Tidal coupling of a Schwarzschild black hole and circularly orbiting moon

TL;DR

This paper investigates how a Schwarzschild black hole responds to a distant orbiting moon's tidal forces, revealing gauge-dependent ambiguities and a time-varying induced quadrupole moment, with implications for gravitational-wave observations.

Contribution

It provides a detailed analysis of tidal coupling for a non-spinning black hole under a distant orbiting mass, highlighting gauge dependence and the dynamic nature of induced quadrupole moments.

Findings

No static induced quadrupole moment in Schwarzschild coordinates

Orbiting moon induces a time-varying quadrupole proportional to the tidal field's derivative

Tidal field causes a horizon bulge that leads the tidal perturbation by a small angle

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.