Linear Instability of the Reissner-Nordström Cauchy Horizon

Abstract

This work studies solutions of the scalar wave equation \[\Box_gϕ=0\] on a fixed subextremal Reissner-Nordström spacetime with non-vanishing charge $q$ and mass $M$. In a recent paper, Luk and Oh established that generic smooth and compactly supported initial data on a Cauchy hypersurface lead to solutions which are singular in the $W_{loc}^{1,2}$ sense near the Cauchy horizon in the black hole interior, and it follows easily that they are also singular in the $W_{loc}^{1,p}$ sense for $p>2$. On the other hand, the work of Franzen shows that such solutions are non-singular near the Cauchy horizon in the $W_{loc}^{1,1}$ sense. Motivated by these results, we investigate the strength of the singularity at the Cauchy horizon. We identify a sufficient condition on the black hole interior (which holds for the full subextremal parameter range $0<|q|<M$) ensuring $W_{loc}^{1,p}$ blow up near the Cauchy horizon of solutions arising from generic smooth and compactly supported data for every $1<p<2$. We moreover prove that provided the spacetime parameters satisfy $\frac{2\sqrt{e}}{e+1}<\frac{|q|}{M}<1$, we in fact have $W_{loc}^{1,p}$ blow up near the Cauchy horizon for such solutions for every $1<p<2$. This shows that the singularity is even stronger than was implied by the work of Luk and Oh for this restricted parameter range.

arXiv commentSubmitted as MSc thesis, University of Cambridge, 2016

References in corpus (13)

• A proof of Price's law for the collapse of a self-gravitating scalar field
• Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space
• Analysis of linear waves near the Cauchy horizon of cosmological black holes
• Cosmic Censorship: As Strong As Ever
• A scattering theory for the wave equation on Kerr black hole exteriors
• Proof of linear instability of the Reissner-Nordström Cauchy horizon under scalar perturbations
• The wave equation on Schwarzschild-de Sitter spacetimes
• Phase Space Analysis on some Black Hole Manifolds
• Linear waves in the interior of extremal black holes I
• Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime
• Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space
• The interior of charged black holes and the problem of uniqueness in general relativity
• Instability results for the wave equation in the interior of Kerr black holes