Boundedness of massless scalar waves on Kerr interior backgrounds

Abstract

We consider solutions of the massless scalar wave equation ${\square }_g\psi =0$, 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, $|\psi |\le C$, 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--Nordström backgrounds, the analysis depends on weighted energy estimates, commutation by angular momentum operators and application of Sobolev embedding. In contrast to the Reissner--Nordström case the commutation leads to additional error terms that have to be controlled.

54 pages, 13 figures