Boundedness and decay of scalar waves at the Cauchy horizon of the Kerr spacetime

TL;DR

This paper proves boundedness, regularity, and decay of scalar waves near the Cauchy horizon in Kerr spacetime, extending techniques to show polynomial decay without symmetry assumptions, and complements existing blow-up results.

Contribution

It introduces new microlocal and scattering methods to establish decay and boundedness of scalar waves at the Cauchy horizon in Kerr spacetime without symmetry assumptions.

Findings

Scalar waves are bounded and decay polynomially near the Cauchy horizon.

The results hold for subextremal Reissner-Nordström and slowly rotating Kerr spacetimes.

The decay is shown relative to a Sobolev space of order slightly above 1/2.

Abstract

Adapting and extending the techniques developed in recent work with Vasy for the study of the Cauchy horizon of cosmological spacetimes, we obtain boundedness, regularity and decay of linear scalar waves on subextremal Reissner-Nordstr\"om and (slowly rotating) Kerr spacetimes, without any symmetry assumptions; in particular, we provide simple microlocal and scattering theoretic proofs of analogous results by Franzen. We show polynomial decay of linear waves relative to a Sobolev space of order slightly above $1/2$. This complements the generic $H^1_{\mathrm{loc}}$ blow-up result of Luk and Oh.