Spacetime Singularities - Roger Penrose, Dennis Lehmkuhl and Melvyn Bragg

Good evening. My name is and I go, really and I'm in charge of external relations for the Mathematical Institute tonight for the Oxford Mathematics public lectures. We have a very special event. But before we proceed. A special thanks to a sponsor market because markets are leading quantitative driven electronic market maker with offices in London, Singapore and New York. The ongoing support is crucial in providing you with quality content.

No, it is usual to say that a particularly prestigious speaker does not need an introduction. This is especially true for Roger Penrose, code winner of the 2020 Nobel prise in physics, for his work on black holes. Tonight, we will join the celebration and explore its work and life in details. The event will have three parts revolving around Roger black holes and singularities. As Sherlock Holmes once said, Singularity is almost invariably a clue, and much of Roger's work follow that clue.

First, we will hear from Dennis Lam from the University of Bonn. It will introduce the general concept of spacetime singularities and black holes and Chris Rogers work in a story called Context Segun. We will hear directly from Roger about his own work. As you probably know, Roger is also an extremely talented public speaker, and I've always enjoyed his lectures. Finally, the lectures will be followed by an interview with Roger by Melvyn Bragg men.

Melvin Bright does not need an introduction, either. As you probably have listened to his wonderful BBC radio programme in old time. On a personal note, I've had the great pleasure and privilege to interact with Roger the Mathematical Institute. I would like to echo the voices of many in the institute and say that, of course, I've always been impressed by his brilliant intellect, but also by swarms as a human being.

So one more time congratulation Roger, and a very much look forward to tonight's event. Denis, please start. No. Space, some similarities amongst the most fascinating ideas in the history of science. They are also amongst the most puzzling. What are they are they objects that could actually exist in nature, or are they artefacts of our mathematical reasoning? And how are they related to black holes?

At different points in the history of general relativity, ideal for short, different answers to these questions have been given. Major milestone of this story was Roger Penrose, a singularity theorem of 1965, which changed our entire thinking about space, time singularities and indeed our thinking about air more generally.

The purpose of this lecture will be to put this theorem into its historical context and to prepare you for selecting his own lecture on how he found the theorem and on how his thoughts on the topic have developed since then. Everything starts with three questions. The first question is, how does gravity work? And in 1915, Albert Einstein gave an entirely new answer to this question.

In 1920, Arthur Eddington answered the question how to start work and the two questions together, as they developed over the next decades, brought about a third question which was answered and the work of Roger Penrose does the gravitational collapse of stars lead to singularities in spacetime? Now, how did all that happen? Here's a. Snapshot of Rogers paper from 1965, which bears the title gravitational collapse and space, something it urges.

In the next half hour, I will try to show you how this phenomenon became related. And here are a few sentences from the introduction Do not worry, if not, everything becomes clear immediately. It's the purpose of this lecture to help with it. So it goes most exact calculations concerned with the implications of gravitational collapse have employed the simplifying assumption of spherical symmetry.

The general situation with regard to factory symmetric body is well known when sufficient thermal energy has been radiated away. The body contracts and continues to contract until a physical singularity is encountered at the centre of the body.

As measured by local court moving observers, the body passes within its transfer radius, the question has been raised whether this singularity is in fact simply a property of the high cemetery assumed it will be shown that deviations from Straker symmetry cannot prevent space time singularities from arising.

And what we're going to do in the next half hour is to essentially introduce the core concepts of these claims and to make it visible how much of a game changer the result at the end really was. So we're going to talk about the notion of gravitational collapse and how it's related to spherical symmetry, which was a major assumption in earlier work on gravitational collapse by Oppenheimer and Snider, criticised by Wheeler and referred to by Penrose in this footnote number two.

We're going to discuss about different notions of singularities, clarify what a sponsored radiuses connected to the so-called sponsored singularity. And we're going to put into a wider context why it was such a game changing result. A deviations from sorry symmetry cannot prevent spacetime singularities from arising.

Now to start with here is the general structure of panels, this thing clarity here and indeed Penrose is theorem served as a blueprint for all later singularity theorems and completely changed how physics thinks of what a singularity actually is, and whether accepting G.R. means that you have to accept the existence of singularities in spacetime. He has the structure of the 1965 theorem in particular, and I'm going to argue that almost every part of it.

What came before it on its head is the first assumption which says that local energy is not negative and we're going to see this if this is a far more general assumption than any assumption of collapsing matter, that it can be flat. We're going to talk in detail about the newly introduced concept of a track surface and why it really changed so much in the history of China. And even the conclusion. The conclusion that incomplete duties is.

We'll follow from these conditions and bring about a singularity in space time was new because it was a new way of thinking, but what a singularity really is. Yes, we're going to see. But to appreciate how radical these changes were, how much of a game changer the first singularity theorem actually was. We have to understand what came before it. We have to understand how we got from Einstein's derivation of the perihelion of mercury to that of sponsored singularity.

From there to had a mass catastrophe and ancient argument that the catastrophe would never, ever happen. And we have to understand Oppenheimer's and Snider's thoughts on the gravitational collapse of a star before we can understand the significance of Penrose as newly introduced concept of a trapped surface in spacetime. And what follows from it? Now we're going to structure the whole thing by looking at the three questions I mentioned at the beginning.

First, how gravity works, how stars work and whether they might be able to collapse. And finally, this will point us to the question of what is space time singularity is and whether it's linked to gravitational collapse and the existence of black holes. OK, first, the question how does gravity work?

The whole thing starts in 1915, when Albert Einstein introduced, what would you take while the Einstein fielded questions, the new gravitational law that was supposed to replace Newton's gravitational law from hundreds of years before. The equation links the gravitational field on the left hand side associated to the curvature of space time and the mass energy of matter on the right hand side.

Essentially, the equation says that the more mass and energy you haven't given region of space time and depending on what form and shape that mass and energy has, will create a particular gravitational field, the particular curvature of time in that region.

Now you have the two equations again. The full ancient equations with mass energy of matter as a source and the special case, when no matter is present and when you have a pure gravitation of fear, which is described by solutions to this equation. In fact, every solution to these equations represents a universe or a part of the universe that is possible, according to Jia.

And in 1915, in the same month in which he found defeated questions, Einstein set out to describe a particularly important part of our universe, namely, he wanted to drive the path of Mercury around the Sun. That's a problem that has caused some trouble for Newtonian gravitational theory. And Einstein thought that his newly introduced theory of gravity should be able to do better.

And in setting about to solve this problem, he took the linear sized vacuum equations the nearest because he knew that the gravitational field of the Sun was comparatively weak. And so he could use the linear equations. He was also only interested in the exterior gravitational field of the Sun, not in the Sun itself. And so he took a solution to the vacuum equations. The Sun itself did not need to be modelled in order to know how its gravitational field influences matter.

This is going to be important. He further made three assumptions about the gravitational field that the Sun would produce. First, he assumed that the field would be strictly symmetric because the Sun itself is approximately spherical symmetry. Second, the gravitational field would be static because the sun is approximately static doesn't change very much, at least as far as Mercury is concerned. And third, that the gravitational field would be some trophic effect.

But far away from the Sun, the curvature of space would tend towards flatness because the gravitational field becomes weak. Now, Einstein did not really expect that an exact solution to his complicated equations would be possible, and the approximate solution was good enough to give a precise calculation of Mercury's path. But only a few months later, co-sponsored astronomer. And the exact counterpart to Einstein's 1915 approximate solution.

And you see the solution down here, every solution of the ancient equations gives an interval that is formed by the metric tensor, which gives the distance between any two points in spacetime. And transit solution had two surprising properties. Well, at least one surprising property, it had a single entry in the centre. And that Centro Singularity was not quite unexpected.

This fairly symmetric solution to Newton's gravitation equations had a singularity at the centre to the point where the gravitational field, while the components of the metric tensor tend towards infinity. Schwartzman and Einstein interpreted this singularity as a placeholder for the Sun that had not been included in the model. But there was also a second singularity, and that came as a bit of a surprise. Plus, it had a ring like shape. So you see, here's is the centrist singularity.

She has the ring shaped second singularity that has no counterpart in Newtonian gravitational theory. Both ancient and fashion were a bit puzzled about this, but I it eventually convinced the two of them that it wouldn't really be a problem because for any realistic star, that ring shaped singularity would be inside of the star, not really accessible to any possible measurements. So there was no real problem without what I would call the centrist energy.

It was the so-called sponsored singularity ActionScript, but some by others so-called with a drink with its ring shape that puzzled everybody. And discussions about the nature of this structured singularity came to a head in April 1922, when Einstein gave a series of lecture at the collection of class in Paris. So you see him here discussing something that looks pretty similar to what I just said in the previous slide.

Turing's that might well be one of them. The outer surface of the Sun and the second the swan should singularity with its ring shape. Now, at this conference on the 5th of April 1922, Jack Harman's great French mathematician asked Einstein what if a star were to be so massive that it sponsored Singularity ended up outside of the star announced and immediately answered. This would be a catastrophe for the theory.

Now, only two days later, Einstein came back and said, well, had a mass catastrophe would never, ever happen. Calculated that before a star could accumulate sufficient mass to be entirely contained within its own structure to reduce the pressure at the centre of the star would become infinite under such conditions. He said clocks would not run and thus time would stop preventing the drama catastrophe from ever developing since nothing develops without time.

Now, why would it have been a catastrophe in the first place? Einstein was fine with introducing singularities to stand in for matter. Some. As long as singularities could be interpreted as a placeholder for matter, rather than something that actually exists. Everything was fine, fine. That's why he did not worry about the central singularity in the transport solution.

But singularities were not supposed to actually appear in nature, and that's what was at stake in the debate with Hatami Harriman. Asked if a star could be so heavy that its fractured singularity would be outside of the star and actually appear an empty space? Now, discussion during the 1920s kept focussing on the nature of the sponsored singularity and how it should be interpreted. Here's a quote from Arthur Eddington from 1920.

He said about the structured singularity there is a magic circle which no measurement can bring us inside. It is not unnatural that we should picture something obstructing our close up post and say that a particle of matter is filling up the interior. So for Eddington, it seemed like a barrier and other names introduced in the 1920s, there was a real cottage industry introducing ever new names for this wretched singularity when in the same direction.

Schwarzer himself called it the discontinuity caller called it the barrier while quoted sometimes the sphere, sometimes the gravitational reduced pricing pressure quarter to the hole in the world. Quite prophetic, given that as we would see, the structured singularity will eventually be related to the event horizons of black holes.

Clearly encoded the singular frontier, the catastrophic frontier, the limits circle, and not one the person who took the notes at the Paris meeting simply quoted the death. Now, I have not seen anybody comment on this, but it's a bit surprising, it's a bit ironic that the person who introduced to discover the sponsored solution and the sponsored singularity that would eventually be called would eventually be related to the event horizon of black holes.

Was cars fashioned who if you translate his name into English, the name would mean black shield. Accident of history. All right. So much for now about the development of the theory of gravity in the 1910s and the 1920s. Now let's turn to our second question. How do stars work? And first answer to this question. This promising answer to this question was given by Arthur Haddington. In 1920, Eddington suggested that stars shine because of the energy produced in the nuclear fusion of hydrogen.

The resulting outward radiation. The resulting outward radiation pressure. Is what keeps the staff from gravitationally collapsing? Because if there was no pressure to the outside, then an object as heavy as a star would simply collapse under its own gravitational attraction under the gravitational attraction of its parts.

But this immediately created the question What would happen when the star runs out of hydrogen to burn and thus out of the fuel that makes the gravity radiation pressure to the answer happen? Then only the inward gravitational attraction would remain. Would the star collapse? And if so, would he find a new equilibrium state become a smaller star, less bright but still stable?

Well, in 1935, Chan Rasika calculated for the special case of a neutron star kind of star that in principle could become the smallest possible star. That's if a neutron star is heavier than a particular limit. There will be no new equilibrium state. This suggested that the star will just continue collapsing to ever smaller regions of space time. This line of thought was further strengthened by a calculation of Oppenheimer and Snider in 1939,

who quote that paper on continued gravitational contraction. Oppenheim and Snider constructed an explicit it's symmetric model of a collapsing star. They predicted it collapsed collapse continuously without ever achieving a new equilibrium condition. Collapsed within the ring formed by its fractured singularity. They also predicted that once this happened, what lies within the boundary formed by the structured singularity would not be accessible to an outside observer anymore,

and the outside would not be accessible to the observer. Within the boundary formed by the structured singularity ensured they predicted a collapsing star to turn into an object that would soon be called a black hole. And they predicted that as a result of this collapse, the central singularity of the Schwartzel solution would form. So if Oppenheimer and Stein are correct, then what Einstein and Rothschild thought of as a placeholder for matter could actually come about in nature?

Now, did Oppenheimer and Snider falsify Einstein's reasoning of 1922? I think not, because Oppenheimer and Snider make idealising assumptions that the pressure inside the star vanishes. Westphal Einstein the assumption of a non vanishing pressure played a major role in his argument.

Thus, Oppenheimer and status analysis rests on two very specific assumptions that the gravitation of fear of the collapsing star is exactly spherical asymmetric, and that the metal of the stars made off can be idealised as pressure less dust. Now, as we're going to see, Penrose would show that the major results of Oppenheimer and stator that of continuing collapse and that of a formation of a centre singularity do not depend on these assumptions.

This brings us to our third topic where gravitational collapse and singularities come together. Relativists analysed the Swatch institution by using ever new coordinates systems, some of them got rid of the puzzling and now famous sponsored singularity. And in them, it looked like a completely normal part of spacetime. Such corporate systems were found by Eddington in the 1920s. The Matt and Robertson in the 1930s and by David Finkelstein in the 1950s.

But the interpretation of the school and its systems remained controversial and knowledge of them did not really catch on, even when Finkelstein rediscovered arrangements for that system in 1956. Many people spoke of the structure of singularity. Shortly before Finkelstein had published on his governance system, John Wheeler had entered the study of GM. Remember that witness paper was sited next to that of Oppenheimer and state owned panels opening paragraphs?

Wieder was a nuclear physicist and was lured into jihad by studying the work of Oppenheimer's group from the 1951 onwards. Willa Harrison and Marcano reconsidered. Oppenheimer started in 1957 using some of the earliest computers to model the gravitational collapse of a star. Way into the 1960s, when I was very sceptical of Oppenheimer's nineties prediction of continued collapse.

He expected that general activity was not trustworthy beyond the shots at singularity and would need to be replaced by a quantum theory of gravity. Yet to be found. Still, it was Wheeler who made the term black hole stick for the type of object that Oppenheimer and Snider had described and who spent decades of his life researching the physics of black holes.

Now, also in the early 1960s, the Russian School of Racket Activity took aim at the central singularity that Oppenheimer and Snider had claimed would form. They wrote a paper so salvage and in particular, Lifshitz and Kalashnikov wrote a paper that argued that Oppenheimer's and standards analysis of gravitational collapse is not generic and in particular that if you did not have perfect spherical symmetry of the collapsing matter. A central singularity would not fall.

And that suggested that the fact that the centre singularity did form and Oppenheimer Cincinnatus analysis was an artefact, a mathematical artefact produced by the assumption of symmetry. Now, so far, everything I've said was mostly about theoretical G.R. and how gravitational collapse could in principle happen to stop. There was no so far no relationship to any heavenly bodies.

This all changed in 1962 and became clear, especially in 1963, if the first Texas conference, which brought together mathematical service and astrophysicists because quasars had just been discovered. Here's a picture of the quasar 3C 273, the brightest quasar on the night sky. And a quasar. It's a fascinating object, Iguazu can emit more energy than our entire Milky Way galaxy. Now you might ask, how could it be that these objects were only discovered in 1962 if they're so energetic?

The answer is because all that energy is met. It is admitted in the radio spectrum and we don't see radio waves with our bare eyes, nor with our optical telescopes. And so radio telescopes had to be developed before quasars could be observed. And this picture of a quasar was discussed at the 1963 Texas conference. Here's a picture of the same quasar the brightest one in the sky from 2013 from Hubble, but a pretty good picture. Now, how is all this related?

The idea quickly came up that the source of all that energy put out by crisis could be the gravitational contraction of a star, the death of a star and produced an enormous outburst of energy. And well, so suddenly, all the calculations by mathematical relativists about gravitational collapse became immediately relevant to astrophysics. And everybody was happy. Tommy Gold, at the after dinner speech at the Texas conference, summarised up the spirit of the conference and said everyone is pleased.

The relativists who fear they are being appreciated, who are suddenly experts in a field they hardly knew existed. The astrophysicists for having enlarged the domain, the empire by the annexation of another subject, general relativity. Roger Penrose was present at this conference and was inspired to revisit his thoughts on gravitational collapse and the formation of singularities from years before, especially in the light of new mathematical tools that he had developed in the meantime.

Now, one can interpret Eddington Limit and Robertson as having understood that the structured singularity is not a real singularity, but a coordinated singularity, an artefact of the court and system chosen. But it took Finkelstein and Rentschler to fully realise that though the fractured singularity might not actually be a singularity, it is something else. In 1956, Greenblatt called it an event horizon defined an event horizon that's a boundary in space time,

which divides all events into two non empty classes. Those that have been, are or will be observable by a given observer and those that are forever outside a given observe as possible power of observation. While the Schwartzman's solution, this meant that no observer outside of what was formerly called the sponsored singularity can observe what is going on inside and no observer inside this event horizon can ever observe what is going on outside.

Penrose understood all this, and he would tell you himself how Finkelstein and Rentschler influence just work. But in a way, all this meant that those strange the structured singularity was not as big a problem as physicists had long thought it to be. In contrast, the centrist singularity of the structures solution that originally nobody was worried about had become more and more threatening with Oppenheimer Sunstein as omniscience of gravitational collapse.

Well, there it looked as if such a singularity could actually arise in nature as the result of a physical process, namely gravitation. A contraction of a star wheeler, slovic, Lifshitz and Kalashnikov had argued that this was an artefact of the spherical symmetry assumed by Oppenheimer and. And this is where Penrose this theorem entered the fray. We're now in a position to understand the project the panel's outlines at the beginning of its 1965 paper. We understood what came before.

What kind of gravitational collapse had been envisaged and how in the work of Oppenheimer and Snider and criticised by Wheeler, spherical symmetry played a major role. And especially how the spherical symmetry had been argued as being maybe responsible for a physical singularity being produced at the centre of the body, according to Oppenheimer and Snider, it was to speak spoken about the structured radios, which used to be called the Schwartz with singularity.

But by now, when Penn Resort had been re conceptualised as an event horizon, especially by Rinella, even though the battery conceptualisation would only really catch on after the work of Penrose into the 19 late 1960s, and we can also now appreciate why it would have been such a big deal, why it was such a big deal that deviations from spherical symmetry cannot prevent space time singularities from arising

because that went entirely against the development that had been going on in the previous 20 years when Willa Lifshitz, Kalashnikov, everybody expected that the formation of a centre singularity was only an artefact of produced by the stroke of symmetry. Assumption employed by Oppenheimer Stop. We are also now in a position to see that at least two of the premises and the concept used in the conclusion of Penrose as 1965, there were utterly game changing. Regarding the first assumption.

If you think back to what I said, all previous treatments of gravitational collapse had assumed a very specific model for the collapsing, but Penrose only assumed that whatever the collapsing stars made of whatever symmetries the matter has, and no matter whether it is subject to pressure or not, the only assumption needed is that its local energy is everywhere, not negative.

It's an incredibly minimal assumption about the matter concerned. Translated into a condition on the curvature tensor, this brings about a focussing of enthralling light rays, and Roger himself will talk more about this. Now, the concept of a chat service that is introduced in the third premise was truly revolutionary, or it was one of the first global concepts in GIA, and it managed to ship around all the previous problems involving code and systems and symmetries.

I will say more about what it is in a moment. Finally, you've seen that previously, whenever anybody talked about a singularity, what they referred to was that the metric tensor or the curvature tensor would tend towards infinity. The problem was that you could never really be sure whether this behaviour was subject was because of the coordinated system chosen or whether it reflected something deeper.

The idea of moving away from thinking about whether the metric of the curvature tensor becomes inverted and towards thinking instead about whether the path of a light rail is complete or instead suddenly stops, which is the incomplete GTC referred to here. And to take this as a sufficient criterion for that to be a singularity was another game changer. Now it is today hard to fully appreciate the ingenuity, but the concept of a chat surface demanded at the time.

Note that it needed a coordinate system like that of Eddington and Finkelstein to be brought about. And yet the concept itself does not need any reference to a cornered system, and it does not rely on the space time having any symmetries. It boldly allows for a global property of space time. Why the consensus for half a century had been that only local concepts are to be trusted. That had been the whole push to once shy away from Newtonian theory.

So here's the basic concept basic idea of a track surface after the stars collapsed within its transit radius, which has now been understood to not be a singularity. The sphere can be found in the empty regions surrounding the collapsing matter, and the sphere is an example of a trapped surface. Light rays, which means that light rays emitted from within that sphere, no matter in which direction they are shot.

Will always converge in what's. And here is A. This conclusion from the 1965 film about how Jack surfaces and singularities are related. He says the existence of a trapped surface applies irrespective of symmetry that singularities necessarily develop. And the singular behaviour that he speaks of here is exactly at the path of light race cannot be extended further, but it suddenly stops and the case of the structured space time, this happens exactly at the centre.

So if this is the track surface laden race, I might add somewhere here, they will all converge inwards towards the singularity where their path suddenly stops, and this is taken as a sufficient criterion for the presence of a singularity. In the case of the structured solution, the outermost track surface is located at the structured radius, i.e. at the event horizon. Thus, what I said and swatch of thought of as the swatch at Singularity is actually something entirely different.

It's how been reconceptualize as an event horizon. Something very different from a singularity and a concept that had not been available to Einstein instruction. OK. So Einstein and Schwartz should have thought of the centre of the structured solution as harmless.

Of the centrist and hilarity of this fractured solution as a placeholder for the start itself, but for general showed that four collapsing star and without making any assumptions about the symmetry of the style about what it consists of, a singularity like this will actually arise in nature. If it is right, then such a singularity is not just a mathematical placeholder, but something that could actually exist in nature.

It would come into existence as a result of collapsing matter and the formation of a black hole. Now, and this was brought about by Richard Penrose, this would. But what about Einstein? A. Singularity argument his argument against the paramount catastrophe? Well, remember that Einstein's argument went against the possibility that the Schwartz should singularity could ever end up outside of Oppenheimer, and Snyder showed that this is possible and thus that black holds impossible.

But Finkelstein and Penrose showed that the sponsored singularity is not a singularity, but an event horizon. Now, would Einstein be reconciled by this? I think not for the centrist singularity to to appear in nature would have been equally catastrophic for him. Now that it can't be interpreted as a placeholder for matter, but must be seen as the result of collapsing and something that actually appears in the universe.

And what about wena scepticism, about space, time, singularity, superior nature? Does the existence of black holes necessitate the existence of space, some singularities? Well, yes. If she is trustworthy all the way down into a black hole, Penrose, a singularity theorem shows that singularities an inescapable consequence of G.R. and that they will inescapably form in gravitational collapse,

as described by Gaia. But as structure would be the first to admit and indeed did admit already in the 1965 paper, Wheeler's dream of complementing classified air with a quantum theory of gravity that kicks in beyond the event horizon and might stop singularities from happening and coming into existence, remains an open and viable project. Now, I would like to finish with two quotes. The first is a quote by Griffith and Podolski on a textbook on the exact solutions of ancient equations.

They wrote It appears that it is much easier to find a new solution of ancient equations than it is to understand it. Well, and given as we've seen that it took that it took structured only a few months to find the Schwartzman's solution. And yet it took more than half a century to properly understand sponsored solution, which is the simplest exact solution of the answer to any questions. I think Griffith and Polonetsky have a point.

Here's a second quote, which is from pretty patient. He's writing about the history of quantum mechanics, but I think the case of Paciencia. He writes how difficult where these advances, if we now acquaint ourselves with some of the detours, errors and psychological blocks. It is not to console ourselves with the thought that even the great physicists do not always move in a straight line, but rather to grasp in some measure how difficult it really was.

Well, this points us to the question, how did Roger Penrose actually come up with a singularity theme and all of the new concepts and ideas that went into the Singularity theme? If you think back to what's the last half hour, you'll see that haven't really told you how he came up with it. I only told you what was there before and showed you how all these concepts that he introduced put the entire state of the discipline on its head.

Well, with that, I turn over to Sir Roger himself, who can tell you how he came up with this year and how his thoughts on the topic have developed since then. Thank you. With this talk is sort of my Nobel lecture, but a little bit different. This first slide shows. Four dimensional space, that is to say, I've got three space axes and a time axis. Now this space time is actually you two in Kosky rather than Einstein.

But I didn't like the idea when he first heard of it, but nevertheless you got used to it, and then he it was crucial to his theory of general relativity. Now in the next slide, I'm going to show you what a light rail looks like. So imagine a photon which is whizzing along with speed of light. Now an order that shouldn't be just sort of in the special direction pretty well. I want to make the axes. So the speed of light is comparable to actually.

So if I if the time is in seconds, then the space would be inspected in light seconds. If it's years, then it would be not years. So it's 45 degrees or some reasonable angle. Okay, so that's that's the photon. I'm really interested in the hope possible photon trajectories that gives us the nail cone. So the red there you see the can, and I'm more interested in the outcome. And in the particular light red, because it really represents all light rash.

So that should move the light right, impacting into moving the axes. We don't need that. The crucial thing is the light that's the most important structure in spacetime. It's almost all of the structure I'll come to later. Well, completed to be more of a structure, but it's almost all the structure now. In general, relativity and special relativity light cones will be all uniformly arranged.

But in general, relativity, where the spacetime is curved and now represents, represented it in this picture, where the looks like a surface. But you must imagine that the surface there is. Four dimensional and that the cones also have another dimension to them because the end of the cone looks like looks like a circle and looks like a circle flattened out, but it really should be a sphere. So you've got to imagine if you can beat the cone really has a sphere at each end of it.

OK, well, let's go on to the next picture. Now, the important thing about space time is it represents the history of things. Now if you have a. Particle, which is moving the speed of light on the left hand side, you see a massive particle. Its trajectory is always within the cones. It's travelling less than the speed of light, if you like. That's what that means. Whereas a photo travels with the speed of light, it's always tangential to the curve.

Now you see these little cones represent usually alcohol and are now cones, but the lichen, if you like you have imagined that a flash of light occurs at this point in the middle. Just note on the left, the suppose there's a flash of light there, and then the light worked its way out. And because the space-time is curved, it may do funny things later on. It starts out quite reasonably, but then it may start crossing itself.

So you see at the top right hand side, you see the light rail start crossing over and we get what's called a cross section, the crossing region. And it's important to realise that these things happen because in the general space, time light cones can behave in this sort of way. OK, now let's move on now. I want to describe a little story here because I first got interested in general relativity, but my brother told me a few things about it.

When I was young and when I was an undergraduate, I bought a copy of Schrodinger's little book Space-Time Structure. Very nice little book, where he described the ideas of general relativity. So I had some little bit of a feeling for the subject and also a little bit later. I think when I was an undergraduate, I was listening to a.

Some lectures given by Fred Hoyle. I think there were five of them or something like this, and he was talking about it, first of all, he asked and they went out to Solar System and further and further out. And then he started talking about cosmology, and he was describing the particular view of cosmology known as the steady state model and the steady state model.

You have the universe expanding all the time. The problem was at that time that there were two observations which seemed to contradict one. No one was certain clusters of stars called globular clusters, and they seem to be very, very ancient. And the age of the universe is calculated from the Sun. Certain stars called shifted variable stars seem to say that he was actually less than the age of these clusters, which is a contradiction. How could the globular clusters be older than the universe?

It was ultimately resolved because there was an error and people had looked at two different kinds of these certainly variable stars. And actually, the universe was considerably older than people thought. But at the time, there was this apparent contradiction. So the point of view, known as the the steady state model was put forward by the gold boundary, and Hoyle and Hoyle was describing this model in this lecture.

So what we have here in this picture are the light cones in this model in the central strip going upwards. That would be the part of the universe that we can see. This is us up here somewhere and we are looking back. And the way Hoyle described it was that the galaxies get further and further away. They expand away from us. There's always hydrogen being produced to sort of keep the the density of the universe constant.

But the galaxies drift away, and when they get faster than light, then they disappear from view. That was what Hoyle said. And I find this very puzzling because I drew pictures like this one here, and I imagine us looking back and these cones sort of point outwards. Notley's and you see our past the light rail as it follows the cones. Will always intersect the the world lines of the galaxy, so let me go to here.

Here we see some galaxies. The Wilsons are the galaxies are always within the light cones. They don't travel faster than light locally, but you see relative to us, they travel faster than light. And so it was arguing that they would disappear. And I don't know, they don't disappear because here's us at the top and we look back on our past like sort of creeps along inside this well horizon between collide and we would always see any galaxy which had crossed the cross you associate.

It may fade out gradually and become stranger and stranger. Now I did believe, but I didn't believe I just vanished. So I happened. To be visiting my brother at that time was a graduate student in Cambridge, and I complained about this and said I didn't see how the galaxies disappeared. And he said, Well, I don't know much about cosmology, but over there is Dennis Sharma sitting at the table and [INAUDIBLE] explain it all to you.

So I went and I talked to Dennis and I gave him this argument and he said, Well, I'm not sure about that. And he went and said, I'll talk to a friend about it. And that's the main point about this was that he realised that that he thought I would be good cosmology I was doing. When I went to Cambridge, I was actually studying mathematics. I'm studying geometry. But Dennis decided he would teach me physics, and he thought that maybe he'd get some input from talking to me.

He was very good at bringing people together, and at one point he suggested this. I think this a little bit later on here we here we have the extended statement again. You can see our past like intersecting of the range, the whereabouts of all the galaxies. Anyway, when I went back to Cambridge later on as a research fellow, I think it was and I was at St John's College Cambridge and Dennis suggested that.

I might accompany him to go to hear a lecture by David Finkelstein that was at King's College London in 1958. And David Finkelstein was describing basically what this picture here is the Oppenheimer Schneider picture of a collapse to a black hole at the bottom. We see material you have to see. Time is always going up the picture in my pictures here. So here we have the bottom. Some matter which is considered to be sort of impressionist fluid is the way Oppenheimer described it, and it's neither.

It was what they call dust, which has no pressure anyway. Never mind, so collapses inwards. And then there you have this situation in the future where you have these light cones pointing in now because I wasn't describing this particular. But he was describing, look at the top part. This is a picture of the structural solution. Misfortunate solution was introduced shortly after Einstein introduced his general theory of relativity.

Karl Schwarzschild solve the equations for historical body such as the Sun, something like that of star. And the way that the metric is based on metric, including the light cones would behave was given by his equations. But the way he wrote the formula down, you came into a problem just at this place, which is the boundary. This sort of synergy that you see going up was its boundary, where it becomes tangent to the light cones in the coordinates that Georgia was using.

It looked as though it was singular, so you had a place where things go to infinity. Singular really means that something goes wrong. Your metric seems to go to infinity, and that doesn't make any sense when that happens. So. So you give up at that point. But so people thought that the structure of singularity, which is this boundary, was a place where things blew up and you got nonsense.

However, various people showed that by a change of coordinates, you could actually see that it wasn't a singularity that you could. As long as you had a coordinate system which enables you to to describe these like crimes pointing inwards. And David Finkelstein in his store was describing, I think of the top part of this picture.

Forget about the collapse at the bottom. That was the picture that Finkelstein was describing, and I was very impressed by this because I really was this most amazed by how you could get rid of this seeming singularity, and it wasn't there at all. Nevertheless, at the centre, there you have this squiggly line in the middle. You really do get a single act. This is where the space-time carriages become infinite. And I came away from this lecture thinking.

But somehow you could manage to get rid of the same thing if it was a singularity, but you're still stuck with it. Maybe there's some general theorem which tells you whatever you do, you can't get rid of this nasty thing in the middle. That look quite wise. And you couldn't. But it seemed to me that somehow this is a problem that maybe you have generalisations of. This is not disparately symmetrical case with irregularities and maybe the stuff falling in the pressure or something like that,

which might make a difference. A much more general situation, which is qualitatively similar, but not exactly the same. And do you still get the singularity or do things sort of squish around and come squirting out again? And then I thought to myself, Well, I don't know much about general relativity. If I was going to try and prove this, a theorem like this, this is about nineteen fifty eight.

What do I do? Well, the only thing I knew that I thought might be unfamiliar to general relativity period of time was about two components spinners. I think a few months earlier, I'd been, of course, due to Paul Dirac great quantum physicist. And this was his second loss of his. He had an elementary course and a more advanced course, and this is in his more advanced course. And he deviated from his usual course, which was rather an unusual thing for him to do.

And he described two components steps. I was trying to learn about them for other reasons. Previously, I just found a very puzzling and I didn't understand them at all. The direct lectures made them very clear. And this is the sort of picture which I sort of out of this imagined myself on the right hand side. We see this little flat there. It's we have the future cone and the flagpole is along the cone. So it really points out towards as a as a light rain would go.

You can imagine also the pass pulling back on the light cone so that if you look out at the sky, we're looking at photons coming in from infinity. And these photons coming in would be a sphere. So this sphere on the left hand side, you can call the celestial sphere if it's if it's the passcode cone, that's the actual celestial sphere of light coming in. But we can still call the celestial sphere of a light going out to infinity.

And the the little two component spinner describes a single point on this celestial sphere is also about a little flag on it. And this is the just characterises the description of the two components. There's a little problem about the sound of it, which I won't reach at the moment. OK, now what's that got to do with this? Well, not much, but I started thinking about this problem and seeing what I could do with general relativity,

and I found that some things worked extraordinarily well. If you wrote them in terms of these two components finished. You take the curvature of space time and it automatically splits into these two parts. One of them is called the viral part and the other is the Ritchey part. And the Ritchie Pass is what Einstein's equations directly determine. So if you have a massive distribution, this massive distribution gives you the Ritchie curves.

That is the Einstein equations. It just tells you that the richer curvature is given by the massive distribution stress-free. That really means you're looking at the light rays and not the whole thing, but never mind too much about that. But what the richer curvature does is to focus the light rays inwards and a sort of uniformly where it's like a positive lens, which magnifies things. So if you're the observer at the top you looked at, this was slightly magnified.

What you're looking at now, what is the violent curvature do? It distorts. So it's purely stigmatic. It's a lens which is entirely asymmetric and it magnifies in one direction and magnifies in the other direction. Anyway, I got a good clear picture of how light rays were affected by the curvature, and at one point I remember thinking about the steady state model.

I was thinking about it, I think at the time where it had lost some credence because people had in the meantime discovered the microwave background, which showed that the universe had only stage in which it was extremely hot so that this was more in accordance with the models of the universe, which expanded out from a Big Bang. Whereas the steady state model didn't really agree with this at all, Dennis, when he was convinced that these observations were correct.

He changed his mind and he said, Look, I was wrong to promote the steady state model. And I thought about it a bit myself because various people were still going on trying to promote it. And I had a sort of thought, can you show that if the energy density is positive, even if you wiggle so, so steady state model, we wiggles in it with that.

Can you have such a model? And I convinced myself using focussing arguments, you simply couldn't have one that would be an impossibility as long as the energy flux was not negative. So this is something I had in the back of my mind. And later on in the early 1960s, when quasars were observed, these are very, very bright things which initially people thought were stars.

I think they were radio sources initially that they noticed which were extremely bright, something like a hundred or a thousand times brighter than an entire galaxy. Yet they seem to be pretty small because they're they're and sort of the order of a week or so, which said that they couldn't be really bigger than the Solar System. So how could it be that bright have that much energy in them and yet be so small?

So various people did start wondering whether something like the Oppenheimer saga collapse might actually be involved and that you were looking at things maybe as matter gets smaller and smaller would produce a lot of energy, and this engine might. You be a product of these things. Probing their short shot radius, I, a friend home in particular, was keen on idea Is this the shot?

And there's a strong possibility, but this made many people think what happens when that collapse and they were thinking again about the iPhone, and it's not a collapse which I just showed you before. Yeah. The Oppenheim, not a collapse. And here you get a singularity in the middle. But let's suppose that the matter falling in is very irregular. So it's a lot of irregular things swishing around in all directions, and maybe you just square swirls around and some switching out again.

And I at this time, there were some papers. There was a paper by Lifshitz and classmate Kalashnikovs, two Russians who seemed to show that in a general situation, we would look at singularities. And so I remember John Wheeler worrying about this a lot, and he asked me whether. You know, it can give me insights into this, and I remember I looked at the decision that was planned across paper and I didn't find the arguments completely convincing.

There actually was an error in the paper, which I didn't notice. I hadn't gone into deeply enough to see that. But nevertheless, I thought it was worth thinking about the collapse problem anyway. And I remember walking in the woods and seeing trying to think, Was there some way when I remember thinking that had to be a non-local description that you couldn't because of the way things scale, it couldn't be something that was local. And it just got too big.

And then it sort of swelled out again, and it had to be some more global criterion. And at a certain point when I was talking to a colleague and I had this idea came to me how you could characterise it collapse, which had reached a point of no return. Now this is the opposite of collapse. Picture more or less. It's actually pay attention to the picture from a paper that I drew. I drew the picture, which describes the paper, which eventually got the Nobel prise, apparently.

And here we have a collapsing material at the bottom of the picture. And as we go upwards, we see this little black ring going around it. You have to remember there's another dimension here, so that isn't really a ring. It's a surface. But you picture as this little ring going around, it's a surface going round the collapsing material.

And what's funny about it is that if you imagine a flash of light going all over that surface, that the flash of light converges on both sides on the top right hand picture here you see a little bit of a surface and you imagine a flash of light over that surface on the concave side that will converge on the convex side and diverge. What we have in a trap surface is a situation where it converges on both sides that looks a little strange. It's not actually that all that strange in relativity.

The bottom situation we have to pass light cones, and as they intersect, you find a little bit of surface which has this property. It converges on both sides. But the thing about that is it's not closed up, it's not fiscal compact. And what's funny about the picture here is that you have the surface which the light waves are converging all the way around and it is compact. And it was that contradiction between not being a compact surface will is the fact that you look at the future of that.

So you remember I had this picture of me, the future of a point, now a future of a surface, something complicated than that. But the way that it behaves and the fact it has light rays on it, which finally close up, and you could show that this was the boundary of that region, which is the future that when I call the trapped surface, that's the little ring in the middle there. That track surface had to be a compact region.

And then you sort of project that backwards. I had a rather clumsy argument at the time. Charles Misner had a better way of describing it, which I hadn't thought of at the time. But the argument was I was thinking of GDDR6 coming and hitting the initial surface. I should say the surface. The bottom is what's called a koshy surface. This is what people are doing in physics.

You consider that one moment you say, well, what everything is doing and then your equations tell you what that stuff is doing later on. It's called a koshy surface. You have enough data on your initial surface to predict the future. Now, in this situation, I'm imagining a local thing. So the Kushi surface could be assumed to be non compact.

It goes off to infinity in all directions. And when you project back to your compact region at the top, which is the future of the surface, the boundary of the future of the surface, you simply arrive at a contradiction in that contradiction. Something is wrong as long as the energy doesn't ever go negative. That would spoil the argument, but as long as it remains not negative, you have this problem and you get it.

What was it? Whatever definition you have, a singularity makes a difference, but you have something goes wrong and this is the singular stage. So what it's telling you is that you can't have a general collapse of the Oppenheimer Schneider type of body, which is sort of reasonably, reasonably compact in itself and can be very, very irregular, actually can consist of a lot source. You see, a lot of people used to worry maybe a realistic situation.

Could you could you avoid having a track surface? Do you necessarily have a track surface? Well, it's not so easy to see. So I used to often use a different argument. The same theorem applies. Suppose you have a point. The point at the bottom of the concert is its future lies. And as you can see it, following up on the boundary of its future. With this light rain start to converge again, none is necessarily going to be a contract region, and that's good enough.

You don't have to have a truck surface as long as you can imagine that they have a point where the future light comes at a light rail start to converge at some point. And it's not so hard to see that if you have a large cluster of stars, let's say an individual star, there's probably most people worried that you'd have individual stars. The density might get so huge. And do you trust your equations when the density gets very large?

But if you consider the collapsing collection of many, many stars, then it's very easy to see if you have enough of them. It's not hard. The density doesn't have to be the average density. It can be less than the density. Now, it's not much problem getting enough stars doing that. So it seemed to me that it was a very general phenomenon, and there's no problem about whether you can have a trapped surface. You could say that this condition is there just as the serum works just as well.

OK, well, let's move forward a bit. Now we move on to the Big Bang. You see, that's another thing that actually and Stephen Hawking, I gave a talk at King's College where I describe my techniques. This was in 1964 and the paper was published, which got the Nobel Prises in 1965. It was published in 1965. I wrote it in 64, and then Sharma suggested I would give a repeat of this in Cambridge. The movie suggests that Stephen Hawking was present at my lecture in London, which is not true.

He wasn't there. But however, he was there at the talks in Cambridge, and we had long discussions. He and George Ellis and I discussed the techniques I was using, and Stephen very quickly picked up on the issue and started to generalise the techniques and use them in reverse time directions. So you could ask whether if you have a Big Bang, which was not could be complicated and not not not nice and symmetrical, as they usually have pictures of the universe, then might you avoid the singularity?

And the Big Bang could be the result of a previous collapsing phase which bounced and came out again. And he generalised these arguments very much and had several papers, three papers in the Royal Society. Then later on, we got together and proved a very general theorem, which which will encompass things that happened before. This is not a picture of the universe. I remember I was puzzled by the fact that people did not discuss in cosmology and serious cosmology.

I'm talking about the cosmologists, not the people who worked on general symmetry theory. I'm talking about the people who actually don't observational cosmology. And they looked at universities and I asked them, I remember talking to James Peebles, who was sitting in the car about to go off to a meeting at Stevens Institute, and I went off on a different car before he drove off.

I asked him, Why is it that cosmologists, you cosmologists don't ever study these other complicated singularities and loads and loads of them? You just concentrate on the one case where you have a very symmetrical initial state? Well, why don't you look at these other me? She looked at me and said, Because the universe is not like that. I thought, Oh my gosh, it isn't. This is his argument I think was based on the cosmic microwave.

Background is very uniform, and in this picture, I've drawn it as this very symmetrical you'll notice at the back, there's some sort of irregularities that's just so that the universe could be closed or open. It's a bit hard to draw a picture of the universe if it's not closed, so I pretend that it's closed for a bit and then it goes off to infinity around the back. But I'm not really prejudicing the issue as to whether it's actually open up close.

Now, one of the arguments that people use in cosmology is that right there, tucked in at the Big Bang in order to see it, we need a magnifying glass, a really powerful magnifying glass. Very, very powerful magnifying glass because it's way down in sort of the first 10 to the minus 32 of a second.

And what you apparently see, according to the theory, is this thing called inflation that is another exponential expansion of the universe, which may be which moves the universe, and it does other things which are more technical about the cosmic microwave background. But one of the arguments we're trying to make is that that Big Bang, if it had this inflationary phase, it would smooth out.

And that's why you don't see all these irregularities. Now, I was not very convinced by this for the following argument. Let's imagine that time is going the other way. And so you have a model which is collapsing. And in this model, you've got. All sorts of irregularities, including black holes and things forming. And they they will converge on some horrendous next event.

Now that is the cause of arguments from Stephen Hawking and Beck and so on and so on, who show the entropy of a black hole was it shows that this is an extraordinarily special state not to have these black hole constituents in a collapsing model so that a very general situation wouldn't be anything like what we see. It would be much more like this. Yeah, we have black hole singularity or congealing in a great mess.

It's much more like the sort of model that Russia in the costs and charts and came up with later on. When corrected, the mistake was corrected and they found very complicated solutions, which would be nothing like what machine in the Big Bang and I should say. One feature of solutions is the vile curvature becomes enormously large, absolutely huge. Now that is a much, much more likely initial stage. Why don't we have that rather than what we seem to see, namely this?

Why do we have a very, very regular universe? This is the problem posed by Jim Peebles and previous Nobel prise winner last year, and he had very good reasons to believe that it was like this from observational signals. Now, I find it very strange that we seem to see a singularity in the bang. The arguments were sort of similar back in time and forward in time.

So why do you get this very, very great difference when used to think that these singularities would be resolved by some wonderful theory of quantum gravity? Quantum mechanics modifying gravity a small scale small sizes coverage does get very large. But the trouble is, if that theory explains how to get rid of the singularities in the future, why doesn't it apply to the Big Bang? Why did we get this very, very peculiar situation? So I just formulated a theory wasn't much of a theory.

It's just hand-waving to say that whatever quantum gravity it is must be a very tiny, symmetrical model where the violent curvature is somehow necessarily zero. In past such singularities like the Big Bang was future type singularities that one we've just seen they need to be zero at all. Very, very strange kind of quantum gravity. This is all tied up with a major problem in physics.

Another man called the Second Law of Thermodynamics, which tells you that the entropy increases with time entropy as measured randomness. And here we have in the top three pictures the gas in the box we have on the left hand side of the gas in a little box and then we open the box up and the gas spreads out. So this is what you find with entropy increasing for gas in the box. But what about stars and a huge box? They behave the other way around that they would address if they are spread out.

This represents a low entropy state and as time progresses, disaster to clump and then you got black holes and things like that where the entropy just goes shooting up. So in both pictures, the top three and the bottom three, we have entropy increasing from left to right. In the top pictures is ordinary matter and the bottom picture is gravity. So gravity behaves very differently. What we do see in the early universe indeed, is a situation like the top right and the bottom left.

In other words, uniformity. What does that mean? That means high entropy in matter, and that's confirmed by the microwave background and low entropy in the gravity, which is also confirmed by the microwave background because you see that it's uniform. But it tells us that the Big Bang was a very strange thing because it seems to be no entropy in gravity. It's the single outgrow. It is something special. Now that way, I started thinking about this.

Later was due to an argument or a suggestion due to Paul Todd, my graduate student. Then at the time, he became an instructor at Oxford, and he had this suggestion that maybe what was peculiar about the Big Bang, not just say that our curvature is zero, but to say that it's if you could formerly stretch it out, it becomes regular. And the fact that you can squash down infinity and make that regular was something I knew about before.

This picture is an Escher picture. Why don't you set limits and it describes a geometrical, hyperbolic geometry? Don't worry too much about the geometry. What I'm trying to say is a conformal representation you can see in the eyes of the fish. They're all circles. And as you get towards the edge, they remain circles, the shapes or the small shapes or angles are preserved and this match of the fish at the edge.

Don't think they're any different to the ones in the middle. But this conformal compression is what Asher has used in this model. It's a Beltrami representation of geometry, and I want to try and apply the same idea to cosmology. That's something I'm used to do a lot with this question and finish. It's very useful. I'm talking about gravitational radiation. But the idea is to use that same argument in the opposite way to talk about the Big Bang.

Now what does control mean in general relativity when it really just means the light cones you see? Think about the light cones. That's not quite the whole of the metric. That's nine out of 10 components in the metric. If you want the tenth component, you have to think of these little pill shaped things or bow shaped surfaces. What do they represent? They represent the ticking of clocks.

Yeah, I've got the two most famous equations of 20th century physics Einstein, Jacobs and C squared Max Planck new news frequency. First one tells you energy and mass are equivalent. That's Einstein. Planck tells you energy and frequency equivalent. Together, this tells you that frequency and mass are equivalent. So if you have a massive particle that has a very definite frequency given by these equations.

Now, if you imagine now there were lines of massive particles, so here we have two two particles. Travelling with large speed, but not the speed of light, the red one intersects these bulkhead surfaces that stick number one, tick number two, tick number three, the blue one is six and pick number one pick number two three particles of fix of a definite stable particle size. Definite mass make wonderful clocks. So they are perfect clocks if they have definite masses.

Now that's this. But if you don't have mass, then you have photons and photons don't even see these surfaces, so photons don't register time at all. From the origins of the photon to the where it gets finely absorbed, if it does is as far as the photons concerned, no time at all. So it gets right out. To infinity, these lichens in our conformal structure. They don't depend on the mass they don't scale. The Beltrami picture or picture we just showed you is a conformal picture in space time.

It's the lichens that we're talking about now. OK, now let's think about the whole universe. On the left hand side, we see the model of the universe with the Big Bang, and then we see the exponential expansion, which is a feature of Jarod's again, another Nobel prise early in the 20th century 21st century where people discovered that there was a cosmological terminations equations or equivalent.

And this seems to show that when you issue squashed this down, you get a finite boundary in the future. I can see my error in one screen and those in the other. Unfortunately, that's try that. Yeah, we have a remote future, a nice space like boundary. Despite the Big Bang, there's also space like so you get you can squash down infinite and stretch out the Big Bang. Now, I was very struck by the fact that when you do that, they look very similar to each other.

You see, you might say they're very different topic. Well, what about the remote future's very cold, very rarefied? What about the Big Bang? So hot and very concentrated? But if you use this conformal squashing it not only just make things smaller, it makes things hotter. Next, the energies go up. What happens to the Big Bang when you stretch it out?

It makes things colder as well as less dense. So you something which looks as though they're pretty similar in the remote future, in the remote past. And in fact, now here's the exotic model that I put forward a few years ago, which I call hot conformal cyclic cosmology.

Is that our. Universe is only one aeons and calling it of infinite only extended can succession of these models where our Big Bang was the conformal continuation of the remote future of a previous aeon, our remote future will be the Big Bang of the next year. Now, I guess I gave lectures about this lot thinking nobody would ever disprove me because there was no observations,

and then I started worrying about this because there were some observations. So let me go and see certain signals can get through. In fact, if you have something with no mash, then it can get through. You might argue, Well, how about in our actual universe? Why it doesn't mass for a picture with a remote future? It's partly because it's mainly photons running around, and they don't have any mass. You have to have some other condition that the mass gradually fades out from massive particles.

But I don't want to go into that for the moment. What about the Big Bang? Well, there the argument is that things are moving around so fast that the energy in particles is almost entirely in motion and not in their mass. And so that therefore they are effectively mass us because they're so hot, so that mass things at both ends. And if it's like a photon, it can actually get through from one side to the other. What else could get through? Well, gravitational waves.

And yeah, I have one of the ideas that I proposed that if you have black holes in the previous year and actually before this horizontal surface, the previous part, this is the previously on the red part is already on the top, opposes the previous year and I have a number of black holes in a galactic cluster, supermassive black holes running into each other. So there will be several of these, and that cluster eventually becomes one big black hole and that sits around for an awful long time.

But in the process, it will emit these huge gravitational wave signals. And can you see them? Well, my colleague Vijay Guzelian, who actually did an analysis of this, I won't go into the details of it. I show you this one picture, which is an analysis where he looks for rings. You see, when you look back into the previous Ian intersection between our past, current and the future of the purple cones is really a ring because you need another dimension, of course, to get sense of that.

And so what I did was to look at rings and look at rings, which had low variance. That's the sort of signal that you might get from these colliding black holes. Little variation as you go around, but that wasn't very strong signal. What you had to look was for at least three concentric such rings as you. As I said, you get several of them in these collisions. And this one I want to point out is a remarkable signal because he didn't look for whether the temperature was warmer cooler.

But the the red ones are with warmer temperatures are actually the more distant ones and the blue ones are with coat cooler temperature, the closer ones they actually with. Well, we call on with a partial icon, the red ones. You wouldn't even see the galaxy because they're outside that partial like you can do that because you look into the period you see on and the current spreads out further than it does in us. So we see these very curious, very concentrated regions.

That's what's so puzzling about this picture. Whatever it is, it's producing his signals is very, very, very non-uniform. And that's a great puzzle for cosmology because people think the universe is very, very uniform. This tells you, is on a very large scale. It's not all that uniform and you have this very, very super duper clusters of galaxies.

Of course, that may be another explanation. Maybe that's not the real reason in the model I'm proposing in the previous year, and you have great clustering and that will undoubtedly come through to you. The second observational feature was more to do with my Polish colleagues Christoph Meisner, Pavel Nawrocki and Daniel, and who did some analysis on, well, another pole originally.

But Daniel, and in the subsequent measurements, which I'm talking to you now, this is another thing that Hawking predicted is that black holes not only have entropy, but they have a temperature. And this temperature is very, very, very, very cold. So you don't see much of that. But in the very remote future, when the universe expands and expands and expand, the black hole becomes the hottest thing around and it gradually evaporates and finally evaporates away.

And it just goes away, maybe with what I call a pot. But whatever happens, all that radiation, because you remember the fish getting so concentrated in the in the present in the picture, yeah, everything gets so squashed down. All the radiation in the in the final picture gets squashed into a single point. Now here we have a. A picture from our paper published in the Monthly Notices of the Royal Astronomical Society. Daniel Han did the analysis detail computer analysis?

Kristof also has a particular criterion for the looking for the use of hawking points, she said. I'm calling you hawking points here in the bottom part of the picture. No, just the horizontal line at the bottom that is the cross over from one year to the next. I'm imagining that there is a black supermassive black hole finally evaporating way way just underneath that crossover surface. As we call the hawking point, the all that energy comes pouring through and that will come pouring through.

You don't see it immediately because it takes 380000 years before the light can ever get out. That's standard stuff. James stable and his colleagues very good work that they've done on the universe 380000 years, and that spreads out to what is about eight times the diameter of the full moon so that all their energies spread out to eight times that.

I mean, you see at the top the sort of distribution of temperatures that you have in the middle, larger, larger increase in the temperature and it stays off as you go to the outside of a size, which is eight times the names of the moon or slightly less. If you're not quite in, quite seeing it head on. But these we do seem to see with a confidence level of ninety nine point nine eight percent. This is in the Planck data. This is a satellite which we originally looked at.

Then we compared it with the older satellite, the map data. And for the five strongest points in this Planck data, actually where they are is not such confidence level, but as you see them in both. That seems to be pretty strong evidence that there are genuine hawking points that the five strongest points are seen in exactly the same places in the map and the Planck data,

so I think they're pretty reliable. There's another one in the map data, which you see also in the same place in the Planck data. So there does seem to be something there if somebody else has another explanation. Yes, I'd like to see it, but it's not. It seems to me this is a pretty strong indication of something like the model that I'm putting forward. It certainly is something I hope people will consider seriously. Thank you very much. Well, welcome to this final session.

And welcome to Roger Penrose, who I'm going to talk to you for the next half hour or so. Roger, when you think of your background, your family, there were artists, there was a chess grandmaster as a geneticist mathematician. Do you feel you were fed by that multivariate academic atmosphere? I think that's undoubtedly true. I mean, my father worked in human genetics. Both my parents were medically trained and actually they assumed I was going to be asked of me and my two brothers.

I was the one they decided was going to follow them in the medical profession, and I believe this myself right up until I think I was in my. Second year, I think at school, if you had to decide what in the last two years, what subjects we would do and each one had to walk up and talk to the headmaster. I remember as I was walking up to talk to the headmaster, I was going to be a doctor because that's, you know, it was sort of established I would follow my parents professions.

And then he said, Well, what subjects you wants to do in your final two years? And I said, I'd like to do biology, chemistry and mathematics. And he said, No, you can't do that combination. If you want to do mathematics, you can't do biology, you want to do bios, you can't do mathematics, make your choice. So I said, well, mathematics, physics and chemistry. And that was because I didn't want to lose the maths.

I realised there was my medical career going down the drain, and when I got home, both my parents were really annoyed with me because that was the end of my actually they one in the end because I had a little sister who was eventually became a doctor herself and married to one as well. So they really wanted me and not with me. Was this? Was this something that you've mathematic, which is the thing you had to keep a grip on?

I think it was just I just fell in love with it. Hmm. No. I certainly had a great fondness for mathematics from quite early on. Certainly, when I was nine or something like that, I can't remember the exact age of the earlier, but I hadn't thought as a career that really came gradually. First of all, with this when my parents would be very disappointed, which I ditched biology.

Of course, these days you could do biology and mathematics because in those days it was very much more restricted what was allowed to do. But later on, when I decided to do mathematics, just just mathematics at University College for my undergraduate degree, my father was again very angry about that. He considered that people who just did mathematics are rather strange characters, you see, and he thought I could do other things as well as maths.

So I should be more broad in what I did. But I was very determined. And I think, yes, my father had one of the lecturers, a man by the name of Kesselman, who was very good. He was extremely good lecturer and I didn't know that at the time because I wasn't there yet. But he made up a special lot of test questions. I think there were six or 12. I can't remember quite. And they were very unusual and I would never come across before. I respect respected him a lot. So taking all this trouble.

But anyway, he made these questions up and he gave them to me and said, Well, you can take as long as you like, and if you only do one or two, that's fine. But then he came back later in the day and I done all of them. He was quite impressed by that. So I think that results converge as my father, he allowed me to do mathematics. Yeah. Did the MLA wasn't actually extremely well ordered household amongst your relations than their friends?

Well, that certainly worries me when you that it was a bit chaotic, but go on. Yes. I mean, did you did you get something from the artist side and from the other side of that side? Yes, I think definitely from the artist side to now because my father was a he was a very good artist and he was one of four brothers, all of whom but one of them particularly distinguished as an artist. That was Roland, who was a younger brother of his.

And Roland became a big figure in the art world and surrealism, and he was familiar with Picasso and Maxence and various things like that. And he then became a big figure in surrealist art and in Britain. But there's a plaque to him. I pass most days. Oh yes. See? Well, yes, but you see my grandfather, my father's father was a professional artist. He was. He was very skilled, extremely skilled, but he sort of very religious pictures.

He was a Quaker, and my father was also a Quaker, but also the religious Quaker. He was more moral Quaker, you might say. And he used to go sketching, and I sometimes went with him sketching. He was very good at sort of green pen and ink, I think was this particular scaly hand. But I did more wild things which were not necessarily realistic, and I don't think he approved of that very much.

But the idea of drawing and drawing as part of the intellectual and the life you allowed was just that from the start, that was very much true. In my case, I. I don't think my brothers were too much on that side. My sister picked up on it rather. But certainly my father's side. Yes. And I got a lot from him on the artistic side. I think that's true when you are at school. You said that you, you realise your mind worked in an unusual way.

You know, I thought maybe when I go to university, I'll find people who think like me, but it wasn't like that at all. When I would talk to someone about an idea, I found myself not understanding a word that they were saying. Gladstone remark, which was sort of that I could pick up what the gist of what they were saying without really understanding the words. It was quite strange in a way. Obviously, that thought processes were very different.

So when I went to university, I found that in fact, people thought very differently from me. On the whole, I think it was mainly the visual on visual divide, which is quite a significant one. I mean, there are lots of other differences, but primarily that one. There are people who tend to think visually and those who tend not to think visually. And I was very much on the visual side.

I seem to remember when I was at university, they were only one or two people who thought very much on the visual side. One of whom was me for two or three. I guess I can remember one other particularly, but I mean, other people could do it, but that wasn't their primary way of thinking. So it was a lot the variation in ways people thought about things where it was quite striking.

And as I say, when somebody might be talking to me describe some idea, and I wouldn't really follow the actual words or sentences in any detail. I'd pick up the gist of what was being said and maybe come to the conclusion that was required at the end, but not exactly following the details of what was being said to me that stay with you. Did it resolve itself in some way? I think it stayed with me. I think. No, I think it's very true.

But I think it was true in relativity. I think that was a distinctive feature that I thought about the subject very much in pictures and how light rays behave and things like that happened rather than the equations do the equations when needed. Although even then I would have a very different way of looking at equations often. And this came about when I was a graduate student, mainly, and I was just doing I was supposed to be doing algebra geometry under the Hodge.

William Hodge was a very distinguished algebra geometry that he I don't think he was very visible in his thinking, but he gave lectures on differential geometry and he was sort of equations all over the board and disorganised lecturer. And I had a great deal of trouble following in detail what these things meant.

And so I developed this notation where instead of having symbols you see in the subject of differential geometry or relativity theory, you have symbols with lots of indices and indices at the top and indices on the bottom. And then you have to see which letter here corresponds to which letter over there. And that was much too complicated for me. So I just draw a line connecting these things, and I developed this diagrammatic way of expressing these equations.

And it wasn't the way anybody else, but it was very helpful to me in trying to sort out the problem that he'd sent me any work. Yes, it. I didn't solve the problem because the problem didn't have a solution in the form one expected this was a striking thing because I use this method to show that there was no solution to this problem. In the most obvious way, you'd expect you're trying to construct something you had to to object to and you had to combine together to make a third object.

I want to go into the details of that. And these were very complicated algebraic expressions that Jim Jarmusch is simply looking for at the intersection of two two geometrical figures. And that was clear to visualise what you're trying to do, but to do it algebraic, it was very, very complicated. And in order to do this, I developed this notation and I had these pictures with lines all over the place, and I remember showing them to my supervisor and I don't think he he followed what I was doing.

He thought about it very differently. So at that point, I didn't show anybody for a long time. So, but no, it helped me sorting out this problem and to see that there was no solution in the most direct way and you had to go in directly to find a solution. But when you were when you were still a graduate student, you developed the Penrose Triangle. What was that and why was it such a breakthrough? Not a breakthrough when it plays you so much?

Well, this was a complete digression from what I was doing. You see, I was at Cambridge at the time. I think it was in my second year and there was a conference at the International Congress of Mathematicians was taking place in Amsterdam, in the Netherlands, and I and a colleague decided we'd go to this conference. And I remember finding a lot of it was very mysterious and difficult to understand, but lectures by him and violet remember, which was very impressive and so on.

And I then I think I was getting on a tram at one point and one of my lecturers, Sean Wylie, told me about algebra, algebraic topology. And he had in his hand a catalogue. And this was a catalogue of the an exhibition that was being held in the. And one of the early the museums in Amsterdam. And he said, you might be interested in this. It had this strange picture on the catalogue of these now famous picture.

One of these it's called night and day. One of the pictures with birds flying and their one went into the night and the other way it sits in the day and the gaps between the birds become the birds and the other side. And it's very clever. And I'd never seen anything like that before. And so I went to this exhibition that was in the Van Museum, and they had a special room full of these as your pictures,

as they thought mathematicians would be interested in that. He wasn't, as she was not famous at all at that time. And I was absolutely stunned by these things, most particularly one called relativity, which is the picture where the force of gravity was in three different directions and people were walking upstairs and at right angles to each other and that sort of thing. And it was absolutely fascinating.

And I went away thinking that I hadn't quite seen something of the nature which I had in mind, which is a picture which was locally consistent, but as a whole inconsistent because there is a sort of ambiguity in the local picture about how far away it was from there. And I drew pictures with bridges and rivers and roads going different ways, and then I simplified it to this thing that people refer to now as the tribal three rods, each at right angles to each other.

But you you can't construct locally, you can. So you can have a local picture that makes sense locally. But since the distance away from you gets inconsistent at some point, then it's an impossible figure. And I remember showing my father and he showed it to his colleagues, and they were so ill looking at these things. And he then developed that into impossible buildings. He had when he called an impossible college, I remember.

And then he produced a staircase which went round and round and round, and we then wrote a paper on this the. Sorry. You and your father? Yes. My father and I and we we wrote a paper on this and we couldn't decide what the subject was. Where do we send it, you see? Well, my father said, Well, I have to know the editor of the British Journal of Psychology, so I'm sure he will accept it.

So we decided the subject was psychology. And he said this paper, the British psychologist, it was duly accepted. And then we sent a copy to Asia. We acknowledge the, I think, the catalogue to the museum so interested should have. In the meantime, assured produced a picture of Belvedere in which which does illustrate the same thing.

But then he picked up on this, and he developed the staircase into the one called ascending and descending and the triangle into a vertical waterfall, you said of the triangle and its impossibility in its purest form. Why did that attract you and why did you use the word purist? Well, you see, it's really. It's not there's no kind of elaborations to it. I mean, it really is just just the impossibility.

That's what I meant. What I didn't know was that there was this Swedish artist Oscar Rosfeld, who previously drawn a very similar thing with cubes going around, I guess, which I didn't know about. And there are old pictures you can find. There's a Broyhill picture with called the Gallows. I think it's called a jack on the gallows isn't so subjective, and it's joined up differently in the top of the box.

So it's quite deliberately. I mean, some people thought, Oh, he's made a mistake. No, no, he didn't. It was absolutely deliberate. You can see that. Did this attract you partly because of your interest in drawing as well as you interested in in the pure impossibilities? I think so. Well, you see, we did other things together that wasn't actually the first thing we did together.

The first thing, as far as I can recall. As we were on a train going to Switzerland, where my mother's mother, my grandmother on my mother's side lived and we were going to visit her and my father came up and he said, perhaps you could help me developing an idea because there was no.

The question is whether you could build a machine which would reproduce itself. And the great mathematicians on the island had proved some theorem, but with some several thousands or millions of pieces, you could make this machine, which would copy itself. My father was he knew about this, but he wasn't very pleased. I thought, Well, you know, life has got kind of come about from something simple.

So there must be some simple thing which will reproduce itself. He suggested using some something with gadgets, which would. Copy themselves and so that to solve on the train, on the way to Switzerland on the night, it was a night train, you see, and so we had bunk beds or something, and I remember to produce, I produced my solution and he produced this solution. This was better than mine. I think he appreciates his mind, but I think he got his respect. I think it was best.

So he then wrote an article for Nature, which got published between us, which was called self produced, self reproducing analogue or something. And it's had these two little pieces and you haven't got a track and you shake the track. And then they just hit each other and they don't combine. But if you link two together and you can mix them two ways this way or this way, think this way, then two other pairs will make up the same way.

If you let them this way, then other peoples will link up that way the same way. So it was a very simple, very simple example of something. Which you could imagine some molecules doing something similar, so his life could begin with something extremely simple, not these complicated, vastly complicated things that so moment would introduce. So I thought the idea was was a very good one, not in detail and seemed like chemistry or what life would be like.

Then my father then went on to develop much more complicated things with. He was trying to model DNA, that sort of thing. So he got a little bit, I think, too far into making this workshop in the lab, in the back garden that he would plug away with his petals, jigsaw and death and make dozens and dozens of these machines. And in 1964, you were said to have, I'm quoting, revolutionised the mathematical tools that we used to analyse properties of Space-Time.

That's quite a mouthful. It must have been quite a it's quite so unravelled for us. I think it is what you were saying earlier. I was thinking about the subject differently for most people who see most people at that time.

When I said at that time, this was a is bit before 1964, it was when the quasars were being seen and they were suggested that they were gravitational collapse sort of bodies coming together and producing who knows what and producing these very bright signals, which became known as the quasars, which are extremely bright sources in the radio signals.

And they were very puzzling to people. But you see. The two ways people might think about general relativity at a time, either the mathematicians who are just trying to solve the equations and you could only solve the equations of a very simple symmetrical, had some very simple characteristic which you could use as a as a gadget to allow you to to find exact solutions in the equations. I could do that sort of thing, but I wasn't a particular expert at it.

The other thing people might do is put things on computers. At that time, they haven't got very far with with how modelling things that we know called black holes. But I was thinking about this in a different way, which is much more geometrical. So I had this way of trying to visualise what was going on all the time, how light behaved, and it was very much my style of thinking. Were you making drawings? Oh, absolutely is drawings all the time.

And that was how how light rays focus. That was the main point. And there's a really close analogy on the edge of physics at school, because you see, one of the things you did at school was was blue with optical benches, and you have a logical bench and you have these lenses and have different distances and you see how they focus lies and all that. And then you could have pure lenses, which are another stigmatic. So the focussing is the same in all directions.

Or you can have a stigmatism where the focussing in one direction is different from the focussing other direction, and a purely stigmatic lens is when is completely positive. One direction, the negative in the other direction and the curvature due to pure gravity is of that kind. So I could understand how the focussing of light rays behave from having this experience in doing physics at school. We're playing with optical benches, so that was useful to me.

So that's not just drawing pictures that was having some idea of what physics was involved. Says You got lenses now that the kind of lens that gravity. That's how it acts, a light rays of a purely positive focussing lens convex lens, which is positive focussing is what matters. And I just come back to the quotation now that the summary of another child is said to be, the two words revolutionised the mathematical tools. Not. Well, on those two revolutionising tools for the minister, too.

I'd be very grateful. It's a hard question to answer, it's just the people weren't thinking that way. As I say, they were either looking at partial solutions, which didn't tell you much about the general case. There was this model of collapse to a black hole, as we would not call it mutual Oppenheimer and Schneider, which was put forward in 1939, just before the Second World War. And that was extremely symmetrical. So you could actually solve the equations.

You would write down the solution on a piece of paper. That's the actual solution for the empty space and the Friedman solution for where the matter was. So you could combine these two together and simply write it down on a piece of paper what the equations were. But that was only possible because you assumed exact spherical symmetry. So it's absolute the same all the way around a sphere. So the collapse was exactly symmetrical in the fact that it had no pressure, nothing to stop it.

If you like the falling in matter and the fact that it was symmetrical, is it? What is it got to go? There's nothing to stop it. So bits in the centre. And so the density would become infinite centres. You get this infinite density singularity where you, your equations give up because things go to infinity. And what do you do? So that was the problem. But most people thought, Oh, well, that's not realistic because things will be swishing around in a complicated way.

They won't be in the middle. They won't be aimed at the middle. So what do you do? But I didn't try to solve the equations that was hopeless. I didn't know much about how to programme computers, and so that was hopeless. But I had my way of thinking, but just to see how how light rays focus and how they behaved and what is actually you see. There were two problems I'd worked on before which were helpful to this.

One is one I described in my lecture, which had to do with the steady state model, which is one of the things that got me interested in doing cosmology and my friendship within the show, which is important to me. The other was quite a different problem. I was working at the same time a little bit earlier in 1964 on the problem paper, which I submitted to the Royal Society, which had to do with the asymptotic property of gravitational field.

So say you're looking at gravitational radiation. And how does it behave and what is the curvature look like in detail as you go to infinity? And I had a way of looking at this, which meant squashing down infinity to a finite boundary. So I have in my head took this picture of me. Yes, your picture of the situation that you've got this boundary. So limit boundary edge. But that's a particular instance of what I was doing. We squash infinity down to make it look like a nice boundary.

And I was able to prove certain things and I had one of the problem was that I needed that infinity look like a sphere. So this means that as you look out in the sky, you see the celestial sphere now the opposite direction. When you're looking at the future, incidentally, does it necessarily have this structure of a sphere up there? Or could you have something more complicated? And so in the appendix to this, I don't think there's any sensible person would just assume it's just showing up.

But I wasn't satisfied that. So I set about, Well, can you prove it, Sofia? So I thought about what the light rays do when they get out. And I had a sort of argument. I think it was reasonably good in the appendix. It wasn't part of the main paper I slaves as a way of thinking. I was wasting my time all the time. And I developed a lot of these techniques of looking like what is the boundary of the future of a particular point with its future?

And what's the boundary look like and how do the rest cross over? And you look at that after they've crossed you. Just look at the outer boundary and what step ahead in life and so on. And I developed enough of an understanding there that when I looked shortly afterwards, not very long afterwards at this problem of gravitational collapse and you knew about the solar in the paper, which seemed to show that you didn't get singularities. And I wasn't totally persuaded by that.

Then I remember going walking in the woods and trying to think of what condition could you have to show that these singularities? I don't know why. I pretty well thought those singularities were genetic. I guess that was an intuition I have. I don't quite know what I was toying with both ideas that it might bounce ideas or it might be generally singular. But I think I thought it was probably generally singular in opposition to what the Russians seem to believe.

And then I realised you needed something non-local, just a local condition that says things blow up because of the equations work just because because of the symmetry, you could blow everything up. And there was nothing of the scale to tell you that it could be local. So I was pretty persuaded it had to be something more global. The condition, which is sort of more global. And then it was later on this story, I keep telling people.

About us, I think I was I had this lectureship at Birkbeck College in London, and I think I was going to my office in the neighbourhoods and there there Euston Square and I was talking to a colleague who is an Englishman pull over Robinson. He actually was. His job was then in the US, but he was a very good person. It was one of these people very skilled at producing exact solutions. He had a real skill doing that.

I don't know what he was talking to me about how there were very these also skilled with with verbal if if you talk to him, he would tell them this wonderful way. The Americans loved him because he had this flowery way of talking in English. It was a beautiful skill. So he was talking to me like this. He was doing most of the talking.

And then we got to this point. We were across the street, and as we crossed the road, he stopped talking and we were looking at the traffic and then we got to the other side and he talked again. And then after he left, I had this strange feeling in relation to it was, this is this I must have had something during the course of the day visit to admit. And so I'm rather fussy about this sort of thing. So I went right through starting from the beginning and wanted to have some breakfast.

And then what was going on during the day and then it came to me crossing the street. I God, that was it. I had this idea as we crossed the street of a characterisation of a point of no return, which was a sort of global characterisation. This is the thing I call the track surface. And this was a surface which you see as the matter falls inwards, you surround it. We're still now probably in the vacuum region outside the margins and collapses.

And you could draw the sphere around it. And it has a curious property that if you had a flash of light, imagine instantaneous flash of light on that sphere that the light rays converge both ways. So if you imagine this convex surface, the massive flash of light on that surface, on the concave side, the light rays will converge on the convex side. They will diverge on this track surface, they converge on both sides. Now, actually, this sounds strange, but there's nothing wrong with that.

If it's only a local region, what's wrong with it is when it goes all the way around the sphere and it's converging all the way around. And I call that a trap surface. And I realised from what I've been doing on this other problem that I knew, what if you looked at the future of that region that it had to have a boundary that was what's called compact, closed upon itself because of this focus? And then you could see that was inconsistent with the initial surface being vertical.

So that could you see there? I didn't have the best way of doing at that time. There was a slightly better way that Charles Misner, who and I read a lot from in Princeton and certainly a lot of things to do with the collapse issue here. One thing that I remember him telling me about this again, this question of whether it's a local, you see, one of the things that people worried about is when stars collapse. See, there's a lot of there's a lot of work.

I think Dennis Lim Cohen and so talked about this, the history of the collapse to black holes and a lot of this work was on stars. And when they collapse down to this what's called the short short radius, the density could be enormous. And so you might say, well, we don't really know enough about densities. And so maybe that's the problem that you see. But Charlie, isn't it? It's crazy, but it's not. There's nothing to do it. You've got a big, big city with lots of stars coming together.

And the overall density of stars coming together can be very small, much less than density of air. And it was still that this condition of a trapped surface. It this take you to working with Stephen Hawking. It did. And what happened in that collaboration when you see it was I. Well, I think it was probably December six before I gave a talk on this. I just I just thought of this idea with the track surface. I'm not sure where they'd actually written the paper on that point.

I think perhaps ahead had, but I gave a talk at King's College London at which the distinguished. Well, it's A.J. Helsing, whom I had learnt a lot from, he had a very geometrical way of thinking about relativity, and I learnt a lot from his books on that. But he happened to be in the audience, which is I was so proud of. I gave his talk on the singularity theorem, which was the one that eventually got the Nobel prise.

I have to talk about that. According to the film Stephen Hawking was in, there was sparks coming out of his head and being inspired by this talk. He wasn't there at all. That was just a fabrication. But Sir Dennis Sharma, who is in Cambridge at the time and he heard about the stock and he said, Well, would you like to come and give a talk or piece if you were to Cambridge? And that was early, perhaps January, in 1965, and I gave a talk there.

And Stephen Hawking was in the audience, and Dennis was very keen that I would talk to Stephen. So he organised a private session that I had with Steven and George Ellis, who was collaborating with Steven on trying to find some sort of a theorem about the Big Bang. But it was much more limited using techniques. It was I was using. So I gave a little private talk to them on my techniques. Stephen very quickly, quickly picked up on the techniques and used my theorem in the reverse time direction.

In the very distance it was, it wasn't quite so. Well, the argument was to show that my theorem could be used in reverse time direction to show that the singularity of the Big Bang was generic too. So you could imagine a situation where the Big Bang was not the beginning, but there was a previous collapsing universe and this switched around and did something complicated,

then come swirling out again, and that's what the Big Bang was. So I think that was quite a picture that was presented by people. I don't remember when that came out of you and Stephen Hawking working together it. I can't remember whether that was the prevailing view that people have had, but you see that was what Steven worked on. Yes. I mean, he was using the techniques which I'd use for the black hole collapse, but as applied to the whole universe.

But now in the reverse direction, he then developed these techniques much further than I had to use them and introduced other ideas, which, yeah, I had a condition that I've tried to describe. It probably would being a little technical, but it's more or less saying that you can have an initial. This moment, if you like, where you know what's going on in space time and then you can evolve your creations into the future. Now that's what's called the Koshy surface Kristie the famous.

Mathematician and philosopher Koshy had a hand the view that you could. Rockets that were shot across the coast, she was using the idea of developing equations to into the future, when can you do that? So we call this a koshy surface is very appropriate to consider these things. So that's the surface that you start from. When I say it's a surface of three dimensional space, surface in space science as a moment in time, if you like.

And that koshy surface evolution then gives you what happens in the universe now. Might it be like that? But it might be. Some other thing comes and joins on to your universe. So you have to consider those other possibilities. And so Stephen develops ideas of what you call Koshie Horizons, which are a very important idea coming from extending the things I did by quite a lot.

And then later on, we got together as a curious story to go together and wrote a paper together which generalise pretty well all the techniques we've had previously into one theory. Can I take you back to the Big Bang? The the saying that the Big Bang didn't start from nothing and it eased in previous universe, but that was an idea. Can you developed that? Because that's fascinating said there was a previous universe before us.

Was there a previous universe before that previous universe? And I think we shouldn't take this as a theory at the moment. I think it was more that people. I mean, yes, there were certainly people who discuss that kind of thing. Evidence is the word. So what evidence is there for this previous universe? None whatsoever. Oh. You see, it was more. I mean, there is now, but that's that's jumping ahead at that time. There was none whatsoever. It was just this wild speculations.

People were wildly speculating about all sorts of things you see. You see that Einstein had this wonderful theory. And people, this was a wonderful playground for making cosmologists usable, and this was what people were doing. When I say people love the mathematics mathematicians who maybe didn't know much cosmology or physics, or that maybe they were cosmologists who were trying to see what the universe might be like?

I mean, this is a simple model. You see the chart, you produce this early solution, which is the thing which describe Oppenheimer. It's not a use and so on. And the initial models are circles symmetrical black holes came from that. But there are also people working on cosmology. Where was this Russian mathematical statistical? Friedman and Friedman, who discussed various different kinds of global structures.

The universe might be especially close in, especially flat to much and spatially hyperbolic. That's the kind of geometry in my picture I showed you. So these are all possibilities that people considered, and they might be there was a Big Bang, or maybe you could have a solution, have a Big Bang. And so maybe other things. There are all kinds of things in the steady state model. You didn't have a Big Bang. So certainly there was a lot of discussion about different things that could happen.

And the point about the work that Steven was doing there was it narrowed these down that any of those models in which you had an initial collapse which bounced to an expansion and suddenly you could have Friedman type models, which which behave that way, whether people took them seriously. I mean, it was one which was called the oscillating, especially close, and it bounced like, this is happening. So the number of repetitions was quite a well known and studied model.

And so maybe something like that was true of the universe. So you are you. Which where are you in your thinking about it? Then do you think there was a previous universe of the one in which we let's go, let's go inhabit to inflation? How to think of the history here? Because initially you see, when I went to Cambridge and I got involved, the Danish German people, and they were all very keen on the steady state model.

And so I got quite keen on that. It was philosophically very nice because the universe was there all the time and it was there. And then there, you see, and so it was philosophically satisfying.

And I thought so, too. But then when they, Penzias and Wilson really discovered that there was this Big Bang and the evidence for the early stage of the universe in which it was extremely hot in the microwave background was just discovered seem to indicate that there was this early stage and that really put paid to the steady state. And I greatly respected Dennis because when he was convinced that this was true, he says, Look, I was wrong.

And he went around lecturing and saying, Look, now take the Big Bang seriously. There it was this initial stage where you might have to consider quantum gravity and all these things over that, and that was all going on.

But you see, Stephen was really arguing against the possibility of a previous collapse bouncing into it into an expansion, and that was that you could use the theorems in future directions and maybe strengthening my black hole results by maybe getting rid of the Koshy surface condition. And sure, you could do things like that. But I think he was primarily concerned with proving that there was a Big Bang. Yes, I was going to write John, but they said it didn't matter.

I'm going to keep going on like they can and brilliantly later. There's two or three questions. One is why are we talking about beginnings, Big Bang? You've described yourself as an atheist, but you've also said I quote, I think I would say that the universe has a purpose. It's not somehow by chance. Yes. You know, it's difficult to interpret what I was saying that I certainly didn't believe that there was no man with a beard who is one of them to this universe or anything like that,

a conscious being producing. And that didn't help at all. I mean, where does that come from? I don't believe any of that, but there is something in its in the connexion with mathematics, which is so deep in physics. The more one probes reality, the more one sees how it tied off properties with mathematics.

But it's not the kind of thing that you see. I guess it may have to do with my views on consciousness and that there is something in the time of mathematics which involves we haven't really seen that yet in our physical theories. But you have to prove that kind of mathematics to understand what it is that governs conscious beings and things which go beyond computation and things like that when I call it a purpose.

I suppose I would say in order to be a self-respecting universe, the thing which you might call a universe has to have beings in it that can perceive it, that something like that if I can try to expand upon what I was saying. So it's not that the purpose is to produce conscious beings, if you like. Maybe I could say it that way, but it's in a certain sense that it wouldn't even be universal if they weren't conscious beings in it, because otherwise there's been nobody to see it.

It's like, I don't know, it's a little bit contorted talking about coming to this and talking about it, about consciousness. There's a discussion going on or disagreement going on about the connexion. Excuse me, about the connexion between fundamental physics and the human consciousness going on. Thinkers like Minsky believe that humans are machines, and there are a lot of people who gathered around his case You're in the minority.

And as I understand it, you say that physics are inadequate, inadequate to explain the problem of consciousness. Could you give us some idea of the battle that's going on? We see again, a lot of these things go back to when I was a graduate student and I saw an important thing was I think I was a graduate student. I went to lectures and general about funding by Dirac on quantum mechanics and of course, by a man called Steam on Mathematical Logic, Steam Steam.

And he explained Girdles Theorem. And I found it completely stunning because what he said is if you have a system of proving results in mathematics, it's called a formal system and you could put this in the language of computers. So if you could put this formal procedure on a computer, and he told he talked about Turing machines and all that. So I wrote about all that. If you could put it on a computer, then all the things you could do.

How do you know what's true in mathematics when you have a proof and what is a proof? Will you follow certain rules, axioms and rules of procedure? Perhaps. And then if you follow them correctly, that's a proof that what girls showed is that there are certain statements that you can't prove this way, but that's not so much. What struck me, what struck me was that you could see, by the way, that girl constructed that particular statement reproduced is true.

Now how do you see it's true. You true? You see, it's true by virtue of your belief in the formal procedures that you're following. If you're simply following these procedures, why do you trust them? Why do you know that gives you truths when you understand them? Now, if you understand them, then you know, not just things that you can get using them, but you have things that you can get using your knowledge that they're correct. That was remarkable. That's what goes in does.

It uses your belief that these procedures give you truths and that gives you a truth beyond the procedures if you like faith? I wouldn't say so. It's like understanding what they've done is a word. You see, you sort of give them this thing and you and you follow. I dropped it, and it's a good question. It's not. That's exactly what it's not like. You see, it's like, understand you understand something.

I thought, What does this tell us? It tells us that understanding is not computational is not a computation, and understanding is something you do with your consciousness or consciousness is. It's very it's part of consciousness, as I understand, and it's that part of consciousness, which is a you enabling you to do something which transcends the computation, but the rest of your ego and it's nobody is really.

I the loss of complaints endlessly. But the argument which one can produce, it's not quite the argument, which tends to concentrate on that because I got myself all wrapped up in complications as a much simpler argument. But it's quite clear. I think it's quite clear to me and I don't know why people don't follow me very much. The understanding that God was a this understanding is something which is not a computation, and that thing is what consciousness is doing.

OK, you can make these machines rich and go incredibly well at chess or something, but they don't understand a damn thing. They don't understand anything. They've just either. They played zillion played itself a zillion times, and that's made passive recognition. And it says, well, it doesn't understand why such and such a thing works. And it's that thing which is not encapsulated without understand.

Why do you think that the opposition is a minority position in this argument, but the idea that that we can become we can be analysed like machines and provided we keep at it as it were, has being accepted by an awful lot of people as I understand it. I quite understand why it's a minority position because it means going beyond current physics. You see, you could and I had this view already when I was a graduate student, and it goes above beyond Newtonian mechanics.

It goes about special relativity, it goes beyond generals that you can put it on the computer. See these things like how you get these little simulations. Yes. Yes. Yes, yes. What about quantum mechanics churning equations? That's computational. But what about the amazing thing that I staring out of the window in Dirac came during this first lecture? Why is it that you can't have a piece of chalk in two places at once?

I didn't follow his reasoning because I wasn't this long, and I realise it's probably I could because there isn't a reason. That's where you have to go beyond quantum mechanics. So you see, it's a very outrageous thing I'm saying. I'm saying that what is going on in our brains, which is making us conscious, is going beyond current physics. It's not outside physics, it's outside currencies.

And it's what happens, this is the position I held then and still is that is what's involved in the collapse of the way function. Now the collapse of the way function, is this something which people brush under the carpet when they do quantum mechanics shrouding Delphine said Homer's homelessness by trying to understand how that can make sense. And he was saying, You know, I wish they had nothing to do with quantum mechanics. If I hadn't understood what was jumping, that has to go on you.

The jumping is you follow the shortening equation. Suddenly, it jumps. That's something because this or this or this, that's not following this equation. That's what's called the collapse of the wave function. We need a theory of the collapse of the wave function. When we have this theory and when we know how to harness theory, then maybe we can make a conscious thing. I hope I'm not alive when this happens, and I'm not sure I'm in any danger of that being the case.

But is this? Sure, it's part of physics. It's not part of current says. It's not part of computational physics. Why don't you have, you know, the low? I do not think it's something rather also about away. I think if it was there, I've changed my mind. So yes, I would shake his hand and say, Must you? Sure. But there is something rather rather frightening about this. I think, given that this machine, the idea of human beings actually being because.

You see, I don't think any of these machines have any consciousness whatsoever. It's not even the tiny bits, it's just muscle because they don't incorporate this thing, which they need to know if they built a device. There's no code in the machine. A device which could take advantage of you can harness the collapse of the wave function or what really is going on in the collapse of the. Then there is the potential for it to make a device, which is conscious.

And I certainly would accept that they're not going to do it with with computers. They're not going to do with quantum computers. Quantum computer doesn't harness the collapse of the wave function. So sure, it's got to be something else. But people who will be around when these things come along, we'll be quite happy to say shake its hands and say, yes, OK, you take over your business. That's all right. No, I don't mind things being better than I am.

I don't want to. The claim to be conscious when they're not at all. That's a wonderful place to stop. Thank you very much, Roger, that. And I was terrific. Thank you very much indeed. Very much and generous and your explanation? Thank you. Thank you.