I am interested to know what is the best we can say about stability of Schwartzschild 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 that maybe there is a way to show the stronger stability requirement by means of linearized perturbations that satisfy the equations of motion..)