Black Holes: The Key to Understanding the Universe

7

The Kerr Wonderland

In 1963, New Zealand mathematician Roy Kerr succeeded in finding the unique solution to Einstein’s equations for a spinning black hole. Perhaps you might have expected that adding spin to Schwarzschild’s 1916 solution should not be particularly taxing, but the fact that it took almost half a century to be achieved is testament to the complexity Kerr discovered. Kerr’s solution, like Schwarzschild’s, corresponds to an eternal black hole: an immortal twisting in empty space. But unlike Schwarzschild’s, it is no longer spherically symmetric. Like most spinning objects, including the Sun and the Earth, a Kerr black hole bulges at the equator and is symmetric only about its axis of spin. This lack of symmetry has dramatic consequences.

There are two main types of Kerr black hole, which differ according to how fast they spin. We’ll consider slowly spinning black holes first and get to the faster ones later. A slowly spinning Kerr black hole is illustrated in Figure 7.1. Compared to the Schwarzschild black hole, it has three new features. Firstly, the singularity is a ring.* The plane of the ring is aligned at a right-angle to the spin axis, which means that only trajectories in the equatorial plane will encounter it. All other trajectories will miss it. An astronaut could therefore fall into a Kerr black hole and dodge the end of time. Secondly, the hole has two event horizons, which we’ve labelled the inner and outer horizons. Thirdly, there is a region outside of the outermost horizon in which space is being dragged around so violently that it is impossible for anything to stand still.† This region is known as the ergosphere.

To appreciate the wonders of a spinning black hole, let’s once again follow the adventures of an immortal astronaut. On descending towards the black hole, the first new feature our astronaut encounters is the ergosphere.‡ The outer surface of the ergosphere is the place where a light ray travelling radially outwards will freeze. In the Schwarzschild case, this is also the event horizon of the black hole: the place where the river of space is falling inwards at the speed of light, trapping the outward-swimming photon ‘fish’ forever. For the Kerr geometry, this place does not correspond to the event horizon – the place of no return. Our astronaut could pass into the ergosphere and then decide to turn around and escape back into the Universe. How can it be that a light ray heading radially outwards cannot escape, but an astronaut can? When the astronaut enters the ergosphere, it is impossible for her to avoid being swept around in the direction of rotation of the black hole. Space is being dragged around with the spin and no amount of rocketry can prevent the astronaut, or anything else, from being dragged around with it. This drag is the reason why an astronaut can beat a radially outgoing light ray and escape. We investigate this in more detail in Box 7.1.

Figure 7.1. Schematic representation of a slowly spinning black hole.

BOX 7.1. A place where it is impossible to stand still

Figure 7.2 illustrates a rotating black hole, viewed along the axis of spin. Look at the little circles with nearby dots. The dots correspond to places where a flash of light is emitted, and the circles show the outgoing light front a few moments later. Far away from the hole, the dot lies pretty much at the centre of the circle, but as we move closer to the hole the dots become increasingly displaced. The circles are shifted inwards and also in the direction of the spin. Inside the ergosphere, the dots lie outside of the circles, and that is the important feature. For a Schwarzschild black hole, a similar thing happens inside the horizon: the dots lie outside of the circles, but in this case the circles are only pulled inwards. For a Kerr black hole, they are also ‘dragged around’ in the direction of the spin.

A way to picture this is to appreciate that, for a non-rotating black hole, a dot (flash) on the horizon produces an expanding spherical shell of light that must fall inwards because none of the light can ever travel beyond the horizon. In the language of the river model, the light is being swept inwards by the flow of space. For a rotating black hole, the same is true, but there is also a swirling effect which drags the circles around. It is possible for a dot to lie outside of a circle, as shown in the diagram, which means that it is not possible for someone who emitted the flash to stand still at the position of the dot. If they did so, they would have outrun the light they emitted. They therefore are forced to swirl around with the black hole. This is the same idea we discussed when considering observers falling into a non-rotating black hole. In that case, observers cannot stand still inside the horizon. In the rotating case, the same role reversal is true in the ergosphere, which is a region from which it is possible to escape.

Figure 7.2. The ergosphere of a rotating black hole.

We can see how it is possible to escape from the ergosphere by focusing on the circle that straddles the ergosphere (the third circle from the left). A little portion of the circle lies outside the ergosphere. If we think of the circle as the future light cone of the person who emitted the flash, then we see that it is possible to draw a worldline for that person that crosses the boundary of the ergosphere and heads back outwards into the Universe beyond.

On crossing the ergosphere, our astronaut decides to continue inwards and cross the outer event horizon. As for the Schwarzschild black hole, this is a featureless place that marks the point of no return. The astronaut must now head inwards, and must cross the second, inner, event horizon. But there is a fascinating difference between the Schwarzschild and Kerr cases. For the spinning black hole, the astronaut regains her freedom to navigate once she has crossed the inner horizon. The singularity does not lie inexorably in her future, so time does not have to end. We can appreciate all of this by considering the spacetime diagram in Figure 7.3. Close to the outer event horizon, in region I, the geometry is similar to the Schwarzschild case. On crossing the outer horizon into region II, the light cones tip inwards, which means that the astronaut inexorably travels towards the inner horizon. However, on crossing the inner horizon into region III, the light cones tip back again and our astronaut can navigate around without encountering the singularity. Forever is still available. What, then, becomes of an astronaut who chooses to dodge the end of time?

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.

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.

If a piece of the in-falling object breaks away inside the ergosphere and carries negative energy into the black hole, the energy of the black hole will decrease. Energy must be conserved overall, however, and this means the outgoing part must carry away more energy than was thrown in.

Figure 7.8. Black hole mining. From Misner, Thorne and Wheeler’s 1973 book, Gravitation. (Figure 33.2 from Gravitation by Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, page 908. Published by Princeton University Press in 2017. Reproduced here by permission of the publisher.)

In Figure 7.8 we reproduce a figure from Misner, Thorne and Wheeler showing how an advanced civilisation living around a Kerr black hole could exploit the Penrose process to dispose of their garbage and generate electrical power for their civilisation. The ultimate green energy scheme.

We’ve spent quite some time exploring the ergosphere because there’s an important postscript. It transpires that the area of the black hole’s event horizon always increases after a Penrose process. At first sight this is surprising because a rotating black hole loses mass through the Penrose process. We might therefore expect the horizon to shrink. We are considering rotating black holes, however, and the area of the outer event horizon can increase even if the mass of the black hole decreases, provided that the spin of the black hole also decreases. Using the equations of general relativity, it can be shown that the spin of a black hole always decreases in a Penrose process, and by enough to guarantee that the area of the outer horizon always increases. This ‘area always increases’ rule doesn’t just apply to the Penrose process. In 1971, Stephen Hawking proved that, according to general relativity, the area of the horizon of a black hole must always increase, no matter what.†† This is a result of great importance. It is our first encounter with the laws of black hole thermodynamics.

Before we delve into this important subject, we will take a step back from the purely theoretical to explore the formation of the real black holes we observe to be dotted throughout the Universe.

* The radius of the ring is J/c where J is the angular momentum of the hole divided by its mass and c is the speed of light. For the Earth J/c is roughly 10 metres and for the Sun it is roughly 1 kilometre.

† With respect to the distant stars, say.

‡ ‘Encounter’ is a bit misleading because the freely falling astronaut won’t notice anything as she crosses into the ergosphere – as ever, spacetime for her will be locally flat.

§ What our astronaut experiences in region III is not something we can appreciate from the Penrose diagram, but this is what the equations tell us.

¶ A horizontal, spacelike singularity like the one in a Schwarzschild black hole.

** Using notions of energy and time defined by an observer far from the hole.

†† It will be allowed to decrease when we come to consider quantum physics.