Geodesic structure and quasinormal modes of a tidally perturbed spacetime

Tidal perturbations play an important role in the study of the dynamics in the classical two-body system. Understanding tidal effects in strong-field regions may allow one to use gravitational-wave or electromagnetic observations to locate or constraint the location of possible companions. Here, we investigate how timelike and null geodesics of a Schwarzschild black hole are affected in the presence of a companion. There is a panoply of new effects. In some limiting cases, we find analytical solutions for closed null or timelike geodesics. Our results show that light ring period as measured by a far-away observer can be eiter shorter or longer, depending on the location of the companion. We also show that there are closed lightlike trajectories which are elliptic (for equatorial companions), and that timelike particles are affected in a similar manner. Finally, we attempt at estimating the ringdown from tidally perturbed geometries. Our results indicate that there are two stages in the relaxation of such geometries, one associated with a prompt decay of waves around the deformed photonsphere, and a later relaxation of the global geometry. These results are consistent with previous, full numerical studies.