# 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?

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The perturbations of a black hole are given by the so-called quasi-normal modes. The "quasi" is because unlike the regular normal modes, a black hole's oscillations are damped. There is a very nice paper on this by @Lubos Motl ref. Though for a intro to the basics you might want to start with the review paper by Kokkotas and Schmidt. – user346 Apr 9 '11 at 7:55
Deepak Vaid: while typing my answer your comment appeared - I think you could easily have upgraded it to an answer; anyway, I upvoted your comment... – Daniel Grumiller Apr 9 '11 at 8:11
Thanks, @Deepak, your generosity is appreciated but your compliment is undeserved. ;-) I didn't discover the concept of quasinormal modes or the fact that the ringing modes describing the exponential approach to the static black hole solution at later times. I (and, later, we with Andy Neitzke) just analytically calculated the quasinormal frequencies (of the highly damped modes) for some common black holes. However, the long-term behavior of the deviation from the sphericity etc. is governed by the low-lying quasinormal modes with a small Im(omega) - I haven't found anything new on them. – Luboš Motl Apr 9 '11 at 8:14
@Daniel - thanks. – user346 Apr 9 '11 at 8:15

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.

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Thanks a lot, @Daniel, but it is very undeserved. I and we with Andy just calculated highly damped quasinormal modes (with high $Im(\omega)$ of the frequency) of the Schwarzschild in 3+1 and then any dimension and the Reissner-Nordstrom one, with some incomplete results on others. The long-time behavior is dominated by the low-lying quasinormal modes (those that are not highly damped) with the lowest Im(omega), those that are hard to calculate. In the Schwarzschild time, $\delta g_{\mu\nu}$ then goes like $\exp(-Im(\omega)_{\rm min} t)$. – Luboš Motl Apr 9 '11 at 8:17
Moreover, the main reason why my and our analytic calculation became more famous than it deserved was that it proved the appearance of $ln(3)$ in the frequencies, in agreement with a heuristic guess based on numerical calculations, and this constant has been used to defend "discrete models of black holes", including loop quantum gravity, before it became very clear that the result doesn't generalize to other black holes and the "discrete interpretation" is misguided for many other reasons, too. See also motls.blogspot.com/2005/03/… – Luboš Motl Apr 9 '11 at 8:20
Lubos: I agree that for almost all applications one is interested in the opposite limit than the one you considered. I should have pointed this out in the answer. I still like your calculation, even if it does not lead to a "killer application". It always gives me a warm fuzzy feeling when something can be calculated analytically that previously was accessible only by numerics. – Daniel Grumiller Apr 9 '11 at 8:34
lol, +1 for warm fuzzy feeling :) – Marek Apr 9 '11 at 12:21
Hi, thanks for the good answer. I was wondering about the review article; they arrive at the equation (20) by simply stating they assume the $h_{\mu \nu}$ is of the form of eq. (20). How do they justify this as a starting point? – Otto Mar 23 at 2:26