On the Wave Theory of Light in General Relativity, I: Path of Light

On the Wave Theory of Light in General Relativity 0 I Rzth oj Light - - 0 0 Research Institute for Theoretical Physics , Hiroshima UniversitJl (Takehara-machi, Hiroshima-km) Progress of Theoretical Physics, Vol. 10, No: 4, October 19s1 In the general theory. of relativity the problems of optics have been restricted ,mostly to those concerning the path of light as null geodesic, and the chal'acter of light as wave has scarcely been treatd. Therefore we intend to investigate the nature of light as wave in genel'ar relativ'ty. In thisht'st paper we shall research the relation between the method of wave optics and -that of geometric~l optics cOncerning the propagation of' light in a curved space-t;me. Introduction eu the Wave Theory 0/ Lig-htin Ceneral Relativity; ! The propagation of light* Relation between the wave front and the path of light Kab=-l, (a, b=l, 2, 3), and the wave equation takes the form: (x,y, 2', t= x1,.:I. 2,X3,X4). Let 2] : So (x,y, 2', t) =0 be the four dimensional expression of the above stated moving surface of discontinuity. This hypersurface is the 2J treated in the last section and the following relation holds on this hypersurface: On the Wave Theory of Lght in emeral Relativity, ! In the general theory of relativity Along the path of light we have g ij dx i d x4 where a comma denotes covariant derivative. * = 0 from which we have Since each hypersurface of this family is to be the characteristic surface of the wave equa tion, the equation must be of the form3l On the Wave Thqory of Light in General Relativity, I The condition null: f{J. f{Ji=O. pi; P;j=O, (P;j=ffu, n), Conclusion APPENDIX Hyoitiro TAKENO and YoshioUENO v;=p./" From these equations we easily obtain (2) 'Where Eijlm is the E-tensor ana Pfj=Vri, .1]. -Proof. From v.1 V;'j=O and v.1 f'j=O' we have 11.1 Pij=O Pi; VIc4-~k 1l i +Pki Vj=O. (...truncated)