Abstract
We consider solutions of the scalar wave equation \({\Box_g\phi=0}\), without symmetry, on fixed subextremal Reissner-Nordstrom backgrounds \({(\mathcal{M}, g)}\) with nonvanishing charge. Previously, it has been shown that for ϕ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast on the event horizon \({\mathcal{H}^+}\). Using this, we show here that ϕ is in fact uniformly bounded, \({|\phi| \leq C}\), in the black hole interior up to and including the bifurcate Cauchy horizon \({\mathcal{C}\mathcal{H}^+}\), to which ϕ in fact extends continuously. The proof depends on novel weighted energy estimates in the black hole interior which, in combination with commutation by angular momentum operators and application of Sobolev embedding, yield uniform pointwise estimates. In a forthcoming companion paper we will extend the result to subextremal Kerr backgrounds with nonvanishing rotation.
Highlights
The Reissner-Nordström spacetime (M, g) is a fundamental 2-parameter family of solutions to the Einstein field equations coupled to electromagnetism, cf. Fig. 1 for the conformal representation of the subextremal case, M > |e| = 0, with e the charge and M the mass of the black hole
The purpose of the present work is to extend the investigation to the interior of the black hole, up to and including the Cauchy horizon CH+
We first state an analogous result to our Theorem 1.1 for general subextremal Kerr black holes
Summary
As a first attempt towards investigation of the stability of the Cauchy horizon under perturbations without symmetry, we employ (1) on a fixed Reissner-Nordström background (M, g) as a toy model for the full nonlinear Einstein field equations, cf (10). 3 we give a brief review of estimates obtained along H+ from previous work, [6,15] and [41], for φ arising from sufficiently regular initial data on a Cauchy hypersurface In subsection 4.4 we propagate the estimate to the region RV I up to the bifurcation two-sphere, and obtain a bound for the energy flux globally in the black hole interior completing the proof of Theorem 1.2. 5. We first state an analogous result to our Theorem 1.1 for general subextremal Kerr black holes (to appear as Theorem 1.1 of the follow up paper). We will discuss what our results suggest about the nonlinear dynamics of the Einstein equations themselves
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