Abstract
A review of the current status of the linear stability of black holes and naked singularities is given. The standard modal approach, that takes advantage of the background symmetries and analyze separately the harmonic components of linear perturbations, is briefly introduced and used to prove that the naked singularities in the Kerr–Newman family, as well as the inner black hole regions beyond Cauchy horizons, are unstable and therefore unphysical. The proofs require a treatment of the boundary condition at the timelike boundary, which is given in detail. The nonmodal linear stability concept is then introduced, and used to prove that the domain of outer communications of a Schwarzschild black hole with a non-negative cosmological constant satisfies this stronger stability condition, which rules out transient growths of perturbations, and also to show that the perturbed black hole settles into a slowly rotating Kerr black hole. The encoding of the perturbation fields in gauge invariant curvature scalars and the effects of the perturbation on the geometry of the spacetime is discussed. These notes follow from a course delivered at the V José Plínio Baptista School of Cosmology, held at Guarapari (Espírito Santo) Brazil, from 30 September to 5 October 2021.
Highlights
The Kerr two-parameter family of metrics is arguably the most important solution of Einstein’s equations
This review is organized as follows: in Section 2 we briefly introduce the ideas of mode decomposition of linear perturbations based on background symmetries; in Section 3 we review the proofs of instability of the Schwarzschild and Kerr naked singularities (NSs) instabilities, and of the instability of the region beyond the Cauchy horizon of Kerr black hole (BH); in Section 4 we review the proof of nonmodal stability of the Schwarzschild (Schwarzschild de Sitter)
This is done in detail in the paper [14], where the initial value formulation for the even linearized Einstein Equation (LEE) is dealt with, and it is shown that generic initial perturbation data have a projection on the unstable modes and develop an instability
Summary
The Kerr two-parameter family of metrics is arguably the most important solution of Einstein’s equations. Kerr and Schwarzschild metrics, are stable under perturbations Otherwise they would be irrelevant solutions of General Relativity. This review is organized as follows: in Section 2 we briefly introduce the ideas of mode decomposition of linear perturbations based on background symmetries; in Section 3 we review the proofs of instability of the Schwarzschild and Kerr NS instabilities, and of the instability of the region beyond the Cauchy horizon of Kerr BHs; in Section 4 we review the proof of nonmodal stability of the Schwarzschild (Schwarzschild de Sitter). The discussion is centered on the effect of perturbations on the background geometry
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