Comparing Criteria for Circular Orbits in General Relativity

TL;DR

This paper analytically compares two criteria for identifying circular orbits in general relativity using a thin shell model, confirming their equivalence and clarifying their application in binary black hole solutions.

Contribution

It provides a transparent analytical comparison of two criteria for circular orbits, supporting their consistency in complex numerical solutions.

Findings

Both criteria yield equivalent results for circular orbits.

The analysis clarifies the criteria's application in binary black hole models.

Deviations in numerical solutions are not due to the criteria differences.

Abstract

We study a simple analytic solution to Einstein's field equations describing a thin spherical shell consisting of collisionless particles in circular orbit. We then apply two independent criteria for the identification of circular orbits, which have recently been used in the numerical construction of binary black hole solutions, and find that both yield equivalent results. Our calculation illustrates these two criteria in a particularly transparent framework and provides further evidence that the deviations found in those numerical binary black hole solutions are not caused by the different criteria for circular orbits.