Black Hole Interior in Unitary Gauge Construction

TL;DR

The paper advocates a unitary-gauge (holographic) construction of black hole interiors, where the exterior, high-entropy degrees of freedom (hard, soft, and far modes) encode interior physics through coarse-grained dynamics. It provides explicit operator constructions (mirror microstates, canonical and globally promoted mirror/infalling operators) and an effective interior theory erected at boundary times, clarifying state dependence and intrinsic ambiguities via an error parameter and their relation to quantum error correction. The approach yields a semiclassical interior within a finite domain, with well-defined procedures for computing interior correlators in the in-in formalism and a framework that addresses firewall-like puzzles while generalizing to young black holes and Minkowski space. Overall, it links UV/IR relations and holographic error-correction to the emergent interior, suggesting robust interior descriptions that do not depend on microscopic quantum gravity details.

Abstract

A quantum system with a black hole accommodates two widely different, though physically equivalent, descriptions. In one description, based on global spacetime of general relativity, the existence of the interior region is manifest, while understanding unitarity requires nonperturbative quantum gravity effects such as replica wormholes. The other description adopts a manifestly unitary, or holographic, description, in which the interior emerges effectively as a collective phenomenon of fundamental degrees of freedom. In this paper we study the latter approach, which we refer to as the unitary gauge construction. In this picture, the formation of a black hole is signaled by the emergence of a surface (stretched horizon) possessing special dynamical properties: quantum chaos, fast scrambling, and low energy universality. These properties allow for constructing interior operators, as we do explicitly, without relying on details of microscopic physics. A key role is played by certain coarse modes in the zone region (hard modes), which determine the degrees of freedom relevant for the emergence of the interior. We study how the interior operators can or cannot be extended in the space of microstates and analyze irreducible errors associated with such extension. This reveals an intrinsic ambiguity of semiclassical theory formulated with a finite number of degrees of freedom. We provide an explicit prescription of calculating interior correlators in the effective theory, which describes only a finite region of spacetime. We study the issue of state dependence of interior operators in detail and discuss a connection of the resulting picture with the quantum error correction interpretation of holography.

Figures (4)

• Figure 1: The effective theory of the interior can be erected at a boundary time $t_*$ in the original unitary theory which has a one-sided black hole. It has an effective two-sided black hole geometry and describes physics in the causal domain of the union of the zone and its mirror region on the spatial hypersurface corresponding to the $t_*$ hypersurface. The effective theory is intrinsically semiclassical and cannot describe dynamics occurring above the string scale $1/l_{\rm s}$ (although space below $l_{\rm s}$ exists to accommodate an object kinematically squeezed by a large Lorentz boost).
• Figure 2: The contour of integration $C$ for a time-ordered $n$-point correlator $\left\langle T\{\tilde{\Phi}_{a_1}(x_1)\, \tilde{\Phi}_{a_2}(x_2)\, \cdots\, \tilde{\Phi}_{a_n}(x_n)\} \right\rangle$ in the Schwinger-Keldysh formalism. Here, we have assumed $x_n^0 < \cdots < x_2^0 < x_1^0$ for illustration purposes.
• Figure 3: In the $e^{S_{\rm tot}}$-dimensional space ${\cal M}$ spanned by orthonormal vacuum microstates, we can build infalling mode operators that cover a subspace $\tilde{\cal M}$ in a state independent manner. The choice of $\tilde{\cal M}$ is arbitrary as represented by regions with different colors. Furthermore, the space $\tilde{\cal M}$ can be made larger as represented by the graded red regions, but only as long as ${\rm dim}\,\tilde{\cal M}$ is sufficiently smaller than ${\rm dim}\,{\cal M}$. This implies that a single $\tilde{\cal M}$ cannot cover a significant portion of ${\cal M}$. Note that the figure is only a schematic representation of the situation.
• Figure 4: (a) To describe the fate of a falling object that reached the stretched horizon at $t = t_*$, we may erect an effective theory of the interior at $t = t_*$ (blue) or some time later $t = t_* + \varDelta t$ with $\varDelta t > 0$ (red). (b) The same is depicted in the coordinates adapted to each effective theory, $\tilde{U}$ and $\tilde{V}$. The blue and red arrows represent the trajectory of the object in the effective theories erected at $t = t_*$ (blue) and $t = t_* + \varDelta t$ (red), respectively.