Symmetry Transformations in Cosmological and Black Hole Analytical Solutions

TL;DR

This paper explores how a broad class of static metrics in cosmology and black hole physics can be transformed to reveal various coordinate systems, including well-known ones like Kruskal-Szekeres and Lemaître, through symmetry transformations.

Contribution

It introduces a general method for transforming static metrics into different coordinate systems, encompassing many known and new transformations in black hole and cosmological solutions.

Findings

Derived relations for metric parameters enabling symmetry impositions

Demonstrated transformations to FLRW, near horizon, and isotropic coordinates

Outlined a procedure to obtain Kruskal-Szekeres and Lemaître coordinates

Abstract

We analyze the transformation of a very broad class of metrics that can be expressed in terms of static coordinates. Starting from a general ansatz, we obtain a relation for the parameters in which one can impose further symmetries or restrictions. One of the simplest restrictions leads to FLRW cases, while transforming from the initial static to other static-type coordinates can lead to near horizon coordinates, Wheeler--Regge, and isotropic coordinates, among others. As less restrictive cases, we show an indirect route for obtaining Kruskal-Szekeres within this approach, as well as Lema\^{\i}tre coordinates. We use Schwarzschild spacetime as a prototype for testing the procedure in individual cases. However, application to other spacetimes, such as de-Sitter, Reissner-Nordstr\"{o}m, and Schwarzschild de Sitter, can be readily generalized.