Physics of near-extremal black holes II

discussion with your questions, okay? So, we have a computed this uh We have everything that we need to compute the absorption probability uh uh using the full uh quantum theory of JT gravity uh that describes the quantum black hole. Uh the results were written there the blackboard I erased them, but uh out of the those results, first thing that uh you can do is uh verify that in the regime the quantum regime where uh the classical regime where these are larger than the energy scale at which uh quantum effects become important you can verify that that result using that uh hyperbolic cosine and uh the sinh for the density of states that this goes to the correct uh absorption probability that uh we had before. And I think uh yeah, I don't know whether I'm putting the factors always right. Yeah, this is correct. This and uh Yeah, so we recover also the absorption cross section equal to the area. So, that's a check that the quantum results reproduce the classical results where they should. But of course, the interesting thing is the regime where these uh energies are comparable, okay? Where the emission or absorption of a single quantum has an important effect on the black hole field on the black hole state. So, the results are explicit. We have formulas that we can plot. We can plug in all of the absorb the full absorption cross absorption probability into this formula and compute what is the emission rate. The emission rate of the black hole as a function of uh or for a given frequency per unit frequency that's uh essentially this this factor over here we can plot it. We can plot it as a function of the frequency of the field divided by the energy above externality of the black hole. Then the Hawking calculation gives you a typical black body spectrum thermal emission. So, this is Hawking semiclassical. Which we know it can't be right because this is when the when the frequency of the quantum that you're emitting is is of is equal to the energy that's available to radiate. So, this uh leads you to the puzzle that the black hole would be emitting or radiating more than it can radiate. So, this curve can't be right. This is a problem that we discussed the other day. Okay? Now, if you instead of uh using this result, this is what you get by inputting this. If instead of this you take the full quantum result then the curve is uh something like well, it's not above, it's always below. Something like this. So, this is the full quantum result, the quantum black hole emission rate. So, you see that it's much smaller which perhaps shouldn't be surprising because you have fewer states that can emit radiation, okay? We saw that the spectrum of the states is uh depleted at low frequencies uh due to these large quantum fluctuations. So, you have fewer states that can radiate and then this gives you smaller radiation. And then also and this spectrum now doesn't go into the region where it shouldn't go. So, it's cut off at the maximum uh available energy, okay? Now, this uh this means that the black hole is not emitting thermally. So, the Hawking calculation that gets it uh badly wrong. This uh spectrum is not uh is not thermal. You have a breakdown of ensemble equivalence. Uh you know that ensemble equivalence between say microcanonical and canonical that's uh valid when you're in the thermodynamical limit where the fluctuations are small. Fluctuations here are not small. So, if you consider a black hole that has where you specify say the temperature then there's a wide distribution of black holes with a given energy that you're exciting that that your ensemble contains, okay? This also means that if you begin if you start your system, you begin with a black hole say in a canonical ensemble and then you let it radiate then as the black hole radiates, it's not going to remain in the canonical ensemble. It's going to go to some other ensemble. The ensemble has been worked out. That was work by Anna Bics who followed the evolution of an evaporating black hole in this quantum regime. Then what she found that there's some other ensemble, a new ensemble that describes how well an evaporating black hole evolves. The ensemble of black holes evolves, okay? This is uh the picture that solves or the result that solves the the puzzle that we discussed at the beginning of the lectures. That this is the available energy. This is the energy of the Hawking quantum. This was the three halves of T that we got which tells us at least that we uh have been uh energy where we can still radiate, but we're not radiating thermally, okay? So, radiation from this uh part of the spectrum that's not uh not thermal. So, this solves a puzzle, but then we can also ask another question. What happens to the absorption cross section? Say that now we have the black hole over there and then we throw a wave we scatter a wave off the black hole and then we measure the cross section. That's uh in the classical regime, this gives you uh the result that you get of the after absorption cross section if you measure it and then this gives you a measure of the area of the black hole. And then uh in the semiclassical regime, you could say well, this experiment that I'm performing so, we're performing a scattering experiment of a black hole you can say that this is probing the states that the black hole the the the black hole states that uh make up uh the black hole, okay? So, you measure in the in the semiclassical regime the absorption cross section is equal to the area. And then a measure of the absorption cross section in this regime is giving you a measure of uh the density of states. Okay, so this is kind of an operational way of measuring the number of black hole states. Now, this is semiclassical. If we now go to the quantum regime, you may ask yourself well, if I perform this scattering experiment, am I also getting something that I would reflect uh directly the number of uh the number density of states of black hole states that I have? Well, we have this result in the quantum regime. We have the absorption cross section. And then you may plot it. And so, this has some dependence on omega. You may plot this absorption cross section as a function of the frequency divided by the energy of the initial black hole. Then the semiclassical result would be that at low frequencies, but as the frequencies go higher, this will change, but at uh low frequencies, the semiclassical result is that the absorption cross section is equal to the area. Now you have this result that we have computed in the lecture and then you plot it now. And then you might expect that since you have a fewer black hole states then the absorption cross section should be smaller. Okay? So, expectation naive expectation would be that quantum sigma absorption should be less than the area. Okay? Because the area overestimates the number density of states, so that we have this spectrum. This is the what the area would give you, but the quantum result is smaller. So, this is what you might expect. But the result, when you plot it, and you can see that I've drawn this low, is that actually the result is higher. So, this is for some energy that's uh say uh 0.1 1 eV. So, in a regime where you're deep into the quantum regime, the absorption cross section is larger than one. So, sigma this is wrong. It's larger than the area. Okay? And the closer you are to extremality, the larger this becomes. Okay? So, this may sound paradoxical, because what it's telling you is that you have fewer states, but the black hole is absorbing more. How can this be? Well, what happens is that the calculation of the absorption cross section folds into two effects. One of them is the density of states, but also the probability or the ability of these states to uh uh to induce the transitions. It turns out that in the quantum regime, uh the quantum fluctuations enhance the ability of the black hole to perform transitions between two states. And this effect is something that outweighs the suppression of the depletion the depletion the number of states that uh are radiating. So, you have fewer states, but they are better at performing transitions. They if you induce uh if you excite the system with this uh quantum with this external field, then the transition amplitudes between states they become larger. And this is what makes this effect. Okay? So, you have fewer states, but they are uh more easily excited. And that's something that we could have anticipated, because it's known that the two-point correlation function Okay? This is what is telling you about uh what you excite the system at some instant, then you measure the response at a later instant at a later time. In a typically in a black hole, this is going to decay. In a semiclassical black hole, this is going to decay exponentially with a factor that's controlled by the surface gravity or the temperature of the black hole. Okay? So, this is what you get in the semiclassical regime. But quantum fluctuations, instead, make this decay like a power law. But this is a quantum fluctuations. So, this means that at late times, these excitations die away less quickly. They last longer. And late times corresponds to uh small frequencies. So, then this means that at small frequencies, the black hole is more able to make jumps to make the transitions between the two different states. And that's the effect that drives this quantum cross section larger. Okay? So, then you can think that well, this could be an observational way, if you can perform this experiment for these black holes, uh that you have some black hole out there, you measure the charge, because you can measure the charge through scattering experiments with the electrons, then you can measure the the mass also with the or dynamometer or then by also throwing particles around it. That's where you have a an estimate of what the mass and the charge of the black hole is. So, that's pretending here that I'm an experimentalist, that I have the object down there, I'm measuring this, then you find that you make a very precise measurements, and then you find that M is almost equal to Q. And then you say, well, how can I know whether this is a classical or a quantum object? Well, you can perform this scattering experiment, you send a wave, then you measure the absorption cross section of the wave, and then if you find that this is larger than pi Q squared, then you say, well, that object is behaving like a quantum object. You find that this is Q squared by Q squared, then this means either that we got something wrong about the gravity or simply that the black hole is not close enough to extremality. Okay? But that's in principle an operational way of observing large quantum fluctuations of of the black hole. Is this something that we think that we can observe in black holes out there? Uh unfortunately, the answer is no, because the quantum regime is a regime that's where the difference between energy levels is really tiny. And then this means that since the difference between energy levels is very tiny, the temperature of the black hole is tiny, the wavelength of the radiation that's probing this is extremely long, and we're never going to be able to measure this, at least in astrophysical black holes. Primordial black holes might be a different matter, but they they also come with other issues. Just to give you an idea of some numbers, which I haven't given so far. So, the quantum effects become important at this energy scale that we're calling eV. And this energy scale is of the order of the black hole inverse of the black hole radius times the entropy uh the naive black hole entropy. So, if you want to this is Q squared by Q squared. Okay? So, what is this number for an astrophysical black hole? If you have an astrophysical black hole, the radius is of the order of a kilometers. You have black holes that LIGO is observing have a radii of the order of say 100 km, 10 km or so. Okay? So, if we translate this into a wavelength, so this would be the wavelength of the radiation that the black hole is emitting, the wavelength is the wavelength of this radiation, the wavelength of the radiation that we're using to perform our scattering experiment. Okay? So, this is a wavelength that's really very long. A wavelength of a kilometers for an astrophysical black hole that's or for a astrophysicist, that's very long. Okay? I mean, that's something that they are not going to observe. That's uh beyond the what uh yeah, typical astrophysics would give you. But the wavelength that this scattering experiments is uh uh using, this is a wavelength that's not the order of the kilometers. It's the order of kilometers multiplied by the entropy of the black hole. And the entropy of the black hole of an astrophysical black hole, this entropy for a black hole that has kilometers in size is of the order of maybe 10 to the 77, 70 something. So, the wavelengths that are involved here are of the order of 10 to the 70 something kilometers. That's larger, of course, much larger than the size of the visual universe. So, you would need to wait for the universe to expand, to cool down. The CMB, of course, would be a massive uh distortion here. So, you have the universe has to expand to leave enough room. The uh temperature of the CMB should uh get below also these scales, and then that's when you can perform the experiment. So, don't wait to to do this, because I don't think that we have any chance or at least in astrophysical black holes. Of course, it's not surprising that we cannot observe large quantum effects in astrophysical black holes. Astrophysical black holes are objects of as I say, kilometers in size. You don't expect on something that has kilometers in size to observe large quantum fluctuations. Okay? We don't even Hawking radiation. So, this is something that uh if you want to look for it observationally, then you have to yeah, wait for or look for a different scenario. Another indication of how difficult these effects are to to observe is the following. As I already mentioned, we can also consider there are also other modes that enter at the same scale. We have the social modes, but we also have the rotational modes. And as I mentioned yesterday, then the gap in energy or the typical energy of adding uh angular momentum to one of these black holes, so, this is of this disorder. So, this was J J plus one eV. Okay? So, if we have a say, what is the difference between uh the spinless black hole, which has zero energy, what is the gap between the spinless black hole state and the uh black hole that has one unit of angular momentum? So, this is of the order of eV. So, we're talking here about the measuring differences in black hole states, where the difference is the same as you have the spinless black hole, you add one unit of angular momentum, and this is the energies that are involved in this experiment. Of course, we don't expect to observe a difference between a spinless black hole and a black hole with one unit of spin in any astrophysical setting. So, that's why this is something that it's a theoretical ex- uh thought experiment, and I think that they are interesting because they are telling us that we can have a quantum control or we would control over quantum gravity effects, black holes, and that's something that I think it's a uh something that uh well, so far we haven't had. So, theoretically it's very important or very interesting. Uh observationally, I don't think that we have any chance. Okay? But, as I say here, I mean, this is a school on theory. Uh we can come then to I mean, I want to make some remarks that kind of go back to the what I said at the beginning of the lecture, and that's something that I didn't say yesterday, but I try to summarize now. Um Okay. Just to uh >> [clears throat] >> final remarks that I want to make are the following. The main theme of these [clears throat] lectures is that in this regime of very small temperature black holes are highly quantum objects. That maybe it shouldn't be surprising. Okay? Another theme of the what I'm saying is that the black holes are objects that follow the laws of quantum mechanics. Black holes behave Black holes behave as quantum mechanical systems. This is not to say that we're solving the information problem or the unitary problem. That's another problem. I mean, the unitarity problem is also asking whether black holes behave as quantum mechanical systems. But, we had already argued at the beginning of the the lecture that there are problems with the description of a black hole as a black hole as a conventional system that appears at low temperatures. So, it looked like uh black holes at uh low T low temperature are they quantum mechanical good conventional quantum mechanical uh systems. We found that the fact that they have non-zero entropy at the classical level and that they have a uh radiation can't be uh would seem to violate QFT in curved space-time. Okay? So, that the radiation spectrum that we get from quantum field theory in curved space-time didn't make sense. So, this is uh something that would tell you that well, there's some problem with uh black holes at a low temperature, and it's unclear whether they are going to behave like conventional quantum systems. These two problems we have solved. What we have done, so now we know that the entropy or the density of states as a function of the energy, goes to zero when the energy goes to zero. And radiation is calculable and not thermal. But, to cal- calculate it to compute it, we've used the just conventional quantum mechanics. So, black holes are conventional quantum systems. Black holes are not outliers. We can bring them into the fold of uh physics as we understand the rest of the world. Okay? I think that this is only a very important conclusion. So, as I say, the black hole information problem is also asking whether black holes are conventional quantum systems, whether they behave according to unitary quantum mechanics. What we're doing here doesn't address the unitary problem, but it gives us confidence that black holes should also in that case behave as conventional quantum systems and follow unitary quantum mechanics. Okay? So, that's uh one of I think uh the good news of this all of this analysis. Okay? Now, >> [clears throat] >> again, going back now to things that we have to revise. What does this mean? Well, it means that as I was saying the extremal black hole is not an object in physics. The classical extremal black hole is not an object in physics. Despite what we've been teaching and learning for many years. Let me see why it's not So, we found that the corrections to the entropy uh were went like log T. Okay? Of course, this factor log T. The leading uh term in the entropy that's the area of the black hole, that's 40 H bar. Okay? So, what we have is that uh delta S over S up to the factor of G Newton that goes like H bar log T. This is what I want, and there's some factors here. Factor depending on the area. The point that I wanted to make is that the classical extremal black hole is the result of taking the wrong order of limits. You take the When you consider the classical extremal black hole you're taking first H bar goes to zero to go to the classical theory and then you take T goes to zero, or if you want, kappa goes to zero. Okay? And we see that that's wrong. If you send H bar to zero and then well, H bar goes to zero would kill this completely. The right limit The correct way of doing correct procedure is you keep H bar fixed and it can be small and then you lower the temperature. And then as you do this this is small but not zero, there's going to be some value of the temperature at which these corrections are going to be large. Okay? And there's always a temperature at which smaller than one. So, this is wrong. The classical extremal black hole doesn't exist, and we should uh say this clearly. Okay? Now, let me just say one more thing about the quantum corrections to the entropy. As I was saying in general you have this expression for the Bekenstein-Hawking expression for the for the entropy. Then quantum corrections, we've seen that they give us this effect. But, then you may have heard about the other log corrections to the black hole entropy. Those would be some constant here log area. Log area corrections, they appear universally as quantum corrections to to black hole entropy. So, both of them come there with a log, but they correspond to different physics. So, this is what comes from quantum fluctuations at the scale of the horizon radius. That's the width of the throat. Okay? And this is the scale that enters enters here. This is log R plus. This is a coefficient that in general you have to work out. That's something that requires some work. People have been doing this for different kinds of theories and black holes, but these corrections are always small because log area is always much smaller than area. Okay? But, these ones these are quantum fluctuations at the scale of in terms of a length scale, the order of one over T. And this can be much larger. Okay? So, this is the the difference. So, we have both of them. These are always small, but these are the ones that we could have anticipated these effects that should be there. But it's only more recently that we've understood that we can compute them with with control. Okay? So, then that's a uh entire picture. These come from different contribution from the one with determinant, different scales. These are the ones that we obtain from the Schwarzian, and these are the ones that tells us that tell us that the extremal limit in in general it's undefined. I think that I will stop here and then uh well, ask you for questions um in our If you have any questions, then I will we can discuss them now. Okay? >> [applause] >> Okay, and we can start discussion.