Linear perturbations of a Schwarzschild black hole

Abstract

We firstly numerically recalculate the Ricci tensor of non-stationary axisymmetric space-times (originally calculated by Chandrasekhar) and we find some discrepancies both in the linear and non-linear terms. However, these discrepancies do not affect the results concerning linear perturbations of a Schwarzschild black hole. Secondly, we use these Ricci tensors to derive the Zerilli and Regge-Wheeler equations and use the Newman-Penrose formalism to derive the Bardeen-Press equation. We show the relation between these equations because they describe the same linear perturbations of a Schwarzschild black hole. Thirdly, we illustrate heuristically (when the angular momentum (l) is 2) the relation between the linearized solution of the Einstein vacuum equations obtained from the Bondi-Sachs metric and the Zerilli equation, because they describe the same linear perturbations of a Schwarzschild black hole. Lastly, by means of a coordinate transformation, we extend Chandrasekhar's results on linear perturbations of a Schwarzschild black hole to the Bondi-Sachs framework.