Black Holes: The Key to Understanding the Universe

8

Real Black Holes from Collapsing Stars

‘In my entire scientific life, extending over forty-five years, the most shattering experience has been the realization that an exact solution of Einstein’s equations of general relativity, discovered by the New Zealand mathematician, Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the universe. This shuddering before the beautiful, this incredible fact that a discovery motivated by a search after the beautiful in mathematics should find its exact replica in Nature, persuades me to say that beauty is that to which the human mind responds at its deepest and most profound.’

Subrahmanyan Chandrasekhar24

The black holes we’ve explored so far have inhabited the mathematical landscape of general relativity. These remarkable universes were known to and broadly speaking dismissed by physicists for a large part of the twentieth century, Einstein included, on the very reasonable grounds that we shouldn’t conclude that something exists just because a physical theory allows it. If black holes are to exist in the real sky rather than the mathematical one, Nature must construct them. Real black holes, formed from stellar collapse, are the focus of this chapter. We will learn that the solutions to Einstein’s equations of general relativity discovered by Schwarzschild and Kerr are of extraordinary significance in the real Universe because they are the only possible solutions for the spacetime in the region outside of every black hole. Nowhere else in physics is something apparently so complicated as a collapsing star reduced to something so simple and with such precision. The Schwarzschild solution depends on just one number (the mass) and the Kerr solution adds a second number (the spin). Knowing these two numbers alone, we can compute the gravitational landscape in the region outside of real black holes, exactly. That is an astonishing claim – it does not matter what collapsed to form the black hole, nor does it depend on how it fell in. All that remains outside the horizon is a spacetime perfect in its simplicity. This is what moved Chandrasekhar to write the powerful prose quoted above. Quoting Chandrasekhar again: ‘The black holes of nature are the most perfect macroscopic objects there are in the universe … and since the general theory of relativity provides only a single unique family of solutions for their descriptions, they are the simplest objects as well.’

John Wheeler, as ever, found a more pithy phrasing: ‘Black holes have no hair.’ In his memoir Geons, Black Holes and Quantum Foam, Wheeler recounts an exchange with Richard Feynman in which the often irreverent Feynman accused him of using language ‘unfit for polite company’. ‘I tried to summarize the remarkable simplicity of a black hole by saying a black hole has no hair. I guess Dick Feynman and I had different images in mind. I was thinking of a room full of bald-pated people who were hard to identify individually because they showed no differences in hair length, style, or colour. The black hole, as it turned out, shows only three characteristics to the outside world: its mass, its electric charge (if any) and its spin (if any). It lacks the “hair” that more conventional objects possess that give them their individuality … No hair stylist can arrange for a black hole to have a certain colour or shape. It is bald.’

A series of papers throughout the late 1960s and early 1970s established the magnificent simplicity of black holes, as viewed from the outside. Once formed, according to general relativity, the horizon shields us from the complexities within. Even if something as large as a planet or star falls across the horizon, the American physicist Richard H. Price proved, in 1972, that the black hole quickly settles down again into oblivious perfection. For a Schwarzschild black hole, the horizon will reassume the shape of a perfect sphere, and any disturbances caused by the in-falling body will be smoothed out by the emission of gravitational waves. The conclusion is that the spacetime outside of all black holes in the Universe is either Schwarzschild or Kerr.*

What, then, happens in the case of a real collapsing star? Is it possible or even inevitable that a dense enough lump of matter will fall inwards to create a horizon and ultimately disappear into a spacetime singularity? The first attempt at addressing this question came back in 1939, when Robert Oppenheimer and Hartland Snyder showed that a star will collapse to form a black hole under certain assumptions. Specifically, they considered a pressureless ball of matter with perfect spherical symmetry. You may well baulk at this: the interior of a star is certainly not a zero-pressure environment, and the collapsing matter isn’t a perfect sphere. Perhaps the Oppenheimer–Snyder conclusion that black holes can form in Nature is associated with the assumption of perfect spherical symmetry. If everything is falling towards a single, precise point in the middle of the ball then no wonder something weird happens. The more realistic case will have matter swirling around and involve all the complexity of real stars. Maybe that leads to a collapse that does not generate a spacetime singularity. For many years, the possibility that black holes do not form out of collapsing stellar matter remained a mainstream view.

The publication of Roger Penrose’s paper in January 1965 essentially resolved the issue. He showed that the complex dynamics of the stellar collapse does not matter, and black holes must form if certain conditions are met.†

Penrose demonstrated that the formation of a spacetime singularity is inevitable once a distribution of matter has become so compressed that light cannot escape from it. Figure 8.1 is taken from Penrose’s paper (it is hand-drawn by Penrose himself) and provides an intuitive way of picturing the collapse of a star to form a black hole. Time (as measured by someone far away from the star, labelled ‘outside observer’ on the diagram) runs from the bottom to the top, and one of the three space dimensions is not drawn. The surface of the star is therefore drawn as a circle on any horizontal slice through the diagram. For example, on the slice labelled C3 at the base of the diagram the star’s surface is represented by the solid black circle. The dotted circle inside represents the Schwarzschild radius of the star‡ (recall that for the Sun the Schwarzschild radius is 3 kilometres). All the complex physics of the stellar interior plays out inside the solid circle and the beauty of Penrose’s argument is that the details of what’s happening there do not matter once the star has collapsed inside its Schwarzschild radius.

Figure 8.1. The spacetime diagram for a collapsing star, from Penrose’s 1965 paper ‘Gravitational Collapse and Space-Time Singularities’. (Reprinted figure with permission from as follows: Roger Penrose, ‘Gravitational Collapse and Space-Time Singularities’, Physical Review Letters, vol. 14, iss. 3, page 57, 1965. Copyright 1965 American Physical Society.)

We can follow the collapse of the star by moving upwards on the diagram. Each horizontal slice corresponds to a moment in time. As time passes, the circles representing the surface of the star get progressively smaller and the stellar surface traces out the cone-like shape on the diagram. The interior of the cone is the interior of the collapsing star and is labelled ‘matter’. The vertical dotted lines mark out the Schwarzschild radius. They become solid lines when the black hole has formed and then denote the event horizon. There is a lot of detail on this diagram that we don’t need – it is after all taken directly from Penrose’s published paper. It is worth looking at the light cones, however. Once the stellar surface has passed through the Schwarzschild radius the light cones inside all point towards the singularity. The outside observer therefore never sees the star collapse through the horizon. They see an increasingly slow contraction of the star as its surface approaches the horizon. For anyone inside the horizon it’s easy to see that the singularity lies inexorably in their future, although they never see it coming. Again, we see that the singularity is a moment in time.

While Penrose drew his diagram for the Schwarzschild case of zero spin, his theorem is more general and also applies to Kerr black holes, or to any conceivable collapsing distribution of matter. The theorem is concerned with what is happening at the solid black circle labelled S2, which forms before the singularity. This imaginary surface is known as a ‘trapped surface’, and it’s the key element in Penrose’s argument because he demonstrated that not all light rays will continue to propagate forever if the spacetime contains a ‘trapped surface’. So what is this trapped surface?

Figure 8.2. A trapped surface.

Figure 8.2 illustrates the idea. Picture some blob-like region of space and imagine lots of pulses of light flashing from the surface of the blob. For a blob in ordinary flat space, half of the light will head outwards, away from the blob, and the other half will head inwards. That is illustrated on the left of the figure. We’ve only shown five flashes of light, but we imagine many more. The black wavy lines represent light heading out and the grey wavy lines represent light heading inwards. The shaded region is the volume between these two sets of flashes and it will grow with time as the flashes head outwards and inwards at the speed of light. Since nothing travels faster than light, any matter initially sitting on the surface of the blob must stay in the expanding, shaded region. So far so good (hopefully).

On the right we’ve drawn a trapped surface. In this case, both the grey and black flashes are heading inwards. This happens inside the horizon of a black hole due to the curved geometry of spacetime. The converging of the light rays spells trouble. As before, any matter sitting on the trapped surface must stay inside the shaded region because nothing can travel faster than light. But now this region is shrinking down to nothing. In Penrose’s diagram the shaded region labelled F4 corresponds to the shaded region in Figure 8.2.

You might suppose that this is obvious since all matter inside the trapped surface is destined to get squeezed down to nothing, but we should be careful when wielding our dodgy intuition like this. As we’ve learned in the case of the Kerr black hole, matter might slip through a wormhole to explode into an infinite spacetime ‘on the other side’. What Penrose demonstrated rigorously is that at least one in-falling light ray will terminate. The mathematical techniques Penrose employed in his 1965 paper opened the door to a series of successively more wide-ranging singularity theorems, developed mainly by Penrose in collaboration with Stephen Hawking. Significantly, they managed to extend Penrose’s original theorem to include all particles (not just rays of light). They also applied the theorems ‘in reverse’ to show that in general relativity the Universe must have a singularity in the past which, to repeat the quote from the beginning of this book, ‘… constitutes, in some sense, a beginning to the universe’.

As something of an aside, it’s notable that the singularity theorems alone do not guarantee that a black hole will form in all circumstances. Black holes are not just singularities; they are black holes because their interior is shielded from their exterior by an event horizon. As we’ve seen, there could conceivably be singularities that are not shielded by a horizon such as the naked singularity in a fast-spinning Kerr black hole. To avoid that possibility, we also need the cosmic censorship conjecture as discussed in Chapter 7.

Naked singularities aside, the only way to avoid the conclusion that black holes must exist in our Universe is to argue that it’s not possible for matter to be squashed down sufficiently to form a trapped surface. That does not seem likely since we know from the work of Chandrasekhar that no known physics can halt the collapse of sufficiently massive stars. One might try to argue that some dramatic and unanticipated astrophysics or some new force of Nature steps in to halt the collapse before a trapped surface forms. Perhaps sufficient matter gets blown away as the gases swirl or as the collapsing star implodes. That might happen, but it is unlikely to be the case for every possible collapsing system. To emphatically illustrate the point, Penrose’s theorem applies if a large number of ordinary stars are close enough to form a trapped surface around them, such as could conceivably happen in the centre of a galaxy. In that case, the stars could still be very far apart so that the average density of matter is far less than the average density of a star, and we understand physics at these densities very well. Nevertheless, the theorem tells us that the stars are doomed to collapse.

The singularity theorems of Penrose and Hawking marked a change in the way physicists regarded black holes: combined with the work of Chandrasekhar, the theorems served to convince virtually all physicists that, in the words of the Nobel Prize committee, ‘black holes are a robust prediction of general relativity’. Today, we don’t need to rely on the theory alone because our twenty-first-century technology has allowed us to take photographs of supermassive black holes and to observe black hole collisions using gravitational wave detectors.

The Penrose diagram of a real black hole

When we encountered the wormholes and wonderlands of the eternal Schwarzschild and Kerr solutions to Einstein’s equations, we said that the portals to other universes were located in a part of the Penrose diagrams that would not exist inside a black hole formed by the gravitational collapse of a star. What, then, does the Penrose diagram of a real astrophysical black hole look like?

In 1923, the American mathematician George David Birkhoff proved that the spacetime outside of any spherical, non-rotating distribution of matter must be the Schwarzschild spacetime. This remains true even if the matter is in the process of collapsing. With this extra piece of information, we can draw the Penrose diagram corresponding to an entire spacetime in which a lone black hole forms out of a collapsing, spherical shell of matter.

Imagine a thin spherical shell of matter in the process of collapsing. A real star will of course not be just a shell, so we are simplifying a little. Outside of the shell, according to Birkhoff, the spacetime will be that of Schwarzschild. Inside the shell, there is no gravity. This is also true in Newton’s theory of gravitation, as Newton himself proved in his Principia Mathematica. Inside, therefore, the spacetime is flat. All we need to do to make the Penrose diagram therefore is to stitch together two bits of spacetime corresponding to the interior and exterior of our shell. We do that in Figure 8.3.

The top left diagram is flat (Minkowski) spacetime as depicted in Figure 3.10. The blue shaded region is the region inside of our collapsing shell. The curving black line is the worldline of the shell. We assume it starts collapsing in the distant past (the bottom of the triangle, which corresponds to past timelike infinity) and that it shrinks to zero radius at some finite time. The interior of the shell, i.e. the blue shaded region, is Minkowski spacetime, because there is no gravity there. The exterior of the shell (the unshaded region) is, according to Birkhoff’s theorem, Schwarzschild spacetime. That is illustrated in the top right diagram by the shaded red region. The curving line is the worldline of the collapsing shell again, but this time drawn in Schwarzschild spacetime. The complete spacetime must be Minkowski inside the shell and Schwarzschild outside, which means it must look like the lower diagram in the figure, which is obtained by patching together the blue and red portions of the upper diagrams.§ The disappointing result is that there is no wormhole or white hole anymore – the region inside the shell is ‘boring-flat-old’ Minkowski spacetime.

We can get a different picture of what’s happening during the collapse of the star by drawing some embedding diagrams, just as we did for the wormhole inside the eternal Schwarzschild black hole. In Figure 8.4 we show a series of embedding diagrams which represent slices through the Penrose diagram at progressively later times as the shell collapses. Time runs from top to bottom. Initially, the shell is very large and not particularly dense and barely makes a dint in the otherwise flat spacetime. We’ve also shown an astronaut named ‘A’ who decides to follow the collapsing shell inwards. The astronaut sees the shrinking shell below them. The second row shows the situation at some time later. The shell is now smaller and denser. When its radius shrinks inside the Schwarzschild radius, a black hole is formed. The astronaut, unbeknown to them, has also passed through the event horizon, but they don’t notice anything out of the ordinary. They still see the shell below them. The lower diagram is close to the moment of the singularity: the super-dense shell has distorted space dramatically. Even though A is very close to the singularity, they still see the shell way down below. In a sense the shell is blocking up the wormhole that would have been present in the eternal Schwarzschild geometry. The singularity is the moment when the space gets infinitely stretched and infinitely thin, at which point the astronaut and the shell cease to exist.

Figure 8.3. The Penrose diagram corresponding to a collapsing shell of matter (bottom). The inside of the shell is Minkowski spacetime (blue) and the outside of the shell is Schwarzschild spacetime (red).

Horizons: shuddering before the beautiful

Black holes, then, exist in our Universe and we are forced to confront their intellectual challenges. As Chandrasekhar wrote, ‘this incredible fact that a discovery motivated by a search after the beautiful in mathematics should find its exact replica in Nature, persuades me to say that beauty is that to which the human mind responds at its deepest and most profound’. And yet there is something at first sight that is deeply disappointing about the way Nature has chosen to realise this beauty, because it appears that the real treasure will be forever hidden from us. Black holes have horizons, and the horizon would appear to ensure that we must forever remain blind to the details of the collapsing material that falls into the singularity. That blindness is what gives rise to the remarkable applicability of the exterior Kerr and Schwarzschild solutions that Chandrasekhar refers to. Every black hole is identical, save for its spin and mass. Black holes have no hair. On the one hand this is a beautiful thing, but on the other hand it is bad news because it means we cannot observe the singularity to learn more. Cosmic censorship is highly desirable if we wish to maintain predictability, because we have no idea what the laws of physics are at the location of the singularity. By stashing the singularity behind a horizon, Nature appears to protect physicists living outside of a black hole from their ignorance of the singularity. But physicists don’t want to be protected. We want to learn about what happens when the known laws break down and must be replaced by the holy grail of theoretical physics – a quantum theory of gravity. To date, there is no proof of cosmic censorship, but if there is such a censor, we may never have direct access to the clues we need to explore quantum gravity.

Figure 8.4. A collapsing shell of matter is one way to make a black hole. The shell shrinks and increasingly distorts the spacetime nearby. The side view on the right helps us to see the curving of space. Astronaut A falls in behind the shell and always sees it way below them, shrinking as it recedes due to tidal effects. The ball in the middle column is what they would see in three dimensions. The singularity is the moment in time when the ‘throat’ becomes infinitely long and infinitely narrow. The third row corresponds to a spatial slice just before the singularity. You can see that the shell is still way down below the astronaut. Neither shell nor astronaut actually hit anything as they fall to their doom. Rather, the singularity is a ‘pinching to nothing’ of the space at a moment in time.

This was the view of many theoretical physicists until very recently. In the last few years, however, there has been a realisation that the clues to quantum gravity may not only reside near the singularity. They may also be found in the physics of the horizon. This came as a very welcome surprise, because it was long assumed that whatever happens close to the event horizon of a black hole, the physical processes should have nothing at all to do with the extreme conditions at the singularity where we expect quantum gravitational effects to be important. The horizon, after all, is a place in space through which an astronaut can happily fall and suffer no ill effects. This assumption now appears to have been too pessimistic. The study of the thermodynamics of black holes, which began in the 1970s as an investigation into quantum mechanical effects in the vicinity of the horizon, has opened an entirely unexpected window into the deep mystery of quantum gravity. And it is to black hole thermodynamics that we now turn.

* Strictly speaking, black holes can also carry electric charge but astrophysical black holes are electrically neutral.

† The waters were muddied by a Lifshitz and Khalatnikov paper of 1963 which claimed to prove no singularity would occur. Following work with Belinskii, Lifshitz and Khalatnikov withdrew their claim in 1970.

‡ On the diagram, the Schwarzschild radius is at r = 2m where m is the mass of the star because Penrose is working in units where G = c = 1.

§ Strictly speaking this logic works only for the region outside of the horizon and we are guessing somewhat as to what happens inside the horizon.