Physical-space estimates for axisymmetric waves on extremal Kerr spacetime

TL;DR

This paper derives physical-space-based energy decay estimates for axisymmetric waves on extremal Kerr black holes, simplifying previous methods and extending results to the extremal case.

Contribution

It introduces a new classical-current method to obtain Morawetz estimates for extremal Kerr, simplifying and extending prior frequency-based approaches.

Findings

Established integrated local energy decay for axisymmetric waves on extremal Kerr.

Simplified the derivation of Morawetz estimates using classical currents.

Extended sub-extremal methods to the extremal Kerr case.

Abstract

We study axisymmetric solutions to the wave equation on extremal Kerr backgrounds and obtain integrated local energy decay (or Morawetz estimates) through an analysis \textit{exclusively in physical-space}. Boundedness of the energy and Morawetz estimates for axisymmetric waves in extremal Kerr were first obtained by Aretakis through the construction of frequency-localized currents used in particular to express the trapping degeneracy. Here we extend to extremal Kerr a method introduced by Stogin in the sub-extremal case, simplifying Aretakis' derivation of Morawetz estimates through purely classical currents.