Black holes without firewalls
The postulates of black hole complementarity do not imply a firewall for infalling observers at a black hole horizon. The dynamics of the stretched horizon, that scrambles and re-emits information, determines whether infalling observers experience anything out of the ordinary when entering a large black hole. In particular, there is no firewall if the stretched horizon degrees of freedom retain information for a time of order the black hole scrambling time.
I Introduction: complementarity or firewall?
Black hole complementarity was introduced in Susskind et al. (1993) in terms of three postulates for black hole evolution:
The process of formation and evaporation of a black hole, as viewed by a distant observer, can be described entirely within the context of standard quantum theory. In particular, there exists a unitary S-matrix which describes the evolution from infalling matter to outgoing Hawking-like radiation.
Outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations.
To a distant observer, a black hole appears to be a quantum system with discrete energy levels. The dimension of the subspace of states describing a black hole of mass is the exponential of the Bekenstein entropy .
These postulates refer to observations made outside the black hole and provide a basis for a phenomenological description that is consistent with unitarity. A further key assumption, based on the equivalence principle, was made in Susskind et al. (1993) and can be expressed as a fourth postulate that applies to observers who enter the black hole:
An observer in free fall experiences nothing out of the ordinary upon crossing the horizon of a large black hole.
The combination of this assumption and the three original postulates requires one to give up the notion of spacetime locality. In particular, the fate of observers entering a large black hole is very different depending on the frame of reference: In their own rest frame they pass unharmed through the horizon and only come to harm as they approach the curvature singularity, while from the viewpoint of distant observers they never pass through the horizon at all but are instead absorbed into the stretched horizon and thermalized before being re-emitted along with the rest of the black hole in the form of Hawking radiation. It was argued in Susskind and Thorlacius (1994) that no low-energy observer can detect violations of known laws of physics even if information carried by infalling matter appears to be duplicated in the outgoing Hawking radiation.
The stretched horizon is a surface outside the global black hole horizon that remains timelike. Outside observers ascribe non-trivial microphysical dynamics to the stretched horizon that serves to absorb, thermalize, and eventually re-emit the information contained in infalling matter. The usual thermodynamics of black holes is assumed to arise from a coarse graining of this (unspecified) microscopic dynamics. From the point of view of outside observers, no information ever enters the black hole in this description and the stretched horizon is the end of the road for all infalling matter. In that sense it is indeed a firewall. According to the fourth postulate the story is very different for an infalling observer. The spacetime curvature is weak at the horizon of a large black hole and an infalling observer should not notice anything out of the ordinary upon crossing the horizon. In a recent paper Almheiri et al. Almheiri et al. (2013) claim, however, that the first two postulates imply that an infalling observer must also see a firewall 111Closely related earlier work appeared in Braunstein et al. (2013).. In other words, that the fourth postulate is inconsistent with the others.
The microscopic stretched horizon in Susskind et al. (1993) was placed at a proper distance of order the Planck length away from the global horizon. More generally in the present work, we require the stretched horizon be placed at some large fixed redshift from asymptotic infinity Thorne et al. (1986). According to the second postulate, physics outside the stretched horizon is described by semi-classical field equations of some low-energy local effective field theory. The definition of a low-energy theory includes specifying a cut-off and in the black hole context this means that the spatial slices, on which the effective theory is defined, terminate at an effective stretched horizon located outside the microscopic stretched horizon. For concreteness, let us consider a quasi-static spherically symmetric black hole as shown in figure 1. The effective stretched horizon can then be taken as the surface , where fiducial observers (who remain at rest with respect to the black hole) would measure a local temperature equal to a cut-off scale. Equivalently, is the surface such that a radially outgoing massless particle is red-shifted from the cut-off energy at to the characteristic energy of Hawking radiation at infinity. In this approach, anything that is inside the effective stretched horizon is represented by degrees of freedom living on the effective stretched horizon. This includes the entire black hole region and the region between and the global horizon, as indicated in figure 1. If the cut-off energy is taken very high, close to the Planck energy, then approaches the microscopic stretched horizon of Susskind et al. (1993). For lower values of the cut-off, the dynamics on the effective stretched horizon is in principle obtained from the dynamics on the underlying microscopic horizon by renormalization.
The argument of Almheiri et al. (2013) proceeds as follows: At very late times we have by supposition a pure state consisting only of outgoing Hawking radiation. Since we know the laws of physics up to the stretched horizon we can evolve this state back mode by mode from late times to the stretched horizon. The authors of Almheiri et al. (2013) then seem to introduce the hidden assumption that even beyond the surface we can still evolve the mode back all the way to the global horizon, ignoring the stretched horizon degrees of freedom. At that point the argument can be reduced to that of a single mode, since by locality such a mode very close to the global horizon does not have time to entangle with the stretched horizon, and thus cannot entangle with other outgoing Hawking modes emitted at later times. Thus, once one has evolved this single late-time mode back to a point very close to the global horizon, unitarity and local quantum field theory prevent any entanglement between the outgoing mode and the black hole state. This is sufficient to guarantee that an infalling observer will see high energy modes. In fact, the states obtained in this manner are finite excitations of the so-called Boulware vacuum, which is well-known to have a divergent stress energy tensor on the horizon Candelas (1980). Vacuum states regular on the horizon include the Hartle-Hawking vacuum or Unruh vacuum Candelas (1980), which requires entanglement between outgoing Hawking modes and negative energy modes inside the black hole.
The firewall argument of Almheiri et al. (2013) is flawed because the stretched horizon has been dispensed with. In the effective field theory description of Postulate 2, Hawking radiation is emitted from the stretched horizon, which is a boundary of the spacetime. The fact that at late times the state of the stretched horizon is maximally entangled with the early Hawking radiation is no more of a problem than the corresponding statement about the remaining embers of a burning lump of coal that started out in a pure state. The firewall problem only arises if one attempts to extend the semiclassical description to the region inside the stretched horizon. If there is no entanglement between the outgoing modes and the state of the black hole, as is argued in Almheiri et al. (2013), then the state of the field is given by an excitation of the Boulware vacuum, which has a stress energy tensor that is power-law divergent as a function of proper distance as the global horizon is approached Candelas (1980). In that case, infalling observers encounter drama already outside the stretched horizon, in violation of black hole complementarity. We will give a counter-example below, where the semiclassical description outside the stretched horizon is compatible with unitarity and locality and the expectation value of the stress energy tensor remains finite in that region. This is achieved by making a different assumption about the semiclassical state of the system. Our semiclassical construction involves a firewall but only inside the stretched horizon, where the effective field theory of Postulate 2 no longer applies. The presence of a firewall, both in our example and in , is at odds with Postulate 4 but this is hardly surprising. Black hole complementarity was after all put forward to address problems arising from applying semiclassical theory in a region extending inside the stretched horizon.
Observations made by infalling observers are only well described in the local effective theory of Postulate 2 as long as they remain outside the surface in figure 1 and not after they pass through it. An alternative description should be possible, involving an effective quantum field theory on time slices where an infalling observer has low energy in the local frame of the slice Lowe et al. (1995); Lowe (1995). However, to describe an infalling observer crossing the global horizon of the black hole requires time slices that extend past the location of the stretched horizon in the original effective field theory, making it difficult to map states and observables from one low-energy theory to the other. Moreover, it has been argued that, due to the large relative boosts involved, the effective low-energy description on time slices, such that both an infalling observer who has entered the black hole and the outgoing Hawking radiation are at low energy, cannot be a local field theory Lowe et al. (1995); Lowe (1995); Lowe and Thorlacius (1999); Giddings and Lippert (2004).
With some new assumptions about the stretched horizon dynamics, and taking care with the application of the semiclassical approach, we will argue in the following section that information may be recovered without introducing a firewall for infalling observers. Various alternatives or modifications of the firewall scenario have appeared in Nomura et al. (2012); Mathur and Turton (2012); Chowdhury and Puhm (2012); Avery et al. (2012); Banks and Fischler (2012); Giddings et al. (2012); Hwang et al. (2012); Hossenfelder (2012) but for the most part these are also alternatives to black hole complementarity. The argument in the present paper, on the other hand, is consistent with black hole complementarity, as formulated in Susskind et al. (1993), with additional assumptions about the dynamics of the stretched horizon.
Ii Emergence of information
Let us begin by revisiting the setup of Hayden and Preskill Hayden and Preskill (2007), making explicit some of the relevant timescales. Alice throws her diary into an old black hole, where more than half the entropy has been emitted in Hawking radiation, and Bob measures the outgoing Hawking quanta, having first faithfully recorded all the quanta previously emitted by the black hole. That Bob can manipulate the state of the Hawking radiation in a relatively short time can be argued as follows. On average, a black hole of mass emits a Hawking particle every units of time, with an average energy of Hence the total number of emitted quanta during the lifetime of a black hole will be , and the total lifetime will be This timescale can be thought of as ’boxes’ of length , and distributing the emission times of the Hawking particles into these ’boxes’ gives Bob access to roughly
different states; more than enough to differentiate between the microstates making up the black hole.
In this time an average of Hawking particles will be emitted, leading to a possible problem: since these Hawking particles can be entangled with the diary before the black hole scrambles, there may be a modification of the high frequency quanta, and hence an infalling observer may see a firewall. On the other hand, as argued in Hayden and Preskill (2007), if the entanglement only appears after one finds compatibility with the postulates of black hole complementarity.
One can use information theory arguments to place a lower bound on the time scale of information retrieval. This is computed in Hayden and Preskill (2007) in eqn (1). They find the probability for failure to decode a -bit message in the diary satisfies
where is the number of bits that Bob reads after the diary is thrown in. Now this seems to imply the information comes out faster than the scrambling time if , from which one might infer a firewall. There is, however, an error in this train of logic, since the authors of Hayden and Preskill (2007) assume scrambling has already happened when they make the estimate in (1). Prior to scrambling the emission rate of quantum information is not governed by (1) but rather depends on details of the stretched horizon dynamics.222If, for example, the message was unentangled with the bits received by Bob, he would still have a probability of success of of decoding the message by measuring the first bits. However this does not improve as additional bits are measured, so does not correspond to a useful measurement of information in the diary.
Let us try to model these effects in more detail to determine their implications for the stretched horizon theory. Consider an old black hole prior to the diary being thrown in. It is fully entangled with the train of Hawking radiation that has already been emitted, and its state can be written as
with indexing the basis states of the black hole, and being the corresponding string of Hawking radiation recorded by Bob. Tracing over the Hawking radiation, the black hole density matrix is diagonal, with uniform entries due to the maximal entanglement,
Into this state Alice throws her diary, which we initially take to consist of bits in a pure state. Without loss of generality, we can take the state to be Immediately after the diary is inside the black hole, but not yet scrambled, the state and the black hole density matrix are given by
where the density matrix is a matrix, written in term of blocks above. The vanishing blocks of correspond to states involving , over which Alice’s diary does not have support yet. This is to be compared with the maximally entangled matrix
In modeling the stretched horizon dynamics, we assume that there is one degree of freedom per unit Planck area, consistent with the third postulate. The transverse wavelength of modes emitted from the stretched horizon then ranges from of order , for the low-angular momentum modes that make up the bulk of the Hawking radiation that reaches distant observers, to of order one in Planck units, for high-angular momentum modes that never emerge far from the black hole and are re-absorbed by the stretched horizon.
A simple model for a long transverse wavelength, Hawking particle emitted from the stretched horizon is an operator close to the identity operator in the dimensional Hilbert space, acting on all the stretched horizon states with approximately equal weight
so these operators do a good job of making the emitted radiation look thermal, regardless of the stretched horizon state.
On the other hand, a model for the emission of a short transverse wavelength mode would be to pick an operator such as
In this case, immediately after the diary was thrown in, we would find
so we see the answer is highly sensitive to the state of the stretched horizon. The danger is that operators of the form will lead to strong modification of the high-frequency near-horizon Hawking modes, giving rise to a firewall. For this to happen, the short transverse wavelength modes coupling to such operators would have to themselves scramble and become entangled with the early Hawking radiation on a timescale that is short compared to the black hole scrambling time . If, however, the stretched horizon dynamics is causal, then short transverse wavelength modes are unable to scramble (i.e. achieve approximate global thermalization with respect to the measure described in Hayden and Preskill (2007)) in a time . The fastest a localized signal can causally traverse the stretched horizon is in time using time measured at the stretched horizon. This then redshifts to when measured using Schwarzschild time. We therefore introduce another new assumption about the dynamics of the stretched horizon theory – that it be local and causal. Without this assumption the stretched horizon dynamics can in principle contaminate the causal physics outside, violating Postulate 2.
The bounds on information retrieval time placed in Hayden and Preskill (2007) are lower bounds. The diary will scramble most efficiently if it is coded into modes with transverse wavelength of order , as exemplified by the operator (4). Due to the long transverse wavelength, such modes couple globally to the stretched horizon degrees of freedom, and there is no causal bound preventing an scrambling time. If, on the other hand, the information in the diary is present in short transverse wavelength modes, such as (5) then it may be emitted more slowly, on a timescale of order or longer.
It is, however, necessary to assume there is a genuine information retention time during which no “prompt” information is emitted from the stretched horizon while these long transverse wavelength modes scramble. This distinguishes the dynamics of the stretched horizon from, for instance, an accelerating mirror which would indeed look like a firewall from the point of view of a freely falling observer. The dynamics of the stretched horizon must be such that the reflection coefficient vanishes, the information is retained for a time , at which point it is then primarily emitted in long transverse wavelength modes.
It is important in the argument of Hayden and Preskill (2007) that the diary be much smaller than the black hole. This can also be seen from the following estimate of the maximum number of degrees of freedom that can scramble fast enough. We ask that a causal signal from a cell of size on the stretched horizon overlaps with a neighboring cell after and assume that this is sufficient for scrambling, with respect to the measure of Hayden and Preskill (2007), to take place. This is rather strong assumption about the efficiency of the scrambling dynamics so the resulting number of fast scramblers is likely to be an overestimate. The above condition implies , in which case the number of independent fast scrambling degrees of freedom is of order . Sending in a larger diary than this will compromise the rapid rate of information retrieval, as more generic short transverse wavelength modes scramble on a slower timescale of order .
There is a finite time delay during which information scrambles on the stretched horizon after the infalling diary is absorbed. An early infalling observer (see figure 2) sees no substantial difference from the Unruh or Hartle-Hawking vacua in this time interval as they cross the global horizon. However an infalling observer crossing the outgoing mode after this time interval sees a mode that has had time to spread a distance at least of order from the stretched horizon. This mode is now entangled with the diary.333Note that entanglement depends on the time-slicing in this context, and is a quantity not accessible to local probes. Nevertheless for a given time-slicing the state undergoes unitary time evolution and all external observers agree on the state.
Next let us consider the argument of Almheiri et al. (2013) which essentially replaces the infalling diary by a set of vacuum Hawking modes. The difference is illustrated in figure 3. An observer outside the stretched horizon must see entanglement of the outgoing mode with the early Hawking radiation according to Hayden and Preskill (2007). At the same time, an observer crossing the global horizon sees the mode entangled with interior modes, not with the early Hawking radiation. Because the timescale separating infaller 1 and infaller 2 can be much shorter than the scrambling time, this creates an apparent paradox. In the work of Almheiri et al. (2013) it is argued that the radiation on the outside must be some purely outgoing mode at infinity. As noted above, this can be viewed as a finite excitation of the Boulware vacuum. The Boulware vacuum has a continuous divergence outside the global horizon. A freely falling observer will see temperatures of order the ultraviolet cutoff scale as they cross the stretched horizon. This then leads to a violation of the postulates of black hole complementarity, since the physics outside but close to the stretched horizon is no longer described by a conventional theory.
However it suffices to show the firewall need only ever appear behind the stretched horizon to exhibit the flaw in the reasoning of Almheiri et al. (2013). To do this it is helpful to work with the Hilbert space separated as shown in figure 4. The Hartle-Hawking vacuum involves an entanglement of the modes on each side of the horizon Unruh (1976). However the left modes never propagate into the external region on the right. Both sets of modes propagate into the interior of the black hole, and their entanglement is essential for the absence of drama for an infalling observer.
In this picture, the exterior modes are a superposition of infalling and outgoing modes. Consistency with figure 3 then demands that modes representing Hawking particles emitted after a time of order the Page time be maximally entangled with the earlier radiation. From the exterior viewpoint, there is no contradiction with unitarity and locality. The problem arises when one considers an infalling observer. Following the argument of Almheiri et al. (2013) the exterior mode cannot be simultaneously entangled with the early Hawking radiation and the interior mode.
We can model a state where the exterior modes have no entanglement with the interior modes by simply placing the left interior modes in their vacuum state. The argument is cleanest if the exterior modes are placed in a thermal density matrix, rather than a pure state. Unlike the Boulware vacuum, this does not change the expectation value of the stress energy tensor in the exterior region 444One may also place the exterior modes in a pure state where the expectation value of the stress energy tensor remains very close to that of the Hartle-Hawking vacuum. Note such a state involves an infinite number of infalling and outgoing exterior modes, so the state may be picked so fluctuations in such a local observable are very small.. It does however produce an infinite firewall on the global horizon. However this is a crucial difference, because now we have a counterexample where the firewall only need appear behind the stretched horizon, and the expectation value of the stress energy tensor need not depart by a substantial amount from that obtained in the Hartle-Hawking vacuum, even if the stretched horizon is Planck scale. Therefore we can conclude that outside a Planck distance from the global horizon there is no sign that the postulates of black hole complementarity break down. At or inside the global horizon, all bets are off for a conventional description of the quantum theory, as emphasized in Lowe et al. (1995); Lowe (1995); Lowe and Thorlacius (1999).
It is also interesting to estimate whether the outgoing Hawking radiation leads to an observable deviation in a local quantity outside the stretched horizon, such as the expectation value of the stress energy tensor. Along the path of the early infalling observer in figure 2 the result will match that of Candelas (1980), who found, for example, a contribution to the trace of the stress energy tensor near the horizon due to Hawking radiation in the Hartle-Hawking vacuum. The later infalling observer will see the same phenomena, with small differences due to the interference with the earlier outgoing Hawking radiation. Since the outgoing radiation at this point is maximally entangled, any fluctuations away from the thermal expectation value for the stress energy tensor are expected to be down by an extra factor of or more.
Iii Entropy subadditivity bounds
Finally, we note that one of the arguments for the firewall of Almheiri et al. (2013) is based on entropy subadditivity bounds Araki and Lieb (1970); Lieb and Ruskai (1973). They divide the system into , the early Hawking modes, a late outgoing Hawking mode, and the interior partner mode of . They claim the entropy subadditivity bound
is violated. Let us analyze this bound, first in the stretched horizon theory of Postulate 2, and then in a model with timeslices that extend inside the stretched horizon but terminate on the global horizon.
In the effective field theory of Postulate 2, it is incorrect to view Hawking radiation as the formation of a maximally entangled pair and outside the stretched horizon, with the negative energy mode subsequently absorbed by the stretched horizon. Introducing degrees of freedom outside the stretched horizon, and insisting that they are maximally entangled with the outgoing modes, indeed leads to violation of entropy subadditivity as we will see momentarily. It amounts to cloning of quantum information, and by assumption the effective field theory of Postulate 2 is a local quantum field theory where such cloning cannot occur. Rather in the effective field theory of Postulate 2 only describes the and the modes. In this theory the stretched horizon is a boundary of spacetime. It is a hot surface from which the Hawking radiation is emitted. At late times the state of the stretched horizon of the remaining black hole is maximally entangled with , the early Hawking modes. When a late Hawking mode is emitted, the size of the stretched horizon Hilbert space gets reduced accordingly, and both and the new stretched horizon state are separately maximally entangled with . There is, however, no entanglement between and the new stretched horizon state. This is entirely in line with what happens at late times for burning lump of coal that starts out in a pure state. Entropy subadditivity reduces to the statement Araki and Lieb (1970)
which is close to saturated at late times .
We now turn our attention to a description where timeslices extend inside the stretched horizon and modes are included. If we take that description to be a conventional local quantum field theory then we will run into problems with entropy subadditivity as pointed out in Almheiri et al. (2013) and these problems can indeed be avoided by introducing a firewall for infalling observers. It is important to note, however, that in this case we are no longer considering the effective field theory of Postulate 2 but have made the further assumption that local effective field theory can be extended to the region inside the stretched horizon.
For the sake of argument, let us instead consider a model where modes are included and the physics inside the black hole region is described by some quantum dynamics on the global horizon (rather than on the stretched horizon as in Postulate 2). A pair appears in a pure state due to a quantum fluctuation and we assume that the mode then scrambles with the state on the global horizon in a time of order . There are two limits where the entropy subadditivity bound can be easily analyzed, before scrambling has had a chance to occur, and after the scrambling time. Prior to scrambling, remains in a pure state independent of , so , and . Substituting into (6) yields
At first sight this seems similar to the analysis of Almheiri et al. (2013). However before is scrambled, we expect the entropy of the outgoing radiation to increase
rather than decrease as stated in Almheiri et al. (2013) because one simply has one additional thermal Hawking particle. We conclude because and are independent prior to scrambling.
It is helpful to break the interaction of with the black hole, described by some Hilbert subspace , into two steps. First interacts with the black hole, reducing the number of degrees of freedom there. This process requires working in some infinite dimensional Fock space, as is usual in second quantized field theory. However immediately after this interaction, there will be a dimension subspace of the global horizon Hilbert space that is entangled both with and with .
After scrambling things become simpler. Scrambling mixes with all the other horizon degrees of freedom. will become maximally entangled with , so and the entropy of the external radiation decreases, . After scrambling it is no longer true that . Rather if the dimension of is much larger than we expect and to become independent, with . Likewise the system will be close to maximally entangled with after scrambling, so . Substituting into (6) yields
which is satisfied.
It should be noted that the reduced density matrix
is independent of unitary transformations that act within the subspace. Therefore the only way to accomplish such a change of entanglement described above is via a non-unitary transformation. However if we follow the picture described in the previous section, such a non-unitary horizon theory is only needed on the global horizon rather than the stretched horizon, and so remains consistent with the postulates of black hole complementarity.
We find no violation of entropy subadditivity implied by the postulates of black hole complementarity. The correct description of the stretched horizon theory does not allow for a description of interior modes that is independent of the outgoing modes. By assuming that pairs form outside the stretched horizon, Almheiri et al. (2013) unnecessarily clone information and this leads to the claim that simultaneously (as in our global horizon model prior to scrambling) and (as in our model only after scrambling).
The main point of this paper is that a firewall for infalling observers is not an unavoidable consequence of Postulates 1 and 2 as is claimed by Almheiri et al. (2013). We do not claim that such a firewall is impossible. In fact, we have considered several examples where firewalls do occur. In each case, including that of Almheiri et al. (2013), the firewall follows from making assumptions about physics in the region inside the stretched horizon that do not follow from Postulate 2.
It is interesting to further consider the history of the infalling observers in figure 2. As a model for the dynamics inside the horizon, let us imagine we use the simple non-unitary model described in the previous section. The early infalling observer, smoothly passes through the stretched horizon according to Postulate 4. However once an interval of order the scrambling time passes, the entanglement between the and the modes will change, and this observer will no longer experience a vacuum state. By this time they will have passed a distance at least of order inside the stretched horizon. This, however, coincides with the location of the curvature singularity. This picture, where information cloning was prevented by a firewall located near the classical curvature singularity was advocated in Lowe and Thorlacius (2006), and supported by AdS/CFT computations in Lowe (2009). These works focus on resolving cross-horizon complementarity issues, rather than the outside-the-horizon issues that are the main focus of the present work. Scrambling therefore allows the early infalling information to be safely annihilated before the later infalling observer, who has access to the information from the exterior Hawking radiation, is able to enter the horizon and potentially see a contradiction.
Acknowledgements.We thank J. Polchinski and the authors Avery et al. (2012) for helpful comments. The research of K.L. and D.A.L. is supported in part by DOE grant DE-FG02-91ER40688-Task A and an FQXi grant. The research of L.T. is supported in part by grants from the Icelandic Research Fund and the University of Iceland Research Fund.
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