I am absolutely delighted for Roger Penrose. He has contributed so much to mathematical and theoretical physics, it is a surprise to me that he hadn’t won the Nobel Prize sooner.
When I was first getting into mathematical physics twistor theory, originally proposed by Penrose in 1967, was a topic that I had been obsessed about, particularly as it then came to twistor string theory and the amplituhedron. It remains a notebook project for the future, but the great enthusiasm and interest I developed for the twistor programme I think aided my focus in mathematical physics before arriving at university. The geometry of twistors and the twistor description of massless fields, as well as his description of Grassmannian space and Grassmann algebra were things that left me inspired. In hindsight it is nice to think about how, as twistor space is chiral and treats the left and right handed parts of physical fields differently, my current thesis is building on what I think is a wider project involving the chiral string which is not so far removed.
In the world of modern mathematical and theoretical physics it seems one is never too far from a Penrose contribution. It was not too long ago when I was thinking about the Ward-Penrose transform, which has a wonderful relation to the geometry of strings. Then of course there are the beautiful Penrose Tilings. But I don’t think the discussion is complete without also recognising the importance of his views on different scientific and philosophical matters. Whether one agrees or disagrees, Penrose’s views on quantum mechanics and objective collapse, with his theory of dual fields, serve as more than sampling food for thought. Additionally, I’ve personally taken a lot from his views on mathematical Platonism in which he relates the Platonic mathematical to the physical and the physical to the mental, and so on. The topic of a separate essay, there is quite a lot here to unpack and develop in the context of the wider history of mathematical thought.
For the layperson or even the engaged student of physics, The Road to Reality is one popular book that I would not hesitant to recommend. It is a proper masterclass, which goes into many important ideas in modern physics and does not spare on the mathematical details. The stories of Am-tep and Amphos in the prologue followed by the discussion on the roots of science is perhaps one of my favourite introductions to a book.
Having said all that, it is not possible to write a post celebrating the work of Penrose without mention of the singularity theorems for which the Nobel Prize was awarded. He showed that, as a consequence of General Relativity, black holes are a deeply general phenomena predicted to emerge in cases of gravitational collapse irrespective of symmetry. A key concept here is that of a singular manifold and trapped surface as related to geodesic incompleteness.
The proof of the Penrose singularity theorem is quite nice. There isn’t enough space here to detail it in full, but a summary of what the proof looks like might be a nice way to conclude the present entry (should one like to work through the complete proof here is a good text as well as notes and review).
First some definitions. Let us consider some spacetime , which we may think of as being singular if it is not geodesically complete. Now, in a global analysis, let us define to be a globally hyperbolic spacetime in which we denote the Cauchy hypersurface and some future pointing unit normal vector field. For a compact 2-dimensional submanifold with unit normal vector field , the proposition is that is trapped if the expansions and of the null geodesics are negative everywhere on . The initial conditions for the null geodesics are assumed to be and , and they satisfy the null energy condition.
The proof then goes something like this. Let be a global time function such that . The integral curves of are timelike, and they only intersect the hypersurface once and once, where is an open set and defines the chronological future along each point of the compact surface. What we obtain is a continuous injective map with an open image . What is really cool is that if we have a point that chronologically precedes the point , with the future-directed chronological (timelike) curve from to , following a number of definitions and corollaries, it can be shown that should then in some neighbourhood of are images of points in . In the full proof one can then show that there is a contradiction in the intersection of the integral curves of , and since is trapped there exists some such that the null geodesics and orthogonal to satisfy . One can complete the proof from this point, finding that indeed should be a connected globally hyperbolic spacetime, where the Cauchy hypersurface is non-compact and satisfies the null energy condition typically written , then if contains the trapped surface the spacetime is singular. Like I said, it is quite nice!
Before this post grows too long. I would also like to leave a special note congratulating Reinhard Genzel and Andrea Ghez in sharing the Nobel Prize in Physics. I cannot profess to have read their papers, but a review of their history and extensive work shows that for decades they have made significant contributions to long-term scientific research focusing on Sagitarius A*! The astronomers in the blogosphere will be able to speak more to the methods both Genzel and Ghez have developed, as well as the technicalities of the brilliant techniques they have prioneered along the way to providing the most convincing evidence to date of the presence of a supermassive black hole at the centre of our very own galaxy! For myself, I look forward to reading a few of their respective papers. Meanwhile, here is a 2017 article discussing some of their research.