The second reason why the Kerr wonderland shouldn’t exist is illustrated in Figure 7.5. The short green curve is the worldline of someone moving without drama towards future timelike infinity in one of the region I universes. They send light signals to our astronaut inside the hole at regular intervals but, as we saw in Chapter 3, there is an infinite amount of time compressed into the tip of the diamond. This means that the light signals pile up along the upper edge of the diamond and into the black hole along the inner horizon. This represents an infinite flux of energy (one of these signals is indicated by the purple wiggly line) which will result in the formation of a singularity, sealing off region III, the ring singularity and beyond. ‘The inner horizon marks the last moment at which our astronaut can still receive news, but then she gets all of the news.’23 Worldlines will end on the singularity¶ so no extension is necessary or possible into the region containing the ring singularity, time machines and the infinite tower of universes.
Fast-spinning black holes
If the spin (J/c) of the hole is bigger than one half of the Schwarzschild radius, the Penrose diagram is not that of the Kerr wonderland. The spacetime is vastly simpler, but involves what is known as a naked singularity, as illustrated in Figure 7.6.
Figure 7.6. The Penrose diagram for a fast-spinning black hole.
The event horizons have disappeared, leaving just a ring singularity (the wiggly line), which remains a portal to an infinite space where gravity repels instead of attracts. A naked singularity is a singularity from which the Universe is not protected by an event horizon. Naked singularities are an anathema to physicists. So much so that Roger Penrose was moved to introduce the ‘cosmic censorship conjecture’, which asserts that no naked singularities exist in our Universe other than at the Big Bang. The trouble with naked singularities is that they contaminate spacetime with ignorance; the world becomes hopelessly non-deterministic. Perfect knowledge of the past would be insufficient to predict the future. To see why, imagine any event in the spacetime of Figure 7.6. There will be light rays that can reach this event, and therefore influence it, coming from the singularity. The singularity, however, is a place where the known laws of physics break down. This means that every event in the spacetime can be influenced by an unpredictable region of spacetime, and this is a nasty situation for physicists who are in the business of predicting the future from a knowledge of the past. Having said that, Nature is not obliged to make physicists’ lives easier.
In 1991, Kip Thorne, John Preskill and Stephen Hawking made a famous bet that the laws of physics would forbid naked singularities. Hawking thought that naked singularities would be forbidden in all circumstances, but this appears not to be the case. In 1997, Hawking famously conceded the bet ‘on a technicality’ and his concession made the front page of the New York Times. The technicality is that computer simulations can produce them, although the models are highly contrived. Having said that, they do not require anything too exotic beyond the known laws of physics. Thorne, Preskill and Hawking therefore renewed their bet with modified wording. No naked singularities will occur naturally in our Universe without the need for the intervention of some unimaginably advanced civilisation that could arrange to fine tune gravitational collapse. Hawking paid out by giving Thorne and Preskill t-shirts ‘to cover the winner’s nakedness’, in accord with the wording of the bet. The t-shirts were so politically incorrect, in Kip Thorne’s words, that they were forbidden from ever wearing them.
What prevents a Kerr black hole naturally acquiring sufficient spin to produce a naked singularity? One could easily imagine making a black hole spin faster, such that even though it started out spinning slowly, it ends up spinning fast enough to expose a naked singularity. For example, why not drop a spinning ball (or maybe a star if we want to get serious) into the hole and arrange things so that the spin is in the same direction as the spin of the hole. That would increase the hole’s spin, potentially pushing it over the critical value. This calculation can be done in general relativity, and it turns out that the hole pushes the spinning thing away. This ‘spin-spin’ interaction is a nice example that illustrates how the theory of general relativity appears to be constructed such that cosmic censorship holds. It seems that naturally occurring black holes always have their singularities tucked away behind horizons.
You may be disappointed that Nature does not appear to permit wormholes and Kerr wonderlands to exist, but your disappointment may be too pessimistic. The message is that general relativity has a richness that permits a remarkable gamut of spacetimes. Perhaps some of this extraordinary potential is realised in Nature? We will return to this cryptic statement later: the answer isn’t a straight ‘No’.
Return to the ergosphere
While the interior geometry of the Kerr black hole may be eliminated by the in-falling matter of the collapsing star that formed it, this is not true of the ergosphere which sits beyond the outer horizon. Spinning black holes do exist, and the external spacetime is described by the Kerr solution. So let us return to the ergosphere; the region just beyond the outer horizon within which it is impossible not to get swept around by the flow of space. Recall that, inside the ergosphere, space and time swap roles (see Box 7.1), but it remains possible to escape. Roger Penrose first appreciated the consequences of this space and time role reversal inside the ergosphere; it makes it possible to extract energy from a rotating black hole. The idea is illustrated in Figure 7.7.
Imagine throwing an object into the ergosphere, where it breaks into two pieces. One piece falls into the black hole and the other piece comes back out. This is possible, because the ergosphere lies outside of the event horizon. The surprising thing is that the piece that comes out can carry more energy than the original object carried in. How does this magic come about?
The important insight is that inside the ergosphere objects can have negative energy from the perspective of someone outside the hole. Outside of a black hole, objects always have positive energy. Inside the ergosphere, however, it becomes possible for negative energy objects to exist.** This possibility arises because of the swapping of the roles of space and time and because energy and momentum are intimately related to space and time.
Figure 7.7. The Penrose process.
To appreciate the link between space, time, momentum and energy, we need a brief detour back to 1918 and to Amalie Emmy Noether. Noether was, in Einstein’s words, ‘the most significant creative mathematical genius thus far produced since the higher education of women began’. Among her many achievements, Noether discovered that the law of conservation of energy is a direct consequence of time translational invariance, which when the jargon is stripped away means that the result of an experiment does not depend on what day of the week it is performed (all other things being equal). Likewise, the law of momentum conservation is a consequence of translational invariance in space, which means the result does not depend on where the experiment is performed (all other things being equal). This means that the role reversal of space and time in the ergosphere is accompanied by a corresponding reversal of the roles of momentum and energy. In the Universe outside the black hole – our everyday world – momentum can be positive or negative because things can move left or right. Inside the ergosphere, the switch means that energy can likewise be positive or negative.