Black holes or firewalls: A theory of horizons

Abstract

We present a quantum theory of black hole (and other) horizons, in which the standard assumptions of complementarity are preserved without contradicting information theoretic considerations. After the scrambling time, the quantum mechanical structure of a black hole becomes that of an eternal black hole at the microscopic level. In particular, the stretched horizon degrees of freedom and the states entangled with them can be mapped into the nearhorizon modes in the two exterior regions of an eternal black hole, whose mass is taken to be that of the evolving black hole at each moment. Salient features arising from this picture include: (i) the number of degrees of freedom needed to describe a black hole is e A/2l 2 P , where A is the area of the horizon; (ii) black hole states having smooth horizons, however, span only an e A/4l 2 P -dimensional subspace of the relevant e A/2l 2 P -dimensional Hilbert space; (iii) internal dynamics of the horizon is such that an infalling observer finds a smooth horizon with a probability of 1 if a state stays in this subspace. We identify the structure of local operators responsible for describing semi-classical physics in the exterior and interior spacetime regions, and show that this structure avoids the arguments for firewalls-the horizon can keep being smooth throughout the evolution. We discuss the fate of infalling observers under various circumstances, especially when the observers manipulate degrees of freedom before entering the horizon, and we find that an observer can never see a firewall by making a measurement on early Hawking radiation. We also consider the presented framework from the viewpoint of an infalling reference frame, and argue that Minkowski-like vacua are not unique. In particular, the number of true Minkowski vacua is infinite, although the label discriminating these vacua cannot be accessed in usual non-gravitational quantum field theory. An application of the framework to de Sitter horizons is also discussed.

Entropy of a vacuum: What does the covariant entropy count?

“…Recently, however, it was argued that complementarity cannot be a consistent story [8][9][10][11]; the argument essentially boiled down to the incompatibility of the uniqueness of the (infalling) vacuum and the distant unitary description [12,13]. We disagreed with this conclusion [14][15][16]. We argued that the apparent problem had arisen from an overly simplistic picture on how classical spacetime emerges in a full quantum theory of gravity.…”

“…(21) is purified by the (intrinsically quantum gravitational) degrees of freedom located on the stretched horizon; namely, the stretched horizon degrees of freedom play the role of the second exterior region in the Unruh-Israel description [16]. Item (ii) above then implies that there are e ≈A/4l 2 P different ways in which ρ ext is purified by the stretched horizon states:…”

“…[23]: a smooth horizon requires a near-maximal entanglement in two sets of basis states, which was not the case in Ref. [16].…”

“…[16] (and particularly emphasized in [21]) that the number of degrees of freedom relevant to describe the black hole interior is a tiny fraction of the total number of degrees of freedom available for a black hole. The explicit realization of the idea, however, is different in this paper.…”

“…In Ref. [16], it was considered, following [22], that a variety of black hole vacuum states |ψ k is allowed because of a freedom in the way the near horizon and stretched horizon modes are entangled; specifically, we considered a freedom in quantum mechanical phase factors appearing in the entangled states. This approach, however, leads to the following issue.…”