Tidal coupling of a Schwarzschild black hole and circularly orbiting moon

TL;DR

The paper develops a perturbative framework to quantify how a Schwarzschild black hole responds to the tidal field of a distant orbiting moon, within the LISA EMRI context. It shows that the dynamical induced quadrupole moment scales with the time derivative of the external tidal field, $\mathscr{I}^{\text{ind, DP}}_{ij} = \frac{32}{45}M^6 \dot{\mathcal{E}}^{\text{ext}}_{ij}$, and that the static part of the response is gauge-dependent and ambiguous. A gauge-invariant horizon phase shift is found: the horizon shear leads the tidal field by $\delta_{\text{Horizon}} = 4M\Omega$, highlighting a dissipative-like tidal interaction at the horizon. The work also discusses the limitations of using static polarizability and lag-angle concepts for black holes and outlines future extensions to spinning holes and more general orbits.

Abstract

We describe the possibility of using LISA's gravitational-wave observations to study, with high precision, the response of a massive central body to the tidal gravitational pull of an orbiting, compact, small-mass object. Motivated by this application, we use first-order perturbation theory to study tidal coupling for an idealized case: a massive Schwarzschild black hole, tidally perturbed by a much less massive moon in a distant, circular orbit. We investigate the details of how the tidal deformation of the hole gives rise to an induced quadrupole moment in the hole's external gravitational field at large radii. In the limit that the moon is static, we find, in Schwarzschild coordinates and Regge-Wheeler gauge, the surprising result that there is no induced quadrupole moment. We show that this conclusion is gauge dependent and that the static, induced quadrupole moment for a black hole is inherently ambiguous. For the orbiting moon and the central Schwarzschild hole, we find (in agreement with a recent result of Poisson) a time-varying induced quadrupole moment that is proportional to the time derivative of the moon's tidal field. As a partial analog of a result derived long ago by Hartle for a spinning hole and a stationary distant companion, we show that the orbiting moon's tidal field induces a tidal bulge on the hole's horizon, and that the rate of change of the horizon shape leads the perturbing tidal field at the horizon by a small angle.