Microscopic Origin of the Entropy of Astrophysical Black Holes

We construct an infinite family of microstates for black holes in Minkowski spacetime which have effective semiclassical descriptions in terms of collapsing dust shells in the black hole interior. Quantum mechanical wormholes cause these states to have exponentially small, but universal, overlaps. We show that these overlaps imply that the microstates span a Hilbert space of log dimension equal to the event horizon area divided by four times the Newton constant, explaining the statistical origin of the Bekenstein-Hawking entropy.

Figure 1

Penrose diagram of the time evolution of a microstate of an eternal one-sided black hole. The semiclassical state is defined at the time reflection-symmetric Cauchy slice $\mathrm{\Sigma }$. The exterior geometry extends between the future and past horizons ${\mathcal{H}}^{\pm }$ and the conformal null boundaries ${\mathcal{J}}^{\pm }$. The interior contains a thin shell $\mathcal{W}$, which divides the geometry between a region of flat space $<$ inside the shell, and a region of black hole geometry $>$ outside the shell. The zigzag lines at the bottom and top are the white hole and black hole singularities where time starts and ends. The semiclassical state on $\mathrm{\Sigma }$ is nonsingular and perfectly well defined.

Figure 2

Euclidean continuation of the spacetime geometry of the microstates along $\mathrm{\Sigma }$. The Euclidean section consists of a Euclidean black hole (right), and a region of a Euclidean flat space (left), glued together along the trajectory of a thin shell. The shell starts at the asymptotic spatial infinity, bounces back at $R_{*}$, and gets back to $R=\infty$. The Euclidean times ${\tilde{\beta }}_m,{\tilde{\beta }}_m^{\prime }\le \beta$ depend on the mass of the shell.

Figure 3

Euclidean wormhole contribution to the second moment of the overlap. The wormhole has the two inner products as its boundaries. It consists of two Euclidean black holes in flat space, glued along the two shells.