The Relativity Theory o f Plane Waves.

Transcription

1 95 The Relativity Theory o f Plane Waves. By 0. B. Baldwin and G. B. J effery. (Communicated by Prof. L. N. G. Eilon, F.R.S. Received January 15, 1926.) Weyl* has shown that any gravitational wave of small amplitude may be regarded as the result of the superposition of waves of three types, viz. : (i) longitudinal-longitudinal; (ii) longitudinal-transverse ; (iii) transverse-transverse. Eddingtonf carried the matter much further by showing that waves of the first two types are spurious ; they are merely sinuosities in the coordinate system, and they disappear on the adoption of an appropriate co-ordinate system. The only physically significant waves are transversetransverse waves, and these are propagated with the velocity of light. He further considers electromagnetic waves and identifies light with a particular type of transverse-transverse wave. There is, however, a difficulty about the solution as left by Eddington. In its gravitational aspect light is not periodic. The gravitational potentials contain, in addition to periodic terms, an aperiodicterm which increases without limit and which seems to indicate that light cannot be propagated indefinitely either in space or time. This is, of course, explained by noting that the propagation of light implies a transfer of energy, and that the consequent change in the distribution of energy will be reflected in a cumulative change in the gravitational field. But, if light cannot be propagated indefinitely, the fact itself is important, whatever be its explanation, for the propagation of light over very great distances is one of the primary facts which the relativity theory or any like theory must meet. In endeavouring to throw further light on this question, it seemed desirable to avoid the assumption that the amplitudes of the waves are sm all; terms neglected on this ground might well have a cumulative effect. All the solutions discussed in this paper are exact. When the amplitudes are not small it is no longer true that any wave may be resolved into waves of Weyl s three types. We may nevertheless discuss these types as important particular cases. In the earlier part of this paper we show that Eddington s results are still true for waves of finite amplitude ; longitudinal-longitudinal and longitudinal-transverse waves are spurious, and transverse-transverse waves are propagated with the velocity of light. * 4Itaum, Zeit, Materie, 4th edition, p. 228 ; English edition, p t Roy. Soc. Proc., A., vol. 102, p. 268 (1922).

2 96 O. B. Baldwin and G. B. Jeffery. In 5 the theory of the propagation of plane transverse-transverse waves is shown to depend on the solution of a single differential equation with five dependent variables. Since we deal only with particular solutions, it is not possible to state the result with certainty, but a strong presumption is created that an infinite plane electromagnetic wave (or pulse) cannot be propagated without change of the wave-form. There is little doubt but that this result arises from the infinite character of the wave-front, and it seems that the fuller working out of the relativity theory of light must come through the study of divergent waves. 2. The Field Equations. The equations to be satisfied in space devoid of matter and charge are (1) = 8flyEM>,, where GrM is the contracted Riemann-Christoffel tensor, y is the constant of gravitation, and EM is the electromagnetic energy tensor; together with the electromagnetic equations (2) ^ { W < - 4 = 0, where FM*' is the electromagnetic force tensor. If BMV is the Riemann-Christoffel tensor, we have (*^) 9pB/i» (ppav) and (4) = BM.a = (ppav). The Christoffel four-index symbols are given by (5) (wov) = i { 3fe-+ p e - - p f I V ) L oxpcxa-ox^gxy oxpoxy oxmdx<rj cf? [pa, a] [pv, p] + a/3 [pv, a] [pa, p], in which the Christoffel three-index symbols of the first kind are given bv [pa, a] = fisa* + ^ \ OX,, If /emis the electromagnetic potential vector dxa, (7) v / ± Tp ~, CXy dxp and the electromagnetic energy tensor is defined by (8) E / = - V +

3 Relat ivity Theory o f Plane Waves. 97 For plane waves propagated with velocity V in the negative direction of xx we assume that the components of all tensors are functions only of = x q_ Vx4. The electromagnetic equations (2) then give three independent equations which a t once integrate to show that (F14, F21 + VF24, F31 + VF34) ( g) are constants. These are components of a tensor density defining the force per unit mesh on charge moving with velocity V with the waves. By the superposition of an appropriate static homogeneous field these constants can be made to be zero.* We accordingly take (9) F14 = 0, (10) F21 + VF24 = 0, (11) F31 + VF34 = 0. Denoting differentiations with respect to by accents, and writing (12) *4' = kq we have (13) Fm = 0, K f *o' V e Using these values, we have *2 * s> 0, 0, - V *2' 0, 0, -V * 8 ' Ko > V*2, V*8#, 0. -i Fn(SF ^ = F41 + *2' {F21 + VF24} + < {F31 + VF34} = 0 by (9) (11). Hence (8) reduces to (14) E / = - F mof - 3. Longitudinal-Longitudinal Waves. For these waves (15) 91V 0, 0, , - 1, 0, 0. 0, 0, - 1, " 0, 0, 9w * A similar situation arises in the ordinary wave theory of light, when the integration of Maxwell s equations gives in the first instance that Ex, H, + VE;/, Hj, - VE are constants. These are taken as zero for ordinary light, non-zero values of the constants arising in the theory of the Zeeman and Stark effects. VOL. CXI. A.

4 98 O. R. Baldwin and G. B. Jeffery. The determinant of (16) and (17) (hi 9w o, 0, o, 9> 0, 0 0, 0, 9> 0 9w 0, o, 9 The three index symbols of the first kind vanish unless each index is either 1 or 4, and (18) [11, 4] = gu '- 1 [14, 1] = V [11, 1] = V W 5 [44, 4] = V (14, 4] = \ V#44'. Prom (5) we see that the four index symbols vanish unless each index is either 1 or 4, and, owing to their anti-symmetry, the only independent surviving symbol is (19) (1414) = { ^ 7 p ^ ) ( V 2!/ 11-2 V s,i4 + <, )}. Prom (4) we have (20) Gu = - g (1414), G14 = the remaining components of GM vanishing. Prom (13) and (17) we obtain 9 F'J 0, (gu Y g u W, ((h i V g u W, (gu Ygu ) «2', 0, 0, ( g u Y g (gu v</u) ky, 0, 0, (gu Ygn)* Ko, (gu Ygn)** (gu Yg 0. Substituting these values in (9) (11), we have either (i) Kn 0? or (ii) /c0' == 0 and gu 2Y-f- Y 2gix 0 = 0, and therefore GM = 0. Hence from (20), (1414) = 0. In the second case (1414) = 0 by (19). It follows that for any longitudinal-longitudinal wave all the components of the Riemann-Christoffel tensor vanish. Accordingly, such waves are spurious in the sense that they disappear on the application of an appropriate transformation of co-ordinates.

5 Relativity Theory o f Plane Waves. 99 In this case 4. Longitudinal-Transverse Waves. - 1, 9x21 9x21 0 9x2-1, 0, 92X 9xz;> 0, - 1, 92X 0, 92X1 92x1 1 The contravariant ( f \ the 3-index symbols, and the 4-index symbols are more complicated than in the last case, and it is convenient to proceed somewhat differently. From (8) we have E 2V= F 2aFia = F 21F l2 F^F"4 = - ^ '(F "1 + VF"4) = 0 for v = 1, 2, 3, 4 by (9) (11). Similarly, 3* = 0. It follows that (23) E 2 = 0, E3, = 0, and the only non-vanishing components of EM are E ll5 E 14, E41 Hence, from (1) we have, in particular, (24) G22 0, G33 = 0. We will first calculate the 4-index symbols necessary to determine G22 and G33, viz.. (2per 2) and (3pcr 3). The non-vanishing 3-index symbols are (25) [11, 2] = g12',[11, 3] = g f, [44, 2] = Ygf, [44, 3] = V (26) [14, 2] = 1- {gf + Yglt'), [14, 3] = + V [12, 4] = [24, l] = (</24'-V < 7 12') [13, 4] = [34, 1] = t(g ' - V g u '). From these we obtain (2112) (2142) _ (2442) 9U 4 (3113) = (3143) = (3443),9 44 y u 911 i (9u ~ I (9sx V</13')2, all other symbols of the type (2per 2), (3per 3) vanishing. Hence (27) i (9U944 (9U)2} {92,'~ Vfir12'}2 i (9119U (9U)2} {9* - Vsr13'2. h 2

6 100 O. R. Baldwin and G. B. Jeffery. Now g l l g U _ {gu)2 <7n > g 12, g l z, g14 0, 1, 0, 0 0, 0, 1, 0 g u, 9*, g M, g Multiplying by the determinant g, we obtain $ fore can never be zero. Accordingly, from (27) we must have (28) $24' - V$12' = 0, gu - = 0. If these are satisfied, all the 3-index symbols vanish except (29) [11,2] = glt', [14, 2] = $24', [44, 2] = [11, 3] = gxs, [14, 3] = $34, [44, 3] = V$34. It may now be shown that all the 4-index symbols vanish. In (5) the terms involving the second differential coefficients will vanish unless, of the four indices, one is 2 or 3 and the remaining three are 1 or 4. It is easily verified that in this case these terms vanish by reason of (28). The terms involving the 3-index symbols will vanish paless all four indices are 1 or 4. Hence the only possible surviving symbol is by (28). (1414) = [11, a] [44, p] + g * [14, a] [14, p] = ($22$ 2 /+ $2SW ) ($2/ v$i2') + ($23$2/ + gz%i) igu Vgn) = 0 Hence for any longitudinal-transverse wave all components of the Riemann- Christoffel tensor vanish ; the wave is spurious and may be transformed away by an appropriate change of co-ordinates. 5. Transverse-Transverse Waves. For these waves (30) - 1, 0, 0, 0 0, g 22 g , g 23> #33> 0 0, o, 0, 1. The determinant of is (31) g = g $22$33 and (32) g t T = g, 0, 0, 0 0, - ~g 33> g 23> 0 0, g 23 > g , 0, 0, g

7 Relativity Theory of Plane Waves. 101 The 3-index symbols vanish unless one index is 1 or 4, and the other two 2 or 3. We find (33) V [12, 2] = - V [22, 1] = - [22, 4] = [24, 2] = V [12, 3] = V [13, 2] = - V [23, 1] = - [23, 4] = [24, 3] = [34, 2] = i V<723' V [13, 3] = - V [33, 1] = - [33, 4] = [34, 3] = The eleven 4-index symbols which do not vanish identically are % (34) (1212) = - \ g 2 2 ' + i{g22g /^22^2 (1213) = ^ 92z" + 4 { f Z ( + + 9^ 9239f}> (1313) = - 1 gf' + l {g (4242) = V (1242) = V2 (1212), (4343) = V (1343) = V2 (1313), (4243) = V (1243) = V (4213) = V2 (1213), (2323) = i (1 - V2) ( ~ 9232)- From these values and (4) we have (35) Gn = - g2 (1212) - 2 g2*(1213 (36) G44 = V G14 = V2 Gn, (37) G22= (1 - V») {(1212) - - f c ' 2)}, (38) Ga = (1 - V») {(121S) + J g ( f e ' - (39) G = <1 - V*) {(1313) - i f > (ga 'g ' - -»)}. For purely gravitational waves tifli, = 0. If V2 ^ 1, we have, from (37) (39), m (1212) = \ g*3 (g22' g ^ g,,22), (1213) = - i 92S (gn 9* ~ gzt (1313) = \ g22 (g22 gs3' g ^'2). Substituting in (35), we have [g22g33 _ (023)2} _ g^2) = (# Hence, from (37)-(39), (1212) = (1213) = (1313) = 0, and we see from (34) that all components of (jxporv) vanish and hence the Riemann- Christoffel tensor vanishes. Purely gravitational waves are spurious, unless

8 102 O. R. Baldwin and G. B. Jeffery. the velocity of propagation is unity, in which case the equations GM 0 reduce to the single equation Gu = 0. Passing to the case of electromagnetic waves, we have from (13) and (32) (41) g F ^ 0, K 2 #2 3 *3 > ~~ (#2 3*2 # (. 933K 2923 K? ') ^ (933 K2 I 923 ^2922 K3> 6, #22 From (9)-(ll) we have k0' = 0 and 9Ko ', V (#33*2' #23**), v 3 g22 (1 V2) (g33 k2g23 k3) = 0, (1 V2) (g33 k 2 #22 ^3) 0. If V2 1, these give k 2 = k 3 = 0, since # 22 #33 = # ^ 0 ; we are led back to the purely gravitational waves already discussed. Hence, for electromagnetic waves, the velocity of propagation is unity and «0' = 0. Making this simplification and calculating EM from (13), (14) and (41), we find (42) E n = E 14 = E 44 = (#33 K K 2K3 I- #22 the remaining components vanishing. Thus, in view of (35) - (39), the equations (1 ) reduce to the single equation Gn = 8 7ty E n, that is to (43) # # # ^ [#33 (#33 #22' #22 #23'2} - 2 #23 {# ~#23 (#2 + #22 {#33 #23'2-2 #23 #23' #22 #3s'2} = 16 Try (033 «2' 2 2 # 23 *a' *3' *3'2) which may be written more concisely as d2 (44) # # 2 3 #23" + # 2 2 #33" - # ~ (log 0 ) = 327iy (033 /c /C2 /C3 -)- 022 /C3 ). This differential equation represents the only condition imposed by th

9 Relativity Theory o f Plane. 103 general relativity theory on the five variables </22, k3 for the pro pagation of plane waves. It possesses a variety of simply periodic solutions in which the variables differ from constant mean values by terms of the type A cos p + B sin, where A and B are different for the different variables. But these solutions have no physical significance since in every case they imply the periodic vanishing of the determinant g. If the g ^ differ from their Galilean values by small quantities the squares of which may be neglected, equation (44) becomes (45) g22 + 9zz (KP + which agrees with the equation given by Eddington, who discusses the particular solutions in which g22 + #33 = const, (purely gravitational waves) and g22 g33 (light waves). The corresponding exact solutions may be obtained as follows : Assume (46) g22 = 1 + 0, g33 = 1 0, g23 = 0, = k where 0 < 1 but is not necessarily small. Equation (44) becomes which integrates to give 200" (1-02) + 0'2 (1 + 02) = 0 (47) = xx + Xi = It follows from this that, unless 0 is constantly zero, it will steadily increase (or decrease) to ± 1, and will attain one of these values for a finite value of t,. Hence, there can be no plane gravitational wave of this type for whiclitthe determinant g does not vanish for some finite value of. For electromagnetic waves, we generalise Eddington s result by assuming (1^ ) 92 = 9zz 1 H- Equation (44) may then be written, rt1 (49) V (! 6) ^ i V ( 1 e) = 4:ry + «3'2). Suppose that when = 0, 0=0, and O' = 0, then, since the right hand side is negative and in general not zero, d/dt, s f (1 0) is zero for t, 0 and decreases at a finite rate as increases. It follows that 0 will attain the value 1 for a finite value of 2*. A slight modification of this argument shows that the conclusion is not substantially affected if the wave is a pulse, so that k2' and k3 vanish except

10 II. Baldwin and G. B. Jeffery. for a small range of values of E,.The root of the diffic propagation of an infinite train of waves, but rather in wave-front of infinite extent. It appears therefore that the relativity theory of light must be approached by way of the study of divergent waves. The importance of the problem hardly needs emphasis.- If light cannot be propagated without change of wave-form we should know the measure of that change in order to ascertain whether it is likely to give rise to observable effects in nature. Electronic Orbits on the Relativity Theory. By 0. R. Baldwin and G. B. J effery, University College, London. (Communicated by Prof. L. N. G. Filon, F.R.S. Received January 15, 1926.) The problem of this paper is to determine the possible circular orbits of a charged electron about a charged nucleus. From the point of view of the older theories the problem is very simple. Let the mass and charge of the nucleus be m and e, and of the electron m and e', the masses being positive but the charges either positive or negative. Write ( 1 ) X 0 yf _ C m/d m'/d where k is the constant of gravitation. The effect of the charges may be regarded as a modification of the masses, and the two bodies will attract according to the inverse square law as if the product of their masses were mm' (1 XX'). If the charges are of unlike sign, or of like sign and such that XX' < 1, this effective mass product will be positive and a circular orbit will be possible for every radius ; otherwise no circular orbits are possible. On the relativity theory the inverse square law is no longer accurately obeyed, particularly for small distances, and the problem is much less simple. In its exact form its solution seems to be impossible in the present state of knowledge, and the present investigation is based on the assumption that the effect of the reaction on the nucleus may be neglected, so that the latter may be regarded as fixed in position. This is a serious limitation, but, pending a more exact solution, the results may serve as an indication of the kind of results