What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?

Question

I received an email today about the award of the 2020 Nobel Prize in Physics to Roger Penrose, Reinhard Genzel and Andrea Ghez. Roger Penrose receives one-half of the prize "for the discovery that black hole formation is a robust prediction of the general theory of relativity." Genzel and Ghez share one-half "for the discovery of a supermassive compact object at the centre of our galaxy". Roger Penrose is an English mathematical physicist who has made contributions to the mathematical physics of general relativity and cosmology. I have checked some of his works which relate to mathematics, and I have found the paper

• M. Ko, E. T. Newman, R. Penrose, The Kähler structure of asymptotic twistor space, Journal of Mathematical Physics 18 (1977) 58–64, doi:10.1063/1.523151,

which seems to indicate Penrose has widely contributed generally to the mathematics of general relativity like tensors and manifolds. Now my question here is:

Question What are contributions of Sir Roger Penrose, the winner of the 2020 Nobel prize in physics, to the mathematics of general relativity, like tensors and manifolds?

We may motivate this question by adding a nice question which is pointed out in the comment by Alexandre Eremenko below where he asks: Is Sir Roger Penrose the first true mathematician to receive a Nobel prize in physics? If the answer is yes, then Sir Roger Penrose would say to us "before being a physicist you should be a mathematician". On the other hand, in my opinion the first mathematician to be awarded several physics prizes is the American mathematical and theoretical physicist Edward Witten. This seems to meet Sir Roger Penrose in his research such as cosmology and research in modern physics (Einstein general relativity).

Related question: Penrose’s singularity theorem

Answer 1

It seems (as mentioned by Sam Hopkins above) that the Singularity Theorem is the official reason for the Nobel Award.

But that is by no means the only (and perhaps not even the most important) contribution of Sir Roger Penrose to mathematical physics ( not to mention his works as a geometer and his research on tilings, and so many other things).

In Physics, his grand idea is Twistor Theory, an ongoing project which is still far from completion, but that has been incorporated in other areas (see for instance here for its connection to Strings Theory, and also there is another connection with the Bohm-Hiley approach using Clifford Algebras, see here ).

But his influence goes even beyond that: Penrose invented Spin Networks in the late sixties as a way to discretize space-time. The core idea was subsequently incorporated in the grand rival of String Theory, Loop Quantum Gravity. As far as I know, all approaches to a background independent Quantum Theory of gravity use spin networks, one way or the other.

Moral: Congratulations Sir Roger !

ADDENDUM @TimotyChow mentioned that my answer does not address the ask of the OP, namely Penrose's contribution to General Relativity. I have mentioned two big ideas of Penrose, namely Spin Networks and Twistor Theory. The first one is, as far as I know, not directly related to standard relativity, rather to "building" a discrete space-time. It is not entirely unrelated, though, because the core idea is that space-time, the main actor of GR, is an emergent phenomenon. The ultimate goal of spin networks and also of all theories which capitalize on them is to generate a description of the universe which accommodates Quantum Mechanics and at the same time enable the recovery of GR as a limit process.

As for the second theory, Twistors, I am obviously not the right person to speak about them, as they are a quite involved matter, with many ramifications, from multi dimensional complex manifold to sheaf cohomology theory, and a lot more.

But, for this post, I can say this: the core idea is almost childish, and yet absolutely deep. Here it is: Penrose, thinking about Einstein's universe, realized that light lines are fundamentals, not space-time points. Think for simplicity of the projective space: you reverse the order. Rather than lines being made of points, it is points which are the focal intersection of light rays. The set of light rays , endowed with a suitable topology, make up twistor space (it is a complex manifold of even dimension).

Now, according to Penrose, relativity should be done inside Twistor Space, and the normal space-time can be recovered from it using the "points trick" and the Penrose mapping which transforms twistor coordinates into the lorentzian ones. What is more is that twistor space provide some degree of freedom for QM as well. How? well, think of a set of tilting light rays. Rather than a well defined space-time point you will get a "fuzzy point". But here I stop.

Answer 2

I answered about the incompleteness theorem in the other thread. Let's talk about some of his other contributions here. (This list is definitely incomplete^*, but just some stuff off the top of my head.)

1

The "black hole" theorem (incompleteness theorem) is closely related to, yet subtly different from, the Hawking-Penrose Singularity Theorems. The Hawking Penrose theorems again prove the geodesic incompleteness of spacetime under certain cosmologically reasonable assumptions. The difference is in the interpretation. The Penrose theorem proves the genericity of black hole formation; the Hawking-Penrose Theorem guarantees, in some sense, the genericity of the Big Bang.

2

Penrose made significant contributions to how we understand causal geometry of space-times. A particularly interesting paper is Kronheimer and Penrose, "On the structure of causal spaces" (Proc. Camb. Phil. Soc. (1967)). In this paper they abstracted the relation between two space-time events (as being time like or light like) into a partial order. From this one is naturally led to study the ideals and filters, and their principality. This leads to a beautiful description of what the idealized "boundary at infinity" should look like for space-times.

3

The GHP Calculus (named after the authors Geroch, Held, and Penrose of the 1973 paper "A space-time calculus based on pairs of null directions" (Journal of Mathematical Physics)) and the more general Newman-Penrose formalism ((1962) "An Approach to Gravitational Radiation by a Method of Spin Coefficients" (Journal of Mathematical Physics)) are some of the most common ways to perform symbolic computations in GR.

The calculus is a version of the Cartan formalism (or a special way of looking at Ricci rotation coefficients), but taking special advantage of the four dimensionality of space-time and the Lorentzian structure of spacetime.

4

The Penrose inequality is a conjectured (and partially proven in many special cases) relation between the area of the apparent/event horizon of a black hole space-time with the mass (as observed at infinity) of the corresponding black holes.

This inequality actually lead to a lot of interesting recent works in Riemannian geometry.

5

Also, he formulated and named the Strong and Weak Cosmic Censorship Conjectures.

6

Penrose is also responsible for suggesting his namesake process for extracting energy from rotating black holes through backscattering. The process, combined with some putative nonlinear feedback mechanism, gained popular fascination under the martial name of the Black Hole Bomb. In the literature this is called the superradiant instability and has been proven to work in certain linearized matter models around rotating black holes (such as the Klein-Gordon model for massive scalar waves).

An interesting modern mathematical discovery is that the superradiant instability does not apply to massless scalar fields. Understanding how this works for tensor fields, especially for those solving the linearized Einstein equations, is a massive undertaking and crucial in the current effort to demonstrate nonlinear dynamical stability of the Kerr black hole.

7

One way to probe the nonlinear effects of gravity is by understanding how gravitational waves can interact. Our experience from Fourier theory suggests that it can be useful to start with the interaction with plane wave pulses. This was treated first in Khan and Penrose "Scattering of Two Impulsive Gravitational Plane Waves" (Nature, 1971). The impact of this collision still reverberates to this day. (The state of the art, as I understood it, is that we can now understand a bit about what happens when we collide three waves. Four is still somewhat out of reach.)

8

Finally, something a bit more whimsical, since I don't know anyone who actually uses it: the Penrose notation for tensor computations. I tried to use it for a few weeks when I was in graduate school, but gave up mostly because they are impossible to type up.

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^* Pun very much intended.

Answer 3

A very interesting contribution (not directly related to relativity) is joint with Moore on the so-called Moore-Penrose inverse or generalized inverse, which is crucial in inverse problems theory and ill-posed problems.

Answer 4

I would say Penrose is a mathematical physicist and I don't think he can be considered (at least not primarily) to be a pure mathematician. For example, his argument for the Penrose inequality is a plausible but non-rigorous physical argument.

The main contribution of Penrose and Hawking and the one cited was that they showed (roughly speaking) that if one makes some physically reasonable assumptions, the existence of a closed trapped surface implies that the evolving spacetime contains a black hole. This was built on by Schoen and Yau in 1983 who proved that black holes form when matter condenses into a sufficiently small region following on from their work on the proof of the positive mass theorem in 1979 - 1981. Essentially, Schoen and Yau show that if an initial data set is asympotically flat with a large mass density on a large region (large region being suitably defined), there is a closed trapped surface in the initial data.

Theoretical physicists can generally only win the Nobel Prize when there is undeniable evidence from experimental data for the correctness of their theoretical work. The concrete experimental discovery of a supermassive black hole at the center of the Milky Way by the other winners gave a nice cohesive reason to give the award to Penrose as one of the recipients.

Finally, Nobel Prizes are not generally awarded for bodies of work in the way that a Fields Medal is. The Prize is not a statement on the correctness of any of his other work in physics (some of it outside of relativity being very controversial). The Prize was awarded only for his theoretical work predicting black holes as stated in the official citation.