Pieter Blue

Phase Space Analysis on some Black Hole Manifolds

Abstract The Schwarzschild and Reissner–Nordstrom solutions to Einsteins equations describe space–times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space–time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L 6 norm in space decays like t − 1 3 . This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an ϵ loss of angular derivatives.

DECAY OF THE MAXWELL FIELD ON THE SCHWARZSCHILD MANIFOLD

We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild radial coordinate takes values in a bounded interval away from the event horizon, we obtain decay for all components of the Maxwell field at a rate which is bounded by the inverse of the standard time coordinate. We use vector field methods and no not require a spherical harmonic decomposition. In outgoing regions, where the Regge–Wheeler tortoise coordinate grows at least linearly with the time coordinate, we obtain decay rates for each of the null components. These rates are similar to the rates in flat space but weaker. Along the event horizon and in ingoing regions, where the Regge–Wheeler coordinate is negative and the outgoing, Eddington–Finkelstein null-coordinate is positive, all components (normalized with respect to an ingoing null basis) decay at a rate which is bounded by the inverse of the outgoing null coordinate.

Second order symmetry operators

Using systematic calculations in spinor language, we obtain simple descriptions of the second order symmetry operators for the conformal wave equation, the Dirac-Weyl equation and the Maxwell equation on a curved four dimensional Lorentzian manifold. The conditions for existence of symmetry operators for the different equations are seen to be related. Computer algebra tools have been developed and used to systematically reduce the equations to a form which allows geometrical interpretation.

Spin geometry and conservation laws in the Kerr spacetime

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

Decay of solutions to the Maxwell equation on the Schwarzschild background

(2,0)

Hidden symmetries and decay for the Vlasov equation on the Kerr spacetime

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.

HIDDEN SYMMETRIES AND DECAY FOR THE WAVE EQUATION ON THE KERR SPACETIME

A new Morawetz or integrated local energy decay estimate for Maxwell test fields on the exterior of a Schwarzschild black hole spacetime is proved. The proof makes use of a new superenergy tensor

Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzschild Space

H_{ab}

Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates

defined in terms of the Maxwell field and its first derivatives. The superenergy tensor, although not conserved, yields a conserved higher order energy current

The wave equation on the Schwarzschild metric II. Local decay for the spin-2 Regge–Wheeler equation

H_{ab} (\partial_t)^b