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.