 Welcome back everyone. Thank you for joining us for today's 150th low physics webinar. My name is Alejandro and I'm going to be your host. Today we are celebrating our series of webinars with a truly extraordinary guest speaker Juan Maldacena. Renowned as one of the most brilliant minds in theoretical physics, Juan Maldacena has left an indelible mark in all understanding of the universe. As the mind behind the game-changing concept of the ADS-CFT correspondence, he has illuminated several connections between gravity, quantum field theory and string theory. Maldacena's passion for discovery and relentless pursuit of knowledge led him from his studies in Argentina to numerous academic positions at prestigious institutions. With countless honours included the Breakthrough Prize in Fundamental Physics, Maldacena's groundbreaking insights continue to reshape the landscape of modern physics. So please be prepared to be captivated as he takes centre stage in this celebratory webinar, sharing his unparalleled wisdom and recent discoveries and perspectives. So as usual, remember you can ask questions over email through YouTube channel or Twitter and then the questions will be read at the end of the talk. So without further ado, we will turn the time to Juan and thanks for joining us. Okay, thank you for the very kind introduction and let me start by sharing my screen. Can you see the screen? Perfect, yes, you can go full screen now. Okay, wonderful. So yeah, it's a pleasure to be here. We will be discussing the entropy of Hawking radiation. So black holes have been in the news recently. We are in a golden age for black hole observations through various interesting observations. But this talk will be mainly about quantum aspects of black holes. So there are more theoretical aspects, as you will see. So in particular, we will discuss recent progress on the black hole information problem. And this talk is based on a review that we wrote and you can find it in FTH. And it's also this, this, this recent progress was based on two important papers in 2019, one by Pennington and the other by And there are also many previous and follow up papers. And there's another interesting development that I will not review, but it's somewhat similar in spirit to what I will talk about. So the only end of this talk is first I will remind you of how the black hole entropy is equal to the area of the horizon. We'll discuss another black hole entropy formula, which is the fine grain gravitational entropy formula, which will say that the entropy is equal to a certain minimal area. So it's the area of some other surface. And then we'll use that formula to compute the entropy of radiation coming out of black holes. And the bottom line is that we'll get a result that is consistent with information conservation as opposed to information loss. That's the outline of the talk. Now this talk will not be historical, but hopefully it will be pedagogical. Now, the simplest black hole solution is the Schwarzschild solution or the Schwarzschild metric that Schwarzschild wrote soon after the discovery of general relativity. It has the following four dimensional geometry. And the most important result about the black holes, about quantum aspects of black holes, is that black holes are hot, black holes have a temperature. So they have a temperature which is proportional to 1 over 4 pi r s, where r s is the typical scale size of the black holes. The black holes are characterized by a certain scale called the Schwarzschild radius, which is given in terms of the mass through this formula. And that length scale fixes the length scale associated to the temperature. So the radiation coming out of black holes will have a wavelength which is proportional to r s. So in particular, this phenomenon is so surprising that it can lead to white black holes. So if you have a small enough black hole, the temperature will be high enough, let's say like the temperature of the sun. And that black hole will look white to our eyes, so we have a white black hole. Now, let me just remind you how, let me remind you of the derivation of this formula. And before we discuss black holes, let's just, I would like to remind you of a connection between finite temperatures and circles in Euclidean time. Let's imagine you have a quantum system and you're considering the partition function of the quantum system is given by the trace of e to the minus beta h. And you can think of this, of this canonical partition function as evolution in Euclidean time on a circle of length beta. So if we had here it h that would be ordinary Lorentzian evolution. But because we don't have the eye, we have that's the same as going to imaginary time. And so we have an evolution in imaginary time over a time of order beta. And we are taking the trace over identifying the initial and final state and summing over all of them. And that's equivalent to considering the theorem a circle on a Euclidean time circle of length beta. Okay, now this is true for any quantum system. It's true for a harmonic oscillator for it's true for, you know, two level system for for any for any system. It's true if you take a quantum field theory and so on. So the point is that the theory on Euclidean circle is related to a system in thermal equilibrium at a certain temperature, which is given by one over the size of the circle. So that's a general feature. So now let's consider Euclidean black hole. So let's consider, let's start with the Lorentzian black hole the metric that trial wrote which is written here. And now let's consider the Euclidean time version of that. So we all we are doing is just simply taking t to it Euclidean. And now it has the result of changing the sign in front of this term from this minus disappears and now we have a plus. Now, here we see that the where the horizon was before so now we have a situation where this the coefficient of this term in the metric is shrinking to zero. But similar to what happens in Euclidean time, sorry what happens in just Euclidean space if we write it in terms of polar coordinates that the length of the let's say angular direction could shrink to zero. And that can be non singular. If we adjust properly the length of the Euclidean time direction. So we can adjust the length of the Euclidean time direction if we make beta. So if we adjust the length of Euclidean time equal to two pi RS, then we find that this metric is completely non singular. And the radial and time directions combined in order to give a space which has the topology, essentially of a cigar so at each position at each radio position we have a circle. So our length is given by this proper length square is given by this formula. And then when we get to our equal to RS, it shrinks smoothly, as if it was the origin of the Euclidean plane. So this is a four dimensional geometry and over each point of this two dimensional space there is also a two dimensional sphere whose size is also changing and has a minimal value minimal non zero value here at are equal to RS. So that's the Euclidean black hole. And given that we have the circle we can the idea is to interpret this geometry as the geometry of a black hole in thermal equilibrium with some system outside at temperature better so the fact that this non singular is interpreted as saying that if we you put the you put the black hole in contact with a thermal that's a gas at temperature better this will be a thermal equilibrium situation. So this is the simple way to derive the entropy of the temperature of the black hole. This is the derivation discussed by Givens and Hawking. Now, once you have the formula for the temperature, you can derive a formula for the entropy by using the first law of thermodynamics so that the small variation in the entropy is given by small variation in the energy of RT and the energy is equal to the mass. There is a connection between the mass and the temperature. So we can integrate this formula to find that the entropy is equal to the area of the horizon over 4G Newton. Of course, historically previously Beckenstein had conjecture similar formula for the entropy without the precise numerical factor, but Hawking's derivation of the temperature gave us the precise formula for the entropy of the black hole. H bar and C and Boltzmann constant equal to one. And this was down to, you can also rewrite this as the area divided by four L Planck square and L Planck is very small. So this entropy is very large for a microscopic black hole. So we see that the black hole is the thermodynamic object and the base the loss of thermodynamics and this is definitely surprising. So now we'll discuss a bit more in detail the, in a bit more detail the geometry of a black hole. So let's discuss the geometry of a black hole made from collapse and this is something that was first discussed by Oppenheimer and Snyder in the 1940 roughly. So they consider a star that was collapsing into a black hole or more precisely, let's say a ball of dust that collapses into a black hole. And we represent these geometries through something that is called the Penrose diagram. So these are some diagrams that they characterize the geometry. So now this talk will be considering spherically symmetric geometries. So, and we will forget about the directions of the sphere so at each point on this diagram there will be a two dimensional sphere. But these diagrams will only include the radial and time direction. So these diagrams represent the radial and time direction of the four dimensional geometry. The, that geometry has been rescaled so as to fit it on the page, but in such a way that angles has are preserved and the direction of light rays are at 45 degrees so the light ray and outgoing radially. So the light ray that goes out in a radial direction would be moving here at 45 degrees and incoming light ray will be moving in this direction. So the vision here corresponds to the region that was infinitely far away in the four dimensional space, and we rescale all that infinitely far away distance to some finite location in the diagram. Here, the point are equal to zero which would be the center of the space. That's the only that's a place where the two sphere shrinks to zero but in a smooth way. So that's, that's this vertical line. So here, roughly speaking this direction is roughly like time and this direction is roughly like space. And the surface of the star in this diagram follows this trajectory and ends up in a place which we call the singularity so singularity is a situation is a place where the curvature of the space time goes to infinity. So we cannot continue to vote the questions beyond this point. So this is a point where general activity breaks down. And then, if we consider outgoing light rays, so if you have a light ray that starts from here and goes out so it can go out through the surface of the star and go all the way to infinity. But if you have a light ray that comes out from here, and it starts going out it will end up at the singularity. The space time is divided into regions from which you can send the signal to the outside such as this region down here. And a region where you cannot escape to the outside where even if you go out of the speed of light, you cannot avoid falling into the singular or get into the singularity. And the singularity is really into your future so it's a bit like a cosmological big crunch singularity in this region of the space time. And there is a surface and let's say imaginary surface nothing special happens to you when you cross the surface. But there is the surface that we call the horizon, and it's generated by all the by the light rays that neither go to infinity nor fall into the singularity so this is like a separate in surface between the light rays that go to infinity and the light rays that fall into the singularity. Okay, so that's the geometry of a black hole made from collapse. And as I said this was understood in 1939. Now there is something interesting that happens with this geometry, which is that if you if you look at the area of the horizon. So here, this is r equal to zero the area of this surface. This is a. If you look at the surface at some moment in time, it's a two dimensional surface covers the two sphere, but this two sphere has a zero size here and it has a size which steadily increases until it gets to the surface of the star. And then after that it remains constant. And it's given by the area of the horizon of the just fragile geometry which is given in terms of the fragile radius. We see that the start with small area it increases and then it reaches a maximum. Now it turns out that it was proven that the area of a black hole horizon always increases not only in this simple situation, but even in complicated situations such as for example to collide in black holes and so on. And it's a consequence of the questions of general relativity that the area of the horizon increases. And this is in agreement with the second law of thermodynamics of interpret the horizon as some entropy, then Einstein equations are implying the second law of thermodynamics. Now that the full entropy. So it was a concept of generalized entropy that was introduced by Beckenstein. And his idea was that the whole entropy of the universe outside the black hole would be the entropy of matter, which is the obvious contribution to the entropy, plus the area of the horizon. The total contribution of the entropy if you are outside, in particular, you're not including the entropy of matter inside the black hole horizon, but you have to include this area of the horizon formula. Okay. And now one question you might have is the following so when a black hole emits hooking radiation, it loses energy so its area becomes smaller. So let's ask what happens to the entropy naively would think that if the entropy is dominated by the area and the area becomes smaller. Then you might be in danger of violating the second law of thermodynamics. Now, for this you need to remember that the full formula includes also the entropy of matter, and this entropy includes the entropy of quantum fields outside the horizon. And when you have the process of hooking evaporation. It's true that the area decreases, but the entropy of the fields outside because of the radiation, it increases, and it turns out that the entropy of radiation increases more than the decrease of the area. So actually this full entropy increases through the process of black hole evaporation. So it's an irreversible process which increases this thermodynamic entropy. In fact, one can prove one can prove that the this full entropy obeys the second law of thermodynamics and this was done relatively recently by Aaron wall in the 2010s. And the only thing you need to assume to prove this is that the matter obeys, it's relativistic and it's given by a relativistic quantum field theory. There was a long discussion from the time of Baconstein till 2010 actually continued after 2010 because some people did not read this paper of trying to have imagining that the second law of black hole thermodynamics might impose some constraint of matter or imagining thought experiments that might or might not violate the the second law and they would find that well there's a miracle that makes it obeyed but now we understand all of that and it's all a consequence of relativistic quantum field theory. Okay, so these results have inspired the very influential hypothesis that we are going to call the central dogma in the study of quantum aspects of black holes, and it's the following. So it's the idea that if you look at the black hole from the outside, then a black hole can be described as a quantum system with s degrees of freedom or s qubits where s is the area over 40 new. And, and furthermore, this degrees of freedom evolved according to unity revolution as seen from the outside. So that's, that's this idea. Now, we are calling the dogma dogma means hypothesis and it's something that is cannot be well we don't know how to prove it from the equations of general relativity. It's some some assumption. So it requires a certain, well, certain degree of faith also in the sense that you might believe or not believe in this and some people think it's wrong and some people think it's correct so. Now, in other words, if we include some area over 40 Newton mysterious qubits, then the black hole can be described as an ordinary quantum system so more precisely if you imagine you have some. imaginary surface that surrounds the black hole, we can replace the black hole and the whole space time within this imaginary surface by some system of ordinary qubits that are evolving according to some Hamiltonian. So the only thing we know about the Hamiltonian is that it's a Hamiltonian that is acting on these qubits. Of course, it's an open system because it's this degrees of freedom in here can interact with the outside, but the non trivial hypothesis is that there are no extra degrees of freedom somehow behind the black hole horizon that this qubits can interact with this qubits take full account of the degrees any possible degrees of freedom behind the horizon. Well, or at least any degrees of freedom that are necessary to describe the black hole from the outside. So that's the, that's the idea. So, now this, if you make this assumption, then you can reinterpret this computation that given some Hawking did. They computed some the action of gravity, and you can think of that the action, the gravitational action of this geometry as giving you the partition function of those mysterious degrees of freedom with that mysterious Hamiltonian that we don't know. So the left hand side is somehow some hypothesis. And the idea is that this gravitational computation, which is a purely geometric computation than solving Einstein's equations has the interpretation as the trace over some mysterious Hilbert space. So this gives us the answer but it does not tell us exactly what this black hole so called microstates are. Now let me just tell you some evidence for for this hypothesis. So there is some evidence from the so called entropy counting. So there are some special black holes in special theories that have supersymmetry that can be counted precisely using string theory using deep brains. And this counting reproduces the area formula and not only the area formula but also corrections such as this entropy of the fields outside and further corrections that exist to this formula. This was initially done by stronger and buffer and there were many other papers and lots of recent developments on this and this very precise matchings of this kind. This is for typically is for special extremal black holes that you can get most control over this calculation. Another piece of evidence is the ADSC of the correspondence, which says that if you have a physics in ADS is the same as field theory or conformity theory on the boundary. And what that says is that if you have a black hole, that black hole is described by hot fluid of particles at the boundary. It's a situation similar to what we discussed before, except that in this case we actually know what the Hamiltonian is so in some special cases we have very explicit description of the Hamiltonian. And it's a situation where that cut off surface that we discussed before is pushed all the way to infinity in space so all the way to the boundary of the ADS space, and we are describing. Not only the black hole, but the whole space time around it using this, this degrees of freedom so you can view this as a special case of what we did that central dogma we talked about before. However, so these are some evidence in favor, but there was also some evidence against. And so in 1976 Hawkins said that this couldn't be true. And so let's review his argument so his argument is based on considering the geometry of an evaporating black hole that was made from collapse. So this is the penrose diagram is very similar to the diagram that we were discussing before. So we have again the star that collapses and now we are going to include the Hawkins radiation. So we can think of the Hawkins radiation as arising from entangled pairs that were there in the vacuum so the vacuum. If you take the full slice the full geometric slice let's say on one of these green surfaces. It's a pure quantum state. The horizon devices into two parts device that state into two parts, and at the short distances, the vacuum of quantum field theory has some entanglement, and you can think of that as arising from pairs of particles that are entangled with each other. And so one of the members of the pairs can go out to infinity and become the Hawkins radiation, and the other member of the pair will go into the singularity. The two members together are forming a pure state but if you only see one of the members, you'll find a mixed state. So if you so on each of these green slices you have a pure state, but if you the black hole evaporates completely, we think that after black hole evaporation. The pendulum diagram will look like this which this is looks basically the pendulum diagram flat space where this vertical line is just equal to zero. Then a slice drawn through that part of the diagram will only intersect with the, with the members of the Hawkins radiation and will not have anything, but will not contain the interior. The entropy on the slice of the quantum fields will be non zero while the entropy on this initial slice could have been zero if the star was in a pure state. Even if the star is not in a pure state this entropy would be much bigger, it would be an entropy which is proportional to the area of the horizon chest, but it's even bigger than the horizon after the black hole forms, the entropy of initial start so we have some net increasing entropy. This does not violate the, the, of course, the second law of thermodynamics, but it does violate the idea of unitarity that if this entropy is really the full fine grain entropy of the system. We went from a situation where we had low entropy to a situation with large fine grain entropy. This is another point of view of on the same of the same stuff so this arises by thinking of this somehow slightly complicated diagram in the following way so we had the matter produces a second baby universe so that the large large times. We have the original universe filled with some Hawkins radiation. And then we have a second universe, which would be this whole interior, which collapses collapses into a singularity so it's a baby universe a bit very big. We can call it the teenage universe. So it's a big baby universe. Cost is some problems for the parent university some problems about the unitarity and Yeah, and so, so from this point of view you say well, this is an evolution from some single universe into two separate universes, and well they could be entangled with each other so that if you take both of them there in a pure state so the full evolution is very different but if you only look at the parent universe, then you have some net increase in entropy. This is similar to a particle that the case into two particles that are entangled if you only look at one particle you find a non serentropy. If you look at the two particles together you have serentropy. So from this point of view, it seems rather natural that you could have some increasing entropy, and this is what Hawkins point of view. So now we'll make a slightly better statement of the problem that is due to page. And the idea is to compute the entropy of the radiation as the radiation comes out of the black hole. So we start equal to zero the black hole forms, and then we'll start collecting the Hawkins radiation. Next we have a small amount of photos and then we'll have more and more. And as we have more and more the entropy of the Hawkins radiation will steadily increase until the black hole evaporates completely. Now that need not be a straight line might be a curved line what's important here is that that it increases monotonically and it stops increasing after the black hole evaporates completely. And that's what Hawkins calculation tells us. On the other hand the thermodynamic entropy of the black hole which is given by the area of the horizon. After the black hole forms it will be given by the area of the horizon, and then that entropy starts decreasing, and it decreases all the way to zero. Okay. Now that area of the horizon, it's a measure of how many degrees of freedom the black hole has. So, at this very early times, it might be that the entropy of Hawkins radiation is increasing because it's entangled with the black hole degrees of freedom. But there will be a contradiction at this point, where the entropy Hawkins radiation is bigger than the area of the black hole horizon. So this entropy cannot possibly arise from entanglement with this black hole degrees of freedom. So the contradiction with that central dogma central hypothesis doesn't happen when the black hole evaporates completely, but it happens already at this point this point is called the page time. When the entropy Hawkins radiation becomes equal to the thermodynamic entropy of the black hole or the area of the horizon. So that's the point where we have a contradiction. So what page said was that, if you need charity is to be preserved, then the actual entropy of the radiation should not follow the green curve, but it should follow this purple curve that we see here. So it should, it could rise up to here so that would be consistent but then it necessarily has to go down, and it can be at most equal to this quantity. Okay, so that's, that's the idea so this here we assume that the black hole was formed with a very low entropy state or zero entropy state. So this is the purple curve is what's back in for me. Now, it's important that this problem involves understanding the fine grain entropy. So I remind you that there are two notions of entropy. So there is the fine grain entropy or final man entropy. And then it's constant under time evolution is given by minus trace of roll over raw is the density matrix of the system. So the system is evolving, according to the evolution. This doesn't change. And then this will be the entropy that will be mainly talking about in the rest of the talk. This is also the course green entropy and sometimes called thermodynamic entropy or Boltzmann entropy. And this is the entropy that obeys the second law. So it was introduced by Boltzmann in his discussion of the second law. And in some sense it arises from some sloppiness is the fact that we are forgetting to measure some things in the system. It's somewhat subtle to define it precisely and we're only here mentioning it to distinguish it from the star of our show, which is this actual entropy, the front line manager. It has some precise definition that I'm not going to discuss. Now, we will let's just see the difference between these two entities in an example. These two entropies are typically different in out of equilibrium situations. So let me let me show you an example. Imagine you have a big box and within the big box you have a small box and you have some gas in the small box. Then at the time equal to zero you open the small box. This is a unitary process and the particles come out of the small box and feel the big box. Okay. Now, if we had some density matrix describing the initial particles, then the final density matrix is just a unitary transformation of the initial one and the final entropy will be equal to the initial entropy. That's because you can, you can remove this you and you minus one from this formula using the cyclic property of the trace. On the other hand, if you compute the thermodynamic entropy of of a gas in a bigger box it will be bigger. Okay. So thermodynamic entropy is assuming that well you know that the gas is in the big box but you don't know that it came from this particular state so it's possible for the gas in the big box and in a smaller box. That's why it increases and I'm calling it thermodynamic here because that's the entropy that increases according to the thermodynamic second law. Sometimes also called as I said boss man entropy. So at the moment we'll be talking about the entropy of the black hole as seen from the outside. And so this would be the entropy of the quantum system that appeared in this central hypothesis or central dogma that we mentioned before. So that's the entropy we're going to be computing in the next few slides, the phone I'm an entropy of that. The horizon area is computing a thermodynamic entropy of both man like entropy. Just, and we can ask how to compute the fine grain one. Now that the reason that the horizon areas computing the thermodynamic entropy is that we saw that when a black hole collapses. That's a fairly rapid process. That's essentially unitary because there is no much chance of emission of Hawking radiation and so on. And we saw that the entropy increases dramatically because the area of the horizon increases from zero to something big. Okay. So the question is, we have some nice formula for the horizon area and this well known formula for the black ball entropy. And the question is where there is one that computes the finite man entropy. Now the interesting thing is that there, there is actually a formula for the candidate formula for the final man entropy. And it's somewhat similar to the black hole entropy formula in the sense that it involves an area and some entropy outside that area. So basically there's some area of some surface which we are calling going to call X. And there is, let's say this is a surface is a point on this diagram but remember that each point on the diagram is a two dimensional sphere. So that's a two dimensional surface. And then we consider the three dimensional slice outside the surface, and we have this blue, blue line here represents this cut of surface outside the black hole so we're surrounding the black hole with a big surface and we're going to move everything that is inside. So we go all the way up to some surface x we consider the entropy of all the quantum fields here or any kind of matter entropy we can have here. That's this piece of my classical entropy of the quantum fields and then we add the area of the horizon. So the full quantity then is minimized over the choice of the surface x. So now we are going to start moving the surface x, both in the space direction and in the time direction, keeping this surface space like, and we're going to find the point in where it maximizes this. So typically it's a minimum in this direction and maximum in the vertical direction, the time direction. So we extremize and we find some extrema. And there might be situations where they're small than one extremum. And then we're supposed to minimize over all those extreme. So that's the procedure. So, that's the procedure we're supposed to follow. So it's a bit like, it's very similar and for reasons we'll discuss later to let's say finding the extreme as fun action and then minimizing the action. Anyway, this this formula was developed in a bunch of papers and the last paper has now this final formula that we're going to use. Now we'll discuss the derivation of this formula later for now we will just use it and I'm going to show you some examples on how to use it. And you should be somewhat you should be surprised by the claim that there is a formula for the fine-grained entropy because for ordinary physical systems we don't have a simple formula for the fine-grained entropy. It's something quite difficult to compute. So, and difficult to measure also. But let me show you some examples. So imagine that you have the full extended Penrose diagram of the Schwarzschild solution. So that's the solution that describes really two black holes. So there is as one exterior, this is exterior one black hole, this is exterior of the other black hole, and they're connected through the interior. Sometimes through the so called Einstein-Rosen bridge but anyway, so that's the full geometry in that case. And if we surround one black hole by a cut of surface and compute the backdrop inside, the extremal surface will be here when the past and future horizon meet, and the area will be just the area of the black hole. So this is a situation where the surface that we find is actually at the horizon. So in this case the two notions of entropy agree with each other. And that's also consistent with the idea that this geometry, if you look at only one of the black holes, represents a black hole in thermal equilibrium. So thermal equilibrium, the two notions of entropy agree with each other. Now we can go and look at the case of a collapsing black hole. So this is again the same diagram that we had before of the collapsing black hole. Now, if we try to minimize the generalized entropy, so the area plus the entropy outside, we find that there is a minimum where the area of that surface is actually zero. So that area might actually be zero. And in that case, we have no contribution from the area, but we have some contribution from the entropy of matter fields. In this case, you see that if the star had some entropy, the entropy on the surface would be the entropy of the star. So in this case, the fine grained entropy is giving us the entropy of the star. Notice that in this particular case if we were to calculate the area of the horizon at this point, it will give us the area of the horizon which is in general much bigger than the entropy of the star. If we compute the thermodynamic entropy or the Beck Einstein Hopin entropy, that will increase and will be large. While if we compute the fine grained entropy is equal to the entropy of the star, the same entropy we would have had if we had computed it before the black hole forms. So this formula for the entropy is manifestly in agreement with the idea that the entropy doesn't change under unitary evolution as we have discussed before. So now we'll we'll consider an evaporating black hole. And so, when we have an evaporating black hole, we can start computing this entropy at later times. Now there will be Hawking radiation the Hawking radiation will leave this cut off surface of the entropy inside to start changing. And we will always have a solution which is the solution with vanishing entropy, and that will give us some entropy for the black hole that will start increasing similar to what we saw before for the radiation. But these two new papers contain a new observation, which is the idea that there is a second extremal surface that is kind of close to the horizon. It's a surface that arises due to some equilibrium between the gradients, let's say of the area and the gradients of the entropy of the quantum fields. And it's kind of close to the horizon. And it's serious is close to the area of horizon. So if we consider now the two candidate surfaces. So there are two extremal surfaces one was this one that had vanished in area that exists for all the times. And in this case what happens is that as the Hawking radiation leaves the system, then we start getting a larger and larger entropy that is coming from the entropy of these partners of Hawking radiation. Okay, so that if you catch somehow, for example, on this surface, we here have one of the members, one of the members of the pair, and we're missing this member so we have some non serum entropy. On the surface we have two members were missing this one so the entropy will start increasing. On the other hand, here we have. We have this other second extremal surface which which which will follow track basically closely the area of horizon, and so it will give us this other possible entropy, which actually will decrease in time. So the, the, the prescription is telling us that we should choose the minimal one so for some time, we choose this one, and when the two become equal, we choose the other one so we get some curve that basically looks similar to the page care but in this case we're computing the entropy of the black hole. So we get something similar to the page care for the black hole but we really wanted the page care for the radiation. So the radiation lives in a region where the quantum gravity effects could be very small. It could have left this antecedent space it could be collected in a far away quantum computer. However, since we obtained the state using gravity, we should apply this fine grain gravitational entropy formula to compute its entropy. And so in particular, so we had this cut off surface we have now the radiation lives in the region outside this cut off surface. We try to compute the entropy of this region. But the idea is that if we want to compute the entropy of this region here outside, we in principle should allow some other surfaces with connected to another special slice. And now compute the enter when we compute the enterprise radiation we should also include the entropy of the quantum fields on this life plus the area of the surface. Now, you might wonder why might it be convenient for you to include this region. So the idea is that if there is a lot of entropy here but this entropy comes from entanglement with quantum fields that live in the interior. Then we could decrease the entropy by including this portion of the slice, this part of the slice, we pay a price which is the area of the surface. But if there is a lot of entropy here, this price to pay might be good enough in the sense that the area of the surface might be smaller than the entropy of the fields outside so we might get some reduction in the total entropy or this minimum there could be important. So we call this region here an island, this is a region that arises purely in the calculation of the entropy. And then, for the same reason we found an extremist surface in the previous calculation we also find an extremist surface here close to the horizon in this calculation. And this formula there are two entropies one is the entropy of the semi classical fields in this background. And the other one is supposed to be the entropy of the exact radiation state so there's some exact radiation state that exists in the full theory of quantum gravity. And the idea is that the we can get an approximation approximate formula for that exact entropy by computing by using this formula. And this formula is computed using the leading order semi classical geometry, which is this black hole solution. Now if the initial matter state is pure than the quantum extremist surfaces or these surfaces that extremize the that appear in the definition are the same as the ones we discussed before, and then we get the page care for the Hawking radiation. So we had success in computing the entropy of Hawking radiation, in a way that you get a result that is consistent with your entirety. Now, the skeptic will complain, and we'll just say well this is just an accounting trick. They would say, I have always said that if include the black hole interior, then the state is pure. And with this prescription we are including the black hole interior so. That seems to be a problem, and the information problem arises because you don't have access to the interior so that's what they would say. However, we should not view this as an accounting trick but a bit like an oracle so it is some formula that can be derived from the gravitational path integral. I didn't discuss the derivation I'll discuss it in a second. So we will discuss how we can derive that the formulas that we discussed so far from a first principles gravitational path integral discussion. And it's a oracle in the sense that it gives us the true fine grain entropy of the exact state, but, or at least an approximation to that entropy, but only using the semi classical state. So let me now say a few words about the diving this formula we that we've been talking about. The idea is that it is conceptually similar to the derivation of the black hole entropy using the Euclidean black hole. So, I remind you first of the derivation given by given some cooking of the Euclidean black hole. So, the idea was to start with this Euclidean black hole, and then compute the gravitational action, which is given by the action of gravity and then the action of quantum fields living on this geometry is completely smooth and clear. And then you interpret this as e to the minus beta h. So if interpreted this way, then the entropy associated to this partition function is just given by one minus beta d d beta logarithm of C. So that's the standard thermodynamic formula. And then it can be argued by using the questions of gravity. I'm not presenting the argument but can be argued that if you change the length of the circle, and you form this combination, the change in the gravitational action will basically come mainly from some near this near the horizon and will give us a formula equal to the area of the horizon. And this semi classical partition function of quantum fields will give us the entropy. So this this formula will reduce to the entropy of the standard entropy of the semi classical fields on this background. So we get the formula that we were discussing before. We can think of this formula in the following way so imagine that we do the Euclidean path integral on a circle but not a closed circle and but an open circle. So we can think of this as defining a density matrix that has the two entries of the left and right entries of the density matrices are the these lies here in the past and the future. This is not something we know how to do very precisely, but we could do this computation it's an unnormalized density matrix that that's what the tilde means. And then we wanted to calculate the entropy. We could take n copies of this density matrix, raise it to the power and now when we when we raise it to the power and and we close the circle then that has a well defined geometrical description. And then if we manage to analytically continuing and we can then calculate the entropy so formula similar to the one we had before. So, now if in this case that we have a you want symmetry this is just identical to the previous formula we discussed before changing betta is the same as changing and continuously. And with this description we can also consider situations which are not symmetric. So let's say a situation which differs by a little bit, compared to the previous one where we do a little bump here in the evolution little bump means we change the boundary condition for the fields or for gravity and so on we introduce extra, let's say gravitational waves or waves of scalar field into the geometry so it gives a time dependent geometry. So this is a new clean and time we could consider continuations of this to Laurentian signature if we wanted. But in this case, we could define a new density matrix and we could then define also integer powers of traces of integer powers of this density matrix by considering the same path integral with the same boundary conditions but let's say repeated three and with the condition that in the interior, the geometry is non singular. So, similar to the condition we had for the Gibbons Hawking discussion. So in this case we can do this for any integers, we can even analytically continue this computation for for fractional powers here and then do this computation and when we do this we find that the entropy computed in this way reduces to the entropy that that we had before. So roughly speaking what is happening is that when you analytically continue in this power, you get a small conical singularity here, which is a bit like a cosmic string whose action is proportional to the area. And that's where roughly where the area term comes from. But so that the bottom line is that this formula that looked a little bit mysterious can be derived from this discussion and the fact that here we should minimize and minimize and so on. It's related to the fact that we get the kind of cosmic string here. And we need to a part of imposing the Einstein equations is sort of minimizing the the energy of the the action of that cosmic string, and that involves minimizing the area or extremizing the area. So that's where this extreme minimum come from. So, in the same way that the Euclidean black hole gives us the entropy this so called replica trick gives us the gravitational fine grain entropy formula. And if the state was prepared by Euclidean path integral, then, and it has dynamical gravity only in some regions we should apply. We should allow various topologies in that region. So in particular, the interiors could be connected by replica formulas, and that gives rise to that island formula that we discussed. So that is also derived from, let's say, the gravitational path integral. Let me see, do I have time to discuss this so Well, let me not discuss it. Let me skip all of this. So part of the conclusions is that we reviewed the gravitational find an entropy formula and we applied it to the computation of the entropy of Hawking radiation, and we obtained results that are consistent with new entity. And at late times most of the interior is part of the radiation is not not really part of the black hole degrees of freedom. Now the question is what was black Hawkins mistake. In the case that he was not using the fine grain gravitational entropy formula, he was using something that was more similar to the thermodynamic entropy. So a lot of what was discussed was derived by thinking about aspects of ADSFT which itself involves some string theory, but you only need to know gravity as an effective theory to apply these formulas. So you don't need to know anything particular about ADSFT or string theory, you only need to know the gravitational path integral to apply these formulas. And so this is an amazingly deep connection between gravity and quantum mechanics. Now the question is whether the information possibly sold. But one aspect of it is it isn't it sold which is the aspect of computing the entropy. So from a gravitational computation purely gravitational computation, we computed the entropy. But however another aspect which is to understand what state comes out of the black hole evaporations we have a black hole and evaporates, we like some formula, purely using gravity of what the state is, and that we don't know how to derive from gravity. We could derive it using ADSFT but we don't know how to describe it from gravity so from that from gravity we don't have a complete understanding. So we can compute the entropy of the radiation but not exactly the state that the radiation is in it's a little funny so the computation for the entropy does not go through first computing the density matrix and then computing the trace but it involves through expressions that involve computing the trace of powers of the density matrix so you never have access directly to the density matrix. Now, as it as was the case for the black hole entropy, this is a bit like an accounting oracle that gives us the explicit. The explicit representation of the states is still mysterious. And the semi classical solution is representing only some aspects of the state, but enough aspects that allows us to compute the entropy itself. Now for the future, we can ask what further lessons this is teaching us about the black hole interior and the singularity. And we hope that this new understanding might have some implications for cosmology, though that remains something to be done in the future. Okay, thank you. Thank you very much one for this amazing webinar. Let me just see we have some questions. Something about doing this. Okay. So before I open the my for people here. Let me I see there's already a question. Let me just read this that I received, could you please explain again what are these quantum extremist surfaces and if the they are unique. Yes. Okay, let me. Okay. Go back to the definition. They are not unique, because we saw that some cases there were two choices. But the quantum extremist surfaces the surface that minimizes this quantity. So the area of the surface plus the entropy in this slice that goes from the surface to some kind of surface far away. Okay. This is some quantity. So this thing here within the brackets is something that depends on what the surface is, and then you can extremize this quantity if you find an extreme moon that's called quantum extremist surface. Thank you there is a delay so I'll let you know if there's a follow up question. So I see. Anna Romano, you can open your mic, I think, and ask the question. Okay, thank you. Can you hear me. Yes, yes, yes. You're very interesting book and I have a question regarding your final remark about connection to cosmology. So, so the, the, the so called continuity equation can be interpreted as the first flow of thermodynamics. Is it is the kind of connection which could make when you're on when you were talking about this idea. I didn't understand. So, so that there is of course a connection in the center space between the area of the horizon. And the center horizon or even cosmological horizon in general. First of all, based the second law and and perhaps could be representing some entropy, let's say at least perhaps the same thermodynamic entropy or Boltzmann entropy that is accessible to an observer who lives to an observer in that space. But we don't know what whether whether it represents let's say fine grain entropy, or so a similar notion for fine grain entropy is not known. So, let me just say that way. Okay, so, if you think about the fine grain entropy for that observer you would say it's zero. And a similar prescription would tell you that the surface, so let's say you find a surface around in yourself as an observer and then you can shrink it to zero in the other in the other. And in the anti code of the spatial section of the cedar space, which is a three sphere. And so you get you would get zero. Okay, so in that case the cosmological horizon would play the role of the structure rights on the blackout. Yeah, would be similar to the horizon for black hole. Thank you. Okay. Thank you. Let me say in a slightly different way so in the case of black holes we have a fairly well accepted sort of central dogma or hypothesis of how we are supposed to think about black holes. We don't have a similar hypothesis for the case of the cedar space or cosmology in general. And that there being various hypotheses people have made that they are not as solid I would say are not as we don't have as much evidence as we have for the case of black holes. So, you could make the hypothesis that you could make the hypothesis that in the cedar space, all the physics is described by a system whose entropy is proportional to the area of the horizon. And, well, that that might be something you might hypothesize. As I said the computation of the finder and entropy seems to contradict that but perhaps still is compatible so it's a little market less clear. Okay. Thank you. I have a question. What was the use of considering a collapsing star. We do this type of calculations with just a simple blood hole or other geometries. So the point of the collapsing star is first that is a simple initial condition that does not contain a black hole that then forms a black hole. Second is that this something that occurs in nature, perhaps not the simple spherical collapse but somewhat similar. And that historically was, you know, shown to show shown to be a mechanism by which a black hole can form when trash I found his solution it wasn't clear that that's an object that can form through a reasonable process. And the last part was so can you do these calculations with a simple black hole or other geometries. Well, I'm not sure what you mean by a simple black hole so the, the, these calculations involve an evaporating black hole or they're not trivial for an evaporating black hole. And as I mentioned, you can, so if you consider a simple black, the simplest black hole is the, the charge solution, including the semi classical corrections. And we have said that in that case, we can do the calculation. And indeed, we find that the fine green entropy is equal to the area of the horizon. So we don't get something fundamentally new in this case we get the same thing that you know was calculated by given some hockey and so on. But you can do the calculation. And the interesting thing is to do it in other situations, situations where you have an evaporating black hole or the black hole that from collapse or other situations that are not in equilibrium. Okay. Okay, thank you. Let's see if this is a fault if there is a follow up question. Are there. Okay, I'll see one question over YouTube. So, how does the temperature vary in the semi classical approximation. Does it change for different frames and different types of black holes. Well, the, the, the temperature is defining the restaurant so even for an ordinary fluid, the temperature is defined in some breast frame. So, the, if you have a black hole, and you have radiation coming out of it you can this radiation will have some kind of black body spectrum that you could use to define the temperature the statement is not completely precise because there are some great factors. So, if you want to define the temperature in a completely formal way in a more precise way, you would have to use some kind of fluctuation dissipation theorem and think about the various correlation functions as a function of time, the relationship. Okay, that's how you practice you define the temperature. So, how do you get for the temperature well that the temperature will be inversely proportional to the radius. So as the black hole operates the radius becomes smaller the temperature becomes higher for black hole. So, I mean, some of some aspects of the question could be raised even for the temperature of, you know, a radiating piece of cold, right so you have some hot piece of cold it's radiating and then you can ask how you do you measure that temperature. So that that's, that's involved the same conceptual difficulties as measuring the entropy for black hole. Okay, thank you. There is this other question. As you say, a posteriori one doesn't need string theory nor adseft to compute the entropy. No, to know this to know the state of the radiation do you expect to need string theory adseft or not. Well, currently the adseft is the only way we know how to compute that state. I hope that we'll understand the bulk theory well enough that eventually we will not need it or that will have another description that is purely in terms of bulk quantities and bulk concepts that allows us to compute the states themselves. So this is one, let's say this is a hope. Now, some people have the idea that you'll never be able to do that and that the bulk concepts are intrinsically approximate that there is some approximation that you do when you talk to talk about the bulk concepts and you, you will be able to have another description. That's, that's another point of view. I hope the first one is the, the better one, but the first one is the one we'll have and hopefully and then because that if we had that then that's would be something that would be able to generalize to other situations like the seat there. You know, that are more relevant to our universe. Thank you. I see. I think Nicholas or not as a question. Yes, thank you very much one for the very nice talk. So, so the black holes are evaporating right so they're supposed to a small one. They're shrinking the temperature increases that some point you have to stop right. I mean, if the temperature increases all the way to the to the plan scale, right, and that happens when the black hole has a massive order the plank mass. And here we're making so in the diagrams that I made and so on. We, one makes the hypothesis that when it gets to this point it very quickly evaporates and it emits some final number of quanta which is have an energy for the plank and the black hole disappears completely. So we made that like hypothesis. But by the time you get to this point the black hole doesn't have a lot of entropy. Now I should emphasize that that's so here I'm kind of apologizing a little bit, but that was the advantage of this page discussion. So in order to, to have this the page discussion, you don't have to say exactly what happens at the end. So we are now discussing the endpoint of the black hole of operation. But page pointed out that we have this contradiction when the black hole is could be very big. You know, here the black hole might have shrunk to a fraction of its initial size, but it's a finite fraction maybe it's half or maybe a quarter of the initial size. So it could be very big. And so semi classical physics continues to be reasonably good approximation this whole region. And so forth and then the discussion we had about computing the anthropological radiation is also valid in this region. The formulas we discussed are not valid in this very last points, this is very last regions, but they are valid everywhere here. In particular, they're valid after the time where we expected to find the contradiction. And this formula is resolved that contradiction. When I have a question since you are here. Oh, sorry, Nicholas. No, thank you. So there are there are a few times scales that you have mentioned. So one of them is the scrambling time that is pain. And then we know that the second external surface appears after this time and then there is the page time. But then why is at a time in between them is when you make a transition between considering the first external surface to the second or how are those times. Yeah, I didn't quite describe I didn't quite talk about the scrambling time. So I mainly this talk talked about this page time which is when the two are equal. That scrambling time appears in the details of when the other extremal surface exists. Okay, so here. We discussed that sometimes there is the surface I didn't I didn't give the details of where the surface was. But if you if you take a light ray that comes out of the surface and goes to the discount of surface. And you ask, what is the time difference between this time and that time. It turns out that that that difference is a time which is logarithmic in the entropy. And so it becomes large when the entropy becomes large but not not very fast. So somehow only logarithmic only. So, so it's not from the point of view of this talk is not a very long time. So I completely ignored it in this discussion. But if you are care about the details and people do care about these details and they're important for some other things which I didn't discuss here. Then that's the definition of that time, but it didn't play a role in this discussion. Is there any other question here we are way past the hour. I don't see any other question. Okay, I'm going to ask a general question I typically ask which is what are your recommendations to younger generations what should they study in order to like make contributions in this field, and any general advice might want to give to young scientists. Well, I think you should be curious and understand the basic. The big picture of the problems you're working on your you'll probably be working in a very specific problem very narrow area and usually that will almost always be the case that you're always working a very specific problem. But you should always keep in mind the big picture of how that problem fits into the whole, you know, scientific enterprise to the bigger bigger and bigger areas. What, how we're planning to advance knowledge. Don't don't. Certainly you have to concentrate on that problem to make progress but don't concentrate to the extent that you completely ignore where that problem fits in the whole building of science. Wonderful. Okay. Thank you very much for your time. Thank you everyone for joining us today. This is the, as I said, our women are 150th that finishes our current season and then the next system will start pretty soon. So stay tuned. We have talks by Ken Bantulbi, Ren Souz, Steven Hanmeier, Pisa Vendu-Berman, Rebekah Linn, Diego Portillo and others, night enough for the next season. So thank you very much. Thank you everyone and see you next time.