Tidal coupling of a Schwarzschild black hole and circularly orbiting moon

Abstract

We describe the possibility of using the laser interferometer space antenna (LISA) 's gravitational-wave observations to study, with high precision, the response of a massive central body (e.g. a black hole or a soliton star) to the tidal gravitational pull of an orbiting, compact, small-mass object (a white dwarf, neutron star, or small-mass black hole). Motivated by this LISA application, we use first-order perturbation theory to study tidal coupling for a special, idealized case: a Schwarzschild black hole of mass $M$, tidally perturbed by a ``moon'' with mass $\ensuremath{\mu}\ensuremath{\ll}M$ in a circular orbit at a radius $b\ensuremath{\gg}M$ with orbital angular velocity $\ensuremath{\Omega}$. We investigate the details of how the tidal deformation of the hole gives rise to an induced quadrupole moment ${\mathcal{I}}_{ij}$ in the hole's external gravitational field at large radii, including the vicinity of the moon. 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, and we contrast this with an earlier result of Suen, which gave, in a very different gauge, a nonzero static induced quadrupole moment with a sign opposite to what one would get for a fluid central body. 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, ${\mathcal{I}}_{ij}=(32/45){M}^{6}{\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\mathcal{E}}}_{ij}$ and that therefore is out of phase with the tidal field by a spatial angle $\ensuremath{\pi}/4$ and by a temporal phase shift $\ensuremath{\pi}/2$. This induced quadrupole moment produces a gravitational force on the moon that reduces its orbital energy and angular momentum at the same rate as the moon's tidal field sends energy and angular momentum into the hole's horizon. As a partial analog of a result derived long ago by Hartle for a spinning hole and a static 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 (i.e. the horizon shear) leads the perturbing tidal field at the horizon by an angle $4M\ensuremath{\Omega}$. We prefer to avoid introducing an ingoing null geodesic, as Hartle did in his definition of the phase shift, because the moon is in the central body's near zone ($b\ensuremath{\ll}1/\ensuremath{\Omega}$) and thus should interact with the horizon instantaneously, not causally. We discuss the implications of these results for LISA's future observations of tidal coupling, including the inappropriateness of using the concepts of tidal polarizability and tidal lag or lead angle, for the massive central body, when discussing LISA's observations.