Sir Roger Penrose - 'Einstein's Amazing Theory of Gravity: Black Holes and Novel Ideas in Cosmology'

it now gives me great pleasure to welcome professor Sir Roger Penrose Roger is currently emeritus rass ball professor of mathematics at the mathematics Institute at the University of Oxford as well as an emeritus fellow of Wadham College he's best known for his work in mathematical physics in particular for his contributions to general relativity in cosmology his deep work on general relativity has been a major factor in our understanding of black holes his development of Twister theory has produced a productive approach to the classical equations of mathematical physics his tilings of the plane underlie the newly discovered quasi crystals rogers long career has received distinction through a considerable number of prizes and awards including the 1988 Wolf Prize for Physics which he shared with Stephen Hawking the Copley medal of the Royal Society and the DeMorgan medal of the LMS in addition to his scientific work he's been instrumental in popularizing mathematics and engaging the public with work of mathematical scientists it's indeed a privilege to have him as our guest speaker this evening by a wonderful coincidence in addition to the Society's hundred and fiftieth anniversary this year also sees the hundredth anniversary of Einstein's theory of relativity on this very day in fact and it's in celebration of these two auspicious and historical occasions that Roger is here this evening Rogers presentation is entitled Einsteins amazing theory of always checking case they've changed the title Einsteins amazing theory of gravity black holes and novel ideas in cosmology Roger the floor is yours well thank you very much I certainly feel very honored to be able to celebrate both the 150th anniversary of the Society and of course also Einstein's general relativity subject which I have devoted a good deal of my life to I suppose people will have gathered something about general relativity or black holes or something from the film interstellar I may say I was particularly disappointed because it was touted as being something where all the science was worked out and exactly rides and everything like that and I must say I was a bit disappointed in that particular respect most particularly I was pointed disappointed by the black hole which I thought didn't look like when I thought a black hole was supposed to look like when I thought a black hole was supposed to look like was that or something like that you see here we have well that black thing here is the black hole but then we have a disc surrounding it now as far as I can make out there was a thin line across the middle of what was supposed to be a black hole in the film and there was no indication of the gross distortions that you get in the presence of a black hole and this is an early one based on what's the sort of an astronaut sketch according to the much more accurate illuminated in 1978 I think Brandon Carter was the one who suggested the idea and here you're looking at the black hole from about ten degrees above and you see light you see this is black what's that black thing there well you see that's the black hole again and in between that you see the disc you see there's a disc which is supposed to be in here and I think that was the thing that there was this thin line across the middle but there were no distortions that I could see at least maybe I missed them all of the star patterns and things like that which would be changing all the time and here we have this is the ring again they're seen the light gets deflected right around the black hole and you see things all over again you might see a lot more complicated things as well now black holes were thought to be in existence ever since I think Cygnus x1 was the double star system or double fing system one was a blue giant star I think that's why it's blue in the picture a blue giant star which was going around something and the something it was argued to be a black hole this was caught thing called Cygnus x1 the X referred to the X rays which were emitted these x rays occur this is a sort of close-up here you see there's the black hole and this there's the biskits what's called the accretion disk of material which rotates around the black hole and it gradually sucks it up but not sort of immediately take quite awhile and as the material gets close to the black hole gets extremely hot and you see it as x-rays so that was the idea and this is this edge on view as I suppose that was what they were meant to be showing in the in the film that I don't know why they didn't do the distortions which have made it look much more interesting and more accurately now I want to say you've probably also seen pictures explaining general relativity to the general public which probably have a rubber sheet and then there's a big ball sitting in that and a little ball sitting and the big roll makes a big dent in it in a little warm accident and this is in illustrating curvature of space-time I never was very impressed by these things but it seemed to me if the ball is sitting that's being pulled by gravity anyway and so the gravity is being caused by the earth and that's what's making it dip and that didn't seem to me quite appropriate but never mind it's a good first shot at these things so you'll get a feeling for the space being curved in some sense but I think you don't get much of a feeling of space being curved I want to give you a better feeling by trying to get you to think in terms of space-time because that's really well it goes to this idea I should say space-time that time should be another dimension in addition to the three of space was really not Einsteins idea at all or you know and Stein picked up on it a bit slowly he didn't like the idea first it was an idea to do two minkovski who is a Polish Russian German something who worked with Hill but very very distinguished mathematician who died young but before I think it was just before he died he produced a remarkable paper in which and a remarkable lecture which he gave on this in which he was explaining that space individually and time individually doomed to fade away and you only the combination of the two to which is what has an absolute existence it actually sort of invalidates the terminology to some degree because we think of relativity the name relativities is more or less saying and well things are relative to other things well it's true of time you see time is relative to how fast you're moving if you like but the space time is an absolute thing and this is what Einstein's general relativity was in terms of and I think Einstein came to regret the terminal terminology in later life particularly I think it was picked up by philosophers and people in a way which understand didn't like at all but let me show you how this geometry works it's the beginning of it and I should explain you see we have three dimensions of space my time tends to be going up the picture so you have a progression of time which would be something like a plane which would represent three space you move just move it up the the picture so here we have time going upwards and these are three world lines so that would be three observers if you like our clocks let's say there's three clocks this clock is going straight from A to C this other clock is taking a detour it goes to B on the way and then it goes to C all the time you have to think of these speeds as being less than the speed of light and this is explained in the second picture where we have these things called light cones they'll play an important role in what I want to say these cones represent how light would travel let me show you a picture of the light cone by itself it's perhaps good thing to do at this stage here we have the coordinates X Y and Zed you see you've got to have three coordinates going horizontally so a little bit of imagination is needed there and then the time going upwards and so here I use my picture before you see it slices through the cone and it's time moves upwards it makes a slice which where you see it look like a circle through this cone but you have to imagine there's another dimension so that's a sphere coming into a point and then moving out again as a sphere and that represents light converging in on that point and then coming out again so that's the picture that's what the light cones do and therefore you have to think of if these are actual clocks or observers or material objects then the world lines would have to lie within the cones they're not allowed to travel even as fast as light if they've got mass and photons would be allowed to go as fast as light but not faster and this means that if these are if that's at one o'clock and this is the other clock then this clock has to be have a time light world line that means it's its world line lies within the cone with the cone all the time smaller than the speed of light likewise this and likewise that but this clock would is this is what people refer to as the twin paradox you have somebody stays on the earth and somebody goes out to a distant star and comes back again that's the sort of image we have here always traveling less than the speed of light but quite close to it would be the idea and the distance in Minkowski geometry is the measure of the time registered by the clock so here you see this clock is you think is the sort of length of that line and that's the length AC call that a and that's C and a b a b c now if that we're in ordinary Euclidean triangle that you see that would be longer this Plus this would be longer than that but in this funny geometry admit cos keys this sum of that side and that sides are always less than that so you have to get used to that that's a funny idea but that's the way the geometry works and it explains or it if you like it illustrates the so-called clock paradox or the twin paradox that the clock which makes the root accelerates at this point so you can smooth these things off and make that generation and the the moving clock will register less of a time than the one which is going directly now you could have a curved line and then you have an integral along the curved line I won't bother you with details here and that gives you the length which is really a time so space-time geometry is really chronometry that was a name introduced by singh and i think it's a good name and here we have the space traveler going out and coming back again and having a length which is really the time as registered by that astronauts clock is less than this one even though it looks looks longer but it's actually shorter in this geometry now one of the wonderful things about general relativity it's now become an extremely accurately tested theory and this where you use people used to talk about general relativity by imagine little rulers drawn on the space-time but rulers are very bad for measuring the distance which is relevant in relativity theory they're very bad because they don't actually measure the distance here then the rulers are really strips you see and you have to see what point you're measuring on the edge of the strip that's not good at all wears clocks really do directly measure the geometry of the space-time and if you know what how clocks behave you can tell how distances are so here we have a clock going from this point to this point and a light signal goes out there and then these light signals come back so that they coincide when they get to the clock again then that tells you how far apart in spatial distance that point is from that point that's using units where the speed of light is 1 so they meet a rule in Paris which used to be the definition of a metre is not nearly accurate enough now we have much more accurate kinds of clocks I'll come back to that later it depends on the two most fundamental equations of 20th century physics put together and you see that the mass of a particle is has a natural frequency and that frequency frequency makes it a wonderful perfect clock so that's the idea and you have these extremely accurate clocks and they are well the clocks are so precise now that if you think of this time from the Big Bang to now I don't mean actually because the clock wouldn't survive in the Big Bang but just imagine that length of time that clock would keep time to a precision which is less than a second with them if it were a modern clock so you can get some feeling for how precise clocks I think probably that's out of date too and it's even more precise than that they're so precise the that's the effects of general relativity which I'll come to in a minute but just let me say something the effects of general relativity see if I have a clock on this table and a clock above somewhere then one of the predictions of generals if it is that they would run at different rates the one down here will be slightly slower than that but so much slightly that people didn't worry about it much but now if you just just move it about that far that far from the table to there we are so accurate that they can tell that difference thing the clock down here runs more slowly than that one to a degree that can be measured directly by clocks which is so precise and these precision in clocks is is of great practical importance because if you want to I don't know how many of you use the GPS to get here general one of these the position general positioning systems they depend upon general relativity for their precision if you didn't bring in general TV they we hope to still be very bad special relativity certainly you need but you also need general relativity where the space-time curvature comes in I'd better say something about space-time curvature because that's very important in the theory everything I've said really so far has been special relativity so you don't actually need to know about space being curved however this is the same picture as before but now imagine here's the light-cone you're looking sections through it it represents a flash of light here and here we have now the three dimensions worth there's the flash you match the colors and see how these pictures relate to each other and here we have an incoming flash focused on that point now you it doesn't mean there is actually a flat right there it's just that every point in the space-time has sort of painted on it just imagine that one of these cones and that cones tell you that those cones tell you what light would do if there were a flash of light then the light or the photons if you like would have to travel along these lines which are in these pictures tilted at 45 degrees because I've calibrated the space and time so the speed of light is 1 and that's a natural thing to do in relativity theory so if you're talking about seconds for time that means you're talking about light seconds for space years for time that means you're talking about light years for space ok and this is the flat Minkowski space the light cones are all nicely uniformly arranged a photon has to have a world line which is along the cones massive particle has to have a world line which is within the cones and in general relatively those rules hold just as well it's just that the cones are not so uniformly arranged there can tilt over in various ways now why did Einstein have to introduce this curvature into his theory it's one of the amazing things that he realized something like that but he didn't have the mathematics it was very fortunate that some hundred years earlier the really great mathematician bernhard riemann had initiated the ideas he really it was him and his followers who were able to have a very beautiful calculus which enables you to work out things in these curved geometries started from ideas due to Gauss but these were really only in two dimensions and the difficulties of doing it in higher dimension didn't arise and tell you until riemann injury well he was showed how to deal with these difficulties which was the great thing okay well why was Einstein wanting to do this sort of thing well it really goes back to the Galileo's principle if you like the principle of equivalence and here we imagine Galileo of course historians of science say he never did this but let's imagine he did this is the Leaning Tower and he's dropping a big and a little rock from the Leaning Tower and of course he says if there were no air resistance they would fall together and we imagine therefore a little insect on one rock looking at the other one and this rock would seem to hover in front of it and they gravity's been canceled out Einstein sort of raised this into a principle but Galileo was really perfectly aware of this I mean he described things like fireworks you have a firework like that and it produces the sphere of Sparks and they retain their spherical shape and he knew that perfectly well because he was aware of this feature here it's a property only of gravity I should say other forces don't have this principle of equivalence but this is the imagined astronaut and there is some futuristic Space Station and even though the earth is sitting right there they seemed not to have any as though gravity had been canceled out and it's just like the insect looking from one rock on to another well it's not quite canceled out it's not quite canceled out because the gravitational whether Newtonian terms the force has different directions and different strengths depending on how far you are from the earth and it's directed towards the earth and this means that when you try to get rid of the gravitational field by falling freely that's fine at one point but if you were trying to make that work in the neighborhood of that point you find that you can't do that in a flat space and that's really why Einstein was driven to UM he didn't look at it quite this way but it's basically the idea why he was driven to thinking about curved spacetimes and the analogy you must think of is something like this in two dimensions suppose you have two lines which initially parallel if the curvature is positive K means the curvature then they come together like this they're focused inwards if the curvature is negative is spewed out from each other so that's a generalization of this idea in four-dimensional space-time is the way one has to look at gravity and Newton Einstein's theory and here we have placed an astronaut out here and that astronaut is surrounded by Galileo's fireworks if you like Galleria suit Galileo assumed that the gravity was unified she's not quite because he did consider that the force would go in different directions but for firework the effect would be very small and here we have there's the astronaut freely he astral can be in orbit if you like as long as the particles are following around in orbit and then the force apparently due to the earth here according to the astronaut would be slightly inwards out there slightly stronger there than there and so therefore there will be distortion effect on a sphere so Galileo sphere wouldn't be quite a sphere it would start to get distorted into elliptical shape pointing in the direction of the earth if however used your sphere was so big that it's surrounded the earth altogether then the sphere would simply keep its shape and the the difference between those two situations is in the what the mathematicians would call the viol curvature and the Ricci curvature Ricci followed up on the ideas of Riemann and introduced a sort of simplified version of the curvature and Hermann vile introduced this opposite kind of curvature really the point is these are combinations of these focusing in different directions and if they're always in the same direction always in say so the focusing is just inwards that's the Ricci type if it's a stigmatic squashes and one direction stretches in the other that's the vile type and so I've Illustrated that here this is a very important feature of the way things work in general activity and suppose you have a star here and observer here where the star is going to be the Sun this was the famous observation that Eddington made in well there were two places but one was in the island of príncipe over the coast east coast of Africa west coast of Africa and he was able to see the effect as predicted by Einstein and I have a picture of it here let's imagine that the Sun was transparent and so photons who come right through and not with any refractive index so they just go taking a straighter path as they can then you see the Ricci part would be like on magnifying lens it would magnify the star field behind it and the viol part you'd see outside here which would be a distortion so these effects are well I Phyllis trated than the corner down there the distortion is what's called the conformal curvature let me not worry too much about that but I do want to show you pictures you see this was initially an idea Einstein suggested that you might see such effects but he thought it was way out and that nobody would ever see them the effects were much too small in modern astronomy cosmology it has become a very important effect let me just show you an example here I hope it shows up nicely this is one of the most impressive ones I think I'm going to show you the black and white one because even though I like the colors of course they're artificial colors but they give you an impression a bit the black and one right one I think shows the picture just a little bit better except it's got some scratches on it but ignore that I think you could see there is a sort of tendency for things to be just usually very sharply distorted there and there and there are other places there is a great mass in the middle layer which is stretched out the field of vision all the way around and that tells you there must I mean what you do is a statistical analysis you see galaxies unfortunately they tend to be elliptical and if you see one galaxy and you see it's elliptical you have no idea that if whether that's its shape or whether it's got distorted but when you see a general tendency but this distortion all the way around you can do with the statistics and work out yes that is a lensing effect and these lensing effects give you a very good indication of mass distributions which you had no other way of measuring so it's an important way of telling when there is mass out there somewhere in the universe you look at things which are further back way out further than that and the distortion just the effect I've been describing tells you what how much mass there is there in the middle it's an incredible thing this is this is a very important tool in cosmology and you may well have heard of the idea of dark matter dark matter can be seen and tracked using this method okay now what about the black holes that's where the matter gets very concentrated may well be a black hole in here but you don't see it directly probably in the middle of all this distortion there is a black hole a very big one I just suspect but we don't have direct evidence of that at least I don't know what the evidence is but I should describe a black hole in the way I like now this is not the picture you see of course in in the cellar this picture is something which you is very useful to tell you what's going on it's a space-time picture the time is going up picture as indicated here but I may have a little wiggle on the line because the time is not totally well-defined but you see it distorts the light cones to such a degree that they become I'm this cylinder thing that looks like a cylinder here that is the horizon of the black hole and you see the light cones are tilted inwards and that tells you because of what I said earlier that signals can't get up because if buzzes says some poor astronaut had fallen in here and said help then that signal isn't going to get across because it's got to follow within the light cones or on the light cones and it can't do that because the light cones are pointing inwards so that poor fella is doomed unfortunately and so this gives you a clear indication of that sort of thing now there's something else in this picture which is indicated namely this rather nasty thing in the middle that's called a singularity Einstein was not at all keen on these things he in fact never really accepted black holes even though he was at the same Institute as Oppenheimer and his students Snyder who wrote it wrote a very beautiful paper illustrating a very symmetrical collapse down and showed that that you could get that through what we now call a horizon and there would be a singularity in the middle now this singularity is a place where the equations go here why basically the densities become infinite and more importantly the curvature becomes infinite so I was talking about curvature this distortion that is that the gravitational effects are and these become infinitely strong down here see people sometimes use the term spaghettification that means you see if you were unfortunate enough to fall in the thing then if your feet were you went in feet first let's say then your feet would be stretched and your head stretched in the opposite direction and you'll be squashed in the other direction so you'll be drawn out into inter spaghetti so they say not not very nice spaghetti I'm afraid but that's the picture that these tidal effects will get very strong in fact they get very strong you don't have to get to a black hole in things called neutron stars it would already be extremely strong ok these singularities I saw had various ways of arguing he didn't think this would happen and this was one of the things that people worried people in the round about I think particularly in 1962-63 I think it was when Martin Schmidt Dutch astronomer had noticed that there seemed to be objects which were so immensely first of all immensely powerful I mean this thing was brighter than an entire galaxy but it was also small comparable at least to the size of the solar system because it varied is a certain rate you kept in varying intensity more or less and more or less like this suddenly regularly but this seemed to indicate that from relativity considerations it couldn't be huge it couldn't be bigger than about solar system and the two facts the fact that was meeting so much energy and still small seemed to indicate to people that it must be of a general size that would collapse according to this picture so something like that would happen and so people started worrying and I also started worrying up at that stage it was John Wheeler I think who persuaded me I should be worrying about it and the question was was the model that the Oppenheimer and Schneider has suggested was that likely to be realistic or was that a fluke of the symmetries and so I was able to argue that this isn't a fluke of the symmetries and that you do get this situation very generally and then Stephen Hawking developed this to the cosmological case and he proved similar results theorems I should say these are the original models due to in 1922 Friedman the Russian mathematic mathematician and and then the maitre and various other people showed that there were solutions of this kind where again times going up the picture and according to the three possible kinds of general symmetry here they are you are again assuming exact symmetries in these models otherwise the equations are too hard to solve these are the three kinds of geometry you can get to two dimensions but it's the same in three dimensions which are uniforms these three uniform geometries either the flat case that's the Euclidian case or the spherical case that's the positive curvature or the negative curvature case these are all very beautifully illustrated by the Dutch artist MC Escher and I want to say a little bit more about those later on one of them I Scott lost track of the time when it actually starred sorry I've got half an hour good yes okay now see Iceland didn't like these things the singularities and I think at first he was very bit rude to poor old Friedman saying you know that okay your mathematics is all right but you don't understand physics or something like that which is a little bit unfair but he came around eventually I think particularly la Mettrie convinced him that we sign originally there's an interesting story here because originally Einstein proposed a model in which the universe did not expand you see if it expands see as time goes up this means expansion if it expands then you will see there could be a problem at the beginning go backwards in time looked as though all matter is focused on this point at the bottom and that gives you a singularity now Einstein thought well let's assume that the universe is static unchanging with time so it looks like a cylinder in this picture this is still in the cylinder going up up the picture and that was his model in order to have that model he needed to introduce well I just repeating what I said here you get infinite densities and temperatures but he introduced what he subsequently regarded as his greatest mistake that is the cosmological constant now the cosmic cosmological constant I'm it was reported by Georg gamma that in a conversation with him Einstein had said introduction of this thing which is often called lambda it cosmological constant was something he introduced this so you could have his static model you see but then Hubble and various other people began to make it pretty clear that the universe was actually expanding and so Einstein the mistake he considered he'd made was by introducing this he could have a model which didn't expand and since it really does expand he sort of missed predicting it now it's quite curious historically curious that this particular family of solutions well something like them do seem to be close to the actual universe in fact it does seem that this mistake of Einstein's was actually a brilliant idea and it represents actually what a feature of a actual universe I'm afraid Einstein didn't live to see that it's the Nobel Prize I guess it was in 1998 now that's where they the prize was later is in 2000 something but they made these observations Perlmutter Schmidt and kerchner made these observations on very distant supernovae stars exploding stars and so they could tell any feeling for the geometry in its expansion at very great distances and this seemed to indicate that the universe you see the previous models look very much like this ones here they start off pretty well the same but then the universe starts to do this accelerated expansion and people now refer to this as the mysterious dark energy or something but actually Einstein's Landa introduced he thought as a mistake but actually for the wrong reason yes but it turns out actually to so or something like it seems to be true of our world and that's actually very important for something I want to say I don't run myself out of time too fast you notice that the future behavior is whereas in the previous cases it's sort of modeled the space so if the space was closed then the time is closed you see it expanded and then collapse back down again so there was a big bang and what they call a Big Crunch at the end if the universe was open in either these two cases then it would simply continue indefinitely open in time as well however with the cosmological constant this nice feature is removed and provided the value of lambda compared with the density and so on is reasonably large it's very small but still reasonably large you still get this exponential expansion whichever model you take so the picture it's nice in a way because you can encompass everything in one picture positive cosmological constant lambda the solutions all look sort of like this now you see I've put this kind of frilly stuff at the back not because I think there's a frilly stuff at the back but just because I think that the universe might be closed it might be open it might be hyperbolically open we just don't know it's very close to being flat that's certainly true but that doesn't tell you that it's not one or the other or the other okay so this is that my sort of Universal picture of cosmology as we now understand it ah one thing the experts will tell me I forgotten to include is this thing called inflation which is argued to take place in the very early universe well the two reasons I haven't put it into the picture one is maybe I have because it would be so small you wouldn't see it in this picture the other is that I don't believe it now that the second one's a bit tricky but the first one let me say something about let's suppose it is there in order to see it I'm going to bring out my magnifying glass which I have to suppose is a very powerful magnifying glass and what do we see well we see something like that according to the inflation as' and this is an exponential expansion which was supposed to have taken place in the early universe you see it's very similar to what is observed to be taking place now and that's the idea it was supposed to be in there was a sort of cosmological constant which was much much huger that took over in the very early stages and then for some reason turned itself off although that's always been a weak point of the theory - how to get it to turn off let live it nevertheless that's the general picture ok well now here we have that picture again whether or not there's a inflation there I'll come back to this in a minute one thing I should include in this picture since I'm putting singularities in are the black hole singularities so that is something like the picture that we actually have ok schematic in many respects now this does illustrate an important thing I can't go into one into it here but there's an important law of physics known the second law of thermodynamics and that second law of thermodynamics tells you roughly speaking things get more and more random as time goes on well that's fine we see that and that's certainly a very powerful feature of the world as we see it however there's a huge paradox or something you see one of the reasons that we believe now that there is a big bang of clear evidence for it that is this radiation coming from all directions called the microwave background and that microwave background is argued to be a remnant or at least the close to the Big Bang let's say close it's something like 300,000 years after the Big Bang but that's pretty close in this picture right down there somewhere this radiation was created in the hot universe and it cooldowns the universe expanded so that now it's only about 3 degrees or whatever it is above absolute zero but nevertheless it's an important feature that you have it have and that seems to indicate that the Big Bang was there and inflation was sort of posed to be saying well the uniformity of the universe which we also see in this radiation very very uniform to something at one part in 100,000 the uniformity is indicated by this or temperature variations being very slight as to go around the sphere of the sky and so it DS the universe is very uniform now inflation was supposed to explain this you see it's supposed to say well maybe a universe wasn't that uniform to begin with but then this enormous thing smooth it all out trouble is that doesn't work doesn't work for it rather sort of obvious reason which I'm a little surprised why people tended to think it should work see what I'm doing here I just turned the picture upside down you have to imagine that time is now going up the picture if this is a solution of Einstein's equations then that is also because the equations work both way in time however if the second law is working in this one you expect the entropy to be going up here and this way well if it's going up what do you expect to see what you expect to see is all black holes forming congealing with each other and producing one unbelievable mess at the end now if that could happen according to Einstein's equations inflation or not it doesn't matter whether you put the equations of flashing in there or not this is going to happen if the universe was collapsing with these irregularities and so the big puzzle is why was it not like that it would have been like that we wouldn't have had a second law of thermodynamics it would be a great mess nothing like the world we see okay so my view is inflation whether or not it's there can't be the answer there's got to be another answer now in order to explain its other answer I have to give another idea to to you here which is I should say that everything I've said up to this point is absolutely conventional general relativity I want to say something which is not conventional general relativity and so people complain about it however it does have various advantages one of them is the second law problem and the uniformity of the universe is another one now but there's no inflation in this picture now what I want to try and persuade you of is that a good way of looking at the geometry of space-time remember I was talking about light cones and unfortunately I've probably forgotten oh here we are no no we I have forgotten well I should have put this out so I would keep track of it and I have okay here we are good remember the light cones they describe a good part of the geometry of space-time in fact most of it and I'll quantify that in the moment but not all of it the cars remember I started off by talking about clocks so we need to see what clocks do as well as the light curves okay what the clocks do well I put a clock in here and that clock well what happens is the ticks of this clock or of other clocks that might be going through that point at different velocities they form sort of bowl-like surfaces here and muckman hill alike services on that side those represent where the ticks first stick second tick third tick and so on so it gives you a scale so they'll worry about the details of that but the point is that the full metric of space-time that you that answered our needs for his theory is not just the like cones but this scaling as well which tells you how how fast clocks tick okay so that's the full geometry that is just the partial what's called the conformal geometry conformal geometry I've got some words here to say what it means basically it's the geometry of angles but in relativity you can switch the geometry of light cones that's the same thing okay now conformal geometry is respected by matter which has no mass in particular photons so photons are interested in the conformal geometry they're not especially interested in the full geometry and this picture here sort of illustrates that in fact the top part here gives you a bit of physics which I just mentioned a little earlier why are clocks so wonderful in measuring time nowadays well they depend on the two most basic formulae of twentieth century physics of course one of them has to be Einsteins e equals mc-squared the other one is e equals H nu or probably people call def or something now that's a frequency Einstein tells us energy and mass are equivalent there's a constant there but energy and mass are equivalent Max Planck even though earlier told us that energy and frequency are equivalent and that again is a constant that's Planck's constant but the two together and you see that frequency and mass are equivalent so you have a particle of a definite mass the stable particle that behaves as a very perfect clock and okay you you're tapping into the we act in more enormous precision there is in the ticking of clocks because of these four basic formula so that's where it comes from so mass gives you precise clocks on the other hand the reverse of that is if you don't have mass you don't have clocks and you can see that in this picture here see these clocks are whizzing by and they tick you can see their ticks is the intersect these surfaces a photon goes zipping along and never hits the first one that means if you were a photon looking out of the world the beginning in the end of your world line would be instantaneously following each other there'll be no time in between okay now you need mass therefore to measure time and that's an important thing this is illustrated in particular by the wonderful equations of James Clerk Maxwell who showed the equations of electromagnetism he demonstrated what they were and photons are basically things which are the quantized version of these equations of Maxwell that's what they are there Maxwell's things but sort of bubbled up into a photon according to quantum mechanics but the point is that Maxwell's equations don't need or they don't need the scaling they don't care about the scaling the nine components which tell you where the light cone is are all it needs not the tenth which gives you the scaling C the metric of Einstein's theory a thing offer a G a B or G mu nu or something tells you the geometry at any one point has ten numbers to define it nine of them tell you where the cone is and the scaling is the tenth one that gives you a lot but lots of physics only need the nine and photons physics without mass where gravity is a special case because it needs to know mass for its source but anything else if it has no mass then it doesn't need the scalings it just needs to conform all geometry now what I want to do is to use this idea to the whole universe right back near the beginning the Big Bang here you say well part all's whatever their masses are right way down near the Big Bang are going to be effectively without mass because the energy that's around will completely dominate the energy in that particles mass so it must well be considered not to have any mass so this you can use conformal geometry for in the remote future there's a little bit of a problem with black holes but they will disappear according to Hawking evaporation let's not worry too much about them the universe will smooth out and it'll be dominated by photons so you could say again it's the geometry the conformal geometry which is the important one now this is useful for a couple of tricks these are two mathematical tricks which we played with for a long time one of them is stretching out the Big Bang so it looks like a surface it looks like a line here but bear in mind is a three-dimensional surface it explains an awful lot of things and if you look at it that way all sorts of things about horizons and so on very nice way of looking at the Big Bang and if that's what the physics is doing down there might as well be like just starting off from a surface here future infinity well this is the old picture that I just showed you of ashes which keep nicely illustrates it conformal geometry you can represent infinity as a finite boundary and that's what this nice picture of Escher's actually shows you so conformally you can think of infinity as a boundary okay that's all nice and okay a little unusual but it's perfectly all right physics well it's perfectly all right you can squash down infinity to get a smooth future boundary there are theorems to tell that that's good stretch out the Big Bang singular to get a smooth initial boundary well that is a huge restriction in fact my colleague Paul Todd in Oxford said now this is a wonderful condition mathematical condition to impose that is to say you could stretch it out so that it's nice and smooth and you could continue it mathematically but who knows what that would mean so don't do it physically just imagine it mathematically and that's a nice way of expressing the conditions that we need on the Big Bang that it has low entropy in exactly the form that we seem to see so these both for quite different reasons are good things to have okay well there's still nothing outrageous about what I've said here I'm now going to say something outrageous except I might have lost my slide now I haven't here we are my register is that indeed you can extend in both directions and that extension is not just a mathematical fiction it's real but physics really has that property that the remote future of our I'm going to call this our eon you see I'm saying that what we think of as the entire history of our universe without the inflation you see I don't want the inflation particularly for all sorts of reasons but particularly because well it's there in another what's the inflation you see in the current theory right after the Big Bang in ten to the minus thirty six seconds or something that was this inflation took over for no good reason I mean you can paint an equation I mean what they do is the sketcher a curve for the potential function and say well oh no that isn't good let's try this one I mean there's no theory which tells you what that stuff should work like I think they call the in photon field here you've got an inflation of a kind already given it was in the previous Eon you see it was before the Big Bang not after it and this is the kind of inflation the exponential expansion that seems to be supported by observations but not tucked in here it was actually before and the continuation can only be understood if you're looking at the conformal geometry of massless things so the argument is that there's nothing with any mass here or here so you get away with it that's the idea that's the idea and the claim is not just that it's the idea I had this idea about ten years ago and I thought nobody would ever detect whether this is right but I did wonder about this a bit maybe signals could get through from before this is Ariane say get through from here to here and then I thought what's the most violent thing I can think of that might have a chance of getting through most violent thing I can think of well think of our galaxy our Milky Way galaxy in its center is observed to be a black hole of about four million times the mass of the Sun now our galaxy is observed to be seems on a collision course with the Andromeda galaxy okay well it'll be awhile before we hit it a few thousand million years but that's not the point the point is that we've probably hit it it has a black hole which is about two orders of magnitude bigger it's much bigger than our black hole I forget the exact figure now that people claim for it what's likely to happen is not that they will crash into each other but they will miss but feel each other out so they will spiral found each other and a few more thousand million years probably they will swallow each other up in some great violent event maybe they are miss then and the galaxies go through each other and slow down and then they've come back and have another go and they'll keep doing that until they settle down as one big galaxy comprising both of them stuck together and this big galaxy will have a backhoe eventually as they spiral into each other of much larger size than either of them right now but the point here is that in this process they will emit there will be one great bang it won't be a kind of bang you can hear it would be the bang in the form of gravitational waves ripples in space-time which we believe should be there this sort of observes to be there already although the tests which are more direct that should indicate whether or not they're there but the indications are that they must be there then those gravitational waves will have an effect at the crossover on the microwave background which we should be able to see they would be effects in the form of circles which will actually form concentric sets because in a cluster of galaxies the point that they reach in the crossover will be about a center of what I'm saying you will see rings and these rings can be detected in the slight variations in temperature as you go around and Armenian colleague of mine waha Gossage on claims to have seen these things already and a Polish group headed by Kristoff Meisner in quite independently seemed also to have seen rings that using completely different method it seems though they are there if they are there it's hard to see what explanation in ordinary cosmology is going to give you such effects but they are predicted on this scheme I think we just have to wait and see people having a lot of trouble believing them I can understand that my minion colleagues observations were repeated by a sceptical American group and they did see them but seem to argue that they could somehow come about from the conventional explanation I didn't understand their explanation I think we just have to wait and see until more specific tests are made which may be much more definitive one way or the other thank you very much your kid Johnson got some questions worked out so we have time for a couple of quick questions literally just to okay who would like at the back yes good question yes you probably entropy supposed to be going up and up and up and up and up and up indeed why isn't it in this picture well it's a subtle thing according to this picture where you could ask what is the largest entropy in the current universe the largest entropy is by far in big supermassive black holes so black holes is where the entropy is gone as the stage of our own Eon now according to Stephen Hawking these black holes will gradually evaporate away it takes an awful long time the biggest ones that will take something like 10 to the power 100 years or Google years north a long time but in that time they'll eventually disappear now Stephen also originally said that the information gets swallowed by the black holes now I think he was right however later on he changed his mind because this violates certain physical principles to do with quantum mechanics that people hold dear and prefer not to be violated so personally I think those principles have to be violated and this is a good example of why because the reasons that Stephen gave originally I think are pretty powerful that the black hole swallows information and since that's the major contribution to the entropy that gets swallowed away now it doesn't violate the second law it's a kind of subtle thing but the degrees of freedom that that black hole swallows are somehow removed from consideration so you would say well in calculating the entropy of the universe should we call as consider those degrees of freedom or not when the backhoe is gone is hardly any point say okay we won't count them that means you've now renormalized or changed your entropy definition to be much smaller than it was before so it's a quite a subtle thing the entropy can come down only because you change your mind about how you define you to be whether or not you take the swallowed degrees into consideration it becomes stupid to take them in consideration when you get very late here and cross over into the next Eon so the entropy the new entropy is lower than the old entropy but the new that the entropy as a whole if you like is still going up but it's it's a subtle issue it took me a long time to come come to that conclusion I think that's the right answer and clearly it's a very important point that you raise because second law was a major reason for considering this scheme in the first place one more quick question we've we we're about to break for a drink for which you will need your little yellow coupon I have six and Roger will have six as well in a minute but before we have the drinks let's thank you very much indeed for a fantastic talk you

Channel: London Mathematical Society

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Keywords: Roger Penrose, Mathematics, London Mathematical Society, Academic Lecture, Science Musuem

Id: xAcvNnSrkcM

Length: 56min 22sec (3382 seconds)

Published: Mon Jan 04 2016