Lunar atmospheric tide at twenty-seven stations 323 References Bartels, J. 1939 Gerl. Beitr. Geophys. 54, 56.
324 H. McManus The Lorentz theory is unsatisfactory because it is not relativistically invariant. If, however, we proceed to the limit when the electron radius becomes zero, and compensate the resulting infinite electromagnetic mass by a negative infinite mass of some other origin, then the higher terms disappear and the approximate Lorentz equations can be considered as exact. As terms involving the electron structure have been eliminated, the theory is then invariant.
Classical electrodynamics without singularities 325 The four velocity of the electron will be denoted by v = z, where the dot represents differentiation with respect to the proper time. This and its derivatives obey the usual relations » , . _ . , v2 = 1, v.v = 0, v.v + v2 = 0, . (2*2) With this notation the equations of ordinary classical electrodynamics can be obtained from a Lagrangian principle, which is, for a single electron, *ijMraj a* + T ^ /^ x ) *™(x) dx<4> + J j*(x) dx<4>[ = 0, (2-3) where m is the mechanical mass of the electron measured in energy units, and j*(x) is the charge-current density of a point electron j*(x) = e f 8(x- z(s)) v{(s) ,ds = 8{x0) £(aq) (2-4) J —00 In the present theory the first two terms of the Lagrangian are unchanged, but the interaction term is replaced by the double integral JJ>(X' ) F[(x -x')2] 4,(x) dx<4>dx'<4>. (2-5) Thus the interaction depends not on the product of the potentials and the charge- current density at the same space-time point, but on the product of the two quan tities at different points with a weight factor which, to be relativistically invariant, can be a function only of the invariant distance between the two points. We suppose this averaging factor to be large only when the distance is small, and to fall rapidly to zero when the distance much exceeds a quantity r0. In this way we have introduced a parameter corresponding to the radius of an electron in relativistic fashion.
326 H. McManus which, Tiding the Lorentz condition dAk/dxk = 0, become r+oo □A*(x) = 477’eJ vi(s) 47rJ*(x), (2*10) _ a _ a a a a 2 2 2 2 2 where □ dxkdxkd x% dy% dz2' If we now vary the electron co-ordinates zi} keeping the potentials fixed, we obtain the equations of motion = etJVw(x) F(R2) dx<4>, (2-11) where the integration is over all space time and 2 is given by (2*7). The field to be inserted in (2*11) is, of course, the external field plus the field produced by the electron itself.
Classical electrodynamics without singularities 327 negative value in a short distance of the order of magnitude r0 along the world line through the retarded point. Thus F (R2) falls to a very small value in the vicinity of the retarded point, and, if it is an odd function, is roughly equal and opposite on each side of the light cone. Consequently in an integral like (2-8), at large distances from the world line, the contributions from points in the vicinity of the light cones, where F is appreciable, largely cancel, and the resultant charge-current density is negligible except for points whose spatial and temporal distances are separately within a distance of the order of magnitude r0 from the world line.
328 H. McManus Also jy — jz = 0 ,jx = fijQ, and from (2*8) expressing F as a Fourier integral, and using the above notation, = p = _ £2) dk^dk0dt', which becomes, on integrating over t', Jc0 and the directions of k, P - r'^T-~P)l ? Siakr'lli- k*)dk’ where r' — {y2 + z2 + (x—/3t)2/(l—fi2)}i.
Classical electrodynamics without singularities 329 Thus the second term in equation (3-1) is r+oo \e\ vi(s)A(x — z J — 00 which is half the difference between the retarded and advanced potentials of the usual theory. It has been shown by Dirac (1940) to be a solution of the Maxwell equations without sources and is introduced with the positive sign because of the choice of retarded potentials. The full expression for the retarded potentials thus becomes r+00 Ai(x) = vet(s) [ + *A(R)] da, (3-3) J —00 the first term of which represents half the sum of the retarded and advanced potentials. This is given in the ordinary theory by the expression r+* e v^SiR^ds, J — 00 which contains the Coulomb potentials. It is singular on the world line and gives rise to the divergences of the usual scheme. In the present theory the singular function 8(R2) is replaced by the regular function G, which removes the divergences.
330 H. McManus 3*3. If we take the potentials given by (3*3) for a field point very distant from the world line, G(JB2) falls to a very small value in the immediate neighbourhood of the light cones, and the main contribution to the integral will come from the region where I?2 is zero. Thus, taking into account the condition (3-5), G can be replaced without appreciable error by £(I22) and the potentials reduce to the retarded poten tials of a point electron on the usual theory.
Classical electrodynamics without singularities 331 the world line. This quantity has been evaluated by Dirac and results in the radiation damping term fe^ + v ^ ), where the dots denote differentiation with respect to s. Thus the equations of motion become, after integrating by parts the first term of (4-1), the integro-differential equations mvt- f e*(«« + ir\) = 2- A) - »$(*, - *5 )}^?* ' + er’ jVf/(x) *)ix®, (4-3) where cr2 = (z(s) —z($'))2 and the field J^-(x) refers to the external field only. The fact that integro-differential equations of motion are obtained is satisfactory, for differential equations, if they are to give an account of radiation damping, have too many solutions, whereas integro-differential equations may have a much more restricted group of solutions than differential equations of any finite order; so there is a possibility that only one solution of the equations of motion of a free electron exists. The equations would be simpler if the integration over the world line extended ffom minus infinity to the point s on the world line considered, instead of over the whole world line. This would be the case if one could construct a function H which was zero in the future light cone of the time-like vector z' — z for then H would be zero for s' > s. However, although we are free .to construct an averaging function different in the three invariant regions of space time, this does not give a corre sponding freedom in the choice of H. For let us denote the Fourier transform of the averaging function by gr(K) to show its dependence on the division of Fourier space into three invariant parts. Then the functions H(R), H1(R) become J H(R) = — 4tt g(K)^(2 K) er-iK.KfK(4)5 Hl(R) = 4:7T2ijg(K) K) e-“ dK<4>.
332 H. McManus 4*2. The force exerted on an electron by an external field is that due to the space- time average of the field around the world line. The principal contribution is made by the field within a distance rQ from the world line, so one would not expect the field to be altered very much unless it changes appreciably in a small region near the electron.