general relativity - Perturbation of a Schwarzschild Black Hole

If we have a perfect Schwarzschild black hole (uncharged and stationary), and we "perturb" the black hole by dropping in a some small object. For simplicity "dropping" means sending the object on straight inward trajectory near the speed of light.

Clearly the falling object will cause some small (time dependent) curvature of space due to its mass and trajectory, and in particular, once it passes the even horizon, the object will cause some perturbation to the null surface (horizon) surrounding the singularity (intuitively I would think they would resemble waves or ripples). Analogously to how a pebble dropped in a pond causes ripples along the surface.

Is there any way to calculate (i.e. approximate numerically) the effect of such a perturbation of the metric surrounding the black hole?, and specifically to calculate the "wobbling" of the null surface as a result of the perturbation,maybe something analogous to quantum perturbation theory?

Or more broadly, does anyone know of any papers or relevant articles about a problem such as this?

Answer

Your intuitive picture is basically correct. If you perturb a black hole it will respond by "ringing". However, due to the emission of gravitational waves and because you have to impose ingoing boundary conditions at the black hole horizon, the black hole will not ring with normal-modes, but with quasi-normal modes (QNMs), i.e., with damped oscillations. These oscillations depend on the black hole parameters (mass, charge, angular momentum), and are therefore a characteristic feature for a given black hole.

Historically, the field of black hole perturbations was pioneered by Regge and Wheeler in the 1950ies.

For a review article see gr-qc/9909058

For the specific case of the Schwarzschild black hole there is a very nice analytical calculation of the asymptotic QNM spectrum in the limit of high damping by Lubos Motl, see here. See also his paper with Andy Neitzke for a generalization.

Otherwise usually you have to rely on numerical calculations to extract the QNMs.