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.
Highlights
The quantum mechanics of a black hole [1,2] has been a confusing subject
The gist of this paper is to present the unitary gauge construction of the black hole interior as explicitly as possible and analyze its salient features
We stress that the prescription of obtaining interior operators described here does not require a detailed knowledge of microscopic dynamics of quantum gravity. This insensitivity to the microscopic physics puts the unitary gauge construction on an equal footing with the approach [9,10,11,17,18] based on the global spacetime picture
Summary
The quantum mechanics of a black hole [1,2] has been a confusing subject. On one hand, a naïve application of the semiclassical method leads to a violation [3] of unitarity, a fundamental principle of quantum mechanics. A fully quantum mechanical treatment of the problem, such as the one based on the AdS=CFT correspondence [4], indicates that unitarity is preserved, while it seems to be at odds with the existence of the black hole interior [5], a consequence of the equivalence principle of general relativity. The other approach to the problem is to begin with a manifestly unitary description This corresponds to viewing the black hole from a distance. One may view that the degrees of freedom outside (and on) the stretched horizon comprise the entirety of the system [19,20].1 In this picture, the challenge is to understand how a description based on near empty interior spacetime emerges [5,21,22]. The unitary gauge construction deals with the issue in the most intuitive way: the resulting picture is mostly local, it leaves some residual nonlocality
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