Black Hole Evaporation Unitarity & Final State Projection

0:00 Okay, Daniel Goddardson from the ProModern Institute will kick off this morning's session with a talk on black hole evaporation unitary, and final session. Thanks. So we've heard a fair amount of some mention of the information loss of black hole evaporation so far. A few people have talked about it at some length, but nobody has talked about it really in great detail. No one's given an overview. So that's one of the things I'm going to do today. For the first half of the talk, I'll talk about a variety of different proposed solutions to this black hole with intervention law problems, and some examples of solutions. And then in the second half of the talk, I'll describe some more recent work involving a new proposal for how to resolve this problem. It's worth pointing out, I'm from the Perimeter Institute. For those who don't know where it is yet, it's in Waterloo, Ontario, Canada. But by the end of today, you'll all because out of the four regular talks and two blackboard talks today, one of those is from people from primitive. But we have people from both the quantum information and quantum gravity side. So that's why we're well represented at this workshop. So anyway, so black hole evaporates. Well, you certainly heard about that. So the main points of choice, black holes emit fog and radiation. And over time, it will slowly evaporate. black hole, but it happened sooner or later. And they have an entropy associated with that temperature. So black holes are governed by, well, the laws of black hole thermodynamics, which then you make a direct connection to because of its entropy and its temperature, to the regular laws of thermodynamics. So for instance, this solves the problem of what happens to the second law of thermodynamics if you throw stuff into a black hole. Entropy does not disappear down the black hole, it gets absorbed into the black hole variety. And actually, generally, the entropy will increase when that happens. And another point that's been mentioned, but I wanted to bring up again, because it's going to be important in the second half of the talk, that the outgoing poppy radiation is not alone. It has pairs. It's part of a pair. That's why the outgoing radiation is thermal, because it's entangled with other radiation that's falling into the black hole. We can imagine that the skin-falling radiation has negative energy, the outcome of radiation has positive energy, and that ensures that energy is conserved,

2:30 so that the energy leaving the black hole with the softening radiation is cancelled by the negative energy of radiation that's falling into the black hole and making it not smaller. So here's the basic points about black hole evaporation, but this brings up the information loss problem. First of all, you have a big black hole before you can form a big black hole before you have a moderate-sized black hole from any kind of matter at all. And then you wait, and this black hole evaporates, and eventually, perhaps, it's completely gone. Well, you can ask the question, according to this calculation that Hawking did, this semi-classical calculation, the radiation that comes out of the black hole is thermal. Whereas the stuff you put in was certainly not thermal, it was maybe even in some pure state, or half an entangled state, where you kept the other half outside. In that case, you have to wonder, well, when the radiation gets out, is the overall, what happens to the information about the stuff that I put in to form the black hole, or stuff that I threw in after it was formed? Is that gone? Or is it somehow hidden in correlations among this thermal radiation? And then there's very closely related questions, so perhaps it's not quite identical. hole. You can add for instance black hole evaporation sorry, quantum mechanics need to be lifeline to describe black hole evaporation. That's really what's at stake here. Quantum mechanics is exactly correct the radiation that comes out should have a direct relationship to the stuff that you put into the black hole. A one-to-one. I think it's fair to say that despite and perhaps we can discuss why later in the talk. And another way of framers is to say is black hole evaporation unitary? Is there a unitary relationship between the radiation that comes out and the matter that got put in? So, okay, so probably many of you are familiar with this, the Sherlock Holmes quote. When you've eliminated the impossible in whatever remains, however improbable, must be the truth. So this is perhaps a good motto to apply to the study of black hole information loss problem. However, there's a complication, which is that, as you can maybe see from these questions, some very basic things are on the table.

5:00 And we have to question even some assumptions that we otherwise would take as very strict rules. And so it's really very hard to eliminate anything. It's impossible. So instead, we're in the position of eliminating improbable and then we're left over with nothing at all so then we have to go back and try to argue about which of those improbable things was less improbable than the others so okay so first proposed solution kind of the easiest one the one that that that's the most obvious consequence of walking type addiction information is lost so it's very simple when a black hole evaporates, the information is gone. So this has a very simple advantage, which is that it has an obvious relationship to this calculation that says that pocket radiation is thermal. You don't have to really modify it at all. Maybe there's some small correction to the back reaction, but basically it's still thermal. The stuff that comes out is a mixed state. Anything you threw in, too bad. It's gone. But there's a lot of problems with it. serious than others, and some people feel them to be more serious than others. So one problem is there's this argument due to Banks, Peskin, and Susskin the loss of unitarity in a situation like this is somehow associated with problems of creating a theory that has conservation of energy, strict conservation of energy. I can't say I've ever been particularly convinced by this argument, so that's why I put a question mark there. But I thought it was worth saying, because It's a fairly standard argument that we'll hear. So two things that seem kind of more serious to me. So one of them is this, is that either if information is lost in black hole evaporation, you have to choose either between time reversal and variance of the fundamental laws of physics or predictability. That's because the time reverse of black hole evaporation would be, I don't know, anti-evaporation, where a bunch of radiation gets together and forms a black hole, or really a white hole in this case. out of it. And if black hole evaporation loses information, then anti-evaporation should be able to create anything at all coming out of this white hole, even though all you're putting in at the beginning is the same thermal radiation. I don't know, there's probably ways around that. I mean, time and crystalline variance is maybe not as sacrosanct as I would

7:30 like. And really here I'm talking not just time reversal, but CPT, so it's really a full reversal of everything. But, you know, in the real world, we kind of see violations of time reversal invariance. Normally we think that's due to boundary conditions, but maybe it's really due to something more fundamental. Certainly lots of people propose that for other reasons. The other thing is that if it's possible that a black hole can evaporate in a non-unitary way, well, that means that gravity is allowed to act In particular, you have to assume that quantum gravity can also act in a non-unitary way. So quantum gravitational processes are presumably happening all the time, all around them, at very small scales, at the Planck scale and at the Planck time. And if black hole evaporation can be non-unitary, well then why can't those processes also be non-unitary? Obviously, in this case, there's no fundamental law of nature that says that they have to be unitary. So, why not? So maybe there's some process that forbids that, except in very special cases, or suppresses it very heavily. But that's something that you have to explain, then, in your quantum theory of gravity, why you don't see this all the time. And in particular, why don't we see them in atomic physics, right? Where we know quantum mechanics falls to a very high degree of precision. and someone who's worked a lot on quantum error correction getting rid of decoherence to use in quantum computers I can tell you it's not a trivial thing to get rid of decoherence once it's introduced into your system and I think it's probably worth making a variance even if the linearity of quantum mechanics wasn't true quantum error correction could still effectively make quantum mechanics linear in the sense that we could still build a quantum computer even if quantum mechanics broke down in some level, so long as we could get down below the threshold-limited fault-tolerant direction. Right. So, okay. So in some sense, it doesn't really matter for the big program of building a quantum computer. No, no. I'm not referring to building quantum computers. So first of all, the evolution density matrix to

10:00 to thermal density makers is linear, it's non-unitary. So linearity is not really at stake here. Although, of course, it could be. We just don't know about that. The question is unitarity. And so I'm not really worried about building a quantum computer. I don't think this affects it one way or the other, particularly. Although if the non-unitarity is particularly bad, maybe it still would. The question is, how does quantum gravity get rid of its decoherence to become coherent then, again, So, I want to talk a little bit about a variant of the solution, which is that the information is not really lost, but it's a black hole singularity, there's a baby universe form. Incidentally, I think that would be a good title for some children's form, Cosmo, the baby universe. I have no idea what it would be about. Anyway, so the baby universe. The baby universe, the singularity, I don't know, inflates into some baby universe that sprouts off from our own. And all the information that's lost with the black hole ends up in this baby universe. And the event, from our point of view, it looks like the information is lost because once the black hole evaporates, the baby universe, I don't know, is born and goes out on its own, and we have no further contact with it. But, of course, overall, if you count our universe plus the baby universe, the information is not gone, it's still there. The evolution of the two combined is unitary. You're saying that the baby universe still remains after the black hole evaporates. Well, it remains. I mean, it doesn't mean it's separate from us. So it remains somewhere, whatever that means. It's not in our universe. But evaporation doesn't have to kill the baby universe. That's right. The evaporation doesn't kill the baby universe. It kills kind of the umbilical cord that connected us to the baby universe. work with the black hole. So, as I said, this preserves overall unitarity. It also solves the problem of time reversal, because really then the correct time reversal of black hole evaporation would be to take baby universe plus big universe and somehow put them together to create a white hole. And then what comes out of the white hole is what was ever in the baby universe. But it doesn't solve this last problem, non-unitarian atomic physics, right? Because it's still true that, well, if black holes

12:30 can produce topology change, then why can't regular quantum gravity effects? Why don't mini-beta universes split off all the time? Yeah? If I think of the information going to black holes, real information, and I take some hand-wagon argument that's the minimum size for a black hole, I can be able to use very, very, very high energies to construct these bases of these networks so it's even to get big off one that would be an intervention. Well, okay, so it's, I mean, obviously there's a lot of hand-waving involved. It's not really clear. I mean, it doesn't mean that in general relativity it doesn't generally make sense to talk about the energy of the whole universe. So there's no particular reason I know of that a big black hole would have to be associated with a big So it doesn't mean why it's hard to construct these baby universes and why you might not see them in a... Right, I mean, so again, yes. So as Seth points out, you can certainly make excuses. You can imagine a solution where Planck-scale processes do produce baby universes, but they're very small baby universes. And so they don't carry on much of the data. Or maybe somehow they recombine with us very quickly, something like that. Again, as I said, we're eliminating the improbable here. argue about what's most of the problem. So, okay, another solution, which doesn't actually seem to be that popular nowadays, but has certain things to be said for it, is the idea that the black hole doesn't completely evaporate, but it leaves behind some kind of remnant, say, of a few plank masses inside. And so the advantage of this is, well, the semi-classical calculation can still be exactly right because thermal radiation emitted from this black hole. And all that's left behind, the semi-classic calculation is right, except in the case where you get to very small black holes, and then it gets modified. But we expected that anyway. Semi-classic calculation only implies when the curvature is small, when the mass of the black hole is much, much larger than the fine cell. And you also don't need to modify quantum mechanics because there's a lot of different kinds of possible remnants. In fact, one different kind of possible remnant, for each possible way, you can make a black hole. And so the overall evolution is unitary.

15:00 You're going from, you know, in fallen matter to radiation entangled with this remnant state. On the other hand, these remnants have to be very, very vulnerable. I mean, for a very large black hole, there's an enormous number of possible states you could have used to form that. black hole is very very high and so you have to have an enormous number of remnant states and in fact the mass could be arbitrarily high of the initial black hole and so there has to be an infinite number of possible remnant states so we don't know anything like that I mean in fact it violates the holographic principle but maybe the holographic principle I mean that's a more recent invention perhaps perhaps that's wrong and another another argument that's been made here is well if we have this infinite species infinite number of species of black hole remnants why don't we see that in particle physics right you could have in some collision you could have a a remnant remnant production okay we haven't gotten up to the plank scale but maybe but there could still be virtual effects from virtual remnant remnant production they can form and then uh then then uh rejoin collide and so so the energy can be But with an infinite number of species, you have to sum over an infinite number of terms of including these. And so that seems like it should have a big effect. Well, again, maybe you can make excuses. Maybe remnants with a lot of information have much smaller coupling than remnants with only a small amount of information. Okay. And so then to the third class of solutions. the information actually escapes in the Hawking radiation. So to a lot of people, this is a very attractive solution, that somehow these thermal photons are not exactly thermal, but there's correlations between them, maybe very subtle correlations spread out over very many photons. So if you only look at a few photons, it still looks exactly thermal. We're very, very close to thermal. But if you look at many, many photons comparable to the mass of the black hole, you can start to reconstruct some information about the objects And if you have all the Hawking radiation, maybe you could reconstruct this information exactly. Okay, so the obvious advantage of this is that it preserves unitary. You don't have to worry about what quantum mechanics is doing. Because clearly, we discussed another very complicated

17:30 physical unitary transformation. You can somehow describe this whole system using the same underlying laws of quantum mechanics. And another advantage, which really the other explanations don't have, is this gives a very good explanation for black hole entropy. Black hole entropy becomes just like any other kind of entropy, as a fine-braining of the system. Black holes, you've heard, have no hair. But that doesn't mean there's not lots of different ways to form a black hole of a particular for its annual momentum. And then the black hole entropy corresponds to all the different kinds of black holes you can form with that set of parameters. And in fact, in your favorite theory of quantum gravity, you can hope, and people have in some cases done this, to sit down and actually calculate the number of possible states of your quantum gravitational theory that give you a black hole of a given type. And when people have done this, frequently And so that's a very hopeful sign, that it means that the black hole entropy really does represent microscopic states of the system, which in turn suggests that somehow the information really is conserved in the whole process. But the difficulty, actually of all the proposals, this one faces in some sense the most severe conceptual problems, because how do you get the information out of the black hole? Okay, I'm going to describe this in more detail, but looking very closely at this, or even actually not that closely, it seems like somehow the information has to get cloned between the imploring object and the Hawking radiation, and we know that quantum cloning is impossible. Or the alternative, of course, would be faster than light travel, that somehow, once it's inside the black hole, information escapes. Yeah? So you would think you preserve unitary as seen by outside, but there's something that's non-unitary going on at the singularity, right? That's what you're going to do. Well, so this is just a kind of general class of proposals. The very specific example I'm going to give later does have that property. That's a good story. That's something I don't understand, because what people think we want to do is preserve unitary, and I think when you get 20 and I don't see how it's supposed to stay okay

20:00 and it's supposed to come outside, but you still don't have to come back to the future. Maybe I can come back to this. Hopefully, I mean, we'll have a lot more to talk about certainly in this point later. So you can definitely avoid the task of the mindset by having these actions at the horizon. Right, yeah. Right, that's right. So I'll talk about that too briefly. Okay, so you saw Penrose diagrams yesterday in Penrose's talk, but he didn't really explain what they were. So if you've been for the point of information people, I can say that to understand the rest of the talk, you need to know a little bit about gravity and a little bit about quantum information. So I'm afraid I'm going to do some fairly basic stuff, but I promise that I'll fight for both communities are part of the time. I don't have a cell before you eat for it. So, okay, so Penrose diagram. So this is a picture of the universe when it's flat. Flat spins. And I've kind of bounded it in a nutshell here, this little diamond. So let me start to label it. So first of all, time in this picture goes up. And in this picture, I've written things throughout my views along 45 degree angles. So, of course, I've only drawn universe one space one time but you can imagine extending this to a quarter micro universe so light moves along 45 degree lines so it's probably that way or this way and uh matter has to move more up than to the side so for instance massive objects could follow trajectory like that. Yeah, hopefully this is not, yeah, that's less important. Okay? And then the edges of these things represent things that are infinitely far away. So for instance, this very top point represents future time-life infinity. So if you wait, if you sit in one place and wait an arbitrarily long amount of time, that's where you'll be. Sorry, an arbitrarily long, an infinitely long amount of time, that's where you'll be. Yeah. So, is this modified at all by, I mean, if I think about teleportation and I think of sharing EPR pairs between Alice and Bob, then, and if they go through some non-trivial background space-time, so that, you know, entanglement is some sort of Fermi-Walker

22:30 transport in that way, and if I think about, if I try drawing the sort of path of the Fermi-Walker transported particle from the beginning to the end of the teleportation protocol, and seem to follow a path which is limited to, you know, 35 degree angles, because that doesn't seem to make any important difference because you still have the right kind of limitation but in some sense it modifies it. So this is, perhaps it, let me put it this way, so this is the way, when I draw conjectories like this, you can think about the conjectories of particles, say, tracking energy or something like that. So, tracking information is kind of a more subtle question, not something that's really covered in this diagram. Right, so future time-like infinity, you said you want to place it where you'll go. And in fact, even if you move, that's where you'll go, so long as you don't go at the speed of y. So if you have mass, you can't tell, but end up with future time-like infinity. And then correspondingly, this bottom point is past time-like infinity. So if future time-like infinity is where you're going, this is where you came from. And again, any massive particle is coming from past time-like infinity. I would prefer to find a path. This, of course, is flat space, so there's no big bang right now. And then these two points on the corners here, on the sides, are space-like infinity. So if you're somehow able to move faster than life, and you head it off for an infinitely long time, That's, again, where you'd end up with one of these space-like infinity points. So all space-like lines will spread between these two points. And then these diagonal lines, the sides of the square, are past null infinity and future null infinity. So this is where light comes from and where it goes to. It's perhaps a little surprising side effect in these diagrams that all massive objects go to just points, whereas light gets a large choice of where it goes to. But that doesn't really affect what we're doing. In practice, all we really care about is what's happening on the inside. The edges become less and less important

25:00 space is compressed into a smaller and smaller amount of diagrams. It's like a Mercator projection. Kind of. The opposite. Right. It's like the Penrose diagram projection. Well, it is the Penrose diagram. What's quite more familiar is you may have seen this Escher picture with a circle where In the middle, the fish are big, and you get closer and closer to the egg, they're smaller and smaller fish. So that's actually a picture of a hyperbolic plane that's very similar to this. Okay, and so the point of these Penrose diagrams is it makes it very easy to understand causal structure. Because you can just look at these 45 degree lines to see where light moves, and then massive objects have to go within the light room. Just one really stupid question. Why isn't an object starting infinitely far back coming from infinitely far away? So why isn't... The point is because... So it is coming. It could be coming from infinitely far away. It doesn't have to be, of course. But the point is it's coming... Because it's moving slower than life, it's coming from further in the path than it is in space. And so that all gets compressed to this single point. anything that has that property. And anything that's further away in space than there's in time, and when it's equal, then it's one of these lines. Okay, other questions? Okay, so now let's go to the black hole. This is the Penrose diagram of the black hole that forms. I haven't seen the evaporation here. so we have the same kind of points as the corners we have future time-like infinities this point up here space-like infinity this point over here in the corner half-time-like infinity down here at the bottom so one small modification I've made that's not incredibly consequential is instead of having a whole diamond I've cut it in half and gone to spherical coordinates so this vertical line on the side is r equals 0 So before the black hole formed, or if you did this in flat space, there would be nothing

27:30 special about this line. If you head towards r equals zero and you keep going, well then you'll go of course right through and start increasing r again. So you kind of bounce off this edge. And of course there's nothing that prevents light or massive objects or anything from doing that. something that's perhaps more interesting is the black hole singularity and so once the black hole forms this wiggly line is created well, it's created in the future and that represents the causal structure of the black hole anything that falls into the black hole will eventually hit the singularity so the singularity is in their future that's why it's an important safety tip don't fall into a black hole So we draw it up here at the top. And the other important structure that's added here is the black hole event horizon. And this is, in some sense, unnecessary because you can just deduce it from looking at the rest of the borders of the diagrams. The event horizon is defined as the boundary of the region that cannot escape to infinity. So what you can do is you can look at this point, future time-like infinity and project backwards along 45-degree lines and say, when can we no longer reach it? And you see that this line divides those regions. Anything in here can move even slower than light and reach future time-like infinity. Anything in this region over here cannot. It's always going to hit the singularity. Everyone see that? Yeah. So, this diagram suggests that if you do fall past the event horizon, then the smartest thing to do, in other words, to give yourself the largest amount of time before you hit the singularity, is to do nothing. And if you actually try, you know, zipping off in any direction, you'll just meet your fate faster, according to your own clock. Actually, that's not true. So first of all, these diagrams are very good for getting causal structure, but as you might have guessed from the discussion about compressing space, they're very, very bad for guessing distances. And so in fact, the best thing you can do if you cross the event horizon and you realize too late that you've done that and you want the most time possible is to head straight out of the black hole as fast as you can.

30:00 And while you're still eventually into singularity, but it'll take a little bit longer. Even by your own clock. Even by your own clock, that's right. Pray. Well, yes, I guess you can hope that maybe the information is not really lost and someday someone will be able to reconstruct you from the thermal resin. That's your best hope? Or you could try to, well, okay. I don't know if that's true. That's your best hope? In the calculation, yeah. That's your best hope? Yeah, that's your best hope. the way to delay things. Oh, you mean there might be some other trajectory? The trouble is you become closer and closer to null, which is zero length. Yeah, that was my point. As you get faster, then you just... So it's not clear to be that that's the best. No, I think that is. I think Jim Harville wrote a paper about this. I'm not sure. The proper time, the new years of experience. And I think that they also take this conclusion is the same in this report. I certainly have not done the calculation, I've heard this in discussions in the past and hadn't really thought about it. But it's certainly not obviously true that it's a correct way to do it. Probably there is something you can do that will take you a little less. Okay, other questions about the diagram? Okay, so let's go back to the information loss problem. So, okay, here I've gone...oh, wait, sorry. So, yeah, this is an example here... Oh, it's kind of left out. Here in green is the space-time trajectory of an object which essentially stays outside the black hole. and then here is an object that was crawled into the black hole. Nothing really subtle about these. The main thing is that the way I've drawn the diagram, it looks like you have to kind of do something to stay outside the black hole. I guess that's kind of true. I mean, the black hole has... I mean, the space kind of has a black hole and nothing else. Everything will eventually be drawn into the black hole. But in practice, if you're very far away, when you're using radial coordinates like this it's no longer true that

32:30 all light lines projection of a three dimensional thing and it's no longer true that light lines of 45 degrees here because it may not it may be going in a two dimensional direction so this is just in the RT direction yeah so I mean I guess what you have to really imagine is this kind of a sphere at radius attached to each point in this diagram. If you want to imagine the whole quarterback in this case. OK, so now, black hole information. OK, so here we have the black hole that forms, and then an object falls into it. So I said that getting information out, it seems like it requires some kind of component. So why is that? Well, this object is, the event horizon is, for a large black hole, is very close to flat. It has curvature that's very small. So we would expect, again, at the level of illuminating the improbable, we would expect that nothing really spectral happens to something that's falling into the event horizon. It's falling across the event horizon, particularly if it's freely falling. Freely falling, an object won't even see the Hawking radiation. So it seems like that object should feel that it's still itself. It should still have a copy of whatever information it represents. So that's going to be our first copy. But at the same time, this population is coming out of the black hole. and somebody at later times by our assumption to be able to reconstruct information about the original stuff that fell in. And so that's a second copy. Okay? So we have one copy here, one copy here. And in particular, there is ways we can consider a single time slice. So we say, so this slice here is everywhere a space slice. So we could write down coordinates that is equal to t equals 0, or t equals whatever. In fact, we can foliate this whole space-time with things like this. And if we draw a line in the path here, it looks like there's just one copy. But if we draw this line, there's two copies of this information. So that seems like quantum clinic.

35:00 And of course, if we want quantum mechanics to be exactly correct, can't do that with quantum mechanics. And so it seems like here is a series of time evolution of space-like slices in which quantum mechanics would have to be violated. Kind of ruin the point of restoring quantum mechanics overall. Let's see. I forgot if I was going to say anything more about this. Yeah, OK. So then let me make a couple of additional comments. course, you can start to make excuses. There's a lot of excuses you could make. For instance, as the rest of the topic suggested, you could say that there's some kind of interaction that goes on here between the Hawking radiation and the infalling object that somehow eliminates this copy of the information and moves it into the Hawking radiation. But again, it's a little puzzling why that would happen at the horizon, which is not spectral. Why is the object feel this when it doesn't even see the Hawking radiation at all. And so maybe you have to make another modification, an idea called black hole complementarity, that well, the infallible object thinks it has the only copy of its information. And the Hawking radiation, well, somebody outside thinks the Hawking radiation has the only copy. And that the object, well, it's gone past the vent rise, and it's gone. You don't have to worry about it. So there are arguments you can make that will get around this problem. but really to make them work I think you require some kind of fairly radical new physics do you disagree with that? so some kind of radically new physics, so again when a situation is having to make a big modification to our existing physical laws to describe what's going on so okay so now I'm ready to move on to the new stuff, the idea of the final state, but before I do that I have to, yeah, maybe just one suppose someone on the outside reflected the Hawking radiation back in so that the copy and the original were actually inside the event horizon then you could actually do a test so you've fallen through and then you suddenly see that you've violated the no-cloning theorem and so you know monomechanics is indeed messed up however it's too late so sorry, did anyone hear that question? the question is what happens if you collect

37:30 all this information with Hawking radiation and throw it back into the black hole and there's a paper by Lenny Susskind about this, that it turns out that to collect all this information, you have to wait a very long time, until the black hole is almost gone. And then if you threw it into the black hole, well, this object probably would already hit the singularity. And if you were in cahoots with that object, so it decided that it was going to try to wait around as long as possible, not proper time, but kind of your time, it is, and had headed out of the black hole, which we had to do that very, very soon after crossing the horizon. In fact, we didn't say something at the order of the time length of the horizon at proper time. So it would require accelerations that were really immense at the level of quantum gravity. So somehow quantum gravity would have to resolve this problem, would have to allow quantum energy. But you couldn't see it without that, without going into that regime. So quantum teleportation. So now it's time to board the quantum information people. So this is a picture of a quantum circuit diagram that describes a process known as quantum teleportation. So in quantum teleportation, we have two people, Alice and Bob, and Alice wants to send her quantum state to Bob. The problem is they don't share a quantum channel. They can only send classical information. And sending a quantum state over a classical channel would destroy it, would decoherent. So this picture is kind of the sideways, not up. It goes to the right. And Alice has two quantum bits. So one is a quantum bit that she wants to send. The other one is half of this Bell state, 0, 0, plus 1, 1. Bob has the other half of this Bell state. So Alice and Bob share a tangled quantum research. classical track. And Alice then will take her one quantum bit, her qubit, and her qubit from the EPR pair and make a bell measurement, a tangled measurement. She'll get out of these two classical bits. And she sends those two classical bits to Bob and based on the results of those bits, A and B, Bob does one of the four poly operations. Right, two bits, so there's four possibilities. 0, 0, 0, 1, 1, 0, 1,

40:00 and four polymatrixes, sigma x, sigma y, sigma z, and the identity. And no matter what this result is, Bob then gets back the original quantum state sign. So the black hole's final state, this is a proposal by Horowitz and Malaschino last year to use something very closely related to quantum teleportation to get the information out of the black hole. And so it's really a very clever idea. It says that, well, instead of having to do something special at the quantum, at the van horizon of the black hole, we're going to do something special with the singularity. And that's somehow less objectable, because the singularity we already knew had to involve quantum gravity somehow. So let's say we don't understand quantum gravity, things can happen in the singularity, as long as everything is normal elsewhere. So the idea, in particular, is that the black hole singularity only admits one state to touch it. And so whenever it's falling into it, it projects on this particular state. And anything else is eliminated from the universe, or never existed, really. So the idea here is that, in particular, it's projecting onto a maximum integral state, The matter that fell into this black hole, formed it, was thrown in later, whatever. And the half of the Hockney radiation that's falling into the center. Remember I said that the Hockney radiation came in pair. Half of it went out, half of it went in. So this is the half that went in. You see, this acts a lot like quantum teleportation, right? Because here's your EPR. So the singularity here is playing the part of Alice. The EPR pair is the inbound Hawking radiation for Alice, and the outgoing Hawking radiation for Bob, who's outside the black hole. And then the state that's been teleported is the state of the matter. Yeah? What's the dimension of these systems? Well, the dimension is really huge, because the dimension is the dimension of the set of things that fell into the black hole. if you want harmonic oscillators into the dimension. I'm worried about the fact that you've tangled space. Yeah, so, so, let's, yeah, I mean, so, so, so, I guess, I guess this could go into a long discussion,

42:30 but let's say the black hole has some fixed mass. So that, that kind of cuts off the number of states in the harmonic oscillator, for instance. And the black hole has, you know, therefore fixed dimensions, so it's a fixed size. So basically you're regularizing everything. is probably a finite-dimensional space involved here. Right, but for this purpose, I don't have to worry about the dimension of this stuff maybe in the distant past before block-leaving form. It was almost flat space. Okay, so as I said, this acts like quantum teleportation, but because we're projecting on just a single final state, this is unlike Alice's Bell-Meckren. Alice's Bell-Meckren can have multiple outcomes. But the singularity always picks a particular outcome. Yeah? But how does one outside know what that state is, and if he knows, he already has some information? Well, I'm glad you asked. The object outside knows what, I mean, it doesn't have to know what it is, but there's only one state that it could be. the singularity just forces it to be a particular state so it might be that it's a very very complicated state, in fact it probably will be a very very complicated state but that means that there's only one state that can come out based on the state that went in so it's a unitary transformation but isn't it like then I already have the information that you say is escaping by knowing what the state is well no, I mean in teleportation the bits that Alice gets by measuring are completely random. They're uncorrelated. So it's the same thing here. It's always, I mean, it's like you did this bell measurement, you always have the outcome zero, zero. Yes, but, I mean, the thing is that so Alice makes a measurement that she has four possible outcomes, and if Rob outside, or all

45:00 outside, doesn't have access to those who have conclusions, where he sees this complete thermal statement, he has no information. into what state then he would be able to obtain a whole state where he gets all the information but he leads those two classical bits by the one outside knowing already into what state the similarity is going to is already having those two classical bits well that's right, so if you could do this in a flat space it would allow you to communicate faster than life but this is a very spectral situation This projection is only happening at the black hole singularity. So to find out kind of what state was thrown into the black hole, you have to look at the Hockney radiation. It takes a very long time for the Hockney radiation to come out. And in fact, it's going to come out after the object that went into the black hole did so. So you don't really have to worry about that particular problem, although it'll come up again in a minute. So let's hold off until I get a little bit further. Let's see. So I could think of doing this with many black holes. And there are two possibilities. each black hole projects to a unique and potentially random maximally entangled state, in which case it doesn't help repeating this exercise, or each one is determined by the same physics somehow and it projects to exactly the same maximally entangled state, in which case maybe by some heroic ethics of tomography, you know, very long time scales, I could work out what that is and then, you know, from then on actually convert this into a faster-than-light machine or... So again, you can't, even if you know exactly what unitary is performing, and it does perform in unitary because we're not obviously projecting on the regular Bell state, it's going to be something much more complicated. So it does some unitary from the matter that falls in, the Hagen radiation that comes out. You still can't find out what the thing that fell in until you get the Hagen radiation Even just a little bit of the Hagen radiation is not going to tell you anything. You need basically all of it, or almost all. And that's going to take a very long time to get. So I think in this case where we're projecting on a maximum entangled state, there really isn't a problem of causality. It is possible there's some very clever way you can arrange things, but I really don't think so. But maybe by this tomographic method, we could actually retrieve the information in detail

47:30 if it was repeatable. Absolutely, in principle. The original causality, you're putting on a future condition. Yeah, exactly. It's a boundary condition. Causality is a boundary condition. Well, yeah. future boundary conditions or causality is not an appropriate concept. Well, this is a kind of subtle question. Certainly, if you're putting future boundary conditions in arbitrary places, causality would be in trouble. But again, you might hope that by putting it just at the black hole singularity, you can somehow preserve the regular laws of physics outside the black hole. You can certainly hope to do that. Whether you succeed or not is another question. I think that's not a completely a reasonable thing to do. Particularly since the way this is set up, this projection always has a constant probability of success if you were to use this kind of regular final mechanics and do that projection. The chance of getting this particular final state would be independent of the state that you throw in. It does this actually if you use different states. Essentially zero would look. it's very very small that's why I said essentially zero you couldn't tell the difference between it and zero without doing any experiments if you do this measurement yourself you essentially never get that particular outcome but nevertheless somehow the singularity can do it in this proposal OK, so these are the basic points of this proposal. And here's another picture of it. I need to help fund this name for people, but there should be more specific. Here's the in-plot of matter, you've got the top rate. So it really couldn't be 0, 0, plus 1. It could be a sum over the whole 2 to the black electropy, basically, in different states. But it looks something like this. And then the black hole makes this final state projection. And because the outcome is predetermined, you don't need to communicate anything back outside of the hole. The final state outside of the Hawking radiation is unitarily related to the original, by depending on exactly what maximum entangled state you project on here. So now the problem with this. Well, the problem that was pointed out by John Pressel and myself

50:00 is that this assumption only holds true if you're projecting on a maximally entangled final state. And in particular, you have to worry what happens if there's some kind of interaction between the inflowing clocking radiation inside the black hole and the matter that's falling in. Right? There's no reason that those two could interact before they hit the singularity. And the problem is that if they do that, then unitary would be lost. I love to hear the moment. Actually, we got this idea because we had heard talks by Ben Schumacher and Charlie Bennett about their idea of simulated time problems, which I think we're going to hear about tomorrow. So, well, I guess it's appropriate that I'm talking about first. Simulated time problems. So anyway, the reason this class of loss of unitarity is that you can absorb this interaction into the final state. and so the final state you're projecting on is no longer a maximally entangled state so here's an example of how bad it could get if the interaction were some kind of maximal entanglement it could completely undo the entanglement from the final state and it would be like projecting on the state of all zeros so this final state projection would take this input state and project on zero and it would take the inflong hogger radiation to project on zero that means the outgoing hogger radiation zero. No matter what input state you put in. So that's a complete loss of information. So let's see, since the in-falling matter of psi is freely falling, I suppose, then it doesn't see quantum effects, so it doesn't see also the in-falling Hawking radiation. However, I guess what you're saying is that that's effectively breaking down when you get really close to the singularity, then the curve just is so strong that you'll end up seeing those quantum effects. So they will interact. Well, so, okay, so First of all, I don't want to get really close to the singularity. I want to, because once it's really close to the singularity, maybe this final state condition is already kicking in. So I want something that's happening further away from the singularity. But obviously, in this peak example, where it gets completely untangled, that's an extreme case. There could be some small corrections, though. And so it's not unreasonable to expect there's some kind of interaction between the infalling matter and the infalling operator. be small, but it means that unitary is not going to be preserved exactly.

52:30 What's the difference between infalling matter and infalling operating? There is only one thing. Well, yeah, exactly. That's the point, is that there isn't this pure separation, whereas this proposal does require pure separation. I don't understand how the proposal requires a pure separation either. Oh, okay. So, the point is, The pros of the project for separation is that this kind of projection has to be max entangled. When you do this separation, when you split it up between object and radiation. The point is that the radiation... Why is it entangled between what? Um, the, the, the, there's, there's, the, the, the, the, the, the Hilbert's, so take the Hilbert's space, of everything that's involved, the original matter and the marked radiation, You get Sun-Hilbert space, right? The same thing. Well, whatever. The same Hilbert space. Yes, exactly. So there's Sun-Hilbert space of these two things combined. And the hot radiation outside has some entanglement with the stuff that's inside. Okay? And so that gives you a special subspace there. And that's what you can call the infalling hot radiation. This is without the bond state projection. You can do it. Okay, and without the interactions. Whatever. This is their proposal, okay? And it makes this distinction. You can argue about the distinction, and that's kind of what we're doing. In the final state, you think that the final state is totally disentangled to the healthy radiation outside, right? That's why you have it as the maximum pure state. Sorry, the final state here? You mean? Okay, so the final screen prediction is entangled to not have any radiation outside, because you say like it's a maximum entangled pure state. Does it have any entanglement to radiation outside? Well, so in that proposal, they're projecting on a pure state. I mean, I suppose you could make varying proposals where they're projecting some kind of weak measurement or something. well I was trying to know exactly what that would mean in this case but the idea is yeah it's just on the inside over space so you don't care what happens the relationship to the outside over space well I mean what happens to the outside

55:00 obviously you care very deeply what happens to the outside over space but that's deduced solely by applying this projection So, okay, so two things to point out. One is that this is completely non-unitary because arbitrary initial state goes always to the same final state in this particular example. And more generally, if it's not a matriletangal state, there'll be some loss of information. It's also true that in this case, this is putting a restriction on the states that you can throw into the black hole. It doesn't accept the input state 1 because the probability of this final state projection is 0. It's not really clear what that means, actually. but it meant if you say put in a superstition of 0 and 1 you'd really only be putting in 0 and so that has another problem associated with it as was pointed out by your center talking is that if you have some loss of unitary to do this, you could do you actually could do fast and light communication outside the black hole in fact you could do it before the black hole evaporates so the need here is you have two people Alex and Bob that want to communicate Alice wants to send just a practical bit to Bob, or in this case really even just a fraction of a practical bit. And Alice and Bob share a regular EPR pair. Bob is far away from the black hole, and Alice is right near the black hole. So Alice has a poise to make. Either she can keep perhaps an EPR pair outside the black hole. In that case, when Bob measures his bit, he'll sometimes get zero and sometimes get one. It'll be a mixture. Or Alice can throw her qubit into the black hole. And if that happens, well, suppose we have this extreme final state projected on 0, 0. Well, if that happens, Alice's state is not allowed to be 1. Alice's state gets projected on 0. And, well, we don't really care what happens to the Hoss radiation here, but what we care about is what happens to Bob's qubit. Since they started with 0, 0, plus 1, if Alice's state is projected on 0, so is Bob's. And so Bob, when he measures it, will always get 0. Okay? So you can tell the difference between these two actions of Alice. If he sometimes gets 1, then Alice kept her Cupid outside. If he always gets 0, then Alice threw it in. So they could do this many times and actually communicate,

57:30 assuming they understood exactly what was going on inside the black hole, of course. They could actually communicate faster than life. And of course, there's no real reason that Bob has to be far away in a space-like separation. He could be sometime in Alice's past, so they could actually communicate backwards in time. So the problem is that once you allow these interactions, once the final state is not maximally tangled, you start to have this non-unitarium leak outside the black hole. And it can be due to all sorts of horrible things like that. Okay. So, to summarize, well, we've had a lot of discussion already. Black hole information, is it lost or not in evaporation? Is black hole evaporation, right? Black hole information, unitary or not? Black hole evaporation, unitary or not? And then there's this final state, black hole final state proposal by Harwitz and Rolacena, which seems like a good idea. the new physics to the singularity. But it does suffer some problems because it's very delicate. It should perturb the situation precisely if the final state is not an act for a tangent state, you get non-unitory evolution. In fact, a non-unitory evolution can lead to much more mundane situations. Thanks. Yet more questions. I think what you see is again the interaction in England there's all the parts of our arcs whatever you send in affects the arc going by the arc even if the fact is very weak at the horizon it gets blown up so no matter how we can if you compute the gravitational interaction even if it's very weak at the horizon it doesn't matter into the big effect. So that's, well, we have to calculate it. That's because the, if you follow the optimization back to the path, as you showed in the old lecture, it gets squeezed, right? It's squeezed onto the horizon. So conversely, if it gets out of the horizon, it stretches. So there's a minute gravitational effect which displaces the motion practically by a very small amount, the depth gets stretched when it gets out. So, if you throw nothing to the black hole, then you have to turn off the black hole,

1:00:00 you throw nothing in, the Hawking relation comes in a certain and parallel state. After only one single state, you get the Hawking line across the network and it goes in. Now it throws something into the black hole, no matter how easy and light it is, it puts the Hawking relation into a different state. So now, if you consider a whole set of states the Hawking Pythagoras are in, you get a complete set of all possible states, provided you throw in Pythagoras in a complete set of possible morphs. That way, the ingoing Pythagoras can affect the outdoor morphine radiation in a one-to-one manner. And if you encounter this, you get quite a reasonable unitary amplitude. The only thing is that it gets higher if you want to include the transverse motion and which is the reason why one doesn't get the black hole and at least you do answer some of these questions. Yeah, so this, I guess, is another example of, as I said, you can't really rule out anything if it's possible. Basically, all of these explanations have components that will explain what disadvantage is that I've given you. So, originally, I wrote here, in fact, we don't know. And then I crossed it out and decided I couldn't do that people out here will tell you that they do know. But they all have different answers. So I feel very safe in saying there's no consensus on what the solution is here. I should note that Chris Adani recently at least close to the paper, stipulated the emission of use in the constant radiation from the in-going matter. But he didn't get all the information going out, only some of it. And so then you still are left with this particular problem of what happens with this stuff that goes in. Let's hear from someone who we haven't heard from before, Adrian. Just a comment about the simple little singling. Sorry, the what? The simple little singling. Okay. See, this comes up in general, you know, past and future, and you might think that it leads to a terrible grandfather paradox. But I believe there's a loophole in that argument, which is good. whenever you try and set up a paradox, unless you have a good decision in the license, in the second-hand license, it's your paradox. So you can't explain the way around it. No, that's completely correct. The rule that gives you the final state projection

1:02:30 tells you how to resolve all these paradoxes. I don't want to say too much about that because I think Charlie's going to talk about that tomorrow. Yes, but not much more than you've done today. So feel free. No, but yes, that's true. The main point I want to make is that physics outside the black hole is behaving very differently than you would expect, which was trying to avoid it. So this is what I think that happens. I mean, the one that's outside doesn't know into which state the black hole projects the state. So he has to average over all possible final states, and then he gets just terrible. Well, no, I mean, so you have to distinguish between he doesn't know because of some fundamental reason, or he doesn't know just because he doesn't know enough about the black hole. In this proposal, if he knew enough about the black hole, he would know. Yeah, but there's no way you could ever know about it. I mean, I can always make an experiment and say, oh, now I know, now I need it. every time I try to probe it and to see into what the final state of life exists, every time I get to determine what it is, it's not. No, no, no, that's not right. In this proposal, again, this is not standard physics, this is new physics. In this proposal, you could, in principle, even calculate what that unitary transformation is based on first principles. In principle, obviously, in practice, that would be a very, very difficult thing to do. I'll discuss it. Charlie? This negative energy Hawking radiation is invisible to somebody who is freely falling towards the singularity. So under what conditions can there be some interaction of the kind that spoils the Horwitz and Maltusena? if I am on a rocket and I'm accelerating away trying to prolong my life while my rocket fuel is living a shorter life then I'll feel this stuff and I could interact with it but if I were to just really fall I wouldn't care about it because I wouldn't see it it's probably not exactly invisible even if you're freely flying you'll still have small interactions with it you can detect vacuum fluctuations For a freely falling observer, they just look like ordinary black static fluctuations, and those are detectable coordinates. Okay, may I just note, as the chair I pulled this off,

1:05:00 so I want to look at this question of how much information gets lost, and you find that for typical interactions between the stuff that falls in, just assume some random interaction, that almost all but half a bit of the information will escape. So you get a slight deviation from unitarity, be very tiny, in particular for a large black ball, which could have, you know, 10 to the 180 bits in it, or much more. And losing half a bit seems like a rather small price to pay. Not from the perimeter institute, so-called because it guards the dangerous perimeter of the United States of America. Our next speaker is, I'm sorry, your first name is Christoph. Christoph Bellard, who will be talking from CMS, who will be talking about the pathological approach to the black hole information problem. Christoph, maybe wait for just a minute because I can just jump in. I'm sorry? Go ahead, go ahead. Go ahead and do another shift. I think Hawking's here. Oh, good, good point. Let me see if he's coming in. I think he just came. Yeah. Roll on. Sorry, it is very difficult to hear you. Could you use the mic? Better? Very slightly, yeah. Maybe lift the microphone. I like a bit. Like that? Can you hear me? Yeah? Yeah, a little bit. Thanks. All right, so I'll do it old-fashioned. I'll give you the plan of the course. There it is. Well, I'll first start to introduce what are asymptotic gradient spacetimes. Not all of them. I'll take two. And I'll try to describe them a bit. Then I'll introduce what bat integrals mean,

1:07:30 the fields in the first place, and then its generalization . We'll get down to the semi-classical approximation and see what we can say about it and try to get beyond this semi-classical approximation. I'll give a list of references at the end where all these matters can be. Thanks. OK, so what are asymptotically antidecitor space signs? Antidecitor space is one of them. So let's see what it is here. Antidex-terospace is a solution of Einstein equation with negative and can be seen in many ways, one of which is as a hyperbola in one higher dimension with a special metric. But we don't need all this now. We just need to know that there are many ways to describe antidex-terospace and And we can put some metrics on these spaces. These two could cover the whole of the hyperboloid. We could have it as a hyperboloid. And this one is just given as an example of a metric that doesn't cover the whole of the super-supers. In this coordinate system, L is a distance that comes in with an ADS. It's the ADS length. And in this coordinate system, we can see that rho goes from 0 to pi over 2. That is, that if we forget about this factor here, this metric could be continued over pi over 2. Actually, this part of the metric, this one which is here, conformally related to that one, is . And we can see, if we stay within that part, that the boundary at infinity for the rest of the talk I'll include that I won't talk much about that this is the metric I'm going to do But most for the rest of it.

1:10:00 This is a metric that works, the mass term that is here. It's not exactly the mass, it's related to the actual mass of the black hole seen from infinity by the volume factor. But let's keep it in this one with n called the mass. All right, so let's short that idea. We'll see later that when r is big, the only theory that counts here is the r square over b square, and here is b square over r square, that is a minus. Which is exactly the same as happens here. That's what we're going to call . Okay, so let's have some information about anti-decedor space. For instance, gravitational potentials. So let's split the Klein-Gorgen field in the . That's the Klein-Gorgen equation. The field is max s for a . There is also d minus 1 symmetry in radius, which we can factor out, plus some other factorization equation to become a particular equation for this field five, given in a particular, for this radial factor, given in a particular radial coordinate, which is called the Willard-de-Witt coordinate, or tortes. This yL that is here is the SOD minus 1 symmetry. It's the the parameters on the T minus 1's field. We do it like that, and get an effective potential for the field that looks like this. R star is our radial coordinate. It starts at 0, which corresponds to R equals 0 in ADF space. And it goes all the way to R infinity, which is not plus infinity. It's a finite distance. It's a finite parameter. The potential, you can see, blows up at our infinity. Basically, that means that ATS acts like a gravitational shock. Anything that's in there will not be able to go too far away. All the things that do get far away, when the potential is not like that,

1:12:30 could depend on the mass. For instance, light could get to infinity. I didn't give too much detail about that. Let's say for everything that goes to infinity, we impose a boundary condition at infinity here. So there's no energy . We want to do some quantization in this space. We don't want some obvious energy going out of our space to continue. So it's reflected back in. But for most of the fields, massive fields, this would be reflected back anyway because of the gravitation of the tension. So the alien space adds a bit like a gravitational box. Now, if we look at Schwarzschild's aliens, we reflect the same instructions. Instead of, you can look at master skills in there. It's only the same f, of course. But we can do the same construction. We find another purpose coordinate with this new factor in here, which will make a difference for the domain of definition of r star. R star, the perfect coordinate, goes to minus infinity at the variety, that's where we saw. The attack is potentially in this time, in this case, it looks like that. With the variety being the variety infinity in that direction, it's still being asymptotic infinity, as before. Now, this is quite interesting, because if we are in a black hole in there, there are lots of problems. Because black holes won't be able to be in equilibrium in there. As long as they radiates, either the radiation will increase, or if matter begins to fall in, the equilibrium will not be possible. So the best way to say that is that the specific heat of black holes in asymptotic class phase is negative. You need to put the black hole in the box in terms of thermal properties. In AES, you don't need that to show this potential.

1:15:00 So if ever something wants to get out of the black hole, which we don't know yet. If something was to get out of the black hole, it could be reflected back at . And equilibrium is possible in the whole space. Something does come out of the black hole. It's called a working radiation. That can be reflected back from . in the orbit boundary condition appears as a dual radiation, and it actually is a fact that the canonical ensemble can be defined in such a space-time, which is made yet. Not for all black holes, black holes need to be big enough for this to happen, for these things to happen. But you'd still consider this as a technical trick, which you need, I mean... It's not a technical trick, but it is the idea of space that has its property. No, I do understand, but do you believe that real space has its property? If I believe we are in idea of space? Yes, that's all right. That's the question. Let's put it that way, that observations tend to change every 15 years, so maybe 15 years will be in idea of space again. No asking religious questions, please. It was a technical question. For the moment, observation stands for the other sign for the cosmological council, but it's so small. But I mean, you don't intend, say, to take lambda equal to zero after you've done the calculation. Yeah, we can talk about that. The whole point of having this cosmological constant here, putting aside all of these considerations, is that these ensembles are well-defined in these cases. So I'll kind of be able to make something clear. The black hole has to be in the order of the radius of curvature of the antithesis. Bigger than that. Bigger than that. We'll get back to that. So for a small line that has to be really huge. For what? It has to be really huge. It's true. I'm taking one case. All right? It's more, it's actually more precise. For what I want to do, it doesn't matter. You will see that .

1:17:30 So basically, yes. What is important in that example is that the canonical ensemble is going to be well-defined. Just to give you a hint, once the canonical, if micro-canonical ensemble, for instance, it's very fine, we can have all the temperatures possible with a distribution in temperature. And this temperature, large temperatures, or large beta, or large energy black hole will happen with maybe a small water dissipation, and we will get back to that. OK, so what to say again. in this case that thermodynamic considerations will be useful. Now, thermodynamic considerations can be looked at from the Euclidean point. What does that mean? And I'm sure most of you know about that, but I'll just review it. If we want to see the correlation function of phi1 taken, phi1 is, say, the field at x1 and the field at x2, with the evolution operator that brings that field, which is at t1, to that field, which is at t2, we look at this correlation function. Now, if we take this little trick, we change the time, the imaginary time, This correlation function, written like that, looks like this. H here is the Hamiltonian. That's something that works in a unit here. That's the evolution of quantum . Now, here, this, we can do some tricks about this object. For instance, we can take the trace over a basis for the Hamiltonian. And when we do that, when we take the trace, we get this thing, where bn is the energy for the phi m eigenvalue of the Hamiltonian. But this thing turns out to be the partition function for the field theory of temperature beta. That's the canonical ensemble. In thermodynamics, from these quantities, any thermodynamic quantity can be calculated, all right? Now, we've just seen a bit before that the canonical picture for Schwarz-Scharnin's anti-decyther actually works.

1:20:00 I didn't prove to you that it works, but it does. So let's have a look at all this from a Euclidean point there, and see what we can say about this. Right, so the two space-time that I introduced before were Schwarzschand Antideciter and Antideciter. Let's look at what they give us in Euclidean. So that's called a reprotation. So we reprotate the time to imaginary values. The metric for Schwarzschand and Deciter becomes this thing. It's only pluses here. It's, let's say, a positive, plus, plus, plus, plus signature. And the Euclidean section will be defined when the signature stays all plus. And that's only when R is bigger than R plus, which is the first zero of this function. When that becomes negative, we're not in the Euclidean section anymore. So this is defined only for R bigger than R plus. OK, when one does that, one realizes that actually this tab, this imaginary time, this imaginary time, it's called like that, has to be seen as an angular variable. And not only as an angular variable, but it has a fixed periodicity if we want the Euclidean space to be complete. As all angular variables, if it stops before the 2 pi rotation, there is a wedge missing in your space. So for no wedge to be missing, one has to identify the period with a certain value, v prime of r, v being this term, which is here. So the periodicity depends on the mass of the back vortex. Imposing that periodicity in imaginary time is called a KMS condition in thermal theory. And one can show that assuming a Euclidean periodicity in time is exactly the same as having a field theory, or a theory that's a quantum theory, at finite temperature, the temperature being given by the inverse of that beta.

1:22:30 And that, indeed, works, because when one calculates that, one is the inverse of . So we're getting some. Now, for Euclidean . We do the same reproduction. The metric is like that, only plus is everywhere. This time, this doesn't go negative anywhere. This is always positive, that as well. So r is bigger than 0, and there is absolutely no constraint per ADCC on the imaginary time tower. So if we want a thermal theory, we need to identify the period of time in a certain given period. That here's a yard theory. You decide what thermal, what temperature you want to put in your theory. That's still the KMS condition, OK? Now that we have our last, we can no back-to-back integrations. That was some sort of introduction for the spaces in which we're going to do some back-to-back integrations. It's like a question. If you had children at a different time, say proper time, would you have gotten a similar condition? Let's say I'm using this metric. I know, but let's say you use another metric, another notion of time, say proper time. Okay, so I'll say it differently instead of answering like that. In what I'm going you can use metrics that are asymptotically ADS, OK? And there is a way that you can show that saying asymptotically ADS means that you get rid of this one and this one here, all right? And the metric is asymptotically ADS, roughly, if it looks like that in some coordinate system, OK? So that's what I'm going to use from now. and we're in the Euclidean space here proper time considerations well you could have started with a different Lorenzian time of course and then try to continue I'll use that for them alright so I'm going to try and answer your question I hope we think what it was For quantum information people, I introduced Alice and Bob.

1:25:00 So, what's the point of path integration? The point is that when there are two paths that are possible, leading to the same place, then there is no reason why both paths shouldn't be taken into account. So that's the superposition principle of quantum mechanics, that there are different ways for a particular process to happen, and the amplitude for the process to happen is the coherent sum of the amplitude for each ways for the process to happen. So how can we demand that in principle? What paths are possible? Which way to give them? the whole point of the path integration approach to quantum field theory, that's it. That all paths should be taken into account, simply all of them. And the weight for each path is something to be defined. And one way to define it is to remember that semi-classically, well, classically, when you don't have quantum effects you would like your theory to reduce to a classical field. So, that's a parallel, all right? Here, we have all the fields, all the configuration space, and we're putting a measure on them in this space of fields, in the configuration space. What measure to put? We can talk about that later. This part here is what weight each part is going to be given in that sum. And s, which later on will become i, this s is the action. All right? You have to remember, actually I didn't write it down, but there is a 1 over h bar in here. And since there is a 1 over h bar, when we want the classical limit, we take h bar goes to 0. And this oscillates very, very fast. The only paths that do not be killed by these weights, by these factors, are the ones that are stationary. That's a big heuristic argument, but in quantum mechanics, actually, it works quite well, all right? So that's for the path integration.

1:27:30 Now, first, there are some problems with this integral. So first, this classical limit I told you, in the Laurentian space, in a classical mechanism, let's say in the two-slit experiment, we know that there are two paths, let's say, classically, that the particle could take. But in general, these classical solutions have to be solutions of hyperbolic equations. Maybe we can boundary conditions. We don't know if there are solutions. If there are solutions that don't depend continuously on the boundary conditions, it's not that way to find. But, in some cases, they can take. Second, and maybe even more important, that is oscillates. So it is absolutely not to know whether it converges or not, from a practical point. Now, if you remember that in quantum field theory, but I might introduce an integral point of view on correlation functions. You have to introduce an I epsilon prescription for the integral to converge and not to have time ordering problems. OK, so that I epsilon prescription is kind of equivalent to introducing a complex time in the final point. Your time direction, which was real, you open it up to the whole complex plane. Your epsilon is real, so you actually rotate by phi over 2, like k. That's your real line, and that's your complex plane. So you recrotect that way. That's what we do here. integration, which looks like that, now looks like this, which feels defined in the equilibrium space. And this is looking better, much better, because paths that are away from classical paths are going to be exponentially suppressed. Now, this actually can be made to look like a very nice Gaussian form if the action is quadratic in a field. We can integrate by part or a thing like that. It can be made to look good. And the differential operators in Euclidean theory are like elliptics, boundary conditions in the defined solutions. It's a well-defined problem .

1:30:00 Another very nice argument in favor of this approach is that once we rotate back the other way around, like that, you get the time-order coronation functions. Finally, correct, automatic. Now, can we generalize that to quantum gravity? It would be nice. So in quantum gravity, the boundary conditions that we have are not . What was important, actually, in the path integration is to define what fields to integrate over. Which are they? If you take no field at all, if you take zero measure, you don't have a theory. If you take all fields with all different boundary conditions possible, you won't get a theory either. What you want is something that describes what you see. So you would like to have some boundary conditions. And the boundary conditions, which can be total as some definition of what fields to be specified, actually a part of the back integral point of view. So now, if we want to introduce gravity as well, these are where the boundary conditions have to be proved. We start here on the hyper-surface, and we want to have another hyper-surface here. What we know is the induced metric on that hyper-surface fields that live on it. Here, the same. The path integral says that you get from here to here and the correction function, we have to take into account absolutely anything that happens in the middle. What happens, I don't know. But all the spaces that are in the middle. There we go. That is the partition function once rotated in the video use to be able to define the theory we're looking at. Which way, that defines the boundary conditions given here? What ways? The semi-classical limit tells us that the ways is the action, if we want to generalize it that way. As soon as we're going to view the information, where we're going very soon, the action, which is here,

1:32:30 as a part for the matrix, as a part for the cube. That's the Einstein-Liberg part of the action, and we have to add the macro action, depending on the theory one is working at. We get a generating functional, then j. Just add a source j in the action, OK, so that's the definition of the path integral approach to constant grads. Now, what can we do with this? It looks a bit hard. And if you expect me to do that integration, you're going to be pretty disappointed. So the first thing to do is check the semi-classical approximation. And for the semi-captical approximation, whatever approximation we're going to make, the first thing we need to do is to impose the wrong recognition we're going to look at. And I'm going to put as boundary conditions that our spaces are syntactically antithesis, which can be said as dimension in some coordinate system looks like that at R very, very large. Okay, and in order to have the platform inside as well and to look about the thermodynamics and what can happen, we're going to impose periodicity for the imaginary time. Okay, so that we know that the two metrics we've been looking at before, that's lucky, they correspond to these requirements. The thermal ADS, which is that one, which we saw before, and the fraction ADS, which is that one, both obey this boundary condition. It's going to minus 1, but the PR is true. Yes, sorry. Sorry about that. Now, we saw earlier that for partial ADS, the periodicity is imposed by this factor and depends on n. Whereas for thermal ADS, the periodicity in time could be . But at infinity, we want those two to be the same boundary condition. So that we're going to take the periodicity in the imaginary time,

1:35:00 in the eukedian time, for the thermal ADS space as the one for Schwarzian A. Fixed. That's one of our boundary conditions. Now, in the semi-classical approximation, we're going to check which one of those two metrics has the highest weights in the sum of our path integral. And we're going to stick to that. That's the semi-classical approximation. That's the leading contribution. And that was done by Rooking and Ledge a long time ago. And they did the calculation to check the difference between the interactions between those bases. So there we go. That's the result. R plus is the radius at which we equal to the g t t coefficient in the metric goes to zero. That's the actually that's the length rise. The temperature is given by that, and we have the mass, which is given by that. They classified what we can get according to the temperature, and they saw that we have four cases. And as we increase the temperature, the black hole gets more and more likely to happen. When the temperature is very small, then thermal radiation, there is absolutely no reason for it to collapse. As we increase the temperature, they found out that two black holes can have the given temperature, the given mass. Well, two black holes can be in equilibrium with the temperature, but one is stable and one is not. Anyway, once we reach high temperatures, there is no equilibrium without the black hole. The black hole has to fall. And this depends on the radius of the radius. So that's kind of the proof of what I was saying before that in ABS, because we have the boundary conditions at infinity that make matter that go out, come back in, this gravitational box, black holes can be stable. So no equilibrium without a black hole, black holes can be stable with thermal radiation there.

1:37:30 They're called eternal black holes. From now on, and let's otherwise specify, I only take into account the B temperature, when the black hole is favored. OK, this picture is in the canonical ensemble. That picture you can only have in a symbolic idea for the reason I gave, but you can also have the micro canonical ensemble where you fix the energy of your space rather than the temperature. OK, now from the Euclidean path integral formulation, we know that sub-leading contributions to the path integration are exponentially suppressed. Okay, that's here. Okay, so semi-classically, if we want to have a look at correlation functions, we just pick the leading contribution. Large black of large temperatures, the leading contribution is the partial ABS. And one can calculate what these correlation functions in ABS. Yes, A here is the Laplace box, OK? Let's take a very simple example. Trotia is a 2 plus 1 dimension. One can calculate with Gibbs, and the correlation function actually decays exponentially with the difference of time, q1, and q2. These are, sorry. This calculation has been done in Euclidean. You recreate, you get the Laurentian two-point function, which does decay. In comparison, from 2, it's going to be 1 over beta. And that gives you the time in which it decays. So when one looks at that theory, then I see that function. That actually goes to infinity. This goes to 0. It decays all the way to 0. Now, one can calculate exactly the same correlation function background, and we'll see that this is periodic in delta t. So even though this, even though the Schwarzschild space sign is the leading contribution after a long time, ah, since this goes to zero, some other contributions should be taken into account.

1:40:00 So let's have a look at beyond the semi-classical approximation and some over instantons. So instead of approximating the whole Euclidean-backed integration with just one solution, we're going to take, let's say, the two ones we found, all right? Which are the two classical solutions, thermal ideas and partial ideas. Those are weighted by the action, and there is a normalization factor. For large mass, this term is much bigger than this one. So we can forget about that, and that looks like this. We will recortate that back to Lorentzian spaces. And we see that the two correlation functions for an observer at infinity, I'll specify that in a second again, looks like that. Now, at small x times, this is the leading contribution. But since that goes to zero, the contribution at large times is coming from the alias space, but there are late. The fact is that we have a late time recovery of information. It looks like information that an observer at the beginning gets comes at late times from another instance. So it's not from the leading one, but from the sub-leading one. So that's what we are going to look at. Does that mean that unitarity is recovered by remembering that in the path integration, we should sum over topology? Well, here is an argument that's been made. I'm not going to get too much details about that. It will take us too far. But within the path of the direction. Sorry, I just missed what the first term was. This was just short-share or it was black hole in the US? Throughout the talk, I'm not talking about asymptotically flat spaces. Okay, so it's always asymptotically. Yes, okay. And so the first, I just can't see what the first and the second term was. I'll write bigger next time. Yeah, so it's all ideas. All the things end by ideas. The first letter says either S for Schwarzschild or T for Thermal, OK?

1:42:30 OK, so this can be done through a, OK. The best check one has so far, I think, for the unitarity to check if this is unitary or not, is using a holographic conjecture between ABS and the conformal field theory. So this I'm not going to get too much into, but I'm just going to say that those two are conjectures to be dual theories. If one has a process in this area, one can have the equivalent process in that one. And it just turns out that those two don't happen at the same regime, so that when one comes to calculations here, one can here, and vice versa. That's the strong conjecture of . Now, the point is that in this space, quantum-pointary recurrences happen. because the CFD leads on the finite volume at the bound. And Poincare recurrences can be checked as unitality, and one can check what kind of Poincare recurrence one gets by the sum of our instances that we used before. And there is a mismatch, just telling you, but there is one. There is a mismatch, but, as we've seen, something does come back. zero we get from , some information comes back from the other instantons, OK? So the question actually is, did we really expect to get unitarity back from the sum of our instantons? So let's summarize what we've done so far. Usually, in path integration, one starts with some sort of classical picture, which are the paths. Some paths are classical, so that you can imagine what those are, like in the two-slip experiment, where you can see particles going everywhere under two paths that are flavored. And that defines the quantum field. So this, for instance, when you fix the background, you find the quantum field theory with a fixed background.

1:45:00 defines quantum gravity, okay? Now to calculate correlation functions, what we did so far was to do, from the full quantum gravity correlation function, we approximated that, we, sorry, first we rotated it to Euclidean, where it makes both sense, then we approximated with the sum of our instantons, and we could, we rotate back to get the correlation function, Lorentz. This sum, I'll write it like that. One part comes from non-singular matrix, and one part of non-singular space signs, I'll say, and one part comes from singular space signs. We can specify. There may be just one in each. It doesn't matter. I'll just specify. And once we have that, we can ask, what does an observer that is at infinity look, see, what does one, what can one measure, okay? So this brown line here is the asymptotic boundary of ADS spaces. That's the boundary condition of all the fields that we're integrating over in the path integration. That's my impression. We observe that infinity do not have a clue what space space lines are in here. They don't know, OK? That's their boundary condition. Since they don't know, they have to take into account all the possibilities that are in here. That's what this means. You know the boundary conditions. To get your correlation function, this is really schematic. The delta t doesn't really have this direction here. It's really a schematic, OK? So we're going to integrate over all the space lines And I know I've fixed the time, so the temperature so far,

1:47:30 but equally good to look at the hybrid canonical ensemble for what I'm going to say now. So to look at large energy, that means that the time will be favored again in the practicing function. If you look at that, this can be approximated in the instant-on approximation at this time. So what do measurements between X1 and X2 give for the scale of use in this equation? The delta is small. The non-singular spaces do not contribute much compared to that Remember, the weights are proportional to exponential minus positions in action, okay? So the singular space signs are the ones who are going to have the highest weights. Again, there may be just one that contributes more than all the others, but I'll keep it written on that, okay? And in here, for instance, in here, would be Schwarz-Charles-Idias. Maybe other things, I don't know what, but at least Schwarz-Idias, which will make all these look small, OK? This goes to here. Since the ,, they will get this as their leading fraction function as small . That means that inside these spaces, for them, it looks like there is a black hole, even though it's an average, even though it's calculated by summing over geometrism, or what It turns out that a small delta is just a semi-carsical approximation coming into a gap. And we get a black hole inside. Now let's have a look at later times. Later time. This goes to zero in these spaces, and this oscillates. At least the ADS1, we know it as in C. So these do not contribute anymore. And the observers here, between x1 and i2, see that their correlation function as in C. So that's not possible in the black hole space time. It looks like there is no black hole in the average

1:50:00 of our geometries or whatever. In the approximation, it looks like there is no black hole anymore. Now, the question is, once you've seen all that, is there really a black hole in there at any time? Because if there was one, even though the correlation function goes to zero, one can ask the question, two black hole spaces carry information in the sun. Does the path that information takes within singular space carry information? If yes, then that means that in this instanton approximation at delta t goes to infinity, information is lost. This is gone. It is not linear. If no, So why consider brackets at all in the path integration? We don't carry information. We don't need to include them in the path integration. So that is the question that we raise. In order not to get, because this is all approximations to quantum gravity, because the path integral that we're starting from is supposed to be quantum gravity. There is the ADS-CFT correspondence that makes us believe that this is unitary. Otherwise, if we look at this instant approximation, we see that this is not unitary in the case black holes carry information. So I consider, if we still want a unitary theory, let's forget about black holes. And in our path integration, some only over a non-singular space now. If we do that, then we choose what I want to say. If we do that, then let's look at what they see. They see for a small time a black hole in there, but they find no black hole. So the question could be, is there some way to see that all the way through, that does such a process happen? Can we see it won't be in general relativity, because that we know doesn't happen.

1:52:30 So it has to include quantum effects. It has to include some quantum effects in it, because starting with a black hole cannot lead to something that opens up. So that's still an open question, and that's work that is in progress. And the question is, for instance, is there a rebound that is possible? Okay? Some indications that there may be something that is going on there. This is not the full resolution of the whole pattern direction. This is some approximation to it that, again, but I can't believe that, is an example of of spaces that are not semi-classical, and which contributes to the idea that no black hole can afford it. No black hole should be included in the back of the equation. OK, so we take the gravitation of the collapse, which we can follow through, and stay with the outer bounds of the collapse in the shell. If it's a collapse in the shell, we're looking at part of the higher side of the collapse. At some stage, there are regions that will be of high curvature. At these stages, quantum effects can't be ignored. And that's what was used in the trace anomaly with an inflation, for instance. As soon as there are large curvature, quantum effects cannot be ignored. It has to be taken into account and can lead to a repulsive term in the equation, which help bring the collapse to a rebound, to an expanding. calculations are a bit difficult to generalize exactly like that. But again, from a holographic brain world picture, we can see that it looks like there is, this is the potential for a self-gravitating, say, self-gravitating collapsing Oppenheimer-Snyder scenario on the red. We don't need those two minutes there. effects can bring a minimum in a year, which would mean that there would be a rebound. This is work in progress, and nothing to discuss about that. I just wanted to mention that as an example to see how no black hole formation

1:55:00 could be seen from observers at infinity using one particular non-classical solution. So let me, before I do the reference, I wanted to thank my boss for his work on such subjects, which are quite fun. I'm Christiane with her manier with a last inside of my eyes. So here are the references that I use. Well, thanks. Here are the references that I use. If you need more information about them, please. That's a good one. Okay. Like that. Here's the . And just one single funny thing that I saw on a short point in the connection, people would be happy to see that. That was the boundary at infinity of our space time. And when you really close me with a telescope, you can see that Alice and Bob were there. Thank you. Questions for Christoph? So I'm a bit confused, I mean in the earlier part you seem to be doing these thermal calculations and the result that you're getting basically the way I would interpret it would be to say that the probability of having a black hole in a thermal equilibrium is small. It is not small if the energy is large. Even in the micro-dominant... The subdominant, those things are subdominant. The black holes are subdominant. They're not. They're not. There are not in all regimes. In high energies or at the temperature that I was talking about, they can become dominant. So I'm using the regime in which there is a black hole so that at least the most likely semi-classical approximation gives a black hole looks even stronger. If the backboard only has a remote probability to come in, all this would still apply, because if there is information going into it, it would contribute a little bit in the beginning, and not at all at the end.

1:57:30 So, basically the idea is that the backboard semi-classical effects, that they appear as instantons in the semi-classical approximation. in the full continuity theory, there shouldn't be that. It's kind of coherent. You could say that the snake eats its tail, but you can stay in this argument. Is a preprint of this work going to be available soon? I hope so, yes. Can you make predictions as to how soon? It depends on the path, I guess. Okay. Jonathan? you can apply it, I mean you read it, you might be able to apply this argument to say Schwarzschild where it would seem to know whether you have a black hole in there or not. The point of what I was doing all the time in this talk is that the observers are at infinity. So whether you're in Schwarzschild or not, they don't know if there is a black hole in it. It's like in the two-slit experiment. If you put an observer at the slit, you know the particle one there, you won't guess. And then you don't need to say that the particle goes back, it doesn't, it just goes back. So in a non-equilibrium situation where you think it's just going to be a really good population, like the good word of black hole, you shouldn't sum over all space time, you should just keep one, you turn that off. Whatever black hole you start with, it looks like something at infinity, whatever it is. If you start with a spacetime, which has a boundary at infinity, you find a way to define this. Pick one space, pick one process in there, that will be the mark at infinity. Don't forget what happened inside, keep these boundary conditions, okay? It's not sure at all that there is just one solution that fits in these boundary solutions. But the point is to sum all the possibilities, all the solutions we can. Actually, it's not just solutions. Solutions of equations of motions and Einstein equations only happen at the semi-classical level. Just like in time and path integration, you have to take into account all paths, however where they look.

2:00:00 And if my boundary is, if on my boundary has popping radiation, are there solutions which are trivales? Sorry, say again? If my boundary has popping radiation, then it's like, are there? That's what happened with an ADS, the same boundary condition, where it's in both things. So, because you're doing an instanton approximation, you're approximating the partition function. What guarantees do you have that this approximation doesn't make the rip-rotated back into real-time propagated non-unitary. I just showed that I guess in general, so to reveal my ignorance of quantum field theory, suppose I have a theory that is unitary and then I look at the stationary points of the action and then I do this approximation. Do I still get a unitary theory? Basically what you do is that when you sum over all paths, roughly it's equivalent to having a filter, a wave. If you forget about sun, then you approximate your theory. So it should be approximately . Approximately. that's right. That's a good question. It's quite hard to see straight how information loss can occur in this space. So there are different ways to do it. One is to look at IDS-CFT again, and you can compare what you get from both sides, and you see that in the black hole space, and you don't get these unitary bounds that you get in the quantum theory, which we know is the quantum theory. So there is a mismatch, OK? So to get this information back, you need to set over in sentence that was the next approximation, and I just showed that it's not enough. Now, for cooking evaporation, that's one of the first

2:02:30 There is cooking evaporation. That's what allows us to introduce this thermal ensemble. It is eternal, yes, because it is in equilibrium with a box. It's in equilibrium with third gradation, that's why it just really works. I'm a little confused, I mean, you're talking about high temperatures, but at high temperatures I assume it's a small black hole, and those are unstable, okay? They either roll or shrink away. I'm not using the same. I mean, with high temperature, the black hole has to be very small, much smaller than the radius of curvature of the space-time. I thought it was the big black holes and the very low temperature ones that were stable. In which case are you for us to see. Let's see the, everyone can talk about it in a minute, but I, I think that's, that's right, these are, okay, I can check out, but that, that, that, that's correct, I'm, I'm, I'm in that text, you can see here, please, please, please, please, please, please, please, please, please, please, please, please, please, please, please,

2:05:00 What gives you confidence that this object you call the partition function exists beyond this instanton approximation? Sorry. What gives you confidence that the partition function you write down actually exists in any even hand-waving sense beyond your instanton approximation? Yeah, so for that, this rebound I'm talking about at the end would be if or when it is found and if the insuring works, that would be a good example of that. There could be a less unitarity as well. I don't have any explicit check that it works here. So you're relating this correlation function at infinity to the information loss for information But what you get for infinite times must be extremely tiny since you have infinite minus It's zero, as it's . I mean, for the pool, for the pool, but no, it doesn't matter. What comes from the ideas of space, that's . But with the sub-down, it's very, very small. It's tiny, what you get for . Yes, but that's why it's not enough to get genitality back. Because that's . let's see it's usually put in a very strong terms that if unitarity is really true presumably the original calculations of Hawking and the semi-classical calculation have to be seriously misleading us about the causal structure of a large black hole in what way is it seriously misleading us maybe you can is a semi-classical approximation, OK? So that within semi-classical approximation, there's no way you can get out of the hockering operation. Of the, of the, there's no way.

2:07:30 We need a new temporizer. Let's say you get a new temporizer. If you want to go beyond that, you will find out that there is no singularity. No, no, no, that's fine. But somehow the whole causal structure of space-time be seriously misleading, presumably, in the . Well, yes, in a way. But to have a similarity or not to have one makes quite a difference as well. Maybe that's a quantum theory of gravity. That's my point of view. Probably a few other people as well shouldn't have similarities. No, but the problem is occurring just Yeah, but if you don't have a similarity, well, in classical general relativity, I mean that variety forms, then you will get a similarity. To include quantum effects, you should be out of these. If by quantum effects, you kind of violate the assumptions that were made in these for similarity formation, then you can get out of it. Does that answer your question? So despite the fact that it's a massive black hole, then the coptic- Even just to say that is a semi-classical approximation. You're within it. As soon as you have a black hole with a similarity, you are in a semi-classical approximation. don't ask the question well if I have some compact object I don't know I don't want to give it a name now I'll be told that I'm looking in a semi-classical I just have some compact object out there and I throw refrigerators and camels and carrots and things into it and you're going to tell me that it's okay, that the causal structure described by the, you know, that I'm thinking I'm getting and sort of classical general relativity

2:10:00 is actually not a problem. What I'm thinking when I think that there's going to be some sort of event horizon is that I'm just completely wrong. Yeah, I think so. Why I don't think even horizons will occur. Right. I think that first, OK, it's a bit theological now. First of all, well, we'll talk about that for a long time, But even varieties are defined retrospectively from . If you have one, you should include it in your . That's a bad trick. But, well, OK. But let's say I'll semi-classically and say that You know, your calculation specifically does not look at the region where these black holes are actually evaporating. So it's actually rather hard to look at exactly that. This is to assess the problem with information. We look at one place, which is the boundary at infinity. We're staying there, and we're saying, what can happen in the back? And that's it. What you've shown so far is that information is lost. In the semi-classical approximation. But from what you've done so far, you've found that information is lost. In the semi-classical approach. That's all you've done. The rest of it is straight. No, but semi-classical approach. Yes, in that sense, that's correct. In an instant on approximation to the path integration, you get what seems to be the information of the structure within that framework. unless there are any more questions

2:12:30 that seems to be a good place to Steven do you have a question sorry yes so the organizers have asked me have pointed out that now it's time for the conference picture so before heading off to lunch could you just head out onto the grass out there and I'll say are you doing the honors today Thank you. Thank you.