Explicit solutions of Maxwell's equations on a space of constant curvature

AbstractAn explicit solution is given to the Cauchy problem for the source-free Maxwell's equations in a vacuum on a space-time of the form R1 X M3, where M3 is a 3-manifold of constant curvature. This solution satisfies Huyghens' Principle, that all electromagnetic radiation propagates at exactly the speed of light. The solution is obtained by harmonic analysis on M3, and in the process a generating class of plane wave solutions is found. These solutions approximate the flat-space plane wave solutions in a neighbourhood of a point, but their global properties are somewhat different. The solutions obtained are easily transplanted to the Robertson-Walker models of General Relativity by re-scaling the time variable