Black Holes, Information, and Hilbert Space for Quantum Gravity

TL;DR

This work presents a locality-preserving, unitary framework for black hole formation and evaporation based on a covariant Hilbert space ${\cal H}_{\rm QG}$, where complementarity is realized as unitary reference-frame changes. A black hole's late-time state is generically a superposition of macroscopically distinct geometries, with information encoded in relative amplitudes and phases across branches, while individual observers perceive a semi-classical world due to decoherence. An $S$-matrix description applies to the full covariant space rather than a fixed background, and the interior of old black holes emerges only after a frame change, with a true interior/exterior map involving superpositions. The firewall phenomenon is shown to be exponentially unlikely unless the initial conditions are extremely fine-tuned, reinforcing a coherent, information-preserving picture of black holes consistent with GR for classical observers.

Abstract

A coarse-grained description for the formation and evaporation of a black hole is given within the framework of a unitary theory of quantum gravity preserving locality, without dropping the information that manifests as macroscopic properties of the state at late times. The resulting picture depends strongly on the reference frame one chooses to describe the process. In one description based on a reference frame in which the reference point stays outside the black hole horizon for sufficiently long time, a late black hole state becomes a superposition of black holes in different locations and with different spins, even if the back hole is formed from collapsing matter that had a well-defined classical configuration with no angular momentum. The information about the initial state is partly encoded in relative coefficients---especially phases---of the terms representing macroscopically different geometries. In another description in which the reference point enters into the black hole horizon at late times, an S-matrix description in the asymptotically Minkowski spacetime is not applicable, but it sill allows for an "S-matrix" description in the full quantum gravitational Hilbert space including singularity states. Relations between different descriptions are given by unitary transformations acting on the full Hilbert space, and they in general involve superpositions of "distant" and "infalling" descriptions. Despite the intrinsically quantum mechanical nature of the black hole state, measurements performed by a classical physical observer are consistent with those implied by general relativity. In particular, the recently-considered firewall phenomenon can occur only for an exponentially fine-tuned (and intrinsically quantum mechanical) initial state, analogous to an entropy decreasing process in a system with large degrees of freedom.

Figures (8)

• Figure 1: The Penrose diagram representing a black hole formed from a collapsing shell of matter (represented by the thick solid curve) which then evaporates. The left panel shows the standard "global spacetime" picture, in which Hawking radiation (denoted by wavy arrows) comes from the stretched horizon. To obtain a consistent quantum mechanical description, we must fix a reference frame (freely falling frame) and then describe the system from that viewpoint Nomura:2011rb. Quantum states then corresponds to physical configurations in the past light cone of the origin $p$ of that reference frame. Here we choose a "distant" reference frame; the trajectory of its origin $p$ is depicted by a thin solid curve. With this choice, a complete description of the evolution of the system is obtained in the shaded region in the panel. In other words, the conformal structure of the entire spacetime is as in the right panel, when the system is described in this reference frame.
• Figure 2: In the left panel, we show the result of simulating the black hole displacement $|{\bf x}_{\rm BH}|$ at the Page time, $t_{\rm Page} \sim M_0^3$, as a function of the initial black hole mass, $M_0$. We find the behavior expected from the general argument, $|{\bf x}_{\rm BH}| \sim M_0^2$. In the right panel, we show the probability distribution of the displacement $|{\bf x}_{\rm BH}|$ for a fixed $M_0 = 5000$, obtained by performing a larger number of simulations, $N_{\rm total} = 10000$. The distribution takes the form expected from the central limit theorem; see Eq. (\ref{['eq:x-distr']}).
• Figure 3: Typical paths of the black hole drifting in the three dimensional space ${\bf x}_{\rm BH} = (x_{\rm BH}, y_{\rm BH}, z_{\rm BH})$, normalized by $M_0^2$.
• Figure 4: A schematic depiction of the evolution of a black hole state formed by a collapse of matter. After long time, the state will evolve into a superposition of terms representing the black hole to be in macroscopically different locations, even if the initial collapsing matter has a well-defined macroscopic configuration. The variation of the final locations in the evaporation timescale, $t \sim M_0^3$, is of order $M_0^2$, which is much larger than the Schwarzschild radius of the initial black hole, $R_S = 2M_0$.
• Figure 5: The left panel shows the standard global spacetime picture for the formation and evaporation of a black hole, with the shaded region representing the spacetime region described by an infalling reference frame. (The trajectory of the origin, $p$, of the reference frame is also depicted.) As discussed in the text, this is the entire spacetime when the system is described in this reference frame, so its conformal structure is in fact as in the right panel. Here, the wavy line with a solid core represents singularity states.