ar X iv : g r - qc / 0 31 00 66 v 2 1 4 A ug 2 00 6 The wave equation on the Schwarzschild metric II : Local Decay for the spin 2 Regge Wheeler equation

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 > 32 , there is a constant C, depending on the initial condition through the energy norm, so that ∫ ∞