New metric reconstruction scheme for gravitational self-force calculations

Abstract

Inspirals of stellar-mass objects into massive black holes will be important sources for the space-based gravitational-wave detector LISA. Modelling these systems requires calculating the metric perturbation due to a point particle orbiting a Kerr black hole. Currently, the linear perturbation is obtained with a metric reconstruction procedure that puts it in a “no-string” radiation gauge which is singular on a surface surrounding the central black hole. Calculating dynamical quantities in this gauge involves a subtle procedure of “gauge completion” as well as cancellations of very large numbers. The singularities in the gauge also lead to pathological field equations at second perturbative order. In this paper we re-analyze the point-particle problem in Kerr using the corrector-field reconstruction formalism of Green, Hollands, and Zimmerman (GHZ). We clarify the relationship between the GHZ formalism and previous reconstruction methods, showing that it provides a simple formula for the “gauge completion”. We then use it to develop a new method of computing the metric in a more regular gauge: a Teukolsky puncture scheme. This scheme should ameliorate the problem of large cancellations, and by constructing the linear metric perturbation in a sufficiently regular gauge, it should provide a first step toward second-order self-force calculations in Kerr. Our methods are developed in generality in Kerr, but we illustrate some key ideas and demonstrate our puncture scheme in the simple setting of a static particle in Minkowski spacetime.

Quasinormal Modes from Penrose Limits

“…In both cases, it would be valuable to connect more closely and in more generality to the large literature on solutions to the wave equation on a rotating black hole background [89,96,97,115]. This could include using higher-spin fields and making contact to the Teukolsky equation [116], as opposed to the scalar wave equations used here, considering other modifications to the wave equations, as for example in modified theories of gravity or black holes [68,[117][118][119][120][121][122][123][124], the introduction of sources [125,126], investigating how to describe caustics as well as the propagator and scattering on the black hole background more generally [127][128][129][130][131].…”

Quasinormal modes from Penrose limits

“…In both cases, it would be valuable to connect more closely and in more generality to the large literature on solutions to the wave equation on a rotating black hole background [89,96,97,116]. This could include using higher-spin fields and making contact to the Teukolsky equation [117], as opposed to the scalar wave equations used here, considering other modifications to the wave equations, as for example in modified theories of gravity or black holes [68,[118][119][120][121][122][123][124][125][126][127], the introduction of sources [128,129], investigating how to describe caustics as well as the propagator and scattering on the black hole background more generally [130][131][132][133][134]. To do so, it will be important to understand the matched asymptotic expansion that embeds the plane wave into the full spacetime, and the respective interplay between the 'far-zone' outer spacetime and the 'near-zone' plane wave.…”

Metric perturbations of Kerr spacetime in Lorenz gauge: circular equatorial orbits

“…non-vacuum) case by augmenting the metric perturbation with a so-called corrector tensor, which is determined by solving certain decoupled ODEs by integrating over the outgoing Kerr-Newman radial coordinate. Toomani et al [46] have shown that the gauge singularities arising in the GHZ approach can be softened by moving to a 'shadowless' gauge, in order to obtain a metric perturbation suitable as an input for second-order calculations. This approach is certainly promising, and is under active development.…”