Black Holes: The Key to Understanding the Universe

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.

In Chapter 6 we met the maximally extended Schwarzschild spacetime, which can be interpreted as representing two universes connected by a wormhole. We noted that, sadly, large traversable wormholes of the sort beloved by science fiction writers do not reside inside real black holes because the interior contains matter from the collapsing star. We did, however, say that ‘microscopic wormholes could be part of the structure of spacetime’. It is now time for us to follow that thread.

Figure 14.1 shows the Penrose diagram of an eternal black hole that is very similar to the maximally extended Schwarzschild black hole we explored in Chapter 6 (see Figure 6.2). The difference is that this black hole is sitting in AdS spacetime. One might ask why we don’t focus on a universe more like ours rather than on an AdS universe. The answer is that we would if we could, but we don’t know how to yet, and Maldacena’s AdS/CFT correspondence is the most well understood model we have to hand. The majority of experts in the field at the time of writing believe that the underlying ideas should also be valid in our Universe.

The upper and lower triangles represent the interior of the black hole,* bounded by the event horizon and the singularities in the future and the past. The edges of the diagram, labelled L and R, are the boundaries of the AdS spacetime. Just as for the Schwarzschild case, the left and right triangular regions are the entirety of the spacetime outside the black hole, and they are linked by a wormhole. The AdS/CFT correspondence tells us that we can describe the interior of this Penrose diagram by two quantum field theories (CFTs†) located on the left and right boundaries. In the jargon, we say that the interior spacetime is the holographic dual of these two quantum theories.

Figure 14.1. A two-sided black hole.

Now here is the big deal: the two CFTs must be maximally entangled with each other to describe this spacetime. If the two CFTs were not entangled, there would be no wormhole. Instead, there would be two disconnected black holes in two separate universes. The wormhole appears when we allow the two quantum theories to become entangled. In other words, entanglement builds the wormhole connecting the two universes together: quantum entanglement and wormholes go hand in hand. This is a profoundly important connection between quantum theory and gravity.

To explore this connection further, we’ll introduce one of the central ideas to have emerged from holography: the Ryu–Takayanagi conjecture.46 Discovered by Shinsei Ryu and Tadashi Takayanagi in 2006, the RT conjecture has been demonstrated to be correct in a wide variety of different scenarios. It is important because it makes the connection between quantum entanglement and spacetime geometry calculable.

Figure 14.2. A snapshot in time illustrating the wormhole in Figure 14.1. Points on the boundaries, L and R, are circles and the interior is a two-dimensional surface. The horizon is the smallest curve that divides the wormhole in half. The length of the horizon fixes the amount of quantum entanglement between the two quantum theories living on L and R.

In Figure 14.2 we have drawn an embedding diagram representing a slice through the middle of the two-sided black hole of Figure 14.1. This is just like the wormhole diagrams in Chapter 6. The two CTFs live on the circles labelled L and R (these circles are points on the left and right vertical lines in Figure 14.1). RT says that the entanglement entropy between the quantum theory on L and the quantum theory on R is equal to the size of the smallest curve that divides the interior space into two. In other words, if there is no entanglement, the entanglement entropy is zero and there is no dividing curve and no wormhole. In that case, the two quantum theories are disconnected and there is no space linking them. With maximal entanglement, the wormhole appears and the smallest curve is as drawn in Figure 14.2: it is the horizon and it wraps around the wormhole at its narrowest point.

This result should ring bells. If we make things a bit more realistic by adding another dimension of space (we cannot visualise the wormhole now), the CFTs live on spheres that bound a three-dimensional space. The entanglement entropy between the two CFTs is equal to the area of the throat of the wormhole that connects them, and recall from Chapter 6 that, when the wormhole is at its shortest, this is equal to the area of the event horizon of the black hole. This sounds very much like Bekenstein’s result: the area of the event horizon of a black hole is equal to its thermal entropy. Here, the RT conjecture is telling us that the area of the event horizon is given by the entanglement entropy between two quantum theories.

This conclusion is worth repeating. We started out with two isolated quantum theories, each describing a bunch of particles. If these theories are not entangled, the two theories describe two disconnected universes. As in the previous chapter, the two theories still each have their own holographic dual but they are otherwise entirely disconnected. If instead we set up the mathematics so that the two theories are entangled with each other, holography tells us that the dual description is a wormhole. And the RT conjecture relates the entanglement entropy between the two quantum theories, which we can calculate using quantum theory alone, to the geometry of the wormhole – and in particular to the area of the wormhole at its narrowest point, which is also the area of the event horizon of the black hole.

Entanglement makes space

Although these links between entropy, entanglement and geometry were initially discovered in the context of black holes, they are now understood to be much more general. In his 2010 prize-winning essay entitled ‘Building up spacetime with quantum entanglement’, Canadian physicist Mark Van Raamsdonk writes that ‘we can connect up spacetimes by entangling degrees of freedom and tear them apart by disentangling. It is fascinating that the intrinsically quantum phenomenon of entanglement appears to be crucial for the emergence of classical spacetime geometry.’ By ‘degrees of freedom’, Van Raamsdonk is referring to particles, qubits or whatever are the ‘moving parts’ of the quantum theory, and by ‘connecting up’ or ‘tearing apart’ spacetime, he means that entanglement is not only related to geometry – it underlies it. Here’s the idea.

Figure 14.3. As the entanglement between the two halves of the sphere decreases, the bulk space stretches apart and eventually splits into two disconnected regions.

At the top left of Figure 14.3 we show a sphere, and on the boundary of the sphere is a quantum theory in its vacuum state. The vacuum, if you recall, is highly entangled. We’ve split the boundary into two parts, labelled L and R. The vacuum on the left part of the boundary is entangled with the vacuum on the right. The RT conjecture says that the amount of entanglement between these two regions on the boundary is equal to the area of the smallest possible surface (known as the ‘minimal surface’) that correspondingly divides the interior space. This dividing surface is shown as the shaded disk. There are no black holes here – just space bounded by a sphere. Suppose now that we could reduce the amount of entanglement on the boundary. According to RT, the area of the surface splitting the two regions must also reduce, which means that the two halves are joined together as in the top right picture. If the entanglement is further reduced to zero, the interior splits into two disconnected regions, as illustrated in the bottom picture. Space exists only in the interior regions of each bubble; there is no space connecting the bubbles. Thus, we see that the geometry of the interior space – the bulk – changes as we change the amount of entanglement in the quantum theory on the boundary of the space. But, as Einstein taught us, the geometry of space is gravity. This is the remarkable essence of the RT conjecture: gravity is determined by entanglement.

As an aside, the RT conjecture also delivers insight into something we said in the context of the AMPS firewall paradox. The essence of the paradox concerned the effects of breaking the entanglement of the quantum vacuum across the event horizon. This, we claimed, would be tantamount to tearing open empty space. We see this effect in a different guise in Figure 14.3. Here, if we switch the entanglement off between two regions of the quantum theory on the boundary, we slice the interior space in two.

We may also be glimpsing something even more deeply hidden: a new way of thinking about quantum entanglement. Let’s think not about entanglement between two CFTs on the boundary of space, but the much simpler case of entanglement between two particles. The idea, which has become known as the ER = EPR conjecture, asserts that we can picture these two particles as being linked together by something akin to a wormhole. This 2013 conjecture, due to Juan Maldacena and Leonard Susskind,47 follows nicely from Van Raamsdonk’s work. The ER side of the equation refers to the Einstein–Rosen bridge (wormhole) and the EPR side refers to the famous analysis by Einstein, Boris Podolsky and Nathan Rosen in which they attempted to make sense of quantum entanglement.48 Just as the Einstein–Rosen bridge between two black holes is created by quantum entanglement, so, in Maldacena and Susskind’s words, ‘it is very tempting to think that any EPR correlated system is connected by some sort of ER bridge, although in general the bridge may be a highly quantum object that is yet to be defined. Indeed, we speculate that even the simplest singlet state of two spins is connected by a (very quantum) bridge of this type.’

Islands in the stream‡

It’s now time to return to black holes and an important thread we have left hanging. Maldacena showed that it is possible for information to emerge from black holes, and that was enough for Stephen Hawking. However, at the time Hawking conceded his bet, nobody knew how the information emerges, and, in a closely related question, nobody knew what was wrong with Hawking’s original 1974 calculation. Until 2019, this was how things stood in the theoretical physics research community. The breakthrough came from two independent groups who were able to derive the all-important Page curve using 'old fashioned' physics (general relativity and quantum mechanics).49 The calculations support the holographic notion that the distant Hawking radiation and the interior Hawking radiation are two versions of the same thing. It is remarkable that the Page curve can be derived using ‘old fashioned’ physics, and another hint that Einstein’s theory of gravity knows much more about the fundamental workings of Nature than we might otherwise have given it credit for. The laws of black hole mechanics that we met in Chapter 10 are another striking example of this hidden depth of general relativity. They reveal to us that the theory knows something about the underlying microphysics, since Hawking’s Area Theorem is the Second Law of Thermodynamics in disguise.

The big idea in the 2019 papers is that, for an old black hole (one that is older than the Page time) part of the interior of the black hole is really on the outside. The full ramifications of this inside-outside identification remain to be understood but, as we will now see, both RT and the ER = EPR conjecture play a role.

In Figure 14.4, we show the interesting part of the Penrose diagram of an evaporating black hole.§ The Hawking radiation streams along 45-degree lightlike trajectories heading towards future lightlike infinity and the partner particles head along similar trajectories inside the horizon. In Einstein’s theory, these partners would be destined to hit the singularity, but in the new calculations something more dramatic happens: the partner particles behind the horizon end up outside.

Let’s first see how this leads to the Page curve. Imagine someone far away from the black hole sitting a fixed distance away and collecting the Hawking radiation. This observer follows the wiggly line on the Penrose diagram (you might like to check that by looking at the Schwarzschild coordinate grid on Figure 5.1). Suppose that our observer collects all the radiation emitted up to some time, t. We will refer to this radiation collectively as R. Our interest is in knowing the entanglement entropy between the radiation they collect and the black hole. If t is large enough, the black hole will have evaporated away, and the observer will have collected all the Hawking radiation. In that case, the entanglement entropy should have fallen to zero if all the information came out. This is precisely what happens in the new calculations, and precisely what doesn’t happen in Hawking’s.

The essential difference between the two calculations lies in the shaded region inside the horizon marked ‘island’. The island is a special region of spacetime. Its existence, and where it is located, is the subject of the 2019 papers. It turns out that the location of the island is dictated by the amount of radiation our observer has collected. If the time t is smaller than the Page time, there is no island. After the Page time, the island appears. How does the island lead to the Page curve?

Figure 14.4. Part of the Penrose diagram corresponding to an evaporating black hole. The wiggly arrows denote Hawking particles and particles of the same colour are entangled partners (one is outside the event horizon and its partner is inside). The Quantum Extremal Surface corresponding to the radiation R is indicated, along with its island (the shaded region). Interior partners in the island should be considered to be part of R.

At the right-hand tip of the island is a point on the Penrose diagram that we’ve labelled the Quantum Extremal Surface (QES). As with all the Penrose diagrams we’ve drawn, this point corresponds to a spherical surface in space.¶ The modern calculations give us a formula to compute the entanglement entropy in terms of the area of this surface:

SSC is the entanglement entropy of the Hawking radiation just as Hawking computed it, with a very important difference. The calculation mandates that we should also include Hawking partner particles inside the island in the calculation. This is the big new idea. In Hawking’s original calculation, he missed the existence of the island. For times before the Page time, when we have less than half the radiation, there is no island and Hawking’s calculation is correct. This gives the rising part of the Page curve, illustrated again in Figure 14.5. After the Page time, the island appears, with its QES very close to the horizon.

The new idea tells us that when we calculate the entanglement entropy, we must include the Hawking particles inside the island. These particles, inside the black hole, are ‘reunited’ with their partners outside and, once reunited, their combined contribution to the entanglement entropy is zero.**

The result is that, once the island has formed, the overall entanglement entropy of the Hawking radiation, SR, is given mainly by the first term on the right-hand side of the equation, which is simply the area of the QES (divided by 4). But this is approximately equal to the Bekenstein–Hawking entropy of the black hole because the QES lies close to the horizon. Now, since the area of the horizon shrinks as the hole evaporates, so too does the area of the QES. The entanglement entropy therefore starts to fall, and it goes all the way to zero when the black hole has evaporated away because the area of the QES (and the horizon) vanishes then. In this way, the Page curve starts to fall after the Page time and the calculations deliver the correct Page curve. This is a brilliant piece of physics.

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.