The Common Solution Space of General Relativity

TL;DR

This paper uncovers a shared mathematical structure in the solution spaces of various static, spherically symmetric solutions in General Relativity, revealing a common geometric and algebraic framework that simplifies their analysis.

Contribution

It demonstrates that diverse static solutions in General Relativity share a common solution space due to an underlying Lie algebra symmetry, enabling a unified approach to their analysis.

Findings

Solution space is governed by a common three-dimensional Lie algebra.

Field equations can be transformed into null geodesic equations in conformally flat geometries.

This algebraic structure simplifies the construction of analytic solutions in gravitational physics.

Abstract

We review the solution space for the field equations of Einstein's General Relativity for various static, spherically symmetric spacetimes. We consider the vacuum case, represented by the Schwarzschild black hole; the de Sitter-Schwarzschild geometry, which includes a cosmological constant; the Reissner-Nordstr\"{o}m geometry, which accounts for the presence of charge. Additionally we consider the homogenenous and anisotropic locally rotational Bianchi II spacetime in the vacuum. Our analysis reveals that the field equations for these scenarios share a common three-dimensional group of point transformations, with the generators being the elements of the $D\otimes_{s}T_{2}$ Lie algebra, known as the semidirect product of dilations and translations in the plane. Due to this algebraic property the field equations for the aforementioned gravitational models can be expressed in the equivalent form of the null geodesic equations for conformally flat geometries. Consequently, the solution space for the field equations is common, and it is the solution space for the free particle in a flat space. This appoach open new directions on the construction of analytic solutions in gravitational physics and cosmology.