The relativity theory of plane waves

Abstract

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. Eddington 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 transverse-transverse 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 aperiodic term 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 small; terms neglected on this ground might well have a cumulative effect. All the solu­tions discussed in this paper are exact.