 Well, good morning to everybody. Let's start. We always say the good things have always to end, also. So this is the last lecture of Juan. So we'll be very pleased to have one extra hour of a table for black holes. So Juan. OK, so we'll continue in this talk with this theme of traversable wormholes. And this talk will be based on two papers that we wrote during the last year. And in the first one, it will be some construction of traversable wormholes in ADS-2 and SYK. And the second part will be how to use those lessons to construct similar solutions in four dimensions. So we'll see that these two-dimensional, let's say, toy models can be very directly lifted to certain four-dimensional configurations. So the first one was done with the Siola and Chi. And so I remind you of the metric of ADS-2. So this is a metric we discussed yesterday. We have the Penrose diagram is a strip parameterized by the coordinate sigma going from 0 to pi. And the coordinate T, which is going from plus to minus infinity, this is what we normally call global coordinates of ADS-2. In these coordinates, the metric has two boundaries. So this is a space with two boundaries. The boundaries are the places where sigma goes to infinity. That's there the G00 factor of the metric or the GTT factor of the metric goes to infinity. And while the proper distance to the boundary can be calculated by integrating this sigma over sigma along a space like trajectory. That's also infinite. So it has two boundaries. And these two boundaries are causally connected. So you can send the signal from one boundary to the other. And so in order to understand the propagation of any field here in ADS-2, of course, you have to put in some boundary conditions. Now, if you have a massive particle in ADS-2, it would follow an oscillating trajectory of this kind. And basically, this particle is feeling a gravitational potential that pushes the particle to be close to the center of ADS-2. So you have a big gravitational potential that is essentially this 1 over sine square sigma potential. And that's why the particle is oscillating. Of course, this space also has a group of isometries, which is an SL2R group. And one of the isometries is manifest in these coordinates. It's just the global time translation. So it's just time translations under this time. The other two are not manifest in these coordinates, but they're also present. In fact, by acting with those isometries, we can take any geodesic like this one and move it to the origin. So this particle moving along this trajectory would feel the same as a particle that is just sitting here at the center. So those are the so-called global coordinates. Now, this can be viewed as a kind of traversable wormhole because it has two boundaries. And we can send a signal between the two boundaries. So this is just a space time. And whether it's a solution of some theory and so on, we'll discuss later. Now, the same ADS2 has another set of coordinates, which are the thermal or Rindler coordinates, where we make manifest another SL2R symmetry, which is the boost around this point. So that's another SL2R symmetry. So we can choose these coordinates in ADS2. And now, the manifest symmetry is time translations under this time t. So this is what we call Rindler time. So these coordinates look a bit like Rindler space for flat. The analog in flat space would be Rindler space, but this is the ADS analog of Rindler space. And in these coordinates, the metric looks like it has a horizon at the rho equal to 0. That's where G0, 0 vanishes. And the geometry locally is similar to that of flat space. So that looks very much like a black hole horizon. And in these coordinates, if you go from t equal to plus infinity to minus infinity, you cover only a portion here of the boundary. And if you imagine that only that portion is, let's say, physical, then you find that, again, you have two boundaries. But now the boundaries are closely disconnected, because if you try to send a signal from this point to some point here, you cannot do it. So the signal will go into this region of the boundary. But if we, for some reason, define that the physical region is only this portion, then we have something like a horizon. And indeed, if you take the limit of a near-extremal black hole, you will get ADS2 in this coordinates. And the physical region will be this region of the boundary. So in this case, we can think of this as describing a non-traversable wormhole. So now, in both of these cases, we had exact ADS2. But it turns out that quantum gravity with exactly ADS2 boundary conditions does not make sense. So it does not allow any non-zero energy states. So it actually makes sense if you are only interested in the extremal entropy. And indeed, you can think about that problem of just counting the number of extremal states. But if you're interested in any dynamical question, then you cannot impose the exact ADS2 conditions. So we need to break slightly some of the ADS2 isometries. And so we should think about some space which is sort of nearly ADS2 space. So as opposed to higher dimensional ADS theories, we cannot preserve the isometries of ADS2 exactly. So we've talked in previous talks about, well, yesterday, about nearly ADS2 space, which has this T isometry, so the isometry of Rinder space. And we obtained that from near-extremal limits of black holes. And also, we saw that the SYK model also had a picture like that. So now, we could wonder whether we could do a different breaking of the isometries of ADS2 that preserves the global time translation. So a different type of nearly ADS2 where the global time translation isometries preserve and we break some other isometries. So those would be different boundary conditions for ADS2, different ways of setting up a problem that would make sense, that would preserve that isometry and would break others. So first, I'll recall some of the things we said yesterday. Perhaps since we said them yesterday, I probably don't need to recall them very much. So we need to, in order to impose these other boundary conditions, we need to consider an action which contains an extra field, which is the so-called dilaton. From a higher dimensional perspective, we would be related to the volume of the internal manifold. And it gives the leading gravitational dynamics. So if this is coming, this ADS2 is coming from a four-dimensional black hole, phi 0 would be some constant, that's the extremal entropy, and phi would be the deviation from the extremal entropy and the extremal area. So the area of this neck is growing slightly away from the extremal value, and that's this little phi. And then the gravitational dynamics, we said that it can be, that comes from this action, is equivalent to a dynamics of a boundary, of a boundary particle. And this boundary particle is physically telling us how to cut off the ADS space and join it to some higher dimensional space. So the only part of the ADS2 space that we keep is the piece inside these two boundaries. That's just recalling things we had before. So the whole Hilbert space of quantum gravity in this ADS2 with this description is the Hilbert space of all the bulk particles. So these yellow things are bulk matter fields. So we could have, for example, a fermion field living in the bulk. We could have a boson field, a free boson, or an interacting boson, or whatever we want to have in the bulk. That lives in a rigid ADS2 space. So this is a quantum field theory in ADS2, a standard quantum field theory in ADS2. That's the yellow line. And that is this part of the Hilbert space. And then we have the Hilbert space of the right boundary particle and the Hilbert space of the left boundary particle. And all of these Hilbert spaces, so with the full Hilbert space, is a tensor factor of all three. And there is a gauge symmetry, which is an overall translation of these three systems. This is a bit like the Mach principle that tells us that any excitation here makes sense only when it's, I mean, we have to refer to the distance stars, which are, let's say, this boundary of ADS2. And only in relation to that distance, their positions are defined. So this is just a gauge symmetry, it's an exact symmetry, but be quotient by it, so it's not physical. The final theory does not have, is breaking the physical ADS2 symmetries. So one should not confuse this gauge symmetry, which is translating everything, with physical SL2 transformation of the boundary time. So there's a boundary time and you can do an SL2 transformation of the boundary time. That SL2 transformation of the boundary time is not a symmetry of the full problem. Anyway, so that's the story. And now if we just have pure gravity, the only solution with phi, this field that grows towards both boundaries, is this term has the isometries of Rindler space. So in order to have a solution that has the other isometries, we need some kind of matter. But in order to have phi to grow in both directions, so the equations of motion for phi, one of them implies that the derivative of phi is equal to, well, the second derivative of phi is equal to the stress tensor, and once you integrate that, you get this type of integral. You are integrating this between the left and right boundaries. This equation is the same as the four-dimensional, if it arises from higher-dimensionals, this is the same as the Rachel-Dury equation, so the change in area as you follow a light ray. And so you find that in order to have phi growing into both boundaries, you need that the integral of the null component of the stress tensor should be negative. Now, this is typically forbidden for classical matter and ordinary matter, so we'll have to do something special to get this. So the special thing that we'll need to do is we'll have nearly ADS2 gravity and we'll have some matter, but the matter will have to have some boundary conditions that connect the two sides. So if we have matter in ADS2 space and it has, let's say, direct boundary conditions on both sides, independent boundary conditions, or any boundary conditions that do not relate the two sides, then this integral will always be positive. People have shown that it's always positive. And so in order to have it negative, you need some trick similar to the one we discussed yesterday for the traversable wormholes, this Gauss-Jeffery's proposal. And in this context, that would be similar to saying that you put an extra interaction term in the matter lagrangian. So we have chi is a matter field that lives in the bulk and we'll add an interaction term that couples the boundary values of the fields on the two sides. Now here we put this by hand. So we said that in order to fix the dynamics of the bulk fields, we need to put some boundary conditions and we are free to put whatever boundary conditions we want and in particular we can put the boundary conditions that relate the values of the fields on the two sides. This looks a little funny from the bulk point of view because you're saying that, let's say, an excitation that comes out of one side can immediately appear on the other side. Looks a little funny, but we'll see later how this could arise from higher dimensions. So this seems to be an acausal interaction in this two dimensional space, but we'll see that it will be consistent with causality in a higher dimensional space because these two sides of ADS2 will be connected in the higher dimensional space. So for the time being, we'll just put it by hand and then we'll see how it arises from a higher dimensional point of view. At the point is that by having this type of interactions we can allow negative energy and we can now have a solution with global time isometry where phi grows towards both boundaries. And we'll analyze this problem in parallel. So this ADS2 problem in parallel to with DSYK model because actually the mathematics and so on it's exactly the same because in order to find the solution we have to understand the solution of the Schwarzian action. We yesterday discussed the gravitational dynamics of ADS2 and the dynamics of a particular mode in SYK is the same and so we'll see that here too. So we discussed the SYK model yesterday, this model with interacting fermions I won't go through it again. And the only difference now is that we'll have two copies of SYK. So the left and right system with identical microscopic coupling. So in SYK we had these random couplings and so once we choose them for the let's say left system we take identically the same couplings for the right system, okay? So we don't have two separate fluctuating couplings on the two sides, so only one. And then so we take two identical then SYK models and we put some interaction between them. So now these sides are the fermions of the SYK model. So we are adding this interaction and actually I forgot an I here, there should be an I for this operator to be her mission. And this interaction, it's a relevant perturbation of the SYK model. So naively you would say, well, you have this interaction. Before the interaction you had some kind of conformal, almost conformal theory and you add the interaction you will generate some kind of mass gap and that's it. So that's the naive intuition. And this intuition is correct but in fact this gap system has still many properties of the conformal theory. In fact they will have the conformal symmetries realized in some interesting way. But in order to analyze this, so we'll analyze this system and it will be important to understand in order to solve the system, first we'll solve the decoupled system, so with mu equal to zero. And in that case, some low action part of the dynamics is given by this two decoupled Schwarzian theories that those are the ones we discussed, we discussed those yesterday for each SYK model separately. And now at low energies, we can include the effects of this interaction and we include the effects of this, I'll try to discuss this a little bit more in a second. So the way to include these interactions is to first neglect these reparameterizations, assume that F is just constant, just proportional to U, so it's on the identity map between the two times, the boundary, so U is the boundary time or the time in the boundary theory in the macroscopic time of the real time of this SYK model and F is some kind of infrared time, some effective time. In the ADS language, it will be the time coordinate of ADS that we were talking about in the previous transparencies. It's actually the Poincare time, very precise. And so we'll evaluate this, we'll just simply approximate this by its expectation value, the expectation value in the thermo-field double state and that's just one over T minus T prime, so that's this conformal limit that we were talking about before. And then we'll include only the fluctuations along these reparameterization directions and that will reparameterize that two point function into this function of left and right times and now we get this, that induces this new term in the effective action and this is an approximation to the action of the full theory and then once we have this action we can just take it as an action on its own and then calculate the equations of motion and find the solutions and we can find the solution and so on. So let me try to, so here I'm going to explain the same thing but from a slightly more abstract point of view to explain exactly what we are doing so this approximation to solve in the theory. So yesterday we discussed that the SYK model is equivalent to a theory in terms of some function G of two variables. So we can go from the SYK model to at large hand to an effective action or an action in terms of a function of two times. So this diagram symbolically represents the space of functions of two times. So it's a big space of functions and in this big space there was a solution that corresponds to the thermo field double, okay? So that's the particular solution of the full theory, the full SYK model. In terms of this function G, there is one minimum of the action, okay? Then we saw that at low temperatures there is some slice in this space of functions that is parameterized by functions of one variable. So functions f of one variable, these reparameterizations of the solution over here and along, this is a kind of valley in the space of, in the action, the space of functions and the action has a very low value along this valley, okay? So if you think of this as some kind of landscape but there is given by the action and there is a global minimum at the point x but there is a very shallow valley along the red line, okay? And the shallow valley corresponds to these reparameterizations. So those are functions of one variable and to first approximation, the valley is completely flat but then when we remember that this reparameterization symmetry is actually broken then we get the little potential along the valley which selects this vacuum. So that was for the thermo field double configuration of the coupled system. Now what we are doing is we are adding some interaction term in the action, okay? So we are adding a new term in the action and the way we are solving the model is we are simply evaluating that term using the solution. So we're evaluating that term not everywhere but only along this valley. So we take that term and we evaluate it along the valley, okay? And so we get some new term, so some new let's say extra force along this valley and the new solution will be somehow displaced along this valley by a relatively large amount which will be the new solution for the function F but will be displaced in the directions orthogonal to the valley by a small amount because the walls of the valley are sharper, okay? So that's what this is in words what we did in the previous transparency. So the approximation we used in the previous transparency is conceptually what we're trying to explain here which is a common thing to do in physics. It's roughly like the Bonaparte-Hammann approximation for the atoms, right? So the distance between the two nuclei of let's say the hydrogen molecule that would be like the function F and the wave functions of the electron is like integrating in the directions orthogonal to this valley. Okay, so then evaluating along the valley is the thing that taking this particular term which in general is this term as a function of the function of two variables. So it's evaluating the function at two points and integrating over time. Then that evaluated along the valley becomes this function of the reparameterized time. So here I express the same function we had before but in terms of reparameterization of the global time. Ah, okay, maybe here was a picture. So now we have a new potential along the valley that will have a minimum at this other position, okay? So it's a position that is far along the valley but very close to the minimum of the valley in the orthogonal directions. So that in pictures is saying that we have these two boundary particles and there is some interaction between them that comes from the fact this is now the ADS picture. So we have the two boundary particles and there is a force between them that we get by evaluating this interaction term, this mu times the boundary values of the field in the vacuum that we had before. So in the vacuum corresponding to just ADS space, the quantum field in ADS space with the previous boundary conditions. And then this gives this term the same term as before and we get a simple solution. So a simple solution to this action. So here this interaction term and it turns out that the simple solution is to say that the function, the time t, t was the time, the global time in ADS as a function of the boundary time. It's just some, it's just they are linearly proportional and with some constant which we call t prime, t derivative of t with respect to u and t prime is just a constant. And so this t prime sets the rescaling we have to do between the boundary time and the global ADS time normalized, let's say in some conventional way. And so that gives us the ratio factor between the boundary time and the time at the center of some observer who sits at the center of ADS, please. Yes. Yes, yes, yes. Which were decoupled. Yes. And now you are introducing a coupling. Yes. Shouldn't there still be some flat direction? Because now if I only look at the right, then the coupling to the left is introducing a new potential but can't I move them together? Yes, yes, yes. You're absolutely right that there will be flat directions but these flat directions are in, so this theory, okay, I went over this but the theory has some SL to our gauge symmetry and all the flat directions of this action are gauge modes. So once you put the gauge conditions then, so the flag directions correspond to taking the two boundaries and moving them inside the ADS and that's an irrelevant, that's a gauge symmetry. So we just quotient by that. And once we do that, then this theory has no flat directions. And in fact, so this is the lowest energy solution and then you have also excitations where you could make the boundaries oscillate around this minimum. So this is like the minimum energy configuration and then there are other solutions of this action. There is one other physical mode which are small harmonic oscillator motion, at least for small fluctuations, there are oscillations around this point. So it's a stable minimum. So yeah, so that's the solution and so this is an answer for a solution and then we put it into the action, we demand this gauge charges are, I mean this gauge charges are not anything mysterious, they're just simply the usual reparameterization constraints of general relativity. So it's just the Hamiltonian constraint and stuff like that. So in this case, we impose this Hamiltonian constraint and then we get this term has two pieces, one comes from the, let's say each individual Schwarzian action which has roughly this form and then a piece coming from the interaction. So this is essentially the value of the two-point function, the expectation value of the two-point function as a function of this time t. So it depends on the time t because the time t gives us a t prime gives us a rescaling of how far we are in ADS so the two-point function changes so that when we go further and then we set them equal and then we get some value for t prime. So t prime value which in sort of yeah, which is given in terms of the value of the interaction. So when the interaction is small compared to the scale that appears in the SYK action, so for small values here, then we will always find a solution as long as delta is less than one. Delta was a quarter for the standard SYK model. And when delta for example is a quarter which is between zero and a half, then this new scale that we get, so this is some kind of energy scale and this energy scale is smaller than the J of SYK. So that justifies, a posteriori it justifies the approximations that we've made because we made the approximation that the SYK model was in its low energy regime so that we could use the Schwarzian approximation and indeed we could. And then it turns out that then this value, then this mu is also smaller than this t prime and this is implying that the effect of this perturbation is relatively small. So by relatively small, I mean that in the directions orthogonal to the valley wall, so in the directions of the walls of the valley it's a small thing so we are not displaced too much in the directions of the valley walls. So it was correct, a posteriori again it was correct to use this description of the valley and so on. So this is somehow, well if you didn't follow this, we are just a posteriori checking that the approximations that we did find the solution were correct. So that's the final picture is that we have some negative energy here in the middle that is somehow holding together these two walls. We can think in terms of a force between the two walls, the two boundaries and that's roughly the final picture. As there could be particles, also we could add particles in the middle, they will oscillate, this would make the walls oscillate a little bit, the walls can also oscillate and you can find the spectrum of all these excitations. So we have quantum field in the bulk, it will have a spectrum that is given by SL2R representation, so delta is the, well that's the minimum energy of the particle in the bulk at the center of the bulk and then we have all the sand. So the main point is not the precise formula, if you've never seen this formula, don't worry about it, this formula is just saying that the spectrum has a piece which is determined by the SL2R symmetries of the bulk of ADS2. This is a physical SL2R symmetry which is almost a symmetry of this problem. So we said that this theory is nearly conformal invariant and this spectrum has a leading piece which is exactly conformal invariant, which is this piece and then there will be higher order pieces which will not be exactly conformal invariant. In fact, it has also a piece that has to do with the motions of the boundary. I briefly talked about this in answer to Merdad's question of the two boundaries could be straight or there could be small oscillations and those harmonic oscillators, that's the spectrum. That's the energy of a single harmonic oscillator and M is some integer, so it could be occupied any number of times. So that's the spectrum of the model. So it's a curious thing that you take something that is a conformal field theory and you deform it by a relevant perturbation and in the new, let's say, gap phase, you can still find the full spectrum in terms of the conformal limit. So in some sense, the conformal symmetry is still present in this and that's something very special for this system. This is not true in other conformal field theories in higher dimensions and so on. So it's, okay. Anyway, sorry. Now in S-Y-K we could do everything we were saying but we could also solve the large hand equations exactly. But the large hand equations can be solved numerically and there are, well, I didn't talk about the large hand equations before, they have some form, take this as impressionistic, don't, I'm not expecting you to follow, but it's some concrete set of equations you can write down and then you can calculate, you can solve these equations either numerically or there is some way to solve them by changing the S-Y-K model and instead of having four fermion interactions, you can have q fermion interactions so there's some variation, some family of models and in some large q limit you can solve them analytically. So that there is just some technique for solving the equations in various approximations. And so in that case you can solve the equations without this mu over j that we discussed and we can also study the finite temperature situation. And so the finite temperature means you can take this global S-Y-K model and fix now, so before we were considering the ground state so we're going to zero temperature but now we could be at finite temperature and sum over all these possible excitations that we could have and when you do that, you find basically a first order transition so there's a zero temperature phase which is given where the free energy is given by this straight line. That corresponds to basically this empty ADS-2 space with some excitations and the number of excitations that you can have is not too independent so you basically at low temperatures you won't excite this bulk excitations very much so you have basically a free energy which is given by the ground state energy of this global model. So you'll get something with constant energy and no entropy so that's the straight line. Then if you raise the temperature very much then you'll have another phase which is this phase here and this corresponds to a situation where you have the two separate S-Y-K models at high temperatures so you can think of a single S-Y-K model at high temperatures and then the other one will also be at high temperatures each of them will be related to a kind of black hole with a horizon so the bulk picture will be two separate black holes and that's the high temperature phase and then there is the two lines crossed here and there will be first order phase transition between the two so this is something that is standard when you consider ADS in global coordinates for higher dimensional conformity ADS phases it's also true it's called the Hawking page transition discussed by Hawking and Page and the same happens in this context. Now the interesting aspect the interesting feature of this problem is that at least in the S-Y-K model you find that these two phases are actually continuously connected by a phase which would be unstable in the canonical ensemble so they are continuously connected and actually if you think that instead of fixing the temperature since naively you would not be able to access this other phase when you are in the canonical ensemble because it's unstable as I just said but if you are in the micro canonical ensemble where you fix the energy then you find that the behavior of the entropy is completely smooth and so these are all results in the S-Y-K model where we can solve the equations and so it's complete it is a smooth curve and we expect we will kind of conjecture that if you did it in the micro canonical ensemble this phase will be stable so you would... and now we have actually there is also a gravity picture for what these phases are so these constant phases like taking this ADS-2 space which was a bit like a wormhole and compactifying the time direction so the geometry of the Euclidean manifold would be roughly like this two circles and joined by let's say Euclidean wormhole between the two and then here we will have two separate disks two separate standard finite temperature black holes Euclidean black holes that's the phase here and here in the middle we have basically the same picture as here but with some extra matter in the middle and for a given so there are certain points here for example for a temperature here so the temperature fixes the length of the boundary circles and for a given length of the boundary circle you can have these three geometries so the difference between these two geometries is that the length of the circle here in the middle is larger in this one sorry, shorter in this one than it is in this one okay well I could explain a little more maybe I'll stop here so then you can also do some numerical analysis of equations you find and so on now one application of this discussion is that we could make the thermofill double state so we talked a lot about the thermofill double state before but we never discussed how easy or hard it is to make it and you could be left with the impression that making that thermofill double state where you have to match all the energies of the left and right moving systems might be something exceedingly fine tuned and that would be very very difficult to make but in fact you can get a state that is fairly close to the thermofill double by this construction so you first start take let's say two SYK systems you couple them and then you couple the whole thing further to a heat sink and cool the system down to its ground state and after that you find the ground state of this couple system and once you do that then at time equal to zero you could turn off the coupling and if you turn off the coupling the subsequent evolution of that system will be close to the evolution of the thermofill double first of all the state that you get is close to the thermofill double state and its evolution is also going to be close to the thermofill double state so that's one way to get a state that is close to the thermofill double state and as I said there you could imagine if you could simulate an S by K model in the lab you could simulate two of them if you are really powerful you could do two of them you could couple them you could study their spectrum you could make the thermofill double state you could do all these things now I have a small comment is that here we have assumed that the microscopic couplings were identical between the two sides but even if you don't make them identical you make them slightly different the situation will be the same so you don't need to have a perfect matching between the couplings or even a perfect matching between the energy eigen states of the two models to have a state that is close to the thermofill level so the conclusion of this the first part is that we had this variant of the Gauss-Wald and Gauss-Jafferys and Wald proposal of teleportation proposal so here we had an interaction that is present for all times we found this eternally traversable wormhole or a state in the S by K model that can be interpreted in ADS 2 or is similar in ADS 2 to an eternally traversable wormhole we can analyze it completely in context and we discuss some thermal aspects and so on and we saw how we could realize a state that is close to the thermofill double as the ground state of the coupled model okay so now okay you could say well these are all some very two-dimensional models I'm only interested in the four-dimensional world and so I don't care about anything you say so then we'll see how all of these can be can be lifted to four-dimensional discussion this is based on some work we did to Princeton students and this this is related so what we are going to discuss is the construction of a solution that I guess was first ambition by John Wheeler like many of the things we are discussing were ambition by John Wheeler and he he made this drawing two charges and the flags going from let's say could be like say two magnetic charges and the flags does not have any explicit source he called this charge without charge so there is the flags goes between the two charges through a kind of wormhole in space time okay now this is of course a configuration that is possible in the path integral gravity and so on you have these configurations and his idea I guess was that you could have maybe charges and physical matter actually comes from geometry in this way now that might well be true but what we'll try to do now is to find an actual solution of the equations that has this form a classical a mixture between classical and quantum solution that is under control so we'll find a solution that has this shape as the spatial geometry so spatial geometry would be this and the whole thing will have a time translation symmetry so it would look like this for all times okay and so this will have the geometry of a traversable wormhole because we could enter one mouth of the wormhole and come out at the other end of the wormhole okay now first we have to recall that we expect an expectation that there are no science fiction wormholes so what is a science fiction wormhole the science fiction wormhole is a wormhole that will allow you to travel faster than the speed of light in the ambient space so it would be a wormhole that where you go in this mouth and you come out this mouth before you would have come out if you were traveling in the outside space at the speed of light so this is the kind of wormhole So those wormholes, we believe, are not allowed by the loss of physics, as long as this loss of physics obeyed these two conditions. So one is the Einstein equations, and the other is the acronal average non-energy condition. And this condition is supposed to hold, and it's been proven in some cases, it hasn't been proven in all generality, but it's probably going to be proven in all generality soon. There is, of course, no counter-example. And what this condition says is that the integral of t minus minus along, let's say, some light ray that is along the x minus direction will be positive. So it's important we are integrating over an infinite line, and it's also important that the line is acronal. So acronal means that it is the fastest line in the geometry. So if, for example, we had a traversable wormhole, then we would have, this would be the fastest line in the geometry, so it would be an acronal line. And then the idea is that if you take quantum fields and you integrate t minus minus along this line, you should get something positive. But then Einstein's equations imply that you cannot have such a wormhole, OK? OK, so that's similar to what we, yeah, so that's, so we cannot have science fiction wormholes, OK? But this argument does allow other kinds of wormholes, which are not shortcuts in the fabric of space time, but they're sort of long detours, OK? So you can have this line could be longer somehow than the, I mean, if you go along this null line, you could arrive later than you would have arrived if you were moving in the ambient space. So those things are allowed, but it is not possible in classical physics, because in classical physics, classical physics doesn't care about these global conditions, and in classical physics t plus t minus minus, the integral of t minus minus always positive. So we need some quantum effects to find a solution of this kind. So and in quantum physics, you can have negative energy and negative t minus minus, integral of t minus minus could be negative. If the line is not a chronal, and one example of that is Casimir energy, OK? So it is possible to have these longer wormholes if you find the suitable Casimir energy. But the question is where you can do it in a controllable way. So because ordinarily Casimir energy is a small quantum effect that has to be competing against some classical effects, and so it's not clear that you could do it in a controlled way. So before we say how to do it, let me just remind you how you could get negative null energy. So imagine you have a two-dimensional conformal field theory, so one space dimension and one time dimension, and the space is a circle. So in this situation, the ground state energy of the system is negative and is proportional to minus 1 over the length of the circle, or minus some constant over the length of the circle. So this is a quantum effect. And the component of the energy at this constant is precisely the null-null components of the stress tensor. And so if you took this line that wraps around the cylinder, and you integrated the null components of the stress tensor, you would get something negative. This is fine because this line is not a chronal. So these two points along the line, for example, are time like separated. So the fastest way to get between two this point and neighboring point is not to go around the circle, but just to not go around the circle. So it's perfectly fine to have, so in other words, it's perfectly fine to have this negative null energy. This negative null energy doesn't contradict the statement we had before. So the condition that is supposed to be true in quantum filtering in general is the a-chronal null energy condition, not this one. And indeed, this is perfectly fine for two-dimensional conformal filters. This is a very well-known result. OK, so the necessary element is that we'll need something looking like a circle to have this negative energy. And we'll also need a large number of bulk fields, of effectively bulk fields, so to enhance the size of the quantum effects so that they can compete against some classical effects. And we can assemble these elements in a few steps. So the theory for which this will be valid, so we will discuss a solution that will be a solution for any theory that contains the following three elements. So it will contain the Einstein term, then it will contain U1 gauge field. We'll use these two components to make sort of near-extremal black holes. And then we'll also need a massless charge fermion, so a fermion that is charged under this field, and it's also massless. Now these components are present in the standard model of particle physics. If we make the whole solution smaller than, let's say, the electroweak scale, because in that case we can approximate all the fermions of the standard model to be massless. So that's why we need the whole solution to be smaller than the electroweak scale. So if we make a tiny little solution of the kind that we will discuss, that will be a solution in the electroweak theory. The U1 gauge field is the U1 of the standard model, the weak hypercharge. So the first solution we'll consider is a magnetically charged black hole with magnetic charge Q. So there is the U1 gauge field. We put magnetic field, and then we'll have a near-extremal charged black hole. That's how this geometry, we were discussing this type of geometries before. So when we are at low temperatures, so we'll have a very long neck, and basically the length of the neck is proportional to the inverse temperature. Precisely, this length is not really the proper length, but it is the retrofactor between the top and the bottom. They're rescaling between the time at the bottom and the time at the top of this geometry. So that's a magnetically charged black hole. Now let's discuss the motion of charged fermions in the presence of a magnetic field. Now we have a magnetic field on the two-sphere, and so if we have a charged fermions, then we'll have a set of Landau levels. And for a massless fermion, there will be a Landau level that has precisely zero energy. And that's a Landau level where the orbital energy and the magnetic dipole energy precisely cancels. And now you can understand that, okay, now this is, I guess, a well-known fact, that you can understand why there should be this precise cancellation from the point of view of anomalies. So if you have a chiral fermion and so on, there is an anomaly, and then that anomaly should persist in the two-dimensional effective theory. So the fermion should be massless in the two-dimensional effective theory that results after compactification on the sphere. And anyway, so this Landau level has a degeneracy Q, where Q is the flux of the magnetic field on the sphere. So the idea is that then from a single four-dimensional massless fermion, we get effectively Q massless fermions in the two-dimensions that are transferred to the two-sphere. So we have the two-sphere, and then we have two other dimensions, which are the time dimension and the radial direction of the geometry. And in those two dimensions, time and r, so the ADS two-directions, we have massless fermions. How many we have? We have Q of them, okay? And we can think of each of them as following, so each of these two-dimensional fermions, we can think of them as Landau level, and those Landau levels can have wave functions with different choices for the wave functions of the Landau levels. Of course, they are degenerates, so you can make different choices. But one choice is to think of them as localized within each magnetic field line, within one flux quantum around the magnetic field line. And so the magnetic field lines are sort of coming out of this throat, and we can think of a massless fermion living along this magnetic field line, and then there is a neighboring one. I here drew only one of the magnetic field lines, of course, and so on. Now, let's go back to, now briefly, well, so we remember that story, and now we go back to ADS two. So ADS two had this global coordinates that we discussed, and then it has also this Rindler type coordinates. And we'll, so this is all of ADS two, but in order to connect it to the four-dimensional geometry, we can cut up these ADS two spaces and connect them to the four-dimensional geometries in different ways. So the idea is we start with two black holes, and we, yeah, maybe we should go back here. So if we take near-extremal black holes, we have two black holes that are connected in such a way that this T isometry is preserved, but now we want to connect the two geometries so that this other T isometry in this coordinate is preserved. And we can do that, so we can take two, two extremal black holes, and then we connect these two geometries in a way that we preserve this global time translation geometry. This is the same as taking this ADS two and then cutting it off along two parallel lines and connecting it to the four-dimensional space. This is not yet a solution. This is just an answer for the geometry, and we'll see what we need to do to make it into a solution. So so far we have an answer for a geometry which roughly has this shape. So we have two, so the geometry near each of them looks similar to the throat geometry, the exterior geometry of a near-extremal black hole. In the interior, it's different instead of having a horizon, we have a connection between the two geometries that locally looks like ADS two in global coordinates. And then we have magnetic field lines that go around this circle in the same way as in this Wheeler drawing. So for example, if we take one particular magnetic field line, the magnetic field line follows a closed circle, and as we said, the four-dimensional fermion will have an eigenfunction, so the lowest energy eigenfunctions which is localized, has a wave function which is localized along this magnetic field line and has excitations that behave as massless fields moving along this circle. So we have now a circle and massless fields moving around the circle, and so we will have some Casimir energy. So even if we don't have the physical field, just the vacuum energy of that field will generate the negative energy in this configuration. We'll imagine a geometry where the length, so we said that in order for this solution to exist, necessarily the length here has to be bigger than the length here between the two black holes, the distance between the two black holes. And so we'll imagine just for the sake of the argument that this length L will be much larger than the distance, and then a posteriori will check that the solution will get this of that form. So if this length is much longer than the length outside, then this Casimir energy will be roughly, well, will be proportional to one over this length, and we'll have a factor which is proportional to the number of fields that we have, and the number of two-dimensional fields is equal to the magnetic charge of these black holes, and all of this is coming from a single four-dimensional field, so we have a single four-dimensional field that single four-dimensional field gives rise to these q-massless fields. So we have this energy, and then we can find the solution by solving Einstein's equations with a stress tensor that has a contribution from this negative energy. We can do that, I won't present that way of doing it. I mean, you can find it in our paper, but I'll give you a qualitative argument, which is sort of an energy minimization argument. Imagine that you find this ansatz for the solution, and now you try to minimize the energy. So there are two competing effects, and so one is the negative energy of the Casimir effect, and the other one is some energy that comes from the curvature of spacetime, so from the fact that when we made these ansatz for the black hole shape, that ansatz was not actually a solution of Einstein equations, but it will contain a little bit of contribution from the Einstein term. And the contribution will scale in this way with the length, and the way to understand the scaling is to remember that, for example, you can understand the scaling by noticing that the energy of a near-extreme, so forget about the solution and think about the near-extreme of black holes. So for a near-extreme of black holes, the energy above extremality is proportional to the square of the temperature, and we said that the temperature is roughly related to the length of the throat between the top of the throat and the horizon. Now, the near-extreme of black holes can be viewed as two black holes connected by this wormhole, and again the length of that wormhole will be proportional to the inverse temperature, so the usual energy of that black hole is proportional to the inverse temperature to minus, so if instead of L we had the inverse temperature, that would be the energy of that wormhole. And so here this solution is roughly similar to a spatial cross-section of the ansatz we assumed. It's very similar to the spatial cross-section of the standard near-extreme of black holes, and so we expect to have the same energy, and that's where this term comes from. So that makes it somehow natural, and if we minimize over L, then we get a value for L, so we balance these two terms, we get a value for L, which goes like Q squared, and then we get that the final binding energy is proportional to one over Q, or one over the size of the, let's say, Schwarzschild radius, or the size of curvature of the sphere of these black holes. So we get a negative binding energy, so an energy which is smaller than the energy of the extremal black hole, and that's the binding energy for this whole thing, and it's a relatively small binding energy, but the binding energy nonetheless, and I should emphasize that the parameter that makes sure that this is a reasonable classical, so that all the approximations that we made are correct is Q, so it's the magnetic charge, so as we make the magnetic charge larger and larger, these approximations that we made are all better and better. So, so far what we've done is to solve Einstein's equations in the wormhole region, so within this wormhole region, but not in the exterior region, and because from the exterior point of view, this solution looks like two sort of close to extremal black holes, so two objects that have mass and magnetic charge, and they actually have opposite magnetic charge, because the magnetic field lines are emerging from one and going into the other, and if you have such a situation, the gravitational and electric forces or magnetic forces will add, and then these two objects will fall into each other, so in order to solve the equations outside, we need to do something, so we could, if we assume that the distance d between the two is large compared to their sizes, then we could approximate them, let's say as point particles and put them in an orbit, let's say Newtonian orbit around each other, and then in this way they will not fall into each other, we'll just be orbiting around each other, and then we still have this wormhole connecting the two sides. Now, okay, so we made a bunch of approximations in discussing this, we assumed the charge cube was large, we assumed various distances were large and small, and so on, and we can find, yeah, I won't go through this in detail, but what this transparency is saying is that we can find the regime of parameters where, for large q, we can find the distance between the two black holes such that all approximations are fine, okay, and so the final solution looks like, from far away, like two near extreme black holes, but if you get close, there is no horizon, you can go just between the two, and it has zero entropy, so if you wanted to have the solution in nature, so we had also the constraint that the distance d had to be smaller in the electro-wig scale, and that puts a maximum size for the charge, which is roughly like this. Now, this would be very tiny black holes, so these are not the black holes that LIGO would be, I mean, detecting, these are black holes that are smaller than the distances that can be explored by the LHC, so. Now, that these solutions can exist does not mean that they are easily produced by either some natural or artificial process, so we only show that the solutions can exist, how easy it is, we didn't give a procedure for making the solutions, so the procedure would look something like the following, so you first make a bunch of magnetic monopoles, then you separate the positive and negative charge magnetic monopoles, you make them collapse into black holes, you somehow hold these black holes near each other, and wait for some time until this one hole forms and the system finds the lowest energy state, I don't know how long that time is, but that is the procedure, so it's not a very practical procedure, but now that's if you want it to make them artificially, if you want them to make them naturally, I think that would be even probably harder, but I think that the point is that this type of solution shows how flexible somehow space time is, or how inflexible it is, because it's hard to make, and it shows something interesting about the concept of a black hole, so you could have a system that you started with these two black holes, but then somehow the black hole disappears and you get this connecting wormhole, so the final solution looks something like this, of course I didn't mention this, but in this configuration that where they are rotating, they would be emitting gravitational waves and electromagnetic waves, so the solution will not have an infinite lifetime, after a while the two black holes will collide and this wormhole will disappear, so the conclusion is that we display this solution that looks like, it's a solution of an Einstein Maxwell theory with massless charge fermions, and it looks like a traversable wormhole in four dimensions, and we didn't need any exotic matter to make this traversable wormhole, so people had discussed in the past traversable wormholes, even science fiction wormholes, but they needed some matter that, you know, it's exotic, it violates some basic principles of quantum field theory, so this is not violating any principles of quantum field theory, and we don't need to postulate any even funny Lagrangian, just the standard Lagrangian that we have for the standard mold works, and it's balances classical and quantum effects, and it has a non trivial topology which is forbidden in the classical theory, but allowed in the quantum theory. Now, it's important that there can be geometries that are forbidden in the classical theory, but allowed in the quantum theory, and this is one example, but there are many other examples, so just let me make us an aside. There have been recent papers claiming that the sitter is not a solution in strength theory, and one of the evidence that is cited for this is some argument that you cannot have the sitter in classical, as classical solutions of strength theory, or 10 dimensional supergravity and 10 dimensional, you know, 11 dimensional supergravity. Now, okay, that's fine, you cannot have it in classical, in classic, as a classical solution of 10 dimensional supergravity, but one of the things about strength theory is that it's not simply a classical theory, it's a theory with, you know, quantum effects, and quantum effects can give rise, in principle, to new qualitatively new solutions, qualitatively solutions that have new qualitative forms. And this was some clear example of a solution of this kind, but of course, all the sitter constructions and many of the evacuees that we discussed in strength theory also have the same feature that you need to put in some quantum effects and the solution exists thanks to these quantum effects. It wouldn't be possible in the classical theory, but it's possible thanks to these quantum effects. So the fact that it's not a solution in the classical theory is no argument, zero, has zero weight. Okay. Now, it only says that it's difficult, we will be slightly difficult to make, but I think it's even more interesting that if it is given by some quantum effect, because, you know, just the mere existence of solution depends on some detail of quantum mechanics and so on. So now this type of solutions have no horizon and no entropy. And we didn't explain exactly how to form it in an even artificial or natural way. Okay, thank you. Thank you very much. It was a wonderful lecture. Questions, Adish? So I have a couple of questions. One about your first part of your talk. So about a swiky. So my understanding is that there is no gap in the anomalous dimension, big gap. I mean, so string corrections are therefore of equally important. So how was it that by just doing a very simple minded to derivative Lagrangian, you got even a qualitative picture which seemed... Yes, yes, yes, yes. Well, yeah, what is... Yeah, so that's, that's one of the... Right, right. We saw yesterday that there is SYK and there is gravity, right? And they had some features in common, like this Schwarzian action was in common. Now we could add a third column, which is string theory, right, in ADS-2 with small string tension, right? So very large alpha prime or very stringy string theory. And that will also have the same feature in common as SYK. This classical action of this Schwarzian and so on, that's true regardless of the value of the string tension. As long as the string tension is not exactly theory, so zero, so that we have a dual that perhaps is free. As long as it's non-zero, then the idea is that the symmetries of all these theories are the same. So we have an approximate reparamentization symmetry that is both explicitly, spontaneously and explicitly broken and so on. And those were somehow the symmetry principles that were underlined, this common feature of the two solutions. Now, so that means that that action, this Schwarzian action will be common to all of these cases. Now what will be different is the bulk theory. So we discuss here in parallel ADS-2, which in the ordinary gravity theory has a bulk, which is standard quantum field theory. If we had SYK, we don't know exactly what the bulk theory is. If we had a stringy theory with small string tension, well, it would be a kind of complicated bulk, but we could explore solutions of this kind. There are solutions of this kind that we can think about, for example, near-extremal BTC black hole with NS, so-called NS background, so you can certainly find exact string theory solutions of this kind for small level that will have small string tension. And, yeah, so those are a little more complicated, but they all share these common features. And to find the solutions, we only needed this common feature. So the details of how the bulk fields propagate between the two sides and so on will depend on the string tension. So for example, other features that we did not discuss here. So for example, imagine you compute the four-point function. So you have this situation, and you have four operators, and you have to compute some Feynman diagram, let's say that looks like this, or might look like this. Now the details of what you find for this four-point function will be different in a local field theory where this is the diagram you should compute, or in a string theory where you have to compute a string diagram, or in SYK. So these four-point functions will all be different in the different cases. But the pieces that give rise to, well, the ground state and the features that the explicit features we discussed are independent. So it's universal, somehow this part of the... Yeah, this part is universal, and not only... No, you can say, well, but you also have the other part. So why is this universal part more important than the other part? Now, it's not an energetic argument. So it's not that this other... The part that I discussed is most important at low energies. It's just that it's the one that has low action, and it's the part that breaks a symmetry. So all these stringy effects in the bulk or the conformal part of SYK and so on, it all preserves the cell-to-art symmetry, the physical cell-to-art symmetry. But this action is breaking that symmetry, and it's the leading term that breaks this symmetry. And that was the fact that was used, that this is the leading term that breaks a symmetry. And that's why we can have a common description for all these cases. And that's why we can focus on all these cases. We focus on terms and effects that break the symmetry. And it was important to break the symmetry in order to find the solutions we discussed, so we wanted the wormhole to have finite length and not infinite or arbitrary length or whatever. So we had this breaking of the symmetry that was important. I have one more question. About the entropy, I mean, if you started with two black holes with entropy, didn't this violate second law of thermodynamics? Yeah, yeah, so that's very good. So this is related to... So that's true, so if you start with... if you try to make this from the procedure we discussed, we would make the two black holes that would have some entropy, and the final configuration would have zero entropy. Now, this by itself does not violate the second law because this system would be cooling, right? So we have two black holes near each other and they would be emitting Hawking radiation, and so the entropy will go into the Hawking radiation. However, the black holes are near extremals, so the temperature is going to be relatively low, and so you will not be emitting a lot of Hawking radiation, at least not the right. So the only way you can get rid of this large entropy of the black hole is to emit Hawking radiation for a very long time. So that means that there will be a very long time until this black hole is formed. So that puts a lower bound on the time that you can... Yeah, so for the time scale of formation of the wormhole. Now, one interesting question is whether... So that gives a lower bound which is polynomial in the charge, so it's some power of the charge. One question is... and then there is an upper bound. So an upper bound is... if you imagine that there is some tunneling process, with probability e to the minus Q to some power, you should be able to make it. So one interesting question is whether the time to form the black... the wormhole is polynomial or exponential in the charges. So these entropy considerations give a lower bound on the time. Another question, please. I should also mention that in the SYK model, the time to make the wormhole is polynomial in N. So you can... it just follows from looking at that microcanonical curve that we discussed for the entropy and assuming that, well, you will radiate some energy. I mean, if you couple that to... couple SYK to a heat bath, then you'll get some time which is actually independent of N. So if you couple it to N fermions, it's independent of N. But well, it goes like some power of the coupling. The case of the black hole is a little different. This extremization of the energy that you did with respect to the length. It looks very reminiscent of the way you get the Bohr radius for an atom using the uncertainty principle. Is there some... does this analogy make sense? Do you see the uncertainty principle in this context? Well, I think... I agree with you that it looks similar, but it is different in detail because we didn't compute it from an uncertainty principle. We computed it by computing two energies, right? Two separate energies. I think... A balance state of something that you tried to... Well, I mean, but it's a balance between two energies, right? It's not that one of them came from a kinetic energy term, as in the case you discussed. So in fact, actually this... Yeah, this is similar to how you compute the potential. It's more similar to a hydrogen molecule where you compute the potential from solving the Schrodinger equation of the electrons for fixed position of the nuclei and then try to minimize that energy. Some sense is also analogous to the Van der Waals force. I think I had a transparency about the Van der... Did this appear? No. Okay, I guess... So let me just mention the analogy to the Van der Waals force. If you have two atoms, right, that are at some distance, let's say, L, then this atom will have some bunch of energy levels, right? And this atom will have also some bunch of energy levels, right? Now the atoms could be exchanged in photons, right? And the exchange of photons will couple the energy levels of each atom to each other. And the coupling has the forms of the Hamiltonian, has the form of an electric dipole moment of the left atom times the electric dipole moment of the right atom divided by L cubed, right? So that's the leading coupling between the two systems. So this electric dipole moment is acting on the energy levels or the low energy states of this atom and this similarly for this atom. So let's say for this atom there is the ground state of the atom, there is some other excited states and so on. Dipole moment sort of mixes, I mean takes you from one state to the next and so on. It changes, of course, the angular momentum. And then when you have this, you will find that the ground state will be an entangled state between the energy levels of this atom and the energy levels of this atom, okay? So that's when you get to second order in perturbation theory, you will find some kind of entangled state and then you get a potential between the two which is proportional to minus one over L to the sixth, okay, the square of this operate. Now, this is completely analogous to what we had here, where we had a black hole, right? So replace the atoms by the black holes and the black holes, well, so instead of exchanging photons, we were exchanging fermions, okay? The fermions that were traveling along the magnetic field and in the outside. So we're inducing on the energy levels of each black hole. So the energy levels of the black hole are all the black hole microstates, right? They are e to the s levels and we're inducing an effective Hamiltonian that looks like psi left times psi right where psi left times psi right is the value of the fermion field in the connecting neck region of the left and right black holes, okay? So we have this interaction Hamiltonian and the entangled state that is the ground state is the state that is close to the thermo field double state, okay? That's analogous to the ground state in this system, okay? So that's... and to lead in this problem, to lead in order is independent of the distance of the black holes, but the fact that we get an entangled state as the ground state and so on is the same. Another amusing thing is that you know these animals called geckos, which are like little lizards that can walk on walls and so on and on the ceiling, blah, blah, blah. So there's a theory that the force that allows them to stick to walls and so on is a version of the Van der Waals force, okay? So this is another example of the connection between entanglement and gravity because they can defy the force of gravity via the entanglement present in these atoms. Any more questions? Let me just answer about this solution. It is crucial that you need fermions or you need only... Why do we need fermions? The fermions were important for getting zero energy Landau levels. So if you had bosons, Landau levels will all have positive energy. So for example, in the standard model the Higgs is a doublet, will be charged under... I mean it's charged under the U1 of hypercharge, of weak hypercharge, but they will all have positive energy. All the Landau levels will have positive energy. So they would not lead to this effect. So let me expand a little more. There is effectively a massive field in ADS-2. You will also have some interaction due to the exchange of the field outside the black hole. So it will induce a similar interaction. But if the field is massive in the bulk of the black hole, then its effects will become very small as the wormhole becomes long. So it's a very irrelevant perturbation of this effective conformal field theory because it would be something that would have dimension bigger than one. And then its effects, when the wormhole becomes long, its effects are not important and do not give rise to anything that depends on the length of the wormhole. Let me say this in perhaps a way that would be more intuitive. So we said that effectively we have a circle of length L and we had a Casimir energy that went like 1 over L. Now imagine that on a circle of length L, instead of having massless particles, we had a massive particle. Then there would also be a Casimir energy, but if the mass is the Compton wavelength is small compared to L, then the effects will go like e to the minus mL. So if the Compton wavelength is small, it means that this will be large and so the effect of all these massive particles do not give rise to any appreciable Casimir energy piece that depends on L, okay? That depends non-trivial on L. And so there will be, of course, in this context, diversion, UV terms that are local and will be proportional to the whole length. And in this ADS context, what that means is that all these massive particles, like the one that comes from the Higgs boson or all the other modes of the field that we have not discussed on the other Landau levels, they all give rise to contributions to the vacuum energy, which are local. They are local and ADS invariant. They are invariant under the SL2 isometry. So the leading piece that breaks the SL2 isometry comes only from the massless fields and gives rise to these effects. And that's why we could concentrate only on those and ignore all the others because they preserve different symmetries. All the other ones that we haven't explicitly discussed preserve the SL2 symmetry of ADS2. And you think of generalizations of the solutions, say, for higher dimensions with the antistatic tensor fields and things like that? Yeah, well, yeah, so there are... Well, if you have something like an S3, for example, instead of an S2, then the issue is that you would need like a massless string or something like this. That doesn't make sense, or at least we don't know how to make sense of it. So I think in higher dimensions what might make sense is something with a bunch of S2s, for example, where you can have UN gauge fields, so to have at least these massless fermion effects. Any other questions? So in your comment about the sit there... Yeah. Joe Polchinski used to argue that, you know, after all, atoms are not stable in classical physics. Yes. So you're stating that the Maldeserian theorem is irrelevant for everybody who claims that the sitter doesn't exist? Yes, yes, I agree. I think it's no evidence that the sitter doesn't exist. It just shows that you need to... I mean, the purpose when we wrote that paper, it was to... Before we wrote that paper, people were trying to find the sitter solutions using classical constructions. So using some constructions in just classical solutions of supergravity. And so one of the purposes was to show that those constructions would not work. But then, I mean, afterwards people found explicit, I mean, some fairly reasonable scenarios where you could have solutions, I mean, KKLT, and where you could find solutions which use the quantum effects by using quantum effects. Very good. Thank you. Is there a last chance to ask any more questions to Juan? Because this is the last lecture. Any brave person, young postdoc or mathematician? It's very brave in this audience with asking, I'm a mathematician. And my interpretation of this Wheeler picture of the construction was that you were trying to glue solutions to systems of PDEs, extremely sophisticated systems, like describing the external part of two black holes with matter, charge and so on, with a neck, which connects the two things. Now, typically, in mathematics, these theorems are extremely sophisticated. I mean, there's a long history of physicists making these constructions and then waiting 25 years to see a proof. My curiosity is, did you write down... First of all, my interpretation is correct. Are you really gluing solutions to systems of PDEs? Yes, your interpretation is absolutely correct. Because of the presence of all the symmetries you described, can you actually write down the actual solution? Yes, you can write down the solution. So we've given the solution in terms of three different approximations. So let me expand on this question. So the... The solution has this shape. And we divide the solution into three different regions. So one is the neck region. So the region that connects this region here. Then there is the solution in this other region, the wormhole region. That's part that I very explicitly discussed. And then, of course, the neck region and so on. And then, finally, there is the solution in the rest of the space. And we made three different approximations in each of the regions. So I discussed quite explicitly the approximation we made in ADS-2. So we had ADS-2 with some boundary interactions and back stress tensor and so on. The solution in the green region is just the standard right-hand and nostrum solution, the charge black hole solution. The standard charge black hole solution. And in the far region we interpret the solution as two point particles that are subject to gravity and electric forces. It's important that the three solutions have overlapping regimes of validity. So in the deep region of the black hole solution of the standard right-hand and nostrum we could connect to the wormhole and the far away region we could connect to the point particle description. And so what we did was to solve the solution in each piece of this solution in the overlapping region of validity. So that's the standard thing that is done when you match solutions of PDEs. Thank you. From that PDE discussion the one thing that was important is that one of the terms in the PDE came from the expectation value of the stress tensor. And the stress tensor needs to be computed using quantum field theory computation. And the quantum field theory computation of quantum fields moving on this geometry. So you first have to make an ansatz for the geometry solve for the quantum fields in this geometry calculate the stress tensor and then solve the PDE. So that's the nature. And again this was done in an approximate way by... that quantum field theory was done in an approximate way taking the 4D field theory so the full field theory in this geometry separating in modes calusal climb modes on this wormhole and arguing that only the master's modes matter only the master's modes give something that depends on the length of the wormhole the rest are terms that are easily taken into account and then those master's fields give a stress tensor that depends on the length of the wormhole. That's the mathematical way that we found this. Thank you very much. Question here. When you're trying to connect you said that you're going to make an ansatz so you... You said that you have the two black holes and you connect, you make an ansatz and do you need in advance to have the same temperature because you said that the length is proportional to the beta which is the inverse of the temperature so you assume that their temperature is exactly the same or you can wait for a period where there will be an equilibrium a thermal equilibrium. The question was slightly confusing. This solution does not have any temperature but we made a symmetric ansatz for the position of where these horizons are. It's just proportionality. Roughly speaking, yes. The answer is yes. But the final solution does not have any temperature. In no entropy, right? In no entropy. Yes, yes, yes. Please. Sorry, maybe I missed that. Let's say the extremal black holes, for example, that you discussed. Could they be supersymmetric, for example? No. Because supersymmetric configurations would have to be black holes that have the same charges. Sorry. If you're in flat space, the answer is no because they would have opposite charges. I think that you could do it if you put it in some other curved space. So there are situations where, for example, if you are in, let me say ADS-3 times, for example, S2 with a supersymmetric solution in ADS-3 times S2. People have discussed in these situations, it might be that you can have a positive charge in the northern part in the north pole of the S2 and a negative charge in the south pole of the S2 and they can be mutually BPS. So mutually preserve the same supersymmetry. And so it might be that in an example like this you might find supersymmetric versions of this. We haven't looked at this a bit. It might be possible. I'm, okay. It might be possible. It might be skeptical because if there is a binding energy it would be like a violation of the BPS bound. So if it is possible then there has to be a separate miracle that this binding energy cancels against some other energy of a scalar field and so on. But I think it is possible that there are supersymmetric versions of this non-trivial solutions with non-trivial topology that exist in this type of setup. More questions? Let me finish with one question just at the end of your talk or your lectures. Since these are SRAM lectures in general how would you explain ER equals to EPR to your children? Well, it's hard if you don't know what ER or EPR is. So let's start with that. I'm going to do it by saying something that I said the other day at dinner and basically told me not to talk about this in these talks. I'm in trouble now. So the analogy is how we transfer thoughts to each other. So we have two people so I have my thoughts in my mind. He has his thoughts in his mind and I want to transfer a thought I have to his mind. So how does this process of transferring thoughts work? It works by I start emitting some sounds and if you are Chinese or whatever you would not know the sounds we emit to each other in Spanish. Nevertheless the thought goes from my mind to his mind. So how can it happen that the thought goes even though it seems to be that we send some random words some random sounds that look random and it works because we have a shared code a shared code for what these sounds meant. So we were sharing this code and each of them had this code that had in our minds saying oh the sound a certain word means something in some correlation to my thoughts and similar other sounds we share these common correlations. These common correlations in quantum mechanics are called entanglement these random words we say some classical information that looks fairly random but because our underlying systems are correlated we can transfer thoughts which are some very abstract representation we don't even know how it's represented and nevertheless can go between us. So we are experiencing EI equal to EPR as we speak hopefully we are experiencing if you are only hearing random words from me then we are not experiencing EPR we are not sufficiently entangled. I wish to receive all your thoughts but that's not telepathy very good ok so this is a good way to finish so let's thank Juan again for such a great lecture.