Figure 7.3. The future light cones outside and inside a Kerr black hole.
To answer this question, we need a Penrose diagram, which we can begin to construct given our experience with the Schwarzschild case. Figure 7.4 is a start. The purple line is the path of the astronaut as she travels from the universe outside (region I) through the outer horizon (the 45-degree black line) into region II and then through the inner horizon (the 45-degree orange line) into region III. Region I and the outer horizon are easy to draw because they are just like the Schwarzschild black hole. The two black lines on the right (ℑ+ and ℑ-) represent (lightlike) infinity and they are bona fide boundaries to the Penrose diagram. Our astronaut enters region II whence she is doomed to pass through the inner horizon, carried inexorably along by the flow of space. This mandates us to draw the orange lines representing the inner horizon. As for all horizons, they must be at 45 degrees. So far so good. Now comes the first novelty. The wiggly, vertical lines denote the ring singularity inside region III. Notice that they are vertical, and not horizontal as in the Schwarzschild black hole. This is because the Kerr singularity is timelike, which means our astronaut can see it (45-degree lines representing light rays starting from the singularity can reach the astronaut’s worldline). This is different from the spacelike singularity inside a Schwarzschild black hole: the horizontal line on the Penrose diagram which nobody sees coming.
Figure 7.4. The Penrose diagram for a Kerr black hole corresponding to the discussion in the text. It is clearly incomplete.
Given our experience with the Penrose diagrams in the last chapter, we immediately notice that Figure 7.4 cannot be the full story. The two upper diagonal edges of the diagram inside the black hole are horizons, not singularities, and they do not lie at infinity. We encountered a similar situation when we investigated the Schwarzschild black hole. It led us to extend the spacetime and discover the Einstein–Rosen bridge. The same applies here. To ensure that all worldlines end either at infinity or on a singularity, we are compelled to extend Figure 7.4. The result, part of which is shown in Figure 7.5, is quite shocking. We already know there can be an infinite volume of space inside the event horizon of a black hole, but in the maximally extended Schwarzschild case we only had to contend with one extra infinite universe through the wormhole. Inside the eternal Kerr black hole there reside an infinity of infinite universes, nested inside each other like Russian TARDIS dolls. The Penrose diagram itself would fill an infinite sheet of paper. This Kerr wonderland is the unique way to extend Figure 7.4 and remain consistent with general relativity.
Figure 7.5. The maximal Kerr black hole. The purple line is the worldline of our intrepid explorer. The part of her journey shown in Figure 7.4 is at the bottom. The wiggly purple lines represent possible paths of light rays.
It would be entirely appropriate to ask ‘What on earth have we drawn here?’ The diagram depicts part of an infinite ‘tower’ of universes, which means there is an infinite amount of space and time tucked away inside an eternal Kerr black hole and no mandatory rendezvous with a singularity to spoil the fun. To see how things play out, let’s resume the journey as our astronaut travels further into the interior of the black hole.
Recall, the astronaut started out in region I at the bottom of the diagram. This is the infinite region of spacetime outside the outer horizon. We might call it ‘our Universe’. She crossed the outer horizon and entered region II. She is now inside the black hole, in between the outer and inner horizons. We can see that, just as for the Schwarzschild black hole, she will be able to receive signals not only from our Universe – region I – but also from another universe – the ‘other’ region I (we have drawn two wiggly light rays to illustrate that point). She could meet astronauts who have crossed the outer horizon from the other universe, but now they are not condemned to rendezvous with the singularity at the end of time because there is no singularity in region II. Instead, she must cross the inner horizon and enter region III. The ring singularity looms, denoted by the wavy vertical line, but she can avoid it. Now the fun starts.
She chooses to avoid the singularity and crosses into a second region II. This region is bounded by a horizon, but it is the horizon of a white hole, a gateway into a different region I – another universe. At this point, she might choose to head out and explore this new and enticing ocean of stars and galaxies, but she doesn’t have to. There is another Kerr black hole outer horizon in this new universe, and she chooses to plunge across it. Once inside this second black hole, the whole story repeats itself until she dives into a third black hole. Emerging into region III at the top of the figure, she is now ready to brave the singularity. She dives through the ring and into another new universe in the infinite tower of universes. This universe is very different from the others, though. In this region of spacetime, gravity pushes rather than pulls.§ It is an anti-gravity universe. It would still be possible for her to swing around and navigate back through the ring singularity. But it is also possible for her to arrange things so that she emerges before she entered it. This is possible because there are paths that our astronaut can take in region III which loop back and return to the same point. It’s not possible to draw such paths on our Penrose diagram because they involve looping around in one of the dimensions we haven’t drawn. These paths are known as ‘closed timelike curves’. Imagine a path over spacetime that begins the day before your birth and, some years later (by your watch) arrives back at the day before your birth. This is time travel. Such paths are possible in the spacetime geometry in region III. This means that the Kerr black hole is a time machine (sometimes known as a Carter time machine, after Brandon Carter who first discovered it).
All sorts of issues now arise. What if the astronaut decided to prevent herself from being born? This isn’t necessarily a paradox if she doesn’t have free will; the universe could conceivably be constructed such that time travel is possible and yet the whole thing remains logically consistent. Perhaps it would be impossible for you to prevent yourself from being born and causing such paradoxes. Musing about free will is (perhaps inevitably, who knows) beyond the scope of this book, but asking questions about possible spacetime geometries is not. The big question is: ‘do spacetimes that permit closed timelike curves actually exist in Nature?’
In the proceedings of Kip Thorne’s 60th birthday party – only the most eminent physicists have their birthdays written up in proceedings – Stephen Hawking writes about spacetimes that allow time machines.22 ‘This essay will be about time travel, which has become an interest of Kip Thorne’s as he has become older …,’ he begins. ‘But to openly speculate about time travel is tricky. If the press picked up that the government was funding research into time travel, there would be an outcry about the waste of public money … So there are only a few of us who are foolhardy enough to work on a subject that is so politically incorrect, even in physics circles. We disguise what we do by using technical terms like “closed timelike curves” which is just code for time travel.’
Although it has not been proven beyond doubt, Hawking proposed a ‘Chronology Protection Conjecture’, which states that ‘The laws of physics conspire to prevent time travel by macroscopic objects’. By macroscopic objects, Hawking means big things like astronauts rather than subatomic particles. The implication is that the maximally extended Kerr geometry should not exist in Nature, and we believe it does not for two reasons. Firstly, as we’ve already mentioned and will see in the following chapter, real black holes are made from collapsing matter. The presence of matter changes the spacetime inside the horizon of a black hole, effectively blocking up the portals into other universes. Both the maximally extended Schwarzschild and Kerr solutions are vacuum solutions to Einstein’s equations – eternal black holes – and as far as we know, no such black holes exist.