Quasistationary binary inspiral. I. Einstein equations for the two Killing vector spacetime

TL;DR

This paper models the spacetime geometry of two mass lines in orbit using Einstein equations and Geroch's formalism, providing a simplified framework for understanding binary inspiral dynamics.

Contribution

It extends Geroch's formalism to derive explicit Einstein equations for a toy model of binary inspiral with two Killing vectors.

Findings

Derived explicit Einstein equations for the toy model

Streamlined and generalized Geroch's derivation

Provided a tensor-based description of the orbiting mass lines

Abstract

The geometry of two infinitely long lines of mass moving in a fixed circular orbit is considered as a toy model for the inspiral of a binary system of compact objects due to gravitational radiation. The two Killing fields in the toy model are used, according to a formalism introduced by Geroch, to describe the geometry entirely in terms of a set of tensor fields on the two-manifold of Killing vector orbits. Geroch's derivation of the Einstein equations in this formalism is streamlined and generalized. The explicit Einstein equations for the toy model spacetime are derived in terms of the degrees of freedom which remain after a particular choice of gauge.