Microscopic origin of the entropy of black holes in general relativity

TL;DR

The paper tackles the microscopic origin of black hole entropy in general relativity with negative cosmological constant by constructing infinite families of semiclassical microstates with distinct interiors behind fixed exterior black holes. Wormhole saddle points in the gravitational path integral generate universal, nonzero overlaps among these microstates, ensuring the Hilbert space spanned by the families has dimension $e^{S_{BH}}$ rather than diverging with the family size. This framework also clarifies the interior geometry of the microstates and shows that Einstein-Rosen bridge volumes can be expressed as superpositions of shorter wormholes, leading to a saturation of complexity-related volume at $ ext{O}(e^{S_{BH}})$. By connecting to ETH-like ideas and the Page curve, the work provides a broad, gravity-based mechanism for black hole entropy universality that does not rely on string theoretic constructions or End-of-The-World branes, and it suggests nonlinear, state-dependent observables for interior geometry.

Abstract

We construct an infinite family of microstates with geometric interiors for eternal black holes in general relativity with negative cosmological constant in any dimension. Wormholes in the Euclidean path integral for gravity cause these states to have small, but non-zero, quantum mechanical overlaps that have a universal form. The overlaps have a dramatic consequence: the microstates span a Hilbert space of log dimension equal to the Bekenstein-Hawking entropy. The semiclassical microstates we construct contain Einstein-Rosen bridges of arbitrary size behind their horizons. Our results imply that all these bridges can be interpreted as quantum superpositions of wormholes of size at most exponential in the entropy.

Figures (14)

• Figure 1: Schematic form of the Euclidean path integral in the boundary field theory, which is two copies of a holographic CFT with Hilbert space ${\cal H}^{CFT}$. The cross section of the tube represents the spatial direction, and the length is Euclidean time. The operator ${\cal O}$ is inserted in the circle at the bottom, and the tube corresponds to propagation in Euclidean time via the Hamiltonian of the underlying CFT, by ${\tilde{\beta}}_{L}$ (${\tilde{\beta}}_{R}$) along to the left (right) of the operator insertion.
• Figure 2: Preparation of the state from the Euclidean path integral. The lower part of the figure (tan colors) shows time evolutions from a past Euclidean boundary with a shell operator (red dot) inserted. In the Euclidean section the horizon is a point, here represented by crosses for the the left and right black holes on either side of the shell. At $t=0$ the Euclidean preparation geometry is matched onto a Lorentzian time evolution (blue and purple). The black hole horizons (thin black lines) become null surfaces, while the shell (red line) continues to move into the future behind both horizons. Spacetime is glued across the trajectory of the thin shell by Israel's junction conditions. Here, we have allowed for the possibility that the black on either side of the shell have different temperatures or masses. The solid black line is the spacetime boundary. For $d$-dimensional AdS geometries this will be a cylinder $\mathbf{S}^{d-1} \times R$ on which the dual CFT lives.
• Figure 3: We show here the Euclidean geometry of the right black hole. This is a disk of circumference $\beta_R$. The shell (red line) cuts the disk, with an associated Euclidean travel time $\Delta \tau_+$. Consistency fixes the state preparation temperature to satisfy $\beta_R=\tilde{\beta}_R+\Delta\tau_+$.
• Figure 4: This is a Euclidean disk geometry (a Euclidean black hole) in between two shells (red lines). The shell travel times $\tau_+^i$, $\tau_-^{i+1}$ plus twice the preparation temperature $\tilde{\beta}_i$ must be equal to the physical temperature $\beta_i$ of this black hole geometry between the shells.
• Figure 5: Example of a multi-shell state of a eternal black hole with ADM masses $M_-$ and and interior geometry with associated mass parameter $M_+$ .