Spin geometry and conservation laws in the Kerr spacetime

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.

30 pages. To appear in the volume "The Centenary of General Relativity" in "Surveys in Differential Geometry", edited by Lydia Bieri and Shing-Tung Yau, in the series "Surveys in Differential Geometry"

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Cited by in corpus (8)

• Stability for linearized gravity on the Kerr spacetime
• Decay of solutions to the Maxwell equation on the Schwarzschild background
• Symmetries of linearized gravity from adjoint operators
• Extended-body motion in black hole spacetimes: What is possible?
• Conserved currents for electromagnetic fields in the Kerr spacetime
• Minkowski, Schwarzschild and Kerr Metrics Revisited
• Conserved quantities and integrability for massless spinning particles in general relativity
• Numerical investigation of the late-time tails of the solutions of the Fackerell-Ipser equation