Physics of near-extremal black holes I

Yeah. Okay. Let us start for first lecture of today. We are happy to have Rabert Empiran with his course on physics of near extremal black holes. >> Thank you. So um today we will have another uh discussion session uh another in the morning. So uh if you have any question now that you want to ask now of course you can do it you can ask it now but otherwise we can postpone it to later of course during the lecture you can interrupt me as you see fit but uh unless there's something urgent I will start right away talking about the problem of interaction with radiation. Okay. So, well, we've seen that uh we have these uh black holes which are close to externality are fully quantum objects and now we want to understand how they interact with uh well radiation field that may live in their presence and this radiation field could be a scalar, could be gravitons, could be photons, could be whatever. Okay. Of course, the phenomena that we want to to study are phenomena where this uh radiation field is emitted by the black hole or absorbed by the black hole. If it's a spontaneous emission, that will be uh hawking radiation. For absorption, we can consider that we're performing some scattering experiment where we throw a wave into the black hole and then we measure the scattered wave. we can measure things like uh amplitudes, reflected and transmitted amplitudes, cross-sections and so on. Okay, so in this way we're going to be probing this uh quantum regime of uh of the black hole. We're interested, of course, in the regime where uh the energy and the temperature of the black hole are comparable to this uh scale. This is a regime where the black hole has this large quantum fluctuations. But then if we if this black hole is interacting with a radiation field say with send wave if this wave has a frequency that's much larger than this scale. Then what will happen is that the black hole will absorb the wave and then it will get excited into a state that's going to be uh back again into the semiclassical regime. Okay, the regime where this energy is much larger than the energy of a typical quantum regime. Okay, so this is not what we want to do. What we're interested in is the case where this is all of these scales are are comparable. Okay? Because then what we will have is the phenomena that we already mentioned that uh the black hole may be emitting radiation or absorbing radiation and this radiation is going to be a significant fraction of the energy above externality. So this is uh in this case the interaction with radiation will have a strong effect on the black hole state. That's uh to say that the radiation the interaction with the radiation will have a strong back reaction on the black hole state and this is something that we cannot describe using the framework the usual framework of semicassical quantum field theory in a curve space time. Okay. So that's what we're interested in something that goes beyond this this framework. The way that we're going to do it is taking this same idea that we've been exploiting when we were studying the black hole that we can divide the black hole geometry the system. We can divide it into a far region which is essent essentially flat or mostly flat weekly curve and then a throat region that's glued into this uh far region. The throat is has a geometry that's a synthetically ads cross s2 and then then it's glued to the synthetic flat region in what we call the mouth. That's the boundary of the ads region. Okay. So we're going to do the same thing for the propagation of the field. We're going to separate the propagation of field into these two regions. Uh propagation in the far region that's simple because there the geometry is not uh fluctuating much. So we can still consider that we have a field propagating in a fixed geometry a classical fixed geometry. Okay. So that uh the field equation that we have to deal in that case is just essentially the client Gordon in a weekly curved background. In the near region things get more interesting because then well we have the black hole propagating in the this road. At the classical level, we can still consider that the black hole is propagating in an ads cross s2 uh geometry and that's something that we can analyze and and solve. Uh but uh since we're solving we're separating the geometry between near and far. Okay, that's near and far. We solve the equation the wave equation in these two regions separately. And then we have to match the two solutions at the mouth at the overlap region where the the two the two the two regions uh meet in the near horizon zone. Well, the solution will be completely determined up to an overall normalization because we're considering linear field. The solution will be determined by imposing a boundary condition at the horizon. Okay. And then uh so here we will have I say we will impose a boundary condition at horizon and then this determines the solution of the wave equation up to overall normalization which as I say in a linear problem that's undetermined. Then when we match this to the solution the near zone then we will obtain the relative amplitudes between the two independent solutions of the wave equation. So then the relative amplitudes between the the two these two solutions what they give us is the relative amplitudes between the outgoing and ingoing waves and that's solving the uh that's the way that we solve the scattering problem. Okay. So this produces ratio of amplitudes or amplitudes of ingoing or incoming and outgoing waves. Okay? And then in this way we solve the full scattering problem. Okay? So that's the idea. This is what's known as the method of match asintotic expansions where you can separate these two regions cleanly. You match them in a common region of validity of the two expansions and then this allows you to solve the the equation in a complete way. This is a technique that's very useful because very often the equations that you find for instance in this the equations if you write down the the Klein Gordon equations in the R no background what you find is an equation which is of a ho type and you know that the ho equations are difficult to deal with. There are many different techniques but one way of dealing with them is to separate them into two regions where the equations become essentially of hypergeometric type and then you can solve them. Okay. So that's a way technically it's it's a way of getting the connection coefficient for the coin equation. We're doing this through a match into the expansion. This mat this separation of the field into these two regions will be valid in the low frequency regime. Okay. We will see that now in the near region. We can reinterpret this uh the solution to the wave equation in terms of source and response. And for those of you who have a familiarity with ADS CFT that will come handy because uh this is a language that's used all the time in ads duality. It is that we're going to solve in this near zone where we have this ads to process to geometry. We solve the wave the wave equation. We can solve it in this ads to resto region. I'm going to use we were using the variable row for the radius. So then we can expand this solution near the boundary of ads2 that's the mouth. When we perform this expansion then we find that the solution takes the following form. Okay. So there are two independent we're dealing with a second order differential equation. So there are two independent solutions. One of them is a solution that grows grows towards the boundary. The other one is a solution that decreases towards the boundary. Okay, here I'm separating. This is the part that grows and then dot the dots here refer to terms that are determined in terms of the amplitude of this uh the leading term and the amplitude of the growing solution. So the solution is determined up to some point. At another point the decrease in solution kicks in. So this is not determined by this uh uh the value of the growing solution and then all of the solution is determined once you give these two asintotic uh data. Okay, the growing and the decreasing part. This delta here, that's a coefficient that depends on the properties of the field. Uh whether it's a massive field or whether it's a it has a spin. And this is an exponent that we will call the conformal dimension of the field. But for us, for now, you can just think of it as some number that depends on the properties of the field. And it's something that comes out directly from writing the wave equation. So we'll see. Okay. So as I said the scattering of the solution in the near region is specified is given uh once we or it's fully determined once we give these two functions. If we impose the condition of regularity on the horizon then this will fix regularity on the horizon. This could be ingoing an ingoing condition on the horizon. then this will fix the ratio between these two amplitudes. So horizon regularity or in going horizon conditions which is what we will impose Okay. And then when we match this solution to the far region then everything is determined by by this I mean the ingoing and outgoing amplitudes at infinity will be determined in terms of this ratio as we will say. Okay. Now so then what this means is that what the far field sees of the near horizon propagation is just this ratio. This is all of the information that we need in order to determine how the field in the far region is behaving. Okay, this solution is what we call the non-normalizable part of the solution of the of the field. That's because if you take the client order norm of the of your scalar field, then that component is going to be non-normalizable. The fact that it's not normalizable in this region, what it means is that this is something that you keep fixed. It's something that you specify. We should think of it as a source. And the other component, the one that decreases. So this grows less uh quickly towards the boundary. So this will be normalizable. And then we think of it as the response. Okay. So the problem of propagation in this field and the presence of the background we think we can think that we're specifying some source at infinity. Then this source interacts with the black hole. The black hole responds by giving a value to the field to this component of the field. Okay. So we should think of this as a problem in say linear response theory. General this source will be time dependent. We can call it usually use the terminal of J for sources. This is the a source for an operator of dimension delta in the person of the black hole. So the action of the source gives an expectation value. This component is going to be the expectation value of the response operator. Okay. So this is something that we're doing just in the it's just a way a fancy uh reformulation of the problem of a classical propagation of the field in the presence of the black hole. Okay. But the the way that the reason that we're adopting this language is because then we can extend it to the quantum regime. Okay. So in the case of the where the black hole is a quantum object then we shouldn't talk about the near horizon geometry because uh there's part of that geometry that's undergoing large quantum fluctuations. So we don't have a good classical description of the black hole in that regime and then uh what we know in that case is the black hole is described by this boundary boundary mode the swartsian mode. So instead of considering that we have this throat with a given metric this is the throat of ads2 cross s2 in instead of considering the field propagating in this region now we don't know that uh or we think what the idea is that this is not described by a semiclassical geometry instead the picture in the quantum regime that we have is that we have the asintotical flat region that's always classical but we're replacing this here with some boundary modes that this is what our field is going to be interacting with this boundary mode we say this is the schwarsian theory that describes the black cover okay and this can fluctuate we saw that the we had a theory that theory describes how this boundary is fluctuating. Okay. So the entire system now using this language. So instead of talking about the geometry of the black hole, what we're going to talk about is an energy state of the swian theory. Okay. Instead of a D geometry what we have some energy against state and then the entire system the action for the system is going to be on the one hand the swassan theory that describes the black hole part and then an interaction term that tells us how this uh uh I'm putting it with a plus or with a minus I don't remember doesn't matter. So the interaction of the external field with the black hole is going to be described by an interaction of this term where we have a source and a response. The response tells us how the black hole reacts to the external perturbation induced by by the field. Okay. So this is the interaction with radiation. This is the black hole system. So we're going to deal instead of with classical propagation this geometry. What we're going to uh deal with in the quantum regime is this. We're not talking about the black hole throat anymore. All of that has been replaced by this. And this is something that we can do very generally. Okay. So whenever you want to study any problem involving the uh physics of the quantum black hole interacting with some external field what you do is this you first uh take the classical problem you rewrite it in this language and then what you do is you replace the classical objects there's a classical uh field expectation values with an operator with a quantum operator. >> I have a question. >> Yes. So in this picture the the operator O is it made with the Schwartzen field or is some extra metal external field that's the external matter. So that's that would be so this O is essentially this part of the field at the classical level what we have this correspondence in the operator the expectation value of the operator that's what you say the classical level is this component of the field. So all of this is the field. But of course this uh response this depends on it's in the interaction of the field with the black hole background. Okay. So the expectation value depends on that but it's made of the of the field. It's the field response to the interaction with the black hole. Okay. So in the quantum uh quantum language what we're using is the ideas are the language of linear response theory. We want to compute you already know this. We want to compute say the expectation value of a field in in the presence of a source. In a quantum theory what I'm going to do is some just some schematic uh description of what linear response theory. You know in quantum field theory this is going to be given. I mean we can expand this. This is the response the operator response to uh an external source acting on the system. We can expand it in linear response theory. We can consider that this is a small source. So we expand linearly. So this is we perform the expansion. It's going to be like this at linear order. This is an obity that we can compute in quantum field theory in terms of a path integral. could be orian I'm not speifying this is the usual expression quantum field theory you add a source to your action then you take a functional derivative with respect to the source uh okay this shouldn't be here because the derivative brings the operator down. So I skipped one step. So then when you take this derivative and then you set this at J equals Z because this is evaluated at G equals Z in the expansion. Oh no sorry this was this was here. It was correctly here. Okay. So we want this uh then this brings another operator down here. We have here the the action of the field at J= Z* J. Okay. and then this is expectation value of the operator multiplied by J. Okay. So what I'm saying is that this ratio between the response and source is given by the two point correlation function. Okay. And the point of this is to say that if we want to evaluate this response this thing or the ratio between the response to the source then this response is given by the two point correlation function of the field. Right? So this is the object that we want to evaluate. Okay that we want to use this amplitude of the response divided by the amplitude of the source. And then in the classical classical propagation proparation the classical proparation gives you this twopoint correlation function. as the ratio between the uh growing and the the ratio between the decreasing and the growing components of the of the field. Okay, so that's in ad CFT that's the way to compute the twooint correlation function of your field. Some background at least in the leading classical approximation. Okay, so classical physics gives you an approximation to this. But then if we want to go now to the quantum regime, what we have to do is instead of obtaining this from the classical propagation of the field, we have to take the twooint correlation function of a primary coupled to JT gravity when JT gravity is fully quantized. It's a fully quantum object. Okay. And this is something that has been done. It has been computed. So we said this is the information that we need in order to fully solve the scattering problem. In the classical case to solve the problem of scattering in a quantum black hole. Then what we have to use the correlation function of the appropriate field operator the quantum JT of theory. And these objects they have been computed. People in the last decade have computed this for uh primaries conformal primaries of arbitrary delta. This is just I'm going to go now into more calculation than this but this is to explain the philosophy of what we want to do. Okay. So we begin with a classical propagation. we reduce it to a problem of computing this ratio and this ratio we reinterpret it as response amplitude to source amplitude. We replace then we by doing this we've integrated out all of the near horizon region all of the information about the near horizon region is contained in this ratio. Once we've uh formulated everything in terms of a problem of interaction with the field of the field with uh theory living at the boundary of the throat region then uh we can forget about this part and treat it. If we have a full quantum treatment of this system then we apply it and then we obtain the full quantum correlation function and the quantum scattering amplitude. Is this philosophy more or less clear? Is it clear enough? Yes. >> Hello. I don't understand this way of doing the expectation value. Uh why you write a J at the end? >> So J here this J is a source that I introduce in quantum field theory will >> the multiplied J. There's a factor of J outside, right? There's a where where are they? uh so they so so here >> I'm taking I'm expanding this for a small j then I'm writing this >> as the expectation >> multip I mean >> it's just a >> okay okay >> is it >> uh and okay then fine then you take a derivative next one >> so I take the derivative >> then this brings well I mean there's a factor factors of I I don't know. Yeah, I mean factors of not this is >> why do you get a zero there? Five of >> I I Why do I set it at J equals Z? >> No, no. Theta of Z O of zero >> theta or this >> down. >> Yes. >> Yeah. >> Oh, well, yeah, that uh uh Okay, not sure. I can it's not uh going to be terribly important. Let me think about it and then I will get back to yeah why I'm putting it at zero. In principle it should be at I mean what I'm going to be assuming at all times is that uh I have my black on configuration of equilibrium. So everything is going to depend just on the difference between the two times. >> Okay. >> That's something that's going to be implicit in everything that I say and that's behind this bar. Yeah. you can have a correlation function of the two different points. But if I'm in an equilibrium state will depend only on the difference of times. >> Okay. >> So now yeah I'm going to begin solving the yeah the classical problem. Yes. So I'm taking that I mean this is going to be linear response. I'm assuming that that's a smaller perturbation. If uh the perturbation is large then this is not valid anymore. Okay. Then when I do the full quantum calculation I'm going to change slightly the the way that I do things because I'm going to go to uh FIA space. I'm going to go to frequency space and then I'm going to formulate it in a slightly different way that makes it uh clear the connection to a problem in quantum mechanics. Yes. >> Yeah. The >> question okay yeah there was a yeah okay over there and then back. >> Okay. Sorry. Um could you clarify what is the action I of fi is not the data >> this this action is the action of the entire system this schematic it's going to be at the is the action that we wrote the other day is the action of the of the black hole system >> okay so that's the daton >> yeah it's the action of the black hole system and also the the the the field so so it's a dilaton gravity coupled to the field but the field is going to be just at the linear Okay, thank you. >> Sorry. Sorry. >> Yes. >> And should you also in this J is time dependent because otherwise you should >> Yeah. So that's uh yeah I haven't been too careful that you wanted to do just something that's schematic to say I mean what I wanted to arrive is at something of this yeah that J in principle that should be time dependent. So the dependence of time that's something that I haven't been too careful about specifying here. Thank you. >> The goal is not that I'm not going to use this this formula as such. What I want to say is that the information for the scattering problem is contained in the twooint function. Twooint function. >> Yeah, I was Yeah, I was asking this file is the deon field, right? This I file. >> No, this file is not the dilaton. This is the scalar field. >> Okay. >> Yeah, I'm using maybe I should use I'm going to later change to a different but this is the the field. So this field here, this is not the diloton. Sorry if I've made it look like that. It's this uh fi is the scalar field that's propagated in the presence of the black colon. That's the same fi. >> So this i of is the shorian then >> uh i of phi. Yeah, I mean here yeah I should put should have I haven't specified. So this is phi and then this is the black hole all the black hole system. Okay. So which is the gravity system or daton gravity if you want JT you have both of them and then this is the interaction between them. So is the idea of coupling it is that you will reparameimeterize t that will have some cost in the schwarzian and then you find the that will find settle point and >> so like when we will try to do this integration so the settle settle point would be like reparimeterizations of t. >> So no the settle point that's not going to be affected. Uh maybe I'm I should have should have uh explained this. This is this was just supposed to be a quick uh uh rough and ready uh derivation of this uh result that uh response uh linear response theory the it's encoded in the twopoint correlation function. I didn't want to yeah maybe I should have done this uh more careful in more detail. Okay. But uh yeah, sorry that I yeah should have done this more carefully. Okay. So now what I'm going to do is uh to solve the problem of scattering in the classical the classical case so we can see how all of this works and then we go to the quantum regime. By the way, I mean since we're interested in studying the problem of uh emission of radiation and that's how hogen emission, what information do we need in order to compute the correct rate of Hawking emission. Well, in general, what we have is the the radiation radiating power uh some given frequency that's given by Okay, this is a sum over angular momentum modes. So this will depending we decompose our field in partial with components treat them separately then we sum over all of them. This is the radiating power of the black hole. The important uh object here, the one that we want to compute is the probability of absorption that the black hole absorbs a quantum of frequency omega. This is it's absorption probability sometimes also known as the gray body factor. Here we assume that we're putting the black in the presence of a thermal bath of radiation. So that gives us this uh thermal factor and then what we need to compute is this uh this number. Okay. How what's the probability that the black will absorb some quantum radiation? We're going to do first the classical calculation and then we will do and that's by just solving the wave equation. in the presence of the black hole and then we will do this in the quantum case. Okay. So for computing this absorption probability what we have to consider as I say solve the wave equation and we're going to do it in a regime of a low frequency that's uh more specifically we want the frequency to be much smaller than the inverse of the radius of the black hole. So we have a small number in our system which is this number. Okay. Why do we do this? Because in this case we can do this uh clean uh this separation between the propagation the far and the near zone. We can do it cleanly. We will see this. Okay. So we want to write down the wave equation for a field that now to distinguish it from the dilaton I'm going to call it. To take it to be massless for simplicity massless minimally coupled scalar field we can decompose this into partial waves. This is the field for a given frequency omega. Okay. So to write down this equation we use that box operator takes this form and then uh we can write down the equation now straightforward to write it in a metric. This one the reson is takes this takes this form. So we can write down the wave equation. It's a So this is the wave equation for uh getting intuition. It's often conven convenient to rewrite it in the form of a rear equation in a in a potential. To do this we change uh variables. Instead of using the pai we introduce another function r and we also introduce the tortoise coordinate which is uh given by r star is of dr r / f of r. Okay. When you do this the wave equation takes the following form. this derivative with respect to rar. So now this uh is the form of shreddinger where you have a second derivative and then we have here a potential this term. So a potential of r or rar. If we draw the potential takes the following form this rstar coordinate at infinity becomes the same as the radial coordinate. So here we're going to infinity. The horizon is at rar going to minus infinity and in between the potential takes a form kind of like this that's a big this region the potential falls off like 1 / r² this is due to centrifugal barriers but also for gra it's also due to gradient barriers because compressing the field in the radial direction cost energy. The height of the potential that's given by this term. So this the height over the potential is of the order of 2 m plus uh l L + one. Okay. The maximum is around this radius is around what's called the photon ring approximately for wave propagation. Okay. So you see you have this potential you want to perform scattering the height of the barrier potential barrier is larger for larger L because uh angular momentum prevents modes from reaching the central region. So then what this means is that at small frequencies at low frequencies scattering is going to be dominated by S waves. What's which is what you would expect. Okay. S waves don't have a have a potential barrier. So it's easier for them to uh penetrate the potential barrier and get to the black hole. So absorption is going to be dominated by S wave. And this is what we're going to consider from here on. Okay. So S wave dominates at the low frequencies and that's what we're going to do. Okay. So now we want to solve the wave equation for uh S waves at low frequencies. And the way to do it as I say if you look at this equation this equation is in principle of ho type but you don't give up and one of the ways of uh doing it is going to be as I say this method of matched asotic expansions. So we're not going to use this form of the equation to solve it. We're going to return to to the previous one and then we sent it we set it at uh L equals Z. So that's f r² f partial r sorry and we want to we're going to assume that we have omega r plus much less than one. So what are the two regions far and near regions here? So we define the far region as a region that where R is much larger than R plus. Okay, here we're taking this F this function. Okay, so that's the region far from the black hole and then the near region we define it as much less than 1 / omega. Okay. And what this means is that there's an overlap region between the near and the far region when omega r plus is very small. Then these two regions have a an overlap uh common region of validity. So overlap is where both expansions are going to be valid. Okay. So we can match the solutions obtained here to the solutions here. So far I haven't uh needed to specify that the black hole is uh close to externality. This is valid also for say black holes far from externality and actually the much of the analysis that I'm going to describe is valid for spherical symmetric black holes with large degree of generality. Okay. But later we will specify that we're close to externality. So when we solve in the far region then what we have is that f is approximately one. So that's essentially the solution in flat space. The equation in this case when you set f to one then the solution in the far region going to be this form. Okay. So, we have two different solutions, two independent solutions. This is the ingoing wave and this is the outgoing wave. and they come with independent amplitudes. If we can determine what is the ratio between the two amplitudes, then we have the uh reflected and transmitted amplitudes and then we're essentially solving the uh scattering problem. Okay. So the scattering problem consists of determining what's the ratio between these two components of the wave. For that we need a solution in the near region. Okay. And in the near region, what I do is I approximate this function by this. Okay, that's consistent. You can verify that that's consistent with the way that I defined it. But by doing this if I take my geometry and I take f to be of this form then the geometry is renderer ads2. Okay. So what I'm going to be solving is essentially the equation the wave equation in renderer ads2. can solve the equation that's uh you can write it down and do it by consulting books on special functions or you can ask Mathematica to do that for you. Then you impose that you have a uh the horizon in going solution that's regular with ingoing boundary conditions. And then this solution the solution the near zone is of this form. to kapa where kapa is the surface gravity. Okay, kapa is now this is of course uh fixed up to overall amplitude. The amplitude will not matter. We can set it to one if we want to. But this is the solution that we need. And now we want to see how this behaves in the boundary of the far region of the near region. Okay. So we expand this for lat r and then the solution. This is form and this is precise of precisely the form that we said earlier. Okay. So here we have a large component and a small component near the boundary. So this is the source and this is the response. This is what we were calling f not call it now here s not and this is s1 and you see that the exponents here correspond to delta equals 1. We had a for a field of conformal dimension delta this would be r to the delta minus one. This would be r 1 / r to the delta. So this is a field of conformal dimension one. Okay. And this as I said this is non normalizable. This is normalizable component. In general what you would find as we were saying is c1 + c2 / r. But here we have determined the ratio by imposing this condition at the horizon. Okay. And as I said doing this is equivalent to determine the twooint function of the operator. Okay. So by solving this we have found we have obtained the two point function of the operator O of delta O1. Now we can match this this solution to this other one. When we take this solution to be at small values of R that's where they will match. Okay. So take now si for a small radi or r much less than 1 / omega. Okay. So then this solution becomes minus i. Uh I don't know why I have this factors of minus i. This should be Okay, again you should verify my science and my my factors but the structure it should be clear that it's the same right when you go to a small uh when this is small then uh you expand this uh this function yeah and you get the factors of y think that they are right so you get a part that is a constant and a part that uh goes like one / So then this we can match to the solution. And then if we set C1 equals to 1. Okay. So this is the ingoing amplitude. This is the outgoing amplitude. Okay. So this is uh the full solution to the scattering problem. Okay. So now we have we can compute uh say transmitted amplitudes and reflected amplitudes without you see that these amplitudes the difference between them is a small number because the difference between them is this parameter and we're assuming this to be small. What this means is that the coupling between the wave and the black hole is small. Okay. So this will justifies linear response theory to begin with. Okay. So this is what we need in order to compute the absorption probability. We have everything that we need. The absorption probability we can compute it. I say this can obtain it. You have a flux of the field. The flux of the field is given by the uh climb more than current. So out of the this measures the current. You can make use this to measure the flux flux at infinity. Obsession probability is the difference between the flux in and out at infinity divided by the flux in at infinity. Of course this this difference between the fluxes at infinity they are also related because this is a conserve current to the ingoing flux at the horizon. So you can compute that in different the two ways in the two places infinity other horizon. So this is is that so the absorption probability we can now plug in the result that we had here and we obtain it. Okay. So final result, what we wanted here is this. Okay. So if you plug this into this expression then you get the result uh all the hawking result for uh radiating power of uh black hole uh emitting hawking radiation. The absion probability you can also convert it using essentially the optical theorem into an absorption cross-section in four dimensions. The absorption cross-section is related to the absorption probability by this expression. This expression theory. So in the case of a black hole as we have computed you plug this in then you find that the absorption pro the absorption cross-section is exactly equal to the area of the black hole. This is a general result that can be derived in marginality for any spherical spherically symmetric black hole absorbing a minimally coupled scalar field. Okay. In a sense, it's a natural result. Okay. So, you're sending a scalar wave into the black hole. The partial higher partial wave components, they are going to be uh mostly scattered off. Only a small amplitude of them will get to the horizon. The component that dominates this S wave. So, you're essentially what what's reaching the black hole is a spherically symmetric wave. On a spherically symmetric black hole, you impose your imposing perfectly absorbing conditions at the horizon. So it's just natural that the absorption cross-section since it's uh absorbing equally well in all directions that this absorption cross-section is going to be exactly equal to the area in the low frequency limit. Okay. So this is a result that's valid to lead in order at small frequencies at higher frequencies because you get corrections to this and you can corre you can compute them perturatively and that's uh yeah something that you can do systematically. Okay. So this is the information that uh this gray body factor or this uh absion probability is the information that we need in order to characterize our scattering problem. Okay, it's not of course it's not all of the information. The scattering amplitude also contains some information about the phase delays that we're not using. Okay, we're only using the part that's uh corresponds to to absorption. As I say this information the way that we have solved the problem we have obtained this two components this uh the amplitude of these two components which as we said this is uh equal up to this philosophically equal to response to source. Okay. And this is the information of the twooint correlation function. The G function is the imaginary part of the range function. If you want the imaginary part and that's what corresponds to absorption. Okay. Any questions up to this point? What are we done here? Now we're going to go to the to the quantum regime. Why do I do this? Yeah. In the notes that I gave to upload in the website there's more detail all the details of the rest of the calculations are also given in notes written in lat. So now we're going to that we've done the classical calculation we're going to do it now for the full quantum system. As I said, we're going to change slightly the language. It's going to be at the end of the day and the idea, the concepts are going to be the same. The information, all of the information that we're going to use, it's going to come from the two point correlation function. But I'm going to reformulate it in a slightly different way so that it emphasizes that what we're solving is the problem of a quantum mechanical system or black hole interacting with an external driving field. Okay, the external driving field is going to be the uh external ex scalar field. Okay, so the far field propagation that uh goes through I mean this is still going to be the same. We have the amplitude, we have these components going to be that depends on So we solve the problem as we we saw before we have to I was denoting this with letters D but that's the same and at the matching mouth what we have is This is some uh field s of t plus a component that goes like this. Okay. So this is the source for delta equal one conformal primary. Okay. We de compose this source frequency components we take them separately. So the interaction Hamiltonian is source times our operator of t. Okay. So we have our system we have our source and black hole our quantum system interacting with the radiation field. So our black hole is say that our black hole is initially in some state of energy either the interaction with the radiation field can lead and drive it to a state of higher energy that's going to be absorption or and lead to emission. This emission can be stimulated or spontaneous. Okay, so that's the problem that we want to to solve. Now we have our quantum mechanical system described by the swashian theory. the interaction described by this term. And then since we've seen that in the regime that we're interested in the system the coupling is weak then we can apply now the firmmy golden rule to compute the transition probabilities induced by the external field. Okay, the firming golden rule. Now I'm going now to the usual stuff that you find in textbooks on atomic physics, right? Quant to mechanical physics. So the transition probability from an initial to a final state, the transition amplitude according to Fermy. given by this formula. Okay, we're going to see that this is I mean I mentioned before that information comes from the two point correlation function. This is going to come from the twooint correlation function. Okay, I'm just uh coaching this in a different language a language that you may be familiar with think that you must be familiar with from your studies of quantum mechanics atomic physics. Yeah. So this is the fmy golden rule that tells you that the transition probability between an initial state where the black hole has this energy and this is the number of quant of the radiation field. Uh this uh state can make a transition to a final state with a given a different energy and a different number of final quanta. This transition is mediated by this Hamiltonian. This is the Hamiltonian that we have here. Okay. the interaction Hamiltonian and uh yeah what we need in order to compute this uh probability this transfer probability is this this information. Of course the if the system can make a transition to a number of final states you sum over all of them and this is what uh appears here the density of final states. Okay, the matching calculation gives us what is the amplitude of uh this coupling. Okay, this is something that uh well you can obtain from the previous calculation. What we have is this is going to be proportional to the number of quant that we have. And if you take care of factors then this uh the amplitude of the source properly normalize according also to a normal that we take for the conformal primary. Uh this is something that comes out of the matching calculation the calculation to the far matching to the far field not giving the details but uh you can see that it's coming from there. So this is a general expression. If we want to compute now these four uh to obtain the absorption probability. This will be proportional to the number of quanta that the field incoming field has. Okay. So this we're absorbing and the probability of absorbing the quanta is given by this expression. This is something that we as I said this has been computed all of these things have been computed using the uh quantum swian theory. Okay. Quantum J gravity emission. You have EF. this by time reversal invariance. This is like this. I think I'm uh should be a minus sign here, I think. Let me check myself. not coming. Well, mission probabilities as I said by through time reversal invariance is directly related to the absorption probability. I think that the correct result in the notes I may have uh not written the right sign there just to verify Yeah, this is correct with the minus sign over here. Okay. So if we want to compute the total absorption probability then we have to subtract the two. Okay. The entire system is going to see there's a probability that the system absorbs but also that it emits. So the absorption probability up to factors that uh you can work out. Okay. So this is the total obsession probability of course we divide it by the number of external quanta. Yes. Um so the first part um we tried to solve for uh you know like the client got an equation for sigh in the background of a fixed classical black hole right so I guess here we are trying to account for the quantum fluctuations in the >> the quantum fluctuations in the near region yes >> yeah so I'm not sure how exactly we are trying to capture that through this calculation >> yeah so it's I think it's will become a little clear now because I said everything in terms of the twooint function this information that we need here I As as I said, I'm discussing now the problem in a different language, the language of quantum mechanics, a quantum mechanical system driven by an external oscillating force. Okay. And that's uh what I'm saying is that so we reduced everything or problem to the perm of interacting the interaction of the swian theory with an external field through this Hamiltonian interaction. Okay. And now I'm treating this I'm just applying now quantum mechanics to a system that I some quantum mechanical system that uh has this uh again states of the energy and then transitions between states mediated by this interaction Hamiltonian are given by that formula. Okay the way that I put it before is that all of this information the information for the transition comes from the twooint correlation function. We're going to see that this uh the twopoint correlation function gives us this matrix elements. >> I see. >> Uh is that uh answering your question or is it modeling the the things uh more? >> Uh yeah, I think I think it's helpful. No, I was just wondering like um is this everything we need to do to capture the fluctuations in the near horizon region? And I mean like I don't >> Yeah. So the fluctuations in the near horizon region are going to come from this twopoint correlation function which I'm going to >> right now and then we're going to be almost done. >> Okay. >> Okay. So what we need because this is something that we already computed for the swan theory. We need these matrix elements. These matrix elements they are given actually by expectation value of the correlation function. Okay. So this is the twooint correlation function evaluated in our state of energy E. You perform the uh FIA transform That's a step that it's given the notes. Don't have time not to do it. But this is ukidian because this I'm doing it in uklidian but then we continue to lorentian if we need to. This is an expression that you can obtain out of this by inserting a resolution of the identity. Insert here a resolution of the identity. So that's where the interal comes from. Sum over the total number of intermediate states. So you see that this information that we need here actually bundled also already with the density of states we can obtain it by performing an inverse laplas transform of the twooint correlation function. Okay. So this information is contained in the object that I said before the twooint correlation function. Okay. And these things have been computed in the literature. Uh well I mean the full expression I I'm giving it in the notes. Well maybe I will do it just write it for a scalar field of or a conformal field of dimension one. This is is this expression written express and this is the cinch function that we me This this is the c of square root of 2i that we have. So we have everything that we needed in order to compute the uh the probability the absorption the pro absorption probability. Okay. So maybe I'll stop now and then I will finish this in five 10 minutes in the discussion session. Right? Because I think it's it's time but we're almost done. Okay. So we have all the ingredients to reproduce that. Okay. So, we'll continue after the break. Okay. Yeah. Unless there Okay.