Interpreting solutions with nontrivial Killing groups in general relativity

TL;DR

This paper explores the invariance groups of solutions in general relativity, emphasizing the role of Killing groups, and analyzes the Schwarzschild solution's invariance properties.

Contribution

It provides a new interpretation of general relativity based on invariance groups, especially focusing on solutions with nontrivial Killing groups and their geometric implications.

Findings

Solutions with intermediate invariance groups exist in general relativity.

The Schwarzschild solution's manifold choice is derived from its unique timelike Killing vector.

General relativity's invariance group varies from trivial to Poincaré group depending on curvature.

Abstract

General relativity is reconsidered by starting from the unquestionable interpretation of special relativity, which (Klein 1910) is the theory of the invariants of the metric under the Poincar\'e group of collineations. This invariance property is physical and different from coordinate properties. Coordinates are physically empty (Kretschmann 1917) if not specified by physics, and one shall look for physics again through the invariance group of the metric. To find the invariance group for the metric, the Lie "Mitschleppen" is ideal for this task both in special and in general relativity. For a general solution of the latter the invariance group is nil, and general relativity behaves as an absolute theory, but when curvature vanishes the invariance group is the group of infinitesimal Poincar\'e "Mitschleppen" of special relativity. Solutions of general relativity exist with invariance groups intermediate between the previously mentioned extremes. The Killing group properties of the static solutions of general relativity were investigated by Ehlers and Kundt (1964). The particular case of Schwarzschild's solution is examined, and the original choice of the manifold done by Schwarzschild in 1916 is shown to derive invariantly from the uniqueness of the timelike, hypersurface orthogonal Killing vector of that solution.