Time-domain evolution of Lorenz-gauge metric perturbations: taming the $\ell =m=1$ gauge instability

Abstract

Calculating the spacetime metric perturbation (MP) sourced by a small "particle" of mass $\mu M$ (with $0<\mu \ll 1$) moving in a Schwarzschild or Kerr "background" black hole spacetime of mass $M$ is a longstanding research area in general relativity. This calculation also has an important astrophysical motivation as a major step in calculating the gravitational waves emitted by an extreme-mass-ratio inspiral system. Here I consider the specific problem of the time-domain calculation of the $\mathcal{O}(\mu )$ Lorenz-gauge MP $h_{ab}$ sourced by the particle. Decomposing the Schwarzschild-background MP into $e^{im\varphi }$ modes, Dolan and Barack [Phys. Rev. D 87, 084066 (2013), arXiv:1211.4586] found that the $m=1$ time-domain Lorenz-gauge MP generically contains an \emph{unstable gauge mode} which grows linearly with time. Here I demonstrate a method for computing a Lorenz-gauge time-domain evolution which is mostly free of this gauge mode. This method computes an "orthogonalized" MP $h_{ab}^{\text{ortho}}$ as a linear combination of the sourced MP and a homogeneous MP $h_{ab}^{\text{hom}}$ (evolved in parallel with the sourced MP). The linear combination is updated "occasionally" to make $h_{ab}^{\text{ortho}}$ orthogonal to $h_{ab}^{\text{hom}}$ with respect to a chosen inner product on MPs. I show that, for a Schwarzschild-circular-orbit test case, the resulting $h_{ab}^{\text{ortho}}$ satisfies the $\mathcal{O}(\mu )$ Einstein equations and Lorenz gauge conditions, remains bounded as $t\to \infty$, and at late (finite) times contains only a small component of the unstable gauge mode. These results hold both with the particle modelled via MP jump conditions and with particle modelled by a "effective source". My numerical code for obtaining all of these results is included with this paper, and will be deposited in the Black Hole Perturbation Toolkit.

37 pages, 24 eps figures, ancillary files include 3 mpeg movies, full C++ source code, and all parameter files and data analysis scripts to reproduce results