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.
14
Islands in the Stream
‘By discovering the AdS/CFT correspondence, Maldacena definitively answered the question of whether information can escape from a black hole. It can. However … we also need to understand what is wrong with the Hawking calculation.’
Geoffrey Penington45
What is it that makes a quantum theory on the boundary spacetime able to encode phenomena in the interior? How does holography work? Remarkably, and as we shall see in this chapter, it is as if the interior space is fabricated by quantum entanglement on the boundary. In other words, current research appears to be stumbling across the idea that space is not fundamental but rather something that emerges out of quantum theory: the quantum gravity puzzle may end up being resolved in favour of quantum mechanics with gravity emerging out of that.