Beyond Schwarzschild: New Pulsating Coordinates for Spherically Symmetric Metrics

TL;DR

This paper introduces new coordinate systems for spherically symmetric metrics, extending beyond Schwarzschild and Kruskal-Szekeres, with potential applications in causal structure analysis and compactification.

Contribution

The paper develops a novel class of pulsating coordinates for spherically symmetric metrics, including Schwarzschild, de Sitter, and Anti-de Sitter spacetimes, expanding the toolkit for analyzing their causal structures.

Findings

Derived a new coordinate set with distinct properties from known solutions.

Showed the new coordinates can represent any causal patch in a compactified form.

Compared the new coordinates with Schwarzschild, de Sitter, and Anti-de Sitter cases.

Abstract

Starting from a general transformation for spherically symmetric metrics where g\_11=-1/g\_00, we analyze coordinates with the common property of conformal flatness at constant solid angle element. Three general possibilities arise: one where tortoise coordinate appears as the unique solution, other that includes Kruskal-Szekeres coordinates as a very specific case, but that also allows other similar transformations, and finally a new set of coordinates with very different properties than the other two. In particular, this represents any causal patch of the spherically symmetric metrics in a compactified form. We analyze some relations, taking the Schwarzschild case as prototype, but also contrasting the cosmological de-Sitter and Anti-de-Sitter solutions for the new proposed pulsating coordinates.