Correlations in Hawking radiation and the infall problem

Abstract

It is sometimes believed that small quantum gravity effects can encode information as 'delicate correlations' in Hawking radiation, thus saving unitarity while maintaining a semiclassical horizon. A recently derived inequality showed that this belief is incorrect: one must have order unity corrections to low energy evolution at the horizon (i.e. fuzzballs) to remove entanglement between radiation and the hole. In this paper we take several models of 'small corrections' and compute the entanglement entropy numerically; in each case this entanglement is seen to monotonically grow, in agreement with the general inequality. We also construct a model of 'burning paper', where the entanglement is found to rise and then return to zero, in agreement with the general arguments of Page. We then note that the fuzzball structure of string microstates offers a version of 'complementarity'. Low energy evolution is modified by order unity, resolving the information problem, while for high energy infalling modes (E >> kT ) we may be able to replace correlators by their ensemble averaged values. Israel (and others) have suggested that this ensemble sum can be represented in the thermo-field-dynamics language as an entangled sum over two copies of the system, giving the two sides of the extended black hole diagram. Thus high energy correlators in a microstate may be approximated by correlators in a spacetime with horizons, with the ensemble sum over microstates acting like the 'sewing' prescription of conformal field theory.

Black holes and beyond

“…Even though the answer to this was not clear, the expression (4.2) was generally accepted as holding for Rindler horizons, and in particular was used to 'derive' Einstein's equation from thermodynamics [42]. What we will now see is that in terms of fuzzballs, there is a logical explanation of (4.2) as a count of states, even though we are describing the Rindler quadrants of empty Minkowski space [43].…”

Evaporating firewalls

“…Thus, they argue that an infalling observer will experience a firewall (if the assumptions hold). 1 Firewalls were also proposed in a different context in [43], following [44][45][46][47][48][49][50][51][52][53]. There, in the nonperturbative setting of AdS/CFT, it was argued that the existence of any spacetime at all behind a general horizon (including Rindler horizons in pure AdS) requires entanglement between the degrees of freedom associated with the region outside the horizon and some other independent degrees of freedom (see figure 1).…”

Unitarity and fuzzball complementarity: “Alice fuzzes but may not even know it!”

“…Recently, Mathur introduced fuzzball complementarity [16]: while quanta with E ∼ T H do not have free infall independent of the details of the fuzzball, more energetic quanta E T H fall in freely, oblivious to the details of the fuzzball. This proposal, further developed in [6,17,23,24], explains in what sense black holes are coarse-grained descriptions of fuzzballs. We review fuzzball complementarity in section 3.…”

“…If black hole evaporation were unitary then a black hole in a typical state would evaporate according to figure 1a. In a recent series of papers [15][16][17][18][19], however, Mathur showed that for black hole evaporation with only small corrections to the leadingorder Hawking process the von Neumann entropy of radiation keeps monotonically increasing in time with no turnover, as shown in figure 1b. To get an evolution as in figure 1a, he therefore argues that the small correction have to be made large and we can no longer trust effective field theory at the horizon scale.…”

“…In a series of recent papers [15][16][17][18][19] Mathur argued that a Hawking-like evaporation process, cannot bend the entropy curve to go down at any point in time (see figure 1b). The argument can be summarized as follows.…”