“Being awarded the Nobel Prize was clearly the most important one, but let me mention some of the people who have impacted me most. Firstly, I have to say my father was extremely important. He used to talk to me about science a lot, such as when we went for walks. Biology was his area, but he was also thrilled by mathematics and used it in his work. I remember telling him that we were going to learn calculus at school, and he immediately taught me calculus! He did the same for my older brother too. It was the thrill of the subject he felt – he wanted to be the one to tell me, and that made a real impression, probably more than what he actually said.

We also explored geometry. We were in Canada during the war years, and I’d noticed a pattern of regular hexagons on a tiled sink. I asked my father if it would be possible for the whole world to be covered by these hexagons. He replied that it wasn’t, but it could be done with pentagons, by which he meant a regular dodecahedron. This was part of my geometrical education as I grew up!

I also learnt a lot from my older brother, who was precocious and 2 years ahead of his class. I was always extremely slow. In fact, in my first year in Canada, my teacher moved me down to a lower grade class. She used to issue mental arithmetic tests, which were too quick for me. A year or two later, another teacher realised that I was just slow, not actually stupid, so gave me as much time as I needed for maths tests. I’d watch people playing outside through the window, whilst I was still struggling through my test! I remained slow but managed to speed up for my later exams. Back in England, as a third-year undergraduate in Mathematics at University College, I specialised in two geometrical subjects. I could see how to do the geometrical problems, but explaining them in words made me slow and I didn’t complete all the papers. In fact, my best paper was in algebra, which didn’t require the same sort of approach.”

“I had been working on something else which used a similar technique. There was a lot of interest because quasars had been observed, remarkable radio signals coming from outer space. The amount of energy being emitted from them was more than an entire galaxy, yet the variation was such that they couldn’t be bigger than the solar system. If they were so small but emitting such energy, people reasoned that they must be about the scale of what we now call a black hole. However, it was not understood by scientists at the time. I had gained my doctorate by then, elected a Fellow at St John’s College Cambridge, and had done some work with John Wheeler at Princeton, who was very interested in fundamental physics, general relativity in particular, and the question of the singularities. People thought they shouldn’t be there in general, and I think that’s what Einstein would have thought too. I never met Einstein. However, my close friend and colleague, the British physicist Dennis Sciama did. Dennis was a great influence on me and kept trying to convert me to cosmology.

Wheeler was really interested in what happened in gravitational collapse. The term ‘black hole’ apparently came from Wheeler, although it was in fairly common usage at the time. The question of the singularities was not clear, particularly because there was a paper written by two Russians, Liftshitz and Khalatnikov (1963), who seemed to have proved that in a general situation you would not get singularities and a black hole would not form. I became intrigued by the problem because of the quasar interest; it was topical. I looked at the Liftshitz and Khalatnikov paper and was not convinced by the techniques they were using. There was a serious error in the paper, which I didn’t notice, but was later corrected after I’d published my result. I’d developed a technique of squashing infinity down, until it became a finite place, just like anywhere else. I wrote a paper for the Royal Society (Penrose, 1965), where, as an afterthought, I realised that to prove this one thing, I had to assume that infinity was like a sphere. The family of spatial directions at point in space forms a sphere, but when you get to infinity, does this family still stay a sphere, or could it have some more complicated topology? I wrote an appendix to the paper, where I had a proof that it had to be a sphere. Looking at the boundaries of futures, you have a point or a region in space-time and you look at what this can reach. What does the boundary of it look like? This was a very crucial part of the argument I used for the appendix in this paper. I began to think that maybe this technique could be used for black holes but I had no way of characterising what a black hole was in general. I remember walking with a friend, Ivor Robinson, near Birkbeck College in London. I had this sudden feeling of elation crossing a street, and it was the idea that I subsequently referred to as a ‘trapped surface’. This was a characterisation of a gravitational collapse that had gone too far and that the boundary of its future had to be compact, which is contradictory to the initial non-compact state. I went back to my office at Birkbeck and sketched out an argument for it. I got a curious reaction from people. People didn’t take it that I was showing that black holes were there; there was real suspicion and it took a while before they were accepted.”

“How to deal with the singularities was considered to be a quantum gravity problem, involving quantum mechanics. However, let’s turn this around and think about how gravity affects quantum mechanics. Many people think that quantum mechanics are about small things, and general relativity is about big things. They would argue that since big things are made out of small things, this makes quantum mechanics more fundamental than general relativity. This is a commonly-held belief, meaning that general relativity has to be brought into the fold of quantum mechanics. In particular, it is commonly believed that the classical singular state of the Big Bang is to be understood by replacing the classical picture by a quantum one. Dennis Sciama was a great proponent of this idea. I had always greatly respected him for publicly disowning his own earlier strong support for the steady-state cosmology, in which there would be no Big Bang at all, but when the observations turned against that theory, he publicly refuted his own earlier beliefs. He then turned his attention to the study of the Big Bang as a manifestation of a quantum-gravity effect. However, I don’t think that quantum gravity works here either, since the Big Bang has a completely different structure from the singularities in black holes.

For a long time, I believed that the resolution of this issue lay in a phenomenon known as the ‘collapse of wave function’. This has to do with what is called the ‘measurement problem’ of quantum mechanics. Is it the conscious observer looking at the state that makes it collapse? I couldn’t make sense of that, so I looked at it the other way round. It’s gravity that makes it collapse. There are good reasons for believing that the general principles of relativity and quantum mechanics are incompatible. I think that’s because gravity is influencing quantum mechanics and the collapse is a gravitational effect. It’s not due to consciousness. We do not know what the physics is, and how the collapse of the wave function takes place. There are new experiments which should be done, but need money.”

“I want to keep thinking about the unanswered questions I’ve referred to, and I’m not giving up. I’m writing two books at the moment. Another ambition would be to connect my notes and write a book about the collapse in wave theory. A young graduate in the US is also going to collaborate with me on a further book about the interplay between the ideas of Dutch graphic artist M.C. Escher and mathematics.”