mif* = 0. Thus in a closed physical system, where the electric forces are balanced by mechanical forces, the moment of force vanishes in any representation and therefore it vanishes for the condenser at rest just as for the condenser moving with a constant velocity.
F. TRANSIENT PHENOMENA 312. We have seen that Maxwell's equations are consistent with the Lorentz principle in the following sense. If a field % can be taken to result from a current distribution 3 , then the Lorentz deformed field
can be taken to result from the Lorentz deformed current distribution 3* =
m i
The dynamical part of the Lorentz principle asserts that as the result of an adiabatic interference a physical system Q changes into a deformed system with configuration O* =
1(0).
Applying the latter result to an electromagnetic field, one expects adiabatic deformations of the type & 3 -
Q*.
(50)
313. So as to analyse what is meant by a deformation of the type (50) we note that, having a field only, we cannot apply any interference to the field itself, but can only interfere with the sources of the field; i.e. having a source distribution Q we can interfere with the matériái, the carrier of charges and currents, and make it change into 3 * . A simple example is e.g. a charged partiele P which is at rest relative to a system of reference K; we may apply outside forces upon P and make it move with a velocity v relative to K. The field of the point charge when it is moving with the final velocity v is given by (64) in 278. Using the retarded potential solutions of Maxwell's equations we can directly calculate the field of the particle. Suppose the partiele is at rest for t < 0, the acceleration starts at t = 0 and at a time t = t the partiele reaches its final velocity v. We may thus suppose that the velocity of the partiele is given by x
, Í0 v(í) = [v
í<0, t>h.
The coordinate vector of the partiele is thus
i
r(r)= j" v(t')dt'
for 0 < t < ^
0
and r(í) = r + yt
for t > r
t
x
ri = rOi) - v í . Calculating the field of the moving partiele at somé time t > t we find three zones in the field (see Fig. 25). x
t
1) There exists a region I surrounding the charge at a time t > t
x
| r(íi) - r | < c(t - h) for r inside region I. Inside region I we find, calculating the retarded potentials of the moving charge, that the retarded times are all t' > t , therefore the action of the charge is only felt from positions where the partiele moved already with its final velocity v. The field in I is thus the deformed field as given in eq. (64) 278. t
Fig. 25. Scheme of regions I, II and III
2) A region II surrounds region I at a time t > t ; we have x
| r | < ct, c(t - íi) < | rí/j) - r | for r inside region II. Inside region II the field is that which was produced by the partiele in the period 0 < t' < t thus II contains the radiation field travelling outward which was emitted by the partiele while accelerated. 3) Outside region II we find region III which for t > í is defined by x
x
ct <
\ T\
for r inside region III. The field in region III is that produced by the partiele at times t < 0, i.e. before the onset of the acceleration. The field inside III is therefore that of the charge at rest. The inner boundary of III expands with a radial velocity c thus it recedes with the velocity of light. 314. The receding of regions II and III can also be interpreted in another way. The electromagnetic field surrounding a partiele which con-
tains currents and charges can be considered as a kind of matériái continuation of the particle. If we now start to interfere with the sources of the field, then the latter will carry with themselves at first only the field in their immediate vicinity, i.e. inside region I. The distant parts of the field, i.e. those inside III, are at first not affected by the interference. In the region II placed in between, we find electromagnetic waves moving from the boundary of I towards that of III and these waves deform the outside field from % into %*. If we compare the field surrounding the partiele with an elastic médium, then we can compare the process of acceleration with the propagation of an elastic disturbance from the centre of the disturbance to distant parts. Indeed, if we were to accelerate an elastic body by means of outside forces acting only upon a region in the centre, then we would find that the outside forces set into motion at first only the centre part of the body and its outside remained for a time unaffected. The elastic disturbance gradually extends to the whole of the body and sets into motion the outside parts. The electromagnetic field of a partiele behaves exactly in this manner, the velocity of propagation of the disturbance being equal to c. The above considerations describe thus the mode of relaxation into the Lorentz deformed configuration, shown by a physical system which contains charges, currents and fields.
CHAPTER X
THEORY OF GRAVITATION
315. In this part of the book we shall deal with the problem of the generál theory of relativity from the points of view we have dealt with the problems of the special theory of relativity. It seems to us that the generál theory of relativity can be taken to deal with the effects of gravitation upon physical processes. Our view that the generál theory of relativity gives in fact a theory of gravitation resembles to somé extent the views expressed by V. A. Fock on this subject.* A. OBSERVATIONAL FACTS 316. The gravitational effects which are successfully interpreted in terms of the generál theory of relativity can be summarized briefly as follows. 1) The deflexion of a beam of light when passing a gravitating mass. This effect can be described phenomenologically by supposing that the beam of light behaves as if it consisted of a stream of particles subject to the gravitational attraction. From considerations somewhat similar to this the possibility of the deflexion of light in a gravitational field was foreseen in about 1800. — These considerations seem to have been forgottén later. The deflexion of light in the vicinity of a gravitating body could also be accounted for by supposing that the gravitational field surrounding a gravitating body acts like a diffracting médium. Thus the gravitational field e.g. of the Sun can be taken to act as a lense surrounding the Sun. The image of the sky viewed through this lense appears magnified and therefore the stars which appear on the sky in the direction near to that of the Sun, seem to shift from the Sun whenever the Sun approaches them. 2) The gravitational red-shift of spectral lines. Atoms change their frequencies when they approach a gravitating centre. The effect was observed when measuring the frequency of atoms in the solar and in stellar atmo* B . A.4>OK, TEOPAANPOCTPAHCTBA,BPEINEHH H TSROTEHHH,H3MATRH3MOCKBA, 1961.
spheres. This effect could also be shown under laboratory conditions with the help of the Mössbauer effect.* It was found that the frequency of certain y-emitters increases when they are shifted upwards. The effect is caused by the change of the gravitational potential of the Earth with height. 3) The anomalous perihelion motion of the planet Mercury. The curved motions of masses in a gravitational field are themselves gravitational effects. However, the laws of Newton which give the form of this motion in a first approximation, are taken usually as classical laws and only the deviations from Newton's laws are taken to be relativistic effects. The orbits of the planets around the Sun can be described with great precision in terms of Newton's laws. The orbits of the planets are in a first approximation Kepller ellipses, but because of the mutual perturbations these ellipses show slow precessions. The orbit of Mercury deviates slightly from that predicted by Newton's theory. The motion of the perihelion of its orbit differs by about 0.4" per year from that obtained from the amount calculated from the perturbations. This latter anomaly could be accounted for in terms of the generál theory of relativity. 4) An important cosmological result of the generál theory of relativity was the prediction of the recession of the distant extragalactical systems by Friedman** which was observed afterwards by Hubble.*** 5) Apart from the observed effects described above, the generál theory of relativity makes use of the fact of the equivalence of gravitational and inertial mass. This equivalence has been proved experimentally to a very high accuracy.**** B. STATEMENT OF THE PROBLEM OF THE THEORY OF GRAVITATION 317. The curved motion of a body in a gravitational field is a very apparent gravitational effect. The deflexion of light in a gravitational field is another gravitational effect which effect is, however, small and can be observed under favourable experimentál conditions only. The task of a theory of gravitation is firstly to find the laws of motion of bodies under gravitational action; the latter laws must give in a good approximation those formulated by Newton. Secondly, one has to find * R. L. Mössbauer: ZS. f. Phys., 151, 124, 1958; R. V. Pound and G. A. Rebka, Jr.: Phys. Rev. Letters, 4, 337, 1960. ** A. Friedman: ZS. f. Phys., 10, 377, 1922. *** E. Hubble: Astrophys. J., 74, 43, 1931. **** R. v. Eötvös: Ann. d. Phys., 59, 354, 1896; R. v. Eötvös, D. Pékár and E. Fekete: Ann. d. Phys., 68, 11, 1922; R.H. Dicke; Experimentál Relativity (Relativity Groups and Topology), Ed. De Witt and De Witt. p. 173, Blackie and Son Ltd., 1964.
the forms of other natural laws valid in regions where gravitational action is not negligible. These laws can be found by extrapolating the formulation of the laws valid for regions where gravitational actions can be neglected. For example, the laws of the propagation of electromagnetic waves can be described by Maxwell's equations in regions where gravitational effects can be neglected. In regions where gravitational effects are noticeable, the propagation of electromagnetic waves can be supposed to be expressed by equations similar in form to those of Maxwell's but such that the parameters characterizing the distribution of the gravitational field occur explicitly in the modified equations. Without giving the exact form of Maxwell's equations which are found to be valid in a gravitational field, we note that one might consider e.g. as a possible form of the wave equations relations
* -^b) = ' 2A
A
0
(1)
where c(r, t) is the velocity of light in a point with coordinate vector r at the time /. In a region where gravitational effects can be neglected we have of course c(r, t) = c = constant, and relation (1) becomes identical with the wave equation derived from Maxwell's equations in their originál form. Similarly, by introducing into the Schrödinger equations the parameters characterizing the gravitational field one can get the laws giving the effects of gravitational fields upon atoms. In particular the frequencies of oscillations of atoms are found to depend on the gravitational potential; therefore the Schrödinger equation adapted to regions containing gravitational fields, must lead to frequencies depending on the parameters of the gravitational field. 318. From a purely mathematical point of view Maxwell's equations, the Schrödinger equation and other physical laws could be generalized for regions containing gravitational fields in a large number of ways. In the following sections we shall give those generalizations which follow from the generál theory of relativity. Sometimes suggestions are made to the effect as if the generalizations of the laws of nature which lead to the forms of the laws in gravitational fields could be obtained from a priori considerations. According to such views the laws thus obtained are logically more or less the only possible ones — and in a paradox formulation one is led to the assumption — as if there existed no other possibilities, and "nature has to obey the laws deduced a priori".
Such considerations are at fault; we shall show in the following that the relativistic laws are based on well-defined physical hypotheses concerning the structure of matter and gravitation. It is a question of facts as to what extent these hypotheses give a correct description of real nature. 1. MATHEMATICAL FORMULATION OF THE PROBLEM
319. In the special theory of relativity only such regions are considered in which light is propagated homogeneously. The laws governing the motion of physical systems inside such regions obey symmetries which can be expressed by the Lorentz principle. In reality light can nowhere be assumed to be propagated strictly homogeneously, as we have reason to believe that the propagation of light is aftected by gravitation and regions entirely free of gravitation do not exist. The Lorentz principle can therefore be taken to be valid only to such an approximation as gravitational effects can be neglected. The question arises how the Lorentz principle should be generalized so as to apply to regions containing not negligible gravitational fields. 2. EXPERIMENTÁL CRITERIA FOR HOMOGENEOUS REGIONS
320. So as to be able to generalize the Lorentz principle — as a first step — we have to investigate how it is possible to decidé experimentally whether a region is or is not homogeneous. The above question is by no means trivial and it can be approached in a way we discuss presently. Consider in a region 9Í a number of points ty , ty with clocks (So, & situated near the points. Suppose 0
N
N
(a) that 9í is homogeneous, (b) that the points ^S do not move relative to each other, (c) that the points ty are either at rest or have at most translational motions relative to the ether, (d) that the standard clock 6 has a uniform rhythm. fc
k
0
If (a)-(d) stand, then we can find representations K(x ) v
=
r„
of the coordinate vectors of the points and establish time measures *(t,) = /, of events (S, occurring in points such that the propagation of light as represented relative to K is defined by a propagation tensor: ^(S) = g = independent of x = r, t.
(2)
We note that provided a straight representation K satisfying (2) can be found, then there exist also other representations ^'(S) = g' = independent of x' = r', t' with arbitrary values of the propagation mátrix g'. 321. In a straight representation the propagation of a signal of light obeys the relation
(x - Xi)g(x - xO = 0 2
(3)
2
where x and x are the four-coordinates of events (5 , i.e. of the departure of the signal of somé point at a time t and its arrival in another point % ti hSupposing that signals are propagated along straight lines we can describe the orbit of a signal of light in parametric representation by x
2
2
x
a t
a
me
2
x(p) = kp + a. where the vector x has constant components and obeys the relation xgx = 0 and a is an arbitrary vector with constant components. 322. If the attempt is successful to determine a straight representation K of the region 9í containing the points ty and clocks E then we may suppose that the conditions (a)-(d) of 320 are fulfilled indeed. A system of reference K can be constructed using methods described in 145, chapt. IV. As it was explained there, coordinate measures r and time measures t can be obtained making use of the exchange of light signals and interpreting the observational data with the help of relations of the form (3). However, the coordinate measures r„ and í„ are obtained as the solutions of a strongly overdetermined system of equations; we can therefore conclude that the region in which we observe the signals is indeed homogeneous provided the overdetermined system admits of solutions. We thus can obtain an internál check of whether the propagation of light in 9t is homogeneous indeed. However, considering things more strictly — when using the method described above — we check the fact whether or not all the conditions (a)-(d) given in 320 are fulfilled. Physically it is mainly of interest whether or not (a) is fulfilled — thus it is of interest whether or not the propagation in 9t is homogeneous indeed. The conditions (b)-(c) reflect only upon the question how the points $ move and how the clocks © are adjusted. The latter questions are of practical importance but are not significant regarding the question of the real mode of propagation of light in 3t k
fc
v
v
v
v
a. AN EXAMPLE
323. So as to give a practical example we note that light in the vicinity of the Earth is propagated homogeneously in a very good approximation. Nevertheless if we take the points ty as points fixed relative to the solid Earth we cannot synchronize the clocks © in a consistent manner supposing the propagation to be given by an expression of the form of (3). Indeed, because of the rotation of the Earth around its axis, the condition (c) is violated and therefore using the exchange of light signals between the points ty using the methods of 145 we cannot obtain a consistent system of reference relative to which (3) holds. The experiment of Michelson and Gale (see 62) showed that the rotation of the Earth around its axis can be determined with the help of interferometric measurements. It makes use of the fact that relative to the rotating Earth light does not appear to be propagated homogeneously. The experiment of Michelson and Gale does not prove, however, that light in the vicinity of the Earth is propagated inhomogeneously. Indeed introducing a system of reference K which does not share the rotation of the Earth, we find that relative to the latter the propagation of light appears to be homogeneous. However, the coordinate vector r of the points relative to K are changing in time. Thus taking into account the non-translational motion of the points ?$ we can introduce a straight representation in which the propagation of light is described by relation (3). Considering on the other hand the deflection of light in the vicinity of the Sun, we have to suppose that the propagation of light in the vicinity of the Sun is inhomogeneous indeed — and there exists no system of coordinate relative to which g(x) = constant in the vicinity of the Sun. k
fc
k
k
k
3. CONSTRUCTION OF STRAIGHT SYSTEMS OF REFERENCES
324. If we want to investigate whether a region is homogeneous or not, i.e. if we want to investigate the question whether condition (a) of 320 holds for a region ÍR and if we want to get rid of the conditions (b)-(d) then we have to use a new procedure. Let us suppose that the points ?$ are situated in a homogeneous region ÍR but we drop the conditions (b)-(d) thus we suppose that the points ?$ may move relative to each other and also relative to the ether in a more or less arbitrary fashion. The clocks S moving with the points should be adjusted arbitrarily and we ascribe arbitrary coordinate vectors to the points S$ . We want to restrict ourselves only to such an extent that we choose "reasonable" coordinate and time measures in the sense that we ascribe to points close to each other coordinate vectors which do not differ to a great extent and we regulate the clocks so that clocks in the vicinity k
k
fc
k
of each other should give readings not differing very much from each other. The latter conditions could be formulated more precisely, but such a formulation does not appear to be important. 325. Using the more or less arbitrary system of reference thus obtained we can establish empirically the orbit of a light signal. Interpolating suitably between the coordinate vectors of the points and readings on the clocks <E we find that the orbit of a particular light signal can be given in a parametric representation in the form k
\(p) = T(p), t(p), where we suppose that the various values of the paraméter p give the time measures t(p) at which the signal passes points with coordinate vectors r(p). We may suppose the representation to be such that t(p) # 0 for any value of p. Considering the orbits of a large number of signals of light we may be able to establish that the orbits of light signals in terms of our arbitrary coordinate measures obey a relation xO>)g(x(/>))xO>) = 0.
(4)
More precisely we may find that in the vicinity of any given four-points x light is propagated homogeneously in a first ápproximation. The propagation tensor g in the representation K may vary with x. a. LOCALLY HOMOGENEOUS REGIONS
326. It must be emphasized that relation (4) expresses already a particular feature of the mode of propagation of light. This feature — at least in principle — can be tested experimentally. Indeed, we can determine empirically the vectors x (p) k = 1, 2 , . . . , « for a number of beams of light passing through a fixed four-point x. Introducing the values thus obtained into (4) we obtain n relations for the elements of g(x). For sufficiently large values of n the system thus obtained is mathematically overdetermined; if the overdetermined set of equations admits of solutions, we can suppose that the propagation of light obeys indeed a relation (4).* We note that one could imagine regions where light is propagated in quite a different manner. We might imagine a region where a signal is propagated so as to occupy surfaces strongly deviating from elliptical ones as illustrated in Fig. 26. Such a supposed mode of propagation of light might be called locally inhomogeneous. k
* Since (9) is homogeneous g(x) can be determined only up to a factor a(x). — See for details L. Jánossy, Foundations of Phys. 1, N o . 3, 1971.
The relation (4) is analogous to the law of propagation of light in an inhomogeneous refracting médium. Thus relation (4) supposes that the propagation of light in the ether obeys a law not unlike the law of propagation of light in an inhomogeneous refracting médium. The assumption (4) can be taken as a hypothesis which is justified by the success of the theory based on it.
Fig. 26. Inhomogeneous mode of propagation of waves
We shall denote a region locally homogeneous if we find that the propagation of light in an arbitrary representation K obeys the relation xg(x)x = 0.
(5)
We note that the relation (5) gives only one necessary condition for the orbits of signals; therefore the orbits themselves in a locally homogeneous region cannot be obtained from (5) alone. b. CRITERIA FOR HOMOGENEOUS REGIONS
327. The question arises whether an extended region 9í inside which the law (5) holds is a homogeneous one or not? If the region is homogeneous, then there exist representations K' diflering from the originál representation K such that relations (5) expressed in terms of K' can be written x'g'x' = 0 with
g' = independent of x'.
The transformation between the measures of K and K' can be written as x' = f(x)
or
x =
f-V)
(7)
where f is a reversible four-function and f its inverse. From (7) we see that a point which is at rest relative to K, i.e. a point with a coordinate vector r = independent of t is moving relative toK'; similarly, if f is a non_1
linear function we see that in the representation K' the rates of clocks Qí differ from the rates which appear in K. 328. Transforming the coordinate measures x according to (7) we obtain the law of propagation of light in the representation K' in the form (6) provided k
S(x)g'S(x) = g(x)
(8a)
S(x) = - ^ - = f(x)oQ
(8b)
with
For given g' (8a, b) represents tett differential equations for the four components of f(x). This system is thus in generál overdetermined. We may therefore conclude that, provided the overdetermined system (8) admits of solutions, then this fact is not an accidental one, but signifies that the region 9í is homogeneous indeed. The representation K' which is obtained from K by the transformation (7) can be denoted a straight representation. 329. In the following we give the explicit condition which g(x) has to obey in a homogeneous region and give an expression for the transformation function f(x) which leads from a curved representation KXo the straight representation K' of a homogeneous region. From the relation (8a) it follows that we can put S(x) =a'A< >cr (x) p
1
(9)
where a' and a can be determined from the elements of g' and of g(x) according to Appendix I; 443 eq. (13). A is a Lorentz mátrix; we note that (p)
P = P(x) the six parameters of the Lorentz mátrix may vary with x. The paraméter p(x) have to be chosen so that (9) should satisfy (8b). However, introducing (9) into (8b) we get a very complicated system of differential equations for p(x) and therefore the solution (9) is of no practical use. 330. In Appendix II we have given in detail the solutions of (8a, b) and have also discussed the conditions g(x) have to fulfil for (8a, b) to admit of solutions. Because of the importance of these considerations, we give here a short account of the problem the details of which are found in Appendix II. Differentiating (8a) into x we find, remembering g' o = 0 (see eq. (29) in 471), (3)
(3)
( l + c - ) ( g S - ( x ) S ) = g(x). 1
3
1
(10)
Applying the operator n = 1 — c 3
1 3
+ c
(3)
to both sides of (10) we find
2 3
(3)
S(x) = S(x)-C(x), with
(11)
1 (3) C(x) = "2 rcsg(x).
(3)
Thus (11) gives a system of linear differential equations for the determination of S(x). Whether (11) admits of solutions can, however, not be seen directly. 331. So as to find the conditions for g(x) which have to be satisfied if (11) is to admit of solutions we differentiate (11) into x and find (4)
Ö)
(4)
(3)
S(x) = S(x) • (C(x) - (24)C(x) • C(x)).
(12)
Relation (12) gives a differential equation for S(x); if the latter admits of solutions, then such solutions can be obtained e.g. by giving an initial condition and integrating step by step. (4)
Since S(x) is to be a third derivative, it must be symmetric in the last three suffixes. The sufficient and necessary conditions for the solution of (12) to possess the required symmetries can be written (see for details 477) (4)
(1 - ( 2 4 ) ) S ( x ) = 0. (4)
Expressing S(x) by (12) we find (4)
S(x)-R(x) = 0 where (4) 1 (4) (3) (3) R(x) = - ; r ( g ( x ) + C(x)-C(x)) 4
is the Riemann-Christoffel tensor. Since det S(x) # 0 w e can also write (4)
R(x) = 0.
(13)
Relation (13) is a necessary condition which g(x) and its derivatives have to fulfil in every point of a homogeneous region. We see therefore that g(x) represents only then a homogeneous region if the Riemann-Christoffel tensor förmed of g(x) and its first and second derivatives vanish identically in the region.
332. The condition (13) is, however, not only necessary but also sufficient for a region to be homogeneous; we can show this by constructing explicitly a transformation function f(x) the derivative S(x) of which obeys (8a) and (8b). To show that (13) is also a sufficient condition for a region to be a homogeneous one, we show that if (13) is fulfilled we can construct transformation functions f(x) which lead to a straight representation K'. To show this we can differentiate (11) successively into x and find expressions of the form (3 +
0
(3)
S(x) = (S(x)-C(x))oQ'.
(14)
Writing down (14) explicitly for / = 0, 1, 2 we obtain a recursion for-
(3+0
mula with the help of which we can determine S (x) in terms of the (3 + /')
(k)
S (x) /' < / and the derivative g (x) k = 2, 3 , . . . 3 + / of g(x).
(3+0 Taking S (x) thus obtained to be the derivatives of f(x) we can develop f(x) e.g. around x = 0; writing 1
(3)
f(x) = (jl + Sx + - S x + . . . 2
with
(3)
S = S(0),
(15)
(3)
S = S(0),
and u. is an arbitrary four-component constant. (4)
If (13) is fulfilled we find that the S(x) are symmetric in the 2, 3, 4-th suffixes; since the suffixes from the 4-th are obtained by successive difW
ferentiation, the S(x) are in any case symmetric in the 4, 5 , . . . , k suffix. (*)
(fc)
Taken together, the S(x) and thus also the S are symmetric in all the suffixes except the first. Inserting (15) thus obtained into (8a, b) and developing both sides into powers of x we find that (15) satisfied the conditions (8a, b) in the vicinity of x = 0. (The considerations can of course be carried out in the vicinity of any four-point x = x„ instead of x = 0.) We see thus that the function defined by (15) gives indeed a transformation to a straight system of reference K'. We note that the transformation (15) contains ten arbitrary parameters, i.e. the four components of u, and the six parameters which can be chosen freely when determining S.
333. Summarizing our considerations we find that observing the mode of propagation of light in a region ÍR, in this region we may succeed in expressing the law of propagation of signals in a form xg(x)x = 0 using arbitrary coordinate measures. If the region around x = 0 is a homogeneous one, then we can transform our coordinate measures according to 1
(3)
x' = f(x) = u. + Sx + y S x + • • • 2
(16)
W
where the S are to be determined with the help of a recursion (14). In the representation K' thus obtained we find Á"(g) = constant. The coordinate measures x' give a straight representation K' of the region, while the originál representation K can be taken to be a curved representation of the region ÍR. 334. If we do not want to carry out the transformation (16) we can check also directly whether or not ÍR is homogeneous. In terms of g(x) = X(g) we may form the Riemann-Christoffel tensor R(x) in ÍR and if the region (4)
is homogeneous indeed, then we find R(x) = 0. We see thus that using signals of light only we are in a position to examine whether or not light is propagated homogeneously in the region we are investigating and if the propagation of light proves to be homogeneous, we are in a position to construct a straight system of reference with the help of signals of light. Returning to the problem raised in 320 we can decidé whether or not (2) holds in a region without regard whether or not the clocks we are using to measure the arrivals and departures of signals obey the conditions (b), (c) and (d). It seems to us important that we need not start our considerations by defining a straight system of reference. Indeed, starting with an arbitrary system of reference we are in a position to find out whether light is or is not propagated homogeneously and provided (inside the accuracy of measurement) the propagation proves to be homogeneous, we can afterwards — with the help of signals of light — construct a straight system of reference. 4. ALMOST STRAIGHT SYSTEM OF REFERENCE
335. The question arises whether it is possible to define in an inhomogeneous region a system of reference which resembles a straight system of reference as much as possible.
In a homogeneous region one can use straight coordinates but one can also use e.g. polar coordinates — which must be taken as a type of curve coordinates. It is reasonable to suppose that in an inhomogeneous region one can introduce coordinate measures which are the analogy of straight coordinates, but one can introduce also such coordinate measures that rather resemble e.g. polar coordinates than straight coordinates. Therefore in an inhomogeneous region one expects that one can distinguish between almost straight coordinates and strongly curved coordinates. So as to obtain a transformation which makes the representations g'(x') of g almost independent of x' we can start from eq. (12). The function (4)
S (x) will not be symmetric in (24) or (34) if the region we consider is inhomogeneous. We can introduce, however, a symmetric quantity (4)
1
(4)
<S(x)>= (l + y
(24) + (34))S
As shown in Appendix II (479 eq. 50) the unsymmetric solutions of (12) (4)
(4)
differ from the symmetrized functions <S(x)>; the difference <5S(x) can be expressed in terms of the Riemann-Christoffel tensor as (4)
1
(4)
áS(x)=- y (l + (23))(S-R) The transformation function
(4)
«5g'(x') = - ( l + ( 1 2 ) ) R ' x ' + 2
+ terms of higher order. We see thus that in the almost straight system of reference the g'fx') is almost constant in the sense that the coefficients of the development are (4)
of the order of R'(x) and its derivatives. 336. The representation K' can be taken to be almost straight in the sense explained above — but it can be taken also as a representation which is as straight as possible in the following sense. Considering any system of reference K' such that in the origin (3)
g'(x') = 0,
(17)
the elements of the Riemann-Christoffel tensor are built as linear com(4)
binations of the elements of g'. The coefficients in the linear combination are of the order of unity. (4)
(4)
The elements of R' may thus be much smaller than those of g' since (4)
the elements of g' with comparatively large numerical values may compensate each other when inserted into the expressions giving the elements (4)
(4)
of R'. However, somé of the elements of g' must be at least of the order (4)
(4)
of those of R' as it would be impossible to construct the elements of R' as (4)
linear combinations of the elements of g' if the latter elements were all too small. We see thus that there exists no representation K' obeying (17) such (4)
(4)
that all the elements of g' are essentially smaller than the elements of R'. Thus we conclude that there exists no representation K' in terms of which (3)
(4)
g' = 0 and the elements of g' are essentially smaller than the elements (4)
(4)
of R'. Thus there exists no representation K'in which the g' are essentially smaller than in the almost straight representation. (3)
337. If we consider representations for which g' # 0 we may formulate our result by stating that the set (3)
(3)
(4)
g'-Hg'og'), g' necessarily possesses elements which are of the same order as the elements (4)
(4)
of R'. Thus there may exist representations where g' = 0 but in such (3)
representations the value of the g' must be comparatively large. 338. Generalizing the above considerations one finds that the almost straight system of reference obtained in 335 has the property that suitable homogeneous expressions built of the elements of the derivatives of the g have orders of magnitude not exceeding those of the elements of corre(*+2)
sponding invariants R . Reversing the latter argument one can define a system of reference to be almost straight if the elements of (*,) (*,) (*,) g""(g ° g ° g) have order not exceeding those of the elements of (k, + k + . . .k„) t
g
1
R
The above consideration does not give strict definition of the almost straight representation. Indeed, representations K' and K" obeying the above criteria may show small differences in the elements of somé of the g and these systems of references can all be considered to be nearly straight. 5. SIMILAR REGIONS
339. Let us consider a larger region 9t with two sub-regions fR and Sí* (see Fig. 27). The propagation tensor in 9t may be given in a particular representation K as 0
0
m
= go(x).
Fig. 27. Similar regions
We denote by x and x* the coordinates of the centres of 9Í and 9t*. Denoting further the four-coordinates relative to the centres of 91 and íft* by \ and we can also write g(5) = gofx + 5) g*(5*) = 8o(x* + \*) Under exceptional circumstances we find that the regions 9í and 9t* appear as transforms of each other. More precisely there exist cases where a function X(x) can be found so that M © g * ( g * ) M ( S ) = g(§) (18)
with M © = x(g)on
We note that the relation (18) gives in generál an overdetermined system of equations for X(x). If the distributions of g(x) taken in the vicinities of x and of x* obey certain relations, then (18) can be satisfied and the regions (3)
(4)
9í and ÍR* can be taken to be similar. Denoting by Mfé) and M(?) the
derivatives of M(5) we find in analogy to (11) (3)
(3)
(3)
M(§) = M(5)-(C(5)-C(5)),
(19)
(19a)
where we have written (3)
(2)
with 1
(3)
c*«*) =
T
(3)
***(§*),
(III)
(III)
and M(cj) is produced from the elements of M(x) like S(x) of those of S(x). Relation (19) admits then and only then of solutions if (4)
(4)
R(5) = R(?),
(20)
where we have written Ü)
(IV)
(4)
R © = (M(E,)R*(!-*)) 340. Relation (20) gives twenty differential equations for the four components of X(Ej). However, M(£) contains six arbitrary parameters, therefore choosing M(£) suitably we are left^with fourteen conditions between the g ( 9 and the g*(§*). Physically we see thus that similar regions are characterized by fourteen parameters (corresponding to the fourteen conditions imposed on the propagation tensor). Further, similar regions can be characterized by a four-dimensional orientation; the relative orientations of the regions 3t and dt* are expressed by the parameters of Mfé). (4)
C. THE GENERALIZED LORENTZ PRINCIPLE 341. According to the Lorentz principle — as it was formulated for homogeneous regions — laws of nature have such a form that provided £i is a real physical system, then a Lorentz deformed version C* = S 0Q) q
js a possible system. Furthermore, to the above principle is added a dynamical part, i.e. if £} is interfered with adiabatically, as the result of the interference, it changes into a Lorentz deformed configuration ö*. The transformations £ can be defined uniquely for homogeneous regions. A generalization of the Lorentz principle may be attempted by extending the definition of the operators 2 to inhomogeneous regions. q
q
1. THE LORENTZ PRINCIPLE FORMULATED IN TERMS OF CURVED COORDINATES
342. If system Q consists of a number of moving points $p\, 5P> • • • > then the system Q* can be supposed to consist of corresponding points ty*> • • • > T h orbits of the points of C can be represented by four-coordinates ?
e
Xfc(/>) = táp),
tk(j>) ik(P)
^0
'«.
= 1, 2,. . ., n.
The Lorentz transformation may be defined by a reversible transformation x* = X(x)
x =
7,-\x*),
thus by supposing the four-coordinates of the points to be given by xj?0) =
H*k(p))-
343. In a homogeneous region — and using a straight representation — we have X(x) = M x + (i (21) t
where M g M , = g. 9
The Lorentz principle in a homogeneous region can also be written down in terms of curved coordinates. If in terms of the curved coordinates we have K(Q) = g(x) = depending on x we have x* = X(x),
(22)
M(x)g*(x*)M(x) = g(x),
(23a)
and with M(x)
=X(x)on,
(23b)
g*(x*) = g(X(x)).
(23c)
and
Relations (23a, b, c) define the function X(x); the system (23a-c) is mathematioally overdetermined, but in a homogeneous region it admits of solutions. In a homogeneous region the Lorentz transformation defined by (22) and (23) is equivalent with that defined by (21) and (22). 2. GENERALIZÁLTON TO INHOMOGENEOUS REGIONS
344. Equations (22) and (23) cannot be taken to express the Lorentz transformation in inhomogeneous regions because the system is overdetermined and in inhomogeneous regions the system admits of solutions only in exceptional cases. Nevertheless it must be assumed that the Lorentz transformation is somehow connected with the systems (22) and (23). Indeed consider a signal of light § passing through somé physical system O. The orbit of the signal can be expressed by four-coordinates x (p) such that s
iÁPtó*(.P)y*ÁP) = 0.
(24)
Submitting the system Cl to a deformation, we obtain a new system ö * which is traversed by a signal 3*. The orbit of g* is expected to be given by x*(/>) = X(x (/>)). 4
Since §* is a signal of light we also expect the coordinate of the signal §* to obey the relation kf(p)g(x*(p))xf(p)
= 0.
(25)
If both (24) and (25) are to be valid, then X(x) has to obey the relation (23). Thus from the hypothesis that in the case of a light signal § passing through Cl the Lorentz deformation leads to a light signal §* passing through Cl* we are led to the condition (23) for the transformation function X(x). Since (23) is overdetermined we see thus that the above hypothesis cannot be strictly correct. a. A PHYSICAL EXAMPLE
345. So as to see in more detail the physical significance of the above hypothesis let us consider a system Cl which is something like a Michelsoninterferometer. Suppose Cl to be a solid carrying a source of light in a point A and mirrors in points B and C. The mirrors should be so adjusted that a signal starting from A should be reflected back into A by them. The distances should be so adjusted that the return signals into A should arrive simultaneously, i.e. the times of fiight A-B-A and A-C-A should be equal (see Fig. 28).
A deformed system £i* can be taken to contain a source in a point A* and to contain mirrors in points B* and C*; we expect that in the deformed system the times of flights A*-B*-A* and A*-C*-A* are also equal. The deformed system £1* can be taken to be the originál interferometer shifted to another position and submitted to a (four-dimensional) rotation. 346. Shifting the interferometer from one point to another we change its gravitational surrounding; the hypothesis of 341 amounts thus in
Fig. 28. Scheme of an interferometer
our particular case to the hypothesis that the interferometer which was adjusted in one position remains still adjusted if we move it to another gravitational surrounding. The above hypothesis is physically unreasonable. Indeed, the gravitational field produces stresses in a mechanical system and the system will deform so as to compensate these stresses. It is unfounded to suppose that the deformations which are produced by changing gravitational stresses upon the mechanical system should give rise to deformations suitable to compensate exactly the effects of gravitation upon the propagation of light signals. So as to see how absurd conclusions we were led to if we were to maintain after all such a hypothesis, let us imagine a Michelson interferometer mounted on a horizontal axis such that the arms move in a verticai pláne. Turning the interferometer round its axis can be taken to be a Lorentz deformation. It is obvious that the interferometer will deform under its own weight when turnéd round. It cannot be expected that the position of the fringes remains stationary, while the interferometer is turnéd round its horizontal axis. In particular we note that the deformations the interferometer suffers on account of its own weight depend strongly on the properties of the matériái of the interferometer. Thus two interferometers built of different materials will deform differently when turnéd round.
347. Taking, however, a small interferometer built of strong matériái the internál stresses may become negligible and the interferometer will behave practically independently of the gravitational stresses. Thus the small and strongly connected interferometer will show no noticeable fringe shift even if turnéd round a horizontal axis. From the above qualitative consideration we may come to conclude that the Lorentz principle in an inhomogeneous region is valid for sufficiently small and sufficiently strongly connected systems. A mathematical formulation which takes the above restriction into consideration will be developed presently.
3. THE LORENTZ PRINCIPLE VALID FOR SMALL PHYSICAL SYSTEMS
348. A Lorentz deformation must be taken in a first approximation to consist of an adiabatic shift, which can be expressed by a four-vector p. and a (four-dimensional) turning round, which corresponds to the deformation obtained by an orthogonal Lorentz mátrix A . The deformation is thus characterized in a first approximation by the ten parameters q and fi.. In an inhomogeneous region a system, if turnéd round, is usually deformed to somé extent because the direction of the gravitational stress changes relative to the system. A shift [x is followed in generál by a change of the gravitational surrounding and this change also causes deformations when the system accommodates itself to the new surrounding. q
a. FIRST APPROXIMATION
349. So as to describe in a first approximation the behaviour of physical systems when subjected to a shift q, (x we consider a small and strongly connected system O in the vicinity of a four-point ty with a four-coordinate x . The points of £i have coordinates 0
0
x (p) = x„ + fc
% (p)k
The relative coordinates íjfcO) are taken to be small. The shifted system £i* has its centre round a four-point coordinate x =x 7
and the points
o o + {*>
with a four-
of the deformed system have four-coordinates
«2 = *í + 5ÍO0-
The transformation function X(x) cannot be supposed to obey (23) exactly, we can, however, require that
M®g*(S*)M® * gflj)
(26)
for not too large values of \ . We have written
M(5) = A(5)og
A(?)=X(x + ?)-X(x ) 0
gflD =
0
g*OÍ*) =
go(x + I), 0
go(x„ + H +
Thus the transformation X(x) connects the vicinity of Xo with the vicinity of x*i = x + (/. and we expect the relations (23) to be approximately valid, when we take these vicinities only. 350. The relation (26) can be taken to be exactly valid for? = = 0; we can thus write Mg*M = g, (27) 0
where we write M in place of M(0), etc. The solutions of (27) are matrices M = a * A„a, - 1
q
where A is an orthogonal Lorentz mátrix and a and oc* are given expücitly in Appendix I. In the above ápproximation we can write q
X(x) or
w
X«(x) = u. + M (x q
Xo)
A(í-) * < > ( ! • ) = M l-. The transformation thus defined contains ten parameters, four parameters giving the four-dimensional shift [Í, and six parameters q defining the fourdimensional rotation corresponding to M . q
q
b. A SECOND ÁPPROXIMATION
351. A better ápproximation of (23) can be obtained if we require that the relations corresponding to the first derivative of (23) should also be satisfied in the point x . Thus we may require 0
[M(5)g*(5*)M(5)-g(?)]on = 0 for 5 = 0 . From the latter relation it follows according to 339 eq. (19) (3)
(3)
(3)
M = M • (C - C)
where C is defined by (19a). We can thus write in this approximation 1
(3)
(3) _
A(?) * A >(E.) = M !- + - M • (C - C)%\
(28)
2
q
q
q
^52. The approximation (28) cannot be essentially improved since a equirement to the effect that derivatives higher than the first of (23) in x„ should give correct relations cannot be satisfied. The above formulation of the Lorentz principle is not yet entirely satisfactory and it must be improved for the following reason. The function A (?) as defined by (28) is not invariant. Indeed, changing from a representation K to K' by a non-linear transformation r
(2)
x' = f(x) we obtain in the new representation A q
2)'
=
f A
<2) -l f
and thus A ' contains higher order terms in x' even if A does not contain such terms in x. If f(x) is a strongly non-linear function, then it can happen that the higher order terms in the expansion of A >' become quite essential. Thus the transformation A defined by (28) may correspond to essentially different deformations, if we apply (28) in different representations. So as to reduce the ambiguity of the definition (28) we note that the relations (23) are best fulfilled in an almost straight representation. Thus we may require (28) to hold in almost straight representations only — in curved representations (28) has to be extended by suitable further terms. That a restriction of this type is necessary indeed, can be best seen considering the homogeneous case. In a homogeneous region the Lorentz transformation is a linear transformation in terms of a straight representation. In terms of a curved representation, however, the transformation function becomes non-linear; the non-linear terms in the expansion X'(x) compensate the curvature of the representation. In a nearly homogeneous region the transformation X'(x') must be expected to deviate only little from a linear transformation, if we use an almost straight representation. However, in a strongly curved representation X'(x') will in any case be an essentially non-linear function — the non-linearity compensating in a first approximation the curvature of the representation. (2)
(2)
(2
(2)
4. THE AMBIGUITY I N THE FORMULATION OF THE LORENTZ PRINCIPLE
353. Restricting thus (28) to almost straight representations there remains still an ambiguity, as transformations between almost straight representations lead to changes in higher order terms. We can thus write l (3) A(5) = % + — M ^ + higher order terms, 2
where in a straight representation the higher order terms should be "small", <*)
i.e. not exceeding the order of the elements of the invariants R k = 4, 5 , . . . . 354. We can write down a definition as how to find transformations A(§) which reduce in a good ápproximation to (28) in a nearly straight representation. Indeed, remembering the considerations of 335 we find that an exact solution of (23) should obey the relation (4)
(3)
(3)
Mft) = ( M © ) • ( c ® - c ( 9 ) o n . (4)
The above relation possesses solutions M(£) but these solutions do not possess symmetry properties required for the second derivatives of a function A(5). (4)
Symmetrizing M(£) by putting (*) i <*) <M(©> = j (1 + (24) + (34))M(5) we obtain a totál differential and we can define e.g. « 1 (3)
(4)
if-
A(l) = M% + - Ml- + - J <M(§')>(? - i'f 2
(29)
o
(the value of the integrál does not depend on the path of integration as (4)
<M(5)> is a totál differential). The latter expression in an almost straight representation contains only small higher order terms — in curved coordinates (29) contains the higher order terms compensating the curvature (4)
and also terms of the order of the derivatives of R(?)355. One might be tempted to regard (29) as the "exact definition" of the Lorentz transformation in an inhomogeneous region. This is, however, not the case. Indeed, the series (29) in an inhomogeneous region is only
almost invariant in the sense that forming the expansion (29) in different representations, we obtain almost equal deformations. Let us write K(2) = A(x) and K'(2) = A'(x') thus let us take A(x) and A'(x') as the representations of an operator £ in K and K'. A detailed analysis shows that (in the terminology of Appendix II) A(x) as A(x) thus the representations of £ transform almost like those of a vector, but not exactly.
CHAPTER XI
APPLICATIONS OF THE GENERALIZED LORENTZ PRINCIPLE A. GEODETIC ORBITS 1. DEFINITION
356. So as to find the form of various physical laws in inhomogeneous regions it is useful to see how the mathematical form of such laws, valid in homogeneous regions, can be generalized. It is a question of experiment to find out whether or not the generalizations which suggest themselves are in accord with experiment. It is a useful guide in formulating the hypothetical forms of physical laws to suppose that in almost straight representations the laws have forms very nearly like the forms of the corresponding laws valid in homogeneous regions formulated in the measures of a straight representation. 357. Let us consider Newton's first law. Consider for this purpose a small closed physical system in a homogeneous region — the latter can be regarded also as a particle. The position of the system can be described by a four-coordinate x'(p). Newton's law in a straight representation K' can be written 5'QO = 0. (1)* We write down the relation (1) in a curved representation K which is connected with K' by the transformation x' = f(x).
(2)
Introducing x = x(p) into (2) and differentiating into p we find x'(/>) = S(p)x(/>)
(3)
where we have written S(/0 = ( f ( x ) o f j )
x=JtOl)
,
* Instead of relation (1) we could also use the relation CPT _
= 0
r = r(/>)
t=t{p).
(la)
The above relation contains more solutions than the relation (1), as the latter relation implies a linear connection between t and p . Since, however, the restricted relation (1) gives already all possible orbits of free particles, there is no need to use the more generál relation (la).
Differentiating (3) once more, remembering (1) we find (3)
0 = S(p)x\p)
+ S(p)x(p).
Multiplying with S (/>) from the left we find _1
(3)
i(p) + e(p)*\p) where we have written (see 472) (3)
=o
(3)
(3)
S" S = g - C = <2. 1
1
2. LORENTZ INVARIANCE OF GEODETIC ORBITS
358. Generalizing the above result we may suppose that the equation of motion of a free partiele in an inhomogeneous region can be written (3)
x + Sx = 0.
(4)
2
The latter relation is Lorentz invariant as we show presently. Taking a number of particles in the vicinity of a point x their orbits obey the law 0
<3>.
5* + <2E. = 0
A: = 1 , 2 , . . .
2
(5)
with (3)
(3)
e = s(5 ) fc
where we have put *k(P)
= x + \ . 0
k
Submitting the orbits in the vicinity of x to a Lorentz transformation we obtain deformed orbits with coordinates 0
x?(/0 = x* + 5*, where according to the definition of the Lorentz transformation (see 358 eq. (28)) { 1 (3) (3) \ 5* = M \$ + — (<S - e)%l\ + terms of the order of q
k
where (3)
(3)
(3)
A I U ) (3)
V3)
e = g~ C and C=|M, C(x*)J 1
Differentiating the above relation twice into p we find (3)
I* = M^
k
(3) .
+ (<2 - <9)
+ terms of the order of
We have included into "terms of the order of
(6)
also terms of the form
(3)
Q\ \ \ the latter terms can be taken to be small compared with the terms not containing as we suppose the Lorentz transformation to apply only to a small vicinity of x . The measures of velocities \ or of the accelerations % need not be small. According to definitions we find k k
0
k
k
(3).
(3)
.
and therefore it follows from (6) (3) .
..
(3).
8 + <2*8 = M,(5 + e5 ). s
2
t
Thus it follows from (5) that (3) .
§í + e*5* = o. 2
Thus the relation (4) is Lorentz invariant in the sense of 351. a. GEODETIC ORBITS AND THE LORENTZ PRINCIPLE
359. The equation of motion (4) has still another aspect. It follows from the dynamical part of the Lorentz principle. that a system Q subject to an adiabatic interference changes into a Lorentz deformed system ö * . If there is no interference at all, then a partiele with somé initial velocity Y will shift during a time t by an amount vr. The latter shift can also be considered a "spontaneous" Lorentz deformation. The analogue to this process is the drift of a system in an inhomogeneous region. The motion in accord with the equation of motion (4) can be taken as a sequence of infinitesimal spontaneous Lorentz deformations — the orbit of the partiele in a gravitational field can thus be regarded as a sequence of such Lorentz deformations. B. EQUATION OF MOTION IN A GRAVITATIONAL FIELD 360. We investigate presently a few properties of the equations of motion. In the latter investigations we suppose g to be known. The connection between g and the gravitational field will be discussed further below.
An integrál of the equation of motion (4) can be obtained as follows. Multiplying (5) from the left by g(p) we obtain an expression which is a totál differential, as can be shown using the explicit expression for the Christoffel symbols. Indeed, one finds (3) . d . %P)g(p)%(p) + UP)C(P)¥ = -jj (&P)%{P%{ )). P
(7)
Comparing (5) and (7) we find as an integrál of the equations of motion Í(p)g(p)Í(p)
= constant.
(8)
In particular, if the constant is taken to be equal to zero, then we obtain the relation valid for propagation of signals of light. Therefore the equation of motion (4) can be taken to be valid, not only for particles, but also for the orbits of signals of light. 1. VARIATIONAL PRINCIPLES
361. The equation of motion (4) was obtained by considerations in the course of which terms of higher orders had to be neglected. Nevertheless, relation (4) can be taken to be an exact relation in the following sense. Describing an orbit by a four-coordinate x(p) we may consider the orbits obeying the following variational principle S f x(p)g(p)x(p)dp
= 0.
(9)
The Eulerian equations which can be derived from the above variational principle can be written 80
d (39)
n
with 9(p) = x(p)g(p)x(p).
(11)
As the result of a short calculation one finds that (10) and (11) reduce to relation (4). Thus the equation of motion (4) can also be replaced by the variational principle (9). We note that the variational principle (9) defines orbits in an invariant way, i.e. independent of the representation. Indeed, the transformation relations for g(x) are so defined that subjecting x and g(x) simultaneously to a coordinate transformation, the numerical values of 6(p) remain unchanged. Thus the solutions of (9) and therefore also those of (4) define
definite orbits independent of representation. Solving (4) and (9) in different representations, we obtain the parametric representation of the same orbits relative to various systems of reference. In the above sense the relations (4) appear to be exact relations in spite of the circumstance that in the course of their derivation higher order terms had been neglected. a. DEVIATIONS FROM GEODETIC ORBITS
362. The fact that higher order terms have been neglected in the equation of motion (4) leads nevertheless to ambiguities in the following way. The geodetic orbits which can be determined in an unambiguous manner must be supposed to be the orbits of sufficiently small and connected objects. If we consider e.g. a planet, then we find that the geodetic orbit gives an excellent first ápproximation of its real orbit. However, taking the problem more strictly, we have also to consider deformations the body of the planet suffers, while it is moving from one gravitational surrounding to another, i.e. we have to consider the tides of the body. The tides themselves produce perturbations of the orbit and therefore the real orbit of the planet will deviate to a small extent from the geodetic orbit. The amount of this deviation is determined by the mechanical properties of the planetary body, i.e. from the measure of the distortions which arise while the planetary body continuously adjusts itself to its gravitational surrounding. Therefore the deviations of the real orbit from the geodetical orbit depend on the internál mechanical structure of the planet. Considering the motion of the planet as a sequence of Lorentz deformations in the manner explained in 358 we can say that the higher order terms of the Lorentz transformation express the deformations of the moving planet. We see therefore that the theoretically expected deviations of the orbit of a real planet from the geodetical orbit are connected with the ambiguous higher order terms of the Lorentz transformation. 2. THE PHYSICAL CONTENTS OF THE VARIATIONAL PRINCIPLE
,
363. So as to make clear the physical significance of (9) we write it down in a little modified form. We introduce the vector x/x = w, 1, 4
where w is the velocity of the centre of Q. We can thus write 6 = ((w + v)G(w + v) -
c )xl, 2
where
ki
xgx = —A = constant, 2
therefore we find x
i
= -
A y/c
2
(12)
=
- (w + v)G(w + v)
(we note that for G = 1, v = 0 we have x, =
.
A
The value of A depends on the scale we choose for p. Let us write A = m , where we take m to be the rest mass of £t in suitable units and the scale of p to be chosen accordingly. We find thus 0
0
Odp = (K-U)dt
(13a)
K = y m(v + w)G(v + w)
(13b)
where
~2 m=
/
^ 1 - ( v +w)G(v + w)/c
2
(13c)*
Let us take K to represent the kinetic energy, U the potential energy and m the mass of Q; we can write K— U = L, where L is the Lagrangian of Ci. In place of (9) we can also write <5
j W í = 0. !
(14)
We see thus that (9) corresponds to the Hamilton principle for the motion for the centre of O. * We note as a curious feature of equations (13) that although they have been derived from the generál relativistic equations of motion, they have a remarkable similarity to the non-relativistic expression. In particular v + w is the velocity of thé partiele relative to the ether and thus (13b) gives an expression very similar to the classical expression for kinetic energy.
We note that the inner structure of the system does not appear in the derivation and therefore we conclude that the orbit of any small closed system is expected to obey the same law. This result reflects on the principle of the equivalence of gravitational and inertial mass. 364. The variational principle can be written also in a different manner. Making use of (13) and remembering dp = díjki we find xgx dp = = — A x dt and have 2
4
ö
.
JJc
2
- (w + v)G(v + w)
=0.
Further writing Jc
2
- (v + w)G(v + w)
we have thus x,
<5 J dx = 0.
(15)
x,
T can be taken to be the propertime of the system and therefore the latter relation has the form of Fermat's principle in optics. The path followed by the system in Q is that along which the propertime is stationary. 365. While (9) can be naturally interpreted as a Hamilton, or as Fermat principle, the usual interpretation given to this relation, i.e. that the system moves along a "geodetic line" seems to us very artificial. We may of course "define" the orbits obeying (9) as four-dimensional geodetic lines — however, it seems to us that it is a play with words if we suppose the geodetic line to be a "straight line in four dimensions". The orbits satisfying (9) have specified variational properties and they can be taken to be a particular type of orbit — however, whatever we suppose them to be, they are certainly not "straight". As we shall see later, the solutions of (9) include among others the Kepler ellipses along which planets move. — If we call those orbits "straight" then we lose completely the meaning of what is usually called straight. The satisfactory procedure seems to suppose that the propagation of a light signal in vacuum is in a good ápproximation straight and an orbit deviating strongly from that of a light signal is noticeably curved. With the help of signals of light we can also construct an almost straight system of reference and orbits in such a system described by x'(p) = constant can be regarded as to define almost straight lines. In the presence of a grav(4)
itational field when R # 0 the concept of the exactly straight line is
meaningless. However, we are in no need of such a concept, as we can construct satisfactory systems of reference without having first introduced the concept of a "straight line" or of a "geodetic line" as this was shown in detail in 334.
C. CONNECTION BETWEEN A GRAVITATIONAL FIELD AND THE PROPAGATION OF LIGHT 366. Relation (4) can be used to obtain the orbit of a small closed system, e.g. a planet in a region, where the propagation tensor g(x) is known. We supposed from the beginning of our arguments that the propagation of light becomes inhomogeneous in a region, where there is a gravitational field. Thus relation (4) gives the equation of motion of a mass in a gravitational field, where the field is characterized by the propagation tensor g(x). Since the motions of the planets are well known the relation (4) can be used to determine the propagation tensor g(x) in a known gravitational field. Indeed, in the vicinity of the Sun using a nearly straight representation we can take the gravitational potential to be given by
*(r)---
ix = MG,
(16)
r
where M is the mass of the Sun and G the gravitational constant. The values of g(x) in the vicinity of the Sun must be such that (4) reduces — at least in a good approximation — to Newton's laws of motion. 1. THE EQUATIONS OF MOTIONS I N A GRAVITATIONAL FIELD
367. So as to carry out the comparison between (4) and the laws of planetary motion, it is convenient to eliminate the paraméter p from the system (4) and to obtain a set of equations with t = x f a ) as independent variable. For this purpose we remember that dt
~dt
4>
With the help of the above relations we can eliminate p of the equations of motion, and we get an equation of the form (2)
(3)
v = A + Av + Av ,
(17)
2
(2) (3)
(3)
where the coefficients A, A, A can be obtained from the elements of (2. Returning to the problem of determining g(x) we may suppose that 0 0 1 0
0 1 0 0
(1 0 g(x) = 0
V>
^ 0 0 -c (r)J 0
(18)
2
Introducing (18) into the expressions 472 of Appendix II, defining the Christoffel bracket symbols we find thus (3)
c\r)C
uk
(3)
=C
kii
= grad
c (r)
k = 1,2, 3.
2
fc
(19)
Introducing (19) into (17) and using the notation drldt = \ we obtain the following equation a dAc c(ír(r) r )^ +, v(vgradc A—_ — ^ _(r))v v = - ^g rgrad vvr 1
2
2
2
y
+
( 2 0 )
where we have denoted with a dot the derivation into time. The second term in (20) is much smaller than the first if we consider velocities v < c(r). Neglecting terms in t? /c (20) obtains the form of Newton's equation of motion if we suppose 2
2
c (r) = constant + 24>(r). 2
(21)
In particular we can write c\ ) = C - ^ 2
T
(22)
where c is the velocity of light at large distances from the Sun. 368. Thus choosing g(x) in the form (18) with c(r) given by (22) the equation of motion of a planet is obtained as
^=-73-+-^ ar
2a(vr)v
(23)
where we have written short c in place of c(r). Neglecting terms in v /c in (23) we obtain Newton's equations of motion and their solutions are the Kepler ellipses. 2
2
Considering the terms in t> /c we obtain corrections and these corrections can be taken as the relativistic perturbations of the motion of the planets we have already mentioned in 316. 2
2
2. INTEGRALS OF THE EQUATIONS OF MOTIONS
369.
So as to integrate (23) we remember •
A
vr = rr
and
Thus multiplying (23) by v we obtain
2
d
vv = - —— v . 2 dt
(v) 2 dt 1 - 2v,2/„2 jc K
1
}
a.r
2
(24)
„2
Neglecting the dependence of c upon r we can integrate both sides of the above relation and we obtain the relativistic energy integrál
2
v
2
\
A
*
where A is the constant of integration. We may also write , = ^-{l-exp{-4^ + ^c }|. 2
2
(25)
Neglecting terms in t> /c the above relation reduces to the classical relation, i.e. 2
2
v = 2 2
j/í +
.
(26)
Multiplying (23) by r in a vectorial manner we obtain d , 2<xr(r x v) — (r x v) = ~ dt cr x
—ka—: 2
- j r (
r
x
v
) =
2
from the above relation we see that the vector P = rxv
(> 27
(28)
does not change its direction, therefore we can write in place of (27)
thus
P _ 2<xr y ~ eV P = aexpi-^1},
(29)
where a is a constant of integration. From (28) and (29) it follows that the motion takes place in a pláne perpendicular to a. Introducing polar coordinates in that pláne we find r = f + rV, 2
P = r q>,
2
(30)
2
where
(3D we can write making use of (30)
rfo-fjsa*,.
JJtl'-P'lr' Introducing P from (29) and v from (25) we have if we introduce further s = 1/r as a new variable of integration 2
l/r
ads Vc /2(exp {4aí/c } - exp {-4-A/c }) 2
2
2
- eV
Developing into powers of 1/c we obtain 2
«,) — f
üdS t
J J2A + 2as - aV + l/c {4ocV - 4^ } + 2
2
+ terms of higher order. a. PERIHELION MOTION
370. Neglecting the terms of higher order we obtain a motion with a period 2a 2n • . = 2n 1 + • 7c -4a /c "V ' e V 2
2
2
2
thus a shift of the perihelion per revolution is given by 4na. ca
2
2
z
However, we can write a = v f, 2
a = vr,
where v is the average velocity, r the average radius vector, therefore we find ác = 4n'^.
(32)
P
371. We note that in a treatment where we modify the classical one by considering the relativistic change of mass with velocity only, we obtain v
2
Aq>
=
2K •
c
-2 5 .
The shift of the perihelion according to the generál theory of relativity as obtained by Einstein is v
2
Acp = 6n • -=-. c
(33)
The latter value seems to be in accord with the observed motion of the planet Mercury. The result obtained from the assumed form (18) and (21) gives therefore the correct non-relativistic form of the planetary motion and gives also a relativistic correction which is at least in a qualitative agreement with the observed deviation from the classical orbit. The value (33) for the perihelial motion of a planet could be obtained from our consideration, if we were to add suitable higher order terms to the expression (22) giving the connection between gravitational potential and velocity of propagation of light. Such a procedure appears, however, completely arbitrary andwe conclude that from the theory, in its form given so far, only the order of magnitude of the perihelial motion of the planets can be obtained. For to obtain precise results it is necessary to find the exact relation between propagation tensor g and gravitational fields. b. THE DEFLECTION OF LIGHT IN THE VICINITY OF THE SUN
372. Considering a signal of light passing the Sun, the deflection A fi of its orbit from a straight line can be obtained using relation (30); we find + 00] n-Ap
= ^~dt.
(34)
— 00 Supposing the signal to move in a first approximation along a straight line, which passes the Sun at a distance with a constant velocity c, we can write r = í + c t\ (35) 2
2
2
and considering (28)
Thus introducing (35) and (36) into (34) we find 2a
4J = -
2
cb
r
4a
dx
J (1 + — (30
2
3
T ) '
4
(37a)
Remember th at th e angular distance at wh ich th e ray appears to pass the Sun as seen from th e Earth is ű = b/r, where r is th e distance between Sun and Earth and a = i?r, wh ere v is the orbital velocity of th e Earth . We find also (37b) Thus •& is th e angular distance at wh ich a star appears from th e Sun and AfS th e apparent sh ift it suffers because of th e deflexion. The value of th e sh ift obtained from th e generál th eory of relativity, which value seems to agree with th e observed one, is twice th e value th us obtained: (37c) We see th at th e simple assumption about g(x) leads to a qualitative correct value of th e deflexion of star ligh t in th e vicinity of th e Sun. c. THE RED SHIFT OF SPECTRAL LINES
373. We consider th e adiabatic sh ift of an atom from a point with coordinate vector r to anoth er point with coordinate vector r* = r + a. The atom in its originál position sh ould oscillate with somé frequency co = l/T; th is oscillation can be described sch ematically by supposing that th e atom emits signals at times tk = t + x > k
k =
x
kT.
The emission of signals are th us events with relative coordinates %
k
= Q,
0, 0,
kT.
Shilling the atom adiabatically from r -* r* we may carry out the shift in such a manner that the atom in its final position should be again at rest, thus we take the transformed coordinates to be given by § t = 0, 0, 0,
T* .
Such a shift can be described by a Lorentz transformation of the form 351 eq. (28). We have thus
:
i r« + lC(r)-
(38)
T
However, it follows from the explicit expressions of 471 given for the Christoffel brackets that (3)
(3)
C ( r ) = :C (r*) = 0. m
444
Since the space components of
are zero we find thus in place of (38^
Remembering further that the space components of % should also be zero we can determine explicitly M,: k
(l 0 M = 0 q
0 1 0 0
0 0 1 0
0\ 0 0
c(r*)
(39)
Thus taking the fourth component of (38) we find with the help of (39) *
T
(> c(r*) * c
T
r
Considering the periods of oscillations we can also write co : co* = c(r) : c(r*). We see thus that if an atom is shifted about adiabatically, then its frequency will change in the course of the motion so that the ratio between frequency of oscillation and local velocity of light will remain constant. In particular, if an atom is brought from the vicinity of the Earth to that of the Sun, its frequency is expected to decrease, since the velocity of propagation of light near the Sun must be supposed to be smaller than that near the Earth. The effect thus described is the second relativistic effect mentioned in 316, i.e. the gravitational red shift of spectral lines.
374. It must be emphasized that from the above result it follows not merely that the frequency of atoms decreases when brought from the Earth to the Sun. Indeed, atoms of a specified type have frequencies which are affected by the gravitational potential. If we have a set of atoms near a point r another set of similar type of atoms near another point r*, then the frequencies of the two sets will be co respectively co*. If we transport an atom of the first set from r to r*, then it will change its frequency from co to co* in the course of its journey and once it arrives in r* it will behave like those atoms which have been all the time near r*. So as to see this in more detail we remember that the frequencies of atoms can be calculated e.g. from the Schrödinger equation. The Schrödinger equation must be supposed to contain implicitly the gravitational potential in such a manner that the frequencies obtained from it depend to somé extent on the gravitational potential and thus its solutions give frequencies depending on the gravitational potential in accord with (38). When an atom is moved adiabatically, then it adopts itself in the course of the motion to the changing gravitational potential and this causes the change of frequency in the course of the displacement.
D. CONNECTION BETWEEN THE SOURCES OF GRAVITATION AND THE PROPAGATION TENSOR g 375. We have seen that the motion of the planets, and the deviation of light in a gravitational field, are obtained, at least to a first ápproximation, if we suppose fi 'o
í°
0 1 0 0
0 0 1 0
1
^ o o . -c(rf) 0
(40a)
and C\T) = c (l + 2í>(r)/c ) 2
2
(40b)
where $(r) \% the gravitational potential. The latter relation is, however, valid in particular representations only. One can suppose, e.g. that (40a, b) hold in a nearly straight representation, where the sources of gravitation are at rest. So as to obtain a formulation valid for moving sources, it is necessary to express (40a, b) in a form valid independent of representation. The latter problem becomes particularly important, if we consider three or more body problems, where the sources of the field are in motion relative to each other.
1. EINSTEIN'S EQUATIONS OF GRAVITATION 376. On the basis of the Laplace-Poisson relation, which must be taken to be valid at least to a good approximation, it follows from (40a, b) (in an approximation neglecting higher order terms in l/c) 4nG S/8u=-y~Q,
(41)
where Q is the density of gravitating matter and G the gravitational constant. We may try to approximate the Laplace-Poisson relation (41), which is valid in particular representations only, by a relation independent of the representation. We show that it is possible to find an invariant relation between tensors only, which relation reproduces in a good approximation the relation (41). So as to obtain such a relation we note that the Christoffel symbols calculated from (40a) can be written
2 8x
dx
k
k
(42) £=1,2,3 all the other components are zero. The non-vanishing components of the Riemann-Christoffel tensor can thus be written (see 482 eq. (63)) (4)
(4)
(4)
1 dg 2 8x dx 2
1 4g
u
k
t
ti
C4)
Ög dx
u
k
Ög dx
(43)
u
t
Expressing g in terms of c(r) u
<> 4
d c(r) c(r)-r—^2
^ 4 4 « = -
k
(2)
fc,/=
cx ox
(4)
1,2,3.
(44)
t
The contraction R of R has thus components (2)
R
= c(r)V c(r) = - V g 2
U
2
c(r)
u
1 - —
2^44
ox ox, k
(grad
g i i
)
2
(45)
Taking thus only the highest order terms in c into consideration, the Laplace-Poisson equation (42) can be replaced by AnG
(2)
(47)
If we take (47) to be the 44-component of a covariant relation we may write thus in free space (2)
(48
R = 0.
So as to obtain the covariant form of (47) in regions containing matter we remember that QC can be taken as the 44-component of the energy 2
(2)
momentum tensor T of matter; thus we might write (2)
(2)
R = - xT x
=
(49)
4nG
Relation (49) gives a possible relation between the sources of a gravitational (2)
field described by T and the propagation tensor g(x). 2. ENERGY MOMENTUM CONSIDERATIONS
377.
From relation (49) it follows that (2)
Div T =
1 x
(2)
Div R
(49a)
(2)
thus in regions where Div R # 0 the physical system described possessing (2)
the energy-momentum tensor T is not strictly conservative. Equation (49a) shows a certain amount of exchange of energy and momentum between the system and the gravitational field. The latter exchange — as will be shown further below — is in generál a very weak one and it does not correspond to the action by gravitational forces of Newtonian type. Einstein was of the opinion that such an interaction does not exist and he suggested to replace (49) by (2) 1 R - j g R = - x T .
(50)
The difference between (49) and (50) is that the Div of the right-hand expression (50) is identically zero (see Appendix II 484), therefore it follows from (50) that DivT = 0 a relation to be expected if T is the energy-momentum tensor of a closed system. Mathematically the relations of (49) and (50) diífer by very small amounts only and the difference of the relations is too small to give such effects that are observable at present. We note that for a region where T = 0 it follows from (50) that R = 0, therefore a difference between (49) and (50) is to be expected only in regions occupied by matériái systems. We come back to this question further below. 378. The question arises whether (50) is the only covariant generalization of (41) which is in accord with (49a)? It was pointed out by Einstein that we can add to the left-hand side of (50) a term Ag without disturbing the relation (49). The relation thus obtained gives in a first approximation the same results as the relation without the A-term. We prefer to write the generalized relation by adding the new term to the right-hand expression. We find thus (2)
i
R - _ g * = -xT-Ag.
(51)
The latter relation can be interpreted by supposing - 8 = T (52) to be the energy momentum tensor of the ether, which tensor signifies the state of stress of the ether.* 0
* The above energy momentum tensor is unusual in the following sense. Provided g has approximately the form (40a), then it resembles the energy momentum tensor of a gas. The first three diagonal components represent the hydrostatic pressures, while the 44-component the energy density. In an ordinary gas one expects that the sum of the first three diagonal elements, i.e. T + T + T of the energy momentum tensor, is less than — T^jc , while we have for the ether n
2i
33
1
5-u +
+ gs > 3
-gjc*.
Thus the hydrostatic pressure is larger than in an ordinary gas. The latter circumstance seems, however, of no importance, as there is no reason to expect the ether to have the same mechanical properties as matériái média consisting of atoms. We come back to this question in more detail further below.
The 44-component of (51) in empty regions can be written with the help of (45) V c(r) = M r ) .
(53)
2
A solution of the above relation which is regular at infinity can be written c(r) = c
0
^ - ^ .
(54)
Thus a A-term with A > 0 corresponds to a shielding off effect of the ether. 379. Relation (51) contains the derivatives of g only up to the second (* + 2)
order. Adding to (51) terms containing the R k = 3, 4 , . . . we can construct covariant relations containing higher derivatives of g. There is no valid reason to suppose that the exact relation between g and its sources should not contain such higher terms. Indeed, under circumstances where observational data are available the invariants constructed with the help of derivatives of g higher than the second appear to be too small to give observable effects. It seems thus very likely that the real connection between g and its sources differs from (51) which gives merely the relation obtained if the exact relation is expanded in terms of the derivatives of g and terms containing third and higher derivatives are neglected. One might suppose that the effects of higher order terms could be noticeable, if more information was available about the properties of stellar objects with extrémé densities. Supposing, however, that (51) represents only an approximate relation, it is very dangerous to extrapolate the results obtained from (51) to too large distances or for too long intervals of time. The danger is of the same type as if we used the approximate formula e~ ~ 1 — x valid for small values of x and if we tried to apply this formula for large x values. Somé of the paradoxes which arise when applying Einstein's equations to cosmical problems might be explained in the above way. x
3. THE SCHWARZSCHILD SOLUTION OF THE GRAVITATIONAL EQUATIONS
380. So as to discuss the planetary motion in terms of the generál theory of relativity, it is necessary to determine the gravitational field of a central body from Einstein's gravitational equation. One can suppose from consideration of symmetry that — at least in an almost straight representation — the field of a small central body should be spherically symmetric, i.e. we may suppose g(x) = g(r)
if
r
P a
where a is a length of the order of the dimension of the central body. Furthermore supposing the central body to be at rest, we may suppose
9kÁ*) = 0. The simplest supposition for g(x) would be that it is of the form given in eq. (40a) 375. However, a g(x) having this simple form cannot satisfy Einstein's equations. Schwarzschild has shown that Einstein's equations can be exactly satisfied by a spherical symmetric g(x) which corresponds to a locally unisotropic mode of propagation of light. 381. So as to obtain Schwarzschild's solution we may suppose that the velocity of propagation in the radial direction diflers from that in the tangential direction. Such a mode of propagation can be written down in polar coordinates r, <j>. We may thus write for the relation giving the propagation of light in terms of polar coordinates Adr + r dd + r sin Mcp + Bdt = 0, 2
2
2
2
2
2
2
(55)
supposing that A and B are functions of r only. Thus we suppose g(x) to be given by A 0 0 V0
0 r 0 0
0 0 r sin d 0
2
o o
2
)
B
(2)
The components of R can be obtained inserting the above expressions into eq. (12) of 331 and (63)-(67) of 4 8 2 - 8 4 of Appendix II. Wefind (2)
(2)
that among the non-diagonal elements of R only R tíontains nonvanishing terms, however, the latter cancel each other identically. The 12
(2)
diagonal elements of R are obtained as > 2 A' B" B' * n = - - — + — - 2&A ( ' (2
A B
(2) ^22=-r—
A
>
AB
2
- 2 + — + rB'jAB,
A
(2)
+ ')
A
22
«
(57b)
(57c)
2
R
(57a)
(2)
= sin &R , (2)
>
2
B» = ^
+
B'
B'
7 ^ - ^ B ^ B
+
A B ' ) .
(57d)
Writing according to (50) and (51) (2)
R = Ag
we find from (57a) and (57d) with the help of (56) (2)
BR
n
7 = —— (A'B + AB') = 0,
(2)
- AR
it
rAíS
thus we have A'B + AB' = 0 or
(58) A = y/B,
where y is a constant of integration. Introducing (58) into (57a) and (57d) we find ylA = B=y[\-^
+ ^kr''
The above expression is found to satisfy identically (57b) and (57a). It represents the so-called Schwarzschild solution of the gravitational equations. The constant of integration y is connected with the velocity of light, we may therefore also write y=-\lc\
(59)
Thus relation (55) written explicitly has the form dr
2
1 1
2 0 1
+ r\d&
2
+
sm -dd
2
1 2
1
kr
1
r
3 c |l2
— r
-~kr \dt 2
= Ü.
2
3
(60)
4. THE RELATIVISTIC EFFECTS IN THE FIELD GIVEN BY SCHWARZSCHILD a. THE PLANETARY MOTION
382. The Christoffel brackets can be obtained from (56) and the equation of motion of a planet in the field of a gravitating mass at r = 0 can be written OL
{
CL
\
2CC
Integrating the above relation in a manner analogous to the procedure given in 369, we obtain eventually for the differential dtp ds
dtp =
Jc l2a 2
2
exp {4asjc } 2
- exp j - 4/c
2
2c
2
Integrating over a whole period, we obtain when neglecting terms of higher order in l/c 2
P c where v is the average orbital velocity of the planet and Atp is the angular shift of perihelion in the course of one revolution. The above result was obtained by Einstein and it is supposed to be in agreement with the astronomical data obtained for the motion of the planet Mercury. b. DEFLECTION OF LIGHT
383. Repeating the calculations of 372 we obtain for the deflection of a signal of light starting from the equation of motion (61) instead of (23) in 368 the value of the deflection 8a where b is the distance of the closest approach of the signal. Thus from the Schwarzschild solution we obtain twice the deflection which we obtained from the simple theory. The reason for the factor two thus obtained is that the Schwarzschild solution of the gravitational field describes a state of the ether where not only the density varies with r, but also the ether is in a state of radial stress giving rise to an anisotropy of propagation. The latter anisotropy increases the deflection of light in the vicinity of a gravitating mass by a factor two. 384. The red shift of spectral lines in the vicinity of a gravitating centre retains its value whether one uses for g the simple form (18) of 367 or eq. (56) of 381. Thus the red shift of spectral lines is not affected by the anisotropic mode of propagation of light in the vicinity of a gravitating centre. E. ELECTROMAGNETIC FIELD AND GRAVITATION 385. The electromagnetic phenomena are described in homogeneous regions by Maxwell's equations on account of the phenomena given in chapts VIII and IX.
The relations valid in inhomogeneous regions can be obtained making use of the procedure discussed in the beginning of chapter X. It is important to remark that the generalization o ' Maxwell's equations for inhomogeneous regions cannot be uniquely determined from mathematical considerations only. We can postulate sets of equations which represent such generalizations and it will be always a question of experiment to decidé whether or not a particular mode of generalization leads to a correct description of the phenomena. We note that formulating Maxwell's equations for inhomogeneous regions, we make really assumptions about the role of g in the equations. Thus such a formulation in effect amounts to a hypothesis as to the effect of the gravitational field upon electromagnetic phenomena. 1. A N INVARIANT FORMULATION
386. A particular assumption as to the form of Maxwell's equations in an inhomogeneous region is obtained, if we start from Maxwell's equations as written down in chapt. VIII 266 in terms of four-tensors. Defining suitably the differential operators these relations are also valid in arbitrary curved representations. One may suggest that the relations thus obtained are also valid in inhomogeneous regions. We may thus write for Maxwell's equations in inhomogeneous regions V¥ = — 47cJ
eff
Div T = 0 DivJ,EFL
(62)
0
j = J + DÍV n eff
F = Rot ¥
Alternatively Div F = 47tJ
eff
(63) Div F = 0 DivJ
eff
= 0.
The operators Div, Rot, L have to be taken in accord with Appendix II. Indeed, the set of equations thus obtained contains Maxwell's equations in homogeneous regions. In inhomogeneous regions using an almost straight
representation the relations (62) or (63) are very similar to the relations valid in homogeneous regions, the gravitational effect upon the electro(4)
magnetic field being of the order of the elements of R. 2. QUESTION OF ELECTROMAGNETIC POLARIZATION OF THE ETHER
387. Relations (62) or (63) thus obtained are, however, by no means the only possible mathematical generalizations of Maxwell's equations; e.g. one could add to the four-current any four-vector which is free of divergence and which disappears in the homogeneous case. One might e.g. replace J by /(2) 1 \ J
e f f
=J + A"F •
R-yg*I.
(64)
The second term in (64) could be taken as the four-current corresponding to the polarization of the ether caused by the electromagnetic field. Whether or not such a polarization term appears is a question which could only be decided by experiment. Such an experiment is rendered difficult, as it follows from the gravitational equations that in free space (2)
R= 0
R = 0.
Thus the extra terms of the form (64) gives only effects for electromagnetic waves propagated through matter. The ordinary interaction of light with matter is, however, so strong that a background caused by the A'-term in (64) could hardly be detected under normál experimentál conditions. 388. A possible mode of detection of the A'-term could be found observing the scattering of light on light. Indeed, in a region containing a radiation (2)
field, the energy momentum tensor is different from zero and thus R and R differ also from zero. Thus the effects of the A'-terms could be felt observing e.g. the radiation of high density. Since g is contained implicitly in Maxwell's equations therefore independent of whether there exists a A'-term, a scattering of light on light is to be expected. From the measure of such scattering information about the A'-term could be obtained—at least in principle. 3. REMARK ON THE CONSISTENCY OF THE GENERALIZED THEORY OF ELECTROMAGNETIC FIELDS
389. Summarizing our considerations concerning gravitational effects we started introducing methods of constructing systems of references with the help of signals of light. We formulated Maxwell's equations by using the systems of references thus obtained.
A necessary check of the consistency of the theory is to show that the propagation of signals of light as obtained from the generalized formulation of Maxwell's equations is consistent with the phenomenological assumptions regarding the mode of propagation of light upon which assumptions all further considerations were based. Indeed, when introducing systems of coordinates we supposed that beams of light bbeyed relations
x(/0g(x(/0)x(p) = 0 ; that the latter assumption is consistent indeed with Maxwell's equations as formulated in 386 follows from considerations analogous to those valid for homogeneous regions; these considerations were given in chapter VIII 272. From the generalized Lorentz principle it follows further that a beam of light follows a geodetic zero orbit, if it obeys the relation (3)
x( ) + e(x(p))i( y P
P
=o.
That the orbits of beams of light obey the above relation could be proved most convincingly, if one were to succeed in giving in inhomogeneous regions the solutions of Maxwell's equations in terms of retarded potentials in a manner resembling that given in homogeneous regions explicitly. The latter procedure meets with difficulties and therefore one may instead use a method well known in classical optics. Making the transition from wave optics to geometrical optics, one can determine with the help of the eiconal the orbits of beams in unisotropic média. The generalized Maxwell equations have a form analogous to the wave equations in an inhomogeneous médium. Therefore, considering the ether as an optically unisotropic médium — the optical properties of which are determined by the propagation tensor g — we can determine the orbit of a beam of light with the help of the methods of classical optics. Such calculations were given by Laue and they prove that in a first ápproximation beams of light which can be derived from the wave equations follow indeed geodetic zero orbits. From these results we conclude that Maxwell's equations as formulated in 386 are consistent indeed with the generalized Lorentz principle. The question as to the existence or magnitude of the non-linear effects caused by the polarization of the ether cannot be decided, as at present there exists no experimentál evidence which gives information concerning such effects. In particular the question whether or not X = 0 presently cannot be decided experimentally.
F. ENERGY AND MOMENTUM RELATIONS OF THE GRAVITATIONAL FIELD 390. Einstein's equations of the gravitational field given in eq. (50) can also be regarded in a different manner. Supposing ^ g = T<'\
(65)
where may be regarded as the energy momentum tensor of the ether, we can also suppose 1 /(« 1 \ -(R- E*]«T« (66) or T
(2)
R/x = T »
(67)
to be the energy momentum tensor of the gravitational field, and we can write in place of (50) X<») + T + = T* (68) (m)
0
where T*"* and T are the energy momentum tensors of matter and of fields other than gravitational fields. From (65) it follows (see Appendix II) identically (W)
DivT<*> = 0, thus the totál energy and moment carried by the ether appear strictly conserved. The energy and momentum stored in the ether are equal to the sum of the energies and momenta of the various fields carried by the ether. The tensor differs from the others in such a respect that according to (66) (or (67)) can be expressed by g and its derivatives only. For this reason "T** could also be regarded as a kind of potential energy and momentum, i.e. the part of the totál energy and momentum density which is stored by the ether in the form of stresses and flows. The remaining part, i.e. T + T , may be taken as the part which appears as the energy and momentum of the fields carried by the ether. (m)
icl)
1. THE GRAVITATIONAL
391.
FORCE
If we take (67) as the definition T**" then we find
DivT<*> = J— - Div gR 2x and according to Appendix II Div T^^-l-Grad-R = f >. (ff
The latter expression can be taken as a force density acting upon matéria! systems by a gravitational field — or it can be taken as the rate of energy and momentum transferred from the gravitational field to other fields. (We can take T also to be the energy momentum tensor of the field of waves of matter.) Since (m)
R = 2k = constant this interpretation can be maintained for regions where T*'^, T*" = 0, i.e. for regions where fields other than the gravitational field do not exist. 392. The force density f is one which is exerted by a gravitational field upon a matériái system. However, the latter force density is not the gravitational force appearing in Newton's theory, but it is rather an internál force, which produces in a first ápproximation an internál stress only. This force is identically zero, if is given by the expression (66) in accord with Einstein's hypothesis. The main part of the gravitational action appears in a different form and quite independent of whether the tensor is given by (66) or by (67). 0
w
2. ANOTHER ASPECT OF THE GRAVITATIONAL EQUATIONS
393. Expressing in the relation (65) the energy momentum tensor and T in terms of g and its first and second derivatives, we obtain a system of equations which can be taken as equations of motions of the ether. Indeed, let us consider somé representation K; we may give T(x) as function of coordinates and time. If we give further (f)
goto = S(J, 0) i.e. the distribution of g(x) at t = 0, then the gravitational equations give a connection between the time derivatives of g(x) at t = 0 and the components of T (r) = T(r, 0) of the energy momentum tensor at t = 0. Since we have ten equations, we can determine in generál ten time derivatives of g(x). 394. The Riemann-Christoffel tensor contains the g k, I = 1, 2, 3, i.e. the second time derivatives of the g . It does not contain g nor the £(£444 thus the second time derivatives of the g or of g do not appear explicitly in (65). The Riemann-Christoffel tensor does, however, contain in the non-linear terms g and also g (k = 1, 2, 3) thus the first time derivatives of g and the g . We can thus determine from (65) the values of the second time derivatives of the g (k, l = 1, 2, 3) and the first time derivatives of the g (v = 1, 2, 3, 4) at t = 0. Thus integrating step by step the g(x) can be determined for times t > 0 from the initial condition goto. 0
km
kl
u u
ki
iU
44
fc44
ki
kl
iv
u
The procedure given above is not quite unique, as the equations from which we determine the first time derivatives of g(x) are quadratic equations. However, choosing once one particular solution, on account of continuity considerations, no ambiguity remains in the course of the integration of the equations. Furthermore, taking T (r) arbitrary, it is not certain, whether the system admits of real solutions for the first derivatives. It seems thus that the gravitational equations (65) give certain restrictions as to the T(x) for which the equations can be solved into g(x). The above restrictions may be real restrictions as to the possible states in which matter appears.* The restrictions may, however, also be interpreted not so much to be real restrictions as to the possible mode of distributions of matter, but it may be supposed that these apparent restrictions show the equations (65) to be incomplete. If we were to include in the latter equations terms containing derivatives of g(x) of higher order 0
(4)
than the seconds (say terms including Grad R ) , then the relations thus generalized may become compatible with arbitrary distributions T(x). 395. The position described in the previous section is not unlike the one we encounter in the case of Maxwell's equations, where — starting from an initial condition — we can determine the distribution of the electromagnetic field — provided the motion of the source densities is known. Just like in the case of Maxwell's equations, also in the case of the gravitational equations the field g affects the motion of T. Therefore, the motion of matter can be derived from (65) only, if apart from the initial condition (66) we give also initial conditions for the distribution of matter, e.g. in the form T(r, 0) = T (r) (69) 0
and besides we have to know the equations of motions for T(x), the latter may contain g(x) explicitly. 3. THE MECHANISM OF THE GRAVITATIONAL FORCE
396. The gravitational force upon a closed system can be regarded not so much as an outside force, but rather as an action of the system by itself caused by internál force. This can be seen from the following consideration. So as to see the essential features of how the force with which a system acts upon itself arises, we may consider an example, an electrically charged partiele placed into a gravitational field. We consider merely the part of the gravitational force which acts upon the part of the mass cor* This restriction may be of the same type as the suggested restriction of Maxwell's equations according to which only retarded solutions of the wave equations are realized in nature (see 263). Or it resembles the restriction according to which wave functions are always antisymmetric. — See also L. Jánossy, Foundations of Phys. (in the press).
responding to the energy of the electric charge. Thus we consider the mechanism of the gravitational force upon Am
=~ c 2
where Am is the mass equivalent of the energy E, the static energy of the electric charge. Consider the charged partiele to fali freely. In a nearly straight system of reference we can represent the freely falling partiele as a partiele which is practically at rest relative to the origin of the system. Calculating the force with which the electrical charge acts upon itself, we can apply in a good ápproximation Maxwell's equations valid in a homogeneous region — as in the nearly straight representation, considering a small partiele, the terms caused by inhomogeneity can be neglected. We thus find that the measure of the electric force with which the partiele acts upon itself is zero. If, however, the partiele is prevented from falling freely, then it shows acceleration relative to the origin of the nearly straight representation, and the force with which the accelerated charge acts upon itself can be calculated in accord with 305 chapter IX. The force thus obtained tends to reduce the acceleration, i.e. to establish the state of free fali. This electric force is equal to the gravitational force which accelerates a free partiele relative to a system of reference where gravitational forces appear. 397. A little more generál formulation of the problem can be given as follows. If we have an arbitrary closed physical system, it is kept together by internál forces which can be supposed to be propagated with the velocity of light. In the absence of gravitation the internál forces are propagated homogeneously and, as a rule, they produce no resulting force of the system upon itself. If, however, the system is brought into a gravitational field, then the mode of propagation of the internál forces is disturbed (just in the way as the mode of propagation of light). The perturbation of the propagation of inner forces disturbs the system in such a way that the forces acting between the elements do not obey any more exactly the principle of action and reaction and a resulting force arises. This resulting force tends to accelerate the system just like the Newtonian gravitational force is expected to do. We may thus suppose that the gravitational force observed phenomenologically is equal to the self force with which a closed system acts upon itself, if the propagation of the internál forces is made inhomogeneous by the gravitational field. We note that in a freely falling particie the propagation of the inner forces is nearly homogeneous relative to the partiele itself and therefore in the state of free fali no resultant self force is present.
-ff
CHAPTER XII
COSMOLOGICAL PROBLEMS A. THE PHYSICAL SIGNIFICANCE OF INVARIANT FORMULATION OF PHYSICAL LAWS
398. In the preceding sections we have given the formulations of various physical laws for inhomogeneous regions. The formulations thus obtained are generalizations of physical laws formulated for homogeneous regions. From the purely mathematical point of view, many different generalizations of a law can be obtained, if we merely require that the generalized law should contain as a limiting case the law valid for homogeneous regions. Among the possible mathematical generalizations of the laws valid in homogeneous regions one usually considers those which have invariant form* and supposes them to give the real physical laws. When considering the invariant generalizations only of given laws we find that there still exist a great manyfold of such generalizations and therefore even if we restrict ourselves to invariant generalizations there remains the question of picking out the correct form. This question was discussed e.g. in connection with Maxwell's equations. 399. In any particular case it remains thus to be decided by experiment which of the generalized forms of the physical law describes correctly the observed phenomenon. However, we have to go further: it is also a question to be decided by experiment whether or not the law describing a particular phenomenon correctly is an invariant one? It is often — incorrectly — argued that the requirement that laws of nature should be expressed by invariant expressions is a trivial one; this requirement is supposed to follow from the obvious requirement that "the laws of nature should be independent of the system of reference we use". While the latter requirement is trivial indeed, we show presently that the usual requirement of invariance contains far more than the trivial fact that laws of nature are independent of representation.
* Invariant form means that the relations can be expressed in terms of tensors and covariant operators.
1. TENSORS A N D DISTINGUISHED MEASURES
400. Linear relations between tensors are automatically invariant because of the transformation properties of tensors. Furthermore, we can define certain types of products between tensors which are tensor quantities themselves. Therefore expressing the measures of a physical system in terms of tensors, we achieve that linear combinations (e.g. sums) of the measures and also certain types of products — being also invariants — can be taken to correspond to significant physical quantities. Thus expressing the measures of a physical system in terms of tensors, we select a distinguished scale for the measures such that in terms of this scale sum and products of the measures are also physically significant. This procedure is much the same as was discussed in chapter III, where we discussed distinguished measures for quantities like length, electric charge, etc. Just as in the cases discussed in chapter III for simpler measures, we note that the fact that certain physical quantities can indeed be expressed in terms of tensors reflects upon physical properties of the quantities thus expressed. The assumption e.g. to the effect that the energy and momentum of a partiele can be expressed with the help of four-vectors can be tested experimentally. 401. Invariant products between tensors (apart from the direct product) can only be taken with the help of the tensor g which must be given together with the system of reference. We supposed in the preceding considerations that the representations of g such as g(x) = *(g) can be obtained by observing the mode of propagation of light in terms of the four-coordinate measures x of K. We see thus that the mode of propagation of light is explicitly being made use of for the invariant formulation of physical laws. 402. The mode of propagation of light has a twofold role for the invariant formulation of natural laws. Firstly, as was already pointed out in chapter X in 325 the assumption that the propagation of light signals in the vicinity of a four-point x can be expressed by relations xg(x)x = 0 (1) is an assumption which can be tested by experiment. The above law expresses a property of light signals and one may take that the above law reflects upon the physical properties of the carrier of light, i.e. of the ether. Relation (1) is in somé analogy to the relations we obtain from the classical form of Maxwell's equation for the propagation of light in inhomogeneous média; the propagation tensor depends on stresses and state
of motion of the médium. Thus (1) expresses the fact — which (in principle) can be verified experimentally — that light is propagated in the ether in a way resembling the way it is propagated in a médium in a state of flow and of stress. Thus invariant formulation of laws using the tensor g makes only sense in regions where g exists, i.e. in regions where light is propagated in a locally homogeneous manner. The latter statement shows that the invariant formulation presupposes an important physical property of the propagation of light. 2. THE PHYSICAL SIGNIFICANCE OF THE TENSOR g
403. A physical system can be described by a number of measures % 33, ©,. . . and the four-coordinates j j , . . . of certain points of the system; the physical properties of a system can be described in an invariant manner by stating somé connections between the measures. These connections are also making use of g. Let us take e.g. the very simple case of a partiele moving freely; we have then in a particular representation 1 (
2
xg(x)x = a
(2)
where a is a scalar. Taking a more generál case a physical law describing properties of a physical system can be described in a form
5(9l,S3,e, ...S1.Í2, ...g) = 0
(3)
where % is a function or functional. A representation of the tensor g can be determined making use of (1) and observing signals of light; g could equally well be determined making use of (2) and observing the orbits of free particles. In principle g could be determined observing any of the phenomena which we suppose to obey a law of the form (3). From the above remark it follows that g represents somé physical field which appears when observing very different phenomena — the propagation of light is only one of many such phenomena. The usually accepted interpretation of g is that it represents the "metric of the space-time continuum". We do not think the latter interpretation to be a fortunate one. We would rather suggest that g represents the state of the ether which is the carrier of all physical fields. The latter assumption makes it clear that in physical laws like (3) g appears explicitly — this circumstance shows that the behaviour of a physical system is determined not only by specific quantities % 2 3 , © , . . . , Zk - • • but also by the state of the ether in the region which is occupied by the system itself.
404. The assumption that we can formulate laws of nature in invariant manner using relations of the form (3) implies also the assumption that the state of the ether is fully determined by the tensor field g. It might be conceivable to suppose that the state of the ether in a region 9t is determined by g together with other fields f. The latter fields might be supposed not to affect the mode of propagation of light — and therefore f cannot be determined observing merely the orbits of signals of light. Treder* has suggested a form of the theory where instead of the symmetric tensor field g a mátrix field with 16 independent components appears. According to this theory the field equations of scalar, vector or tensor fields depend only on g, i.e. on 10 independent combinations of these 16 functions, while the equations of motion for spinor fields depend on all of the 16 functions. 3. A N O R M Á L FORM OF THE PROPAGATION TENSOR
4 0 5 . The measures of the elements of a tensor (or tensor field) can have very different values when given in various representations. For this reason one meets with opinions according to which the one or the other tensor is "void of physical significance". Putting this question, one must be careful. The measures of a physical quantity differ from each other when taken in various representations, nevertheless these measures refíect an objective quantity. Let us e.g. consider the propagation tensor g. In a representation K we have K(g) = g(x), g(x) has for any value of x ten independent elements; applying a coordinate transformation x' = f(x) we obtain a new representation g'(x') of g. However, as the transformation contains four functions only, we can prescribe at most four of the ten independent elements of g'(x') and then find a transformation which transforms g(x) into the required g'íx')- If we have two arbitrary representations gj(x) and g (x') of the propagation tensor, then it is the exceptional case where there exists a tensor g such that 2
*(9) = gi(x) and
/T(g) = g (x'). 2
* H. Treder: Einstein-Symposium (Berlin, November 1965), Akademie Verlag, Berlin, 1966.
As a rulc the two representations g and gó cannot be regarded as representations of one and the same propagation tensor g. Therefore a representation g(x) of a tensor g contains specific features of that tensor which features are common to all possible representations of g and these features express somé objective physical properties of the field g; these features are thus independent of representation. 406. To make this question clearer we give the following consideration. The transformation connecting two representations, say, K and K' can be written x' = f(x). (4) x
As the transformation contains four arbitrary functions, we may obtain something like a normál representation if we impose four suitable conditions upon g(x) = (5)
m)
which conditions reduce the choice among the possible representations. Writing in the usual way G V
V - c
(6)
2
we can e.g. require for the normál representations that V = 0
Grad C = 0.
(7)
We show presently that for any distribution g representations can be found satisfying (7). 407. Indeed, consider a representation K where g(x) is given by (6) so that (5) is not necessarily fulfilled. Changing to a new representation K' by a reversible transformation (4) we may write g(x) = S(x)g'(x')S(x),
(8)
and we may put
(9)
S(X):
where b,=
dx
k
(10a)
8x
k
and a, =
df,<x) dt
(10b)
Inserting (9) into (8) supposing ÍG'
0
constant
(11)
we find G =PG'P-(bob)c
(12a)
2
V = aG'P - ac^b
(12b)
C = ac
(12c)
2
2
-aG'a.
2
From (12a) it follows G' = P
_ 1
GP~
1
+ (bP obP )c. _ 1
_ 1
(13)
From (12b) we have a = (V + ( A J P - V - . 1
(14)
Inserting (13) and (14) into (12c) we can express a in terms of the elements of g together with P, b and G'. We obtain the expression of the form a = a(P, b, c, g), a = a(P, b, c, g).
(15)
If we prescribe the value of c and somé arbitrary values for t = 0 P(r, 0) = P (r) b(r, 0) = b (r)
(16a) (16b)
0
0
then we can determine a and a for t = 0 thus we can determine the time derivatives of the components of f(x) for t = 0. Knowing the time derivatives of f(x) for all values of r at t = 0 we can determine step by step f(x) for any later time t > 0 — or going backwards in time we can also determine f(x) for any time t < 0. 408. Thus prescribing an initial condition of the form (16) we can indeed construct a representation K' so that ^'(9) = g'(x ) /
and
V' = 0,
C = c.
The system K' can thus be regarded as a representation relative to which the ether appears to be at rest. The system AT'(g) is, however, not determined uniquely. In the homogeneous case we know that in the systems of inertia which move with translational velocities relative to each other it is possible to introduce coordinates such that light appears to be propagated isotropi-
cally in terms of those measures. In the inhomogeneous case we find a similar ambiguity. Indeed, considering two representations K and K' so that V = V = 0,
C =
C
c,
we find with the help of relations (10) for the elements of a transformation which connects two normál representations K and K' G' = P - ( G + b o b c ) P 1
2
1
a = (1 - b(G + (bo b ) ^ ) - ! * ) - / 1
2
1
2
(17)
a = c ab(G + b o b c ) - P . 2
2
1
We can choose P and b for t = 0 arbitrarily and find their values for t # 0 remembering the relations and integrating step by step. Since — c a gives the velocity of a point r' = constant relative to one with r = constant, we see that the normál systems of references may move relative to each other just like Lorentz systems of references do. 2
B. PHYSICAL CONTENTS OF PARTICULAR REPRESENTATIONS 1. STATIONARY REPRESENTATIONS
409. If we choose a suitable representation a requirement to the effect that V = 0, grad C = 0 can always be satisfied. Further restriction of g(x) cannot be satisfied in generál by a suitable choice of representation. For example, we may consider a distribution such that ^
= o.
os)
(18) imposes ten independent conditions upon the representation g(x) of g. Given an arbitrary representation g(x) which does not fulfil (18) we may try to construct another representation g'(x') such that ^
= 0.
(19)
Since g'(x') = S-\x)g(x)S-\x)
with
S(x) = f(x)oQ
(20)
(19) and (20) together impose ten conditions upon the four components of the function f(x). The latter system is in generál overdetermined and does not admit of solutions.
In the exceptional cases where g(x) is such that the systems (19) and (20) can be solved we must suppose that it is not a mere accident that the mathematically overdetermined system admits of solutions. In the latter case we have to suppose that the fact that g admits of representation obeying (19) is of physical significance. One may suppose that configurations which possess representations satisfying (19) may be taken to be stationary configurations. We may thus suppose that the ether is in a stationary state if g admits of representations g'(x') where all its elements are independent of t'. Similarly we may consider physically significant other symmetries or particular properties of which appear in somé representation g(x) of g provided these symmetries or restrictions cannot be produced by a suitable choice of representation of an arbitrary g(x). 2. THE ENERGY MOMENTUM DISTRIBUTION
410.
Similar problems are encountered considering the representations (2)
T(x) = K(%) of the energy momentum tensor of matter. It is always possible to find (2)
representations T such that (2)
T (x) = 0 ik
k= 1,2,3
(21)
the latter representations correspond to systems of reference in which matter does not move relative to points with constant coordinates. Comparing, however, the state of the ether with that of the distribution of matter, there may or may not exist a representation where (2)
r *(x) = 0 4
and
g (x)=0 lk
for
k= 1,2,3
(22)
are fulfilled simultaneously. The condition (22) imposes already six conditions upon the representation and therefore starting from an arbitrary representation, in generál, it will not be possible to find a transformation such that in terms of the transformed measures the whole of the condition (22) is fulfilled. In the exceptional cases where we can find a representation satisfying (22) we may conclude that matter moves together with the ether. Considering the distributions of matter at large in the universe, we can suppose in a good ápproximation that the interaction between stellar systems is a purely gravitational one. Considering the stellar systems of the universe like the atoms of a gas, we can suppose that the matter in
bulk behaves like a médium free of inner stress. Thus averaging over sufficiently large volumes we may suppose that (2)
Tyjji)
=
except for
0
v=
n=
4
(23)
(2)
r (x) = nix) > o. 44
(2)
More precisely we may suppose that there exist representations of T = = K(%) such that (23) is satisfied for them. Although (23) is not of an invariant form it nevertheless describes a significant feature of the distribution of matter. Indeed, if we start from (2)
an arbitrary representation T(x) of % then, in generál, it is impossible (2)
to find a representation T'(x') of % obeying (23). If we nevertheless find that applying a suitable coordinate transformation to the originál repre(2)
sentation, a representation T'(x') of Z can be found obeying (23), then this proves that the distribution 2 has particular properties. We may suppose that matter characterized by í£ is free of inner stress. If we find a representation which apart from (23) satisfies also g
ki
= 0
k = 1, 2, 3
then we may conclude that the matter thus represented is free of stress and flows together with the ether.
C. COSMOLOGICAL PROBLEMS 411. In the present section we give a very brief account of somé cosmological aspects of the theory of relativity. Cosmological theories try to give an account of the observed features of the distribution of matter in the universe in terms of distributions in accord with Einstein's gravitational equation or of somé generalization of the latter equation. 1. THE RESULTS OF ASTRONOMICAL OBSERVATIONS
412. Astronomical observation shows that the universe is fiiled with galactical systems which — as far as it can be ascertained — are distributed uniformly in space. Considering sufficiently large parts of the universe, we may consider the number of galaxies per unit volume to be about constant. Considering these galaxies like the atoms of a gas we can suppose matter in the universe to behave like a gas of constant average density.
The proper motions of the galaxies are such that collisions between galaxies must be supposed to be extremely rare events even on a cosmical time scale and therefore it can be supposed that the interactions of the galaxies are given in a good ápproximation by their gravitational interaction only. Taking the hydrodynamical analogy, we can take the universe to be fiiled with a gas of low temperature, which is moving only under the influence of its own gravitational field. 413. The observations of Hubble have shown that the spectra of known atoms emitted in distant nebulae show a considerable red shift. The observations can be summarized by a relation Av — =-kr, v
(24)
where Av is the difference of frequency of a spectral line when observed in the laboratory, respectively when observed in the spectrum of a galaxy at a distance from us. k « 10
_ 2 7
cm
_ 1
x 10" parsec 8
_1
is the Hubble constant; its numerical value can be determined only with great uncertainty, as the measure r of the distances of nebulae can only be estimated by rather indirect methods. Supposing that the Hubble effect is caused by Doppler effect, i.e. supposing that the distant nebulae are receding with great velocities, we can also write in place of (24) v = ckr (25) where v is the velocity of recession of objects at distances r. There exists also somé astronomical evidence upon electromagnetic radiation inside the universe; we do not discuss this complex of problems here. 2. THE SOLUTION OF F R I E D M A N N
414. Many cosmological considerations deal with possible solutions of the gravitational equations (2)
1
(2)
R - y g * =-xT-Ag.
(26)
While attempts were made to find solutions of the gravitational equations which correspond to the observed state of the universe, a solution was looked for where matter in bulk is free of stress and can be represented in a form (23). So as to specify the problem, it was investigated by Fried-
mann whether or not Einstein's equations possess solutions such that in a suitable representation the tensor X is given by (23) while the tensor g is given by IA B 0 0 -C 2
g(*)=L
2
(27)
It is supposed that Grad C = 0
and
-^j-=°>
() 28
while A may depend in an arbitrary manner upon x. The expression (27) for g(x) is a more restricted one than the normál representation (7) of an arbitrary g. Indeed (27) supposes that" G(x) = .4 (x)B(r)
(29)
2
which supposition restricts G(x) and thus g(x). 415. Supposing g(x) to be given by (27) we can obtain the elements (2)
of R. We find as the result of a short calculation <2>
„
d \nA
<2'
3
eA
2
2
From the gravitational equations (26) it follows with the help of (23) (2)
R
ki
(2)
R
it
=0
k= 1,2,3
1 = - — xu + k.
(31a) (31b)
We find thus ö \nA dx dt 2
= 0 n
k
and therefore A(x) = a(0/J(r). Without restricting the form of g(x) further we can include the factor /?(r) into the mátrix B(r) and thus we may suppose A(x) = ^(í).
(32)
So as to determine A(i) explicitly, we note that (2)
DivT = 0 and therefore according to (23) and eq. (37) 476 of Appendix II we find .-£-
c
c - ( - d e , r - " ' 'A ' " ' C
with
C
B
g
(33)
and dt Inserting (33) into (31b) and remembering that according to (30b) and (32) (2)
R
i4
depends only upon t, we find 3 dA _ A dt ~
p + X A
2
2
where
(34)
3
• / ? = 0.
1 dA Multiplying (32) with — A —- we obtain a totál differential. Integrating 6 dt we find IdA'Y
-p-KA
+ XA ^ 3
dt j
3A
where we have written K for the constant of integration. Inserting (35) we obtain explicit forms of A. In particular putting K = 0 and X = 0 we find A(t) = A (t+t ) '
(36)
2 3
0
0
thus A\t) increases with time. We see that according to the initial conditions and the values of K and X we obtain solutions with increasing A(f) or decreasing A(t). Equation (35) admits of a singular solution A = A
0
with
- / } - KA + Al = 0. 0
The latter solution can be shown to represent an unstable configuration, which at the slightest disturbance changes into one, where A(t) varies in time monotonously. The above considerations imply that no stable stationary solution of the gravitational equations exist — i.e. solutions in which both matter and ether are at rest.
D. ANALYSIS O F FRIEDMANN'S SOLUTION 416. We make a few remarks so as to clarify the types of motion which are involved in the configuration corresponding to the Friedmann solution of the gravitational equations. 1. THE
417.
RE CE SSION OF THE
GALAXIE S
Firstly we note that, on taking the Friedmann solution for g(x), the (3)
components of <2, where two or three suffixes are equal to 4 vanish. There fore we find from the equations of motion that a partiele which is at rest relative to the system of reference at t = 0 remains at rest; indeed it fol lows from the equation of motion x(p) = 0
if r(p) = 0 (37)
therefore r(p) = r = constant
is a solution of the equations of motions. We conclude therefore that the matter under influence of its own gravi tational action may float together with the ether. a. THE MEASURES OF INTE RGALACTICAL DISTANCE S
418. The journey of a light signal set from a point P with fixed coordinate vector r to another point P with a fixed coordinate vector r = r + ÖT takes a time 5ty. If the signal is refiected back from P to P the time of to and return fiight is equal to (see 381 eq. (55)) x
2
x
2
x
x
2
1 2
St = őh + őt = Ait^ÖTBŐt) ' ^. 12
(3 8)
2
From the factor A(t) it is seen that the return journey P -* P -> P of the subsequent signals changes in time. In particular if we consider a type of solution (38) we find that the fiight increases in time. If we take the time of return fiight as a measure of the distance between Px and P then we find thus that the distance between the points increases. The points P and P may be regarded as two masses floating freely. Thus Pi and P may be e.g. two distant stellar bodies. 419. If we suppose for the moment that the physical structure of a solid is such that the time of fiight of a return signal between two fixed points of the solid does not change in time then we can interpret the results of the previous paragraph by supposing that the measure of the distance between points P and P which float freely increases in time, when the measure ofthe distance is obtained with the help of solid measuring rods. x
2
x
2
2
1
2
2
x
The Friedmann solution can therefore be taken to imply that the ether as a whole expands and the bulk of matter floats with it. So as to see the physical significance of the above effect it is necessary to investigate the rate of a physical clock of usual construction in terms of the measures of the system of reference K where K(q) is given by (27), (28) and (37). " Considering an oscillating atom, a rotating wheel, or somé similar closed system an ordinary clock, we can suppose that the motion of the latter is given in a good ápproximation by the classical formuláé if we use an almost straight representation. So as to obtain such a representation we have to use a transformation described in 335 and Appendix II eq. (51a); we may put x' = x + F(x)
(39)
with 1
(3)
F(x) = x + - C x + . . . 2
From (27) and (28) it follows, however, that the non-vanishing components (3)
of C in a fixed point x = r , t are 0
(3)
0
0
(3)
C
=- C
kki
= A'(t )(A(t )
ikk
0
a
k= 1 , 2 , 3 .
From (39) it follows dt' <> + terms of order of (r - í ) and (r - r ) . (3)— - = 1 + C Since C dt = 0 we find dt' , = 1 + small terms. dt 3
z
444
0
0
444
(40)
Since the rate of a physical clock must be taken to be constant (apart from small terms) in an almost straight representation, we expect a clock to have a nearly constant rate in measures of /' and therefore according to (40) it must be taken to be nearly constant in terms of the originál measures t also. Thus measuring the return times of light signals between the points P and P we find the point P to recede if we measure the return time with the help of a physical clock of usual construction. 420. Differentiating (40) into the components of r we find X
2
2
17
x
17
x
+
\ W'oWoÜi'
- *°)
wé see thus that the measure of the distance between two points which have fixed coordinates relative to K increases when taken in the measures of K'. However, since K' is an almost straight system of reference, we expect the measures of a solid to remain almost constant when taken relative to K'. We see thus that the distance between two points P and P at rest relative to K increases when the distance is measured with the help of a solid rod. We see thus that in the configuration represented by the Friedmann solution the bulk of matter is streaming radially with the ether. The closed physical systems, however, which can be taken as the "atoms" of the mterstellar matter do riot expand. Thus the expansion can be taken as an increase of the distances between intergalactic systems relative to the dis tances measured on solids. Measures of stellar bodies which are kept together by mechanical inter atomic forces do not increase in time — or rather the increase of inter galactic distance must be taken as an increase relative to the dimensions of measuring rods kept together by interatomic forces. x
2
b. DOPPLE R E FFE CT
421. Consider an atom situated in a point P emitting radiation of frequency v. We suppose thus that wave fronts are emitted at times x
t„ = t + n/v
n = 0, 1, 2 , . . .
0
where the times are taken in the measures corresponding to the point Pi with coordinate vector The signals arrive in the point P with coordinate vector r = T + <5r in times 2
2
t + n/v + A\t„) 0
L
(SrBSrp'/c
+ terms of higher order thus we find n+l
2
/* + -(!+
lAXQAiQ^^lc )
+ terms of higher order and neglecting further terms of higher order, Av
2 A'A c
Av
V* — V.
8r
8r = (őrB őr) 1/2 1
The radiation of the atom situated in i \ arrives in P with a frequency v* < v. The above result describes the red shift of Nebulae. We see thus that the configuration obtained from Friedmann's solution of the equations of motion leads quaütatively to the Hubble effect. 422. The argument can be summarized by stating that the Friedmann solution gives such distribution of g(x) that the coordinates describing the vicinity of a fixed four-point x give almost straight representations. Thus describing the (small) vicinity of x„ by coordinates 2
0
x = x + § 0
then £ can be taken as measures of an almost straight system of reference. Describing a local process like the motion of a small solid, the rhythm of a clock, etc, we can apply— in a good ápproximation — in the vicinity of any point the laws valid for homogeneous regions. Observing with the help of clocks and measuring rods the behaviour of distant objects, we find that the distances of the latter increase in time; further we observe with a local clock the Doppler shift of the frequencies of distant objects just in accord with its apparent radial motion. It is important that the Friedmann solution gives thus automatically two suitable scales: one valid in the vicinity of fixed points and another which can be applied to large distances.
E. MACH'S PRINCIPLE 423. Before the formulation of the generál theory of relativity E. Mach formulated a principle which has somé connection with the theory of relativity. It was thus suggested that the forces of inertia (i.e. centrifugai forces) which appear in a rotating physical system Sí can be taken to be caused by the rotational motion of 21 relative to the systems of fixed stars surrounding 91. It was ventured by Mach that if instead of rotating the system 21 — "we were to rotate the whole of the universe around 21" — the interaction would remain the same and the system 21 surrounded by the large masses of stars rotating round could produce centrifugai forces of the same kind as are produced by the rotation of 21 relative to the surrounding masses. An experiment was attempted by rotating a system 21; a massive wheel was put around 21 and it was investigated whether a reduction of the centrifugai forces can be observed when the wheel was made to rotate with 21. No observable effect was found.
a. THE THIRRING EFFECT
424. It was shown by H. Thirring* that in terms of Einstein's gravitational equations inside a rotating spherical shell one expects forces which resemble to somé extent forces of inertia in a rotating system. Furthermore, it was shown by Thirring and Lense** that in the vicinity of a rotating sphere the gravitational field is affected by the rotation and it is suggested that anomalies of the orbits of the satellites of Jupiter (and possibly of Satura) might be expected to arise asthe result of this effect. The anomalies are, however, so small that they are on the limit of being observable and they have not been found so far by observation. The above effect is known in the literature as the Thirring effect. 425. While the gravitational field of a rotating body differs from a similar one which does not rotate, nevertheless, Mach's principle is not in agreement with the generál theory of relativity as was pointed out by Treder. The problem which arises with the interpretation of Mach's principle can be seen best if we remember that the physical state in a region 9t is given by both the tensor g (representing the state of the ether) and % (representing the distribution of matter and fields). Taking a representation K in which both the g
ki
=0
and
7^ = 0
k= 1,2,3
(41)
we describe a state of affairs where matter is at rest relative to the ether. Making the bulk of matter rotate, we obtain a configuration for which we have (in the originál representation) 8U = 0
T* *0 ki
k= 1,2,3.
(42)
(We indicate by the * the difference between the configurations.) The latter configuration differs physically from the former as was shown in detail in 410. Thus roughly speaking if we make matter rotate, but leave the ether free of rotation, we obtain a configuration which can be physically distinguished from the configuration where matter is free of rotation relative to the ether. 426. From the above considerations the following qualitative conclusions can be further derived. Consider a stationary configuration of the form (41) where gki(*) = g*/(r)
g,i(x) = 0.
* H. Thirring: Phys. ZS., 19, 33, 1918. (See also somé minor corrections: Phys. ZS., 22, 29, 1921.) ** J. Lense and H. Thirring: Phys. ZS., 19, 156, 1918.
If we replace T
ki
by, say,
and suppose that at t = 0 g*(r,0) = g(r)
then we can obtain g*(x) for / > 0 from Einstein's gravitational equation just as discussed in 393. We note that at * = 0 though g = g* because of the change of T we expect the second time derivatives &* and also the first time derivatives g£ to differ from zero. Because of the latter /44
u
(3)
(3)
we find that — even at t = 0 — the C* differ from the C and thus the equations of motion in the starred configurations differ from those in the originál configurations. The latter difference is connected with the Thirring effect. The difference between our remark here and the quoted calculations of Thirring is that in the latter stationary solutions of the gravitational equations were given, which can be found in the inside or the vicinity of a rotating sphere. These solutions were compared with those valid in the vicinity of the sphere free of rotation. We consider here — qualitatively — the dynamic of the process; i.e. we discuss the effect upon the surrounding, if a massive body is being put into rotation. It would be of interest to investigate whether as the result of rotation of the massive body the distribution of g(x) changes indeed into g*(x). As far as we know this question has not been investigated.
APPENDIX I
TENSOR ANALYSIS IN HOMOGENEOUS REGIONS
427. In this book we have used notations and conventions which deviate from the usual ones. We did this in order to simplify mathematical expressions but also we tried to find notations and conventions which are particularly adequate to express the physical concepts upon which we build up the theory. A. SYSTEMS OF REFERENCE 428. We consider a system of reference a means by which we can express events with the help of four-coordinates x = r, t. The orbit of a partiele relative to a system of reference K can be given in a paraméter representation, so that x(/>) = *(/>), t(p)
with
i(p) # 0.
The various values of p give the coordinate vector x{p) at times t(p) of the particle. Dot (•) denotes differentiation into p. 1. THE LORENTZ SYSTEM
429. A Lorentz system of reference is a system K in the measures of which the orbits of signals of light can be expressed by the relation 0
with
xri = 0
fl 0 0
0 1 0 0
(1) 0 0 1 0
^
0
1
0o -c )
lo where we have written x in place of x(p) (the four-coordinate the orbit of the signal).
2
expressing
2. STRAIGHT SYSTEMS OF REFERENCE
430. Submitting the coordinate measures of a Lorentz system K to a reversible linear transformation of coordinates we obtain a new system of reference K' with coordinate measures 0
x' = Sx + s.
(2)
Considering an orbit given by x = x(p) respectively x' = x'(/>) we find differentiating (2) into p x' = Sx.
(3)
x'g'x' = 0
(4)
From (1) and (3) we obtain with g' = s - r s - . 1
1
We denote a system of reference K' to be straight ifin terms of its measures the orbits of signals of light obey (4) where the mátrix g' has constant elements. We can add to the definition that we require g' to be symmetric and to have three positive eigenvalues and one negative eigenvalue. The latter requirement is fulfilled by JT. 3. PROPAGATION TENSOR 0
431. The mátrix g with the help of which we can express orbits of light signals may be called the propagation mátrix. Having systems of references K, K',.. . they can be characterized by the matrices g, g',. . . etc. expressing the propagation of light in their respective measures. We may write q for the propagation mátrix and K(Q) = g, tf'(8) = g',. . . .
(5a)
In particular for a Lorentz system of reference K for various straight representations we have *o(8) = r . (5b) 0
432. Consider two straight systems of references K and K' with coordinate measures x and x' connected by the reversible transformation x' = Sx + s.
(6)
The propagation of light is thus expressed in terms of K ox K' by xgx = 0
(a)
and
x'g'x'= 0
(b).
(7)
From (6) it follows that x' = Sx and thus if we require that the relations (a) and (b) given in (7) should follow mutually one from the other then we have to suppose g = ÖSg'S
0 > 0.
We shall suppose in the following that 9 = 1 and take that g transforms according to g = Sg'S . (8) 4. LORENTZ TRANSFORMATION
433. We denote a coordinate transformation a Lorentz transformation if it leads from a straight system of reference K with a propagation mátrix g to another K' with g' = g-
(9)
From (8) and (9) it follows that a transformation x' = Mx + u, is a Lorentz transformation provided M obeys MgM = g.
(10)
Matrices obeying (10) will be denoted Lorentz matrices. If K and K' are both Lorentz systems of reference then we have in place of (10) ATA = T
(11)
and we can call A an orthogonal representation of a Lorentz mátrix. 434. The Lorentz transformation can also be taken to define a Lorentz deformation. The connection between a system Cl and its Lorentz deform Cl* can be written symbolically 2 0 = O*.
Suppose Cl to consist of points ty k = 1 , 2 , . . . , n the orbits of which are given in a representation K by four-coordinates \ (p). The orbits of the points $*. of ö * are given by four-coordinates k
k
x* = Mx* +
(A
where the M are Lorentz matrices. We note that both x and x* are measures taken relative to the same system of reference K. We can also write tf(S) = M,u. i.e. the deformation 2 is represented by a Lorentz mátrix M and a constant four-vector u. 5. S T A N D A R D FORM OF LINEAR COORDINATE
TRANSFORMATIONS
435. For many purposes it is convenient to split space and time components of g. We use the notations G 8 ~ l< V7
V -c
/"2
where G is a symmetric positive definite mátrix, V a vector and C a scalar. The determinant of g can be written det g = - (C + V G ^ V ) det G; 2
we shall also write
c = y j - det g = J { C
2
+ VG- V)detG x
and consider c to be the measure of the velocity of light in K. We can split g into factors according to g = Sr<x;
(12)
relation (12) does not define o uniquely; however, if we require a
= 0
ik
k — 1, 2, 3
then we find a
_ jG
~|o
1/2
G " V1 1/2
\
(C + VG-V)^/J 2
1
(13)
where —c\ — T . As the result of a short calculation one finds also 44
_
x
"
_ ÍG~
1/2
~[0
-CoG-V/ÍC + VG" c /(C + V G ^ V ) ' 2
2
0
1
2
W '•
K
}
436. Supposing that representations of g in K and K ' are given as in eq. (5), the question arises as to the linear transformation (6) which leads from K to K ' . The homogeneous part of the linear transformation leading
from K -* K' is given by a mátrix S obeying (8). From (8) and (12) we see that the mátrix S = o'"^ satisfies (8) where a is built of the elements of g according to (13) and a ' is built of the elements of g according to (14). One verifies easily that apart from S also matrices - 1
/
S = a'- A<«>«
(15)
1
fl
satisfy (8) where A is an orthogonal representation of a Lorentz mátrix. Thus there exists a six-parameter manyfold of transformation matrices S„ which give coordinate transformations such that g changes into g'. 437. In particular if g = g then (15) gives the mátrix of a Lorentz transformation. We can thus write in accord with (10) ( , )
MW = a-W<úa.
(16)
Relation (16) gives the connection between the orthogonal and skew representations of the Lorentz matrices.
B. VECTORS AND TENSORS 438. It is useful to express measures in terms of quantities with particular transformation properties. Many — though not necessary all — physical quantities can be expressed in terms of tensors (the expression tensor is here taken so as to include also vectors and scalars). The simplest such quantity is the scalar. We denote a quantity a scalar if its numerical value is the same in all representations. Thus if a is a scalar, we have K(á) = K'{a) = ... = a. 439. We denote a four-component quantity a contravariant vector representation if its components transform like the measures of a four-dimensional distance. Thus we may write K(á) =
á,
K'(a) =
and á' = Sa
(17)
where S is the homogeneous part of the coordinate transformation K -*• K'. The sign upon a and a' denotes contravariant representation. We shall use in generál covariant representation K(á) = a,
K'(a) =
»',...
when by definition the covariant representation of a vector is transformed according to a' = S ^ a . In place of (18) we can also write a = Sa' = a'S.
(18) (18a)
From relation (8) we find Sg^S = g -
1
(19)
thus taking (18) and (19) together we find g - V = Sg^a;
(20)
comparing (17) and (20) we find further á = g a _ 1
and
á' = g ^ a '
thus the contravariant representation of a vector is obtained from the covariant by multiplication with g . We shall avoid in generál the use of contravariant representations with the exception of four-distances. - 1
1. TWO-DIMENSIONAL TENSORS
440. We denote a quantity 91 to be two-dimensional if its representations are given by matrices, i.e. #(21) = A where A has elements A v, p = 1, 2, 3, 4. The quantity 91 can be represented relative to systems K, K',.. . tfJL
K(H) = A, #'(91) = A',. . . the matrices A, A',. • • may be given in any arbitrary fashion. We take 91 to be a tensor if its representations transform like those of g, i.e. provided we have A = SA'S, (21) where x' = Sx + s defines the coordinate transformations from K -» K'. From the above definition it follows in particular that g is a two-dimensional tensor.
a. INVARIANT PRODUCTS
441.
From the relation
it follows that ag-'b = a Y - V (22) where a, b and a', b' are the representations of vectors a, b relative to K and K'. We shall use the notation a b = ag- b; (22a) 1
the latter quantity is in accord with (22) a scalar and it will be denoted the scalar product of a and b. 442. Similarly we can form the invariant product between a vector and a tensor. We shall write a-A = a g A = a
(23a)
A-a = A g " a = p .
(23b)
_ 1
and also x
We see immediately that the quantity a and also p defined by (23a, b) are vectors. Finally we denote the invariant product between two tensors A - B = Ag" B = C. 1
(24)
If A and B are tensors, then C defined by (24) is also one. 443. We can form also the direct product of two vectors, i.e. we write aob = A. The components of A are A
r(L
= aJ>
lí
v, u = 1,2, 3, 4
furthermore the vectorial product of two four-vectors can be defined as a x b = » a o b — boa; the vectorial product of two vectors gives an antisymmetric tensor A such that
b. PSEUDO SCALAR
444. The determinant förmed of the elements of a tensor is a one-component quantity; it is, however, not a scalar as the value of the determinant is different in different representations. Taking the determinant of the relation (21) we find det A = det A'(det S) . 2
(25)
Denoting thus det A = a , det A' = d we find 2
2
d = a/det S.
(26)
Quantities transforming in accord with (26) may be denoted pseudo scalars. Thus denoting a a pseudo scalar quantity, we can write for its representations K(a) = a, K'(a) = d, K"(a) = Ű", . . . The representations of a transform in accord with (26). Relation (26) gives more than (25) as it determines the signs of the representations a, d,a" . . . etc. if that of the representation a in one system of reference K is given. 445. We may write ; v
-detg = c
(27)
and take c as a measure of the velocity of light relative to K. In a L orentz system K where g = T the relation (27) gives indeed the measure of the velocity of light in the ordinary sense; the relation (27) is a reasonable definition valid in any straight system of reference. The sign of c can be fixed assuming that in a system of reference K we have c > 0 provided the space part of K is a right-handed system and the time coordinate is chosen so as to make the measure of time to increase with passing time. Using the definition (27) c is a pseudo scalar and any pseudo scalar can be written a = ac, where a is a scalar. 0
C. FIEL DS 446. We have considered so far tensor quantities defined in one fourpoint only. We can define a tensor quantity for all four-points — or at least for those of the points of a region 9t and we thus obtain a field.
The simplest field is a scalar field. Having a scalar a defined for all four points j , we have e.g. in a representation K a(x) = *(a). The transformation of the scalar field quantity is given by the relation a(x) = a'(x').
(28)
Taking the transformation formula of the fourcoordinates into considera tion, we can also write c(x) = o'ÍSx + s) or a'(x') = Ű ( S - V + s+) thus a(x) depends on x in a way different from how a'(x') depends on x'. However, in a fixed fourpoint j all representations of a have the same numerical value. 447. Similarly a vector or a tensor field can be defined as follows. Let us write #(21) = A(x) where A(x) is a fourvector defined for any value of x. We have thus X'(9l) = A'(x') with A'(x') = S ^ x ) ; we can also write A(x) = SA'(x') = SA'(Sx + s). A twodimensional tensor % can be represented by K{Z) = T(x) where T is a twodimensional mátrix. The transformation between represen tations can be written as T(x) = ST'(x')S and also T(x) = ST(Sx + s)S.
1. THE
31 OPE RATOR
a. THE GRAD OPE RATOR
448. Sometimes it is of interest to consider the derivatives of a field with respect to the components of x. We define an operator 91 with repre sentations
=•= oxy
dx
cx
2
3
*'(3Í) = • ' = - £ - , - / T ,
öXi
i -
tfJC
0*3
2
ÖX
4
We shall also write • and • ' if the differentiation is to be applied to the expression at the left of the operator. Using the formai rules of differentiation, we can write
J_ yÍíkJ_. =
dx
dx
v
v
(
29)
dx'p
Differentiating the transformation relation x' = Sx + s into the components of x we find 1 x ~ -
and thus in place of (29) we can also write • = §•' = •'$.
(30)
Comparing (30) with (18a) we see that the 31 operator transforms as a four vector. Applying • to both sides of (28) we find with the help of (30) •ű(x) = SD'a'(x'). Thus putting • a ( x ) = Grad a(x) • ' a'(x') = Grad' a'(x') we see that Grad a(x) = S Grad a'(x').
(31)
Thus Grad a(x) can be taken as the representation of a four-vector. The four-dimensional Grad operation produces a vector field from a scalar field. Sjmilarly we find • oA(x) «= Grad A(x) = S(Grad' A'(x'))S. We see thus that the Grad operator produces from a vector field a tensor field. b. FURTHER OPERATIONS
449.
We can define further
• x A(x) = • o A(x) - A(x) o • = Rot A(x)
(32)
• •A(x) = Q g - A ( x ) = DivA(x).
(33)
and 1
We see easily that Rot A(x) defined by (32) gives an antisymmetric tensor field, while Div A(x) a scalar field provided A(x) is a vector field. The Laplace operator can be taken as the combination of Grad and Div, thus we may write LA(x) = Div Grad A(x) = • • ( • < > A(x)) .
(34)
Similarly we have Grad Div A(x) = • o ( • • A(x)).
(3 5)
Taking the difference between (34) and (35) we find (Div Grad - Grad Div)A(x) = Div Rot A(x). We may also write LA(x) = Grad Div A(x) + Div Rot A(x) where A(x) is a four-vector. The above operations are valid only in straight representations as we have assumed S and also g to be independent of x.
APPENDIX II
TENSORS IN INHOMOGENEOUS REGIONS
A. MORE-DIMENSIONAL MEASURES 450. So far we have restricted ourselves to give a formalism of measures up to two dimensions. This we did so as to give a formalism which is sufficient for purposes of the special theory of relativity. Dealing with the generál theory, i.e. with problems of gravitation, we have to introduce measures that have more than two dimensions. The formalism we give below for such quantities seems to be adequate to our particular physical approach of the problems. In the formalism we shall try to avoid — as far as is practical — the explicit use of sufBxes and upper indices indicating components of moredimensional measures. Furthermore we shall restrict ourselves with few exceptions to covariant representations of tensors. At the same time, we shall consider in detail the symmetry properties of tensors and make use of permutation operators. 1. Ar-DIMENSIONAL MEASURES
451. We denote a quantity 21 to be £-dimensional, if it can be represented by a mátrix of /^-dimensions, i.e. we have
#(21) = A (*)
where A has elements • (*n (*> ' J v , . . . ^ = ^.',.,..., A
t
v v . . .v = 1,2,3,4. a
2
fc
The upper symbol (k) denotes the number of dimensions. k = 0 corresponds to a measure with one component only. We shall omit the upper symbol whenever this can be done without causing ambiguity. In particular we shall always omit it for k = 0, 1 and sometimes also for k = 2.
2. MULTIPLICATION OF MORE-DIMENSIONAL QUANTITIES
452.
(*>
O
We define the direct product of two matrices A and B as (*) (/) (/c+o AoB = C
so that (fc + 0 ^"l'f-'k
(fc) =
(/) -0)1,(1,.. .(!(•
AriVt...Vk
(1)
Another type of product can be defined so that we carry out summations of a pair of adjacent suffixes, thus we shall write
cf( + l) Í/ + 1) A
' here
B = C
(*+/)
4 (i,/i ...fi; t
=
ik + l)
0 + 1)
X o- =
A ,^
B
Vi t Vka
afítfíi
w
.
(2)
l
We shall also use summation over more than one pair of suffixes. Thus we 11 A Bj]=C write where the double bracket indicates double summation and in terms of elements we have <* + /)
+
0 + 2)
Similarly we may use a product where summation over n pairs of suffixes is to be taken. Since it is inconvenient to write down severalfold brackets, we shall write short {A
B J
= C
where the sign (n) over the bracket indicates summation over « pairs of suffixes. Explicitly we have
^ V * i '.
(k+l)
'k+n) tíliL2...M
=
X
0 + ") •Á-r r,...Pi ir «,...a 1
c
l
H
ü f W i.i (
^a ...a,<' n
l
O O V / .I. It is also useful to introduce invariant products in the following way (fc+i) (/+i) (fc+i) (í+i) A • B = A g" B 1
thus the fat dot • denotes that a factor g has to be taken between the factors, where g is the in verse of the propagation tensor. Similarly, we introduce - 1
- 1
Hk + m)
| A • B the above expression stands short for (* + /) ^IV,...! 1 * F»,JI,...|I/~
(fc + m) 'i-»'k
X a,...om TI...Tm
(k + l)
J
= C ; (l + m)
"i..-am8om*m " " '£
A
r,...z
B
m
HI.-.FI;
where g+ are the elements of the mátrix g . Furthermore we shall use a process which produces from a k + 2-dimensional mátrix a fc-dimensional one. We shall write - 1
11 (í(.k
or explicitly
+
YV
2)
(fc)
R g" ]] = R
(3)
1
(k+2)
(AT)
The above process can be generalized in the following way (fc)
/(fc + 2m)
R= 1 R
\(2m)
g(-""j
where we have written g "" = g " o g " . • • ° g (
,)
1
1
_ 1
M FACTORS
B. PERMUTATION OPERATORS 453. We may introduce the following definition. Writing P for a permutation operator, we have (fc) (fc> PA = B (fc)
(fc)
thus the operator P produces from A another mátrix B. The above operation written down explicitly in terms of elements is
(
(*)")
(*)
and P(V!V . . . v ) = uyfr. . .p. 2
fc
k
where / i j / ^ . . . ^ is the permutation of the elements V j V . . . v indicated by the symbol P. 454. The product of two operators P and Q can be taken as a new operator, i.e. PQ = R. (4) 2
k
R is defined so that provided (fc) (*) QA = B,
(*) (*) PB = C,
(5)
then by definition of R (*)
(*)
RA = C.
(6)
(P0A = PleAJ.
(7)
We can also write
Thus we introduce the product of the permutation operator so as to be associative in the sense of eq. (7). 455. We wish to draw attention to another aspect of the properties of multiplication of operators. The relation (5) can also be written down in terms of elements. We find í
ik)\
í (*n
(*)
(fc)
<*)
w
(8)
If we choose P(Hi lh •••Hk) = V! v . . . 2
then we can also write in place of (8) (*)
(*)
(*)
further with the help of (9) (*) (*) CVid—M* = AaiPUitii,...^kn
•
v
k
(9)
From (4) and (6) it follows, however, (fc)
(fc)
Comparing thus (10) and (11) we see that (fc)
ifc)
W =
^Ö[/V,.
.
)]
W
<.PQXIÍ,...^)1
A
=
we see that in generál (*)
(fc)
(fc)
thus the associative law cannot be taken to be valid when applying permutation operators to suffixes of more-dimensional quantities, as seen from the above argument. 1. CYCLIC PERMUTATIONS
456.
We shall make use of cyclic permutation, i.e. c = (1, 2
k).
k
The permutation c applied to k suffixes makes each suffix to shift by one step to the right, with the exception of the last suffix, which moves to the first place, thus k
c-*(viv... ) = v 2
Vk
k
. . . v_ k
v
The powers of c are operators c' which produce shifts of / steps to the right. In particular k
k
4=i.
We can also introduce negatíve powers, i.e. the operator c ' shifts the elements by / steps to the left. We have of course k
k'
c
= 4 '•
An operator c* can also be applied to a set of n > k suffixes. The significance of this operation is that the cyclic permutation is to be carried out with the first k suffixes only, the latter are to be left on their places. We have thus e.g. k
c (viv . k
2
. . vv k
k+1
• • • v„) = VfcVV j a. . . v*_!V* . . . v . +1
B
In particular the transposition P = (lm) produces the interchange of the suffixes on the /-th and on the m-th place.
2. THE TRANSPOSITION OF A MÁTRIX
457. The transposition of a fc-dimensional mátrix can be defined generalizing the usual form of transposition. For many purposes it appears to be suitable to use the definition (*)
(k)
A = cA k
thus written in terms of the elements í
(k)\
(*)
\ k^-)t r,...v
~
c
l
k
(*)
= A,
A,,
kV
Taking the product of two matrices we introduce summation on adjacent suffixes. Sometimes it is necessary to carry out the summation over somé arbitrary pairs of suffixes, i.e. we have to carry out products
(fc + 1)
ik + t)
^>»lf-»t f*l(l/.»W
=
X A, ,.
(í+1)
..,...V
tV
k
a
B
j......>
( tfl f (t/
where a stands between, say, v and v in the first factor and between u, and / Í / + I in the second factor. So as to avoid somé complicated notation for to write down such products we note that (k+i) ( (fc+in / (fc+m Av ...v av,j ...r = \ s+l A )trr,...v l fc + l( j+l A ))p,...v , similarly s
J + 1
=
C
l
s
thus we have
rl
k
C
C
k
ka
(/+1)
• (*I...(t|ff(l/ D
k+ +tt
+
í
<,k + i)\(
y + m
We find a generalization of how to form the transposition of products. Using the rules given above one finds that (*+i)(/+i) A B
r«+i)(fc+in { B A J
(13)
f(/+l)(fc + l)\ [ B • A j
(14)
and (*+l)(í+l) A • B =c'
k+l
In the case of A: = / = 1 we thus obtain the well-known rule for two-dimensional matrices.
3. THE TI, OPERATORS
458.
An operation which often occurs in application is the following
» i - q + cr - ...+(-íy-v'- '-'
•,
1
n
2
1
From the definition we obtain 1(1+
C,- )*,1
1 if / is odd 0 if / is even (3)
Applying e.g. the operator n to a three-dimensional mátrix g we find 3
(3)
(3)
(^g)^, =
(3)
(3)
glfu, - g^í
+ gvlti •
C. THE 9í OPERATOR IN CURVED REPRESENTATIONS 459.
We can define the 3Í operator represented by KQfl) =
8 OX-L
8 8x
•
8 8x
2
8 8x
3
t
The 91 operator produces from a fc-dimensional quantity a (A: + l)-dimensional one. We may write in a particular representation A(X)OQ=
A(x)
which is in terms of suffixes as follows (k) SA
(k + í)
In the following we shall, for the sake of brevity, omit the variable x. 460. Sometimes we have to apply the 9í operator to a product of matrices. It is useful to make use of the following formuláé, which can be verified easily with the help of the definitions given above (.A BJ o Q = A B + c ,_, (cí.íi A )Bj. k+
L
(16)
The above rule can also be extended to /n-fold summation; one finds (A
B )
o• = ( A
B
)
+ c {[c ^ k+l+l
k
m
+l
A
J B ) .
Further, if we apply the 9Í operator to an invariant product of matrices, then we have rtk) CM
(/c) (/ + 1)
ff
[ A - B J c . Q = A - B + c ,_ k+
1
(*) (3)\1 (0|
(ík + l)
iLct+i { A - A - g J J - B j
(17)
where (3)
g(x) = g(x) o
g.
(«
We shall apply the 9í operator to the quantities R introduced in 452 eq. (3) so we have ((lk + 2)
X\
(((
f(fc + 3)
(ft + 2)
(3)1)
~i\
1. TENSOR OF SEVERAL DIMENSIONS
461. The representations of a &-dimensional quantity 91 may be arbitrary /c-dimensional matrices, i.e. <*) #(9Í) = A,
w K'(9l) = A',. . .
We take 91 to be a tensor, if the connection between its representations is of a particular form. Generalizing the definitions of 440 we require in the case of a tensor that its representations are connected according to the following relation (*)
(*)
Mi —fje
where the S are the elements of the mátrix S giving the coordinate transformation x' = Sx + s, VJ1
leading from the system K to K'. Indeed, for k = 1, 2 (18) reduces to relations (18a) and (21) of Appendix I. 462. So as to Write down the above relation free of suffixes, we intro(D duce a 2&-dimensional mátrix, which we shall denote by S with elements (!)
The transformation (18) can thus be written (*) f(f)(*>y*) A = (SA'J .
The reverse transformation can be written as (fc)
'/•()+ (fc)Vfc)
A' = [ S
with
AJ
(D v r ...r í fí _ ...ii
Ll
1 t
kl k
k
l
"Wi^t!,*,
1
• •
•" íít»f J
463. Considering a quantity 91 which can be represented by fc-dimensional matrices (*)
(fc)
K(W) = A, K'QX) = A',. . . (fc) (fc)
etc. where the A, A' are defined somehow, then 21 is in generál not a tensor. (fc)
We can form from the representation A" a quantity (fc) (O) (fc)\(*) A = 1SA'J . (fc)
(*)
If A = A then 21 is a tensor, the difference (fc) (fc) A-A gives an indication of the deviation between 21 and a tensor quantity. 464. The direct product of two tensors 21, 23 built according to 452 eq. (1) is itself a tensor, as can be verined in terms of the definitions. The product (fc + l) (/+l) (fc + O A B = C (fc+D
(/+»
is not an invariant one; even if A and B represent tensor quantities, (fc+/) C defined by (2) is not a tensor. Furthermore one verifies easily that the invariant product of two tensors is a tensor and the process, as defined (fc + 2)
(fc)
by (3), produces from a tensor R another tensor R. 2. SYMMETRY PROPERTIES
465. The permutation operators are invariant operators in the sense that the relation (fc)
(fc)
PA = B
(20)
if valid in one representation K, then it is also valid in any other representation K'. We can thus write in place of (20) also P9t = 33. If a tensor is such that for a given permutation P we have i>2I = 2l then we can take that 21 is symmetric with respect to P. We may also write P s l mod 21 where we denote by 1 the identical permutation operator. 466. There exist tensors so that in any representation we find W (fc) (/m)A = - A : (*)
we denote A to be antisymmetric with respect to the /-th and m-th suffix. In particular a tensor which is antisymmetric in all pairs of suffixes may be denoted totally antisymmetric. An important example is given below. (4)
3. THE ANTISYMMETRIC TENSOR e
467.
(4)
Consider a four-dimensional mátrix e with elements (4)
c if xXfiv is an even permutation of 1, 2, 3, 4 — c if xA/zv is an odd permutation of 1, 2, 3, 4 0
(21)
otherwise
where c = ^J— det g. Transforming e like a four-dimensional tensor, we obtain (4)
am w\a)
e' = ( s
+
ej
.
(22)
From (21) and (22) together with g = Sg'S we find (4) (4) (4) e' = e/det S = ec'/c. (4) The components of e' are obtained by the relation (21) if c is replaced by c'. We see thus that the definition (21) gives the representation of the elements of a tensor.
468.
The operation (4)
(2m (2) TjJ = T
(2) (2) (2) produces from a two-dimensional tensor T another T. T is always anti(2) (2) (2) symmetric. If T is antisymmetric, then T is obtained from T by a certain mode of interchange of the space and time components, as can be seen easily when writing down the relation (23) explicitly. Applying e.g. the above operation to the field strength tensor, we find / F =
0
-B cE{\ Bx cE 0 cE -Bi -cE —cE 0 J B 0
2
3
B \-cEr
2
2
3
2
3
/
0 E -E 0 F = E -Ex V cBx cB 3
-E -cBx\ Ex -cB • (24) 0 -cB cB 0 / 2
2
2
3
2
3
D. TENSOR FIELDS (fc)
469. A A>dimensional field is defined by a mátrix A(x). The field will be denoted a tensor field, if its elements transform suitably, when we change the representation. Consider thus a reversible coordinate transformation x' = f(x). We denote 21 a tensor field, if we have (fc)
#(21) = A(x), with
(fc)
#'021) = A'(x')
Ufc) (fc) («) (fc) M A(x) = [S(x)A'(x')J (!)
where S(x) is built of the elements of the mátrix S(x) = f(x)oQ (t)
just in the manner as S is built of S in 462 eq. (19). 470. We may also introduce a quantity (fc) ni) (fc) y
and we can say that 91 is a tensor field if we find that tt) (*) A(x) = A(x). (*)
We shall apply the 92 operator to the quantities A so we have (fc) Al) (*) \(k) Aon = (SA'j orj.
(25)
So as to obtain (25) in a more convenient form, we note that the right-hand expression represents a sum each of the terms of which being the derivative of a product containing A: + 1 factors. The differentiation • gives thus (« k + 1 terms. One of the terms contains the derivative of A' into the variable x. Making use of the rule
• = 5's we can write (*)
A'oQ=
(*+«
A' S ,
therefore we find for the term obtained by differentiating A'
(.sU'oSJJ
=[stA'SJ|
= [ S
A'j
= A
.
The terms obtained from differentiating the factors of the elements of (i) S give contributions of the form
c * 4 U
A Js-'s].
Taking the k + 1 terms together we find (*) (fc+T) k ív (3)i A o Q = A +X^'LUAJS-1SJ.
(26)
1=1
In place of (26) we can also write (* + l) (*+!) A - A
A*)
W\
= l A - A J o n +
k
U
(k)\
(3)]
E^'IUAJS^S}.
(27)
(fc)
(fc)
(fc)
We note that provided A is a tensor, we have A = A and thus the righthand expression becomes simpler, however (*+i) (fc) A = AoQ is in generál not a tensor, since the sum on the right of (27) as a rule does not vanish. 1. THE CHRISTOFFEL BRACKET SYMBOLS
471. We apply (27) to the tensor g(x). Remembering that (12)g = g we find (3)
g (3)
(3)
(3)\
(
(3)
g O + Í U f U ls ^ s J .
(28)
Since S = (23) S we can write in place of (12) also (12) = (12)(23) = c thus we have (3)
í
(3)
(3>-\
g - g = (I + c - ) l g S - S J . 1
(29)
1
3
Applying the operator 713/2 of (15)
to both sides of (29) we find with the help (3)
(3)
(3)
gS- S = C - C
(30)
1
with (3)
_1
3
1
(3)
1
(3)
0)
C = — n g and C = y 7 c g .
From (31) the reversed relation
3
3
(3)
(31)
(3)
g - O + c ^ C
(32)
can be obtained. In place of (30) we can also write (3)
/-(3)
ü(h
S = S • (C - C J .
(33)
(3)
472. The three-dimensional mátrix C defined by (31) is the so-called Christoffel bracket symbol. The usual notation of the Christoffel symbol is:
In the literature one uses also expressions
where the latter expressions are elements of the mátrix (3)
(3)
e=
- c. 1
g
From the definition (31) it follows that the Christoffel symbols are symmetric with respect to the second and third suffixes. One finds (3)
(3)
(23)C = C . The Christoffel symbols represent three-dimensional field quantities — but these fields are not of tensor character as can be seen. 2. THE COVARIANT DIFFERENTIATION
473. The 91 operator applied to a scalar field produces a vector field. Indeed, applying to the relation Í(X) = Í'(X')
the operation
• • s ( x ) = Ds'(x') = S(x)(rjy(x')).
Writing thus • s ( x ) = Grad Í ( X ) we see that Grad s(\) transforms like a vector, i.e. Grad s(x) = S(x) Grad' s'(x'). We can thus write symbolically ©rab 3 = 81.
Applying the 9Í operator to vector or tensor fields we do not obtain a tensor field, because the differentiation produces also derivatives of the transformation mátrix S. The process of differentiation which leads to invariant fields is considered in more detail below. 474. Writing down (27) for a tensor field 21, i.e. supposing that 21 is <*o (*) (« represented by A such that A = A we find +
(A + i)
A - A
k
[(
'k)\
(3)1
= I^-'LUAJS^SJ.
(3) • (••••••• (3) (3) We can express S S in terms of the Christoffel symbols C and C. Writing the quantities without bar on the left and with bar on the right, we obtain thus the following relation (fc + 1) k [f (k)\ (3)1 (fc + 1) * [Y (fc)\ Öl] - 1
A - Ic -'LUAJ-Cj= A - X^'LUAJ-CJ. k
/=l
•••
-
7=1
Introducing thus a Grad operator as (fc)
(fc)
fc
17
(fc)~\ (3)1
IVLUAJ'-cI,.
GradA = A ö Q -
(34)
(fc)
we find that Grad A is a tensor of fc + 1 dimensions. Applying (34) to a scalar the sum contains no terms and has to be omitted. In the case of a vector we have one term only and find thus (3) GradA = A o p - A - C . (35) 475. If we apply the Grad operator to an invariant product of tensors, then we have A*) (0)
(fc)
(0
Grad \ A • Bj = A • Grad B + c ^ k+l
1
u (*n wi \{c li Grad A j • Bj . k
(*)
In the case we apply the Grad operator to the quantities R introduced in 4 5 2 eq. (3) we have Grad
(C'R 'g- )) = c
(({<#, G r a / Í * }
1
k+1
g" )) . 1
(36)
476, We can define also the covariant form of the Div operator. We may write DivA^ttJGradAJg- )).
(37)
1
It is useful to make use of the following formuláé; which caii be verified easily with the help of (36) and the definitions given above: /Y(*+2)
r\
Div H R g- )J = l k 1
or
Div
í
tó
(* 12)| |
Grad R ( j g H
( ( T g - ) ) = (U+1 \c U Grad 1
k
(K
R '}]
y«)
,
(38)
g^'f'-
The Laplace operator can be taken as (fc)
(fc)
LA = Div Grad A .
(39)
(3)
(3)
G r a d g = g - ( l + (12))g-C = 0 , thus we have identically Grad g = 0
and also
Div g = 0.
(40)
E. CRITERIA FOR HOMOGENEOUS REGIONS 477. A region 91 is taken to be homogeneous, if it admits of representations of Q g' = K'(q) = independent of x ' .
(41)
Suppose g(x) = *(8) to be the representation of g in an arbitrary system of reference K. If region 9t is a homogeneous one, then it exists as a coordinate transformation x' = f(x),
(42)
so that S W S Í X ) = g(x)
(43)
S(x) = f ( x ) c . g .
(44)
with (s)
(?)
From (41) it follows that C'(x') = 0 and therefore also C(x) = 0, and we may write using relation (33) (3) (3) S(x) = S(x)-C(x). (45) Relation (45) is a differential equation with the help of which S(x) and also f(x) can be determined if the region 9í is homogeneous indeed. Whether or not (45) possesses solutions can be ascertained by differentiating (45) into x. We write (3) (4) S o g = S
where we omit to write down the variable x explicitly. We find with the help of (17) (4) (4) r cm W\ (s)\ S = S-C-t-c (c - lS-S-gj-Cj, 1
4
s
(3)
(3)
Inserting S from (45) and g from (32) we find (4) (4) (4) r / (3)) (3)1 S = S-C-C4 cí l - 3~ CJ-Cj. 1
s
c
1
L
(3)
Using the symmetry of C we can simplify the second term and find as the result of a short calculation S = S-ÍC-(24)(C-CJJ.
(46)
(4) If S is to be the third derivative of the transformation function f(x), then it has to be symmetric in the last three suffixes; we expect thus e.g. (4)
( 1 - ( 2 4 ) ) S = 0.
(47)
(4) Since it follows from (45) that S is automatícally symmetric in (23) it follows also that provided (47) is fulfilled it is also symmetric in (34) since we have (34) = (23X24X23). (4) Thus provided (47) is fulfilled, then S is symmetric in the three last suffixes. Inserting (46) into (47) we find (W
(3) (3))
( 1 - ( 2 4 ) ) | C + C - C | = 0. We can also write (4) 1 (4) 1 (*) (1 - (24)) C = - (1 - (24))** = - 2 * 4 * . Defining thus a four-suffix quantity (4) 1 (4) /S) (3) ^ R(x) = y * g(x) + (1 - (24)) (C(x)• C(x)j, 4
(48)
we see that a necessary condition for a region 91 to be homogeneous is that (4) R(x) = 0 inside 91. (49) Indeed, unless (49) is fulfilled the differential equation (45) does not admit of solutions.
478.
The argument can also be reversed. Differentiating (45) successively (2 + /)
into x we obtain a recursion formula for the derivatives S of S(x). Expanding S(x) or f(x) into powers of x using the derivatives thus obtained we obtain a solution of (43)-(44), provided (49) is fulfilled. We see therefore that (49) is a necessary and also sufficient condition for a region ÍR'to be homogeneous. 1. ALMOST STRAIGHT RE PRE SE NTATIONS
479. In an inhomogeneous region the system (43) and (44) admits of no solutions; in such a region one can try to introduce almost straight coordi nates, i.e. one may try to find a.representation K' such that gf(x') is almost constant. For any value of g(x) we are led to (45), giving a differential equa tion which, if it admits of solution, leads to the transformation (42) defin (*>
ing straight coordinates. Differentiating (45) one is led to (46); if S thus obtained is symmetric in the last three suffixes, then (43) and (44) admits of solution. (4)
The mátrix S as obtained from (46) is in any case symmetric with respect to (23); we can introduce a symmetric tensor (4)
(4)
1
< S > = ( l + ' ( 2 4 ) + (34))S. • T
(4)
(4)
The latter is symmetric in all the three suffixes. Since < S > is equal to S (4)
in a homogeneous region, we can take <S> to define an approximate solution of (43) and (44). The difference between the two solutions can be written (4)
(4)
(4)
(4)
1
ŐS = <S> S = j [(1 - (24)) + (1 (34))] S , however (1 - (34)) = (23)(1 - (24))(23) = (23)(1 - (24)), therefore we can also write (4)
1
(
(4)1
SS= - y ( l + ( 2 3 ) ) ( s R | .
(50)
The function <S(x)>, giving the approximate solution, can be obtained by integration; we find (3)
•*•(*)
<S(x)> = S(0) + S(0)x + J «S(x')> (x x') o dxj*
(51)
o
or 1 (3) í r (4)
3
0
(the integration can be taken on any path from 0 to x); the above trans formation function satisfies thus approximately (43) and (44) in the vicinity of x = 0. 480. We may write g(x') = g' + őg(x') for the representation of g in the almost straight systems of reference. We have thus <S(x)>(g' + <5g'(x'))<S(x)> = g(x).
(52)
Differentiating (52) into x we find for x = 0 (3)
W ) = 0.
(53)
Differentiating twice with the help of (51) and (53) we find for x = 0 (
U)\
(?)
1
( l + (12))lgS ŐSJ = ő g , therefore with the help of (52) remembering that (4)
(1 + (12))(23) = 0
mod R ,
we find (4)
1
(4)
ág=3 (1+(12))R and transforming to K' (4)
1
(4)
<5g' = ( l + (12))R' T
thus we have 1 (« g'(x') = g' + - (1 + (12))R'x' + . . . 2
.
The above representation is as straight as possible, since the second derivative of gfíx') must be at least of the order of the elements of R'. 2. TENSOR CHARACTER OF THE RIEMANN-CHRISTOFFEL TENSOR (4)
481. The quantity R(x) defined by (48) is a tensor; apart from a change in convention it is equal to the Riemann-Christoffel tensor. One finds (4)
(4)
át(x) = - ( 2 3 ) R ( x ) , (4)
where <St(x) is the usual form of the tensor. We prove the tensor character (4)
of R using our formalism. We start from a relation which is obtained by applying the operator • to g S S = g; with the help of (16) one finds thus - 1
<í HgS" o • ) = [c,- (g - g S " ? ) ] S" . 1
1
1
(54)
1
Applying • to the relation (30) one finds with the help of (16) and (54) f<3)
ff
(4)
(3)\
(C - Cj c-n = g S ^ S + c ( k
(3ni
f(3) 1
k
(3)1
[g - g S ^ S j J S ^ S J .
With the help of (30) and (32) one can rewrite the second term on the righthand side and find fí(3)
c {---} = c (12)|lc + 1
4
(3V|
((3)
a>\\
CJ-lc-Cjj.
(3)
As the C are symmetric in (23) the expression in the { } is symmetric in (34) and one can write for the operator before the bracket c (12) s (1234)(12)(34) = (24). 4
We have thus A3)
ÖA
(4)
f/O)
(3)1
/(3)
<3)\)
[C - Cj o Q = g S ^ S + (24) {(C + Cj • (C - Cj} .
(55)
(3)
Applying (27) to C one finds (4)
(4)
OA
/(3)
(3) f(3)
<3)\
C-C = [c-CjoQ + C-[C-Cj+ + ca"
1
3
f(S) ((3) L
\í
(3)11
(in my
(imi
C• |c - Cjj + c I k ' c J • IC - C j j . 3
(56)
With the help of (14) we find further (3) /(3)
r/(3)
OÁ
C-{C-CJ
(3A (3)1
= cl[{C-Cj-c\
(57)
and C (3ft f(3) (Sn fffl) (3)1 (3)1 (c - cJ-lC-Cj = c Llc-Cj-Cj. (58) The expression in the []-bracketof (57) is syrnrnetric with respect to (34) therefore we have 1
3
2
o n éöi
|7(3)
c - c [] = ( 2 4 ) [ i C - C j - c J . 1
2
3
(59)
Similarly, the expression in the []-bracket of (58) is symmetric with respect to (12) thus we find cA
=
(14).
We note further ~ f7(8)
m (3)"V (3)1
(1 - (24)(14)) Lic - CJ • Cj = 0 .
(60)
(4)
So as to obtain expressions of the form of R we note that
(4)
1 (4)
( 1 - ( 2 4 ) ) C = -n&.
(61)
Thus applying ( l - ( 2 4 ) ) to both sides of (56) we obtain with the help of (47), (55), (59), (60) and (61) the following relation 1 y
ao
*i [ g -
(7n f ((3) cm (<$> ö n 8 i = (1 - (24)) | - [C + Cj • (C - Cj + (3) H3)
(3)1
/(3)
(3)1 (3)1
+ c-(c-cj-lc-cj-c|.
The terms in the { }-bracket reduce to (3) (3)
(3) (3)
{} = - C - C + C - C and therefore we find 1 2
(«
(<S) OÁ 1 (4) (3) (3) + (! ~ (24))lC-cJ = y n a + (1 - (24))C-C,
or using the notation (48) (4) (4) R=R (4)
thus in accord with 463 R is a tensor. (4) 3 . SYMMETRIES OF THE R TENSOR
482.
We note that (3) (3)
(13)(24) = 1
modC-C
thus (1 - (24)) = i (1 - (24)) (1 + (13X24)) = (T)
y
(1 - (24)) (1 - (13)) = tt
4
(3)
mod C-C and we find in place of (48) also (4) 1 /tt) (3) (3A R = - ( 1 - ( 2 4 ) ) ( 1 - ( 1 3 ) ) U + C-Cj or (4)
]
(62)
(3) (3)\
((4)
R = y JTU + C-Cj. 4
(63)
(62) and (63) are, of course, identical with (48). From the above relations the (4)
symmetries of R can be obtained in a simple manner. From (62) we see immediately that (4)
so
(4)
(4)
(24)R = (13)R = - R (4)
(64a)
(4)
(13)(24)R = R.
(64b)
Further, remembering that according to the definitions (4)
Q7t = -7i 4
(4)
(1234)R = - R .
4
(64c)
Multiplying (63) from the left by (12)(34)(1234) = (13), and remembering (64a) and (64c) we find (4)
(4)
(12)(34)R = R,
(64d)
thus (4)
(12) s (34)
mod R.
Furthermore one finds as the result of a simple calculation that (1 + c + c | K = 0
(modg)
3
therefore (4)
(1 + c , + 4)R = 0 .
(65) (4)
Relations (64a) and (64d) give all the symmetries of R. (4)
483. Taking the symmetry properties of R into consideration, the number of linearly independent elements can be derived. Firstly we note (4)
that because of (64a) elements R x
c a n
o n r
VjlM
v #
x,
y be different from zero if
it # L
(66)
There exist 21 configurations of suffixes obeying (66) and there exist 21 types of components, which do not vanish identically. Indeed, there exist six pairs of suffixes v T 6 K and fifteen quadruplets obeying (66) with v # ii and six with v = n , where we count those configurations only once which give the same absolute value for the element R . We can thus arrange the eleVílxX
(4)
ments of R into twenty-one groups, each group containing (not identically vanishing) elements with equal absolute values. Relation (65) gives a linear connection between three different elements (4)
of R. Because of (64a) this relation will be fulfilled identically for elements (4)
of R expecting those with four different suffixes. For the latter elements (65) gives the relation (4)
(4)
(4)
•^1234 + ^3124 + -^2314
=
0>
which is independent of (64a). The elements occurring in this relation connect three out of the twenty-one groups of elements discussed above. We see thus that out of the 21 groups given above only 20 contain independent elements. Thus the Riemann-Christoffel tensor possesses altogether 20 nontrivial independent elements. (4) 4. THE R E D U C E D FORM OF THE R TENSOR (4)
484. The Riemann-Christoffel tensor R can be contracted and thus we obtain the following tensor quantities
R = ((Rg-i)) = ( ( g - S ) ) .
(67)
(2)
The contraction leads to the same tensor R whether we multiply from the left or from the right by g . Further we have - 1
J ? = ( ( g - R ) ) = (g(- )R) 1
)
2
(4>
where we write g = g og . From (67) we find with the help of (38) ( _ 2 )
(2)
- 1
([
_ 1
\&)
(4XU2)
Div R . [ l g - grad RJ 1
g" ] 1
(
(
(tí\\(i)
= (g<- )[(35) grad RJJ 2
.
(68)
We find also í
(4)1(4)
Div (gR) = Grad R = [g<" > grad Rj . (69) We can reduce (69) to a simple form. With the help of a simple consideration one finds that 2
i
(4)
(5)
Grad R = - (1 - (24) - (13) + (13)(24)) ,
(70)
Y
where
(«)
= ( l + ( 3 4 ) + (35)) It is verified easily that Y
f 1 (S) T
® (4)
(55 (3; (4) (3)
(3) (3) (3)1
g + C-g-(13)g-C-g-C-2C-C-Cj.
(12) = (34) = (35) = 1
(5)
mody.
(71)
Introducing (70) into (69) we can simplify the expression thus obtained remembering that g<- has the following symmetries 2)
(12) = (34) = (13)(24) = 1
modg<- . 2)
(72)
(4)
Therefore we can apply any of the above operators to Grad R — or to any (4)
terms of Grad R without changing the value of the sum. Introducing thus (70) into (69) we obtain four terms; we find that the first and fourth are equal, since í
(sn(4)
(
(
(5m(4)
U^YJ =lg - l(13)(24) jj . We find also that the second and third terms are equal. Indeed, (
2)
Y
(
(
í
«m(4)
(
(73)
<5A(4)
=U- 1(24)YJ
U- 1(13)YJJ (2)
(2)
thus with the help of (69), (70) and (73) we find Grad R = 2 (g<- > ( ( 1 - ( 2 4 ) ) Y ) ) . 2
(74)
W
Similarly, introducing (70) into (68) we find making use of (71) also (
(
<4)Y\(4)
(g<- > 1(35) Grad RjJ 2
<
( 2
Y
- ( g " ((35)(1 - (24))(13) Y ) ) (
(5)U(4)
= lg<- > 1(1 - (24)) jJ
2)
M )
.
(75)
The second term on the right vanishes as can be seen in the following manner. The operator inside the bracket can also be written (5) ( 3 5 ) ( 1 - (24))(13) = (35)(1 - (24))(I3)(35) = (15) - (24)(15) mod Y -
However, under the summation we can replace (24)(15) by (13)(15), further (13)(15) = (13X15X35) = (15). Thus the second term under the sum cancels the first; thus the second term on the right of (75) can be omitted. We find therefore comparing (74) and (75) (2) 1 Div R = y Grad R, and also
/(2) i DivlR-yg*
=0.
1 B2/40 00001492 Fiz. Kvt.
This analysis shows, among others, the experimentál significance of the Michelson-Morley experiment and why it was necessary to use the well-known experimentál arrangement in the form it was actually used. The author has written many articles and gave many lectures both in Hungary and abroad elaborating his views which are deviating in certain respects from those usually put forward. The concepts and modes of treatment have given rise to discussions and aroused great interest. With the new concepts the author's aim is to induce constructive discussion but at the same time his book may sérve as a handbook for physicists, scientists and philosophers interested in the problems. It is an interesting reading and it could also be used as a university textbook.
AKADÉMIAI
KIADÓ
Publishing House of the Hungárián Academy of Sciences Budapest Distributors: K U L T Ú R A Budapest 62. P.O.B. 149
L.
Jánossy
THEORY OF RELATIVITY BASED O N PHYSICAL REALITY The book deals with the subject of the special and generál theory of relativity. It gives the usual mathematical formalism elaborated by Einstein and others. The treatment of the book places great emphasis on the question of how the theory can be derived from the experimentál facts and this is the feature in which the treatment deviates to somé extent from the usual ones. There is also a tendency to make use as little as possible of specialized systems of coordinates. Before writing this monograph the author was engaged for several years in fundamental research concerned with the double nature of light. These researches contained intricate experiments and raised questions which are of interest in connection with somé of the fundamental problems of the theory of relativity, too. The author analyses in great detail the methods of how the velocity of light can be experimentally determined. The controversial question of the carrier of light is also dealt with.