A more artistic rendering of the wormhole is shown in Figure 6.6. Thinking in only two dimensions of space, we can imagine Flat Albert and his friend sliding around the black hole. There is an infinite space inside. We humans can see how this works because we can picture the space curving in a third dimension, but for the flatlanders the idea would seem very strange. Similarly, you could imagine wrapping your hands around a low-mass maximally extended Schwarzschild black hole. Its horizon would be a tiny perfect sphere, but inside your cupped hands would reside an infinite other universe.
Figure 6.7. Embedding diagrams of three other spacelike slices from Figure 6.4. Time increases from bottom to top. We can see how the wormhole stretches open and snaps, leaving two disconnected universes. Nothing can travel through the wormhole before it snaps off.
In Figure 6.7 we’ve drawn three more of the embedding diagrams representing the spacelike slices shown in Figure 6.4. We can see how the horizons move apart as the wormhole lengthens and eventually breaks to expose the singularity. The slice at the bottom occurs at earlier time than the slice at the top, which means that the wormhole evolves in time.¶ It is this evolution that prevents anything from travelling through the wormhole (see Box 6.2 for more detail). We don’t need to draw wormholes to appreciate that nobody can travel from region 1 to region 3, and vice versa, which is what a journey through the wormhole would entail. That much is evident from the Penrose diagram since there is no line you can draw at an angle less than 45 degrees to the vertical that connects these two regions. However, the embedding diagrams of the wormhole provide a lovely picture of how the evolution of the wormhole renders travel through it impossible.
Figure 6.8. An astronaut (the dot) falling into a Schwarzschild black hole. Time advances from the top left to bottom right. Notice how the wormhole grows and pinches off before the astronaut can reach the other side. (Illustrations by Jack Jewell)
Figure 6.8 is a visualisation of the geometry of the maximally extended Schwarzschild spacetime together with an astronaut as he falls into the black hole. The wormholes are constant (Kruskal) time embedding diagrams. In the top left picture, the astronaut is approaching the horizon and the wormhole is open, connecting the two universes together. On the top right, the astronaut is about to pass through the horizon. He is still in region 1, but the wormhole has already passed its maximum diameter and is beginning to close. In the next image, he has crossed the black hole’s horizon and the wormhole is pinching closed, which it has done by the time of the second image on the bottom row. We see that the astronaut cannot traverse the wormhole because it pinches shut before he can pass through it. None of this is peculiar to any particular astronaut or how they manoeuvre on their journey into the black hole. The slamming of the door between universes has nothing to do with the details of the journey, and there is nothing anyone can do to change it. This story is written entirely within the Schwarzschild metric, the unique spherically symmetric solution of Einstein’s equations. How wonderful.
BOX 6.2. Evolving wormholes
Let’s start by writing down the Schwarzschild metric again:
The (1 – RS/R) terms in front of the time and space coordinates tell us about the geometry of the spacetime – how it deviates from flat. These terms do not depend on time outside the horizon, which means the geometry does not change as t changes. The words outside the horizon are important. Inside the horizon, the space and time coordinates flip around such that the Schwarzschild R coordinate takes the role of time. If you recall, a key feature of life inside the horizon is that everything is compelled to move to smaller and smaller R, just as outside everything is compelled to move forwards in time. Why? Because the interval must always be positive along the worldline of anything with non-zero mass. This means that dt2 cannot be zero outside the horizon and dR2 cannot be zero inside the horizon. The ticking of time drives us forwards in t outside the horizon and forwards in R inside the horizon. This is why R is the time coordinate inside the horizon. But the (1 – RS/R) terms depend on R, which means that inside the horizon the geometry is compelled to change, just as inexorably as we are compelled to journey towards tomorrow. This is why the spacetime geometry is dynamic inside the horizon. It changes, and in such a way that not even light can make it through the wormhole.
Although travel from one universe to the other is impossible because the wormhole pinches shut, we have noted that it is possible for someone to jump into the black hole from region 1 and meet up with someone jumping in from region 3 before they both end up at the singularity. That is easy to see using the Penrose diagram. Moreover, someone inside the black hole in region 2 can see things in both regions 1 and 3 because they can receive signals from them. That is also evident from the Penrose diagram. This means that, in the moments between jumping into the black hole and hitting the singularity, our intrepid astronauts would be able to peer through the wormhole and see the universe on the other side.
We must of course ask whether any of this might play out in our Universe. Sadly, the answer appears to be ‘probably not’, at least for the case of astronauts attempting to travel between universes. That is because the maximally extended Schwarzschild spacetime does not correspond to the geometry of spacetime created by the gravitational collapse of a star. Rather, the Schwarzschild solution is only valid in the region of empty space outside of the star. The maximally extended Schwarzschild spacetime in Figure 6.2, replete with wormhole and black and white holes, would be the correct description of a non-spinning, eternal black hole. We are not aware that such things exist.
Why ‘probably not’? Because wormhole geometries are valid solutions of Einstein’s equations. In 1988, Michael Morris, Kip Thorne and Ulvi Yurtsever explored the possibility of keeping the wormhole open.21 ‘We begin by asking whether the laws of physics permit an arbitrarily advanced civilisation to construct and maintain wormholes for interstellar travel?’ These wormholes would not be constructed by collapsing stars, but could conceivably be ‘pulled out of the quantum foam and enlarged to classical size and stabilised, potentially, by quantum fields with a negative energy density’. This is great fun but very speculative. As the authors state, such a wormhole would be a time machine, and the consequences of such things existing are disturbing. ‘Can an advanced being measure Schrödinger’s cat to be alive at an event P (thereby collapsing its wave function into a live state), then go backwards in time via the wormhole and kill the cat (collapse its wave function into a “dead” state) before it reaches P?’ Putting the fun aside, these ideas have resurfaced in the search for a resolution to the black hole information paradox. In particular, the idea that microscopic wormholes could be part of the structure of spacetime is part of the ER = EPR hypothesis we’ll meet in the final chapters. So maybe there are time machines in our Universe after all. In any event, the maximal Schwarzschild extension is a solution to the equations of general relativity, and it is a very interesting and beautiful one at that. It alerts us to the outrageous possibilities that spacetime may have in store for us and, as we shall see next, wormholes aren’t even the half of it.