NNumerical Evidence for Firewalls
David Berenstein and Eric Dzienkowski
Department of Physics, University of California at Santa Barbara, CA 93106
We study thermal configurations in the BFSS matrix model produced by numerical simulations.We do this by adding a probe brane to a typical configuration and studying the fermionic degrees offreedom connecting the probe to the configuration. Depending on the parameters of the probe thereis a region where these fermion modes are gapless. We argue that it is natural to excise the gaplessregion from the geometry and that the black hole horizon is located exactly at the edge of this zone.The physics inside the gapless region is maximally non-local and effectively 1+1 dimensional. Whena probe, considered as an observer, crosses into the gapless zone there is a break down of effectivefield theory where the off-diagonal fermions are integrated out. We argue that this breakdown ofeffective field theory is evidence for firewalls on black hole horizons.
Introduction
Hawking’s discovery that quantum black holes radiateand behave as thermal systems [1] has led to the blackhole information paradox. Recent work on this problemwith a modern understanding of quantum informationtheory [2–4] (see also [5, 6]) has suggested that blackhole horizons (at the very least for old black holes) arenot smooth right behind the horizon. The geometry isinstead replaced by a so called “firewall” where an in-falling observer would be destroyed on contact, beforereaching the classical black hole singularity. The postu-lates that lead to this statement are: i) black hole forma-tion and evaporation processes can be addressed entirelywithin the context of quantum mechanics, ii) outside theblack hole, physics can be addressed semiclassically, iii) to an outside observer the black hole appears as a sys-tem with discrete energy levels reproducing the entropyof the black hole and finally iv) an infalling observer ex-periences no drama when crossing the horizon. Thesefour postulates together were shown to be inconsistent.The gauge/gravity duality gives an example of the firstthree postulates [7] being correct. Thus, the fourth pos-tulate about infalling observers seems to be the one thatneeds to be removed. A rather important question is thenwhat happens to an infalling observer: can we actuallysee what kind of drama it would experience?We will address this question for a special class of blackholes subject to the gauge/gravity duality. Using thegauge theory as a microscopic theory of gravity, all ques-tions will be addressed within the dynamics of the gaugetheory. We study black holes in the BFSS matrix model[8]. Our treatment follows the numerical classical simu-lations carried out in [9, 10]. The black holes in questionare hot and stringy . It is not obvious that our conclusionscarry over to cold semiclassical black holes. We will givearguments for the robustness of our reasoning.The main problem we need to address is whether wecan find a geometric characterization of the black holehorizon in the matrix model. Without such a charac-terization, we can not ask what happens when we crossthe horizon; we can not even tell where the horizon is located to ask such question. We find that the ideas ofprevious work [11] extended to the BFSS matrix modelgive a possible characterization and a geometric separa-tion between the inside and outside of the black hole.The main idea is to consider a point-like probe brane(that is, a D0-brane) in the presence of the black hole.The probe brane acts as an observer which explores an R geometry. We can ask where and under which condi-tions one can integrate out the degrees of freedom con-necting the probe to the black hole. In our setup we caremostly about the fermionic degrees of freedom. The sameproblem has been studied in [12] using mean filed theorymethods where the probe is fully dynamical, but outsidethe black hole. When these degrees of freedom can not beintegrated out, the physics of the observer changes. Wewill see that there is a region of R where the fermionicspectrum of modes connecting the probe to the blackhole is gapless. We will also show that the physics inthis region can be characterized as very non-local andin a technical sense, the spacetime is 1 + 1 dimensional.It is natural to excise such a very non-local region fromthe geometry in gravity. A natural place for a horizonis exactly at the boundary of where the gapless regionappears. We argue that the change in effective physics is very dramatic and the observer can not fail to notice thatthe environment has changed completely. This suggestsa precise description for what a firewall is. Fermionic Observables and a Gapless Region
A typical problem in matrix theory is to describe aconfiguration of matrices geometrically. The generic so-lution for a set of commuting matrices is a set of pointswhich characterize the eigenvalues of the matrices [8].The goal of [11] was to give a geometric prescription toconfigurations of three Hermitian matrices which do notnecessarily commute. The prescription is to measure thespectrum of fermions connecting a D0-brane probe to abackground matrix configuration. The energy spectrumof fermions gives a sense of distance because for strings,their length is proportional to their energy. The minimallength string, corresponding to the minimal energy mode, a r X i v : . [ h e p - t h ] N ov leads to an effective notion of distance. These ideas havea straightforward generalization to any dimension.To apply these ideas, we need the fermionic part of theBFSS Hamiltonian [8] H ferm ∼ Tr (cid:0) Ψ † Γ i [ X i , Ψ] (cid:1) (1)where 1 ≤ i ≤
9, the Ψ are SO (9) spinors, and Γ i thenine dimensional gamma matrices. The constant of pro-portionality depends on the normalization of the fieldsand (cid:126) (in the BFSS matrix model we can trade g Y M for (cid:126) by rescaling the fields). The matrices X i and Ψare in the adjoint of U ( N ), that is, they are Hermitian.Given an N × N configuration of X i we construct an N + 1 × N + 1 configuration ˜ X i by adding a single D0-brane probe with coordinates x i ∈ R in the lowest rightcorner. We only want to consider the fermion modesconnecting the background and the brane probe.˜ X i = (cid:18) X i x i (cid:19) , ˜Ψ = (cid:18) ψ (cid:19) (2)where ψ is an N × U ( N )). The effective Hamiltonian for theconfiguration ( ˜ X i , ˜Ψ) is H eff ∼ ψ † ([ X i − x i I N ] ⊗ Γ i ) ψ (3)This effective Hamiltonian can be considered at each in-stant of time and for each position of the probe. Further-more, we can choose to make the probe fully dynamicalor not. If we make it dynamical, we can call it an ‘ob-server’ and choose a set of initial conditions, that is, theinitial position and velocity of the probe. If the probe isnot dynamical, we can scan over R with the position ofthe probe and label each point by the properties of thefermion spectrum of eigenvalues in equation (3). (cid:45) (cid:45) (cid:45) x (cid:45) (cid:45) Energy
FIG. 1. Fermionic eigenvalue spectrum for a typical config-uration of matrices with N = 47 after thermalization. Theunits are arbitrary. We consider the spectrum of H eff for the thermal con-figurations of [10]. Without loss of generality we may takethe matrices to be traceless. By rotational invariance(the configurations do not have angular momentum), we need only consider moving the brane probe in a singledirection, say x , and we choose to go through the centerof the configuration. The spectrum of H eff for a typ-ical configuration is plotted in FIG. 1. Since the ninedimensional gamma matrices are 16 ×
16 the spectrumof fermions for a rank N configuration has 16 N modes.The spectrum is typically non-degenerate and there arecrossings of zero: a fermion becomes massless at suchloci. Counting such crossings gives a notion of a localindex [11], and this counts the number of strings createdwhen the probe is moved from infinity to a place, givinga generalized version of the Hanany-Witten process [13].The figure shows a clean separation of two regions.In the first region, the eigenvalues of the fermions arewell separated from zero: there is a gap which can beused to measure the distance from the probe to the con-figuration. There is a second region where the typicaleigenvalue separation of the fermion spectrum from zerois similar to the separation of the eigenvalues from eachother. This region is said to be gapless. Remember thatthe simulated matrix configurations are thermal. Thetemperature gives a thermal activation energy kT . Solong as the modes near zero have energies below kT , theycan become thermally active. At large N and fixed tem-perature, the size of the matrix configuration X i scalesas N / (the typical eigenvalue of X i scales as N / T / [10]) and so the number of fermions in the band of energy kT grows as N / (cid:39) N N − / . Thus, in a large N setupthere are a lot of fermion states that could in principleget activated. In this sense, there is no gap. Spectral Dimension and Nonlocality
We wish to understand the nature of the physics insidethe gapless region, and in particular if we can understandthis information geometrically or not. One step in thisdirection is knowing the effective dimension of spacetimethat the matrix configuration describes. Since we areexploring R , it is natural to suspect that the matrixconfiguration in the gapless region is 9 + 1 dimensional.In massless free field theories in d + 1 dimensions, thedensity of states near zero follows a power law ρ ( (cid:15) ) ∼ (cid:15) γ − where (cid:15) is the energy and γ = d . We argue that if wemeasure γ we can measure the effective dimensionality ofthe matrix configuration. The parameter γ will be calledthe spectral dimension of a configuration. We argue thatif we measure γ with a probe we are actually measuringthe dimensionality in standard configurations.Consider a toy model for matrix black holes in whichthe matrices X are essentially commuting and describea gas of D0-branes (this has been argued recently forexample in [14, 15] and references therein). The ma-trices can be diagonalized simultaneously, and we cantalk about the positions of the D0-branes as the com-mon eigenvalues in R . More importantly, we can thinkof a density of eigenvalues in some region of R , call it ρ ( r ). The energy of the fermions connecting a probe at x to an eigenvalue at r will be (cid:15) (cid:39) | x − r | . The num-ber of such states at fixed | x − r | = s for small s is n ( s ) = (cid:82) d r ρ ( r ) δ ( | x − r | − s ) (cid:39) ρ ( x ) s (cid:39) ρ ( x ) (cid:15) . So ina region where ρ ( x ) (cid:54) = 0 we expect to measure a spec-tral dimension of γ = 9. Similarly, we can consider a D2brane background obtained from a fuzzy sphere config-uration. If we put the D0-brane probe in contact withthe fuzzy sphere, it is easy to show that one gets a spec-tral dimension γ = 2, the dimensionality of the sphere asa geometric object. This coincides with approximatingthe fuzzy sphere as a collection of D0 branes uniformlydistributed on the surface of the fuzzy sphere. The spec-tral dimensionality captures the dimension of extendedobjects, or equivalently of the smearing of D0 branes insome region. It is natural to expect that if the matrixconfiguration can be pictured as some extended D-branecontorted to fill the gapless region, one would measure γ = 9 just from smearing into a density of D0-branes.We will now check if this is true or not numerically. Ifit does coincide with γ = 9, we would be giving evidencein favor of the toy model of black holes as a gas of D0-branes or an extended brane filling the region. We finda completely different result.We define γ ≡ lim (cid:15) → d ln( ρ ( (cid:15) )) d ln( (cid:15) ) + 1 (4)The difficulty with this definition is that we can not takethe limit on a configuration of finite size matrices, wherewe only have finitely many eigenvalues in the effectiveHamiltonian. What we need is a fit to a power law bychoosing a few points near (cid:15) (cid:39)
0. The precise way inwhich we choose to do this can give slightly different an-swers. To reduce such problems, we average over manyconfigurations so that we can measure γ statistically.We also need to compare the spectra of eigenvalues ofthe effective Hamiltonian at different values of N . Thisway we can extrapolate to large N and find a value of γ that is valid in the thermodynamic limit. This helps toshow that our result is robust. To put different values of N on top of each other we take advantage of the scalingsymmetry of the classical BFSS matrix model and scalethe X i by some N -dependent factor that is also tem-perature dependent. Since the X i approximately followthe Gaussian Unitary Ensemble for traceless Hermitianmatrices (TGUE) [10], we can scale the matrices suchthat their distributions of eigenvalues can be analyzedin terms of the limits of the TGUE which are semicircledistributions. This is done by fixing the second moment.The width of the associated semicircle is given by 2 √ N σ where σ is the width of the TGUE. The value of σ isgiven by σ = (cid:115) (cid:104) (cid:80) i =1 Tr( X i ) (cid:105) N −
1) (5) where we have averaged over all nine bosonic matriceswhich is allowed by the SO (9) invariance. Energy N o r m a li z e d D e n s i t y o f S t a t e s a t E dg e Edge N =43 N =67 N =97 N o r m a li z e d D e n s i t y o f S t a t e s a t C e n t e r Center
FIG. 2. A plot of the density of states as a function of energyat the center and edge of the gapless region for various val-ues of N averaged over 1000 configurations. Each function isnormed such that (cid:82) ρ ( (cid:15) ) d(cid:15) = 1. The spectral dimension of the field theory when theprobe is at the center of the gapless region is γ = 1 . γ = 1 . ± . | x | = 0 . ± .
03 for the rescaledmatrix configurations. For reference, the matrices arescaled so that average maximum eigenvalue approachesone as N → ∞ . Even with numerical errors the spectraldimension at the boundary is far from nine. If we includequantum corrections, the value of γ should stay close tothe classical physics value: in the BFSS matrix model itis believed that there is no finite temperature phase tran-sition between the high temperature regime and the lowtemperature regime. This is also verified numerically asthe free energy curves as a function of the temperatureare smooth [16–18]. Also, the thermal states in the BFSSmodel are deconfining for arbitrary small temperatures inthe dual supergravity setup [15].How should we interpret this result? The authors of[19] consider a class of theories with fermions defined on afully connected lattice whose links are weighted. They ar-gue that the typical configuration is maximally connectedgiving rise to a theory with one infinite-dimensional sym-plex which is maximally non-local; every site is connectedto every other site by one step. When the weights areGaussian distributed, the spectrum of the Hamiltonianfollows a cumulative semicircle distribution (in the large N limit). Near zero energy the spectrum is linear, thedensity of states is flat, and they conclude that the non-local fermion theory has an effective description in 1 + 1spacetime dimensions, in the sense that the density ofstates is the same as the one of a 1 + 1 dimensional the-ory. We also find a spectral dimension of one and thusconsider ourselves to be in the same universality class asthe models in [19], which is also the universality class of arandom Hamiltonian. By analogy to the fully connectedlattices, we believe that the probe brane is in an environ-ment where is it effectively equally far from the D0-branematrix background no matter where the D0-branes arelocated. Our numerical results are inconsistent with alocal gas of D-branes. If this persists in the fully quan-tum regime, the results of calculations based on modelsof such gases are suspect. The setup is maximally non-local, but it looks as if the dynamics lives effectively in1 + 1 dimensions. Drama at the Horizon
We now argue that we should locate the horizon ofthe black hole exactly at the interface between the gap-less region and the gapped region. This is potentiallydifferent than the proposal in [20]. Consider a string sus-pended between a probe D-brane at fixed position and ablack hole. The energy carried by such a string is givenby (cid:82) r D r T ( r ) √ g tt g rr dr where T ( r ) is the local string ten-sion at r , r is the horizon, and r D is the position ofthe brane. This is finite for typical black holes: the blackhole horizon is at finite distance. In the limit r D → r weget that the energy of such a string goes to zero. In oursetup, the fermionic energies connecting the probe to thematrix configuration have this same behavior. Also, thepresence of a large number of massless modes appearingat the edge of the gapless region indicates that wiggles ona string also get redshifted and produce a large numberof string states with small energies. This is analogous tostrings spreading when reaching the horizon [21].Consider the probe as a dynamical observer movingtowards the horizon. A natural question to ask is if wecan integrate out the modes connecting the probe to thematrix configuration and extract an effective action forthe probe. This is how gravitational interactions andforces between objects are captured in the BFSS matrixmodel [8] (see [22, 23] for a systematic treatment). Inour case the answer to this question is yes, for fermions,away from the gapless region.If we reintroduce (cid:126) and call an eigenvalue of (3) ω ,the energy of such a fermionic mode is (cid:126) | ω | . The lightfermion modes become thermally active when (cid:126) | ω | < kT .In classical physics all the fermions are active, yet wewill assume that the system is cold enough so that mostfermions are not active. After taking the limit where N is large with (cid:126) constant, and rescaling X so that the matrix configuration is of finite size in the probe coor-dinates, the light fermions become active exactly whenwe enter the gapless region. The approximation wherewe can integrate out the off-diagonal degrees of freedomconnecting a probe to the black hole breaks down exactlyat the putative horizon.This breakdown of effective field theory of a probe plusconfiguration suggests that the observer can not fail toobserve that physics has changed dramatically on cross-ing the horizon (see however [24]). The fourth postulateof [3] was that no drama occurs when crossing the hori-zon. Removing this postulate to restore consistency sug-gested the existence of a firewall. We see that the BFSSmatrix model is giving a physical model for the firewall.This in itself does not give a proof that firewalls exist.There could be some other physical effect on the probethat comes from understanding the bosonic degrees offreedom that forces us to put the horizon elsewhere, asin [20]. Bosonic instabilities might be realized only viaparametric resonance and would require a full time de-pendent treatment to be understood.Our analysis so far has been done for stringy blackholes [10]. For such black holes the horizon and the sin-gularity are on top of each other in string units. The1 + 1 dimensional effective physics could be associatedwith the singularity rather than with the horizon. Thiscan be remedied with fully quantum simulations [14, 15].Physics at the black hole singularity is also argued tobe effectively 1 + 1 dimensional in the causal dynamicaltriangulation program, since the effective UV structureof gravity has a different dimension [25] (see also [26]).Also note that this picture, although similar in spirit tothe fuzzball picture [27], is distinct. The known fuzzballsolutions are geometric (non-singular solutions of super-gravity) and they stretch all the way to the horizon. Mi-crophysics in these setups is essentially gravitational. Inour case the inside of the black hole gets replaced by non-geometric, non-local objects whose effective dimension isdifferent than that of the ambient space.The presence of an effective 1 + 1 dimensional fieldtheory starting at the horizon is also reminiscent of ideasespoused by Carlip [28] and suggests that the additionalentropy added to the black hole when the probe is ab-sorbed can be computed using Cardy’s formula. Acknowledgements:
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