 I have never seen this room so full and so quiet, so that means that I don't have to introduce the speaker because I think you all came, you know what you came for. So, but anyway, so this is one of the highlights of the year for ICTP and for Trieste, so we're very pleased to have Keith Thorn visiting and he kindly agreed to give us a colloquium, even though he's in the very, very, very tight schedule, he just arrived at noon and living after the talk, essentially, but he was generous enough to agree to address the whole audience here at ICTP and Trieste in general and we are very happy that he did so. So, usually you have to introduce a speaker and it's hard to find something that you already don't know, but let me say some things anyway, Keith is famous for many things, I'll try to go to some of the highlights, he was a student in Caltech and graduated in 62, then he went to Princeton to work with the great John Wheeler, who is a very big thing because John was one of the greatest thinkers of the last century and in that sense he shared the same supervisor as Feynman, now other great physicist, that he came back to Caltech and again shared time with Feynman, like he was telling us some nice stories about Feynman at that time and he has been a major figure in the field of general relativity. He wrote this famous textbook called Gravitation, the Misner Thorn and Wheeler, some people know it, call it the Black Bible and it has been a very influential book for many generations, in particular for my generation, we started learning many things from that book and then he has been doing many other things like writing popular science books, his books on black holes and time warps was a very lively book to encourage people to learn more about many things about gravity, wormholes, many things that can be addressed with general relativity and some quantum mechanics, but part of it also he tells a lot of the history of the time, how things were working in the U.S., in Europe, in the Soviet Union and so on, regarding that subject, so it's a very lively book and I really enjoyed it many years ago and I'm sure if you haven't read it I would recommend it. Very visionary somehow, Keith just came with supporting the idea of searching for gravitational waves many, many, many years ago and he has been the major figure on that up to the level of the great discovery, probably one of the greatest discoveries of all time in the last couple of years of gravitational waves from the merging of two black holes and that very fairly justified that he shared the Nobel Prize in physics last year. Usually when I give talks, I was used to give that as a prediction of my talks, usually there would be an Nobel Prize 2017, there has been the only prediction that I had been successful so far and Keith is also now a public figure also because he's a participation with a famous movie interstellar and so we don't know if they recognize him in the street because of interstellar, because of gravitational waves or because of Nobel Prize or all of together so that came out also relatively recently and I know that he was telling us that he's also working for a future movie in collaboration with Stephen Hawking so I cannot tell you more about that and he cannot tell you more about that neither so that brings also the figure of Stephen Hawking that we sadly lost this year and that's one of the memories I personally have about Keith is that I consider for my time in Cambridge, Keith probably was the closest friend to Stephen and he came to visit Stephen in very difficult times and he was flying all the way from California to always be with him and that's something that we all appreciated being in Cambridge and I'm sure Stephen appreciated it and so that brings me to an anecdote that I didn't tell you this launch but it's something that that's the only original thing I can say that you haven't heard is that for Stephen's 17th birthday there was a big celebration, it was a major dinner and as usually I forgot to register for dinner and then when I realized he was too late and then the organizer told me okay anyway dress up come to the dinner and if there is someone canceled if someone cancels we can fit you in because otherwise everything is fit so I did that a black tie appeared at the dinner in Trinity College in Cambridge and actually someone canceled and it was Stephen, Stephen was sick he canceled so I was giving Stephen's chair and so who was beside Stephen the top person to be with Stephen so poor Keith had to share time with me instead of Stephen so I don't think he was very disappointed so but that's already illustrates how close he was to Stephen so anyway so I don't take more of the time so please join me to welcome Keith Urn. With a quotation from Stephen Hawking can you hear me always began his lectures but you didn't answer me can you hear me in the back yes okay I want to talk for physics students physics researchers and people with other backgrounds who know some amount of physics about an area of physics that is not often discussed but one that I think is very interesting and is likely to have major discoveries in the next few years this is what John Wheeler called geometric dynamics by which he meant the non-linear dynamics of curved spacetime and so my lecture does begin with John Wheeler this is a photograph I've took of John when he was giving a lecture in Cambridge England Stephen's home area this is Willie Fowler dear colleague of ours with a bald head there he had a little bit of hair around it I don't have I don't share that here and John was talking about the non-linear dynamics of curved spacetime black holes wormholes and other things whose existence in dynamics is owed to spacetime curvature and this was in the 1960s in the 1950s and 60s he was telling us his students and his colleagues that it would be very interesting to study geometric dynamics and learn how spacetime behaves and what I would call a storm and is highly excited and changing very rapidly highly warped changing very rapidly and he urged us to try to solve Einstein's general relativity equations to figure this out and so we tried and we failed before I tell you about the failures and element success let me just remind you that non-linear dynamics elsewhere in physics plays a very interesting roles fluid turbulence tornadoes owe their existence to non-linear dynamics of the fluids phase transitions in condensed matter non-linear optics all of modern optical technology involves non-linear dynamics in some manner uh colliding soliton solitary waves and fluids plasmas and non-linear crystals and optical fibers and in mathematics chaotic maps strange attractors all of these fancy buzzwords their uh aspects of non-linear dynamics and so non-linear dynamics is really very interesting throughout physics engineering and modern technology so the question here that John asked us is what kind of non-linear dynamics do you have in curved spacetime and he challenged us to explore that in initially in the 1950s he wrote a book called geometric dynamics with that title in 62 which was basically a complication of his early papers on this from 1958 to 1998 all we had were analytical techniques with pencils and paper and we made very little progress John foresaw this and he encouraged his students and colleagues to develop numerical relativity developed the techniques to solve Einstein's equations on computers and the first work with numerical relativity was by his students by Richard Lindquist by Charles Misner and by Karen Hahn who was a computer scientist she was a computer scientist at IBM and then Bryce Dewitt took this over and then turned it over to younger generations of people Larry Smarn various younger people but it took four decades to near fruition around 2000 in the late night in the 1990s you began to this began to be mature enough that you could start to learn about geometry dynamics with computer simulations but it took 15 more years to get sufficiently mature by about 2012 that we could really begin to study geometry dynamics in depth in generic situations so I'm going to tell you this history and then look into the future as to where this is going I'm going to talk about four arenas for geometry dynamics metrodynamics near spacetime singularities I'll explain what these words mean in a few minutes nonlinear self coupling of gravitational waves and what's called gravity critical gravitational collapse the collisions of black holes which create what I call a storm in the geometry of spacetime and finally gravitational wave observations of the geometric dynamics that you get when black holes collide so those are the topics that I'll talk about four topics four areas in which we have had major progress even including even now the observational side of things so this is now both the theoretical area subject area and an observational subject area thanks to this discovery of gravitational waves and this was one of my motivations for working on a gravitational wave experiment I could see that this would be the ideal way the only way really observationally to study geometric dynamics so let me begin with geometry dynamics near singularities this is a fairly old story but his story and which I was personally involved but on the sidelines it was my colleagues my friends who were doing this uh it begins with Robert Oppenheimer who who although he was much older than me an earlier generation I did know at the institute for advanced study when I was a student he and his student Harlan Snyder in 1939 gave the first calculation of the implosion of an idealized star really a cloud of dust imploding to form what we now call a black hole with a singularity and a center and a black hole horizon around it and we could look at that solution for the final black hole called the Schwarzschild solution of Einstein's equations and we could see in that in the equations of that solution the thing that you hear about if you hear stories of black holes the astronaut or in this case my wife who falls into the black hole or hangs here and is squeezed from the side by tidal gravitational forces and stretched from head to foot because of course the gravitational pull on her feet is stronger than on her head her head is farther away and so that difference in the gravitational pull constitutes a net stretch on her this is the tidal gravitational force is the same tidal gravitational forces has produced the tides on the earth created by the moon and the sun from the period of 1939 when Oppenheimer and Snyder explored this saw a black what we now call a black hole form until the 1960s there was great debate great confusion great controversy over whether singularities like you have at the center of the black hole where these tidal forces become infinitely strong do they really form in the real universe and what many perhaps most physicists believed in that period from 1939 to the 1960s was Oppenheimer and Snyder studied a very idealized absolutely spherical collapse of a star but if you had a star that are dust and material that imploded in a more generic way so it was not spherical so there was rotation that the particles would go in they would fly around each other and fly back out and no singularity would form and this I think was the dominant view in the from the 39 until the early 1960s Landau had a great school of theoretical physics in Moscow and Russia and two of his young colleagues you have getting lift shifts in Zeiss at Kalatnikov in 1960 completed a long calculation using analytic techniques pencil and paper with the Einstein equations in which they use various approximation techniques perturbation theory type techniques and other analytical approximations to try to figure out what goes on and they came up with an answer that indeed you will never get a generic singularity you'll never get a singularity if you begin with generic arbitrary initial data and you implode some object it will not form a singularity and they codified this in a textbook called the classical theory of fields the addition of the book that I studied as a student it was the second English edition I think the fourth Russian edition published in 1962 there's a section about the absence of singularities in the general cosmological solution really in the general solution of Einstein's equations and a long analysis explaining why this doesn't happen this is a picture of lift shifts and of Landau that I got from lift shifts his wife many many years ago because I spent a lot of time in the 1960s 70s and 80s in Moscow well in 1965 Roger Penrose a young mathematical physicist then working in London introduced a whole new way to try to understand what Einstein's equations say he introduced the field of mathematical subject of differential topology into general activity previously general relativity been studied only using techniques of analysis not techniques of topology that is geometry techniques and analysis techniques but no topology and using these new techniques not new to mathematicians but new at the time to relativity theorists he was able to show in technical language that time like world lines end inside a black hole but that means that you if you fall into a black hole you move along a time like world line necessarily and you die you cease to exist but why you cease to exist he couldn't say he could just prove that this is what happened in a general situation has a techniques couldn't say anything about the details of the singularity where you ceased to exist and he presented these results actually in a conference in London in 1965 that I attended that lift shifts in Kalatnikov attended lift shifts in Kalatnikov were rather disturbed by this because basically Penrose was saying you were wrong your analysis is wrong I don't know why your analysis is wrong but I am giving you a rigorous analysis you had an approximation analysis and you made a mistake somewhere and so from then until 1969 Belinsky a young student of theirs together with Kalatnikov and lift shift struggled until 1969 they discovered the error and I remember I was in Moscow I think it was 69 maybe it was 70 when they had just discovered the error and lift shifts gave me a manuscript to submit in his behalf to physical review letters in that era they could not send a manuscript out to the west until it had gone through the sensors and he did not want to delay sending this through the sensors because he wanted to be the one who had confessed that they had made an error did not want somebody in the west to point out their error and so he wanted this published as quickly as possible to demonstrate that they had found the error and so I carried this out and submitted it to the physical review letters for him and as far as I know he never got into trouble with it this is a photograph of lift shifts Kalatnikov and Belinsky together with me and John Wheeler here the photograph was taken by Charles Misner when John and Charles and I were visiting Moscow a few years later and so what was the mistake well the mistake was basically that they had tried to find the general solution to Einstein's equations near a space-like singularity one that can be reached through a series of space-like slices in space-time and they had missed the general solution they just simply missed it the general solution was much more complicated than they expected it turned out to be a generalization of something called the mixed master singularity it was discovered independently by them and by Charles Misner and this was a generalization of it and the story goes like this as you approach the singularity according to their approximate analysis and you move along this time-like whirl line your friend moves along that time-like whirl line time is plotted up in space horizontally as we do in relativity you become causally disconnected from your friend there's no way for you to influence each other because of the nature of space-time in some sense you're moving farther and farther apart you're losing causal contact with each other but you're each moving along a time-like whirl line and as a result the partial differential equations Einstein's partial differential equations can be approximated by ordinary differential equations in time as you approach the singularity and these could be solved then by standard techniques but the issue was how fast do they decouple how strong is the decoupling and that was where there was a lot of criticism of this the resulting temporal dynamics as I say is what was called mixed master matter has negligible influence on this and so here's the description of this mixed master dynamics it comes from my book from many years ago black holes and time warps so I'm plotting upward I remind you as you approach a a black hole singularity of this sort that was found by Oppenheimer and Snyder as you approach the black hole singularity you get scratched from head to foot and squeezed from the sides but what happens is you approach this general singularity is that the stretch and squeeze oscillate and so in say the east west direction you have a squeeze so I'm plotting stretch up and squeeze down you have squeeze squeeze squeeze squeeze but in the north south direction you have a squeeze then a stretch then a squeeze then a stretch in a solitary way and in the up down direction the opposite stretch squeeze stretch squeeze so the three different axes three different directions behave differently there are principal axes involved in this the mathematics is that of a symmetric trace-free tensor which has three principal axes along which you have the pattern of squeeze squeeze squeeze squeeze or stretch squeeze stress squeeze or squeeze stretch but then there comes the end of these cycles they make up an era and at the end of these cycles the principal axes of stretch and squeeze rotate in some manner that's governed by spatial curvature and then the whole pattern begins all over again and this entire pattern is described by a chaotic mathematical map sometimes called a continued fraction map so analytically they discovered that in geometric dynamics you do have chaos you do have chaotic dynamics like mathematicians and physicists study in other areas of nonlinear dynamics uh and uh so this was quite an amazing discovery but in the west by the west I mean Europe and the United States where perhaps influenced by Penrose and others and many others there was a demand for a higher level of rigor there was enormous skepticism and so this began to be called the Balinsky Caliph Lipschitz Kalatnikov Balinsky Caliph Lipschitz BKL conjecture about how geometric dynamics goes which had been tempted to be proved by heuristic arguments and throughout the English particular French and American literature these phrases were used you would find them over and over again heuristic arguments BKL conjecture it was only when numerical relativity became sufficiently mature that it was possible to solve Einstein's equations to explore this conjecture that a program was developed originally devised by Beverly Berger and Vince Moncreef he was at Yale Beverly was at Oakland University Beverly went on to become our program director for LIGO and for gravitational physics the second program director played a huge role in Washington for LIGO but she's a superb mathematical physicist and she and Vince started this David Garfinkel at Oakland University was the main person who really pulled it off in the end and through numerical simulations showed that indeed the BKL conjecture was almost correct a few things had been missed such as sharp spikes that occasionally occur at the end of a cycle and the rotation of axes could sometimes be different than they had it but the overall general picture was pretty much the way they had it that geometric dynamics near a space like singularity is almost precisely the way BKL figured it out using approximation techniques but we could only know for sure as a result of numerical simulations inside black holes it turns out there are three in general three different kinds of singularities there's a BKL singularity and there are two other singularities and we know this also from perturbation analyses approximation techniques and I like to describe this in the context of the movie interstellar in interstellar you'll you learn from Ramali who's the theoretical physicist who gets killed by the explosion of a robot named Kip I don't not to be confused with me because Kip the robot has two P's on the end of his name so but Ramali says that there are three singularities and that when Matthew McConaughey goes inside the black hole he should try to go into one of the one of let's see what's the phrase that he uses one of the gentle singularities so the BKL singularity is very chaotic and very dangerous this is an embedding diagram of the geometry of space as seen from a higher dimension they bulk or the fifth dimension of interstellar the embedding diagram of the geometry of space time this is just my artist's conception of the BKL singularity once you go through the horizon then the in a rocket ship you're headed toward the BKL singularity and the BKL singularity for those who know about Penrose diagrams sits up here in a Penrose diagram of the external universe the region inside the black hole between the event horizon and the outgoing Cauchy horizon and then the BKL singularity this is just for the experts in relativity who know what this means and what was shown by Eric Plessone and Werner Israel in 1994 I can describe in physical terms this way that after you fall into a black hole time is so compressed inside the black hole that everything else that falls into the black hole after you comes crashing down on you on in a fraction of a second as seen in the inside so there's a shockwave an enormously strong shockwave that consists of everything will fall into the black hole in the entire future of the universe and that constitutes a singularity but it's a gentle singularity in this sense the tidal forces which are a relative acceleration but you between your head and your feet if you do two time integrals of an acceleration you get a displacement so you do two time integrals of the tidal forces and that's finite which means that if I this is just the net squeeze or net stretch in the up-down direction that's the net squeeze in the up-down direction the net stretch is finite by the time you hit the singularity this is very different kind of a singularity turns out to be a null singularity and it is understood however only through perturbation theory thus far and there is an out flying singularity that's similarly caused by everything that fell into the black hole in the past a little bit gets scattered back up toward you compressed into a shockwave where again the second time integral of the tidal forces is finite there have been no computer simulations that I know of yet of the interiors of black holes to see whether or not this picture is correct this is the picture we have from these from these perturbation theory analyses and so that is one of the things I would be working on if I were working in numerical activity is to try to understand this for sure is this really the way things behave or is it not so simulations are greatly needed I want to now turn to the my second example graph geometry dynamics in imploding waves this is famous work that begins with Matthew chalk to it could post doc at the University of Texas when he did an analysis of the implosion of a wave that he wanted to make it as simple as possible so he wanted to be spherical and this wave is to implode toward a center it carries energy and if it implodes toward the center that energy might be enough to form a black hole now the problem with gravitational waves which are the only kind of wave that's just made from curved spacetime that's the only geometrodynamical wave is they can never be spherical it's a thought theorem basically that the only kind of a way that can be spherical is a scalar wave and so chop to it get an analysis of a scalar wave that satisfies the usual scalar wave equation for those physicists who know about this the usual linear scalar wave equation but he then said look we know what the energy density is associated with the scalar wave or the scalar field and so we will compute the energy density and the stresses associated with this scalar wave and we will feed those into Einstein's equations as a source for spacetime curvature and then the scalar wave is going to interact with that spacetime curvature that it itself is produced through its own energy and so he did that analysis and he found something really quite amazing if he began with a scalar wave with an arbitrary shape it didn't matter what the shape was here's a very simple Gaussian shape with some height p if p was bigger than some critical value it would form a black hole the waves came in their energy was just enough to form a black hole a black hole would form and the mass of the black hole was proportional to the amplitude of the wave minus some critical amplitude to some power beta which he found numerically is 0.374 he couldn't calculate it analytically which was all computer simulations and if the amplitude was less than the critical value then the wave would come in interact with itself and then disperse and there would be no black hole forms but he asked then what was the minimum radius of curvature that you could get from this a very very tight curvature or one over that radius of curvature is a length scale you can think of that as a measure of the total curvature of spacetime in that scale this p star minus p to the same beta power so it didn't matter whether you which side of this transition you were on whether the amplitude was big and you got a black hole or small and you didn't get a black hole you constructed something that had the dimensions of length and that thing that you constructed the mass of the black hole in relativity when we set Newton's gravitational constant speed of light to one is like we pathological physicists relativity theorists like to do you found that the scaling was always with this same critical exponent and if the amplitude was equal to the critical value then quite remarkably the evolution was what we call discreetly self-similar basically this way went in and interacted with itself and it send out then an outgoing wave which had some shape and then the same shape was repeated again but on a much smaller scale and then again on a much smaller scale and then again on a much smaller scale faster and faster smaller and smaller scales this is discrete self-similarity I like to think of this as an organized frothing of space time in the outgoing way these behaviors are like what you see in condensed matter in a phase transition critical critical behavior and scaling this is a phase transition when you change your initial data from the initial data the formal black hole to those that don't form a black hole and so in this example then this was our second example of geometric dynamics we were beginning we were seeing the same kinds of behaviors you see in very interesting phenomena of phase transitions in condensed matter physics the next step then was to do the same thing with gravitational waves and this was done initially by abrahams and evans at the university north of caroline and then by Yevgeny sorkin with higher accuracy now they waves because they're gravitational waves they could not be spherical they were actually symmetric they were symmetric around this axis but they had a different shape up here than they had in the equatorial plane and it didn't matter what shape precise shape you put in the behavior was still the same p but the amplitude is bigger than a critical value the mass of the black hole scaled in the same way with the same exponent to within the accuracy of the numerics if the amplitude was less than a critical value the strength of the curvature scaled in that same way with the same exponent and so it's really quite a and there was moderately strong evidence for discrete self-similarity it seemed to be the same regardless of what kind of a wave you imploded so this universality again something that you see elsewhere in nonlinear physics turns out that more detailed studies of other kinds of implosions have revealed that there's not just one universality class there are several universality classes several different behaviors but a discrete number of behaviors in some sense like the for those who know about this the discrete number of roots to turbulence in convection of say oil in a frying pan if you put oil in and you study it very carefully the onset of convection there is a discrete set of roots to of chaotic evolution or evolution into the chaotic state of convection so so this is again like you what you see elsewhere in nonlinear dynamics numerical studies are still in their infancy I think there's a great richness remains to be uncovered okay now I want to go to my third example geometric dynamics in collisions of black holes this is a diagram that I had drew a sketch for and then we had an artist make a nice version of the diagram when we were beginning to work on LIGO together in the 1980s and this was our guess as to what it would look like for two black holes to collide again a scene in an embedding diagram looking in from a higher dimension with the two black holes going around each other spiraling together merging colliding and merging and producing outgoing gravitational waves we could from dimensional analysis and knowing some approximate analyses we knew that the power output of the gravitational waves would be somewhat bigger than the luminosity of all the stars in the universe put together during the collision we now know from numerical relativity that factor is 50 for the particular case of the first gravitational wave burst that LIGO saw it depends on the details but for that particular initial one is you got we got off three solar masses of energy in a time of about a tenth of a second and the power output energy per unit time was 50 universal luminosities quite quite striking and the really remarkable thing about this is there's no electrical charge involved in the collision of two black holes so there's no electromagnetic radiation there's only gravitational waves nothing else so they're our only tool for exploring this except of course if there is an accretion disk around the two black holes it would be disturbed by the collision of the two black holes and maybe we will get enough electromagnetic emission to see that and the electromagnetic astronomers have been looking for that in black in our LIGO LIGO and Virgo black hole collisions that the LIGO Virgo team has seen but there have not yet been any electromagnetic emission found so we're dealing truly with geometric dynamics both observationally and then in computer simulations of this and the key thing for me in the 1980s when I became convinced by the year 1980 that the first thing we would see was what we did see colliding heavy black holes was that the details of the collision would be contained the geometric dynamics would be encoded in the gravitational wave shapes the stretch and squeeze plotted up and down there would be some shape and this is a shape I drew in the early 1980s just drew it no calculation just my imagination of what I hoped we would see something like this and it would contain the information about geometric dynamics and so that was for me one of the most interesting goals of gravitational wave searches was to begin to do observational geometric dynamics so the numerical all of us worked very hard for 60 years to develop the tools of geometric to be able to simulate collisions of black holes for a while I was afraid that they would not have this under control by the time LIGO saw the waves and we would be in bad trouble because in fact the first black hole collision that occurred all the signal power was in the epic of the collision which we cannot analyze we don't know how to analyze in any way except through computer simulations fortunately these superb computational physicists did have the numerical relative to be under control by then and so I'm going to show you now the the collision of two black holes that was the first collision that we saw and so this was from a simulation by Caltech Cornell Canadians from theoretical astrophysics Cal State Fullerton Oberlin Washington State University anyways a collaboration led by Saul Tkulski of Cornell it's called the SXS collaboration for simulating extreme spacetimes and I'm going to then show you the spacetime metric or geometry of spacetime in the collision that produced the first gravitational wave burst that LIGO saw and so we know mathematically that if you have a two-dimensional curved two-dimensional surface it can always be embedded in a flat three-dimensional space that's a theorem that's not hard to prove except it's a theorem only locally not globally and there is a fundamental problem when you have black holes if you go in around this black hole and march around you build this surface that has the same geometry as the orbital plane of the two black holes we're looking only at the orbital plane as a two-dimensional surface we're embedding that in a flat three-dimensional space you go around come back to where you started it will not connect up smoothly and so you have a problem so Harold Pfeiffer one of the young computational physicists in in our collaboration he said I'm going to cheat I'll tell people I cheated I'm going to do a pseudo embedding diagram I'm going to plot upward something that has the dimensions of length it's going to be one over the scalar curvature of this two-dimensional surface that is the equatorial plane of the black hole to the one half power and then I'll put it I'll throw on minus the sign of the scalar curvature to make it look like an embedding diagram would look and so that's what you're really seeing but if you want to think heuristically this is what it would look like to the bulk beings in the movie interstellar looking in from outside at the black hole collision and the color coding is the slowing of time or the so-called lapse function the arrows are the so-called shift function gravity of space in a motion and you see the shape of space this is now in slow motion this is like a storm at sea a huge splash great slowing of time in the red region then an oscillation and gravitational waves propagate out and that is a depiction of geometric dynamics for the first black hole collision except this captures only a very small portion of the geometric dynamics the problem is the too little of the spacetime geometry is depicted in this way first of all we're only looking at two dimensions and really they see a gravitational wave and depict it you need three dimensions you're just stretching along this direction a squeezing along that direction and it's propagating in this direction so you cannot capture the gravitational waves in this way this is just the orbital plane but if these black holes are spinning they drag space into a whirling motion around themselves they create a sort of a vortex in the structure of space around themselves and you miss that entirely there's a lot missing here and so together with a this this is now work that i have done since i retired i retired in 2009 in order to turn the leadership at caltech over to the younger generation in order to be in control of my own time and do what i wanted to do in order to collaborate with students instead of teaching students it's wonderful collaborating with students and having no responsibility for for their education okay and so these are students and postdocs of my successor at caltech uh yanbae chen who's my successor professor at caltech his students and postdocs and i'm collaborating with them and so we devised a way to visualize in vacuum and there's no matter around what's called the riemann curvature tensor which is the thing that describes the curvature of spacetime locally and i'm going to now get a little bit technical and talk about this in the same language that we talked about uh the electromagnetic field okay and i'm going to remind you that if you have an electric field in a magnetic field in one reference frame in special relativity and you change reference frames so say you're moving at high speed in that direction the electric and magnetic fields change the electric field and magnetic field that you the electric field you see is a mixture as you move relative to me it's a mixture of the electric field and the magnetic field that i see and there's an apparently complicated uh transformation and so the electric field the magnetic field depend on your reference frame they depend on how you move through spacetime now we're doing this in curved spacetime and so the analog of a reference frame is a choice of a three-dimensional surface two dimensions on my screen because i have trouble describing three dimensions plus the fourth dimension of time and so i just show you a two-dimensional surface that's really three-dimensional uh it's a space a slice of constant generalized time and time is plotting being plotted upward and when you choose to uh what we call say in technical terms foliate spacetime into space plus time when you choose a sequence of space slices and you think of time as running perpendicular to those space slices as in relativity as in special relativity once you have made that choice of space slices then something called the electromagnetic field tensor which is something that lives in spacetime it breaks up into an electric field in a magnetic field but these electric and magnetic fields don't exist until you have chosen a sequence of space slices the analog of choosing a reference frame but once you've done that then you can draw field lines a magnetic field lines around the earth for example once you have chosen a reference frame then you can draw the field lines that depict the magnetic field and similarly the octrate field well in that same way the riemann tensor that describes the curvature of spacetime it's in vacuum it's the same as something called the vial curvature tensor and so i'm going to talk about in the language the vial tensor for mathematicians and relativists who who know this the slight stickiness of being in in vacuum versus not being in vacuum but think of this as the so-called riemann tensor the time space time space part of this where space is along this surface time is perpendicular is called the tidal field it's a symmetry trace-free tensor and the dual mathematically that i won't go into but for for those who know about mathematical duels the dual of that the time space time space part of the dual is called the frame drag field it's also a trans symmetric trace-free tensor and these two tensors which you can think of as three by three matrices if you wish they embody the entire the entirety everything there is to know about the curvature of spacetime locally so i'm getting into some slightly deep mathematics but it's mathematics that should be in every relativity textbook and is in only one relativity textbook so far it's the relativity section of the thickest physics textbook you have ever seen it sits in your library it's called modern classical physics and i just published it together with roger blanford and that's why this is in there because i because i'm one of the authors of this book so if you want to read about this you have to go to modern classical physics okay uh now this tidal field has the following physical meaning if you have two particles and they're separated by some separation vector uh then the because of the spacetime curvature this particle will be accelerated by some amount delta a relative to that particle they're both freely falling moving freely but there will be a relative acceleration and that example is just the stretching between my wife's head and her feet when she's near a black hole the relative acceleration that relative acceleration mathematically is minus the contraction of this tidal field into the separation vector minus sign is just historical uh and the same thing happens in Newtonian gravity this tidal field in Newtonian gravity is the gradient of the acceleration of the gravitational acceleration it's the double gradient of the Newtonian potential and the same equation holds in Newtonian physics in that way and if i have a gyroscope at each particle then the inertial frames at that particle is described by gyroscopes will rotate relative to inertial frames at this particle is described by gyroscopes and that relative rotation differential rotation uh is described by the contraction of the tidal field into the separation vector and so two pieces of spacetime curvature one stretches and squeezes you the other twists you twists inertial frames makes a relative procession of gyroscopes and that's the entire story of spacetime curvature the tidal field is well known to people not nearly so well known as this differential frame dragging okay so now let me apply this to black holes if my wife hangs above the equator of the black hole and this is a fast spinning black hole she gets stretched if she hangs above the north pole of the black hole she gets squeezed in the grating regions she's not much affected by the tidal field more precisely the normal normal perpendicular to the surface of the black hole to the horizon normal part of this tidal field is what we call the tendicity it comes from the latin word for tendery to stretch tendicity uh it's the relative acceleration between her head and her feet divided by her height uh and uh she then is squeezed by this negative tendicity stretched there by a positive tendicity and as i say down here enn is the normal normal component of the tidal field so this is a little complicated but for the physicists who've studied this kind of things related to this i think it's worthwhile getting some sense that the mathematics is relatively simple it's the mathematics of vectors and tensors uh it just applied to the situation uh we take regions where the stretching is very large on my wife and we call that a stretch tendex a region where the squeezing is very large it's a squeezed tendex so a black hole has a squeezed tendex at its north pole and when it's squeezed tendex in the south pole a stretched tendex at its equator and that's the nature of the radial stretching and squeezing that she experienced outside the black hole we can draw the analog then of electric field lines we call these tendex lines uh we look at the principal axes of stretch and squeeze which i talked about before when i talked about the chaotic dynamics uh near a singularity principal axes along which she is stretched and squeezed are parallel to these field lines the tendex line is an integral curve of an eigenvector of the tidal field there are three eigenvectors of a symmetric trace free tensor there are three then uh tendex lines go through each point and so my wife is stretched along this tendex line that's red she squeezed along that tendex line she squeezed along these tendex lines that spiral up this is the entire pattern of these tendex lines around a fast spinning black hole so that is the totality of the tendex of the tidal field part of uh the spacetime curvature uh the to simplify things i just focus on regions with very large tendicity strong stretch of squeeze and so there is a tendex of very long uh a very large stretching sticks out of the equator of the black hole it's uh actually symmetric so it's basically a cone a cone that goes out of the black hole like this and then there's a squeezing tendex that goes out of the north pole goes around comes back in the south pole and those are the things i'll focus on now as for the the twisting horizon varticity is the angular velocity of a gyroscope at my wife's feet as seen by her head but that's also the same as the angular velocity of her head as seen by her feet if you take a towel a wet towel you squeeze water out of it if you're right left hand sees your right hand going clockwise then your right hand will see your left hand going clockwise and so it's the same mathematics so sticking out of the north pole of the black hole there is a counterclockwise tendex a counterclockwise vortex of twisting space out of the south pole a clockwise vortex of twisting space uh and uh and on the uh on the black hole then there's a horizon vortex of strong counterclockwise twisting here and strong clockwise twisting there outside the black hole the integral curves of the eigenvectors of this of this frame dragging field are then tendex lines and they basically guide the whirling twisting tendex of twisting space that sticks out of the black hole so out of the north pole of the black hole there's a counterclockwise uh tendex that's guided by these uh tend I'm sorry vortex that's guided by these vortex lines and out of the south pole a vortex guided by these vortex lines and here is the entire pattern of vortex lines around a fast spinning black hole the counterclockwise the clockwise twisting vortex lines come out of the south pole go around the north pole and back in the south pole the counterclockwise ones come out of the north pole go back in the south pole and the region's a very strong twist of space there's a counterclockwise vortex sticking out of the north pole a clockwise vortex sticking out of the south pole so I've been kind of complicated this is just to give you some sense of the underlying mathematics for the remainder of my talk I'm going to focus on uh these uh vortices primarily that stick out of the black hole a counterclockwise went in the north pole of counterclockwise twist now to the south pole of clockwise twist we can think of this physically as like a strong tornado of twisting space with opposite polarity sticking out of the two poles of a black hole okay now having understood this mathematically these postdocs and graduate students with whom I collaborated did a computer simulation of the collision of two black holes for simplicity ahead on collision with opposite polarity so there's a clockwise vortex sticking up on this black hole a counter clockwise vortex sticking up on that black hole and so opposite twists on the black holes are identical black holes otherwise the black holes then collide and merge and the the vortices of twisting space retain their individual identity and now we have a black hole horizon this is the horizon of the black hole that has four vortices of twisting space attached to it we began with two vortices one on each black hole we now have four it turns out the black holes don't like to have four vortices that's a highly unstable situation and so they work hard to get rid of least two of the vortices and in this particular case you can see what happens in a this simulation then i'll show you the rest remember that it's the blue vortex up here on the right the black oak lines merges now you have a red vortex on the right now blue now red the vortices fight with each other and exchange vorticity what was originally a clockwise vortex of twisting space has become a counterclockwise vortex of twisting space and that was a big surprise when we saw it in the computer simulations and so looking more carefully at the vortex lines we found that the following is what happens at the moment when the black hole goes green there are no vortex lines sticking through the black hole they have come off so in this case you have a blue set of vortex lines this is a clockwise vortex of twisting space comes out of the black hole around and back into the black hole in the southern end a counterclockwise vortex goes out around and back in so all along the red lines it's a counterclockwise twist going out of the black hole come back in along the blue lines a clockwise twist going out of the black hole come back in and then the oscillation these vortex lines pop off and it turns out they embrace each other and form a torus and this torus of embracing clockwise and counterclockwise vortex lines starts traveling outward at the speed of light every with every oscillation you get a new torus like a smoke ring a sequence of smoke rings produced when the black hole oscillates and it turns out that something called the Bianchi identities of general relativity in a freely falling reference frame a local inertial frame they look just like the Maxwell equations the dynamical Maxwell equations time derivative of the tidal field is the curl of the frame drag field symmetrized this thing is symmetric that has to be symmetric three by three symmetric tensor and the opposite sign in the other way because this mathematics is essentially the same as with Maxwell we know in Maxwell's case if you have a moving magnetic field it generates an electric field so the same thing happens here through the so-called Bianchi identities as these vortices of twisting space travel outward their motion generates tendencies of stretching space along in the long axis of the ring and tendencies of squeezing space around this short axis so now you have these remarkable smoke rings of twisting space both kinds of twists embracing and then they've generated these tendencies of stretching and squeezing space and this in fact is a gravitational wave LIGO only sees the stretch and squeeze it doesn't see the stretching and the twisting doesn't see the twisting only sees the stretch and squeeze we don't have the technology to see the twisting but this is the true nature of a gravitational wave it's all it's a stretch a squeeze a clockwise twist a counterclockwise twist all wrapped up in one one entity produced in this case by these vortices and tendencies that pop on and off of the black hole if we have an orbiting collision it's rather simpler in that case the vortices and tendencies stretch out in spiral patterns like the water from a whirling sprinkler like the spiral arms of a galaxy as the final black hole tumbles around and around but you have very again interesting dynamics but a very different kind of dynamics i'm going to just jump over these and just say that looking at this in greater depth it turns out you have two kinds of gravitational waves created in a black hole collision you have the waves that are generated by vortices that stick out of the black hole and you have waves generated by tendencies that stick out of the black hole they have opposite parity so one is the analog of magnetic dipole radiation the other the analog of electric dipole radiation and this shows up in in very much the same way in gravitational waves but this is the essence in colliding black holes of geometric dynamics as we now understand it and with LIGO now and i think you've most all seen photographs of the LIGO gravitational wave detectors the numerical relativists are doing simulations of generic black hole collisions here's a black hole that weighs 60 times what the sun weighs that one weighs 10 times what the sun weighs this spins this one spins around that axis with a spin angular angular momentum this proportion of the length of the arrow that guy around this axis with such a spin angular momentum that's the direction to earth and the waveform depends on all these parameters on the masses of the black holes and therefore their sizes on the spins directions of spins the magnitude of the spins and the direction to earth and so you can imagine computing these waveforms is not simple and that is this is part of why it took 60 years so it's a small part of why it took 60 years but you get then interesting waveforms out and the name of the game now is from the observations by LIGO and Virgo you get observed waveforms you compare those with waveforms and simulations when you get a very good match you then go back to the simulation and you look at the geometric dynamics if the match is essentially perfect you say well the geometric dynamics presumably is the way the simulation says and so that's what we are doing now we have seen six black hole mergers thus far with a range of masses and distances the farthest source with something like three billion light years the heaviest mass in the collision was 36 solar masses one of the first black holes we saw the smallest one we've seen so far is 7.5 solar masses but we are beginning to build a catalog of black hole collisions and of course bonding that a catalog of geometrodynamical behaviors of curved space time which go hand in hand with the catalog of the waveforms so we have think of ourselves as having a catalog of waveforms and then a dictionary that says if you see this waveform here's the geometric dynamics that goes with that waveform so this is what the SXS collaboration is doing here and so far there's excellent agreement between the observations and the simulated waveforms okay I'm going to finish off now by reminding you that in 1984 I speculated I hope that this is the waveform we would see and the waveforms that we have seen if I go back here the ones we've actually seen are awfully simple there nowhere near as interesting as what I was hoping we would see the actual waveform for example for the first gravitational wave we observed this is a collision epoch was just very simple and the question is why and the answer is that the black hole does not hang on to the gravitational disturbances that it creates very long the disturbances depart very very quickly correspondingly as the black hole vibrates the Q the quality factor of the vibration is very low just a few oscillations and the disturbance is gone that's true of all the black hole mergers that have been seen thus far but work by Huan Yang and Aaron Zimmerman who were students of Yanbei Chen and what used to be my research group together with Louis Lainer at the Perimeter Institute in Canada they have pointed out based on earlier work by Yang and Zimmerman and Yanbei Chen that because that the following happens if you have a slowly spinning black hole the shape of space around it is a very gentle throat if you have a fast spinning black hole you get a long throat in the geometry of space and so quite naturally you can have gravitational waves that are trapped in this throat for a long time so you have normal modes of oscillation of a fast spinning black hole that might be the final black hole from a black hole merger that live for a long time if the black hole is spinning very fast we haven't seen a final black hole that's spinning very fast we hope to at some time over the coming few years when we are by say 2020-2021 we're seeing presumably something like one black hole merger per day instead of something like one per month when we're observing and so the key issue is that from their calculations that they have done combining some amount of computer simulation and a lot of analytical work it turns out that if you have a rapidly spinning black hole this is sort of a canonical fast spinning black hole number for a the black hole spin parameter for those who know about these things over m 0.998 but that kind of a black hole spin which is conceivable we will see the normal modes of oscillation can be held long enough that you get mode-mode coupling two modes interact with each other to produce a third mode and it's that kind of mode-mode coupling that gives rise to turbulence in fluids and in fluids you begin with very large-scale modes eddies think of them as circulating eddies they interact they produce smaller eddies they feed their energy from big circulating eddies to smaller eddies to smaller eddies to smaller eddies described by something called the Kolmogorov spectrum in fluid mechanics what happens here it turns out is an analog of what happens in two-dimensional fluids where the cascade of energy is from small scales to large scales quite the opposite of what you see in ordinary three-dimensional fluid mechanics presumably that's the case here because of the axis symmetry of the unperturbed black hole the black hole is being perturbed but their calculations suggest that LIGO actually LIGO and Virgo actually have hope to see the creation of two-dimensional turbulence in vacuum space time that is these eddies these normal modes interacting with each other and feeding energy from small scales to larger scales to larger scales to larger scales after a fast spinning black hole forms from the disturbances that arise from the black hole collision so this is a example of the future that I'm hoping for observations to match what is seen in these calculations and so let me just finish off by saying that I've talked about four current arenas for geometric dynamics singularities imploding waves colliding black holes and gravitational wave observations and all I suspect we've barely scratched the surface we now have the tools of numerical relativity quite sophisticated now on codes that are becoming public codes that other people can borrow and use so real opportunities to though they're hard to work with it's still early on in this business but it's a community of people exploring beginning to explore geometric dynamics I think the interplay between analytical calculations computer calculations and the observations of gravitational waves in this subject is going to be very interesting over the coming several decades thanks thank you very much for the kids it's a wonderful talk and I'm sure there will be a few questions someone in I'm sure everything was crystal clear so there are no questions yes question please you said we don't have the technology to detect the twisting of black holes but naively one would think that you just set up an interferometer to detect rotation yes so where's the trick the issue is the the sensitivity that you can achieve and you're basically looking to measure an angle of rotation a differential rotation and angles are much harder to measure than just linear displacements and so there was a conceptual design for a gravitational wave detector by Brighinsky in Moscow and a young colleague around 1970 for a detector that works off this twisting of space but the sensitivity that could be achieved just did not look anywhere near good enough and so I don't know any other effort that's gone into it it could be that a really that a good experimenter could design a detector that could actually do it but that small amount of effort that was put in by Brighinsky then did not look at all promising so it's a question of doing the numbers and doing the experiment and I could be wrong I have been wrong many more times in my career than you can possibly imagine but you have been right on at this once more questions yes in the second event that was observed there was also an observation with electromagnetic waves if i'm correct yes and so we were able to find exactly where did it come from so the galaxy which where where the black hole was except in this case there's neutron stars okay okay so but these waves were almost nothing here on the earth but near the the the neutron stars it was a very powerful wave so it must have brought havoc to the star systems which were nearby I suppose is there anything that we can see in that sense so if you look at the numbers by the time you reach if you're in the near zone so the wave has not yet fully developed into an outgoing wave the tidal field the frame drag field are enormous and you don't want to be there but by the time you get to the to the beginning of the wave zone the dimensionless stretch and squeeze the fractional change in your length is something like a part in a thousand or a part in a hundred so it's really quite small it's only in the near zone that the stretch and squeeze are so big that you would be highly uncomfortable or that it would really wreak havoc now if you have two black holes going around each other and accretion disk near it the accretion disk may be near enough that that there is significant influence there will at least be an influence of the loss of mass you have a central body that suddenly has its mass reduced by five percent and the disk has to respond in some interesting way with it if there's just a sudden loss of mass that's holding the disk there that's what the hope is but but that's the only place that I expect from a nearby accretion disk that I expect that there's going to be enough disturbance for us to be able to to measure what are the long-term hopes for space-based observations I mean so Lisa the laser interferometer space antenna which is a ESA mission european space agency mission is very exciting it's currently scheduled for launch around 2034 but there two things have happened since that schedule was developed the first is that LIGO has seen gravitational waves and that has given a big psychological boost to the people who want to do this in space the second thing is that there was a lisa pathfinder mission to test technology in space to test the most difficult of the technologies that are involved in lisa and what was spectacularly successful with results announced just about two months after the first gravity wave detections were announced and so I think there's a big motivation to move forward as fast as possible on lisa lisa is a wonderful mission it looks because you have spacecraft that track each other with laser beams and the separation of the spacecraft is a few million kilometers instead of a few kilometers it's a million times longer separation you see gravitational waves with thousands to millions of times longer wavelength and so you're looking at lisa space-based gravitational wave observations when compared with LIGO and Virgo on the ground it's like radio astronomy versus optical astronomy and we will see completely different things you'll see supermassive black holes we have hopes of seeing the gravitational waves from the phase transition when the universe was 10 to the minus 12 seconds old the phase transition in which the electromagnetic force was born through a a the electroweak force coming apart creating the electromagnetic force and the and the weak nuclear force we expect to be able to map with enormous precision the spacetime geometry of quiescent black holes from the gravitational waves as small black holes spiral around them on complicated orbits so they're just absolutely wonderful science to be done with lisa and it has gotten the highest conceivable ratings in terms of the science in various competitions in the u.s. and i think in europe but finding agencies until now have been afraid of the technology and with the success of the lisa pathfinder mission i think that fear is much reduced and i am quite optimistic that this will move forward really well over the coming few years oh yes we received some questions from people looking at the colloquium online and one of the question is about a possible bet that you did with steven hocking against our physicists and he wanted to know about the status of the bet or does the bet exist and what is the status of that which yeah i think about the information paradox okay the information paradox okay so this is a slightly complicated story first let me comment on the information loss paradox this phrase information loss does not capture does not capture at all what's going on uh the issue is that uh in quantum mechanics if we can make arbitrarily good measurements the best that are allowed by quantum mechanics then if you begin this is using some technical language if then let me put a different i'm going to use technical language then the evolution of a system is always unitary another way to say it as you begin with what is called a pure state in quantum mechanics and it evolves to a pure state another way to say it is information is never lost these are three different ways of saying it but uh steven hocking gave strong evidence back in the 19 mid 1970s when a black hole collide when a object implodes to form a black hole and then the black hole emits hocking radiation due to quantum effects uh and slowly shrinks and disappears that uh the evolution is not unitary information is lost that you go from a pure state to a mixed state there's different ways of saying it but fundamentally what he did was gave strong evidence that the laws of quantum mechanics as we know them are wrong and that's what this is all about are the laws of quantum mechanics as we know them uh correct are they not correct and he gave strong evidence that they are not correct when you deal with the formation and then evaporation of a black hole since formation and evaporation of a black hole is actually controlled by the laws of quantum gravity which we do not understand at all well this is the question of the question are the fundamental concepts of quantum mechanics as we understand them do they apply to uh to quantum gravity that's the the basic issue uh and uh in this bet uh hocking and I said no they don't apply there's something wrong with the laws of quantum mechanics and John Preskell said yes they do apply what standard quantum mechanics is correct Stephen Hawking uh conceded the bet in a famous concession uh speech in uh Dublin Ireland at a big press conference with 50 reporters and tv cameramen at a conference there uh around 19 or around 2005 or so or something like that I was the uh the chair of the session at which he conceded nobody paid any attention to me because I'm not Stephen Hawking but I didn't concede uh Stephen conceded uh and so I am one of the few holdouts who still maintains that it is quite possible that information is lost the standard laws of quantum mechanics are wrong when applied to the laws of quantum gravity the reason that I'm a holdout is because we do know much of this because of work by Murray Galmon and uh and James Hardle but using techniques due to Richard Feynman we know that there's an alternative way to formulate the laws of quantum mechanics which is a for a way due to Feynman called the path integral approach or the sum over histories approach and we know that that approach does allow you to lose information it does allow you to have non-unitary evolution it does allow you to evolve from pure states to mixed states and that does arise if you include in your sum over histories histories with closed timeline curves histories uh with backward time travel you only have some tiny probability amplitude for a closed timeline curve but that's enough to destroy unitary evolution so my personal guess is that uh information is lost that the standard quantum mechanics is wrong and that the correct formulation of quantum physics is Feynman's approach and that uh and that all of the struggle that the community of physicists is having over what they call firewalls and various paradoxes that just are leading to enormous confusion it's because my colleagues are refusing to admit the standard quantum mechanics fails but having said that I again re-emphasize that I have been wrong so many times it's that it's not countable and and I am nowhere near as deep in this subject as John Preskell or Stephen Hawking or the other people who have the opposite view I'm just occasionally I'm just an unnery contrarian uh and uh but I think this is a very deep and very interesting very important question uh and I think I believe that essentially everybody working in the field will agree with me that the final answer is not in it's it still is not not all clear what is going on very good you don't feel bad about close time like uh curves you're okay pardon having close time like time like curves is okay for you uh close time like curves included in a Feynman path integral I have no problem with that but that I think that's that's the essential issue that's the way that you lose unitarity is by including close time like curves in the path integral and uh I think we know that from model calculations um so this is that that's quite separate from the issue of close time like curves on a human scale but the question is in a path integral on arbitrarily small scales uh say in the vicinity of the plank length uh do you have to include close time like curves interesting oh no any other question any other question from the not good okay there are many but okay I'm sorry that was a long answer but I it's something about which I feel passionately but on the other hand I feel very insecure about my answer but nevertheless I feel passionate about it right and uh do you see do you know if there is a better understanding of this chop tweak exponent you know the now or it's only numerical yeah so as far as I am aware there is no understanding as to why it has the the value that is seen but I have not followed that subject closely and there it is as I also indicated rather rich because it turns out that there are several universality classes of behaviors and the one that I described was the first one was discovered but it's only one of several uh but I think even in other places where you have critical exponents uh there is not much understanding of why the critical exponent has the given value as far as I'm aware but again I'm not close to this condensed matter physicist would know a lot more than I do about that very good and these spikes that you said that the medical relativity found that they were missed by by by uh uh Lipschitz et al the spikes that were missed but do they have any implication do they add more to our some understanding or so I don't think we understand the spikes terribly well they show up in the in the simulations they do you can see in the mathematics that they arise due to spatial coupling when you come to the end of a cycle and so this this issue that I tried to highlight at the beginning of that discussion that if you and your friend move toward the singularity side by side uh you become spatially just cup decoupled the ordinary partial differential equations of Einstein become ordinary differential equations uh that's not entirely true and the spikes arise from a coupling between you and your neighbors so the coupling is not complete but it's again it's not a subject that I've looked into in depth I know that those who have tell me that this is where this arises okay and just my last point you said that in the next couple of years we will see like one event per day well roughly we're extrapolating small numbers it's it's impressive just go go play with say even a plus own distribution if you won't have only eight numbers uh it's you can be quite wrong in in guessing what the distribution is but based on what we have seen with those eight it should be somewhere probably between a few a week and one a day when LIGO is and when advanced LIGO and advanced Virgo at their design sensitivity and can you foresee any surprises that you said something completely unexpected that they could eventually the sea there's all this funnels have been seen I do expect the big surprises but that's is only on general grounds that uh every time a new uh kind of electromagnetic waves has uh been uh brought into astronomy and it originally radio waves then x-rays and so for there have been big surprises it's seen through radio telescopes the universe is far more violent than through optical telescopes and so there's a real revolution in our understanding of the universe in that sense in the 1950s and on into the 1960s um and uh similarly with x-ray astronomy so just on these general grounds that gravitational waves are so radically different from electromagnetic waves that you would expect there will be big surprises the LIGO and Virgo team are working very hard to find surprises and the issue is then how do you date do data analysis looking for something you didn't expect uh and so there are many different data analysis channels that have been developed to try to find things that were unexpected uh and I'm pretty confident that will come whether it will come by the time we're at advanced LIGO's design sensitivity or not I don't know I hope I hope but I know that it will come in the next decade or so some huge huge surprise I'm fairly confident just on these general grounds very very good so it's nice way to finish but before we finish since many of you haven't been here before for other colloquia the tradition we have and we will try to impose it today it's very difficult because I can see many people is that when we finish there will be some refreshments outside and everybody is asked to go outside for the refreshments except and only only except for the diploma students of ICTP who will come down and will have a private meeting with keep keep us kindly agreed to do it and they will usually do it and everybody else who can be waiting outside in the refreshment so before everybody leaves so please thank you keep again