Analysis of linear waves near the Cauchy horizon of cosmological black holes

TL;DR

This paper demonstrates that linear scalar waves remain bounded, continuous, and decay exponentially near the Cauchy horizon of certain cosmological black holes, using microlocal analysis and spacetime modifications.

Contribution

It introduces a novel approach by modifying spacetime beyond the Cauchy horizon to apply microlocal regularity and scattering theory to wave behavior.

Findings

Waves are bounded and continuous up to the Cauchy horizon.

Waves decay exponentially fast to a constant along the horizon.

The method applies to Reissner-Nordström-de Sitter and Kerr-de Sitter spacetimes.

Abstract

We show that linear scalar waves are bounded and continuous up to the Cauchy horizon of Reissner-Nordstr\"om-de Sitter and Kerr-de Sitter spacetimes, and in fact decay exponentially fast to a constant along the Cauchy horizon. We obtain our results by modifying the spacetime beyond the Cauchy horizon in a suitable manner, which puts the wave equation into a framework in which a number of standard as well as more recent microlocal regularity and scattering theory results apply. In particular, the conormal regularity of waves at the Cauchy horizon - which yields the boundedness statement - is a consequence of radial point estimates, which are microlocal manifestations of the blue-shift and red-shift effects.