How pure is the tail of gravitational collapse ?

Abstract

Waves propagating in a curved spacetime develop tails. In particular, it is well established that the late-time dynamics of gravitational collapse is dominated by a power-law decaying tail of the form $M{t}^{-(2l+3)}$, where $M$ is the black-hole mass. It should be emphasized, however, that in a typical evolution scenario there is a considerable time window in which the signal is no longer dominated by the black-hole quasinormal modes, but the leading order power-law tail has not yet taken over. Higher-order terms may have a considerable contribution to the signal at these intermediate times. It is therefore of interest to analyze these higher-order corrections to the leading-order power-law behavior. We show that the higher-order contamination terms die off at late times as $M^2{t}^{-4}\ln (t/M)$ for spherical perturbations, and as $M^2{t}^{-(2l+4)}{\ln }^2(t/M)$ for non-spherical $(l\ne 0)$ perturbations. These results imply that the leading-order power-law tail becomes "pure" (namely, with less than 1% contamination) only at extremely late times of the order of $10^{4}M$.

8 pages. This is an extended version of the one published in Class. Quantum Grav. 26 (2009) 028001

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