Boundedness of massless scalar waves on Reissner-Nordstr\"om interior backgrounds

TL;DR

This paper proves that solutions to the scalar wave equation on Reissner-Nordström black hole backgrounds remain uniformly bounded inside the black hole, extending continuously to the Cauchy horizon, using novel energy estimates and Sobolev techniques.

Contribution

It establishes uniform boundedness of scalar waves inside Reissner-Nordström black holes up to the Cauchy horizon, a new result for non-symmetric solutions.

Findings

Solutions are uniformly bounded inside the black hole

Scalar fields extend continuously to the Cauchy horizon

New weighted energy estimates are developed

Abstract

We consider solutions of the scalar wave equation $\Box_g\phi=0$, without symmetry, on fixed subextremal Reissner-Nordstr\"om backgrounds $({\mathcal M}, g)$ with nonvanishing charge. Previously, it has been shown that for $\phi$ 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 $\phi$ 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 $\phi$ 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.