Naked Firewalls Phys. Rev. Lett. 116, 161304 (2016) Pisin Chen, Yen Chin Ong, Don N. Page, Misao Sasaki, and Dong-han Yeom 2016 July 8 Introduction In the firewall proposal, it is assumed that a firewall lies near the event horizon and should not be observable except by infalling observers, who are presumably terminated at the firewall. However, if the firewall is located near where the horizon would have been, based on the spacetime evolution up to that time, later quantum fluctuations of the Hawking emission rate can cause the `teleological' event horizon to have migrated to the inside of the firewall location, rendering the firewall naked. In principle, the firewall can be arbitrarily far outside the horizon. This casts doubt about the notion that firewalls are the `most conservative' solution to the information loss paradox. The AMPS Argument Almheiri, Marolf, Polchinski and Sully (AMPS) argued that local quantum field theory, unitarity, and no-drama (the assumption that infalling observers should not experience anything unusual at the event horizon if the black hole is sufficiently large) cannot all be consistent with each other for the Hawking evaporation of a black hole with a finite number of quantum states given by the Bekenstein-Hawking entropy. AMPS suggested that the `most conservative' resolution to this inherent inconsistency between the various assumptions is to give up no-drama. Instead, an infalling observer would be terminated once he or she hits the so-called firewall. This seems rather surprising, because the curvature is negligibly small at the event horizon of a sufficiently large black hole, and thus one would expect nothing special but low energy physics. The Heart of the AMPS Argument Assuming unitarity, the information contained inside a black hole should eventually be recovered from the Hawking radiation. The late time radiation purifies the earlier radiation, so the late time radiation should be maximally entangled with the earlier radiation. By the monogamy of quantum entanglement, the late time radiation cannot also be maximally entangled with the interior of the black hole. This means that the field configuration across the event horizon is generically not continuous, which leads to a divergent local energy density. More explicitly, the quantum field 2 Hamiltonian contains terms like (@r ') . The derivative is divergent at some r = R if the field configuration is not continuous across R. This is the firewall. The Location of a Firewall Usually it is thought that a firewall lies on or just inside an old black hole event horizon. Of course in quantum mechanics there are no sharp boundaries, and the position of an event horizon should be uncertain, up to perhaps fluctuations of the order of the Planck length. That is to say, a firewall is presumably like a stretched horizon, with the crucial difference that anything that hits a firewall gets incinerated instead of just passing right through. By the nature of the event horizon, if the firewall lies either inside or exactly on the horizon, then it is completely invisible to the observers outside. For a firewall that is not too far outside the event horizon, it is still doubtful that it is perceptible to far-away observers, since it would seem that such a firewall would be well hidden inside the Planckian region of the local thermal atmosphere. Causality Assumption for Firewalls Here we make the assumption that a firewall, if it exists, has a location determined by the past history of the Hawking evaporating black hole spacetime and is near where the event horizon would be if the evaporation rate were smooth, without quantum fluctuations. Then we show that quantum fluctuations of the evaporation rate in the future can move the event horizon to the inside of the firewall location, rendering it naked. Vaidya Metric Assumption For simplicity, we shall approximate the metric near the horizon of an evaporating black hole by the Vaidya metric with a negative energy influx: 2M(v) ds2 = − 1 − dv 2 + 2dvdr + r 2dΩ2: (1) r Here M(v) is the mass of the black hole, which is decreasing as a function of the advanced time v. For a smooth evaporation rate of a spherical black hole emitting mainly photons and gravitons, we shall take (in Planck units) dM α M_ ≡ = − ; (2) dv M2 where α is a constant which, in my Ph.D. thesis of forty years ago, I numerically evaluated to be about 3:7474 × 10−5. Location of the Event Horizon with Smooth Evaporation The apparent horizon is located at rApH = 2M(v), whereas the event horizon is generated by radially outgoing null geodesics, dr 1 2M(v) r_ ≡ = 1 − ; (3) dv 2 r on the boundary of such null geodesics reaching out to future null infinity, instead of falling in to the singularity that is believed to be inside the black hole. For a smooth evaporation rate M_ ≡ dM=dv = −α=M2, the event horizon is given by the solution to Eq. (3) such that it does not diverge exponentially far away from the apparent horizon in the future. If we define u ≡ 1 − r=(2M) and p ≡ −4M_ and assume that n ≡ −d ln p=d ln M is constant, then one can show that the event horizon is at u = p + (n − 2)p2 + (n − 1)(2n − 5)p3 + (6n3 − 28n2 + 37n − 14)p4 + O(p5): (4) The Adiabatic Horizon For a smooth Hawking evaporation into massless particles with p ≡ −4M_ = 4α=M2, so that n = 2, one finds that the event horizon is at 2 3 6 rEH = 2M[1 − 4α=M + O(α =M )]: (5) For a general spherical metric, the covariant generalization of f_ along an outward null direction toward the future is f_ ≡ df =dv ≡ Nα@f =@xα, with outward null vector Nα normalized α so thatr _ ≡ dr=dv ≡ N r,α = (1=2)rr · rr ≡ (1=2) − M=r. Note that dM=dv = −α=M2 implies that d2(M3)=dv 2 = 0, but since r ≈ 2M, we have d2(r 3 )=dv 2 ≈ 0 as well. Let us EH EH therefore define an adiabatic horizon at rAdH by the outer root of d2 @ @ (r 3 ) ≡ Nα Nβ r 3 = 0: (6) dv 2 AdH @xα @xβ AdH The Location of the Adiabatic Horizon The location of the adiabatic horizon is very near where the event horizon would be if the future evolution of the latter followed the adiabatic mass evolution law dM=dv = −α=M2. One can show that rAdH is equivalent to the location where the gradient vector of (1=4)r(r 2) · r(r 2) = r 2rr · rr ≡ r 2 − 2Mr (which, incidentally, α 2 defines M) is in the outward null direction, or N (r rr · rr),α = 0, which gives M_ = (1=2) − (3=2)(M=r) + (M=r)2 and 4M r ≡ : (7) AdH p 3 − 1 + 16M_ When dM=dv = −α=M2 holds, this expression agrees with 2 3 6 rEH = 2M[1 − 4α=M + O(α =M )] to the order given. The Fourth-Root Horizon For fluctuating evaporation, a smoother approximation for the horizon is the fourth-root horizon at rq 4 2 r4rH ≡ (2M) + (32α) − 32α: (8) The inverse of this relation is M = (1=2)(r 4 + 64αr 2 )1=4. If one 4rH 4rH had a model evaporating black hole for which the event horizon (say denoted by r) were exactly at the fourth-root horizon, so that r = r , then using Eqs. (3) and (8) and the definition ≡ 8α=M2 4rH p pp leads to −M2M_ /α = 2 1 + 2[1 − 1 + 2 − ]/, which is p very close to 1 for M 4 α ( 1), but then it decreases to a minimum near 0.954142 at ≈ 0:398463 before increasing to 2 as the mass goes to zero and diverges, with −M2M_ /α always staying finite and remaining within a factor of 2 of unity and hence giving M_ ∼ −α=M2 as M decreases to zero. The Smoothed Horizon An even smoother approximation for the location of the event horizon is the smoothed horizon at Z 1 1+2a y 1+a y rSmH (v)≡2 dyM(v −4M(v)y) 2 exp − − 2 exp − (9) 0 a a 2a 2a for a positive smoothing parameter a. Then if M(v) had a linear dependence on v over the range where the integrand is significant, this would give the extrapolation rSmH (v) = 2M(v + 4M(v)), the radius of the apparent horizon at a time greater than v by the inverse of the surface gravity, which is what the first two terms of 2 3 6 _ 2 rEH =2M[1−4α=M +O(α =M )] give for constant M =−α=M . A plausible value of the smoothing parameter might be a ∼ ln(M(v)) 1, giving smoothing over roughly the scrambling time ∼ M(v) ln(M(v)). Location of the Firewall near the Smoothed Horizon We shall assume that the firewall, if it exists, is close to where the event horizon would be if the black hole evolves smoothly and adiabatically according to dM=dv = −α=M2, as perhaps well estimated by the smoothed horizon for some suitable value of the smoothing parameter a. However, the actual event horizon depends on the future evolution of the spacetime, and not just on that of its past. Therefore, quantum fluctuations in the future spacetime can lead the event horizon to deviate significantly from the smoothed horizon. If the mass loss rate exceeds the adiabatic formula, then the event horizon will be inside the smoothed horizon. As a result a firewall located at the smoothed horizon would become naked, visible from future null infinity.