Back Reaction of Gravitational Radiation on the Schwarzschild Black Hole

We address some of the issues that appear in the study of back reaction in Schwarzschild backgrounds. Our main object is the effective energy-momentum tensor (EEMT) of gravitational perturbations. It is commonly held that only asymptotically flat or radiation gauges can be employed for these purposes. We show that the traditional Regge–Wheeler gauge for perturbations of the Schwarzschild metric can also be used for calculating physical quantities both at the horizon and at infinity, even if the metric components themselves diverges there. In particular, components of the EEMT obey the same asymptotic behaviour as the stress-energy tensor of a scalar field in the Schwarzschild background. We obtain a well-defined inner product for gravitational waves, and show how it leads to a finite normalization prescription. We also use the Gtr equation to compute the monopole contribution to the mass-energy carried by the gravitational waves.