Gravitational Waves on Kerr Black Holes I: Reconstruction of Linearized Metric Perturbations

Abstract

The gravitational perturbations of a rotating Kerr black hole are notoriously complicated, even at the linear level. In 1973, Teukolsky showed that their physical degrees of freedom are encoded in two gauge-invariant Weyl curvature scalars that obey a separable wave equation. Determining these scalars is sufficient for many purposes, such as the computation of energy fluxes. However, some applications -- such as second-order perturbation theory -- require the reconstruction of metric perturbations. In principle, this problem was solved long ago, but in practice, the solution has never been worked out explicitly. Here, we do so by writing down the metric perturbation (in either ingoing or outgoing radiation gauge) that corresponds to a given mode of either Weyl scalar. Our formulas make no reference to the Hertz potential (an intermediate quantity that plays no fundamental role) and involve only the radial and angular Kerr modes, but not their derivatives, which can be altogether eliminated using the Teukolsky--Starobinsky identities. We expect these analytic results to prove useful in numerical studies and for extending black hole perturbation theory beyond the linear regime.

54 pages, no figures. Version 4: fixed typos

References in corpus (10)

• Interpreting the C-metric
• No Dynamics in the Extremal Kerr Throat
• Kerr-CFT and gravitational perturbations
• Gravitational perturbations of the Kerr geometry: High-accuracy study
• Conservative, gravitational self-force for a particle in circular orbit around a Schwarzschild black hole in a Radiation Gauge
• Completion of metric reconstruction for a particle orbiting a Kerr black hole
• Gravity Waves from Extreme-Mass-Ratio Plunges into Kerr Black Holes
• On the Existence of Radiation Gauges in Petrov type II spacetimes
• Extreme Black Hole Anabasis
• New total transmission modes of the Kerr geometry with Schwarzschild limit frequencies at complex infinity

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• Toward Second-Order Self-Force for Eccentric Extreme-Mass Ratio Inspirals in Schwarzschild Spacetime
• Computing spectral shifts for Johannsen-Psaltis black holes
• Metric reconstruction and the Hamiltonian for eccentric, precessing binaries in the small-mass-ratio limit
• Tidal perturbations of an extreme mass ratio inspiral around a Kerr black hole
• The scalar angular Teukolsky equation and its solution for the Taub-NUT spacetime