A Passion for Patterns, Puzzles and Physics

Nicos Starreveld, Raf Bocklandt Interview with Sir Roger Penrose NAW 5/19 nr. 4 december 2018 241

Nicos Starreveld Raf Bocklandt Korteweg-de Vries Institute Korteweg-de Vries Institute University of Amsterdam University of Amsterdam [email protected] [email protected]

Interview Sir Roger Penrose A passion for patterns, puzzles and physics

Sir Roger Penrose is a British mathematical physicist, mathematician and philosopher of Cambridge University). The last talk was science. He is Emeritus Rouse Ball Professor of mathematics at the University of Oxford. about the steady-state theory. In this theo- In July he visited Utrecht for the ‘19th UK and European Meeting on the Foundations of ry he described how the universe remains Physics’ and on this occasion Nicos Starreveld and Raf Bocklandt, editors of this journal, the same and how galaxies disappear had an interview with him about life, the universe and everything. when they reach the speed of light. I did not understand that, how could they disap- Penrose was born in 1931 in Colchester, ten times between 1958 and 1969. pear like that? They fade away maybe, so Essex, England. From a young age he got “I was at school, about fifteen years I went to Cambridge to discuss it with my familiar with mathematics and physics. old, and I had to decide which subjects brother Oliver. His father, Lionel Penrose was a medical I wanted to choose for my final two We were having lunch together and he geneticist, he was a professor of eugen- years. When the headmaster asked me, said ‘I am no expert in that, but a good ics at University College London. He also I told him I wanted to do biology, math- friend of mine is sitting over there, he will had a keen interest into mathematics and ematics and chemistry. He told me it was tell you, he is an expert in cosmology.’ That chess, interests that he transmitted to his not possible to do mathematics and biolo- was Dennis Sciama, so I described this to sons. Penrose’s mother, Margaret Leathes gy together. I then said mathematics, phys- him and he had never seen this argument, was also medically trained, but she did not ics and chemistry. My parents were very he was impressed enough. When I went to practice medicine. annoyed when I got home, they wanted me Cambridge to do research in pure mathe- “My father had a deep interest in math- to become a doctor.” matics we had discussions about cosmol- ematics, we had long walks talking about ogy and physics, he knew everything that nature and mathematics. The physics side Instead Penrose started undergraduate was going on. And this was wonderful ed- came more from my brother Oliver who studies in mathematics at University Col- ucation for me.” was very well read. He used to read to me lege London, where he graduated with a the book Mr. Tompkins in Wonderland.” first class degree. Afterwards he moved to The discussions with cosmologist Den- (Mr. Tompkins is a fictional character Cambridge to do research with Prof. W. V. D. nis Sciama inspired Penrose to make the created by the physicist George Gamow. Hodge, who was a professor in geometry switch to cosmology; he started doing re- In this book he explores a dream where and astronomy. In 1958 he completed his search in geometry and general relativity. the speed of light and the Planck constant PhD, with his doctoral thesis Tensor Meth- In the years that followed, while he held a are different so that you can see relativistic ods in Algebraic Geometry, written under position at the department of mathemat- and quantum effects with your own eyes.) the supervision of Prof. John Todd. ics at Birkbeck college in London, Penrose developed novel mathematical methods Penrose’s older brother Oliver Penrose Moving to physics in general relativity. He introduced the became a theoretical physicist and was a Although mathematics was what he loved calculus of spinors and showed that this Professor of Mathematics at Heriot-Watt the most, physics never ceased to excite. method, when compared to the classical University in Edinburgh from 1986 until his “I was interested in physics all the time. method of tensors, yields simpler equa- retirement in 1994. His younger brother, The thing that set me off in that direction tions when solving Einstein’s equations. Jonathan Penrose, is a chess grandmaster was a series of radio talks by Fred Hoyle This path led Penrose to a much deeper who won the British Chess Championship (British astronomer and cosmologist in understanding of the laws of the universe. 242 NAW 5/19 nr. 4 december 2018 Interview with Sir Roger Penrose Nicos Starreveld, Raf Bocklandt Photo: Wikimedia Commons, Biswarup Ganguly Commons, Wikimedia Photo: Sir Roger Penrose

In 1964 Penrose wrote the epoch-making Then we got to a road where we had space-time. Together, Hawking and Penrose paper where he stated the First Singular- to stop talking because of the traffic. As I proved the gravitational singularity theo- ity Theorem. In this paper he introduced crossed the road, an idea came to me but rems and made the theoretical prediction the concept of a ‘trapped surface’ to char- on the other side he started talking to me that black holes emit radiation, called Hawk- acterize a gravitational collapse that has again and the idea locked up into my head. ing radiation. The first meeting between reached a point of no return. Gravitational Later that day he went away and I was Hawking and Penrose is also portrayed in collapse describes the contraction of a thinking ‘why do I feel this state of ela- ‘The Theory of Everything’ but he warns us massive astronomical object, under the tion?’ I felt happy about something, but I to take that movie with a grain of salt. influence of its own gravity, towards its didn’t know what it was. So I went through “Don’t believe the film! The film has center of gravity. Gravitational collapse is the day: what kind of breakfast I had,... all Stephen going to London to hear me giv- the process that gives birth to black holes! things that had happened, one after the ing a lecture. It shows Stephen listening to The ‘trapped surface’ describes the region other. it and afterwards coming at me, saying he around a black hole where light is ‘trapped’ Then I remembered crossing the street was heavily inspired by this lecture. and can’t escape out of it any more. The and suddenly it came back. I realized what I did give a talk in London, but Stephen formation of black holes as a result of I thought of: a characterization of the col- Hawking was not there, only Dennis Scia- gravitational collapse is described by Ein- lapse in a way that has no symmetries at all. ma was. This talk was about the singular- stein’s general relativity, Penrose’s First Sin- It’s basically light rays converging all around ity in the black hole collapse using topol- gularity Theorem states that this process on a sphere, so you have a topological con- ogy. So Dennis asked me to give a repeat will inevitably lead to the formation of a dition. It was just a rough idea: I didn’t have talk in Cambridge, which I did and then singularity. Penrose still remembers the a complete working argument, it was still I did meet Stephen. We discussed things precise moment these ideas came to him. slightly uncomfortable in some places, but and this started him off on this kind of “I was at Birkbeck College at that time the main elements were all there.” thinking.” and a friend of mine, Ivo Robinson, was visiting me. He was somebody who liked At the same time Penrose’s work on Art and Escher talking, he spoke very elegantly and was space-time singularities was inspiring Apart from his work in mathematics and very nice to listen to. He was talking, talk- Dennis Sciama’s brilliant student, Steven physics, Penrose is also known for some ing, talking to me while we were walking Hawking, who wrote the magnificent essay artistic creations as well. His Penrose tiles along the street. entitled Singularities and the Geometry of adorn the floors and walls of many build- Nicos Starreveld, Raf Bocklandt Interview with Sir Roger Penrose NAW 5/19 nr. 4 december 2018 243

ings all over the world and the famous was and what journal we would submit it impossible triangle was first described in to. My father knew the editor of the British an psychology article by himself and his Journal of Psychology and he said: ‘Well, father. Moreover they both corresponded we’ll call it psychology and we’ll send it to with M. C. Escher and inspired some of his that Journal.’ They accepted it and in this artwork. paper we gave credit to Escher and to the “In 1954, I was in my second year as catalog of the museum. a graduate student in Cambridge doing My father sent a copy of the article to Es- mathematics and the International Con- cher — I don’t know if he had this address I gress of Mathematicians that year was to think it went through somebody else. Any be held in Amsterdam. At one point during way, it did find his way to Escher and he the conference, I was getting onto a bus started to communicate with my father.” and somebody else was just getting off. An example of a Penrose tiling This was one of the lecturers at Cam- Later Penrose also had the opportunity bridge, Sean Wylie. He taught topology, to meet with Escher in person. unique and what is the plan underlying which I liked a lot. He was holding a cat- “I was very bad at correspondence but it?’ So I showed him the plan: you match alog with the picture Night and Day, the once I was driving in Holland with my wife this against, this against this, et cetera. one with the birds flying in both direc- and we passed Baarn, where Escher lived. You can have other designs like this and I tions. I asked: ‘What is that?’ He replied: I was given his phone number from my fa- drew some bird shape. I said it’s not very ‘Oh, there is this exhibition with art from ther’s letters. I called Escher and he was very good I’m sure you could do far better than an artist called Escher and I think you will friendly and said: ‘Come and have tea.’ this. be interested.’ We went there and he was very gener- So he wrote back to me and he showed I had never heard of him but I went to the ous and pleasant. There was this long table me the picture with the ghosts. He was Van Gogh Museum and there were these pic- almost completely empty except some piles not all together pleased with it. I think tures all around the room. Absolutely stun- of prints at the end. He said: ‘This is where he thought he could have done better but ning, I was blown of my feet. It was quite a I work.’ I was expecting staircases going it was probably the one of last ones he big room, and several of them stuck in my out of the windows in all directions, but made.” mind. Particularly the one called Relativity, no it was just a very ordinary house. Then with the stairs in different directions. he said: ‘Look I have this pile of prints. Of This puzzle would later evolve into the I came away from that thinking: ‘Gosh, these ones I’ve not got very many left, so famous Penrose tilings that tile the entire this is amazing. I’d like to try something I can’t give you one of them. But from that plane in a nonperiodic way. Surely Escher myself.’ Something new, just an idea may- pile over there you can take one.’ So, I took would have loved this but unfortunately he be that I hadn’t seen. So I played around the pile and went through it, some of them passed away before he could play around with rivers and bridges, roads going in dif- were quite familiar to me, but then there with it. ferent directions, an impossible situation. was one I hadn’t seen before. “I would say I was too slow to find I simplified it to this triangle, a tripod trian- I said to him ‘This one’, and he replied: these things. It was only just shortly after gle. When I came home and I showed it to ‘That’s very interesting, usually people he died. My father died around the same my father and he asked: ‘What’s that?’ He don’t appreciate that picture.’ It’s called time and both of them would have got a also got caught by this and he produced Fishes and Scales. It’s a square but slightly lot out of them.” some buildings which were inconsistent rounded corners and you have a big fish Another set of pictures by Escher that and then he made an impossible staircase. on the bottom left. Its scales soon begin to are very interesting to mathematicians are We decided we like to write a paper on become other fish and if you follow them his tilings of the hyperbolic plane. The hy- it, but we didn’t know what the subject then you end up at another big fish. Its perbolic plane is an abstract two-dimen- scales are fish too and as you go back they sional space that behaves in almost all become the fish you started with.” respects as the ordinary Euclidean plane except that you can draw more than one That was not the end of Penrose’s rela- line through a point parallel to a given tionship with Escher. Later he would also point. It is impossible to embed the hy- be the source of inspiration of one of his perbolic plane into ordinary Euclidean last paintings. space in such a way that it preserves all “When I visited him he gave me this print distances, but there are other interesting and all I could do was give him some card- depictions of it. In the Poincaré disc model board pieces of a puzzle in return. These all the points of the hyperbolic plane are were all identical shapes which would tile mapped into a disk and all lines in the a plane but in a way which required twelve hyperbolic plane become circle segments different orientations before it repeated. that stand perpendicular on the bounda- He then wrote to me to say: ‘Look, ry of the disk. This model has the special The impossible triangle I found out how it’s done. Is this really property that it is conformal, which means 244 NAW 5/19 nr. 4 december 2018 Interview with Sir Roger Penrose Nicos Starreveld, Raf Bocklandt

that it preserves angles: if two lines make He had found all these models together and we see just after the big bang and the cy- a certain angle in the hyperbolic plane he had figured out the relationship between cle can start all over again. the corresponding circle segments in the them. This was very cleverly done but other “One of the things that deeply troubled Poincaré disk will make the same angle. people didn’t know about it. Poincaré did me for many many decades is the second Escher used this model for his ‘Circle limit’ other things with it which are important, but law of thermodynamics. Excuse me when I paintings. Although the model preserves Beltrami had put the whole thing in a much call that the mammoth in the room. People angles it does not preserve size and this better geometrical picture. say the elephant in the room but it is even is why the fishes, angles and demons The conformal model is the most beau- bigger. The second law of thermodynamics get smaller and smaller as they approach tiful one and that is the one Coxeter sug- tells us that things get more random so to the limit circle, while in the hyperbolic gested Escher should use. And that he did, speak: entropy is increasing. Saying exact- plane all these creatures have exactly the he made an extraordinarily precise picture. ly the same thing the other way around, same size. Right up to the edge you have these tiny back in the past things get less and less “These pictures came about as a result figures and they’re still very correct.” random. of the same conference in Amsterdam. Now go back until you see the earliest Another mathematician at this conference Conformal cosmology things we can observe, the Cosmic Micro- was Coxeter. He was a very distinguished Penrose’s fascination for conformal sym- wave Background. If you look at it, one geometer and he was very impressed by metries also shines through in his work of the most striking features of that is its Escher. He said to him: ‘Look there you on cosmology. He developed a new theo- thermal spectrum. If you compare it to the could do very good things with hyperbol- ry in which the universe goes through an spectral curve of a radiating black body, ic geometry, and particularly the Beltrami endless sequence of cycles, that each start it fits absolutely beautifully. It tells you model.’ You have to be careful about this, with a big bang. But unlike other theories that you have a state of maximum entropy. because most people call it the Poincaré in which each cycle ends with a crunch, Back in time the entropy gets smaller but disk. But the history of it is not quite so Penrose theory predicts that the universe you get to a maximum. straightforward. Poincaré did rediscover it keeps on expanding and expanding. But That’s strange... but it’s not a paradox but originally it was discovered by Beltra- as the last stars and galaxies fade away because you have not taken gravity into mi, an Italian geometer. and all the black holes evaporate, it be- consideration. Gravity tends to work the Beltrami had discovered the conformal comes conformally symmetric. This means other way: entropy goes up when things representation of the hyperbolic plane, that there is no difference between big and clump whereas for ordinary gas in a box which is the Poincaré disk, the projective small, only angles are preserved. From a entropy goes up as it spreads out. Gravity representation, which is called the Klein rep- conformal point of view this vast radiation being universally attractive means that you resentation, and the half plane representa- filled universe looks exactly the same as have a uniform distribution with low entro- tion, which is called the Poincaré half plane. the tiny concentrated energetic soup that py and then the entropy goes up when it condenses. Well that is why we see galaxies, stars, our Sun...”

But this does not explain why the universe started off in this thermal equilibri- um. The standard way cosmologists solve this puzzle is by introducing inflation. This theory proposes that just after the big bang the universe went through a stage of very rapid expansion, in which it grew exponentially by a factor of 1050. So everything we see around us was so close together that before inflation started it all had the same temperature and therefore the cosmic background radiation looks uniform in all directions. But Penrose is not convinced. “Inflation is just not a solution! The eas- iest way to think of it is when you reverse time. Imagine your universe and as it con- tracts, the irregularities will increase and then they will produce black holes. You can put the inflaton field in as much as you like, it makes no difference. That irregular start will not be smoothened by inflation. It’s like a fractal: if you stretch it out it gets Coxeter’s tiling of the hyperbolic plane that was an inspiration for Escher’s ‘Circle limits’ paintings worse.” Nicos Starreveld, Raf Bocklandt Interview with Sir Roger Penrose NAW 5/19 nr. 4 december 2018 245

Instead Penrose came up with a different way to solve the problem. “Some years ago, I had this idea of conformal cyclic cosmology. In this model you have a universe where the Big Bang is the conformal continuation of the ex- ponential expansion caused by Einstein’s cosmological constant. The cosmological constant ensures that the universe will ex- pand exponentially in the far future. In a way it does the same as inflation but at the wrong end of the universe: not at the beginning but at the end. However, if you have a conformal picture and you look at our Big Bang conformally, you can stretch it out and it looks very similar to the other end of the universe. So my contention is that they actually match and that our Big Bang was the conformal continuation of the remote future of a previous universe.”

Penrose proposal has a strange conse- quence. If the universe goes through con- Schematic description of conformal cyclic cosmology, as appeared in Penrose’s article ‘The big bang and its dark-matter content: whence, whither, and wherefore’, publiched in Foundations of Physics 48(10), October 2018, 1177–1190. secutive cycles the second law of thermodynamics tells us the entropy should be increasing all the time. Yet in his model not interested in those degrees of freedom Letters. In it we identify twenty points in each cycle the universe seems to start with anymore. The second law starts with this the sky where there is a source of a huge the same entropy. How does this work? new renormalized entropy and that’s the amount of energy. We call these Hawking “You mean why you don’t have entro- argument. It’s a bit subtle, but I think it’s points. Although we’re little cautious to say py increasing, increasing, increasing,... consistent: you can live with the second what they are, we argue that they are rem- with each cycle? The reason is as follows. law, as well as with cyclic cosmology.” nants of a previous cosmological cycle.” Where is the most entropy in our universe now? Easily in supermassive black holes. But is all this just pure speculation or So even at the age of 86 Penrose is still What happens to those supermassive does he have evidence for his theory? actively exploring and trying to understand black holes? Well they evaporate away and “Just a few days ago, I was in Poland the cosmos. Whether his proposal of cyclic with them some of the degrees of free- visiting my colleague Krzysztof Meissner. conformal cosmology will work out, time dom in the universe get lost. To estimate He and another astronomer, Daniel An, will tell. Lot’s of time, perhaps even 10100 the entropy, you have to have a phase had done an analysis based on my idea of years... s space. But if you lose degrees of freedom conformal cyclic cosmology and they found this means your phase space collapses evidence for this which seems to be ex- down. tremely significant. Event 22 January 2019 You say, okay that black hole has swal- They were seeing little points in the sky In January 2019 Sir Prof. Roger Penrose lowed all those degrees of freedom. It has which seemed to emit huge amounts of will be in the Netherlands for the pre- disappeared, forget it. But up to that point energy. When Meissner mentioned this, I sentation of the Dutch translation of I had been counting those degrees of free- thought: ‘Yeah, I think I know what that is.’ his monumental book The Road to dom in my calculation of entropy. When In the previous eon before the Big Bang Reality: A Complete Guide to the Laws that black hole is gone I change my mind you will have supermassive black holes of the Universe. On 22 January the In- about what I regard as the phase space. and these black holes will after a while stitute of Advanced Study of the Uni- I say okay, with this gone, my measure of start to radiate away by Hawking radiation. versity of Amsterdam will host a public entropy has come down. It’s not that it has This is after a very long time, when the event where Penrose will discuss the violated the second law. It’s just that I have universe has gotten much colder. You have three world model. The three world 100 a new renormalized definition of entropy, to wait 10 years, an enormous time, but model was proposed by Penrose as a which does not include those degrees of then they start to radiate and the entire means to understand how the physi- freedom. That new quantity, which I now energy of these black holes is concentrated cal, mental and mathematical worlds call the entropy, is smaller. into one point in the sky. are connected. The book will be pub- Once you’ve lost all the black holes, the We have just written a paper about this, lished by Merlin Uitgaven. entropy value comes down because we are which we have sent to Physical Review