High-order contamination in the tail of gravitational collapse

It is well known that the late-time behavior of gravitational collapse is dominated by an inverse power-law decaying tail. We calculate higher-order corrections to this power-law behavior in a spherically symmetric gravitational collapse. The dominant “contamination” is shown to die off at late times as $M^2{t}^{-4}\ln \mathrm{}(t/M).$ This decay rate is much slower than has been considered so far. It implies, for instance, that an “exact” (numerical) determination of the power index to within 1% requires extremely long integration times of the order of ${10}^{4}M.\mathrm{}$ We show that the leading order fingerprint of the black-hole electric charge is of the order of $Q^2{t}^{-4}.$