Ensemble from coarse graining: Reconstructing the interior of an evaporating black hole

In understanding the quantum physics of a black hole, nonperturbative aspects of gravity play important roles. In particular, huge gauge redundancies of a gravitational theory at the nonperturbative level, which are much larger than the standard diffeomorphism and relate even spaces with different topologies, allow us to take different descriptions of a system. While the physical conclusions are the same in any description, the same physics may manifest itself in vastly different forms in descriptions based on different gauge choices. In this paper, we explore the relation between two such descriptions, which we refer to as the global gauge and unitary gauge constructions. The former is based on the global spacetime of general relativity, in which understanding unitarity requires the inclusion of subtle nonperturbative effects of gravity. The latter is based on a distant view of the black hole, in which unitarity is manifest but the existence of interior spacetime is obscured. These two descriptions are complementary. In this paper, we initiate the study of learning aspects of one construction through the analysis of the other. We find that the existence of near empty interior spacetime manifest in the global gauge construction is related to the maximally chaotic, fast scrambling, and universal dynamics of the horizon in the unitary gauge construction. We use the complementarity of the gauge choices to understand the ensemble nature of the gravitational path integral in global spacetime in terms of coarse graining and thermality in a single unitary theory that does not involve any ensemble nature at the fundamental level. We also discuss how the interior degrees of freedom are related with those in the exterior in the two constructions. This relation emerges most naturally as entanglement wedge reconstruction and the effective theory of the interior in the respective constructions.

Figure 1

The Fock spaces built on microstates of a black hole of mass $M$, which dominate the microcanonical ensemble ${\mathcal{H}}_M$, do not significantly overlap with each other because of the energy restriction imposed on hard modes. Furthermore, the collection of excited states, $B_{\delta E}{\mathcal{H}}_M$, forms only an exponentially small subset of the microcanonical ensemble, ${\mathcal{H}}_{M+\delta E}$, of the same energy.

Figure 2

Within the spacetime region described by the effect theory of the interior (diamond at the center), the interior hypersurface having the maximal volume ($\mathrm{\Sigma }$ in red) is bounded by the codimension-2 surfaces given by the intersections of the horizon and future-directed light rays emitted from $r=r_{\mathrm{z}}$ and its mirror. The volume of this hypersurface is finite.

Figure 3

Operators (${\mathcal{A}}_{\xi }$, ${\mathcal{B}}_{\gamma }$) acted on radiation states at time $t$ which have the same effect as operators of the effective interior theory ($a_{\xi }$, $b_{\gamma }$) erected at time $t-t_{\mathrm{scr}}$ (or earlier). This implies that one can reconstruct the entanglement wedge ${\mathcal{E}}_{\mathrm{rad}}$ from radiation states at time $t$. The blue shaded region is the interior region relevant to an object that is in the zone at time $t$ and falls into the black hole. The red arrow indicates a light signal propagating from the stretched horizon to the edge of the zone.