The main theme of the 2020 Nobel Prize in Physics is black holes, and it has been awarded to Roger Penrose, Andrea Ghez, and Reinhard Genzel. The Nobel committee awarded half of it to Ghez and Genzel for their pioneering work in showing that there is a supermassive black hole at the centre of our galaxy, the Milky Way; the other half was awarded to Roger Penrose for showing that ‘black hole formation is a robust prediction of the general theory of relativity.’ This three-part series of articles focuses on Penrose’s contribution.
Before I begin my exposition, I would like to mention that I won’t go into complete details regarding Penrose’s theorems here. Even the watered-down version of them can be conceptually challenging to grasp. However, if you have made it this far, then you have done most of the hard work, and you can certainly understand the importance and implications of his theorem. I’ll write about the implication after the exposition, so even if you can’t fully follow it, you’ll still understand why his theorems were important.
Roger Penrose made a name for himself in 1965 after publishing his first famous singularity theorem. Between 1965 and 1970, he along with Stephen Hawking published many famous singularity theorems (both individually and collectively) that came to be known as Hawking-Penrose singularity theorems, along with Penrose’s paper on cosmic censorship. We will look into the statement and implication of Penrose’s elegant 1965 paper Gravitational collapse and space-time singularities where he formulated his first theorem, and we will also glance over ‘cosmic censorship’ and why these results have some profound implications when taken together.
Penrose’s Singularity Theorem (1965)
They are two concepts we need to familiarise ourselves with: what is a singularity and what is a trapped surface?
As we have discussed before, it is not exactly clear what a singularity is. Is it infinite curvature or infinite density? Does it just mean that metric is undefined? Is that where the laws of physics break down?
Penrose bypassed all of that by coming up with a more general definition. He attached the notion of singularity to a property of space-time known as ‘geodesic incompleteness.’ A geodesic is the path traced by a particle on space-time in free fall (with no external force acting on it). Each particle, massive or non-massive, that has a trajectory in some geometry in GR has a corresponding geodesic equation which is the equation of motion for that particle. For space-time geodesic to be incomplete would mean something like the vanishing of space-time description. For instance, imagine a particle moving along a path at some point in time, and then suddenly, the path comes to an abrupt end and there is no space-time for that path to move ‘in’; we would then say that such a description is geodesically incomplete. This is the modern definition of what a singularity is and is still used by physicists working in quantum gravity where they deal with singularities a lot. So, in essence, to say that there is a singularity at r=0 is to say that at r=0 the geodesic of the particle is incomplete (i.e. the space-time description breaks down at that point). A tiny addendum should be made here: although this is how most working physicists talk about singularities, both Penrose and Hawking basically said that a singularity is what occurs when you get geodesical incompleteness at a certain moment in time (which is to say that singularities are implications of geodesic incompleteness, though saying that a singularity exists is a much stronger affirmation than saying space-time is geodesically incomplete) and what that singularity is is still unknown (it most likely means we have reached the limit of our theory and we need something better).
Before we talk about what a trapped surface is, it might be a good idea to get some basic intuition going regarding how a two-dimensional curved surface, embedded in a 3-D dimension works. Imagine we take a slice of the surface of Earth, make a copy of it and project that slice far into the sky along the radial length (where r=0 is Earth’s core) Now, imagine we take another slice but this time project it much closer to Earth’s core. In such a scenario, we will see that the sheet that is projected into the sky has a larger surface area than Earth’s slice, and the one closer to the core of the earth has a smaller surface area compared to Earth.
Now imagine you have someone near Earth’s core with a very powerful torch that goes through the surfaces and they shine a light in the outwards direction. It will hit a certain part of the first surface as it goes through and hits Earth’s slice. We’ll see that the surface area hit by light on Earth’s slice is greater than the one hit near the core; as it travels through Earth’s slice and hits the projection in the sky, the surface area it will hit will be even bigger. So we can see that if we cut an infinite number of slices along the radial length and project an outgoing light through them, the light rays will diverge. Now imagine an inwards light from a source like Sun as it hits the large surface area of the sky projection, then it will hit the Earth’s slice at a smaller surface area and then go onto hit the slice near the core at and even smaller area until it finally hits the core. In this incoming or ‘inwardly directed’ light ray, the light rays are converging towards the centre of Earth, i.e. its core.
Keeping that in mind, we can go onto define a trapped surface. A trapped surface is a surface for which both the ingoing and outgoing light rays converge. Hence they are trapped.
Given all the information we have now, we can start building an intuitive picture of Penrose’s famous 1965 theorem. After gross oversimplification, I would roughly state the theorem as follows: for any global space-time that satisfies some basic energy condition (i.e. the energy density of matter in such space-time is not negative) and it has a trapped surface, then if a light ray originates from surface of a body undergoing gravitational collapse such as that of a collapsing star (the past ‘boundary’ for all light rays — null geodesic — is a non-compact Cauchy surface in this case). All such light rays will always be geodesically incomplete towards the future of the trapped surface, and hence you get a singularity (because the global space-time is collapsing in all directions as you travel towards the future).
That is quite a mouthful, so let’s try to intuitively understand what the theorem is telling us. A star goes under gravitational collapse and, as it does, the matter falls inwards. There will be a point in time when the infalling matter will form a trapped surface, at which point no known force in the universe can stop the formation of a singularity because then the trapped surface will collapse along with all the matter in it (note that for simplicity sake, I am using light rays and matter here interchangeably). What makes this result important is that it implies that the singularity is inevitable, not because of spherical symmetry of the collapse but because of the formation of the trapped surface. The formation of trapped surface does not depend on spherical symmetry, and it can be shown that for small departures in spherically symmetric gravitational collapse a trapped surface would still form.
The main implication of the theorem being gravitational collapse necessarily results in the formation of a singularity. The image below is a modern replication of a drawing from Penrose’s 1965 paper; the image is a pictorial description of what I have stated above (the hourglass-like depictions in the image are known as light cones which I am afraid is too extensive to be covered in this article, but you can ignore them and still get a strong intuitive sense of what is going on).
Rather than going into too much detail, we will give this section a very brief glance. You might recall that when we were discussing Kerr black holes (spinning), then for a special case of spinning black holes, you get what is known as a naked singularity. Well, as it turns out, the cosmos (pretty much like my boring conservative uncle) does not appreciate nudity. Penrose proposed a conjecture for the first time in 1969 that in nature, naked singularities are not physically realisable and are instead clothed by a horizon (for spinning black holes, there are also other physical reasons — in case of large angular velocities compared to their mass — why such special black holes can’t form, so our example of Kerr black hole isn’t something that happens in our universe). It roughly states that given a few assumptions such as gravitational force is always attractive in the region, gravitational collapse does not allow for the formation of naked singularities. This conjecture came to be known as the ‘weak cosmic-censorship conjecture.’ Note that the conjecture does permit the existence of these singularities mathematically, as it is possible to construct a solution to Einstein field equations which have naked singularities; rather, it’s about the fact that you won’t get them in a realistic gravitational collapse (vaguely).
You shouldn’t read too much into the picture since we did not cover the background knowledge needed to interpret Penrose diagrams in this article, but for cosmic censorship, we are meant to infer that an observer at infinity can never receive signals or communicate with the singularity at r=0 because it is screened by the event horizon. This implies that singularities always come with a horizon.
This is still a conjecture because it has not been rigorously proven, but numerical simulations seem to be consistent with it. It can be stated in the most basic form as ‘singularities are clothed by a horizon.’
Although the Nobel prize was awarded to Penrose for mostly his singularity theorem, his original argument for why black holes are real involved the weak cosmic censorship conjecture as well. Given what we all know now, we can construct a version of the argument as follows:
- Singularity theorem implies the existence of singularity (geodesic incompleteness) for massive body undergoing gravitational collapse.
- Cosmic censorship implies singularities are clothed by horizons, so they are not naked.
- A compact massive object that has an event horizon is known as a black hole.
- Therefore, black holes exist.
This is a compelling argument for the existence of black holes that does not rely on any assumptions of spherical or axial symmetry or any other special condition but is instead an extremely general result.
Penrose’s work led to a plethora of other developments. Stephen Hawking, inspired from Penrose’s work, applied similar techniques to show there is an initial singularity at the Big Bang (though this idea was later abandoned by Hawking and Penrose as well); extending the theorem with weaker energy conditions and important discussions on the fate of those theorems in the framework of quantum gravity.
Penrose’s own focus has shifted towards some radical ideas that have drawn him the ire of the community. He has his own interpretation of quantum mechanics and theories of consciousness, and he is an advocate of Conformal Cyclic Cosmology (CCC), his theory regarding the ‘origin’ and far future of the universe that bypasses Hawking’s singularity theorem. He has also been a vocal critic of inflationary cosmology saying that rather than solving problems, it only pushes them back whereas CCC solves them. Though by all accounts, he is still quite humble, beloved in the community, and extremely charitable with his time.
As I wrap this up, there is one point I’d like to make regarding what singularities are. I think it’s wrong to think of them as an actual physical manifestation of infinite curvature; rather, if we get a singularity it means we are pushing a theory beyond its regime of applicability. General relativity, after all, is not a theory about singular space-times and, in fact, many developments in quantum gravity show that the singularities either get smoothed over or are completely bypassed. Having said that, I think it’s best not to think of Roger Penrose’s contribution to cosmology as having shown singularities exist (channelling the cosmologist Janna Levin and others); rather, he showed that general relativity is incomplete and is not the final chapter in the story of gravity, which I think is a more profound achievement.