Saturating The Bekenstein-Hawking Entropy Bound With Initial Data Sets For Gravitational Collapse

It is possible to find initial states for gravitational collapse whose entropy approximately saturates the Bekenstein-Hawking entropy of the final black hole. The prototypical example of such a state is that envisaged by Zurek and Thorne, and also by Susskind: for a black hole of mass $\sim M$ , a number $\sim M^2$ of quanta with energies of order $1/M$ are accreted on a timescale of order $\sim M^3$, an approximate time-reverse of Hawking evaporation. There is lore that all initial states which saturate the Bekenstein-Hawking entropy must involve a formation timescale of this order, $M^3$ , and not the much shorter dynamical timescale $M$ . Counterexamples to this lore have been found by Sorkin, Wald and Zhang, and also by Hsu and Reeb, in the form of semiclassical initial data sets. However the spacetimes that correspond to these counterexamples possess white holes in the past, as well as black holes in the future, which casts doubt on their physical relevance. We modify the counterexamples to eliminate the white holes, yielding formation timescales of order $M^2$ , and argue that the lore is unfounded.