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

TL;DR

This paper analytically investigates the late-time decay rates of electromagnetic and gravitational perturbations along the event horizon of Schwarzschild and Kerr black holes, revealing mode-dependent decay behaviors.

Contribution

It provides a detailed local analysis of decay rates along the event horizon for both Schwarzschild and Kerr black holes, highlighting differences between axially symmetric and non-symmetric modes.

Findings

In Schwarzschild black holes, the $s<0$ component decays as $v^{-2l-3}$, while the $s>0$ component decays faster as $v^{-2l-4}.

In Kerr black holes, axially symmetric modes behave like Schwarzschild, but non-symmetric modes decay at the same rate for both components.

The decay rates depend on the mode symmetry and the spin component of the perturbations.

Abstract

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).