G. Hooft - The hydrogen atom for quantum gravity

thank you very much Costas for your kind introduction indeed my excuse for coming back to this place I like to come here frequently the splendid recollections of earlier visits but today the diet excuse is PFI yes apparent date of retirement and as other speakers this morning have said we hope that he won't take that too literally and that he will only retire on paper but still continue doing physics so yes and while coming here in this institute normally we discuss the things that are most on our minds and what's on my mind these days is black hole physics so I'm going to talk about black holes the title of my contribution is it's the hydrogen atom for quantum gravity I don't I think many people will not be surprised is what I mean by this is that the Schwarzschild black hole is as elementary an object as the hydrogen atom is for a theory quantum mechanics you know when Schrodinger Vulcanus equations one of the first things people did was try to understand how the hydrogen atom really works because news Bohr had a sort of vague idea about about how to make an atomic model out of Compton mechanics or quantum see you each of them cold but now surely returned quantum theory into quantum mechanics so now we had the equations and now we had the equations we can again turn to the hydrogen atom and find that those equations can be exactly solved for the hydrogen atom with a correct decision and in fact we teach that to our undergraduate students nowadays how to do that we want something similar for quantum gravity and my claim is we have something similar for quantum gravity it's called the black hole now black holes can come in a number of shapes very limited but yes you can have different shapes of black holes called the Kara black hole Neumann black hole and and others but the most primary fundamental prototype of the black hole is just a shroud shield back hole shot shield only a few months after Einstein had formulated a steel generativity cartridge shield a well-known astronomer in his day was said well if you write down equations I know how to solve them for the easiest case the spherically symmetric case and that became known as a short shield solution what shut shut shield couldn't have known and unfortunately he died a few months after adding that paper so so he never learned all the commotion that his solution brought about because it turned out that even what he really was describing was the black hole and this only the details of the black hole only discovered many years later by various other physicists who figured out what the thing really was that he was describing with the situations so indeed much later a another gentleman came on the scene while Stephen Hawking is on this picture the figure here he's standing between Einstein and Newton in an episode of Star Trek when walking was once visiting the studios of I think Universal Studios of where Star Trek movies were made he said I want to figure in one of these episodes so they said yes that's a good idea and so in some some time machine or some warp or something or other I suddenly and Neutron are being revived and invited to play a game of poker together with it's not a famous physicist Stephen Hawking and and then one of this of the characters of Star Trek is setting up this poker game because he wants to play poker - the most famous minds of the universe and that's these three other people and what you can you can guess who won the poker game but that's that's another thing so my subject today is now what hawking did to black holes what he discovered was if you apply the known laws of physics you we understand how today particularly we understand quite well how the standard model works blue already in his days it was reasonable to think that quantum field theory describes a world of subatomic particles so he he just took the quantum theory of the subatomic particles and apply it to black holes and at first he was struggling with the problem because something very strange came out he first want to describe empty space what's the vacuum like near the black hole and his equations gave conflicting answers it couldn't really make out the vector we thought to have stupid air I cannot even write down empty space what's happening here he found out that space-time around the black hole can't be empty there must be particles there and he actually at first he didn't believe his eyes but then when he worked out two equations further he found that particles are emitted by a black hole at a constant rate as if a black hole is actually glowing with a certain temperature and when he realized this he realized that this computation was saying something new about black holes which was not understood before the fact that black holes emits particles now all this would have been fine except that the way he described dose made many people believe that this cannot be right there is something fundamentally wrong about what his results were but there was nothing wrong as calculations so something wrong in his interpretation of the calculations or there's something wrong about quantum field theory there's something wrong or generativity because if you combine together the way it should be done according to the textbooks he gets something very strange he gets to the black hole itself does not obey any Schrodinger equation because it turns a pure wavefunction into a mixture of a functions a thermal mixture and in particular in particle physics my colleagues in particle physics said that cannot be right and I soon joined them say this cannot be right there is something that has to be modified and the importance of this is that yes we don't understand how the laws of gravity behave when you add quantum mechanics to them it's called quantum gravity yes we discovered the name of the theory but you haven't discovered to see you itself yet so we don't know how to under how to write down the closed equations for quantum gravity many people believe that string theory gives the answer but string theory says something quite different from what Stephen Hawking said strength you said yes how objects like black holes but they they don't release crazy thing about turning pure States into mixed states no they remain pure as objects so string theorists agree that somewhere somewhere along line talking to something wrong but we can equally put a finger on what he did wrong and not talking himself the Hawking opposed so this led Lenna let me subscribe the particle physicist to get into a battle with Stephen Hawking he calls the black hole war but I am not so much in favor of calling it a war because we are all friends with each other we don't really fight each other as enemies but yeah I in dispute and this was a dispute about whether quantum mechanics survives its behavior behavior of black holes so if you take a black hole can you write down a Schrodinger equation for a black hole if so that's reading recreation will be very similar to the Schrodinger equation for the hydrogen atom so there should be some similarity and I'm going to talk about that similarity which was not found because I think many of my colleagues in particular string theory also another branch of physics also in general relativity they took an attitude which I don't quite agree with so anyway about 30 years ago I started to write papers about this and devised some algebra and since then very little happen so now I'm very puzzled by this why nobody else do with English which could be done 30 years ago only just very recently it was done and what was done was apply the modified laws to whole modified theater explain how to modify the theory in a way which i think is is it does agree vis more than three but you have to think more carefully about how to do it and then you discover that in a very unusual basis of hilbert space now that's quantum mechanics is about hilbert spaces and about states in our space if you use a very special way of characterizing states in Hilbert space then you discover that the black hole is indeed described by a Schrodinger equation and you can write down that's writing an equation and find out that it's very similar it is has very many things in common with a hydrogen atom so but you have to pay a price black holes are not the way they they are thought that they behave but it's some give you some modifications you bring about in a theory I now find those modifications very very natural and there's nothing very strange about them but you have to think about if you don't you you get a picture of black hole so it doesn't make sense logically and you all know that somehow nature manages to make all laws of nature logical so this was the first demand we want to have a consistent theory of black holes you want everything to fit into place and the importance of this is that if that you don't understand everything about quantum gravity not only black holes with everything else as well no black holes play a very important role in understanding the gravitational force so if it doesn't work for black holes we should better it's try and start again so I want the thing that works from black holes and then we can possibly find out how it works under other circumstances and how to do quantum gravity properly so so there was a very striking conclusion so I'll try to explain in this talk which is that you have to modify the boundary conditions and I'll explain what it is so there's some funny things happening with topology of space and time and that's funny thing has to do with the Antipodes if you look at one side the black hole you can look at a point opposite to a black hole and you find there's a relation there that people have missed all the time and not only becomes clear the other thing is a black hole has no interior normally speaking people talk about falling into the black hole and once you pass horizon you're inside well you're not inside but I'll explain that and this is unlike what is called a classical black hole with classical I mean still the general relativity in full swing but but not going to mechanicals so you if you ignore a quantum mechanics the black hole is exactly as a Schwartz shield describe it and the way Schwarz you describe your stretch didn't didn't have the time to describe that way but do we wait the solution was later described by other physicists was that there's an interior region in this next area region the exterior region is like in ordinary world describing the universe of particles surrounding a black hole but there's the interior region as well if you fall into the black hole you enter to the interior into its belly so to speak and you'll die there because the black hole digests you very thoroughly but this is the classical description now we try to apply quantum mechanics something completely different comes out now trying to explain that so all this could have been derived thirty years ago but it wasn't and I don't quite understand why it wasn't but that what I understand why I didn't do it because I was busy other things I did my understanding is limited right no not at all smartest caused us likes to put a me but I also have my shortcoming so I couldn't derives myself and I thought there are thousands of elephants someone who had found it but no and so the the atoms are quite simple and I'll try to explain them here so what Stephen Hawking did was that he described the region near the horizon of a black hole the horizon is a mathematical sphere you draw around it that has a property that months you pass it you can't go back you go over the horizon so to speak and you are lost for eternity at least according to classical physics now when you come to horizon is covered that for a low-profit looks like something else it looks like a point in space but that's because I suppress two coordinates to do two times his coordinates its point to the horizon let's characterize our two angles like a point on the planet Earth is also characterized by two angles so those two angles are suppressed for a moment and then that's just a single point here but as time goes on for a decent observer the equal time lines are these lines all around here but a point that stays at constant distance from the horizon is described in hyperbola to make it a parallel you need acceleration and that's the gravitational force actually so if you undo that then a freely falling observer just made a straight line in this in this plot but this looks like a singular point that singular point is where Schwarz shoots himself thought there was a single arity but it's not a singularity it's not as singular as a North Pole of planet Earth so imagine a polar bear walking on the North Pole that polar bear would couldn't care at all whether there's a singularity in our coordinates frame he would just walk right through it and not even think about notice anything about a singularity so for the polar bear the North Pole just ordinary flat landscape similar to this point here a local polar bear just go right through without noticing anything and that is what made Hawking derive the property of the vacuum space here that you thought this was vacuum space but it isn't if you if you translate what a distant observer would see a distant observer will see particles and that was his the elevation so but you also divide these are something else that he produces a mixed state now this is the one important equation that I hope the physicists know in the Middle East will understand that the mixed state is a so-called entangled State it is the sort of hartle-hawking state and the heart of hawky state says that that has two regions in that closed region one is the outside universe region two is something we don't wanna do is something inside the black hole and in both these you wanna region 2 we can have particles with energies e and other quantum numbers in and then hartle-hawking derived or hawking himself derived that the shape of the wafer is this now if you don't see the things which are in region two then whenever you measure something you have to average over those states and that turns this thing into what we call a quantum mixed state it's not a pure state anymore it's like a thermal state the black hole is like a light bulb emitting light but the light bulb is raised to a certain temperature but electricity current going through it and at that temperature state is not disguised as a single pure contr state but as a probability distribution of different quantum states so that's what hoar he said that the black hole turns into a publicity distribution the states in the region too will never be observed anymore once you're inside like how you can't get out so that was it so there's an inside and outside but there's something fundamentally wrong about this the claim now is this cannot be right we have to reconsider what actually this expression means what the expression came out of solid mathematics about quantum field theory there's no doubt that these expressions basically correct so what is it why are we getting why should the black hole behave differently so quanta walk himself said was too bad for quantum mechanics my black hole itself is not a quantum mechanical thing and you've also explained why that is so the black hole is a corridor between one one universe and other universe as a wormhole so anybody falls into a black hole might really merge to another out of another black hole that is described by region two and the he tried to make sense out of this but particle physicists said that's not the way objects in the view of physical world behave if black holes behave like that then that takes away the the bottom part of our theories of matter and particles this cannot be right something else must be going on and then we turn to super string theory or string to you in general which is the big candidate theory for quantum gravity not the only candidate but I do think the most important candidates your conten gravity is string theory and super strings so what the super string theory tell and that's what all my friends who do string theory are telling me super strings know about black holes there are objects there between all the specs behave like black holes as you can compute their properties and lo and behold the black holes as you get out of that do not turn pure States into mixed States there are just ordinary pure objects so problem solved that's it that's what it is but Cynthia that doesn't tell you why hawking came to quote-unquote wrong answer and so what does the that thing black hole have to do with quantum fluctuations near the horizon what is a horizon in a string all these questions I always ask things first no answer to me why why do you get the contradiction here and they don't get an answer they should say well things you just sell me these are the black holes and if you have problems with space and time just throw space and time away they're emergent and I didn't like that idea at all so there must be something else going on so I said I'm not going to believe things here at this point I'm going to see what else I can do to understand black holes so then came much more recently a paper by 40s for people on mary marvel of polchinski and sally who discovered a difficulty in black holes but that was a difficulty that was quite clear from the beginning and that is the so-called firewall difficulty what's a firewall well hawking predicted that the black hole emits particles but those particles are emitted by the vacuum itself where these particles come from well somehow they come from the past in the black hole they were there all the time but if you do that then the particles which have been queuing up in horizon waiting to get out they form an infinite curtain of energy of particles and the problem they hit upon was that the current disposed of that curtain of particles effect they said the patterns form a firewall a firewall is a defense region that stopped you from falling into a black hole if you try to fall into a black hole you won't get through it horizon because there's this infinite possession of Hawking particles waiting to get out they will kill you so but then you say that you say Einstein was wrong because I never said no no you can use any quadrant for MU like your local coordinate frame will not show you where the horizon actually is so certainly there will not be any firewall so there was a big fight going on is there is not a firewall and again the verdict is that the theory itself that you have today it doesn't answer this question properly so many people believe there's some new physics going technologies here and we don't know what it is this is what is subtly so tremendously important for theoretical physicists we won't understand why what can new physics will that be and how can it improve understanding of how to reconcile gravity with quantum mechanics so if you can't solve it for this case forget it you'll never be able to contact Garrity properly so the price is indeed that there will be new physics and the verdict will be about safety it's thing theory is not foolproof that is the least thing one can discuss perhaps things theory it does say the same things that are which I'm going to derive now but it hasn't betrayed that and that hasn't hasn't come up with his answers but all by itself or sinking you not always gives you the answers that you want to get to something missing there so and so now let me try to explain what you have and I think there are three important observations that you have to make if you don't understand any one of these three observations you won't understand the black hole and the first thing is that whenever a party goes into a black hole yes it interacts with Hawking particles coming out that interaction should not be ignored so many of my friends in particle physics and in general if they say what do you mean is particles interact maybe there's a tiny interaction that will be weak secondary effects like it is in a standard model particles go through each other but every now and then a particle interactions not a particle but that's a secondary effect and if you want to know what state hawking the black hole came in forget these interactions for a moment right we do this in perturbation theory we start with the case that particles do not interact and the interactions introduced as an afterthought later now first you have to understand that's wrong you have to include in the interactions right from the beginning in particular the gravitational interactions why is that well that's precisely because gravity cannot be handled by ordinary theories gravity because a is not renormalizable it means if you go to understand the gravitation force at very short distances then you discover that that gravity group becomes infinite in strength so the gravitational force is divergent it becomes infinitely strong this you cannot ignore you cannot ignore a force that becomes infinitely strong and this is besides the point here if you don't handle gravity precisely correctly you get an infinite force within ingoing particles and outgoing particles in fact the statement that Hawking particles form a firewall already implies this the firewall becomes infinite in strengths so when another particle also goes through the firewall he has to go through an infinite curtain or screen of particles you can't get through that BR without being killed so this is the problem of a divergence in the gravitation force you have to realize this that was point number one point number two is yes you can you can take the defect to account and the effect is linear now general relativity is always portrayed as a nonlinear theory in the gravitational forces fundamental nonlinear that's because gravity carries energy and momentum itself and so Eric gravity is a source of gravity that makes your equations complicated nonlinear however this effect that the ingoing particles interact with the article particles that effect is a component of gravity which happens to be described by linear equations if you make a certain approximation which i think is a valid approximation in that approximation gravity is perfectly linear it's so much so that RH that what you can do is you can make a decomposition of the waves of particles going in a way the particle out call this the spherical wave decomposition exactly the same thing as people do in the hydrogen atom when you say the wave function of an electron can be decomposed into spherical harmonics and to every single circle harmonic mode of the electron in the hydrogen atom you just have a very simple differential equation which which effect you can solve and you can look up in the math math books if you can't solve it but it's very easy so the same thing happens for the black hole by the expanding a black hole into spherical harmonics everything becomes very easy and you can see what happens and what you see is something very very strange you see that the thing makes no sense unless you assume something that wasn't realized before which is a relationship within a point on a black hole and the opposite point you have to assume something there which is out of the ordinary but the more now I've been confronted with this situation the more I like it this is the way it should be and that actually as I'll explain right now it's it's a turn Hawking this argument into something else and that's something else is that pure states who go into pure States there will not not be any mixture from two states anymore but I'll explain why that is so so if you don't understand any theories points you don't understand the black hole and I my complaint is that in most other publications weren't black holes these three points are not even mentions so that's why I don't understand why well in particular things to you it's not mentioned any of these points and so I believe these things is missed something about black holes so let me try to explain now what what the procedure is your first elements understand how in going particles interact with outgoing particles and actually this interaction is rather simple to describe is very elementary what happens with the ingoing particle that particle zone is being dragged along a little bit that dragging effect is well-known in general relativity when you look at photons which come which which come as very short this is from the Sun you look at stars behind the Sun as you know from the Eddington experiment the stars are being shifted a little bit that's general activity causing light to bend but you can also say the light is bent by by the Sun because light is being slowed down a little bit and space and time around the Sun are curved a little bit and these two effects cause the photon which goes by the Sun to be slowed down a little bit so may look at the sky at the other side of the Sun you have to screen off off the light of the Sun of course so the ideal cases would be to look at this when as a solar eclipse so then you don't see the Sun itself but you see the skies behind the Sun that light is being slowed down a little bit you see this the the light of a star is a very small fraction of a second later then you were to see the mother eyes and that is basically this effect called a shabby or effect so the shabiha effect tells you that if there's a very fast particle which is a source of a gravitation field you're in this direction then another particle that causes it is being dragged along a little bit and that distance of which is dragged I call you what you sow you - means that I use light concordance he has a plus direction in the minus action this is the miles direction and the shift is in the same direction as this particle is moving and you can calculate this in flat space the calculations particularly easy although you have to work a bit at it because the factor for inform to suddenly not so easy anymore you can make a mistake here but actually there's a factor for which I could explain if you really want but it comes out of the basic calculation the shift is minus 4 times it's proportional to the momentum P ok often this particle so if this momentum is twice as big the shift becomes twice as big very easy and that this is an easy part to understand also why this has to be so handy with Lauren's event if you can learn some summation the particle gets twice the momentum it also drags this distance by factor 2 that's easy and then there's dependence on the turns since a particle goes here other particles here but they meet each other that transfers this is X tilde X and there's the shift is proportional to the logarithm of that transverse distance so very very close by the shift is very very large so that's this minus sign here the log is a negative and large at life a very large distances log also becomes large but the log is so slowly varying as a function of distance that at large distances you don't notice this but the small distance is log becomes a singularity so when particles causes are very closely the shift effect becomes very big well this is the basic input formula you can modify it for particle in the amount of black hole the only modification comes from the fact that the horizon is a curved it's a sphere it's not a plain sheet and because horizon is a spherical you get an extra term in the equations making equations a bit more complicated but basically it's the same effect oh yeah so now if you don't include that and think of a black hole again horizon and now this punch line is these pink lines here describe Hawking particles coming out to the black hole now what happens if I send the particle in is that this Hawking particles I shifted so it will go back then you look at E at the little circle there the little circle moves up is being dragged along but in going particle now if you wait a minute for the outside observer wait a minute then the only one thing is being lowered and since form so the annoying thing will be closer to the earth to the horizon and the shift will be bigger so you see the the red circle moves to the two upwards and the closer this thing comes to the horizon the faster the red circle moves away so the hawk batters move away from the horizon and the indigo thing moves towards the horizon and this effect now is you should not ignore in fact this is the major effect which changes everything for the black hole so so now this was in night 86 basically the that I proposed this mechanism to be an important mechanism that explains the behavior of black holes and the island was very simple and basically straightforward assume a single black hole which has some sort of past history I don't know what the past history is but let's assume oh everything was made black holes in a pure quantum state so a collapsing star is described using Schrodinger's equation anything and I had a given state that the black hole was put in I call a BH number one that's this in state and that in state gives rise to a final state it emits particles now lets me assume that these particles have to be pure pure that's an assumption but the assumption is logical from a physical point of view in physics you only assume that pure States will evolve into pure States so assume that purity of propaganda States is preserved that'll be a blasphemy hawky would jump up set up Snivy code that this is not right but my friends in in particle physics say yes this must be right things here say yes this must be right let's assume it is right but if you assume this you also have to make some sort of bend some change in the laws of nature because it doesn't come out of me automatically what horn it was correct in principle if you apply long laws of physics it's not a pure state but now I say no we assume this is a pure state something has to give in the equations that you want to write down to get this but let's assume it then what comes out then I say well now I can make a modification I take the same black hole and I put an extra particle in I throw in a teen oh just like that is very soft particle just adopted a black hole nobody sees a thing if it were a neutrino nobody would notice that this intervening would have a gravitational effect on the black hole that sounds outlandish but still it has an effect and you can calculate the Shapiro delay due to the single particle as the full top in well no matter how light the particle is it has a momentum P - and I can compute the effect using this equation what it says here is a describes equation in quantum notation in quantum tation say well the outgoing particles are now being dragged along a little bit I can compute quantum mechanically what this does to the wavefunction it I applied a displacement operator which is the exponent of the momentum P plus of the article in particles and when the P plus drags along the within equation for you - which is the formula I just wrote down before this you - basically or Delta U - it's a shift cost so this is the equation and it means that the black hole the out stated black hole is not equal to the outside of the original black hole but it is shifted our state so it's called BH number - and and now I say wait a minute now I'm going to use linearity I can repeat this process I get thrown out of particle to the black hole and change it again or I can remove one of the particles that went into the black hole and this way gradually built any other black hole states as the initial state of the black hole then I can ask what it what expect in what respect does it shift the article particles well in every case I can compute that inside the exponent you know be informal exponent have to multiply the effects but inside exponents yeah FX is just linear yes AdMob so you have to take the total sum of all moment of the particles going in and then this will shift all the particles and actually it might define the position of all the particle altogether I define the position there are current patios with respect to what they would have been in in the single set vh1 but I can get all the other black hole States this way so this way I can produce all other black hole States I can compute what states the outcome is particles will be in for a new case and this way I get a completely acceptable quantum mechanical description of the black hole it relates the instacharge States in a one-to-one way well if you listen carefully you might notice I skipped a few points in this argument I said that if you know the location of the outgoing particles you know everything this is so far still an assumption but it's a good assumption to that office but if an old location every single particle I know exactly the quantum state it is not quite obvious and in the standard model there are also things such as quantum numbers of black holes suppose I put in a baryon and I I replaced by an antebellum but I don't change the momentum then account this argument the article state will be the same that would mean that two difference in state gives the same our shape that's forbidden in quantum mechanics so I had to watch my steps here and I have to say that apparently the portal and the portal are not the same particles not quite so they gave a slightly different shape of the shifted state later that's detailed has me worked out but apart from that you get basically that the out states are going to be dependent on the in States in a one-to-one way and that will give you a new quantum mechanical theory for black holes that gives me so much scattering matrix the estimate is the thing that Hoff you denied that exists same thing the string theorists say must exist and then I'll show how a pencil you can compute the scattering matrix however there's a little footnote here a footnote which at that time I didn't pick up at all and the food code is yes but you have to allow those are gone particles to be anywhere in particular beyond the horizon so you can shift across the horizon that wasn't taken to account correctly yet so and also the momentum can be positive or negative and you have to take all these things into account so here was still some problems I'm relating the sum of the moment of all the in going particles with the average position of all the article in particles so and though that relationship you want to one relation this led to a linear algebra and the algebra it can then be completed by saying wait a minute and talk about positions and momenta the positions of the argument particles and moment of the incoming particles suppose I want you to talk about the positions of the ingoing particles or the moment of the outgoing particles but then you have to use for everybody does it quantum mechanics to sales positions in momenta do not commute they are non commuting operators and the commutator can be written down and indecent come to theory is basically a delta function there are some complications here and in fact that I'm working on horizon which is a curved surface so you have to make some small calculation of Corrections here and there but I won't bother you too much with that so oh my god here stands short for the solid angle normally given by two angles theta and Phi and now look do that for all the particles go in and all the particles that come out and this generates an ultra and complete algebra now that you can solve the algebra looks amazingly similar to all the wells used in string theory in a sense this is the Vedas are all algebra however it's a lot simpler because I left out difficult parts in string theory one uses algebra but among the other becomes tremendously complicated if you want to understand the theories Lorentz invariant you have to agree on transformations which turn this out which is very nonlinear in the algebra and which make things difficult actually you do have the same thing here so these are just unfinished I there are certain transformations which have to be worked out rotations on the horizon for instance which have not been included in this so they are just linear because I simplified things and that's because I'm not as clever string theorists are just first soft the easy part so these this is the alter and this is the other you have to work with that was in the 1980s that you did all this and we have to say that there a black hole itself is a representation of this all came in the words usually you work for the physics particles are representations of the symmetry group so the black hole is a representation of this algebra and this representation of base various equations so is that all no it wasn't all I had forgotten something and what was forgotten was as a complication I already mentioned briefly the patter can be thrown through the horizon to the other side what happens then this was simply not understood in those days but now let's think again whether we can answer that question and the answer was staring in my face all the time and never did anything with it the answer was these equations are linear this alters the linear algebra so why not find a different frame to work with is algebra so work with a particular basis and what base is better than the base generated by the spherical harmonics so that's also why I call this a hydrogen atom of quantum gravity I'm going to expand the same thing in the same circle harmonics that you're used to in solving the hydrogen atom in atomic physics so let's do that expansion and something marvelous came out that expansion told me that if I take not the position of the article particles elemental didn't go in particles I take the entire distribution over the horizon I expand that in that contribution in spherical harmonics so you have to work with the quantum numbers L and M at every L and M there's a spherical harmonic of ingoing things causing the article things to be displacement displaced and that displacement is also discarding abstract harmonics something marvelous happens with the equations are hold down the modest thing is that the equations are linear and therefore they are diagonal in the L and M basis so a given circle waving going things only displaces the same spherical way for the outgoing things that means I've eliminated the two angles theta and v replace that by spherical harmonic and then I have to work at one L and one M and do the entire calculation the thing that is and what used to be complicated looking functional anything functional differential equations and functional intervals and so on now becomes something very simple just like the Schrodinger equation which is a complicated four dimensional differential equation for the hydrogen atom it becomes very easy if you expand the wave function of the hydrogen atom in terms of spherical harmonics which is all always what we do you write down the L and M of an electron cloud on the hydrogen atom so we do the same thing here and then you find that the displacement of the outgoing stuff is there any proportion to rent on the ingoing stuff times a function depends on L helps VL scalpers in the denominator and there's neutrons constant standing in top because that's neutrons force which caused this relation without taking this interaction G would be zero and there would be no such relation so taking the gravitation force is very important in this whole argument and I find that the U of L of M for the outgoing particles is given by the P atom of the ingoing particles just like in the hydrogen atom and what this also says is that if you look at the wave functions now it becomes very easy these are wave functions in one dimensions every undergraduate student can be taught on a way from one dimension it's just a plane wave go in depth that way that's it so it can't be easier this equation you can solve them in fact the fact of the position of the article particles are controlled by the amount of the ingoing particles means that the wave function of the argon particles is the fluo transform of the wave functions in going particles and that it's the other way around as well so I can replace this P by U of the incoming particles now you get the P here but you get some minus signs he's become different so the outgoing wave is just a fluid transform the ingoing wave and that means that paths go in to do something in out coming out coming particles and the other particles are the foods and swarms of anymore particles this is lovely mathematics which you can use and but then and this is so important why to do it why doing the spherical wave expansion what you discover is something very strange going on the in your path you can be in the region one but the article particle is a fully transformed indigo in particle now if you take a function which is already defined on a half of the real axis you take its free transform it flows over the entire real axis it's non vanishing everywhere so if I take the incoming wave just sitting region one only the outgoing wave spreads over region one and the region to it has to because it's a free transform you can't have a half function and it's three times from also be half function doesn't work that way mathematically it will spread all over so if the in gold particles individual the argon party goes in out is we want and the region - the whole procedure would be beautiful and unitary in quantum mechanical terms except that I have to include both reasons 1 and region 2 sort of something not understood here this is so-called fen nose diagram now I don't know how many of you know what the Penrose diagram is it is a abstract way of describing space and time surrounding a black hole there's a region one described in the outside world there's a region - they just thought to be the inside of the black hole but if you look at the Penrose diagram the inside the black hole looks very much like it's outside it's just a mirror image so this is another black hole somewhere else in the universe that's what's hot you thought that can't be because these two parts now I've seen a conscious equation to talk to each other if you send a particle in in one article particles will be spread over one and two that came out of the equations and I have to quantified now I have the exact equations so you can't fool around to them so something must be such that this region - must prefer somehow to the same black hole so some people thought well it maybe it is the same black hole so we want to describe the same thing you have to fold the whole thing over but if you do that you get a singularity at this point that can't be right either that's again against general relativity something terrible is going wrong here well turned out to be not so wrong after all you have to okay yeah this is the the real bottom several displacement so all the argument particle being displaced so here's a particle which is being displaced the other side and when I did the calculation I made a mistake instead of having a pilot continue this way in my equations didn't realize the particles moving in this direction so this blue line goes here it shifts and then it gets out here well I made a mistake for a good reason if you make that mistake the computation comes up beautifully so the whole thing becomes a perfect unitary quantum mechanical evolution just fine and then I thought oh my god I made a mistake now how to improve that well let's change the whole let's change the whole let's say this mistake was actually correct then it works but say the mistake is correct you modify a little bit the laws of nature and so those something very strange going on so again in 1920s spherical wave expansion showed you how to understand the quantum mechanics of the hydrogen atom now we see that partial wave vacations also understand make you understand how black holes behave and in there's some sort of differences here that the passive waves are used in a black hole are not the wave functions of the particles they are the momentum distribution of the particles so there are some subtle difference between the hydrogen atom and the black hole sufficient subtle to say these are two different things I handle them physically different but mathematically it's very similar actually the strange way which region wanna reach to a hang together in this pendulum it's called the einstein-rosen bridge it seems to be a bridge that connects two different regions of the universe that's what's Hawking all times thought but now this bridge should somehow connect the black hole with itself how can you do that but the only way to make it work consistently is to say that the bridge brings you from one side of the coffin opposite side so this is called the anti polar identification and what is nice about this is that's exactly one solution you get exactly prescribed how to do this and there's only one solution to have a mapping of one side of that hole to the other side black holes such an if you do it twice you get that to the same point again there's only one solution which is the parity transformation you view a place the point is on top or and these two points should be identifiable in this way you generate no singularity at the origin for black holes that works for all of your flat space you can't do that you would get generated singularity so in all our space you can't say two opposite points are identified you run into problems with the laws of nature but for a black hole you can do it without going to any problems locally so there's no difficulty in assuming that you need your two therefore describes the points on the horizon that are the antipodal points of region one now this works and in fact the Schrodinger equation then get on a black hole is very simple it is basically that the Hamiltonian is Menten position of tea in your particle document of the annoying particles or positions in awkward particle size momentum there are current particles that is a physical known as a dilettante operator the rotation operator which expands things to and from the horizon that's exactly but now there's something else which is the article and particles are related to the in going particles and this gave me an expression for the scattering matrix the expression is beautifully unitary with only if you combine a region one with region two so you have a two by two matrix which is a two by two matrix you take but it's not unitary if you leave off the out the off diagonal terms region one talks the region two and vice versa so the black hole has two antipodal points which talks to each other and that makes this thing something very special and so what does it mean well you can think of a black hole as a sphere so now I'm doing good go to get the physical explanation of physical interpretation what I'm doing normally a black hole consists of an outside world and inside world and you believe that the horizon is a sphere and if you fall into the black hole you enter - it's inside inside into its belly if you'll be eaten and then you're eating you can't give away the secrets of your wave function anymore because you're being eaten up that was a standard picture but now it's something else if you enter through the black hole and here I have the arrow saying I'm entering the black hole from that direction you're not being eaten you emerge immediately at the other side so the inside of black holes cook totally empty there's nothing there you go in you reemerge to the other side now at first sight when you see this then you would say this is completely wrong this can't be right because you'd go look locally faster-than-light something that but that's not true because if you go in you're first being tremendously slowed down and today horizon actually do you come out beyond time equals infinity at the other side so actually it takes you a long time to cause a black hole that way the only way in which you can cause a black hole upon a time is do respond to everything that you turn the ingoing wave into a fluid transform to get out at the other side but the free transform comes out size of the black hole and but mathematically you can hide down such spaces rather easily and find the marvelous properties and now I entered your subjective firewalls the firewall is nowadays portrayed in literature as a mystery people don't understand what a firewall is jaurès halted particles accumulate against the horizon which kill everything is wants to go through her eyes and there's something wrong about that well the answer is basic given here and that is that now see we say that in going particles I related to the article particles if a particles into the black hole it comes closer and closer to the horizon with the momentum that increases indefinitely but the article particles then come out with a wavefunction it gets out of the black hole with momentum decreasing exponentially and position increasing exponentially so you interchange from entering position that's of interchanges in without and so give outgoing particles and the closer the annoying part is goes into the black hole making a nuisance of itself making a firewall the other one particle leaves the black hole and here comes the important observation the art going wavefunction is a quantum clone it seems of the wavefunction being in quantum cloning is forbidden occurring to well known laws of quantum mechanics we get impossibilities when parties are entangled so you can't have quantum Long's well here you can and the reason is that the enjoy patchable I cannot be seen anymore by the arts an observer once a particle through the rise it's lost so don't worry nobody has seen it you a quantum floating in fact your reproducing the wave function in terms of the outgoing particle we didn't so this is the one version from the chlorine that is allowed except you shouldn't describe to think twice you sure that both the enjoy particle and the outgoing particle then you're over counting the situation is very similar to what you have if you look in the mirror you have a box with a mirroring wall and you have a particle sitting here the way you describe the comic and you also see the particle in the mirror image of the fire but I should not describe the wavefunction is particle and the wave found for the mirror image no there's only one wave function and the mirror image is a quantum clone of that but that should be thrown away that their image is not real which is virtual it's in your image but it's not really physical so now the claim is that the firewall is the mirror image of particles outside a black hole the closer in Gori particle touches to the horizon the further the article particle is away from a black hole which of the two do you see you don't see the in going particle it's too close to the horizon you can't get there to measure it but you do see the particle outside so now the rule is just only focus on the particles outside don't include the Compton clone that sits on the horizon you remove it it's not physical and this is the procedure that I think resolves the firewall paradox as people tend to call it if the firewall can we remove this way the only thing that this valid is particle the wave front of the particle outside the black hole so it does away with the firewalls and in a very similar way it also does away with matter that formed black hole because the firewalls are a problem for the future horizon but matter which formed the black hole by implosion sits on the past horizon but at time reversal the same icon holds there so now you can see you get a beautiful description of black hole which is symmetric on a time reversal with you all the other picture wasn't and now this basically solves the problem but the expenses yes you do require new physics this identification the ant approach is something is wasn't put in we had to put it in my hand so to speak but it was the only solution that works everything else will make the black hole a violate basic concepts of physics like nonlocality it will communicate other black holes in a way is illegal according to normal physics so this is the only legal way to make a black hole behave and this would be the legal way to proceed to try to do quantum gravity this is just a side remark that the way the pathways I'm looking at all enough to have an odd value for the quantum number l and even values are forbidden that's because the when I look at her Manik functions I need if you go to the other polls the science would switch but then I may only use spherical harmonic function why damage has a property if I go to the antipode I should pick up a minus sign and that's what the older harmonics do if L is odd so I should only use the Earth's natural harmonics this also gives rise to heated discussions people don't believe that that because they thought that L equals 0 is the easiest thing you can consider but L easy equals 0 excluded because it's even I can only allow for all black holes so I so the question is what are the implications for the nature of space-time metric is still mystery so this is only a beginning step and I believe is that once people accept this first step they might be able to apply it to the rest of quantum gravity and see where it brings us we don't know but I see that some fundamental principles have to be formed have to be formulated differently and then you can understand how black holes behave but right now what I see is a black hole can be described as simply as a hydrogen atom I haven't mentioned some of the complications in this argument it's not exactly as easy as I portrayed it now there some some pumps with very large circle harmonics at very large values well you have to remove or only only keep low very cell but that don't take L so large that that things that move in sideways that actually then run in a problem this part of the theory has not yet been solved so there's so many many things to be done much more has to be done which is the last line of my talk so I reached this curly line which means that I've come to the end of my talk again I want to congratulate say care and hope that you see us very fruitful life has been retired and so Alex thank you very much [Applause] so I think I see how this works how this algebraic structure arises if you have a pre-existing black hole yes suppose however I take some state but if I if I imagine taking some tiny ripple and some scalar field at a very large radius locally there's no energy very little energy but it's all headed inward and eventually sometime far in the future if I follow things classically a black hole would form I I think if I were to follow your prescription I couldn't describe that state using how shall I put it the normal approach to quantum mechanics I would have to use your algebra in a situation where the state doesn't yet know that a black hole is going to form and it's going to have a horizon to deal with in the future well comment on that yes well when you when you have situations you describe that at some point a black hole is being formed you start up with a relatively flat domain of space and time that horizon opens up once you enter that flat domain you can't you can't avoid a black hole anymore you're being swallowed by it that's where I now apply the so-called new physics where I just make a mapping locally as flat space-time so a local observer doesn't notice anything it's like a polar bear who doesn't know that he comes through through the horizon or not but but globally one says oh wait I've got to make this transformation like going from position space to mental space you can do it anytime you want in quantum mechanics it doesn't really matter so I'm going from one basis of the other basis my cautious surface is now stretching between both regions and the mapping from States in in the scene by local observer to state by the global server is it one-to-one mapping that's because both regions one reason to mean something so I can always make this mapping for even if even if I don't really know for sure there are black hole forms at but you have already you prepare a thing to the business car to the right way sort of every black hole forms I can take into account what happens so it's not necessarily a contradiction although if you say it's not yet totally clear that I would agree with you and should be able to formulate this better but basically I don't see any you can take the most general initial state you like I do okay one important condition is I describe the black hole as if it is a background metric with all that metric soft particles and with soft I mean their momenta are small compared to the Planck scale the Planck scale plant limit is a value is a limit of the momenta and energies of these particles so as soon as the park has more energy than that it causes curvature of space-time I say done is a quantum clone of another particle being further out here Franchesca V dot a crucial thank you very much for your very clear is position and a crucial point in your argument is how to identify the region one with the region two and your solution is to say okay let's do a parity transformation so my question is why don't you don't we do a time transformation so why don't we reverse just time and let me just give you an argument for this when we work with a wave function of the universe there are two Loki wave function of the universe in order to give consistent boundary conditions we have to take a real function so taking into account the probability of one time direction of the other one so why wouldn't be here be there same if you look carefully here the top figure the the fat blue line is an equal time line so when time goes forward in region one it goes backward in the region two so to make identification pop you first have to switch over region two that's very important so really two is the antipode but when time goes forward in the it goes back with the region two so so you have to flip it over that's the way the identification goes so silly speaking if you go through horizon you having PCT inverters there's a parity switch if you look carefully you find out that they go through the rise to the other side do your enter into your emerging your mirror image and since continuity is only PCT invent but not doesn't any of the other sub advances so then it means that you better also assume that there's a see there's also a tease or in hockey particles also seem to be going backwards in time that we don't see because it's just a factory so it just seems rightly so you would say it goes forward in time but it's a PCT transformation thank you when you do this partial wave expansion and in the end you only keep the odd l that's something we see like if you know if we look at the Laplace room on a sphere and you want to impose norm and boundary conditions on the equator but that seems to single out a special line I mean is there anything like this in your things should be completely irrotational and variant right there's a rather unique but I got into conflict if you take even L so for even I'll for instance you have a particle in at this side and a pocket goes in at the other side with the same sign their effects cancel each other out so I would get two different in states and saying out state that's not allowed so I can't allow for equal L I'm not afraid even though I must exist Arella only odd no I don't know where else this happens of physics or something to be that's a good question you don't know what what other systems have this I I wonder in your picture if you just let your microstate evaporate you know just don't send anything in so that will evaporate in a in a perfectly coherent way you will not lose information so that means that the outcome should not be thermal should not exactly thermal so what what would be the deviation from therm allottee I mean how can you describe the final state of their so now this agent one this really wanted me to now if you to antipodal opposite points so if you look only look at half of the black hole everything looks perfectly thermal because you don't see the other half therefore your average over it and you're back into some description but if you do look at the others Emma sphere it's no longer thermal so for instance if you have imagined that the temperature has become 100 MeV 100 MeV so it's a potent who'd nearly never come out with probability to the minus nine it gives you a proton so it's very improbable but be either minus nine is not such a small number so every now and then a portion comes out in region one then immediately another proton or on the protracted region to not further suppress by another factor e to the minus nine but I factor one so now you see that there's a complete and balanced between the point what parts come out here and all part is coming out at the other side so so that's not thermal because if it were a thermal there would be another factor e to the minus nine for the other proton but it isn't according to this theory so some say this is about stop being thermal but as long as you only look at one half the black hole everything is seasoned perfectly several other why if you look at both sides the the final state will depend on the microstate yes yeah and the other question is what happens to the singularity if you erase the interior there is no singularity anymore there is a similarity second thing that becomes more complicated but that's where R goes to zero yeah it's ratchet so at R equals zero you have to fold authentication generates another conical singularity but I got zero was singular anyway for the Schwartz your black hole so having this on top of it doesn't change much a very naive question but I don't understand why you consider a black hole as a pure quantum state let me say that we we know many Beckles in the universe now is not strange thing and they are formed by matter which coalesce and these matter is not in a pure quantum state so if you say that the black hole is a quantum state then you have evolved from a non quantum state to a quantum state which is the opposite of the problem so I'm a little bit confused what happens to the entropy of the original particles that Qualis into a black hole if you say that the black hole is in a pure state but even if if the original black hole originates from States in a thermal State what you really mean is that you don't know the initial state is sufficient accuracy you don't there's a whole spectrum of pure states that have other probability you know the heart rate the sky mixed age you say it's a pure state which has certain probability of being in this state or that sensor that's in that state now the fact that you do those probabilities there it comes because you don't know which of these states is really describing the right thing just trying to describe a bucket of water I don't know where all the atoms are and I don't know what the momentum so on this I can't describe a bucket of water in a single pure state the best I can do is write on all the pure states that exist which you need a very smart notation of course because I very many of them and then you write out a probability of all these states to be realized it doesn't mean that nature chose that state it just means that we chose that state because you don't know any better about the bucket of water but if you would know it would have perfect information you could describe the bucket of water in terms of pure States of course the question then is what is its entropy well entropy is a kind of of measure of unknown information you have about the system so if you have infinite information the entropy in the formal sense becomes zero but you also know that that's not really what happens in the world even a bucket of water it can still freeze over or evaporate as water does even if it were a pure state just but it's too complicated to describe under all of the circumstances so so I want to scratch the initial state as a pure stage but if you describe the initial state disappear say the final state should be pure as well but in practice of course you cannot write the scratch initial state only as a mixture then the outside the outcome you say to also be in the mixture and it'll look much more like thermal but not exactly and so I still think that there will be this this strange are called well entanglement between the in and our coil particles so that attainment will if all's well not go away completely so when you allow the black hole to evaporate yourself see the entanglement between you enough between the Horten particles alongside in the other side so that's where you could really do an experiment on black holes if you had one you would do an experiment put a detector here I put a detector there see whether the particles coming out are entangled it should be observable one last question could we do the twin paradox here in other words can if I understand you correctly I have a twin pair one of them stays home and one of them rides a spaceship through the event horizon out the other side and presumably can come back and rejoin his fellow and if I understand you correctly the clock on the spaceship will have advanced a finite amount of time and the question is what will the comparison twins clock have done during this this voyage when the two come back together well you can't do this exactly because regions one a region two here describe different domains of the universe so you can have someone sitting in the region one and I say you want him to go to region two unfortunately he can't because he has to go faster the local speed of light so he can't so in the only can do is and Region three have himself Fourier transforms and then come out but I think foods being fully a torrent is battery else No so there's no paradox because a twin can tell me I propose that we break here we think again Cheryl [Applause]

Channel: Département de Physique de l'ENS

Views: 39,102

Rating: 4.805274 out of 5

Keywords: black holes, hydrogen, Planck's scale, Nobel Prize, Hooft

Id: 04d0QQgxULM

Length: 71min 16sec (4276 seconds)

Published: Fri Mar 23 2018

Reddit Comments

[u/MaunaLoona]

Biggest breakthrough in understanding of black holes in 30 years and no one is talking about it.