Late-time decay of gravitational and electromagnetic perturbations along the event horizon

We study analytically, via the Newman-Penrose formalism, the late-time decay of linear electromagnetic and gravitational perturbations along the event horizon (EH) of black holes. We first analyze in detail the case of a Schwarzschild black hole. Using a straightforward local analysis near the EH, we show that, generically, the “ingoing” (s>0) component of the perturbing field dies off along the EH more rapidly than its “outgoing” (s<0) counterpart. Thus, while along r=const>2M lines both components of the perturbation admit the well-known t-2l-3 decay rate, one finds that along the EH the s<0 component dies off in advanced time v as v-2l-3, whereas the s>0 component dies off as v-2l-4. We then describe the extension of this analysis to a Kerr black hole. We conclude that for axially symmetric modes the situation is analogous to the Schwarzschild case. However, for non-axially symmetric modes both s>0 and s<0 fields decay at the same rate (unlike in the Schwarzschild case).