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 deaclass="underline" 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.