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

Abstract

Recently, it has been shown that the wave equation for a scalar field on the exterior part of the Schwarzschild manifold satisfies local decay estimates useful for scattering theory and global existence [2]. The extension for the linearized Einstein equation is considered here. In 1957, Regge and Wheeler investigated spin 2 tensor fields on the Schwarzschild manifold [4]. They classified such fields into two types, which they called even and odd. For the odd fields, they were able to reduce the problem to an equation for a scalar field very similar to the wave equation for scalar fields on the Schwarzschild manifold. In 1970, Zerilli extended their results to include the even case; although, the equation for the even case is significantly more complicated and bears less resemblance to the wave equation for a scalar field [9]. Teukolsky has done a related reduction for the rotating Kerr black hole [6] which has been used to investigate the stability of the black holes [8].This paper extends the local decay estimate for the scalar wave equation of [2] to the Regge-Wheeler equation. Many of the proofs used here follow [2]. We obtain the following for r * the standard Regge-Wheeler co-ordinate and β > 3 2 , there is a constant C, depending on the initial condition through the energy norm, so that Co-ordinates and equationsThe Schwarzschild manifold describes a static black hole solution to the Einstein equation. The exterior of the black hole is most easily described by (t, r, θ, φ) ∈ R × (2M, ∞) × S 2 with the metricTo simplify the problem, Regge and Wheeler ([4]) introduced a new radial co-ordinate, r * , satisfyingThis allows the definition of a space like manifoldThe old co-ordinate r is now treated as a function of r * . In these new co-ordinates, the Regge-Wheeler equation for a scalar field u : R×M → R which determines the behavior of the odd-type tensor fields isü + Hu = 0 (1.4) 1

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