Existence of a smooth family of null hypersurfaces srrounding a black hole in the presence of tails

CARLOS KOZANEH; ALEJANDRO PÉREZ; OSVALDO M. MORESCHI

Varsovia

Conferencia; GR20, 20th International Conference on General Relativity; 2013

International Society on General Relativity and Gravitation

The dynamical description of the late phase of gravitational collapse (described by a black hole spacetime) remains an open problem in general relativity.Any analytical insight into the resolution of this problem is clearly very relevant both physically as well as mathematically.A useful step in trying to solve Einstein's equations in this very dynamical regime is to find a set ofphysically defined coordinates and associated frames that are well behaved both near the eventhorizon and at null infinity.Near null infinity the so-called Bondi coordinates provide a physical frameworkwhere the notion of momentum, angular momentum, and their flux laws adopt a very simple expression.Since the event horizon is approached in the limit where $u$ (the retarded Bondi time) goes to infinity,the Bondi coordinate system can not describe the geometry in a neighborhoodof the horizon. However, one can try to use the same family of null hypersurfaces $u=$constant,in order to introduce a new function $w=w(u)$ which is regular at the event horizon, i.e., the function $w$ should have a finite value as $u\to \infty$.In this work we will show that this idea can be implemented in spacetimes that are sufficiently close(in the sense of perturbation theory) to the the Schwarzschild space-time. We show that the power-law decay modes found in linear perturbations of Schwarzschild black holes,generally called tails, do not produce caustics on a naturally defined family of null surfacesin the neighbourhood of $i^{+}$ of a black hole horizon.This null congruence thus yields a well behaved coordinate system both near the eventhorizon and at null infinity.