Spin geometry and conservation laws in the Kerr spacetime

TL;DR

This paper reviews the geometry of Kerr black holes and introduces new tools like a novel energy-momentum tensor for Maxwell fields, aiding in understanding field decay and stability in these spacetimes.

Contribution

It presents a new energy-momentum tensor derived from a valence (2,0) Killing spinor, facilitating decay estimates for Maxwell fields on Kerr spacetime.

Findings

Construction of symmetry operators and conserved currents

Introduction of a new energy-momentum tensor for Maxwell fields

Outline of decay estimates for Maxwell test fields

Abstract

In this paper we will review some facts, both classical and recent, concerning the geometry and analysis of the Kerr and related black hole spacetimes. This includes the analysis of test fields on these spacetimes. Central to our analysis is the existence of a valence $(2,0)$ Killing spinor, which we use to construct symmetry operators and conserved currents as well as a new energy momentum tensor for the Maxwell test fields on a class of spacetimes containing the Kerr spacetime. We then outline how this new energy momentum tensor can be used to obtain decay estimated for Maxwell test fields. An important motivation for this work is the black hole stability problem, where fields with non-zero spin present interesting new challenges. The main tool in the analysis is the 2-spinor calculus, and for completeness we introduce its main features.