Discussion Session with Don Page, Gia Dvali, Thomas Hartmann and Steve Giddings

I'd like to ask a question to Thomas. Are we sure that we need to include all possible subgroups in the path integral? If you have an integral and you want to evaluate it by a saddle point, there's a procedure that you go through and you can determine which ones contribute. But in the semi-classical approximation to gravity, I don't think we have that. And I think it's even plausible that it depends on the UV completion which saddles... I think it's plausible that different UV completions could include different saddles. I see. So what you showed is that if UV completion says that we need to include all possible saddles, then the unitarity is recovered, but the kind of non-local interaction is required. And on the other hand, if the UV completion tells us that we need to include only Hawking saddles, then the locality is preserved, but the unitarity is broken. Is that what you think? Well, I don't know if the second one is a viable option or not, just because of the information. But that's just the usual information paradox. But yes. And we don't know which is the correct answer from string theory. Well, in examples that we understand in string theory, we know that the curve is unitary, and black holes do behave like an ordinary quantum system. Yeah, yeah, yeah. These are very nice. So the wormholes do have to be included. We want to believe that that is correct. But from the first principle, do we really know that we should include all saddles? If you mean from the point of view of the string theory path integral, then no. I see. Yeah. So it is just a guess. Yeah, I would say anytime we do semi-classical gravity, we try to do our best to do something reasonable and then see if it makes sense. Yeah, yeah. So the result is nice. I agree. And also, I think there is some... Yeah, so the real-time path integral, I forgot the name. Picard-Lefset formalism. And in this formalism, in some cases we know, we can, from first principle, know which saddle we should include. Did anyone try this formalism to compute the entropy to see which saddle we should include? I don't know the answer to that. Not that I know of. I see. If we can do that, then we probably, I mean, in a sense, the first principle answer, maybe. I suspect that requires a UV... I suspect you need a UV completion to implement the calculation that you're talking about. Maybe. Yeah, yeah. But at this moment, the result is just a conjecture. Is that right? Because we don't know which saddle we should include. Yeah, I agree with that. Yeah. I see. Thank you. I think of it as showing that it's consistent with unitarity, not proving that it's unitary. Yeah, so in a sense, maybe including all saddles is motivated by unitarity. And it is consistent with the unitary result. It does not prove the unitarity, but it's kind of showing consistency. Right. Maybe I can answer from a different angle a little bit, this question. Can you hear me? Yes, yes. Yes. So, in string theory, I mean, string theory is an S-matrix theory. So, it is defined through the S-matrix. And so, in string theory, black holes are intermediate S-matrix states. So, in string theory, black holes enter as intermediate states produced in the S-matrix collision of very high energy quanta. And, for example, you can produce black holes in two graviton collisions in string theory or two other string particle collisions, string oxidation collisions, if the energy is sufficiently high. And so, therefore, this is what I was trying to say also in the previous comment. To do string theory computation, you need to know what you are looking for, right? Because you are looking for a certain amplitude. But the amplitude is initial states and final states. There are no black holes in final states in string theory, unless you are producing some extreme black holes by accident. So, therefore, you have to know what is the final state that you are looking for, which would carry information about what black holes are and stuff like this. Of course, there is sort of a guesswork involved. In other words, string theory per se is not enough. You have to supplement string theory with your assumption, what black holes are in string theory. Now, we have done this computation, this is what I was referring to, in the theory in which the black holes are described as saturated states of soft gravitons. So, both in field theory and string theory, you can see that this computation of this particular kinematic regime is fully under control, actually. In string theory, you can compute it. And up to 1 over n corrections, 1 over s corrections, which is the same there. There, what you see is that 1 over s corrections are there to the leading-order result that you would interpret as semi-classical. Semi-classically, we can also say that black holes are formed because all the energy is put within the Schwarzschild radius and stuff. If you match that with string computation, we have shown this explicitly, we have seen this explicitly. There are 1 over s corrections to that. Now, there are also exponent minus s corrections, which you refer to, probably, that's what you're asking about saddles. But this is what I was trying to say, that 1 over s corrections in that language, obviously, they are much less important. So, 1 over s corrections are the ones that essentially give all the answers in this framework. By the way, they also show what Steven's assumption number 2. These 1 over s corrections, they violate this assumption number 2. So, therefore, what he was saying is consistent. It's just assumption number 2 is violated. Steven, you gave some tantalizing hints that maybe some of these non-local corrections could change the wave scattering. But is that necessary? I mean, is it still possible that you could have unitarity? And yet, the black holes, you know, at late times, even after half of the evaporation, that they look pretty much like ordinary black holes. In other words, if you don't, say you don't measure number of quanta, that's at least sort of half the total number coming out. Naively, I would expect that you basically don't see correlations between different modes. Anyway, in other words, I expect the semi-classical approximation pretty good, well, except for the Brownian motion of the black hole, to be pretty good over pretty much all the lifetime, and that the black holes should just scatter waves. Is that consistent with what you're saying? Or are you finding that there necessarily must be some sort of in principle measurable differences in the absorption and reflection of gravitational waves from a black hole? Well, let me try to unpack that. So, first of all, of course, the standard story of local quantum field theory on, say, a semi-classical back reaction, corrected geometry, you know, is not going to solve the problem, we think. Right. And so, you need, say, extra terms that correct that evolution, or represent corrections of that evolution. And so, I've tried to make a principled parameterization of those terms with a certain set of, I think, very general assumptions. And so, generically, those terms will contribute to, for example, corrections to graviton scattering amplitudes, gravitational wave scattering amplitudes off of black holes. Now, that's not necessarily a very drastic thing. And you can ask, you know, what about corrections to other observables? So, say, an astronaut falls into a black hole. How much does it correct what that astronaut sees? Well, you might have a typical momentum check to that, the atoms in that astronaut or whatever, and the typical momentum transfer is 1 over r. And so, let's think about that for a big black hole. Well, 1 over r momentum check for an atom falling into a big black hole is completely negative. So, these are pretty minor corrections. But, again, they can be significant for, say, things with wavelengths of order 1 over r, or, sorry, wavelengths of order r. The really important things we look at or can observe with wavelengths of order r are the gravitational wave signals. You know, for example, event horizon telescopes looking at millimeter wavelength radiation from a black hole bigger than the solar system or something. So, you know, there it's completely negligible. So, generically, you have those kinds of corrections. Whether you can cook up a scenario without those kinds of corrections is a good question. It may well be that you can find scenarios where these corrections are sort of, you know, even more minimal and not visible. But I think the interesting thing is that they could be there and we might be able to look for them. Right, right. No, it would be very interesting if you find it. But, I mean, if we didn't see it, I don't think that, that wouldn't necessarily violate your assumptions, is it? I mean, it's tantalizing that they might be there, but they don't have to be there. Yeah, this should be viewed as an opportunity and it's not yet a situation where we can rule out this whole class of scenarios because we don't see them or something. Right, very good. We haven't quite reached that point, but I think if we more carefully parameterize things like this, these kinds of interactions, then we can start making statements more carefully about what can and can't be ruled out and exactly what constraints there are. Very good. I have a question for all the speakers of the session. So, we have seen a talk earlier in the day from Jeff. It was about a specific realization of a system in which you have horizons, right, an analog gravity system. So, I wanted to essentially ask the following couple of questions. So, which ingredients of the problem, of the information loss problem, do you think that may appear in this kind of systems, in analog gravity systems in which we have horizons? And then, I mean, let's imagine that if we could ask, if you could ask to this experiment specific things that they could measure in order to understand better the problem, what would you ask for? Imagine that we are just, you know, it's like we have the freedom to ask for particular measurements about, you know, correlations between particles, for instance, these kind of things. Or perhaps you think that these experiments are not going to be useful for this problem. So, what I mean is that there is, so the system have the possibility of exploring horizon physics. So, everything that is related to the horizon principle could have an, could be seen in this experiment. I'll make one comment. First, it was too early in the morning in California for me to see it. But, comment. And that is, I think these analog systems are probably useful in confirming some of our expectations about the form of the Hawking state and its behavior and correlations on two sides of the horizon and stuff like that. But, it's not really clear if any of these are going to go beyond to address the more basic question of, you know, what happens when you include disappearance of the black hole and the rest of the structure that we believe happens in that process. It's not really clear yet to me that analog systems will go that far. So, it's sort of confirming our, not only our intuition, but our calculations about, you know, the local quantum field theory description of the evolution of the Hawking state, in some sense, and properties of that. But, the deeper questions in quantum gravity might be beyond the reach of those. Others may have other comments. Yeah, I think so. In fact, let me add a comment here. Probably the most critical thing with these analog models is you're dealing with temporary trapping horizons. You're not dealing with honest-to-God event horizons. You don't have singularities hiding in the system. So, you know, you just have to turn the system off and those trapping horizons go away. So, you'll get a disappearance, but you'll never actually have the major issues that many people are suspecting are relevant to general relativity, which is the actual event horizons. Now, my personal suspicion, actually, is that maybe in general relativity we're being too enthusiastic about using event horizons simply because they're mathematically convenient. But it's not clear to me that they're actually all that relevant, even in general relativity context. Could I comment about this as well? Sure. So, I mean, I had an actual paper on the information loss problem in anorak gravity. And, you see, let's take a BC system. There, you can have region where basically the aerodynamic approximation doesn't talk. So, from the anorak point of view of space-time, there is no notion of classical space-time there. However, that is a pure quantum system. And, of course, the full evolution on the full Hilbert space must be unitary. There is no question about that. However, what we did was to show that when you have an acoustic black hole evaporating, you get, if you are going to a formalism that allows you to track the atoms which are in the condensate, which normally you don't do in anorak gravity. You just simulate a classical space-time with quantum perturbation of it. So, as long as you keep track of these correlators, nothing is lost. And they are not necessarily local in the analog space-time. You see, the problem is that there is a different notion of locality between the fundamental system and the emergent system. So, anyway, what is happening is that the evolution is completely unitary. But then, if you insist that the space-time is classical, then you basically lose this correlator because quantum operators for you become C numbers. So, these correlators disappear and you lose track of all these correlations that you are generating continuously during the evaporation between the atoms of space-time and the Hawking quanta. And these correlators are important like 1 over n, where n is the number of atoms forming the black hole region. So, as the black hole evaporates, the correlators become more and more important. They are always there, but they are more important toward the end of the evaporation. So, this is the picture that seems to at least be embodied in BEC. To me, it seems that as long as you believe that there is unique Hilbert space between whatever is the field that is evaporating in quantum gravity, if there is a unique Hilbert space, this is telling you that the factorization during evolution, even if you start with a state that factorizes in gravity and matter, it will not be factorizable at the end of evaporation. What you get at the end will not be Vinkovsky plus a thermal bath. It will be a highly entangled state between quantum gravity degrees of freedom and the Hawking radiation, what you call Hawking radiation. Then, if you decide that that is a classical space-time and you forget all this, and you basically completely forget about the quantum degrees of freedom forming the space-time, then of course it looks like you have lost information. But the full Hilbert space, the evolution is always unique. This is what you get, at least in an analog gravity model like the analog gravity in BEC. It's a toy model, but it's providing an interpretation of what is going on, a possible resolution. I don't know if there are probably other ways of resolving the issue. Of course, this is it. Okay, thanks. Other comments or questions? I had a comment on that question which was asked about the analog black holes. I think it's important that those black holes may not have the kind of large entropy that the normal black hole has. Just in line with what I was saying in the morning today in my talk, I think sometimes we sort of separate the black hole entropy question from the information paradox in the sense that, okay, the entropy is a number and what do we do with it? But I think at least the way it shows up in the possible resolution of the paradox is that the semi-classical metric, so it has the problem of the Hawking puzzle, so you have to ask why is that not the correct metric? And the reason it's not the correct metric in what we understand now from puzzle balls is because of the entropy. So if you try to write a path integral for the collapse process, there's always a measure term like BG, G is the metrics, and then e to the minus action, which is action over h-bar. And so normally you extremize the action thinking the measure is like an order h-bar term compared to action over h-bar because there's a 1 over h-bar in the action. But then when you compute for a black hole, you find the action term there, which you're trying to minimize, is actually comparable to the measure term because the measure term basically counts the number of states contributing, which is the count of all the states contributing to the process, the measure like e to the s Bekenstein, and there is also something over h-bar. So the essential reason why the black hole is not the semi-classical thing is not the correct answer is because you cannot ignore the measure and just extremize the classical action to get the semi-classical physics like we do everywhere else, but the measure term is comparable to the action term, and then, you know, everything breaks, all hell breaks loose and you get puzzle balls and everything. But you need the big entropy for that, and the entropy which goes like 1 over h-bar. So I was just wondering whether for the analog black hole is ever possible to get to a situation where the action for maybe producing that black hole in some way, I don't know what the number should be, is comparable to the entropy term for that. And then maybe the same kind of paradox resolutions might come up there as well. Well, you can for sure define an entropy, which is the entanglement entropy associated with the quanta that are produced, and that will be proportional to the area, and that will have also a natural cutoff. You don't have to put it by end because you have a Lorentz breaking dispersion relation, so it will be the cutoff of the Lorentz breaking, and bc will be the yielding length. So you will have area over yielding length squared, something proportional to this. But then the rest of your question, I don't know the answer. I mean, you can define that, but how you can use it is a different matter. So would that number be the number of states of this analog black hole? That's what I was wondering. I mean, is there a way to argue the analog black hole has as many states as the area or something? That probably is not true, right? I don't know. Actually, I don't know. It's a good question, and I don't know the answer. Let me ask you something related to this question that you said, that the entropy is the area. I mean, we can always make entropy equal to the area. The question is in which units and whether the scaling holds. I don't think analog black holes, the ones that currently people are discussing producing in laboratory, I don't think they saturate unitarity bound. So, therefore, I don't know. In my talk, I gave this bound that in order to saturate the entropy bound, the system has to saturate the divergent saturating unitarity, truly. I don't think analog black hole systems of analog black holes saturate unitarity, the ones that you produce in laboratory. So, in that sense, I don't think they can really be truly at that level of unitarity. However, I studied some of the different types of analog black holes, the fully quantum, that are at the critical state, more similar to black holes than this naive horizon, only horizon black holes. And you can learn a lot of things about analogies with black holes in those systems. But, unfortunately, I don't think with non-relativistic quantum system we can saturate unitarity bound. That's impossible. So, therefore, there is no analog of the relativistic alpha coupling. Relativistic alpha coupling vanishes. There is an analog coupling that you can come up with. We have done that. So, yeah, the answer is they are useful, but they don't saturate unitarity bound. But in that connection, I mean, again, I'm a little bit confused when we, okay, here we talk about, it's okay, paradox became like a term for an unknown. But what I'm trying to say, what I was trying to say in my talk is that what we are learning is that you can produce systems that have nothing to do with gravity, that literally all the aspects of information are like black holes. So, for example, again, I can give you a baryon in large NQCD, which you give me any parameters of black holes. I can give you a theory of large NQCD, asymptotically free, UV complete, in which the baryon will have literally the same properties as black hole has in gravity. And we can trace those, including the phage style. Now, of course, we know that in asymptotically free theory, there is nothing, it would be crazy to say that there is some paradox there for any loss of information. So, therefore, it's a little bit strange that we are saying that there is a paradox on the black hole side. Because, I mean, just the fact that we don't understand what happens with the black hole beyond phage style, I mean, that's not a, I mean, to say that there is a paradox, I mean, it's a little bit strong statement. I think it's clear that by phage style, the black holes do not operate anything like thermal, because of black reaction. And also, the different microstates of black holes have different external evolutions. I mean, that's what we see whenever we can compute. So, therefore, I don't know, I mean, we can, of course, say that there is still some. Of course, this doesn't mean that we don't have to understand how things work, but I think we have to differentiate between conceptual problems and technical problems. Now, if you ask me to trace the evolution in the oven, that's impossible. But, obviously, we're not saying that there is a paradox in the oven. So, yeah. Well, after a phage time, a black hole can be very far from being, you know, a Planck-scale black hole. I mean, it's not that it's tiny. where you would have said, well, yeah, I need to change completely the rule of the game. I need to introduce quantum gravity and so on. And nonetheless, you know, the basic assumption that Steve was listing, I mean, normally, it would look everything like we can. It's the same problem with Elmeri derivation of complementary. It looks like everything is reasonable. Then you put them together and you find that there is some incompatibility. So, it's not that, it's definitely true that we don't understand something. It's not meaning that there is a problem, a fundamental problem in the theory. It means more that we don't understand. But there is something counterintuitive, no? That somehow we need to revise a little bit our way to look at. I'm just saying that in order to promote something at the level of paradox, we need much more stronger evidence that there is a sort of inconsistency. And I don't think in case of black holes, this is justified. That's all I'm saying. Of course, we can, I mean, here at this conference, we come from different backgrounds, particle physics, gravity, whatever. We understand each other. When we say that there is a paradox, we understand what we mean. And none of us here thinks that, presumably, the things that unitarity is really violated or anything like that. But I'm saying that there are conceptual issues and there are technical issues. I can ask you a question. Then why I'm not saying that there is a problem with baryon in large-angle Sydney? If anyone tells me one aspect, one aspect in which the two are different, then yes. Then I can say, okay, look, black holes are special because they have this additional property, which this other system doesn't have. But this is what we are observing. There is nothing. Literally, I could not come up with any feature that black hole has and this system could not have. So, therefore, I don't know. I take this as a strong indication that there is some universal underlying physics. And it's just a matter of time before we'll be able to compute it beyond page time. Yeah. There is a calculation past the page time. And that is, you know, it's actually one that I sort of outlined in the beginning of my talk. So, take a black hole, let it evaporate to half its original size or whatever. Okay. And then it's a big black hole. And then we're going to study the evolution of its quantum state. It's approximately described by a classical geometry. We're going to study the evolution of its quantum state on, say, a set of slices like I described. And you can see that you just keep, in the usual Hawking calculations, building up entanglement between outgoing excitations and internal states. But you know by that time, from the more general arguments, that there should be information transferring out. And you just don't see that in the local quantum field theory description. Sure. So, that's very good that you are bringing up this issue because this is another important thing. Because, I mean, see, sometimes we use language. And it's very important to understand each other's language in order to find the touching points. And so, what I was trying to say is the following thing. So, the fact that black holes change drastically, deviate drastically from standard Hawking-type extrapolation by pages time, in order to reach that conclusion, it looks like an explicit microscopic theory of a black hole is not needed. More or less, all of us here sort of anticipated there are some corrections by pages time. Now, to do calculations beyond pages time, you need a microscopic theory. I don't think there is any way to do computation without having a microscopic theory. For example, let me give you precisely the answer. You said that by pages time, black hole shrinks to half of its size. Now, there is absolutely no evidence for that. Because you are putting in an assumption that black hole evaporation is self-similar. There is absolutely no evidence, based on any computation, that evaporation is self-similar. That's a calculation in low-energy GR. There's nothing wrong with that calculation. No, that calculation is done in the infinite black hole mass limit. The calculation done in semi-classical limit is calculation in which you keep geometry fixed, and you calculate fluctuations in the fixed geometry. That is the limit of infinitely heavy black hole, in which Newton's constant is also taken to be zero. That black hole is an infinite black hole. Yeah, you can show that the evaporation is thermal as long as the surface gravity is adaptically changing. If kappa dot over kappa squared is much less than one, you can show that each instant of time, you keep evaporating. And if you have a sufficiently large black hole, you will evaporate until the end of the entropy, in a way that is completely controlled by the thermal flux. You don't expect any... You can show that. No, sorry. Who is questioning here that Hawking's computation is correct? The question is, Hawking's computation is done in a very specific limit. Because in physics, to do computations, we need expansion parameters. Yeah, but I'm telling you that you can have a slightly dynamical situation and still get a thermal flux. This is absolutely crucial. In physics, we always do calculations in some expansion parameter. There are extremely limited number of models that we can solve exactly. That usually never happens. So, usually there is an expansion parameter, and we compute things in the leading order in that expansion parameter. My point was that in Hawking's computation, implicitly, this expansion parameter is one divided by the entropy. Why? Because Hawking is taking the limit of infinite entropy, because he takes black hole mass infinite, Newton's constant zero, and geometry fixed. So, now the question is, we have to compute corrections to that. Corrections... This is a very well-defined question. It's a well-defined question. What are the corrections to Hawking radiation in expansion one over s? I have computed, I said, I have computed this in different ways, and one over s corrections are not zero. Okay? So, now... So, that's the question. I mean, the question is, what are the corrections? Are corrections one over s? They vanish? One over s correction to all orders? So, one over s is zero, one over s squared is zero, s cubed, and so on. If you're saying that the entropy is a quarter a times one plus something that goes as one over s, and I would agree, it's probably the entropy is like a fourth a maybe plus log a or something. But I don't really know is, are you predicting that the actual evaporation will be at a much different rate after the halfway time? I mean, I think most of us probably believe that the corrections are one over s, which is relatively small. So, the evaporation will continue more or less the same way. It'll locally look more or less thermal, but of course, it'll have correlations with the previous radiation. So, maybe I've got to say, would your predictions be wrong if it turns out that black hole evaporation just happens at the Hawking rate, and looking thermal, if you only look at much less than half the total number of quanta, would your predictions be wrong if that continued to happen, that sort of approximate Hawking thing after the midway point? Yes, but notice one thing. It's very good that you are asking and formulating the question in that way, because look, corrections per Hawking time is one over s. How many Hawking times are there by the time you reach Page's time? The number of Hawking times is precisely s, order s. Correspondingly, although the correction goes as one over s per Hawking time, the number of Hawking times is also s, and correction by the time you reach Page's time is order one. This is my point. My point is that by the time we reach Page's time, there is no trace of thermality anymore. The deviations from thermality are order one. Of course, oven is also approximately thermal, but deviations are very large. So, that's my prediction. That's right, exactly. But you understand why one over s corrections are not decoupling, because the lifetime of the black hole, half decay time, also increases as s, and s times one over s is always order one. Yeah, I mean, there's certainly, as I pointed out in my first paper, you know, questioning Hawking's loss of information, that there is this Brownian motion of the black hole. So, there's got to be huge quantum uncertainties as to where the black hole is. So, I do expect those sorts of one over s terms to sort of build up by Brownian fluctuation. But what I would be surprised at is if now you, now suppose that there's very late time you find the black hole, and you sort of only look at the part of the quantum state where you now know where the black hole is, and suppose you choose a frame where it's at rest, I would expect as long as the s is very big, the black hole is very big in Planck mass, that it will look, that if you just watch it for a short period of time, not long enough to see the Brownian motion, you'll just see it evaporating more or less, more or less semi-classically. You're also ignoring the correlations between this radiation and the previous radiation. So, the question is, do you have some strong reason for believing that that picture is wrong? Yes, that's right. That's what I was trying to say. In other words, of course, as usual, I mean, we can only predict things that we can calculate. So, when I'm making predictions, I'm relying on the computations that I can do in calculability domain. So, these predictions are based on quantum gravity science. These are computation, these are s-matrix computations, or in fully-fledged string theory, scattering where this black hole state is produced. But on the other side, as I said, in all the prototype systems in which I see that they behave like black holes, their entropy, their area, they have pages time, etc., etc. In all the prototype systems, we see that beyond pages time there, there is, corrections are 100% important. So, my prediction is based on that. Of course, as I said, I cannot exclude miracles, obviously. But based on what I can compute, that's my prediction. Yeah, so maybe that would be, if, as I would think, your prediction is wrong about that, that the black holes do behave in a sort of self-similar way, if you just look at it coarsely, you don't look at the fine correlations, then maybe that would be a way in which they do differ from other systems. Since if you're saying other systems all have some sort of drastic change. That's an excellent point, yes, absolutely, yes. Yes, that's an excellent point, yes. So, yeah, I agree with that, yes, I agree. If a correlation is self-similar, then yes, that's the way to rule out this type of approach. Okay, that's very helpful, thanks. Yeah, I'm going to have to leave in about five minutes to teach my class. But anyway, I really enjoyed the discussion. I'm really sorry I goofed up and wasn't ready at the beginning. I think there are a couple of questions. I don't know if, Luca, you're muted. Yes, I'm Germain, I think there are a couple of questions. Yes, can I share my screen? Sure. These are the kind of experiments we are currently doing, where we reproduce black holes in water wave systems. And here you have an obstacle in an open water channel flow. And here the flow is subcritical and the flow here is supercritical. So there is a black hole horizon here. And I was wondering how we can compute the entropy of this kind of black hole flow. And I have a degree of liberty. If I put a gate at the end of the water channel, I can control the equivalence of the mass of the analog black hole. Here it's a usual flow without central singularity. The waves can propagate on both sides. And here they are completely entrained by the fluid, which is supercritical. So we can look to the correlation by measuring the water depth. And if I introduce a downstream gate, then I see such kind of behavior. So it does not change the black hole region at all. The analog horizon is still here. But we have inside the supercritical region something which I would call the central singularity, which is very, as you can see, turbulent, time-dependent. And of course the analogy of Wunru will break down here. So the acoustic metric is no more valid in this region. It is completely valid here. We can test it by looking to the dispersion relation of the water waves. And the acoustic metric, of course, is recovered at Wunru. But then my challenge for general relativistic people is, how can we compute the entropy of such singularity, singular behavior, which is a kind of mass effect, because you see the accumulation of water. We have a feedback here. And this central singularity is clearly present. Whereas it has absolutely no effect on the horizon, and hence the wave scattering, and hence analog blocking radiations. And of course, if we continue to accumulate mass in this kind of system, we can push the central singularity such that it merges with the black hole region. And we recover the analog wormhole by just changing the water depth. So here it's a case where you will see the analog black hole, the central singularity, and just by using a gate at the downstream end of the water channel, we can see a peculiar behavior where you see a kind of merging of these two regions. The central singularity will disappear and will transform into an analog white hole with an undulation. And this undulation is due to dispersion in this condensed matter system. This is the instability of a white hole, which is predicted in general relativity. And you will see a kind of Hawking radiation at zero frequency. This corresponds to this bump here that you see. So my question is, can we compute this kind of entropy associated to this central singularity? And as you can see, in analog experiments, we can have many type classes, which is very similar to what Stefano was discussing in his talk, the fact that either we have central singularity or analog wormhole, or just pure black hole horizon in this case. So it was just to illustrate the fact that, to answer the question of Raoul, what can we do in analog experiments? Of course, we can turn on and off and we can change the flow rate in this system. But of course, all black holes are eternal, because we still feed the system with the pump. But of course, we can look to the disappearance of the black hole horizon. And if the black hole horizon disappears, I have shown you that there are apparently two scenarios. Either the black hole singularity merges with the black hole horizon, creating a pair of black hole and white hole horizon with the wormhole. Or it just disappears because the two horizons, the black and white horizon, merge and then they disappear. Okay, thanks. Thank you, Germain. Yeah, I think Don has to leave, right? He mentioned he has to go to... You're muted. We cannot hear you. Yeah, sorry. I mean, you can carry on. I'll have to go in about three minutes to walk down to the lecture place and give my lecture about a boost in Galilean relativity. And then the next few lectures I'm going to boost in undergraduate student brief introduction to special relativity. But anyway, I've enjoyed it a lot and I really apologize for sort of messing up at the beginning here. Don't worry about it. Okay. Bye, Don. Bye. Bye, Don. Thank you very much. Bye, Don. Bye, thank you. Bye, thanks for the whole discussion. I enjoyed it a lot. Bye.