Shells around black holes: the effect of freely specifiable quantities in Einstein's constraint equations

TL;DR

This paper investigates how different freely specifiable quantities in Einstein's constraint equations affect the modeling of black hole shells, finding that physical solutions remain invariant despite different gauge choices.

Contribution

It demonstrates that various gauge choices in the conformal thin-sandwich decomposition do not alter the physical content of solutions modeling black hole shells.

Findings

Different gauge choices do not affect gauge-invariant physical quantities.

Analytical solutions are possible for specific gauge choices.

Numerical solutions are constructed for other gauge choices.

Abstract

We solve Einstein's constraint equations in the conformal thin-sandwich decomposition to model thin shells of non-interacting particles in circular orbit about a non-rotating black hole. We use these simple models to explore the effects of some of the freely specifiable quantities in this decomposition on the physical content of the solutions. Specifically, we adopt either maximal slicing or Kerr-Schild slicing, and make different choices for the value of the lapse on the black hole horizon. For one particular choice of these quantities the resulting equations can be solved analytically; for all others we construct numerical solutions. We find that these different choices have no effect on our solutions when they are expressed in terms of gauge-invariant quantities.