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).