Take the A-Metric: Interpretations of Some Known Solutions of Einstein’s Vacuum Field Equations

Charles W Robson,Marco Ornigotti

doi:10.3390/universe8030183

Abstract

In this work, we present a new interpretation of the only static vacuum solution of Einstein’s field equations with planar symmetry, the Taub solution. This solution is a member of the AIII class of metrics, along with the type D Kasner solution. Various interpretations of these solutions have been put forward previously in the literature, however, some of these interpretations have suspect features and are not generally considered physical. Using a simple mathematical analysis, we show that a novel interpretation of the Taub solution is possible and that it naturally emerges from the radial, near-singularity limit of negative-mass Schwarzschild spacetime. A new, more transparent derivation is also given, showing that the type D Kasner metric can be interpreted as a region of spacetime deep within a positive-mass Schwarzschild black hole. The dual nature of this class of A-metrics is thereby demonstrated.

Highlights

We argue that our new interpretation is by far the most natural yet suggested, relating as it does to a local region of the Schwarzschild solution with finite mass parameter
Depending on the choice of this parameter, one finds the well-known Schwarzschild solution (AI), a solution with a negative-Gaussian-curvature hypersurface (AI I), capable of describing the gravitational field produced by a tachyon [14], or the AI I I-metrics; in this paper, we focus purely on the final subclass, as the different interpretations and geometrical features present in these vacuum solutions make them an interesting topic of study
We argue that the deep-radial interpretations of the Kasner and Taub metrics are considerably more natural than the ones suggested by the above limits, as the former avoid the pathologies of infinite positive or negative mass, which are unphysical

Summary

Introduction

A vast number of solutions of Einstein’s field equations have been discovered [1–7]. During this same period, a much smaller number of agreed-upon physical interpretations of these solutions have been established [8–12]. It is arguable that the immediate task is to interpret those solutions that have already been found through many years of great effort [1,2,8]. It is a pressing task as it has been since the early days of general relativity, to find reasonable and useful interpretations of all its solutions and to reassess those with doubtful features

Results

Discussion

Conclusion