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We describe the possibility of using LISA's gravitational-wave observationsto study, with high precision, the response of a massive central body to thetidal gravitational pull of an orbiting, compact, small-mass object. Motivatedby this application, we use first-order perturbation theory to study tidalcoupling for an idealized case: a massive Schwarzschild black hole, tidallyperturbed by a much less massive moon in a distant, circular orbit. Weinvestigate the details of how the tidal deformation of the hole gives rise toan induced quadrupole moment in the hole's external gravitational field atlarge radii. In the limit that the moon is static, we find, in Schwarzschildcoordinates and Regge-Wheeler gauge, the surprising result that there is noinduced quadrupole moment. We show that this conclusion is gauge dependent andthat the static, induced quadrupole moment for a black hole is inherentlyambiguous. For the orbiting moon and the central Schwarzschild hole, we find(in agreement with a recent result of Poisson) a time-varying inducedquadrupole moment that is proportional to the time derivative of the moon'stidal field. As a partial analog of a result derived long ago by Hartle for aspinning hole and a stationary distant companion, we show that the orbitingmoon's tidal field induces a tidal bulge on the hole's horizon, and that therate of change of the horizon shape leads the perturbing tidal field at thehorizon by a small angle.Comment: 14 pages, 0 figures, submitted to Phys. Rev.

Tidal coupling of a Schwarzschild black hole and circularly orbiting moon