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

TL;DR

This work resolves the long-standing challenge of reconstructing metric perturbations from linearized Kerr perturbations by providing explicit, mode-by-mode formulas that map a Weyl-scalar mode $\psi_0$ or $\psi_4$ to a corresponding metric perturbation $h_{\mu\nu}$ in both ingoing and outgoing radiation gauges. The authors chain Teukolsky's decoupled master equations, the mode-inversion procedure, and the Chrzanowski–Cohen–Kegeles reconstruction into a complete, self-consistent pipeline, expressing radial and angular parts through confluent Heun functions. They derive new, general relations between Weyl-scalar modes with complex frequencies, and present detailed, gauge-specific metric components that avoid Hertz-potentials as final inputs, improving numerical stability and applicability to higher-order perturbation theory. The work lays a rigorous foundation for including self-force and second-order effects in black hole perturbation theory, with potential impact on gravitational-wave modeling and Kerr background analyses. The methods are poised to facilitate efficient numerical implementations and to extend perturbation theory beyond linear order in the Kerr spacetime.

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.