Maximal extension of the Schwarzschild metric: From Painlev\'e-Gullstrand to Kruskal-Szekeres

TL;DR

This paper introduces a unified coordinate system that connects the Painlevé-Gullstrand and Kruskal-Szekeres extensions of the Schwarzschild metric, revealing a family of maximal extensions parameterized by energy.

Contribution

The authors develop a new family of Schwarzschild extensions using two time coordinates, unifying partial and maximal extensions and clarifying their relation to existing solutions.

Findings

The Kruskal-Szekeres solution is part of a broader family of extensions parameterized by energy.

The new coordinate system smoothly transitions from Painlevé-Gullstrand to Kruskal-Szekeres.

The family of extensions differs from Novikov-Lemaître, with distinct maximal and partial cases.

Abstract

We find a specific coordinate system that goes from the Painlev\'e-Gullstrand partial extension to the Kruskal-Szekeres maximal extension and thus exhibit the maximal extension of the Schwarzschild metric in a unified picture. We do this by adopting two time coordinates, one being the proper time of a congruence of outgoing timelike geodesics, the other being the proper time of a congruence of ingoing timelike geodesics, both parameterized by the same energy per unit mass $E$. $E$ is in the range $1\leq E<\infty$ with the limit $E=\infty$ yielding the Kruskal-Szekeres maximal extension. So, through such an integrated description one sees that the Kruskal-Szekeres solution belongs to this family of extensions parameterized by $E$. Our family of extensions is different from the Novikov-Lema\^itre family parameterized also by the energy $E$ of timelike geodesics, with the Novikov extension holding for $0<E<1$ and being maximal, and the Lema\^itre extension holding for $1\leq E<\infty$ and being partial, not maximal, and moreover its $E=\infty$ limit evanescing in a Minkowski spacetime rather than ending in the Kruskal-Szekeres spacetime.