No Firewalls in Holographic Space-Time or Matrix Theory
aa r X i v : . [ h e p - t h ] M a y UTTG-10-13 TCC-007-13 RUNHETC-2013-10 SCIPP 13/06
No Firewalls in HST or Matrix Theory
Tom BanksDepartment of Physics and SCIPPUniversity of California, Santa Cruz, CA 95064 and
Department of Physics and NHETCRutgers University, Piscataway, NJ 08854E-mail: [email protected] FischlerDepartment of Physics and Texas Cosmology CenterUniversity of Texas, Austin, TX 78712E-mail: fi[email protected] 20, 2013
Abstract
We use the formalisms of Holographic Space-time (HST) and Matrix Theory[11] to investi-gate the claim of [1] that old black holes contain a firewall, i.e. an in-falling detector encountershighly excited states at a time much shorter than the light crossing time of the Schwarzschildradius. In both formalisms there is no dramatic change in particle physics inside the horizonuntil a time of order the Schwarzschild radius. The Matrix Theory formalism has been shown togive rise to an S-matrix, which coincides with effective supergravity for an infinite number of lowenergy amplitudes. We conclude that the firewall results from an inappropriate use of quantumeffective field theory to describe fine details of localized events near a black hole horizon. In bothHST and Matrix Theory, the real quantum gravity Hilbert space in a localized region containsmany low energy degrees of freedom that are not captured in QU(antum) E(ffective) F(ield)T(heory) and omits many of the high energy DOF in QUEFT.
The purpose of this paper is to sharpen our argument that quasi-local holographic models of blackholes do not exhibit the firewall phenomenon that the authors of [1] (AMPS) claim follows from aquantum field theoretic analysis of Hawking radiation in a regime in which quantum field theorycan be trusted. While we have some sympathy with the arguments of [3][4][2], that the thought1xperiment envisaged by AMPS cannot, for a variety of reasons, be carried out even in principle, ourown view is that the problem lies primarily with the use of QFT to analyze fine grained aspects ofthe quantum information involved in black hole evaporation.In [6] we presented these arguments, but we failed to emphasize sufficiently what the importantpoints were, because our understanding had evolved in the course of writing the paper. We attempt torepair this here. In addition, we present a completely different set of arguments based on the MatrixTheory models of black holes[8] in highly supersymmetric compactifications of M-theory. The detailsare somewhat different, but the conclusions are similar. There is no firewall, though the model hasquantum states that share all of the familiar properties of black holes, and is manifestly unitary.There is also a large body of evidence that Matrix Theory does reproduce the correct scatteringmatrix for low energy effective supergravity.The problem of accounting for black hole entropy is one in which there is an evident breakdownof local quantum field theory in the low energy regime. In the absence of a black hole, local quantumfield theory can account for at most o ( A / ) (in 4 dimensions) of the entropy allowed by the covariantentropy bound in a causal diamond whose holographic screen has area A . In the presence of a blackhole, field theory instead over-counts the entropy, and finds an infinite entropy per unit area, despitethe fact that, in the vicinity of the horizon of a large black hole, the local space-time geometry of thehole is identical to that of flat space. Indeed, the same infinity is found for the entropy encounteredby a highly accelerated Rindler observer in flat space. The infinite entropy per unit area comes frommodes of arbitrarily short wavelength in regions that have a small space-like separation from thehorizon. These modes have low energy from the point of view of an accelerated observer. However, ifwe consider a small causal diamond surrounding a portion of the horizon (Rindler or Schwarzschild)and insist that the state of QFT in that diamond be such that gravitational back reaction is negligible,then the short wavelength modes must be frozen into the Minkowski ground state. In that state,there is infinite entanglement entropy per unit area of the holographic screen, between DOF localizedin the diamond, and DOF an infinitesimal space-like distance outside it. There is an obvious paradoxhere. For the Rindler or Schwarzschild observer, the entropy refers to real excited degrees of freedom,while entanglement entropy is a property of a pure state.Even in QUEFT, this paradox is resolved by noting that the Rindler and Schwarzschild observersuse a different Hamiltonian from the geodesic observer in a locally Minkowski space. The Rindler caseis particularly illuminating because observers with different acceleration see different temperatures.If we make the assumption that observers sharing the same causal diamond must see the samequantum state up to a unitary transformation, we are prodded in the direction of the conclusionthat each accelerated trajectory must have its own Hamiltonian, since they each see a different vonNeumann entropy. This is the starting point for the axioms of HST. We will not repeat the HSTdescription of accelerated observers here, but refer the reader to [12]. However, we do pause to recordthe message of that paper: A causal diamond of finite area ∼ N in a 4-dimensional space-time has o ( N ) (graded)-commuting copies of a super-algebra with a finite dimensional unitary representation.The fermionic generators of that algebra are labeled ψ Ai ( P ) , where the explicit indices are those ofan N × N + 1 matrix. At most o ( N / ) of these DOF have a particle interpretation, and it isonly to this subset that the rules of QUEFT apply. We call the rest, horizon DOF. By contrast, aJacobsonian[5] TH(ermodynamic) E(ffective) F(ield) T(heory) will always encode the coarse grainedhydrodynamics of any Lorentzian space-time in HST . For a geodesic observer in a causal diamond, Jacobson showed that, apart from the cosmological constant, Einstein’s equations follow from the first law ofthermodynamics, applied to a system with an effective space-time description, and such that the entropy seen by amaximally accelerated Rindler observer, near each point, varies along the observer’s trajectory like the area transverseto a bundle of initially parallel trajectories. HST satisfies Jacobson’s criteria, and fixes the c.c. by a boundary conditionrelating the behavior of area vs. proper time in the limit of a diamond with large proper time. he bulk of the degrees of freedom have a coupling of order /N to the particles. They give rise toparticle interactions, but have no long term entanglement with properly prepared incoming particlestates. Accelerated observers have different Hamiltonians, and experience more entanglement: theredshift of particle energies compared to those of horizon DOF couples the particles more strongly andleads, in the large N limit, to the Unruh temperature. The common practice in studies of black holes using QUEFT, has been to cut off the infinity of lowenergy QUEFT DOF in the accelerated observers frame by putting a “brick wall” a finite space-likedistance away from the horizon. Then, the remaining finite entropy per unit area is divided between astretched horizon whose dynamics is admitted to be unknown, and a “zone” where standard QUEFTdescriptions are valid. This is the description used by AMPS. We do not really have a quarrel withthis prescription, if it is viewed as a relatively coarse grained model of the black hole. What weclaim is that such a model cannot pretend to account for the quantum dynamics of the tiny fraction( < o ( N / )) of the DOF, which are all that is necessary to describe particle physics inside the horizonof the black hole. In HST, space-time physics is described in terms of an infinite number of quantum systems, each ofwhich encodes the physics as seen along a particular time-like trajectory, in a proper time depen-dent Hamiltonian. Relations between density matrices for shared information, combined with theholographic connection between Hilbert space dimension and area, enable us to extract the causalstructure and conformal factor of a Lorentzian metric from the quantum mechanics.When we first approached this problem, we believed that the main feature of HST, which avoidedthe firewall, had to do with the fact that different observers had different Hamiltonians. J. Polchinskiand D. Harlow convinced us that the problem could be recast entirely in the Hilbert space of theobserver that is called Alice in most of the firewall literature. Our resolution of the problem in [6] infact relied on different features of HST, but enough of our early thinking survived in the final draftof that paper, that it has obscured the issue. What follows is the description of a decaying black holein HST, purely from the point of view of the detector A , formerly known as Alice. In fact A belongsto a one parameter family of detectors A ( T ), parametrized by the amount of proper time, after theformation of the horizon , before the detector falls through the horizon of the partially evaporatedblack hole. To keep everything finite, we introduce a large time − N , prior to horizon formation,and designate the time of horizon formation by 0. All of the relevant events occur within a causaldiamond of area N . The following occurs in real-time:From the point of view of the A detector, the black hole evaporates for a long time beforethe detector crosses its “horizon”. In HST we understand that the black hole space-time is anapproximate, hydrodynamic description of some of the quantum degrees of freedom in the S-matrixtheory of asymptotically flat space-time. It is appropriate for a description of those causal diamondswhere the detector whose Hamiltonian we are describing receives signals from the black hole horizon.These signals cannot come from particle degrees of freedom inside the horizon. That is the lessonencoded in the classical geometry. In HST, we view those signals as originating from the o ( N ) DOFon the black hole horizon, but, to the extent that the classical geometry tells us that particles insidethe horizon experience ordinary particle physics, we consider the state of those < o ( N / ) particleDOF, to be unentangled with the bulk of the horizon states. Note that the Hamiltonian of thedetector A does not interact with these interior particle DOF until the detector’s trajectory crosses In HST, the definition of space-like slices has to do with a convention of synchronization of clocks for space-likeseparated portions of different time-like trajectories. o ( N ) − o ( N / ) DOF with which it does interact include both those that areassociated with the stretched horizon, and those in the “zone”, in the AMPS description. Cutting offthe infinite entropy per unit area in QUEFT leaves a huge hole through which an entropy sub-leadingin the area can fit.QUEFT describes the single vacuum state in a region in a single causal diamond in terms of a hugeHilbert space of “particle states” viewed by an accelerated observer. We know that in Minkowskispace it is incorrect to view an area’s worth of entropy in this Hilbert space as describing actual,physically accessible states of the system. Yet this is precisely what AMPS do. Their argument isbased on the idea that one can understand the restrictions on the utility of QUEFT in terms of asimple UV cutoff on local wavelengths. In fact, for a geodesic observer in a low curvature region ofspace time, this claim is simply false. Instead, as we have argued above, any entropy in the causaldiamond that is proportional to the area, must consist of states which have very low energy accordingto the Hamiltonian of the geodesic observer, and which therefore cannot manifest as particles in thebulk of the diamond . This is true for diamonds inside the horizon, outside the horizon, or straddlingthe horizon.The utility of the picture of black hole states as partially described by QUEFT with a brick wallcutoff , is due to the ease with which one can understand Hawking radiation in this picture, andwas of course the description used by Hawking in his original argument. This often leads to theerroneous claim that if we give up this picture of the Hilbert space, we will no longer understand thethermal nature of black holes and the calculation of the Hawking temperature. In fact, the thermalnature of Hawking radiation implies that the coarse grained details of Hawking’s calculation will bereproduced by any model of a black hole that exhibits it as an ergodic system , with the correctentropy/energy/size relations.In fact, Hawking’s derivation of Hawking radiation is not valid even for strongly coupled fieldtheories. The reason that we know that the thermodynamic picture of black holes is correct is thatthe Hartle-Hawking state of a black hole is a thermal state, no matter what the field theory is.The entropy and temperature of the black hole are encoded in its classical geometry. We thinkthat the deepest understanding of why this is so comes from Jacobson’s observation that, apartfrom the cosmological constant, Einstein’s equations are the hydrodynamics of space-time, assumingthat space-time emerges from a quantum system obeying the Covariant Entropy Conjecture thatthe entropy of the Hilbert space of quantum gravity is one quarter the area in Planck units of theholographic screens of infinitesimal causal diamonds around every point .The Jacobsonian point of view also sheds light on the following vexing question: On the one hand,a finite entropy black hole can be formed by scattering a finite number of particles in Minkowski space,and on the other hand, it seems to have a distinct space-time metric. The Jacobsonian interpretationof the black hole metric is as a hydrodynamic approximation to the behavior of a large number ofDOF in the Minkowski system, whose behavior is not well modeled as particle physics. .Our goal in this section is to provide a statistical mechanics, which is in agreement with thethermodynamic predictions of this classical metric. For us, these include the behavior of in-fallingparticle systems before they hit the classical singularity. This was done in [6], so we just summarize ithere. We describe the Hamiltonian of the detector A ( T ), which follows a trajectory that encountersthe instantaneous black hole horizon a proper time T after the original black hole horizon forms.The black hole metric is approximated by a time dependent sequence of Schwarzschild metrics witha radius parameter R ( t ) that follows Hawking’s evaporation law. We use the word ergodic very loosely here, not in its technical mathematical sense. There are a variety of hypothesesabout the nature of a quantum system, which lead to certain aspects of thermal behavior. Any one of them will do. Here infinitesimal means much larger than Planck scale but much smaller than the local radius of curvature ofspace-time.
4t any given time, t , we split off π ( R ( t ) M P ) black hole degrees of freedom from the vastly largernumber N , which describe A ( T )’s full causal diamond. For t < T , the horizon crossing time of A ( T )the bulk of these black hole DOF are given a Hamiltonian with a Planck scale time dependence,which is a sum of traces of the matrices (1 − Π) ψψ † (1 − Π). Π is a projection matrix on a K ( t ) × K ( t )subspace with K ≤ ( R ( t ) M P ) / . The rest of the o ( N ) DOF, which are not associated with theblack hole, are given a Hamiltonian, and an initial condition in the remote past that are appropriatefor describing particle physics in Minkowski space. We do not have a complete description of thisbut a class of Hamiltonians that give the right qualitative physics was described in [12]. There areadditonal constraints necessary to guarantee that in the large N limit, the S-matrix becomes super-Poincare invariant, which we have not yet implemented. Our model does not try to describe theformation of the black hole from some particular incoming particle state. Although AMPS assumea black hole formed in this way, they assume nothing about the incoming state besides its purity.Note that, in addition to the rapid variation of the Hamiltonian mixing up the [ √ πR ( t ) M P − K ( t )] states there is a much smaller time dependence of the Hamilton, coming from the time dependenceof R ( t ). The total Hamiltonian is H Mink + H Hor ( t ) + H K ( t ) , where the second two terms act on the black hole DOF. As R ( t ) varies, H Hor ( t ) and H K ( t ) act onfewer DOF, and we add those to the Hamiltonian H Mink . This time dependence is an addition tothe rapid time dependence of H Hor ( t ) , which acts on o ( R ( t ) − K ( t )) DOF.The Hamiltonian H K ( t ) describes the evolution of DOF that will be experienced as particles bythe detector A ( T ). As in all HST models, the time dependent Hamiltonian of the detector splits intotwo pieces H ( t ) = H in ( t ) + H out ( t ) . The Hamiltonian H K ( t ) is part of H out ( t ) until t = T . For t > T it is included in H in ( t ). This is notreally a discontinuous transition. The DOF included in H K ( t ) are, at early times just particles thathave been sent in from past infinity and are initially causally separated from the detector A ( T ). If weconsider the large causal diamond of a geodesic observer that is causally connected to the particles atearly times, but never falls into the black hole, then, at early times, these particles are described bythe geodesic Hamiltonian and initial conditions in that diamond[12]. For simplicity, we assume thatthey don’t undergo any scattering before they enter the causal diamond where A ( T ) will encounterthem.The particles in H K ( t ) are those with which the detector A ( T ) can interact, between the time itcrosses the instantaneous horizon and the time its trajectory encounters the black hole singularity.Thus, for times T < τ ≪ T + R ( t ), the state in the Hilbert space representing the entire black hole isapproximately a tensor product of a state acted on by H K ( t ) and one acted on by H Hor ( t ) . In termsof the matrix DOF ψ Ai , these two Hamiltonians are functions of Π ψψ † Π and (1 − Π) ψψ † (1 − Π).Interactions between the two sets of DOF via off block diagonal matrix elements, are suppressed bythe large number 1 /N , as long as τ ≪ R ( t ).It’s important to stress why we’re making these claims, especially the last one. Our aim is toconstruct a quantum system, whose behavior mimics the classical space-time picture of the interiorof the black hole. In that picture, the particles described by the DOF in H K ( t ) , behave, for a time oforder R ( t ), approximately as they would in flat space.At the time T (within a tolerance ≪ R ( T )) the DOF in H K ( t ) are incorporated into H in ( t ) ofthe detector A ( T ). The Hilbert space of that detector is vast, with entropy of order N . However,our model of this detector’s behavior is that it’s own components, and all the particles with whichit interacts after crossing the instantaneous horizon , are described by the Hamiltonian H K ( t ) . In5ddition to the slow time dependence induced by the change of K ( t )and R ( t ), this Hamiltonian hasa time dependence which becomes extremely rapid as t approaches T + R ( t ). These time dependentterms are traces of products of the full R ( t ) M P × R ( t ) M P matrix ψψ † . They are very small at t = T and become competitive with the ordinary particle physics contributions in a time of order R ( t ). Their effect is to mix up the particle DOF with the horizon, so that the distinction betweenparticles and horizon is no longer meaningful. At a time of order T + R ( t ) the detector A ( T ) has“hit the singularity”.The bulk of the Hilbert space of A ( T ) knows nothing about this catastrophe, either before orafter it happens. The DOF in this Hilbert space interact with the horizon DOF in the matrix(1 − P ) ψψ † (1 − P ). If the infinitely intricate measurements envisioned by AMPS could actually becarried out one would find that the assumption in Page’s discussion[10] of information extractionfrom a black hole was subtly wrong at the time T , when T is of order the Page time. That is, theblack hole is not in a generic state of a Hilbert space of entropy π ( R ( T ) M P ) , because a tiny tensorfactor whose entropy is of order K ( T ) < ( R ( T ) M P ) / is not entangled with the rest of the space.If we synchronize the external clock to the proper time of the in-falling trajectory of A ( T ), then thisfactor becomes entangled with the rest of the black hole Hilbert space at a time of order T + R ( T ).Clearly, neither the thermodynamic properties of the black hole, nor the statement that theevaporation process is unitary are violated by this model. One of the assumptions of Page’s argumentis modified by an amount that is thermodynamically negligible. The centerpiece of the AMPSargument, the representation of the Minkowski vacuum state in a local region of space-time as anentangled state in a certain factorized basis of the field theory Hilbert space, simply does not appearin the HST formalism. Our model clearly treats the black hole as a thermodynamic object with thecorrect energy-entropy-size relations. So it’s simply untrue that one needs this entangled picture toobtain Hawking radiation.We believe that the only valid criticism of our model is that we have not yet shown that it reallyreproduces the results of field theory in conventional situations where no black holes are involved. Weargued that this was the case, to the best of our current ability, in [12], but we are aware that only acomplete calculation of some scattering amplitude will really make the case. In the next section, wewill argue that the Matrix Theory description of 11 dimensional Schwarzschild black holes, gives apicture of black hole evaporation consistent with the one we proposed in HST. There is no firewall,no description of the Minkowski vacuum as an entangled state, a manifestly unitary S-matrix forparticle scattering, and manifest super-Galilean invariance. In addition, it has been shown that aninfinite number of scattering amplitudes in this model coincide with those given by a super-Poincareinvariant QUEFT - 11 dimensional supergravity.In principle, the above criticism might be applied to our claims about reproducing the propertiesof Hawking radiation. However, we demonstrated in [12] that the Minkowski Hamiltonian, whichacts on o ([ N − R ( t )] ) DOF, acts, as N → ∞ , like the kinetic term of a collection of masslessparticles, on a tensor factor of entropy ≤ N / of its Hilbert space. We also argued that in thelarge N limit, pure states of these particles remain pure and that the effect of interaction with thebulk of the o ( N ) DOF could be encoded in particle interactions that were localized in space-time.Thus, our model does describe black hole evaporation as a sequence of quasi-equilibrium states of theblack hole, interacting with a gas of relativistic particles of (potentially) much higher entropy. Thedynamics is explicitly rotation invariant and there are emergent super-Poincare generators, which acton particle states in the large N limit. This is enough to establish the thermal nature of the spectrumof evaporated particles, even though we have not established that the S-matrix of those particles isPoincare invariant. Thus, we certainly cannot claim that our model will reproduce the gray bodyfactors that arise in the field theory treatment of Hawking radiation, but the thermal nature of thespectrum and the correct temperature are guaranteed.6n HST itself, we still have to discuss the consistency conditions between detectors A ( T ) withdifferent values of T . In [6] we showed that consistency between detectors with T ∼ R S and T ≫ R S implied that the late falling detector had to encounter a singularity in a time of order R S afterit crosses the horizon. Consistency of the in-falling detector’s description with that of a supporteddetector implies that the description of the black hole from the supported detector’s point of viewmust include a tensor factor of entropy ∼ ( R ( t ) M P ) / , which is approximately unentangled withthe horizon . Since ( RM P ) / ≪ ( RM P ) , this does not affect the thermodynamics of black holesuntil they are of Planck size. Since R ( t ) goes to zero eventually, there is no problem with unitarityof the S-matrix either. Of course, once the black hole is Planck size, the approximate descriptions inthis paper lose their validity. Indeed, even in Minkowski space, the clean separation between particleand horizon DOF is impossible in small causal diamonds. Indeed, as shown in [12], this featureof the HST description is responsible for reactions that change the number of particles, and theirmomenta. Particles are emergent phenomena in HST, strictly speaking defined only in the limit ofinfinite causal diamonds. The validity of QUEFT, with its implicit assumption of infinite numbersof particle states, is even more restricted. We will restrict attention to the Matrix Theory for String/M-theory in 11 non-compact dimensions.This is the quantum mechanics of the zero modes of maximally supersymmetric SU ( N ) Yang-Millstheory. Similar results would be obtained for compactification of the theory on tori of dimension1 −
3. The Lorentz invariant limit is achieved by taking N → ∞ and computing the S-matrix forstates whose energy scales like 1 /N . However, in [8], the authors argued that one could understandthe qualitative dynamics of black holes of entropy S in the model with N ∼ S ≫ X cl ( t ), whose variation away from the origin of transverse coordinates was bounded by a distance inPlanck units of order one (in the sense of large N counting). The matrices in the solution have rank N . For large N , we can think about them using the correspondence [9] with light front membranetheory. The background defines a (fuzzy) toroidal membrane, whose volume is parametrized by twoangles p, q . The matrices are functions on the phase space [ p, q ] with commutator given, in the large N approximation, by Poisson brackets. There are many such solutions, but in 11 dimensions, thismultiplicity gives rise to a sub-leading correction to black hole entropy.Now write X = X cl + P i x i M i , where M i = X k,l e − N [( p − p i − πk ) +( q − q i − πl ) ] . The commutators of these matrices satisfy | [ M i , M j ] | ≤ e − [( p i − p j ) +( q i − q j ) ] . Taking a distribution of points separated by distances of order √ N (there are o ( N ) such points onthe torus) , we can make these commutators as small as we like. Thus, there’s a basis in which allthe matrices M i are simultaneously block diagonal. The traces of these matrices are o (1). Quantum translation: the time dependent Hamiltonian must mix the particle DOF inside A ( T )’s horizon with thehorizon DOF in a time of order R S . We thank D. Harlow for repeatedly emphasizing this point to us. X cl ( p, q ) and the fluctuations x i gives rise toa harmonic potential binding the x i to the background. The terms bilinear in different x i are of thesame order as the commutator [ M i , M j ] and we drop them. The quadratic potential is X i x i N Z dp dq [ p + q ]( ∇ X cl ) e − N ( p + q ) . The factor of N in front of the integral is the combination of a 1 /N in the translation of tracesof matrix commutators into integrals of Poisson brackets over the membrane, and two factors of N coming from converting derivatives of M i into factors of p or q . For a smooth classical membraneconfiguration, the gradient of X cl is N independent for large N . The harmonic potential thus has anoverall coefficient 1 /N . This is, it will turn out, negligible compared to other contributions to theenergy.The latter come from integrating out off diagonal matrices between the different x i terms. If the x i velocities are small, which is verified self consistently, the effective Hamiltonian is H = X i p i + AG N X i,j ( p i − p j ) | x i − X j | . The coefficient A is of order 1. For a bound system with this Hamiltonian, and large N , the meaninterparticle distance R S , the energy per particle, and the total light front energy (which gives usthe mass) may be calculated crudely by using the uncertainty principle and the virial theorem. Theresult is BG − N R S = N,E per particle ∼ R − S ,M ∼ G − N N . Here R S is the average separation, which is also the size of the bound state. These are the expectedrelations for an 11 dimensional Schwarzschild black hole, with the individual D − branes having thekinematics expected for Hawking particles boosted to the light front frame, and R S the Schwarzschildradius, if N is indeed the entropy of the black hole.The fact that N is indeed the entropy follows from the fact that the D tethered to different positions on the classical membrane, so that they are in fact distinguishable particles,obeying Boltzmann statistics. If we plug x i ∼ R S into the harmonic potential, we find a negligiblecorrection to the total energy. In the second paper in[8], we showed also that the correct Newtonianinteraction between black holes is obtained, if we are careful to note that we are calculating energiesaveraged over the longitudinal circle of Discrete Light Cone Quantization. In the third paper weestimated the rate of Hawking evaporation, and found agreement with the expectations for a thermalsystem with the indicated energy and entropy.The mechanism of Hawking radiation was “snapping of the tethers”: a quantum fluctuation,which momentarily sets to zero the piece of the classical configuration that provides the harmonicbinding for a particular D − brane coordinate x i . That particle then flies out to infinity along theflat direction in the matrix potential. The black hole then re-equilibrates with one constituent fewer.In the second paper of [8] it was pointed out that the analysis of the dynamics of the non-compactdimensions gave analogous results for black holes in all dimensions. However, at that time MatrixTheory technology required one to use a more and more complicated field theory to compactifythe theory on more and more dimensions. For a 5 dimensional toroidal compactification one wasforced to go beyond field theory, and for 6 and more dimensions the Matrix Theory proposal failed.8esults could be established firmly only for compactifications on tori of dimensions 1 −
3, and in thelast of these cases the internal field theory contributed a finite fraction of the black hole entropy.Recently TB and Kehayias[7] have suggested an alternate definition of Matrix Theory, which is asimple quantum mechanics (not a field theory) for every compactification. It would be interestingto return to the black hole problem using this technology. We conjecture that a unified qualitativepicture of Schwarzschild black hole dynamics might result.Be that as it may, the purpose of the present paper is to establish the absence of firewalls for11 dimensional Schwarzschild black holes. Recall that Matrix Theory defines a scattering matrix forasymptotic states along the U ( k ) ⊗ . . . ⊗ U ( k n ) flat directions of the Matrix Theory potential. The SU ( k i ) degrees of freedom are frozen into their unique BPS bound state, and the scattering statesare manifestly those of eleven dimensional supergravitons. If we take k i to infinity, at fixed ratios k i k j then the asymptotic states support an action of the SO (1 ,
10) super-Poincare group. The S-matrixis manifestly invariant under the super-Galilean sub-group of this group, and the existence of theS-matrix in the limit is equivalent to longitudinal boost invariance. It is hard to see what kind ofinstability could make the manifestly unitary S-matrix fail to exist in this limit, because from thepoint of view of the quantum mechanics, it is a low energy limit. In particular, emission of stateswith longitudinal momentum that does not scale to infinity is forbidden by energy conservation.We also know that the S-matrix obeys an infinite number of non-renormalization theorems[13],which imply that an infinite number of terms in the low energy expansion of the hypothetical limit,actually coincide with those expected from the low energy expansion of a super-Poincare invarianteffective Lagrangian. This proves that the limiting S-matrix is not the unit matrix, and stronglysuggests the existence and super-Poincare invariance of all matrix elements.Finally, Matrix Theory provides a definition of finite time transition amplitudes, for processesthat take place over a finite range of transverse distance. These amplitudes manifestly approach thecorresponding S-matrix elements as the time and transverse distance go to infinity. Using these, wecan model the experience of an apparatus falling into a black hole. The apparatus is modeled by a K × K block of the Matrix Theory variables, with 1 ≪ K ≪ N . Initially, we take the transverseseparation between the K × K block and the N × N block, which represents the black hole, to bevery large, and set the center of mass of the K × K block moving slowly towards the transverseposition of the hole. Consider an initial condition for the SU ( K ) variables which consists of twogroups K , of supergravitons coming in from a large distance. For comparison, we take K = K .The first group collides at a time long before the c.m. of the block approaches the position of theblack hole. What this means is that the incoming coefficients of a block diagonal matrix of block sizes p . . . p m , with P p i = K become close enough so that it no longer takes a huge energy to excite theoff diagonal matrices. Non-Abelian dynamics becomes important and we find finite amplitudes forgoing off in flat directions q . . . q n , with P q i = K . We compute the amplitude from the time theinitial separation is L ≪ R S until a time after collision when final separations are of order L . Duringthis entire period of time, the separation between the c.m. of the K × K block and the black holesis ≫ R S .For the K block, we instead time the incoming particles so that they begin to interact stronglywith each other when the c.m. of the block is within R S of the center of the black hole. Since L ≪ R S , the typical distance between particles in the block is much less than their distance fromany of the D − brane constituents of the black hole. Thus, the interactions with the black hole canbe considered a small perturbation of the particle interactions in flat space, over times where thetransverse separation is ≪ R S . Over longer time scales, this is no longer true. For distance scalesof order R S from the K block, a particle we originally considered part of the K block will suffermultiple scatterings with black hole constituents, whose transverse separation from it are smaller9han or of the same order as those in the K block. Since the number of black hole constituents islarge compared to the number of particles in the original event (since N ≫ K ) it is plausible thatthe particle will come into equilibrium with the black hole constituents. That is, interactions withthe constituents will tend to break it up into its individual D − branes and these will equilibrateand become indistinguishable, in a coarse grained way, from constituent D − branes that were in theblack hole before the K block approached it. There is probably a theorem to be proven here in the N → ∞ limit, since in that limit, the constituents of the K block can never escape from the blackhole once they have come within a distance of order R S of it.We claim that this is evidence for the absence of a firewall in Matrix Theory. Particle physics overtime scales smaller than the time to traverse a transverse distance R S is affected only perturbativelyby the question of whether it takes place inside or very far from the Schwarzschild radius of a blackhole. Note also that nowhere in the Matrix Theory model of a black hole does there exist any analogof the high energy particles that are supposed to constitute the firewall. Black hole constituents inMatrix Theory have the kinematic properties of typical thermal Hawking particles. It is probablysignificant that it’s a general property of physics in a light front frame, that the particle theoryvacuum is trivial. This means that the vacuum entanglement that is claimed to be a crucial featureof Hawking radiation by AMPS cannot be a feature of physics formulated on the light front, as MatrixTheory is.The reason that this demonstration of the absence of firewalls adds to the credibility of our HSTargument is that there is much more evidence that Matrix Theory is a systematic approximationto a super-Poincare invariant S-matrix theory of particles, with a low energy effective field theoryexpansion. Furthermore, there is an established construction of states with the properties of blackholes, in a system with a time independent Hamiltonian. The evidence that HST leads to super-Poincare invariant scattering was presented in [12], and is much less extensive.
Both of our models of quantum gravity contain “low energy” DOF, which are not captured byQUEFT, and are crucial to the description of the local dynamics of black holes. In neither of them isthere any apparent hint of the picture of field theory with a stretched horizon cutoff, which pervadesmuch of the literature on the black hole information problem, including the paper of AMPS.It is not our intention here to claim that the stretched horizon picture is completely wrong oruseless. Rather, our position is that the question of whether an in-falling observer encounters largedeviations from flat space physics on a time scale much shorter than the classical in-fall time tothe singularity, involves only a tiny fraction of the DOF of the black hole. QUEFT with a stretchedhorizon cutoff is certainly a grossly thermodynamic description of the system, and is simply insensitiveto these thermodynamically negligible DOF. To make the AMPS argument one must assume thatthe stretched horizon QUEFT description is an accurate accounting of the dynamics at the level ofsingle bits.
Acknowledgments
T.B. would like to acknowledge conversations with J. Polchinski, L.Susskind and D.Harlow aboutfirewalls. He would also like to thank the organizers of the firewall workshop at CERN in March 2013,for the invitation to speak at that conference, and the other participants for stimulating discussions.The work of T.B. was supported in part by the Department of Energy. The work of W.F. wassupported in part by the TCC and by the NSF under Grant PHY-096902010 eferences [1] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?,1207.3123; A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, “An Apologia forFirewalls,” arXiv:1304.6483 [hep-th].[2] D. Harlow and P. Hayden, “Quantum Computation vs. Firewalls,” arXiv:1301.4504 [hep-th].[3] Y. Nomura, J. Varela and S. J. Weinberg, “Low Energy Description of Quantum Gravity andComplementarity,” arXiv:1304.0448 [hep-th].[4] E. Verlinde and H. Verlinde, “Black Hole Entanglement and Quantum Error Correction,”arXiv:1211.6913 [hep-th].[5] T. Jacobson, “Thermodynamics of space-time: The Einstein equation of state,” Phys. Rev. Lett. , 1260 (1995) [gr-qc/9504004].[6] T. Banks and W. Fischler, “Holographic Space-Time Does Not Predict Firewalls,”arXiv:1208.4757 [hep-th].[7] T. Banks and J. Kehayias, “Fuzzy Geometry via the Spinor Bundle, with Applications to Holo-graphic Space-time and Matrix Theory,” Phys. Rev. D , 086008 (2011) [arXiv:1106.1179[hep-th]].[8] T. Banks, W. Fischler, I. R. Klebanov and L. Susskind, “Schwarzschild black holes from matrixtheory,” Phys. Rev. Lett. , 226 (1998) [hep-th/9709091]. ; T. Banks, W. Fischler, I. R. Kle-banov and L. Susskind, “Schwarzschild black holes in matrix theory. 2.,” JHEP , 008 (1998)[hep-th/9711005]. ; T. Banks, W. Fischler and I. R. Klebanov, “Evaporation of Schwarzschildblack holes in matrix theory,” Phys. Lett. B , 54 (1998) [hep-th/9712236].[9] B. de Wit, J. Hoppe and H. Nicolai, “On the Quantum Mechanics of Supermembranes,” Nucl.Phys. B , 545 (1988).[10] D. N. Page, “Information in black hole radiation,” Phys. Rev. Lett. 71, 3743 (1993) [hep-th/9306083].[11] T. Banks, W. Fischler, S. H. Shenker and L. Susskind, “M theory as a matrix model: A Con-jecture,” Phys. Rev. D , 5112 (1997) [hep-th/9610043].[12] T. Banks, W. Fischler, Holographic Space-time, the Unruh Effect and the S-matrix , Journal ofEmergent Physics to Emerge .[13] M. Dine, R. Echols and J. P. Gray, “Tree level supergravity and the matrix model,” Nucl.Phys. B , 225 (2000) [hep-th/9810021]. ; M. Dine, R. Echols and J. P. Gray, “Renormal-ization of higher derivative operators in the matrix model,” Phys. Lett. B , 103 (1998)[hep-th/9805007]. ; K. Becker and M. Becker, “On graviton scattering amplitudes in M theory,”Phys. Rev. D , 6464 (1998) [hep-th/9712238]. ; J. A. Harvey, “Spin dependence of D0-braneinteractions,” Nucl. Phys. Proc. Suppl.68