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 $M$, a number $\sim M^2$ of quanta with energies of order $\sim M^{-1}$ are accreted on a time scale 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 time scale of this order, $\sim M^3$, and not the much shorter dynamical time scale $\sim M$. 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 time scales of order $\sim M^2$, and argue that the lore is unfounded.

Figure 1

This is the plot of $\sqrt{1-r^{2}k(r)}$ versus $r$. As expected, $1-r^{2}k(r)\ll 1$ for $7\le r\le 99/10$.

Figure 2

This is the plot of $\log [S(r)/S_{\mathrm{BH}}]$ versus $\log [M(r)]$ for the example we have constructed here. Notice that the contribution of the spherical shells at $r\ge 7$ brings the entropy given in Eq. (14) close to the Bekenstein-Hawking entropy $S_{\mathrm{BH}}=3.3\times 10^5$.

Figure 3

This is the plot of $t_{\mathrm{H}}-t_{\mathrm{\Sigma }}$ versus radius $r$. The future event horizon is indeed to the future of the slice $\mathrm{\Sigma }$.

Figure 4

Here we plot $\log [1-2M(r)/R(t_{\mathrm{H}},r)]$ versus $\log [M(r)]$ along the future event horizon. Notice that none of the spherical shells are trapped, except for the outermost one, for which $\log [1-2M(r_{\mathrm{out}})/R(t_{\mathrm{H}},r_{\mathrm{out}})]$ diverges.

Figure 5

This is the plot of $\log [n^{-1/3}]$ (solid blue line); $\log [(8\pi m{)}^{-1}]$ (solid violet line, which is collinear with the solid blue line); $\log [(16\pi /3{)}^{1/3}m/|\mathbf{D}m|]$ (dotted brown line); $\log [{\rho }^{-1/4}]$ (green dot-dashed line), and $\log [\mathcal{L}]$ (blue dashed line) versus $\log [M(r)]$ along the future event horizon.

Figure 6

This is the plot of $|\mathcal{L}|$ (solid blue line) and $\mathcal{H}$ (dashed purple line) versus $|\eta -{\eta }_{\mathrm{\Phi }}|$ as we evolve the spacetime backwards in time. Notice that $|\mathcal{L}|$ converges to $\mathcal{H}$ very quickly.

Figure 7

This is the plot of the eigenvalues of the modified stress-energy tensor versus $\log [R(t,r)/R(t_{\mathrm{\Phi }},r)]$ for spherical shells at three different radii.

Figure 8

This is the Penrose diagram of a collapsing object embedded in a Schwarzschild exterior. The ingoing null rays with advanced Eddington-Finkelstein coordinates $v_1$ and $v_2$ intersect the collapsing object at $({\overline{t}}_1,2M)$ and $({\overline{t}}_2,{\overline{r}}_2\approx 2M)$, respectively.

Figure 9

This is the plot of $t_{\mathrm{H}}$ (solid blue line) and $t_0(r)$ (dashed violet line) versus $r$. The future event horizon intersects the curvature singularity at $r\approx 8.657$.

Figure 10

Here we have plotted the solution of Eq. (36) for a number of shells. As can be seen from the figure, shells expand without any crossing.