Surveys in Differential Geometry

Volume 20 (2015)

Spin geometry and conservation laws in the Kerr spacetime

Pages: 183 – 226

DOI: http://dx.doi.org/10.4310/SDG.2015.v20.n1.a8

Authors

Lars Andersson (Albert Einstein Institute, Potsdam, Germany; and Department of Mathematics, Royal Institute of Technology, Stockholm, Sweden)

Thomas Bäckdahl (School of Mathematics, University of Edinburgh, Scotland)

Pieter Blue (School of Mathematics and the Maxwell Institute, University of Edinburgh, Scotland)

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.

Keywords

Kerr spacetime, spin geometry, black hole stability, Killing spinor, Morawetz type estimates

2010 Mathematics Subject Classification

35L65, 35Q61, 35Q76, 83C50, 83C57, 83C60

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