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I am interested to know what is the best we can say about stability of Schwarzschild and Reissner-Nordstrom black holes.

I found some who study the behavior of perturbations that satisfy the linearized Einstein equations and claim that there is stability when these fluctuations become smaller as time passes (or generally as the system evolves).

My understanding is that one should verify that the lowest eigenvalue of the Hessian around the solution is greater or equal to zero (the Hessian is the matrix of the second functional derivatives around the black hole background).

This would tell us that there are no possible fluctuations that make the action quadratically smaller than on the solution.

If we don't check this, then taking the fluctuation in the "direction(s)" of the negative eigenvalue(s) we would obtain solutions with an action lower than on the background we are using.

I have not found any proof that the black hole spacetimes I mentioned are stable in this sense, and I was wandering if this proof exist or what are the current best results.

I am interested in particular in asymptotically flat, static black hole spacetimes, but if somebody proved anything with a cosmological constant I appreciate all the same!

Also if anybody knows of more general results (non necessarily from GR) that could be of help please do tell! (I am thinking to things like a way to show the stronger stability requirement by means of linearized perturbations that satisfy the equations of motion..)

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  • $\begingroup$ Could you tell us what references you've found so far? For instance, googling on "stability schwarzschild spacetime" gives as one of the first hits the following, which may be relevant: arxiv.org/abs/1711.07597 . Have you tried doing a citation search on the papers that proved stability with respect to linear perturbations, such as Whiting, J. Math. Phys. 30, 1301 (1989)? $\endgroup$ – Ben Crowell May 20 at 14:43
  • $\begingroup$ Dear @BenCrowell, from the references you pointed out I understand they restrict themselves to only some classes of perturbations. Moreover the 900 pages review deals with non-linear stability, which is a stronger result than the standard full linear stability that I inquired about. I'll check references in the second paper. $\endgroup$ – AoZora May 20 at 15:28
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Let me just start with saying that the results on the stability of black holes are not complete yet. The question is whether 1) a black hole is stable with respect to linear perturbations, and 2) whether a black hole is stable to finite (non-linear) perturbations. I am just going to tell you that 2) has very few results concerning mainly the Schwarzschild background and special perturbations (you can also "cheat" by making your background de-Sitter, see Hitz & Vasy, 2016), and it will probably still take a while to obtain a more complete picture (it took decades to prove non-linear stability of even something as simple as the Minkowski background).


However, the answer to question 1) is yes for subextremal black holes, and in the exterior and on the horizon (things get funny at Cauchy horizons and curvature singularities). This is known analytically for Kerr black holes (Whiting 1989) and Reissner-Nordström black holes (Moncrief 1974), this naturally includes the Schwarzschild limit (this special case was investigated way back by Regge & Wheeler 1957). A special case is Kerr-Newman, where perturbations do not have separable equations and the results are thus only numerical (but affirmative, Dias et al. 2015). The method used to show this is mode stability, that is, decomposing the perturbation into (tensorial) spherical harmonics and finding frequency eigenmodes. Every frequency eigenmode generally has a real and an imaginary part, corresponding to the oscillation of the mode and exponential decay/growth. One then shows that physical modes can only decay (physicality is expressed in boundary conditions). A physical perturbation decomposed into eigemodes must thus decay as well.

A lot of concern is also spent on which perturbations and with which properties have convergent decompositions into such modes. Ultimately, many people ditch mode stability but rather analyse the general initial value problem by various quantitative bounds and estimates. This approach unambiguously resolves various issues connected to the global structure of the space-time (throat, white hole, black hole,Cauchy horizon,...), but to me as a physicist seeing the exact black hole space-times only as local-in-space-time approximations of the field, mode stability would be sufficient. So far, this program was finished only for the simpler case of a scalar wave equation on the black hole background (Dafermos et al., 2014).


The perturbations fulfill wave equations on the background and these will have an action formulation which itself can be viewed as a an expansion of the full action in functional space to second order in the perturbations (the linear order term vanishes because the background is a solution of $\delta S = 0$). A variation of this linearized action is proportional to a second functional derivative of the original one times the perturbation and the vanishing of the variation can be written as "$ h \cdot \delta^2\! S/\delta h^2 = 0$". So the solutions correspond to zero-eigenvectors of the Hessian of the action. The fact that this equation is fulfilled while a physical solution is unbounded would require that there are singularities and/or degeneracies in the Hessian as well. I do not know how that relates to your statements about the Hessian in the OP.

Ultimately, I believe having a negative spectrum in the Hessian would not indicate a classical instability, but rather a quantum one, as a semi-classical expansion of the path integral would suggest. In other words, the black hole state would tunnel into another state that would possibly not be reachable by a classical process. I am not aware of an analysis of this type for black holes, and it is not clear to me how should the end state of the tunneling look like.

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  • $\begingroup$ Thanks for the effort and the references. From what you say I cannot understand if the equation for the perturbations is generally taken as a constraint over the fluctuations (I saw papers doing that, and it does not make much sense, since you end up with only the zero modes while you should be looking for negative modes). If I understand correctly Regge and Wheeler (luckily) don't do it. Moncrief gives a weaker result cause only does the even sector. If as you say there is no clarity on whether the modes encode all the perturbations maybe the initial data approach is better.. $\endgroup$ – AoZora May 20 at 18:13
  • $\begingroup$ About the last observation, I agree to some extent. The negative mode fluctuations could be seen both as quantum and as classical in my opinion. The path integral does not suppress them more than the zero modes, on the contrary they will be enhanced with respect to the other fluctuations since they make the action smaller. Of course it is an instability so you can expect this. But maybe I am missing your reason to say that they should not have a classic nature.. $\endgroup$ – AoZora May 20 at 18:19
  • $\begingroup$ I must say that looking at it again, Regge and Wheeler also impose the equation of motion on the modes. I don't understand how it is possible to argue stability if one starts by analyzing only the zero modes. By restricting to zero modes there is no control over negative modes: zero modes can bring to whatever good perturbations, but if your system is randomly stirred in the direction of a negative mode it will follow that and leave the solution you are testing. $\endgroup$ – AoZora May 21 at 7:47
  • $\begingroup$ @france95 Equations of motion of the perturbations are always taken into account. I do not understand in which sense you could consider the dynamics of the perturbed space-time if you were not to impose the rules of the dynamics. What you call "zero modes" are simply real classical evolutions. What you call "negative and positive modes" are simply not classical solutions; a classical black hole will never behave like that, Einstein equations forbid it. Moncrief also does odd-parity here. $\endgroup$ – Void May 21 at 8:51
  • $\begingroup$ I see your point. All the same I think it should be important to exclude negative modes, because random or quantum fluctuations could make the system decay to other solutions. Moreover I don't know if you can really claim to have a classical solution if it has unstable directions, since these indicate that the configuration is a minimum only to the first order in perturbations. The analogue with a bead rolling on a saddle is misleading, since the instability is only in the potential. $\endgroup$ – AoZora May 21 at 10:20

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