Schwarzschild Spacetimes: Topology

Abstract

This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric, an SO(3)-invariant metric field, satisfying the Einstein equations. We prove the existence of and find all Schwarzschild metrics on two topologically non-equivalent manifolds, R×(R3∖{(0,0,0)}) and S1×(R3∖{(0,0,0)}). The method includes a classification of SO(3)-invariant, time-translation invariant and time-reflection invariant metrics on R×(R3∖{(0,0,0)}) and a winding mapping of the real line R onto the circle S1. The resulting family of Schwarzschild metrics is parametrized by an arbitrary function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not defined, the Schwarzschild sphere. In particular, the family admits a global metric whose Schwarzschild sphere is empty. These results transfer to S1×(R3∖{(0,0,0)}) by the winding mapping. All our assertions are derived independently of the signature of the Schwarzschild metric; the signature can be chosen as an independent axiom.