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Naked Firewalls Phys

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 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 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 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 . 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 , 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 ϕ) . 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 . 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 , 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 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)∇r · ∇r ≡ (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 2) · ∇(r 2) = r 2∇r · ∇r ≡ r 2 − 2Mr (which, incidentally, α 2 defines M) is in the outward null direction, or N (r ∇r · ∇r),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√ leads to −M2M˙ /α = 2 1 + 2[1 − 1 + 2 − ]/, which is √ 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+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. Quantum Fluctuation for the Mass Evaporation Fromr ˙ ≡ dr/dv = (1/2) − M/r, one can write the mass M = M(v) in the Vaidya metric in terms of the event horizon 1 radius r = r(v) ≡ rEH (v) as M = 2 r − rr˙. Let M1, r1 and M2, r2 be the mass and radius of the unperturbed black hole and of its fluctuations, respectively, with the total mass M = M1 + M2 and the event horizon radius r = r1 + r2. The unperturbed black hole horizon radius r1 is near where the smoothed horizon would be in the limit of very large smoothing parameter a. Now suppose that the unperturbed mass loss would give ˙ 2 M = M1 = M1(v) = (1/2)r1 − r1r˙1, such that M1 ≈ −α/M1 , and that quantum fluctuations M2 = M2(v) and r2 = r2(v) are small compared with the total mass and the event horizon radius, respectively. Then M = M1 + M2 = (1/2)r − rr˙ = (1/2)(r1 + r2) − (r1 + r2)(˙r1 +r ˙2) ≈ M1 + (1/2)r2 − r1r˙2. For simplicity we are making the highly idealized assumption that even with quantum fluctuations, the metric remains spherically symmetric and Vaidya near the event horizon. Departure from the Unperturbed Event Horizon

Now for some particular advanced time v = v0, let us ignore quantum fluctuations before this time, so that M2(v) = 0 for v < v0, and let us define the constant M0 = M(v0) = M1(v0). To 3 leading order in M0  1 and |v − v0|  M0 , the fractional decay of the black hole over the advanced time v − v0 is small, and the 1 negative of the coefficient ofr ˙2 in M ≈ M1 + 2 r2 − r1r˙2 may be written as r1 ≈ 2M1 ≈ 2M0. Then one gets (1/2)r2 − 2M0r˙2 ≈ M2(v). The solution of this differential equation that has no exponentially growing departure of the event horizon r(v) = r1 + r2 from the unperturbed horizon r1(v) at late times is

  Z ∞ 0  0  v − v0 0 M2(v ) v0 − v r2 ≈ exp dv exp . (10) 4M0 v 2M0 4M0 r r r t A Particular Fluctuation for the Evaporation Rate

˙ 2 Since the adiabatic evolution gives M1 ≈ −α/M0 for M0  1 and 3 |v − v0|  M0 , let us consider a quantum mass fluctuation that gives, with θ(v − v0) the Heaviside step function,   ˙ α β(v − v0) M2 = −θ(v − v0) 2 f exp − , (11) M0 4M0 which has two constant parameters, namely f for how large the quantum fluctuation in the energy emission rate is relative to the adiabatic emission rate −α/M2 (with f assumed to be positive so that the quantum fluctuation increases the emission rate above the adiabatic value), and β for how fast the quantum fluctuation in the energy emission rate decays over an advanced time of 4M0 (the inverse of the surface gravity κ of the black hole). Results for the Mass and Horizon Fluctuation

Then with M2(v) = 0 for v < v0, one gets    4α β(v − v0) M2 ≈ −θ(v − v0) f 1 − exp − . (12) βM0 4M0

Plugging this back into Eq. (10) then gives   8αf v − v0 r2 ≈ −θ(v0 − v) exp (1 + β)M0 4M0    8αf −β(v −v0) − θ (v −v0) 1+β−exp . (13) β(1+β)M0 4M0 Consequences of This Fluctuation

This particular form of the emission rate fluctuation implies that the total mass fluctuation away from the adiabatic evolution is M2(∞) = −4αf /(βM0). Then the radial fluctuation in the event horizon radius at the advanced time v = v0, when −r2(v) has its maximum value, is r2(v0) ≈ [2β/(1 + β)]M2(∞). This means that if the quantum fluctuation in the energy emission rate is very short compared with 4M0 (decaying rapidly in comparison with the surface gravity of the black hole), so that β  1, then r2(v0) ≈ 2M2(∞), twice the total mass fluctuation. However, we shall just assume that β is of the order of unity and hence get r2(v0) ∼ M2(∞) as an order-of-magnitude relation. Note that the reduction in the radius of the event horizon at v = v0, where the fluctuation in the mass emission rate starts, occurs before there is any decrease in the mass below the adiabatic value M1(v), because the location of the ‘teleological’ event horizon is defined by the future evolution of the spacetime. Quantum Fluctuations Can Render a Firewall Naked

Therefore, if the putative firewall occurs at a location determined purely causally by the past behavior of the spacetime, and is sufficiently near where the event horizon would be under smooth adiabatic emission thereafter (say near the smoothed horizon), then quantum fluctuations, at later advanced times that reduce the mass of the hole below that given by the adiabatic evolution, would move the actual event horizon inward (even before quantum fluctuations in the mass emission rate begin), so that the event horizon becomes inside the location of the putative firewall. That is, quantum fluctuations that increase the mass emission rate render such a firewall naked, visible to the external universe. A Possible Objection

One possible objection to this conclusion is that for α, β, and f all of the order of unity, the inward shift in the event horizon is by a change of radius, r2, of the order of 1/M, so that the proper distance from the putative firewall near r = r1 to the event horizon at r = r1 + r2, in the frame of the timelike firewall surface outside the event horizon, is of the order of the Planck length. The proper acceleration of an observer that stays of the order of the Planck length outside the event horizon would be of the order of the Planck acceleration, giving an Unruh temperature of the order of the Planck temperature. One might object that effects at such extreme accelerations would make a naked firewall in practice indistinguishable from a firewall at or inside the event horizon. An Answer to the Objection

However, in principle f can be arbitrarily large, in which case the firewall can be arbitrarily many Planck lengths outside the event horizon, and an observer that reaches it but stays outside the event horizon could have an acceleration and Unruh temperature arbitrarily lower than the Planck value. During the emission of ∼ S = 4πM2 particles, the probable greatest value of the proper distance of the putative firewall outside the event horizon is of the order of [(2/π) ln S]1/2 Planck units. For the largest presently observed , of the order of 40 billion solar masses, this is about 12 Planck units, and the corresponding Unruh temperature of the putative firewall is about 1/75 Planck units, which one might think is sufficiently below the Planck temperature that one could in principle view the naked firewall without entering the unknowns of quantum gravity. Disrobing Firewalls by Black Hole Mining

Adam Brown has suggested that one need not wait for the firewall to become naked by a spontaneous quantum fluctuation, but that one could disrobe it by black hole mining. This might be able to cause the event horizon to shrink much further below the adiabatic horizon than would be very probable with spontaneous quantum fluctuations, so that it seems possible that with suitable mining one might be able to make a firewall strongly naked with near unit probability. However, the details of this are work in progress. Hawking Radiation and Traversable asked me about two-sided black holes, and it initially seemed to me that if a sufficiently weak shock wave were sent in from the left CFT boundary early enough, and the right CFT boundary were made absorbant rather than reflective, then the negative energy flux going into the black hole horizon during the resultant Hawking evaporation might cause it to move inward across the weak shock wave from the left, so that that shock wave could traverse the and come out on the right.

However, I was alerted to theorems by Aron Wall that, under reasonable quantum field theory assumptions, imply that the Averaged Null Energy Condition holds and rules out traversable wormholes. As Aron emailed me 2016 Feb. 22, “If we consider only the linearized gravitational response, one needs to violate the ANEC on some generator of an eternal black hole horizon in order to make the black hole traversable. The easiest way to see that this cannot be done is to do a lightfront quantization on the horizon itself, as done in e.g. arXiv:1105.3445.” A Firewall Seems at Odds with General Relativity and

In conclusion, it seems that if a putative firewall is at a location determined by the past structure of the spacetime to be near where the event horizon would be if the black hole evolved adiabatically in the future, quantum fluctuations in the emission rate, and possibly black hole mining, can render the firewall naked.

Therefore the firewall is not hidden in the region with Planckian local temperature; its presence would truly be at odds with general relativity and usual quantum field theory. Firewalls Are Not Conservative

More specifically, being in the exterior of the event horizon means that the firewall could potentially influence the exterior spacetime, so that even observers who do not fall into the black hole could have a fiery experience. In addition, the presence of a firewall well outside the event horizon could affect the spectrum of the Hawking radiation, which means that the presence of a firewall could be inferred even by asymptotic observers. Such a ‘naked firewall,’ i.e., a firewall far outside the event horizon, is therefore problematic, and giving up the no-drama assumption no longer seems like a palatable ‘most conservative’ option. Acknowledgments

We gratefully acknowledge the generous support and the stimulating environment provided by the Yukawa Institute for Theoretical Physics through its Molecule-type Workshop on “Black Hole Information Loss Paradox” (YITP-T-15-01), where the concept of this paper was formulated. In addition, the work of PC is supported in part by Taiwan’s National Center for Theoretical Sciences (NCTS) and Taiwan’s Ministry of Science and Technology (MOST), of DNP by the Natural Sciences and Engineering Council of Canada, of MS by MEXT KAKENHI Grant Number 15H05888, and of DY by the Leung Center for Cosmology and Particle Astrophysics (LeCosPA) of National Taiwan University (103R4000). YCO thanks Nordita for various travel supports.

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