Embedding spherical spacelike slices in a Schwarzschild solution

TL;DR

This paper provides a simple algebraic condition to determine when a spherical spacelike three-geometry can be embedded in a Schwarzschild solution, revealing insights into the structure of spherical superspace and counterexamples to the thick sandwich conjecture.

Contribution

It introduces a straightforward algebraic criterion for embedding spherical geometries into Schwarzschild solutions and explores the structure of spherical superspace.

Findings

Any Schwarzschild solution covers a large subset of spherical superspace.

These subsets form nested domains as Schwarzschild mass increases.

The work provides a counterexample to the thick sandwich theorem.

Abstract

Given a spherical spacelike three-geometry, there exists a very simple algebraic condition which tells us whether, and in which, Schwarzschild solution this geometry can be smoothly embedded. One can use this result to show that any given Schwarzschild solution covers a significant subset of spherical superspace and these subsets form a sequence of nested domains as the Schwarzschild mass increases. This also demonstrates that spherical data offer an immediate counter example to the thick sandwich `theorem'.