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.