Microscopic origin of the entropy of astrophysical black holes

TL;DR

This work addresses the microscopic origin of black hole entropy for astrophysical, non-supersymmetric black holes by constructing an infinite family of semiclassical microstates in Minkowski space, modeled as interior dust-shell geometries with a fixed exterior Schwarzschild geometry. Using Euclidean gravitational path integrals, including wormhole saddles, the authors compute overlaps among microstates and show that wormhole contributions induce non-factorizing, universal correlations, yielding a Gram matrix whose rank saturates at $e^{S}$ with $S = A/(4G)$. Consequently, the degeneracy of microstates within a given energy window equals $e^{A/(4G)}$, providing a microscopic statistical origin for black hole entropy that does not rely on a particular UV completion. The findings highlight that entropy is a coarse-grained, non-perturbative feature captured by semiclassical gravity and wormhole physics, with potential corrections from one-loop effects and UV details, and are compatible with broader quantum gravity frameworks such as AdS/CFT.

Abstract

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.

Figures (4)

• 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 $\Sigma$. The exterior geometry extends between the future/past horizons $\mathscr{H}^\pm$ and the conformal null boundaries $\mathscr{J}^\pm$. 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 $>$ otuside the shell. The zigazag lines at the bottom and top are the the white hole and black hole singularities where time starts and ends. The semiclassical state on $\Sigma$ is non-singular and perfectly well defined.
• Figure 2: Induced geometry at the time-reflection symmetric slice $\Sigma$. The horizon is at $\mathbf{H} = \mathscr{H}^+ \cap \mathscr{H}^-$. The maximum surface $\sigma = \Sigma \cap \mathcal{W}$ represents the position of the shell, at a radius $R_*\geq r_s$ inside the black hole. The geometry inside the shell caps of smoothly, and it is a portion of flat spacetime.
• Figure 3: Euclidean continuation of the spacetime geometry of the microstates along $\Sigma$. The Euclidean section consists of an Euclidean black hole (right), and a region of 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 $\tilde{\beta}_m,\tilde{\beta}_m' \leq \beta$ depend on the mass of the shell.
• Figure 4: Euclidean wormhole contribution to the second moment of the overlap. The wormhole has the two inner products as its boundaries. It consists in two euclidean black holes in flat space, glued along the two shells.