# Guest Post: Joe Polchinski on Black Holes, Complementarity, and Firewalls

If you happen to have been following developments in quantum gravity/string theory this year, you know that quite a bit of excitement sprang up over the summer, centered around the idea of “firewalls.” The idea is that an observer falling into a black hole, contrary to everything you would read in a general relativity textbook, really would notice something when they crossed the event horizon. In fact, they would notice that they are being incinerated by a blast of Hawking radiation: the firewall.

This claim is a daring one, which is currently very much up in the air within the community. It stems not from general relativity itself, or even quantum field theory in a curved spacetime, but from attempts to simultaneously satisfy the demands of quantum mechanics and the aspiration that black holes don’t destroy information. Given the controversial (and extremely important) nature of the debate, we’re thrilled to have Joe Polchinski provide a guest post that helps explain what’s going on. Joe has guest-blogged for us before, of course, and he was a co-author with Ahmed Almheiri, Donald Marolf, and James Sully on the paper that started the new controversy. The dust hasn’t yet settled, but this is an important issue that will hopefully teach us something new about quantum gravity.

**Introduction**

Thought experiments have played a large role in figuring out the laws of physics. Even for electromagnetism, where most of the laws were found experimentally, Maxwell needed a thought experiment to complete the equations. For the unification of quantum mechanics and gravity, where the phenomena take place in extreme regimes, they are even more crucial. Addressing this need, Stephen Hawking’s 1976 paper “Breakdown of Predictability in Gravitational Collapse” presented one of the great thought experiments in the history of physics.

The experiment that Hawking envisioned was to let a black hole form from ordinary matter and then evaporate into radiation via the process that he had discovered two years before. According to the usual laws of quantum mechanics, the state of a system at any time is described by a wavefunction. Hawking argued that after the evaporation there is not a definite wavefunction, but just a density matrix. Roughly speaking, this means that there are many possible wavefunctions, with some probability for each (this is also known as a mixed state). In addition to the usual uncertainty that comes with quantum mechanics, there is the additional uncertainty of not knowing what the wavefunction is: information has been lost. As Hawking put it, “Not only does God play dice, but he sometimes confuses us by throwing them where they can’t be seen.”

Density matrices are much used in statistical mechanics, where they represent our ignorance of the exact situation. Our system may be in contact with a thermal bath, and we do not keep track of the state of the bath. Even for an isolated system, we may only look at some macroscopic variables and not keep track of every atom. But in both cases the complete description is in terms of a definite wavefunction. Hawking was arguing that for the final state of the black hole, the most complete description was in terms of a density matrix.

Hawking had thrown down a gauntlet that was impossible to ignore, arguing for a fundamental change in the rules of quantum mechanics that allowed information loss. A common reaction was that he had just not been careful enough, and that as for ordinary thermal systems the apparent mixed nature of the final state came from not keeping track of everything, rather than a fundamental property. But a black hole is different from a lump of burning coal: it has a horizon beyond which information cannot escape, and many attempts to turn up a mistake in Hawking’s reasoning failed. If ordinary quantum mechanics is to be preserved, the information behind the horizon has to get out, but this is something tantamount to sending information faster than light.

I have always been in awe of Hawking’s paper. His argument stood up to years of challenge, and subtle analyses that only sharpened his conclusion. Eventually it came to be realized that quantum mechanics in its usual form could be preserved only if our understanding of spacetime and locality broke down in a big way. In fact, as I will describe further below, this is now widely believed. So Hawking may have been wrong about what had to give (and he conceded in 2004, perhaps prematurely), but he was right about the most important thing: his argument required a change in some fundamental principle of physics.

**Black hole complementarity**

To get a closer look at the argument for information loss, suppose that an experimenter outside the black hole takes an entangled pair of spins |+-> + |-+> and throws the first spin into the black hole. The equivalence principle tells us that nothing exceptional happens at the horizon, so the spin passes freely into the interior. But now the outside of the black hole is entangled with the inside, and by itself the outside is in a mixed state. The spin inside can’t escape, so when the black hole decays, the mixed state on the outside is all that is left. In fact, this process is happening all the time without the experimenter being involved: the Hawking evaporation is actually due to production of entangled pairs, with one of each pair escaping and one staying behind the horizon, so the outside state always ends up mixed.

A couple of outs might come to mind. Perhaps the dynamics at the horizon copies the spin as it falls in and sends the copy out with the later Hawking radiation. However, such copying is not consistent with the superposition principle of quantum mechanics; this is known as the no-cloning theorem. Or, perhaps the information inside escapes at the last instant of evaporation, when the remnant black hole is Planck-sized and we no longer have a classical geometry. Historically, this was the third of the main alternatives: (1) information loss, (2) information escaping with the Hawking radiation, and (3) remnants, with subvariations such as stable and long-lived remnants. The problem with remnants that these very small objects need an enormous number of internal states, as many as the original black hole, and this leads to its own problems.

In 1993, Lenny Susskind (hep-th/9306069, hep-th/9308100), working with Larus Thorlacius and John Uglum and building on ideas of Gerard ‘t Hooft and John Preskill, tried to make precise the kind of nonlocal behavior that would be needed in order to avoid information loss. Their principle of *black hole complementarity* requires that different observers see the same bit of information in different places. An observer outside the black hole will see it in the Hawking radiation, and an observer falling into the black hole will see it inside. This sounds like cloning but it is different: there is only one bit in the Hilbert space, but we can’s say where it is: locality is given up, not quantum mechanics. Another aspect of the complementarity argument is that the external observer sees the horizon as a hot membrane that can radiate information, while in infalling observer sees nothing there. In order for this to work, it must be that no observer can see the bit in both places, and various thought experiments seemed to support this.

At the time, this seemed like an intriguing proposal, but not (for most of us) convincingly superior to information loss, or remnants. But in 1997 Juan Maldacena discovered AdS/CFT duality, which constructs gravity in an particular kind of spacetime box, anti-de Sitter space, in terms of a dual quantum field theory.

(Hawking’s paradox is still present when the black hole is put in such a box). The dual description of a black hole is in terms of a hot plasma, supporting the intuition that a black hole should not be so different from any other thermal system. This dual system respects the rules of ordinary quantum mechanics, and does not seem to be consistent with remnants, so we get the information out with the Hawking radiation. This is consistent too with the argument that locality must be fundamentally lost: the dual picture is *holographic*, formulated in terms of field theory degrees of freedom that are projected on the boundary of the space rather than living inside it. Indeed, the miracle here is that gravitational physics looks local at all, not that this sometimes fails.

**A new paradox?**

AdS/CFT duality was discovered largely from trying to solve the information paradox. After Andy Strominger and Cumrun Vafa showed that the Bekenstein-Hawking entropy of black branes could be understood statistically in terms of D-branes, people began to ask what happens to the information in the two descriptions, and this led to seeming coincidences that Maldacena crystallized as a duality. As for a real experiment, the measure of a thought experiment is whether it teaches us about new physics, and Hawking’s had succeeded in a major way.

For AdS/CFT, there are still some big questions: precisely how does the bulk spacetime emerge, and how do we extend the principle out of the AdS box, to cosmological spacetimes? Can we get more mileage here from the information paradox? On the one hand, we seem to know now that the information gets out, but we do not know the mechanism, the point at which Hawking’s original argument breaks down. But it seemed that we no longer had the kind of sharp alternatives that drove the information paradox. Black hole complementarity, though it did not provide a detailed explanation of how different observers see the same bit, seemed to avoid all paradoxes.

Earlier this year, with my students Ahmed Almheiri and Jamie Sully, we set out to sharpen the meaning of black hole complementarity, starting with some simple `bit models’ of black holes that had been developed by Samir Mathur and Steve Giddings. But we quickly found a problem. Susskind had nicely laid out a set of postulates, and we were finding that they could not all be true at once. The postulates are (a) Purity: the black hole information is carried out by the Hawking radiation, (b) Effective Field Theory (EFT): semiclassical gravity is valid outside the horizon, and (c) No Drama: an observer falling into the black hole sees no high energy particles at the horizon. EFT and No Drama are based on the fact that the spacetime curvature is small near and outside the horizon, so there is no way that strong quantum gravity effects should occur. Postulate (b) also has another implication, that the external observer interprets the information as being radiated from an effective membrane at (or microscopically close to) the horizon. This fits with earlier observations that the horizon has effective dynamical properties like viscosity and conductivity.

Purity has an interesting consequence, which was developed in a 1993 paper of Don Page and further in a 2007 paper of Patrick Hayden and Preskill. Consider the first two-thirds of the Hawking photons and then the last third. The early photons have vastly more states available. In a typical pure state, then, every possible state of the late photons will be paired with a different state of the early radiation. We say that any late Hawking photon is fully entangled with some subsystem of the early radiation.

However, No Drama requires that this same Hawking mode, when it is near the horizon, be fully entangled with a mode behind the horizon. This is a property of the vacuum in quantum field theory, that if we divide space into two halves (here at the horizon) there is strong entanglement between the two sides. We have used the EFT assumption implicitly in propagating the Hawking mode backwards from infinity, where we look for purity, to the horizon where we look for drama; this propagation backwards also blue-shifts the mode, so it has very high energy. So this is effectively illegal cloning, but unlike earlier thought experiments a single observer can see both bits, measuring the early radiation and then jumping in and seeing the copy behind the horizon.

After puzzling over this for a while we started to ask other people about it. The first one was Don Marolf, who remarkably had just come to the same conclusion by a somewhat different argument, mining the black hole by lowering a box near to the horizon and then pulling up some thermal excitations, rather than looking at the late Hawking photon. This is nicely complementary to our argument: it is a bit more involved, but it shows that if there is drama then it is everywhere on the horizon, whereas the Hawking radiation argument is only sensitive to photons in nearly spherically symmetric states. So if drama breaks down, it breaks down in a big way, with a firewall of Planck-energy photons just behind the horizon.

As we spoke to more and more people, no one could find a flaw in our reasoning. Eventually I emailed Susskind, expecting that he would quickly straighten us out. But his reaction, a common one, was first to tell us that there must be some trivial mistake in our reasoning, and a bit later to realize that he was as confused as we were. He is now a believer in the firewall, though we are still debating whether it forms at the Page time (half the black hole lifetime) or much faster, the so-called fast-scrambling time. The argument for the latter is that this is the time scale over which most black hole properties reach equilibrium. The argument for the former is that self-entanglement of the horizon should be the origin of the interior spacetime, and this runs out only at the Page time.

Actually, over the years many people have suggested that the black hole geometry ends at the horizon. Most of these arguments are based on questionable dynamics, with perhaps the most coherent proposal being Mathur’s fuzzball, the horizon being replaced by a shell of branes (though Samir himself is actually advocating a form of complementarity now).

If we want to avoid drama, we have to give up either purity or EFT. I am reluctant to give up purity: AdS/CFT is a guide that I trust, but even the earlier arguments for purity were strong. Giving up EFT is not so implausible. AdS/CFT tells us that locality, the basis for EFT, has to break down, and this need not stop at the horizon. Indeed, Giddings has recently been arguing for a nonlocal interaction that transfers bits from the inside of the black hole to a macroscopic distance outside. But it is hard to come up with a good scenario: the violation of EFT is much larger than might have been anticipated (it is an order one effect in the two-particle correlator). One might try to appeal to complementarity, since drama is measured by an infalling observer and purity by an asymptotic one, but these two can communicate. Also, the breakdown that is needed is subtle and difficult to implement, a `transfer of entanglement’ (this is particularly a problem for the nonlocal interaction idea). This transfer is reminiscent of an idea that Gary Horowitz and Maldacena put forward a while back, that there is future boundary condition, a final state, at the black hole singularity. Several authors have now proposed that some form of complementarity is operating, but it is telling that some of them have withdrawn their papers for rethinking, and there is no agreed picture among them.

Where is this going? So far, there is no argument that the firewall is visible outside the black hole, so perhaps no observational consequences there. For cosmology, one might try to extend this analysis to cosmological horizons, but there is no analogous information problem there, so it’s a guess. Do I believe in firewalls? My initial intuition was that EFT would break down and complementarity would save the day, but a nice scenario has not emerged, while the arguments for the firewall as arising from a loss of entanglement are seeming more plausible. But the main thing is that I am now as puzzled about the information paradox as I ever was in the past, and it seems like a good kind of puzzlement that may lead to new insights into quantum gravity.