An equation of state in the limit of high densities

We take string theory in a box of volume $V$, and ask for the entropy $S(E,V)$. We let $E$ exceed the value $E_{bh}$ corresponding to the largest black hole that can fit in the box. Several approaches in the past have suggested the expression $S\sim \sqrt{EV/G}$. We recall these arguments, and in particular expand on an argument that uses dualities of string theory. We require that the expression for $S(E,V)$ be invariant under the $T$ and $S$ when $E\sim E_{bh}$. These criteria lead to the above expression for $S$. We note that this expression has been obtained also by a imposing a quite different requirement—that the entropy within a cosmological horizon be of the order of the Bekenstein entropy for a black hole the size of the cosmological horizon. We recall the earlier proposed model of a “dense gas of black holes” to model this entropy, and discuss its realization as a set of intersecting brane states. Finally we speculate that the cosmological evolution of such a phase may depart from the evolution expected from the classical Einstein equations, since the very large value of the entropy can lead to novel effects similar to the fuzzball dynamics found in black holes.

Figure 1

(a) At a small value of the energy $E$, the phase with maximal entropy is radiation. (b) At larger $E$, a black hole has more entropy; this phase continues till the size of the hole becomes of order the size of the box. (c) We are interested in the phase where $E$ is taken to yet higher values.

Figure 2

A box of the same physical size at different times in an expanding cosmology. At an early enough time, the box will contain more mass than required to make a black hole with size equal to the size of the box.

Figure 3

(a) Branes wrapped on different cycles have intersection points; the number $n_{\mathrm{int}}$ of such intersections determines the entropy as $S\sim \sqrt{n_{\mathrm{int}}}$. (b) Branes and antibranes can join up to make effective local objects that do not wrap all the way around the cycles of the torus; in the case depicted, the string winding-antiwinding and momentum-antimomentum modes join up to make an effective string loop.

Figure 4

A pictorial depiction of the configurations that reproduce the entropy (4). Clusters of intersecting branes give the entropy of order the black hole entropy for each cluster. The overall entropy is then the sum of these entropies.

Figure 5

A one-dimensional potential; the particle wave function in the well on the left can tunnel through the barrier into the region on the right.

Figure 6

(a) Minkowski spacetime with an additional compact direction; for simplicity we depict only one spatial noncompact direction. (b) The compact circle can “pinch-off,” creating a “bubble of nothing.”

Figure 7

The fuzzball structure of black hole microstates in string theory. A compact direction pinches off with a twist that creates a KK monopole or antimonopole; these two possibilities are denoted by the $\pm$ signs. Spacetime ends just outside the location where the horizon would have formed in the traditional hole. The different choices of monopole structure at different angular locations give the $\mathrm{Exp}[S_{bek}]$ microstates of the hole.

Figure 8

A graviton is placed in a box with wave number along the $x$ direction. The box contains strings aligned along the $y$ direction. If the number of strings is large, the graviton is quickly absorbed onto the string as a pair of vibrations.

Figure 9

The states at each size $V_k$ can transition with amplitude $\epsilon$ to a band of states in volume $V_{k+1}$, with band spacing $\mathrm{\Delta }E$; thus we get a series of Fermi golden rule absorptions, taking us to larger volumes.