Black Hole Analogues & the Universality of the Hawking Effect

0:00 I have been asked to give a very general introduction since the auditorium or the various subjects, so hopefully I will not spoil you too much if I give you some very similar introduction into the topics I am going to talk about. The idea is going to be to make a relation between a system which we don't completely understand, such as the black hole, and a system which we understand a bit better, which is, for example, some laboratory systems such as Bose-Einstein condensates. Before doing this, I should give a very short introduction into the interesting properties of black hole I'm going to talk about. And this is a very simplified sketch of the collapse of the star to black hole. So let's assume we have some star who has used up only this nuclear fuel and the temperature crops, so the thermal pressure crops, and can't withstand the gravitational attraction anymore. And I'm assuming spherical symmetry, so the only thing which matters is time and the radial coordinate, so time goes up and the radial goes to the side. And this green line is supposed to be the surface of the star. So at the beginning, the star is just sitting here, held up by the temperature, but then the temperature drops and it collapses. But as you know, in general relativity, the question of what type coordinates is a bit tricky. So I've thought that these red lines here, which are supposed to be light rays, in spherical symmetry, these are outgoing or ingoing light rays. And they are physically unique, so it's clear what one is talking about. And at the beginning, this starts just sitting. Here, gravitational attraction is very weak. And the light ray goes out. And the light ray going in are almost the same as in flat space time. So the light ray goes out just like this. After a while, if the star is collapsing, the gravitational attraction becomes stronger. And the light ray going out has a harder time to escape. So it has to bend a bit, and then only at some distance it just goes out again. And as the collapse is even further, the life-way going out doesn't make it anymore.

2:30 The gravitational field is too strong, and even if you're on the surface of the sun, take a flashlight and point upwards, the flashlight, the proton stone, escape to infinity. They are all collapsed to the singularity over here. The boundary between these two light rays, the one which escapes to infinity and the one which can't escape anymore, is this green, sorry, blue line here, and this is just for the horizon which gives the motor line after which you are inside and you can't escape anymore. And after the collapse, the whole thing settles down to some stationary state and this horizon is just given by the short-term radius. collapse, then you have a black hole, and once you are inside, then you can never get out again, at least from this classical point of view, and then let's keep this for the references. After their discovery, or theoretical discovery, I should say, black holes have been investigated quite a lot, since a long time, and what has been realized is First of all, there's something called a new Herb theorem. So the final state of this black hole over here doesn't depend on what matter this star So whether it was a neutron star or dwarf or something else, the final state of the black hole only depends on the mass of the black hole, possibly some angular momentum if it's a rotating one, and maybe the charge, but I'm going to omit charge here. So why is it called no pair? The black hole has no pair. There is no information what was, what the metal is inside. So you only have mass and a minimum charge. Your hand has removed any holes at the end. Yeah. Okay. So at the end, these black holes are described in terms of very few simple quantities. And furthermore, it has been realized that there is a very funny and surprising analogy between these laws of black holes and the laws of thermodynamics. That is, if you make some identification and identify the mass of the black hole with the energy in thermodynamics, which is still a very reasonable choice, but then the next

5:00 thing is a bit more surprising, if you identify the surface area of the black hole, that is the surface area of it, and say this corresponds to the entropy of thermodynamics and the surface gravity of the black hole, which is the gravitational acceleration at the horizon, roughly speaking, and identifying this with the temperature, you find that the laws of black hole dynamics are very similar to the laws of thermodynamics. So first we have the zero slope, thermal dynamics would be just detected, one temperature is constant in equilibrium, and for black holes you can prove that this kappa, the surface gravity, is constant across the horizon in equilibrium. For zero to symmetry that's not very surprising, but even for rotating black holes with charges, it's always constant. Then you have the first law for thermodynamics, that's just energy conservation. Here is basically also energy conservation, but the form, the explicit form of these differentials is very interesting. This shows you why you should make the correspondence between the surface area and the entropy and not, say, the volume of the black hole and the entropy, which somebody might expect. If you have an angular velocity and an angular momentum of the black hole, you can also make some connection to extensive and intensive quantities like pressure and volume, and you get another experiment over here. So this is just energy conservation, but furthermore, what gives you some real hint is that you also have a second law. In thermodynamics, that's just that entropy always increases, and here for black holes, of this black hole, the horizon area, always increases, which is very simple. If you have a black hole, throw something in, then of course it increases. But also if you have two black holes, and the sum of these two areas, black holes collide, and form a bigger black hole, the area of the final black hole is always bigger than the sum of these two areas before. But I should say this is based on some energy conditions. If you assume that the energy is positive and so on then you can prove this. So here you see that really the area instead of say the volume should be something like the entropy of a black hole if you want to make this connection. And finally you also have a weak version of the third law of thermodynamics.

7:30 With thermodynamics you say that you can't reach zero temperature with finite number of steps and here you can also argue that you can't reach a vanishing surface gravity in the finite number of steps. But this is only the weak version, the strong version goes to zero for the energy temperature, we don't have this here for black holes. So this has basically been discovered by Wittgenstein, and he then said that apparently this analogy must mean that nature is trying to tell us something here, and he then made this interpretation that these laws of black hole dynamics are really completely analogous to the laws of black hole thermodynamics. But then you run into some problem here, that the thermodynamics is consistent. It's only working since you have some temperature, and if you have some temperature, it should be radiating. But then if you look here, you have a temperature over here. So this gravity for the black hole, it would mean that the black hole should radiate something. And this was before Hawking discovery, So that was a little flaw in this, in Ekenstein's interpretation. And Hawking then later on looking at this, looking at this problem, really found out that the dichos do indeed emit some radiation, which is exactly given by this corresponding between surface gravity and temperature. So what Hawking did was coming back to this one here. to calculate the quantum properties if you have some quantum fields living in this space-time background. So far, this all has been classical, but what can now consider what happens if you have, say, some quantum electrodynamics, quantum photon field, and I start up in a vacuum state down here and see how these modes behave if the black hole collapses. So then you can follow these red lines, and you see that during the collapse of this black hole, the state is strongly discovered, if you think of this light ray, of the distinct clarity, and just the other light ray over here, you can escape to infinity there. You can imagine that the horizon makes a really big disturbance of the electromagnetic field over here. And if you do the calculation and see if you have a quantum state here, what kind of state

10:00 is the state over the air which is outgoing, then the outgoing state contains a particle number which exactly corresponds to the thermal spectrum which is the working temperature, which is given here. It's inversely proportional to the mass of the black hole, so for realistic black hole masses is very tiny completely negligible but from principle from this point of view it's very important discovery. Now with this Hawking effect the whole thermodynamic interpretation of black holes became really interesting and now you could make a lot of Gedanken experiments and try to violate the second law for example when throwing in some radiation or something else into the black hole or not increasing, and without walking radiation, you could come up with some ideas to decrease the total energy, but if you now accept that black holes radiate, and you accept that the surface area of a black hole corresponds to some entropy, then everything is fine, and the second hole is always satisfied. so this gives us a strong consistency and this is I think one of the basic ideas which also lies at the heart of all these ideas of holography and ADS-CFT correspondence that is that the entropy of some gravitational objects such as their goals is given by the surrounding area in wake sense so this interesting correspondence seems to indicate that nature is trying to tell us something here that there might be a very fundamental relation between these black holes and thermal dynamics and second hole and so on. But there's a catch. If you look again at Hawking's derivation and try to follow these modes, these outgoing Hawking modes, back in time. The problem is, if you have, let's say, red Hawking following coming out here, If you try to follow this back in time and trace it back in time, then it's blue shifted due to the gravitational red shift. So if I take my laser pointer and point this upwards, it's a bit red shifted due to the gravitational field of the Earth, but in the Earth, this effect is really tiny. The point is here a little like all. This horizon is exactly the point where the escape velocity equals the velocity of light.

12:30 So here, this gravitational red shift is extremely strong. so if you had a red photon over here it must have been a blue photon there than let's say an ultraviolet photon over here and you can imagine the whole thing goes if you do the calculations, the whole thing goes exponential and you find that after a very short time 14 photons coming out here originate from extremely short wavelengths in the past but the problem is we know or we assume that our theory is like quantum electrodynamics and classical rotation of fields are only valid up to a certain energy scale, which brings us to the Blanckian problem. So we trust our, the point is that we trust our theories such as quantum electrodynamics only up to scales between say a thousand GeV and some higher scales, where some new physics is expected to come in, but the problem is here, you go exponentially if you increase in your growth, you have an expansion to shift, so you're very fast, very effort leading all these thresholds, and after say the lifetime of what they call one second, the photon which came out here had its origin, which is far above Planck's scale, given by 1019 GeV, and at least at this scale we expected the usual picture of some photon propagating in a background space and it doesn't work anymore. It doesn't make any sense. That means the origin of the smoking radiation, if you do the smoking's original calculations, lies in modes, where we know, or lies in energy ranges, where we know that the theory which we use for calculating isn't valid anymore. But the partial mocking in the later calculation is the Euclidean field theorem, which is immune from this, because you look at periodicity and the... Yeah, for this hard-walking state, then you make some Euclidean, or you go to Euclidean time, but that problem is you're, from the very beginning, setting in a thermal echoluprium In this calculation you get the same amount of photons coming out and going in, so the whole thing is in equilibrium.

15:00 The other point is that this analytical continuation also at some point, you make an analytical continuation, you are very sensitive to very short wavelengths. So at some point this assumption about what happens at very short wavelengths, The Planckian scale here comes into the game. I will later show some examples for which cases it can change things and for which cases it gives the same. So that is the origin of this working radiation which gives us all these nice features such as black hole entropy and his black hole thermodynamics lies in trans-Planckian mode, so we know that the semi-classical, or we expect that the semi-classical treatment breaks down. Now the question is, is this calculation with Schulten did totally wrong, or maybe is it still correct, or can we get some impact of this trans-Planckian physics, or can it be proven that no matter what trans-Planckian physics does, we always get cooperation at the end? The problem is, unfortunately, we don't know what this Transparency in Physics is. We have to find, we have to come up with some other way to get an idea. In this way, it is due to Bill Anru, who originally proposed to look at some systems that have the same problems, and would like to look at the same, same technical, and also get the same problems. But the other system may know what is going on at high, large wave numbers, and we can get some insight here. So for that, let us look at a very simple model system. So let's just look at, say, everyday experience. You have your, you look at the water flow in your drain, or in your kitchen sink, and here's the drain. So you have a water flow, which is spherically symmetric, always inwards flowing. Out here, the velocity of the water is very small, and if you come closer and closer to the drain, the velocity becomes bigger and bigger. And now you look at, say, surface waves or sound waves in this fluid. If you are out here, and you make a little speaker in here, then the sound waves just propagate in the usual way.

17:30 But if you come closer and closer to the drain, at some point the sound waves are distorted. And here, at that point, where the velocity of the fluid equals the velocity of sound, then if you put a speaker here, then the sound phase can't go this way anymore. They are all dragged towards the drain. And if you are further inside, then this is even more. So this blue line here basically gives you the border from which sound can't escape to infinity anymore. And here we see this is very similar to this picture over here. Here you have a border where nothing can escape to infinity anymore. And here is a border where only sound, if you restrict yourself to sound or other waves, can escape to the outside. So this is at the moment just a nice picture. Maybe I should mention you can also make, if you don't restrict yourself to stationary radial flow, there you have some other nice analogies between rotating black holes, but for the moment I'm just going to focus on this example. But as Bill Anuolo showed or found out, this is not just a nice picture, you can prove that if really the equations of motion are completely the same given certain assumptions. So this is the basic idea of these black hole analogs holes than dump holes, where dump here is supposed to mean that they can't speak, so no sound is coming. So if I now do this picture I have over here, and try to do the calculations a bit more properly, I have basically sound waves in moving fluids, and I assume that my fluid is irrotational, so I can write the velocity and also the velocity polarization as some gradient of a potential. Now if I solve the equations of motions here for the sound waves, I get this complicated wave equation where I have the time derivatives, then this is the background velocity of the fluid, which is here going inwards all the time. Density of the fluid, this is the speed of sound.

20:00 And then you have this complicated equation of motion, which describes all these effects I have mentioned here. Now the point is, if you look very closely, you see that you can cast this equation for the sound waves in a form which looks exactly the same way as the equation in the curved space time. We have this curved spacetime that is described by this effective metric, which has here, for this equation of motion for the sound waves, this special form, and in general relativity, this is known under the name of N equals from the metro metric. So if you assume, or if you say, this is my negative space time, the negative metric, then I look at my waves in this curved space time, I see that these waves behave exactly in the same way as my sound waves do in this fluid over here. If you look at this metric closely from the point of view of internal relativity, you find that you get a horizon exactly at the point where the radial velocity equals the velocity of sound. This is exactly this blue line over here. That is what I have described here. It's not just a nice picture, but it's almost one-to-one correspondence between these sound waves in fluids under these functions and other fields of space types. Examples of Bose-Einstein condensate superfluids such as helium. You can also, instead of sound waves, have surface waves to do vibrance. Or you can also consider protons in flowing dielectrics. The main idea is always the same. Here the metric is a bit different but the main idea is the same. And now, since we have the same equation of motion, we can do the same calculation as one thing did. And then you would expect This, for what, a sonic light bulb analog is giving us a walking radiation in this type of sound. That is, if you make a flow in this way, this flow is giving off thermal sound radiation, which is exactly the walking temperature. But here, if you do all the calculations, you find that the walking temperature is determined by the gradient of the velocity. here. And if you think of this gradient, if you have a stationary velocity, the gradient exactly gives you acceleration exactly at the horizon over here.

22:30 Why is this a constant problem now? Why is H-bar in there? Oh, that's always quantum. Yeah, but now he's talking about sun waves. Yeah, but sun waves, I cannot do it. These are quantum sun waves. Okay, very low on this thing. For example, the point is to see anything like this, we need some very low temperature. So for example, in both the Einstein politics, that's the best example. There you have these quantized sound waves. About these mean fields, all of them. So now the question would be, if we have this system, And here we know also, again, at very large wave numbers over here, we know that this negative sound description breaks down, at least if my wave number becomes comparable to, say, the distance of the atoms, then I know sound waves don't make sense anymore. But here I have some hope of being able to solve the whole thing, since in principle I understand the underlying physics. I mean, it's still a very complicated problem, but in principle you can do some calculation. Now the question is how to model the physics which occurs at very high wave numbers. And one idea which I'm going to present here is to say that what happens at very, very large wave numbers is that my dispersion relation changes. If I have just sound waves, then the dispersion relation is linear to the frequency and the wave number. And if I have just a linear dispersion relation, I get the same problems that I have this infinite blue shift or red or red shift for your sound that you get to a higher and higher tone. I assume that the higher at some critical wave number, the dispersion relation changes So you have, say, bands toward this axis, which could be some subluminal or subsonic excluturation, that is, at higher K, if the heat of sound goes down, or you move up, in case the heat of sound goes up at high K. Of course, what really happens at large wave numbers could be more complicated, you could get some interactions or breakdown of the multi-partic picture, but this is how to calculate, so here

25:00 we're just focusing on a change of the dispersion relation. So now let's do this and let's see whether we can do Hawking's calculation by taking into account that something happens at large wave numbers. So the idea to do this, I start again from this metric which I just had on the other And here I just omit any radial, sorry, any spherical directions and just add a 1-plus-1-dimensional radial direction. And I'm going to do this, which is very close to the horizon. So if I do this, then I get my 1-plus-1-dimensional Handelberg-Guestrand filamentometric, which looks like this. And this is just, from the point of GR, it's just some funny parameterization of a decorometer. Now I write down my equation of motion, which is the same as this equation of the sound waves. And I get the same equation which I had on the previous transparency. Here I set the density speed of light, speed of sound to 1. So this is up to here. It's the same equation as I had on the previous transparency. include some deviation at large values of k, and this is to be done with some auditory function of these spatial derivatives over here. I'm not changing the number of time derivatives since I still want to have some hyperbolic equation in order to do some calculations. So, I zoom into this region very close to the horizon, which is here that the speed speed of the fluid, V is just the speed of the fluid here, is minus one, speed of sound is set to one, plus kappa x, kappa is the surface gravity, and the higher orders are not important. And then I'm trying to solve this equation. So I do the usual thing that I'm looking for stationary modes, since this is stationary problem again. My cutoff, where this function becomes important, is of course supposed to be much bigger than V's surface which is clear that the size of the back of all, sorry. Just one question. Why do you still use continuous equations

27:30 when you enter in a regime where supposedly you will see the gradual disruption? Yeah, first of all, this is just a model for what happens. But you can show that the interesting things happen, which still, at scales, which are still much, much larger than the inter-atomic distance. reasonable approximation at this level. Later on I will show you an example where this can be a bit more complicated. So in fact since I'm using these assumptions I'm still in a region very close to the horizon where the speed of the fluid changes only a little bit and there I can still make this approximation. Okay, so the cutoff is supposed to be much bigger than the surface gravity and the frequency, which is clear since I'm considering black holes which are much, much bigger than the Planck scale. Okay, so I'm close to the horizon, but still I'm looking at length scales which are much bigger than the cutoff. Then with all this approximation, I can do a spatial Now, the path transformation of this equation, you just get this one. And I should mention that the idea of a spatial path transformation, in this sense, is from Pauli and Jacobson. So... If I now do the dispersion relation this kind of wave equation that you can just use of the dispersion relation and replace this by omega and this by k so you just get this sense so this is the frequency omega is the frequency in the laboratory frame and this combination is then the frequency in the rest frame of the fluid so it's just dragged away with this velocity here and if you would stop with this term have a linear dispersion relation which is this dotted line here but if you include then your correction then you could also get some subluminal dispersion relation which is shown here so the right hand side is this curved or this curve over here and then if you want to look for solutions

30:00 of this dispersion nation, you should equal this curve over here, with some lines which have an oxide omega, and this slope of 4 expanding to the velocity. So, these solutions are here dots, so you have two solutions which are large k, comparably large k, and two solutions which are very small k, and we can identify what kind of solutions these are. Can you see if I draw here? So this is the horizon. So this is exactly the point where the velocity equals the velocity of sound. And over here, inside the black hole, the velocity is bigger. And over here, outside the black hole, the velocity is smaller. So these solutions are all solutions which are trying to go in the same way, that is opposite to the flow direction. So we have one solution here, and if you look at the group velocity, this group velocity is bigger. So this is my low wavenumber outgoing motion radiation over here. Oh, sorry. This is inside. Walking radiation is outside. And then you have another solution where the group velocity is smaller and this solution is just swept away. So this is the partner particle of the smoking radiation inside the black hole. And then you have two solutions with very high wave numbers. And they are both trying to get out. But the problem is their wave number is so high that the velocity rate is smaller

32:30 and they are to get away with the fluid. So what happens is, if you take this working radiation over here and trace it back in time, it will be a 2-shifted, and at some point you see that it will be 2-shifted in such a way that the velocity then becomes smaller, so it must have come from outside the black hole, except inside, and then escapes as working radiation again. And now the question is, what is the relation between all these modes? that is, if I start with the vacuum state what is the state which comes out do I get what this comes out the point is you have to make some assumptions about the dispersion relation and then you can calculate all these modes using the subtle point method in a complex plane I'm not going to present all the details, but in principle you can you can write down your solution in the Laplace space, and then you have to transform back into real space, and this can be done in this complex plane. And if you are beyond the horizon, if you choose the branch cut in this way, branch cut comes from this pre-factor here, then you can choose the control in such a way, and you see there are basically no contributions. So here we have a solution for the wave equation, there's nothing inside. And then we look what happens, what happens outside. The forming is contour and then we see it. We can identify different contributions. We get a subtle point, two subtle points over here and here. And if you look closer, then these are exactly these high wave number solutions here and there. And then, since we can't close here, there's a branch cut. We have to circumvent the branch cut. And this branch cut exactly gives us this honking radiation. Now we have a solution which relates this outgoing honking radiation to these ingoing high frequent or high wave number modes. And then you can calculate what happens if my infolding state is in a vacuum state,

35:00 then it gets shifted and it goes to the outgoing state. Is it still in a vacuum state or do I have some particle creation? And you see you have a mixing of positive and negative frequencies, so you do get particle creation. And if you do the exact calculation, due to this branch cut over here, you find exactly the same result as in Hawking's original generation, that you get the thermal spectrum of Hawking's heart is coming out, but now taking the current to complete this third generation, and you exactly know what is going on at short wavelengths. The point is, you have this vacuum state at short wavelengths, which is falling in, then it's redshifted, and if it's redshifted enough, then the speed increases again, and what goes out is your Hawking radiation. This was for a system where you accumulate antiphonons. Sorry? Your antiphononon has been created somewhere. Right, sorry, yeah, I'm coming to this. So this is, here I'm just looking, what is, if I look at the outgoing Hawking particle, position into my in-modes. Then you say, okay, if I have my in-going welcome state, how does it look for outgoing? I need two, the working particles and these in-folding partners. I'm coming to this. So what you basically create is always a pair of a working photon and an in-folding partner. In pairs, but I'm coming to this. Okay. So you can do the same thing for a superluminal, for a supersonic dispersion, just means that the large K, and the dispergamation goes upwards. And here the situation is a bit different. Again, you still, you have these two below, or the solutions at low wave numbers are of course the same, since they are here in the linear regime, but at high wave numbers, now you see that your wave packets have a group velocity which is bigger, so they have to come, they come from inside the black hole. So they are basically then here. And in the superluminal case, their group velocity is bigger, so they can overcome the flow. So you have a very high

37:30 wave number mode which comes from inside to the black hole. It's then also red shifted, it, crosses the horizon, and then goes out as walking radiation. So again, you can solve the whole thing with this center point method. And you have basically a very similar relation. now outside outside the horizon there is only walking radiation so if you look at this contour you only have this branch cut over here and then if you deform the contour you will see what is going on inside the horizon then you pick up these two subpoids and these subpoids can correspond and you find same way that you obtain walking temperature. So for both cases, subluminal and superluminal, you get the walking temperature, but the origin is different. For the subluminal case, the vacuum state is basically in-falling and then gets converted to walking radiation. And for superluminal case, the origin comes from inside the black hole. Now I come to your question. what happens with these anti-particles and the point is so far so far we had the branch cut always minus infinity along this axis now we can do the same calculations if we choose the branch cut along the other side And then what we find is that we get the relation, not of the walking radiation, but of this informing partner particle in terms of the in modes. We get the same thing for these partner particles with these popular theta and alpha coefficients. And this can be used to solve for the in modes. That you say, if I have an in mode here, how does it look like in terms of the out modes? And then you have the walking radiation and these partner particles. And these in-modes then can be used to define your creation and annihilation operators which then give you your vacuum state.

40:00 So if you say I have my vacuum state which is in-falling, so the high wave number modes are in their ground state, it falls in. Then how does it look like at final times? And you see that the vacuum state which is annihilated from the annihilation operator initially is no longer an empty state, but it is annihilated by this tiny combination over here. And if you look closely at this state, you see that this state is basically a two-mode squeeze state where you always create in pairs one working particle and one partner particle where the number is always the same. And you get this pre-factor from this one over here. So what happens is then that you are, say in the subcriminal case, you have your ground state which is falling inside into the black hole and then it is converted to a state which contains the same number of outgoing working particles and infolding water particles exactly in this way. That is, you have a strong entanglement between these outflowing Hawking particles and the infulling Hawking particles. And then you can imagine for such a strong entanglement, of course, if you now trace out all these infulling Hawking particles that you can't get into the black hole and only look at the outflowing Hawking particles, you get a density matrix, a mixed state, and if you maybe realize that this is exactly the same expression as in this thermal field, formalism. That is, if you trace over these partner particles, this is exactly a thermal density matrix for these outgoing working particles. So you get exactly the same properties as for a thermal spectrum. So you're probably going to hear about this a bit more, but this directly is related to So you can think about the process, you have this black hole and you are throwing in a lot of matter which may contain a lot of information.

42:30 For example, you are sending in some photons and you encode some information in the polarization that is this way polarized. And what you get out is a thermal density matrix with the working temperature, which of course doesn't contain any information anymore. So the point is if you now think that you repeat this process over a long while, then you see it, that at some point the information we put in cannot get out in the same amount anymore. Or in other words, if you believe this Hawking effect, and if you believe the concept of this black hole entropy, then what happens is you have this black hole where a lot of information has fallen in, then this black hole evaporates, and the Hawking radiation which is going out is just so much, so there is no information encoded in this, if you derivation, and at some point black hole is very small, and then the information on the hub of microspace black hole, which is very small, is also extremely small, so it operates until there is not much, not much probability left on the information, and this of course gives you a problem to account for unitarity, for the same level of this information that's going into. So what we have here with this black hole information paradox is basically a struggle between the second law of thermodynamics, or second law of black hole thermodynamics, this is underlying all these concepts of black hole entropy, and the concept of unitarity that is the conservation of information. If you believe all this, then the second law of black hole thermodynamics is fine, but you have a problem with unitarity. Or if you believe that working radiation or specular entropy isn't correct, then of course you might reconcile this, but you get problems, or you might get problems with the other one. So, what happens to the modes inside of the narrative in that variation? So, you have the partner, so you have your state, it falls in and then one part of the information goes inside and the other part goes outside as radiation but you can't really reconstruct it because you don't have these two parts so then there's evaporation so you always

45:00 have the part outside but what happens to the one thing for example in the sub-dominal case all of the pieces inside they just crush the similarity and then what happens and nobody Because you can have that information which is, if it's lost, it's just crushed in that. Yeah, the point is, if you believe this black hole entropy, then I can calculate how many microstates my black hole of this size has, and then it evaporates until it has, say, 100 microstates left, and with 100 microstates, there's no way to encode all this information which has fallen into. if you don't believe this black hole entropy then I think you're talking about the black hole analogs is there any analog for evaporation from that oh sorry yeah for the black hole analogs you don't have this black hole entropy for the black hole analogs you only have the working radiation the concept of an entropy doesn't work here since you don't have the Einstein equations for these analogs can you measure anything That would be the idea. I mean, nobody has come up with a, oh, that's what's been tried right now. But it's really hard to, if you're blacking some numbers, it's usually about small, you know, or even micro-curving, but that's the idea to make some experiment, yeah. But isn't the reason there's no entropy for this, that it's really an open system? drain. So if I have a microstate inside of it, it just gets washed out. It's singularity and the physical mechanism is clear. That's right. For these analogs, you wouldn't have the idea to define some entropy. For this entropy, you need Einstein's equations. And Einstein's equations, not here. So this is just for the walking radiation. This is good for investigating walking radiation because from this point of view, you can draw some some conclusions so so far we had our models and they all gave exactly Hawking's results and the question now would be is this generic or can we come up with some

47:30 model which gives maybe some deviations from Hawking radiation which is then the concept of whether this Hawking radiation you could maybe see some change if you change your Planckian physics. Of course, the system we investigated so far, this simple dispersion relation is very simple, and in reality there are some holes or some other things, in this dispersion relation is far more complicated. The question is, do you still get Hawking radiation? And now let's look at the alternative derivation of this Hawking radiation. If you consider some trajectory of a freely falling observer or detector which is falling into the black hole. If you write the trajectory as seen in terms from an outside observer, it looks like this. So we have this exponential. That is, it reflects the fact that from outside of the server, you never see the particle falling in, since it's slowed down and it runs, but only from outside. Now you can define the proper time of this in-following detector, that's just taking the trajectory and inserting it in your line elements, and then you find that the modes, the modes where the detector, which is inputting, would click and would define your ground state are given by this expression, where u is this Kruskal coordinate. So if you plug this back in, you see that you get some w exponential behavior. So usually the modes are just e to the i plus minus omega t. But here you have w exponential behavior. And this W exponential behavior exactly gives you then the Hawking radiation in the usual derivation. But we've seen the usual derivation has some problems, since at some point you hit Planck's scale. Which you also see here with this exponential factor that's just exponentially small. So now the question is, what happens, what is the state of the modes if they leave the Planck's scale? We know at some point, we don't know what's going on, but for larger wavelengths or for larger scales, we think we know what's going on. And then the question is, what is the state of the modes in this state?

50:00 That is, in which state leave these modes, the Planck regime? And we see, this calculation shows us that if these modes, which can be very complicated in the Planck regime, So if they leave this energy scale in their ground state with respect to the freely folding observers, then you get 14 radiation. So this might be really general, but you can think of a few assumptions which are not given in all setups. So first of all, you need to see the folding frame, which is in the fluid analysis, just Then you have to say that the modes are in their ground state if they leave the regime. And what you also want is that there are some kind of adiabatic evolution. So if they start off in ground state, they are still in the ground state after some time. And one example where all this is satisfied is the wave equation or the dispersion relation I considered before. But of course, you can also come up with some counter-examples. There we say, okay, let's plug in some Planckian physics, which is not sensible to the freely folding frame, which is the fluid frame, but say to the rest frame, which you also have in these fluid analogs. So the freely folding frame is basically if you sit on the fluid and just flow with it, And then if you have some walls or some containers, this gives you some sort of breath spray. Now if you say, okay, my Planck in physics is just given by some damping tool, but this now with respect to the wall. So I have my wall, and this induces some damping. Then I can plug this in. So this is just the dispersion relation we had before. This is the co-moving frequency or frequency display. Here are some phase velocity, which then can give you the dispersion relation. So if you do this, then we can solve for the frequency over here, and we find this expression. And if you look closely, usually you would just say, okay, there's a negative imaginary part, so the whole thing just decays, which is clear if you have such a damping term. But if you look at the numerator here, if the fluid becomes bigger than the phase velocity,

52:30 say, for example, inside a black hole, then the whole thing changes sign and the frequency gets a positive imaginary part. And this is just instability, which is basically the same instability which generates these surface waves on water if there's wind blowing over where we need the wind velocity must be bigger than the velocity of the waves. And if you have these miles in stability, then of course you get some excitations. And then the modes, which come from inside the black hole, say, are no longer in the ground state. And if they are not in the ground state, if they hit the horizon, then you don't get opening radiation. You can get something else. So this, if you have some interactions which are not with respect to the cooling frame, but to the upper frame, you get deviations from walking radiation. And the same you get, you have some breakdown of adiabaticity, say that you have a disperse foundation which looks like this. It goes up, but then at some point it comes down again. and the problem is here at this point if it comes down again then very small changes can also create particles and this is the same say the same mechanism so if you have little changes in your metric then you can also create particles which extremely high wave numbers and if these particles then come to the horizon then they are not in the ground state anymore and you can get deviations from walking radiations from walking's effect Another example is the black hole laser, where Collian Jacobson showed if you have a black hole with two horizons, which can happen if it has a charge, then you can also get effects for a superluminium dispersion relation that these nodes in between bounce back and forth, and the first time it just emits some Hawking particle, but always with each bounce the quantum state of these nodes changes as well, so after some point the whole thing can either go unstable or settled down to a ground state, if you don't get any hole from radiation at all, from homiums or from rosons, then the hole system will explode. But this is also a highly unstable scenario, as we discussed, but it again shows that you can get deviations from Hawkins' result in some setups. Okay, summary.

55:00 So I hope I could convince you that with these black hole analogs, for example, these You can reproduce major features of black holes, especially walking radiation. And here, since the underlying physics is in principle understood, we can then investigate what happens if you have some deviations at large waves. Then we can look at whether this large cutoff has some effect or impact on walking radiation. And we've seen that you get the walking effect for a large class of models, which are described by the ground state with respect to the freely folding frame and adiabatic evolution afterwards. But we have also seen that there are, you can come up with some exceptions where you don't get Hawking's effect, and then all these problems here with the black hole deformation and black hole thermodynamics get a different twist. But I should mention that here with these black hole analogs you only simulate the kinematics, but not the dynamics, meaning the wave equation of these little south waves is the same as in the curved spacetime, but the equations for the curved spacetime itself, that is the Einstein equations, they are not satisfied in the usual case with this fluid, and then also, of course, since the background fluid, then it's just the solution of the Euler equation or the Euler equation. Maybe just an outlook, which has already been mentioned in the previous talk. If you look at the cosmic microwave background, you get these little ripples which have been shown in the previous talk, and if you trace them back in time, since the universe is expanding, at least in the standard model, say in cosmic inflation, they also have their origin in Franz-Planckian physics, and here is also the question whether we could get some hint for this new physics. If you look at the patterns in the cosmic microwave background, it would be the same. Well, I think in principle, one could apply similar techniques to probe this question, but this is still ongoing research. Thank you for your attention. So, you mentioned that there are some problems by theory, theory, and gravity. So, now, you go to some models where the underlying thing is understood, and you look

57:30 at what happens with the quantum radiation. So, maybe quantizing gravity, we would hope that these problems are solved. And, well, it would be a possibility, and nobody would it be possible to find a way of like quantizing the, you know, like what would be your effective baton? Yeah, in principle there is a way... We also have a semi-classical... For example, if you look at Bose-Einstein condensates, so in principle you know exactly how to do, how to get from this many particle Hamiltonian, how to do all these approximations and finally get to your quantized sound waves. But for Bose-Einstein-Condensate, I think you have a very nice first principle. The problem is whether you can calculate something beyond the diute gas limit or beyond some mean field, correction to mean field, that's the problem. Maybe you find some consequences of quantizing that sort of background in the Bose-Einstein-Condensate. What would you tell us about radiations, for example, and about there? Yeah, I think what this is aiming at is the back reaction problem. So, so far this is just a little sound waves and a given big spectrum. And if you want to see, like walking radiation, what's the back reaction on the geometry of these different stress tensors, then you have to go beyond this. And this, one could also have, or there are works trying to use this approach for this backreaction problem. But this is quite difficult, and one really has to be careful to see how far the analogy extends and where it is useless. So for working effective, I think it's very nice. For other things, one has to be careful, but not clear. So you essentially address this transplanting problem by modifying this special relation. Now, my question is, can you recast this statement about modified dispersion relations in geometrical terms? I mean, you modify this equation, so now, one might think about making a statement or change about the background geometry such that this is kind of automatically incorporated. Because, I mean, from the point of view of gravity theory, I mean, you can make this analog with this fluid, but in a sense, is there a back translation of the change you

1:00:00 then introduce after you? No, I think the point is that if you have deviations from the linear conversion relation, then you can't introduce an effective metric anymore. So as soon as you have deviations from the linear behavior, then the concept of a metric, since in a metric you have this equivalent principle, that everything falls with the same speed. but if this part over here has a different velocity or has dragged away differently from this part then it's not you violate the principle of equivalence with this mode here so that would be if you wanted to translate back to more conventional terms to get rid of this fluid model again would mean you need to introduce extra fields apart from oh yeah okay if you if you introduce extra fields then then you can also write down something like this, and then your violation is then given by the expectation value of these extra fields. No, I'm asking, because you indicate that the solutions do no longer satisfy Einstein's equations, and that also essentially comes from what you say, that there is not even the kinematics still there, because you say you cannot express it in terms of the metric. And the point is, if you have a super-luminar dispersion relation, then I think in the background fields, it doesn't work since even with the background field, you can't propagate faster than C, but this probably can. I mean, in the sub-gluminal, maybe you can reconcile with the background fields, but in the gluminal, it violates the sub-gluminal. This also violates local Loretz invariant. Yes. So the theory can't be a local Loretz invariant theory. Now, one might be able to argue that some sort of non-locality For example, non-vocality could perhaps mimic something like this. At that point, I think one is taking the analogy far more seriously than one should take it. Analogies are only good to certain places, and eventually you run out of what they're good, and it becomes pointless to try and push them that far. Okay, so you're not proposing this as a way to come into more fundamental... It's just a way I'm trying to understand the radiation, You could get some hints what kind of deviations from Planck physics are allowed, still having a hooking effect and what are not allowed, but this is then getting fussy.

1:02:30 That's true. People are trying to simulate inflation, aren't they, in warp drive space times? Yes, this is along these lines. Yeah. You can also come up with some analogs that you have the, say, de-influential inflation represented by some . Yeah, the point... I've seen figures where they need to measure temperatures about 10 to minus 9 or smaller than they... Nanokelbenes. Yeah, the point is, since this is all quantum effects which come from a geometric background, the temperatures involved are usually quite small. So it's, for most of the proposals, it's really challenging to make some experiments. I mean, we thought about some other idea using electromagnetic radiation, and maybe there you can get something like fractions of the Kelvin, but it's still, So it's always hard to do it. I suppose we were thinking about a black hole. I suppose I was a very brave experimenter, and I decided I wanted to test this idea by actually diving through the event horizon, and then getting my whole thermometer to see whether I could measure walking radiation on the inside of the top of the particles. Can I measure something? No, the hooking radiation is for the outside of the zone which sits here. Now actually, the partner particles are also thermal. Yeah, that's right. But the point is, this is the whole thing. There I adopted the particle definition for an outside of the zone. If you're falling inside, then you don't see these partner particles. Because I'm really falling. Yeah, what kind of particles you see depends on your shoulder. completely see nothing, or what you see, but you don't see the same, or you don't see just these polar particles. So, I sacrifice myself in a game. Yes. Is there many experimental evidence from these sorts of animals that there is a temperature? That's what's the idea, or what has been bright, or what's tried now, but as far as I know there's no clear experiment,

1:05:00 but I think maybe in a few years it might be possible, but it's really hard to do. And people are also using giant laser beams to try and accelerate electrons to give out of radiation. Yeah, but the Anu effect is different from the Hawking radiation. Well, the Anu effect is simpler in some sense. I think for, I mean, Bill Anu is the expert, of course, But I think for a new effect, there have been some indirect, some electron beams surfing around, you can see some effects which can be interpreted as the other effect if you're sitting in the frame of the electron. But this is a different effect. It's here in Waterloo, where you're giving off energy and radiation, which is different from the other effect. I don't think that quite what you have to see here, but I hope to do it in a few years' time. You need some detector which is capable of detecting or cleaning, and then it must be accelerated. You get so much, you know, the things at the same time very strong and everything. No, actually, you know, there are some ideas, but this is also difficult, because there's Well, before breaking up, I guess we should thank the speakers again. But before doing that, I'd like to draw attention to lunch, which is out the front door and over to the left, and we're going to get together again at 3 o'clock for tea, and after the There are some informal talks, blackboard talks, which are advertised on the bulletin board just outside the door, above the map of the patient. So now let's return to the principal business and thank the folks for this morning's speech. is that your is one of those talks going to be yours not today good good good because I won't be able to

1:07:30 yeah good because I wouldn't have wanted to miss that and I'd all have to leave this afternoon we've got a full session that guy