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” $\mathrm{}(s>\mathrm{}0\mathrm{})\mathrm{}$ component of the perturbing field dies off along the EH more rapidly than its “outgoing” $\mathrm{}(s<\mathrm{}0\mathrm{})\mathrm{}$ counterpart. Thus, while along $r=\mathrm{const}>2M$ lines both components of the perturbation admit the well-known $t^{-2l-3}$ decay rate, one finds that along the EH the $s<\mathrm{}0\mathrm{}$ component dies off in advanced time $v$ as $v^{-2l-3},$ whereas the $s>\mathrm{}0\mathrm{}$ 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 $s>\mathrm{}0\mathrm{}$ and $s<\mathrm{}0\mathrm{}$ fields decay at the same rate (unlike in the Schwarzschild case).