Boundedness of massless scalar waves on Kerr interior backgrounds

TL;DR

This paper proves that solutions to the massless scalar wave equation on Kerr black hole interiors remain uniformly bounded up to the Cauchy horizon, extending previous decay results and employing energy estimates and commutation techniques.

Contribution

It establishes uniform boundedness of scalar waves inside Kerr black holes, including at the Cauchy horizon, using decay rates and advanced analytical methods.

Findings

Solutions are uniformly bounded up to the Cauchy horizon.

The scalar field extends continuously to the Cauchy horizon.

The analysis adapts techniques from Reissner--Nordström backgrounds to Kerr.

Abstract

We consider solutions of the massless scalar wave equation $\Box_g\psi=0$, without symmetry, on fixed subextremal Kerr backgrounds $(\mathcal M, g)$. It follows from previous analyses in the Kerr exterior that for solutions $\psi$ arising from sufficiently regular data on a two ended Cauchy hypersurface, the solution and its derivatives decay suitably fast along the event horizon $\mathcal H^+$. Using the derived decay rate, we show that $\psi$ is in fact uniformly bounded, $|\psi|\leq C$, in the black hole interior up to and including the bifurcate Cauchy horizon $\mathcal C\mathcal H^+$, to which $\psi$ in fact extends continuously. In analogy to our previous paper, [30], on boundedness of solutions to the massless scalar wave equation on fixed subextremal Reissner--Nordstr\"om backgrounds, the analysis depends on weighted energy estimates, commutation by angular momentum operators and application of Sobolev embedding. In contrast to the Reissner--Nordstr\"om case the commutation leads to additional error terms that have to be controlled.