Black Holes: The Key to Understanding the Universe

13

The World as a Hologram

‘Nobody has the slightest idea what is going on.’

Joseph Polchinski

The entropy of a black hole is proportional to its area, which suggests that all the information concerning the stuff that fell into the hole is encoded in tiny bits spread over the surface of the horizon. In time, those bits break free and end up as correlated Hawking particles. These correlations – quantum entanglement in the radiation – encode the information about the stuff that fell in.* From the perspective of someone freely falling through the horizon, they feel nothing and are oblivious to this magical encoding. Moreover, their fate is to be both spaghettified in the singularity (from their own perspective) and burnt up on the horizon (from an outsider’s perspective). But that is no problem for the laws of Nature because no observer can be present at both events. This is the essence of the black hole complementarity resolution to the black hole information paradox. Is it nonsense?

Today, the evidence is strongly in support of the conclusion that complementarity is not nonsense, but the implication is even more shocking. Complementarity is telling us that what happens inside the horizon is as valid a picture of physical reality as what happens outside. The two pictures are equivalent descriptions of the same physics. The inside of a black hole, in other words, is somehow ‘the same’ as the outside. This idea has become known as the holographic principle.

At first sight there might seem to be no need to be so radical as to invoke the holographic principle because we could imagine that, as something falls through the horizon, it is secretly copied. One copy continues to fall into the singularity to be spaghettified and the other copy gets burnt up on the horizon and encoded in the Hawking radiation. Radical as it may be, copying on the horizon does seem less radical than invoking the idea that the interior is a hologram. There is a serious flaw in this logic though – the laws of quantum physics forbid it. The ‘no cloning theorem’ says that it is not possible to make an identical copy of some unknown quantum state, and in Box 13.1 we sketch a proof.

BOX 13.1. No cloning

Suppose we have a cloning machine that can duplicate an unknown qubit. Specifically, this machine takes a |0⟩ and turns it into |0⟩ |0⟩ and likewise it turns a |1⟩ into a |1⟩ |1⟩. What would it do to the following qubit: |Q⟩ = 1/√2(|0⟩ + |1⟩)? This qubit is a 50–50 mix of |0⟩ and |1⟩ and our cloning machine would turn it into 1/√2(|0⟩|0⟩ + |1⟩|1⟩). But this two-qubit state is not |Q⟩ |Q⟩. In other words, our machine is not a cloning machine after all.

With cloning ruled out, it appears that we are left with holography if we want to respect the foundations of both quantum theory and general relativity. There is, however, another possibility: the black hole has no interior. This radical solution would mean general relativity is wrong because nothing can fall into a black hole, which is a gross violation of the Equivalence Principle. This outrage to Einstein was taken very seriously after a 2013 paper by Ahmed Almheiri, Donald Marolf, Joseph Polchinski and James Sully, provocatively titled ‘Black Holes: Complementarity or Firewalls?’41 The AMPS paper, as it has since become known, found what appeared to be a fatal flaw in the complementarity idea. This led to the proposal that a black hole has no inside, and that anyone unfortunate enough to reach the horizon of a black hole would be burnt up in a wall of fire, even from their own point of view.

Firewalls

In Chapter 12, we imagined loitering outside a black hole and collecting the evidence of an astronaut’s fate as they get burnt up on the horizon. We would then jump into the hole to confront the same astronaut with their own ashes. This would generate a contradiction, because the astronaut would have both burnt up and not burnt up from their own perspective. We explained that this contradiction is avoided because the astronaut will have reached the singularity before we are able to catch them. The more precise version of this scenario involves thinking of qubits and cloning. We can imagine throwing a bunch of qubits into the hole and then trying to determine those qubits by collecting the Hawking radiation and processing it so that we have effectively obtained a copy of the original qubits we threw in. To be consistent with the no cloning theorem, it should not be possible to do that and then jump into the hole and meet up with the original qubits. From our understanding of the Page curve, we know that if the black hole is younger than the Page time, not much information will have emerged and we would need to wait a long time (a silly understatement for young, solar mass black holes) before we could obtain a copy of the original qubits. There should be no contradiction, therefore, for a young black hole.

The situation after the Page time, when the black hole is middle-aged or older, is rather more subtle. In that case, the black hole acts more like a mirror and spits the bits back out again almost immediately. That discovery was made in 2007 by Patrick Hayden and John Preskill.42 Surprisingly, however, it turns out that the time delay is still (just) sufficient to prevent a violation of the no cloning theorem. All appears well in the complementarity camp, but AMPS seemed to throw a spanner into the works by coming up with a similar thought experiment. Their scenario, however, cannot be so easily reconciled with complementarity.

Figure 13.1. Illustrating the firewall. For an old black hole, the Hawking particles emitted when the hole was young, R, are entangled with the recently emitted Hawking particle B, which is also entangled with the interior particle A.

In the last chapter, we saw that if information is to be transferred into the Hawking radiation after the Page time, the Hawking particles must gradually become more and more entangled with each other. This is illustrated in the lower half of Figure 12.2. However, the Hawking radiation is produced as entangled pairs, as illustrated in the upper half of Figure 12.2. And here is the problem: because the Hawking pairs are entangled with each other they cannot be entangled with anything else. This is known as the ‘monogamy of entanglement’, and it is another fundamental property of quantum mechanics.

Figure 13.1 illustrates the problem this causes for a black hole older than the Page time. Imagine two observers, Alice and Bob. Bob, sitting outside of the black hole, collects the Hawking radiation. He processes R, the radiation emitted early in the life of the hole (before the Page time), and distils it into a single qubit.† Now, if information is to be conserved, this qubit is highly likely to be entangled with a Hawking particle B that is emitted late on in the life of the hole – this is why the Page curve goes to zero when the hole finally disappears. Bob therefore concludes that B and R are entangled.

Alice is a freely falling observer who crosses the horizon after the Page time. She will confirm that particle B is entangled with the other half of its Hawking pair, labelled A. To avoid violating the monogamy of entanglement, while still permitting the information to come out in the Hawking radiation,‡ we could suppose that Alice does not confirm A and B to be entangled. This might sound innocuous, but it is not. The consequences of simply removing entanglement like this would be very dramatic: it would create a wall of fire. That’s because it costs energy to destroy entanglement in the vacuum – it is tantamount to tearing open empty space. The resulting firewall wouldn’t merely prevent Alice from entering the interior of the black hole, it would effectively destroy space inside the horizon. The interior of the black hole would not exist.

One might wonder whether a complementarity-style argument might still save the day. Maybe Alice could observe that A is entangled with B and Bob could observe that B is entangled with R with no contradiction because they can never meet to confirm their observations. This is not the case though because there is plenty of time for Bob to confirm that he sees entanglement, and then to dive across the horizon to compare notes with Alice, who would have confirmed that she too sees entanglement across the horizon by the very act of crossing it.

With this chain of reasoning, AMPS appeared to have discovered a genuine contradiction which calls into question the existence of the interior of a black hole. This is possibly what provoked Joseph Polchinski to utter the words that open this chapter. The basic problem can be traced to the fact that complementarity appears to be asking for too much entanglement in order to conserve information as the black hole evaporates and to simultaneously preserve the integrity of the quantum vacuum across the horizon. Complementarity requires the black hole and the Hawking radiation to be in an impossible quantum state after the Page time, by demanding they encode more information than the system can physically support.

In the conclusions to their paper, AMPS mention another way of seeing this information storage limit in action. Why don’t similar arguments imply that firewalls should appear at the Rindler horizons we met in Chapter 3 for accelerated observers? The answer is that Rindler horizons have infinite area and entropy, unlike a black hole horizon, and therefore ‘their quantum memory never fills’. Rindler horizons, in other words, never evolve to become old and can always support any information demands made of them to preserve the integrity of the vacuum.

Holography offers a means to save the Equivalence Principle and render the horizon safe, while also saving quantum mechanics and preserving information. The basic idea is that since the interior of the black hole is dual to the exterior; the early Hawking radiation, R, and the interior particles, A, are really the same thing. Crazy as it sounds, this is the way the firewall problem is avoided in holography. In processing the early Hawking radiation, R, to check its entanglement with B, Bob inadvertently destroys the entanglement with A. This creates a kind of mini-firewall that is just violent enough to prevent Alice from measuring entanglement between A and B, but not so violent that it destroys the interior of the black hole.

The world as a hologram

Spacetime holography was first presented by Gerard ’t Hooft in 1993, and further developed a year later by Leonard Susskind. They presented it as an integral part of their black hole complementarity idea but stressed that it probably ought to be of more general applicability. That’s to say, holography should be a universal feature of Nature, regardless of the black hole issues that originally motivated it. The holographic principle as currently understood even goes so far as to suppose that the entire world as we perceive it is a hologram.43

A hologram as conventionally understood is a representation of a three-dimensional object constructed from information stored on a two-dimensional screen. If you’ve ever seen a hologram, you’ll know that they can look remarkably real. You can walk around them and view them from all angles as if they were the real three-dimensional objects themselves. Now imagine a perfect hologram. What sort of a thing would that be? A hologram would be perfect if every bit of information necessary to reconstruct the three-dimensional object was also encoded on the two-dimensional screen that carries the holographic data. This is reminiscent of the Bekenstein entropy of a black hole, which says that the information content of the black hole can be computed by considering only the two-dimensional surface of the event horizon.

Now, as we discussed in Chapter 9, a black hole has the largest possible information density of any object and, since the information it stores is given by the surface area of the horizon, it follows that there can be no more information inside any region of space than can be encoded on the boundary of that region. This realisation led ’t Hooft and Susskind to argue that the information content of any region of space is encoded on the boundary to that region. The reason we discovered this first by thinking about black holes is that the boundary is exposed for anyone hovering outside the black hole to explore, in the form of the hot membrane close to the event horizon. In everyday life, well away from any black holes, it is rather less obvious how we could gain access to this holographically encoded information since we cannot ‘cut out a piece of empty space’ to reveal that the information in the interior is encoded on its surface.

Holography, then, is a perfect example of complementarity in action. There are two entirely equivalent descriptions of anything and everything, and this is an essential feature of all of Nature and not just black holes. Black holes are the Rosetta Stone, which has introduced us to a new language; an entirely different yet perfectly equivalent description of physical reality. One description resides on the boundary of any given region of space, and the other resides more conventionally in the space internal to the boundary. The implication is that our experience and existence can be described with absolute fidelity in terms of information stored on a distant boundary, the nature of which we do not yet understand. This sounds utterly bonkers, but clinching evidence supporting the idea comes from the most highly cited high-energy physics paper of all time.

Maldacena’s world

Scientific citations count how many times a research paper has been referred to in the literature. Naturally enough, the most important papers tend to get the most citations. Ranked number 13 of all time§ is Stephen Hawking’s 1975 paper, ‘Particle Creation by Black Holes’. The discovery of dark energy is up there, with the two key papers reporting the evidence ranked 3 and 4, and the papers announcing the discovery of the Higgs boson at the Large Hadron Collider are ranked 6 and 7. Top-ranked of all time is a paper written in 1997 by Argentinian physicist, Juan Maldacena titled, ‘The Large N Limit of Superconformal Field Theories and Supergravity’.44 With almost 18,000 citations to date, it is the paper that has, more than any other, changed the face of theoretical physics over the past 25 years. It is also the paper that provides the strongest evidence supporting the idea that the holographic principle is true.

The universe Maldacena considered is not the one we live in, but that’s fine. It is common for physicists to build models of the world with some simplifying features. The real world is complicated, and it’s often useful to do calculations in a pretend world in which things are simpler. The skill is to pick a simple world which delivers enhanced understanding while not being too unrealistic. Engineers make simplifying assumptions when designing things like aircraft and bridges, notwithstanding the fact that the stakes are rather higher. Importantly, Maldacena’s world was not specifically chosen because it supports holography. Rather holography was a feature that popped out of the mathematics.

Figure 13.2. The Poincaré disk projection of a two-dimensional hyperbolic space. Despite appearances, the solid line from A to B on the left figure is shorter than the dashed one. You can see this by counting triangles. On the right is M. C. Escher’s Circle Limit I. All the fish are of the same size and shape, and the lines are shortest lines. The patterns provide us with a visual representation of the metric of the space (we can ascertain distances by counting fish, or triangles) just like the squares on graph paper illustrate the metric of Euclidean space. (Right: M.C. Escher’s Circle Limit I © 2022 The M.C. Escher Company-The Netherlands. All rights reserved. www.mcescher.com)

We can capture the essence of Maldacena’s work by considering a two-dimensional toy universe.¶ The space of the toy universe does not have the geometry of ordinary flat space; rather its geometry is hyperbolic. Figure 13.2 shows a beautiful representation of two-dimensional hyperbolic space, known as the Poincaré disk. This projection was widely employed by the Dutch artist M. C. Escher, and we also include his well-known Circle Limit I. Rather like a Penrose diagram, the space represented in these projections is infinite and there is a great deal of distortion to bring infinity to a finite place on the page. Escher’s fish, for example, are all the same size as they tile infinite hyperbolic space. They appear smaller towards the edge of the disk, which represents infinity, because we are shrinking down space as we head outwards from the centre. The Poincaré disk projection is also a conformal projection, which means that the shapes of small things are faithfully reproduced (for example, the fish-eyes are always circular).

Now let’s add time into the mix. Figure 13.3 is a stack of Poincaré disks, one for each time slice (though we have drawn only two). Time runs upwards from the bottom of the cylinder. This spacetime is known as ‘Anti-de Sitter spacetime’, or AdS for short. For what follows, it helps to think of the AdS cylinder as being akin to a tin-can, with a boundary and an interior. Maldacena triggered an avalanche of understanding by showing that a particular theory with no gravity, defined entirely on the boundary of the cylinder, is precisely equivalent to an entirely different theory, with gravity, defined in the interior spacetime. In other words, the interior is a holographic projection of the boundary. By writing down equations proving the exact one-to-one correspondence for this model universe, Maldacena provided the first concrete realisation of the holographic principle.

To appreciate the key ideas, we don’t need to know the details of what has become known as the AdS/CFT correspondence. The CFT acronym means ‘conformal field theory’, which refers to a class of quantum field theories that are similar to the ones used to build models of particle physics.** It refers to a quantum theory, complete with particles and entanglement and a vacuum state. The quantum theory describes a physical system located entirely on the boundary of the cylinder. If you’d like a picture, think of a gas of particles moving around.

When the quantum system on the boundary is in a pure vacuum state, which means there are no particles, the interior spacetime is just AdS. Now imagine creating particles on the boundary to make a gas. Astonishingly, a black hole appears in the interior spacetime. This is illustrated in Figure 13.3, where the formation and evaporation of the black hole in the interior has a dual description in terms of the gravity-free theory existing on the boundary of the cylinder. Gravity thus emerges as a result of the quantum mechanics of a system on the boundary.

We might ask which of these two descriptions is the real one. Is there really a black hole or is it just a hologram of the boundary physics? Or maybe the opposite is true, and the boundary physics is not real and is just a clever way to describe the black hole. Maybe trying to figure out what is ‘really’ true is to fall into a trap that has long plagued physicists because it leads to navel gazing without revealing deeper insight. There are plenty of people in the world who can perform that function, and too few physicists, so perhaps we should restrict ourselves to explaining natural phenomena and leave questions of ultimate truth to others. Rather, the holographic principle can be viewed as a realisation of complementarity. There are two equivalent descriptions of the world, and because they are equivalent there will be no contradictions: What is true in one will be true in the other. This is the power of the holographic principle, and Maldacena discovered a precise, mathematical realisation of it.

Figure 13.3. The Penrose diagram of Anti-de Sitter spacetime in the case of two space dimensions. The cylinder is infinitely long and the boundary is timelike. Some collapsing matter (at the bottom) makes a black hole that subsequently evaporates by emitting Hawking particles (at the top). The holography idea is that the formation and evaporation of the hole can be described using a gravity-free quantum theory defined on the boundary.

This technique of mapping a problem in quantum physics to an equivalent problem in gravity has proved to be very successful over the past 25 years. Many cases have been found where complicated problems on one side of the correspondence have been answered using methods from the other side. Viewed this way, Maldacena discovered a practical tool that we are learning to use to solve interesting problems in one area of physics using techniques from what superficially looks to be a totally different area. This is one reason why Maldacena’s paper has so many citations; it is very useful. It is also profound, and it answers the question of whether or not information is lost in black hole evaporation.

Figure 13.3 illustrates how the AdS/CFT correspondence demonstrates that information must come out of a black hole. Initially there was no black hole (the bottom part of the cylinder) – just a bunch of stuff collapsing under gravity. The black hole forms and then evaporates away leaving (at the top of the cylinder) a bunch of Hawking particles. Now focus on the dual description. This side of the correspondence says that this whole process can be described by a gas of particles evolving according to the ordinary rules of quantum mechanics on the boundary with no gravity. Because there is a precise one to one correspondence between the boundary theory and the interior theory, if information is conserved in one it must be conserved in the other. Crucially, the boundary theory is a pure quantum theory, which means that information is necessarily conserved. It must therefore be conserved by gravitational processes in the interior – in this case during the formation and evaporation of a black hole. This is what convinced Stephen Hawking to concede his bet with Kip Thorne and John Preskill and accept that information really does emerge from black holes in our Universe. He was persuaded by Maldacena’s AdS/CFT paper.

* This is decidedly not what happens in Hawking’s calculation, which suggests that the Hawking particles are uncorrelated and therefore carry no information, violating a fundamental principle of quantum mechanics. In Hawking’s calculation, the particles come out of a data-less vacuum (in the words of theoretical physicist Samir Mathur). As a result, the entanglement entropy of the Hawking radiation increases indefinitely and never turns around (as it must to accord with Don Page’s curve) because empty space is effectively providing an infinite reservoir of entanglement. It is radical to say that the information encoded on the horizon gets transferred to the outgoing Hawking radiation and incumbent on us to find the theory that explains how that comes about. Providing this explanation is what will constitute a full resolution to the information paradox. Complementarity does not answer the question directly – it supposes that some as-yet-unknown dynamics of the hot region near the horizon leads to large corrections to Hawking’s calculation.

† By some ingenious and complex process that we do not need to know about here.

‡ By which we mean Bob still detects the entanglement between B and R.

§ These citation statistics come from the iNSPIRE database (inspirehep.net) which is run by a collaboration of the world’s leading research laboratories and measures citations in the field of ‘high-energy physics’.

¶ Maldacena’s original calculation was in string theory and involved a ten-dimensional spacetime with five curled-up space dimensions, leaving a five-dimensional hyperbolic space with a four-dimensional boundary. Since 1997, there have been many other examples of the holographic principle involving fewer spacetime dimensions.

** The particular CFT that Maldacena originally considered is similar to QCD, the theory describing the strong interactions between quarks and gluons, and that similarity has been exploited with some success to make predictions in QCD using the dual gravity theory.