The symmetry structures of curved manifolds and wave equations

Abstract

Killing vectors are widely used to study conservation laws admitted by spacetime metrics or to determine exact solutions of Einstein field equations (EFE) via Killing’s equation. Its solutions on a manifold are in one-to-one correspondence with continuous symmetries of the metric on that manifold. Two well known spherically symmetric static spacetime metrics in Relativity that admit maximal symmetry are given by Minkowski and de-Sitter metrics. Some other spherically symmetric metrics forming interesting solutions of the EFE are known as Schwarzschild, Kerr, Bertotti-Robinson and Einstein metrics. We study the symmetry properties and conservation laws of the geodesic equations following these metrics as well as the wave and Klein-Gordon (KG) type equations constructed using the covariant d’Alembertian operator on these manifolds. As expected, properties of reduction procedures using symmetries are more involved than on the well known flat (Minkowski) manifold.