Gauge-invariant perturbations of Reissner-Nordström black holes
Abstract
We present the details of previously published results on the stability of the Reissner-Nordstr\"om family of black holes and discuss, in some detail, the reduction techniques which were used to simplify the perturbation problem. We develop the Hamiltonian formalism for the perturbations of an arbitrary static solution of the Einstein-Maxwell equations (with vanishing magnetic field) and show explicitly that the perturbed constraints are the generators of the coordinate and electromagnetic gauge transformations of the canonical perturbation variables. We show that the perturbed constraints have vanishing Poisson brackets with one another and use this result as the basis for our reduction technique. The canonical transformations which accomplish the reduction and the perturbation Hamiltonian are given explicitly for the Reissner-Nordstr\"om problem. We include a stability result for the (odd- and even-parity) $L=1$ perturbations which were not previously considered.