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.