Figure 14.5. The entanglement entropy of the radiation R according to the island formula. Remarkably, it has the same shape as the Page curve (see Figure 12.3).
At this point, you may be highly sceptical at what appears to be an egregious sleight of hand. We seem to be reassigning particles inside the event horizon as belonging to the Hawking radiation in order to decrease the entanglement entropy. It is as if we are arbitrarily reassigning pieces of the jigsaw (from Chapter 12) from one table to another. That would be fair criticism if the island formula had been written down merely in order to reproduce the Page curve, but this is not the case. Instead, the island formula can be derived using the same basic physics that Hawking originally employed: quantum physics and general relativity. Hawking simply missed a subtle feature of the mathematics that, once included, triggers the appearance of the island.
One might consider the calculation of the Page curve to be a solution to the black hole information paradox. But we don’t intend to leave things at that. We want to know how the information gets out. Recent research suggests that the physics is closely related to both the RT conjecture and ER = EPR.
The meaning of the island
There is a striking similarity between the formula for the entanglement entropy of the Hawking radiation and the Ryu–Takayanagi formula. Is there an entanglement-geometry connection here? The answer seems to be yes. In fact, inspired by the ER = EPR conjecture, we might claim that the formula for SR is precisely the Ryu–Takayanagi result. This is illustrated in Figure 14.6, which also illustrates how it can be that the inside of an old black hole is really the outside.
Figure 14.6. Illustrating how the island idea works. It vindicates both the ER = EPR idea and the Ryu–Takayanagi conjecture. For an old black hole (bottom), most of the ‘inside’ of the black hole is outside. In both pictures, the outside is shaded orange.
The top figure shows the situation for a young black hole. It is an embedding diagram, corresponding to the time slice illustrated in the little Penrose diagram on the right.†† Notice how the blue and red Hawking partners have been linked by a tiny wormhole: this is the ER = EPR idea. The particles are joined by one of Susskind and Maldacena’s highly quantum wormholes. Having made these links, can you see that the distinction between what is outside and what is inside is unclear? The region outside is shaded orange and it seems natural to regard all of the flat region at large distances from the hole as being ‘outside’. But what about the space inside the wormholes? It seems we can travel from outside to inside along a wormhole. The key question is, where do we draw the line between inside and outside? The answer is given by Ryu–Takayanagi. We should seek the smallest area surface that splits the two regions. For a young black hole, that smallest area is (presumably‡‡) obtained by cutting through the wormholes. None of that is too weird. But for an old black hole there are many more wormholes, and the Ryu–Takayanagi surface is now the QES. This is illustrated by the curve marked B in the lower figure. The region between the QES and the curve marked A is inside, and the remainder, shaded orange, is outside. The island is precisely that part of the interior that should more correctly be regarded as outside.
This is the beginning of a physical picture of how a black hole may return information to the Universe as it evaporates. The picture of the singularity that emerges is more speculative still. As we have sketched the interior of the black hole in Figure 14.6, the singularity appears to be replaced by a quantum network of wormholes connecting to the outside. In Chapter 5, we sent a group of intrepid astronauts into the black hole and they all met their doom at the singularity. If we take this new picture of the black hole at face value, we can ask whether the end of time really does lie in the future of the astronauts. Imagine you are one of the astronauts. You fall across the horizon of the black hole without drama and confront … what? According to Figure 14.6, you will meet a network of wormholes, be dissociated by tidal gravity, scrambled up and the information that is you will emerge, through the wormholes, imprinted in the Hawking radiation.
* Actually, the bottom triangle is the interior of a white hole, as in Chapter 6.
† We are going to use the acronym CFT quite a bit in what follows. You can think of this simply as a gas of particles with no gravity acting.
‡ With acknowledgement to T. Hollowood, S. Prem Kumar, A. Legramandi and N. Talwar, of Swansea University, for their 2021 paper (J. High Energy Phys. 2021(11):67). And possibly also to Dolly Parton and Kenny Rogers.
§ A special thank you to Tim Hollowood for his insight and help, especially concerning this figure and also Figure 14.6.
¶ To appreciate that a point on our Penrose diagram is a spherical surface at a moment in time you need to remember that each point on a Penrose diagram represents not just one point in space at a moment in time but all points in space that have the same Schwarzschild R at a moment in time, which is a sphere.
** When one member of an entangled pair is inside some region and its partner is outside then there is a contribution to the entanglement entropy of the region. In contrast, if both are inside (or outside) the region then they contribute nothing to the entanglement entropy of the region.
†† The slice is drawn as a slightly wavy line, to indicate it is not unique. What is important is that it never curves upwards at more than 45 degrees. Otherwise, it would not correspond in any sense to ‘all of space’ at some notion of ‘now’.
‡‡ This interpretation is speculative since we do not understand these little wormholes. The formula for SR does not rely on this speculation.
15
The Perfect Code
‘… every item of the physical world has at bottom – at a very deep bottom, in most instances – an immaterial source and explanation; that what we call reality arises in the last analysis from the posing of yes-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and this is a participatory universe.’
John Archibald Wheeler50
‘… the whole show is wired up together …’
John Archibald Wheeler51
‘… time and space are not things, but orders of things …’
Gottfried Wilhelm Leibniz
Quantum entanglement is turning out to be a key player. We have thought about it so far to keep track of the information coming out of a black hole. In that context, we have seen that entanglement seems to be responsible for creating what we experience as space. What we will learn now is that the way entanglement creates space appears to be very robust. This is just as well for us: we don’t want to live in a space that might be prone to falling apart.
Quantum entanglement is also a key resource for those trying to build quantum computers. At first sight, the construction of computing machines might appear to have nothing at all to do with the emergence of space. In a quantum computer, entanglement is the primary means by which information is encoded in a robust way that is resilient to damaging environmental factors. This topic, known as quantum error correction, is fundamental to the construction of working quantum computers. There are parallels here: it is beginning to look as if space is woven out of quantum entanglement in a manner similar to the way quantum engineers weave qubits together to build quantum computers. The suggestion is that there is a link between quantum computing and the fabric of reality. In this chapter we are going to explore that link.