 So I had a plan to give you some Fermi estimates on this plot, like Fermi trying to get an idea of the energy released in the bomb by just looking at the picture. So I was going to do some dimensional analysis and then do some more calculations. But I think it would be educational and to do some actually more specific calculations. So we're going to do the reservations to some formula that you may have seen or you may have not. So it would be nice to see some calculation. And after we set up these calculations, that will give us the typical binding energy of a binary system and the radiated power, like through the quadruple formula, we will be able to estimate, by looking at this picture, what are the parameters of the source? Which is one of the key goals of gravitational wave astronomy to identify what is the source that produced the gravitational waves that were observed by light. So that was the detection. So what I'm going to do now is do some calculations. Also because when I asked whether some of you had seen this equation, how the geodesic deviation and the change in length based on the propagation of gravitational wave and TT gauge, I got the impression that some of you may have not seen some of these calculations. So I decided to do some basic computations and then we'll see if we can speed up based on how much you have seen before. OK, so the idea is to solve Einstein equations. We're going to do the leading order calculations, which is in the weak field limit. So we have some source, some distribution. In this case, a leading order is going to be some mass that we're going to basically keep the mass distribution, but we have some T min, some stress and it's a tensor. We're going to have the gravitational waves emitted by the time variation of this source. And we're going to write the equation that will give us what is the leading order, which we'll see in which expansion parameters we're going to be computing. The mission of gravitational waves and this TT gauge that eventually is going to be what enters in the LIGO detector or what is going to produce the pattern that we observe in the LIGO detector. So I'm going to be setting this coordinates. I'm in the center of mass. Eventually, this is going to be the binary, but let's make this generic. So some distribution of mass. We're going to sit in the center of mass. We're going to look far away at some distance d. And this viral x prime essentially moves around inside the distribution. And when we have two binaries, we're going to have x1 and x2 only. And we'll see how we're going to treat the fact that the black holes are going to have size. But to first approximation, we'll see that this is will be a delta function of m1 and m2, to leading order. So that's just again ahead of myself. I just basically tell you that keep in mind that we're going to be studying the binary problem, but this is very generic. So we're going to solve these equations with some stress energy tensor and in linearized gravity. So the approximation will be that we're far away. And the wavelength of the gravitational waves in millions are much longer than the size of the object. And this is going to be true for the most part when we can calculate, which is in the regime, for example, in the binary problem in which the velocity is. Sometimes I'm going to be using c's and g newtons. Sometimes I'm not. When it's useful, they will appear. So v over c is going to be less than 1. And then sometimes I'm going to just call it v. So the wavelength is much longer than the separation. And this is going to allow us to do a multiple expansion that is what is going to come out here, the quadruple moment, the octopal moment, so on and so forth. So this is what we're going to walk into. So we're going to solve our instance equations. And here, often, this bar means that we're using the Lorentz gauge. And we use this object that was not an issue when you are in the TT gauge because the trace, we chose it to vanish. Because we're solving the vacuum equations that allow us to give us some more gauge freedom. But in general, you may have seen that there is this guy that is used in this gauge to simplify your life and write this equation out of Einstein, right? And we do in linearized gravity. So we're using around flat space. Good, and as you might know, by using the green function associated with the box, delta t minus the t retarded, t retarded, or c. Then from here, we get the solution for h, which is equal to Newton. You'll see why I'm going to try an effort, because often, when you see this derivation, so you don't see all the factors of genutus and c. I've been trying to keep track of those. Because eventually, we're going to compare with data. I know that this is kind of weird. If we're doing a three-zero course, but we'll try to do that, which means we'll try to get some numbers, which are what they appear here in terms of amplitude, typical amplitude, typical frequencies, and times. And that's why it was going to be useful to see in gravity what are the typical timescales that show up. And remember, we're not doing quantum gravity, so Planck is not going to show up anywhere. So we're not going to use Planck times, Planck masses. Although, when we do calculations in the field theory approach, we will use this idea of five undergrams and the h-bars are going to pop up, but that will cancel in the finite calculation. So you will see in Planck's, probably at the end of these lectures, but that doesn't mean we're doing anything quantum. In fact, we can do quantum corrections and show that they're small, because the charges, in this case, the masses are huge. So essentially, all the three-level exchange dominates over the loop corrections. So that's why I'm trying to keep track of these guys. Hopefully, I didn't forget anybody. All right, so this is what you expect. This is the time. This is the space. And we're integrating over the source. This is nothing different than what you do in electrodynamics. You have an A and you have a J. So you all have seen one way or another this. So now we're going to do an approximation. We're going to say that this is very, very far away. So up to corrections that we'll try to quantify, and I'll show you how we're going to quantify. Essentially, we're going to separate. We're going to approximate the x prime minus x that appears there by the distance, 1 over t. Also, we're going to say that the time variations are slow, which is associated with the slow motion. So we can also replace it here in the time. And we're going to get an equation then from here. That's going to resemble more of what you expect in some approximation that we'll try to set up. So you get a 1 over distance, which is what you expect. And then you get an integral of this. So up to this part, this is just 1 over d, radiation field out of the source. A leading order. I mean, in principle, you can expand this x minus x prime and get the higher corrections and so on. Very good. So how do we massage this expression to get what you expect, which is all these multiple moments? So what we do is we use the conservation of Timinou. And this is going to be key because we're talking about gravity. And now I'm doing the approximation, in which I'm doing linearized gravity. So there's a covariant conservation of the Timinou associated with matter, but a linear order I'm not going to care about those corrections. There is a pseudotensor that is going to include the self-gravitations here. All the forces that are going to bind the system, those are going to gravitate as well. So there's going to be a new Timinou, and I'll show you where it's going to come from. But right now, we're just doing linearized gravity with some stress energy densors. It's just the matter part, which is conserved. And from here, we get something that relates time variations to spatial variations. So the time derivative of this guy. And this you can get just from doing it once. And when the COI, and related the COI to the CO0, doing it again, and then you get something like this. So these two are already giving us a hint of how to massage this expression. We're going to particularize this for the IJ, because ultimately, we want to go to the TT gauge. And in the TT gauge, remember, it's traceless, so this bar is not going to play much of a role. So what we're going to do is, OK, we have this guy, but we're going to do the following trick. We're going to multiply by x i, x j. And we're going to integrate. Now, one of the first lessons I learned in high school, this is not x and x prime. Don't get confused, right? This is just a dummy integral variable. I'm going to show you how to relate this integral to moments of this guy, OK? Even though I call it x here, x prime, right? Very good. So what you do here is you integrate by parts, assuming that you make this integral at infinity, and the source, obviously, is localized. So all the contributions from infinity are not going to show up. So this integral, in principle, is everywhere, right? Now, this is non-trivial when you include the gravitational field of the source into the T minu, which we will. These are key elements of the computation. There will be key elements of the computation. So there are things called tails, which is the gravitational wave emission will talk to the gravitational potential of the whole system far away, OK? Not only here, locally, also far away. Those tails are going to make this argument a little tricky, so I'm not going to talk about it, OK? From the point of view of how we do things, this is not an issue, but it requires a little salty. But here, I'm just using the T minu of the source, so everybody's happy is localized. Or even when we include the short modes of the gravitational potentials, that's also localized, so we don't have an issue with that. So if you integrate by parts, then you see you hit twice this guy. So you get something which is symmetrized, L, I, J. And then the symmetry, then you get the other guy and you get twice the same. So you get T, I, J plus T, J, I. And this is symmetric. And when we include the self-energy, we will make it symmetric too. So this gives you a 2. So if I put a half here, X, I, X, J, then eraser, right? Then we just show. And these are called moment relations. And this is an integral in X, so I can pull this guy out. And this depends on T and X. And I'm taking the derivative with respect to this guy. And I put a half. Then this is T, I, J, D, 3, X. It's an art to do this to higher orders. And you will see what I mean by higher orders. When you have higher order moments of different components of the T, of the T minu, okay? You will use these moment relations often using conservation laws to relate and get the radiative part of the field. Because using this moment relation from here, this integral is exactly the integral of T, I, J. The only difference is the time here. It's gonna be evaluated in the retard time. And therefore, it's right over here, the H, I, J. Let's keep the bar. It's 4G newton over C to the fourth D. And then you have a half, I have D by DT square of the integral of T, C, O, C, O, X, I, X, J, D, 3, X. And this object here, and this is evaluated, sorry. Minus D over C, X, X, I, X, J. And this object here, I put a half in the two here. It's evaluated in T, D, O, C. So this whole thing here is two derivatives. Because now that we did the integral in X, the only thing that has interest here is in T, but this is the variable you need to evaluate it. So you take two derivatives with respect to whatever this depends upon, which is T, and then you evaluate it here, okay? And therefore, H, I, J, it's equal to G newton over C to the fourth D. And this is the quadruple formula. So this guy is this guy, and this guy is this guy. And in the approximation that we're using, that we're neglecting the self-energy, this is just a density. And we're gonna project this, so we're almost done, we have to project this to the TT gauge. So we're gonna define the symmetric trace-free part of the leading-order quadruple, which is where it's gonna enter here. We're gonna use the mass density. All this is what we're gonna use later on for this plot. So a leading-order, this is gonna be just a mass. Remember, we're gonna go to the TT gauge in which is symmetric trace-free. So we're taking the trace out. So this is the usual thing, by the way, that gravitates. If you want to compute the newtonian potential produced by this thing, you will get the one of our Q times this guy, right, x i, x j. The same, if instead of solving the wave equation you will solve in the Poisson equation, you will get something very similar, right? Very good, so what we'll notice is something that we will use later on, something that all the time dependence comes from here. This integral is over all space, but what's gonna happen is that because the source is localized, when we look at the binary, the density is gonna put us on top of the masses, right? So that is gonna give us the formula, it's gonna be a sum over the masses as you will expect. Okay, notice something very interesting here is I'm telling you about separation of scales. So this is a multiple moment associated with a source. It's a moment of a source, right? I integrate over all the space, not space and time, space, and compute a moment in which I map this whole thing into a point with a quadruple, right? So what we did here, we map this interacting with a long wavelength radiation into a point like a source, in this case, the source is the quadruple, interacting with, but this is time dependent. We sort of, in the jargon of field theory, we sort of integrate out the space part, the short modes in space, but the time variation and the scale of the time variation that was happening here and the time variation of lambda is about the same. And therefore our Wilson coefficients, as we will see, terms like this that are gonna address our point-party collection are time dependent, which is not usual what happens. The Wilson coefficients usually are just numbers when you integrate out some, because you're in Lorentz invariant theory. And therefore space and time are the same. Now we don't have Lorentz invariants, we're breaking in the post-Newtonian expansion. It will be recovered order by order, but okay, that's a different story. So there is a split between space and time. And therefore our coefficients are gonna be time dependent. Okay, so to compute the H, to finally compute the H, we need to project into the TT gauge, which is what enters in the equation that give us the displacement. Remember we got the variations. What's that? Symmetric trace-free. Oh, sorry, I said it with words, but I didn't write it down. So it is symmetric, or you can symmetrize it if it isn't. Why we insist on symmetric in this case is obvious, but then a higher order that will depend on the T minu and the T minu will include the correction from the self-gravitational part and so you have to have a symmetric tensor. And trace-free, this is symmetric, trace-free. Free. Thank you. And this is what this is doing for. And why is it because we're gonna go to the TT gauge? So there was a T first for transfers. There is a T for trace, which removes the bar, because now H trace-free is the same as trace because we're removing the trace, which is zero. And here it is symmetric, but let's just put symmetric trace-free. We're almost done, we need one more T. This is our, as I was saying, this is the, actually I can show you again. This is this guy that will give us these equations down there that will tell us in the TT gauge, for example, the plasparization, how is it gonna change distances, which is what we observe here in terms of the perturbation, right? Okay. So we need to do the TT projection. So we need to go now to the transfers, and we're looking far away. So we have some here, some sphere, far away at some distance T and there's some norm. N, H, I over D. We do a projection into the TT gauge in general with this guy, so now we take the trace, but I already did, but okay. We're P, I, J. Projects away from the direction, the N direction, and then the I, J transfers traceless KL double dot. Let's put the two G newton over C to four D, and this is evaluated in T minus D. So this is our leading order expression for the gravitational waves emitted of a source that is changing its quadruple moment in time, and it depends on the second derivative of the quadruple moment, okay? Now there is something that you can ask, why this, you learn in many different flavors across your education, why you have dipole radiation, NEMM, why do you have quadruple radiation here, and if you look at the moments that you can write down, just naively, right? So it would say, what about lower moments? Well, if you write the moments of raw, you get the energy or the mass, which is conserved, so it wouldn't radiate. If you write the moment of X, then you get essentially the center of mass because it's M, X, sum over everything, right? So you get derivatives of that, you get the momentum, the momentum is conserved, so that doesn't radiate either. So then you get the second moment. You can get moments of the velocities, X and V's, with some cross products, so that will give you currents. Currents will give you angular momentum, but the total angular momentum is also conserved, so the current also will not radiate. So but you can have a, okay, we'll see. There are currents, there are also couples to magnetic, like the same way that you can have in NEM, so we'll see a little bit of that, but there's a clear then that the second moment is the one that can have some time dependence, just some conservation laws, right? That's what people often say. It's actually a little trickier, but ask me later to go back to that. How is really the conservation law working here? Because we see that the energy of the system has been lost in gravitational waves, right? So there is an M dot for the system, the energy of the system is not conserved, you're losing energy, okay? So it's a little tricky, but okay. But here, the approximation that we're doing things, this is M and M dot, for example, is definitely zero. At least unless you include absorption where the black holes start growing, which is an effect that you can include and include the conservation of energy in that case, it will give you some constraint, okay? So gravity is much more subtle than, well, GR people obviously don't know this. Okay, so now what happens is with this metric, we can compute the total radiated power. And this is done in the standard way by computing like the pointing vector with the stress energy tensor associated with the metric field, which is just a Einstein's equations and construct the pseudo stress energy tensor, which is conserved that is added to the T minu of matter to have a normal like a war identity. We will rewrite this in a more modern way. Hopefully by the end of the lectures, you'll see how we do this derivations, but I'm gonna do it in the standard manner just to illustrate how it's usually done. One comment here that I want to emphasize is that what happens when you start including this self-gravitational interactions, how would this change? Because here, for example, I define it with the mass. So how would you go on and to include those corrections? Well, how do we get this equation, right? It's a linearized gravity, you have a bulk term and you have a term that is T minu H minu, right? Well, we can just look at that term. We can look at the term that is T minu H minu. And we look in this separation of scales in which we keep the H associated with radiation as our long wavelength field. This is the guy, the virus on a scale of order lambda. And then we can treat whatever happens here, whatever happens here, that we're gonna call actually potentials. We're gonna take those potentials, that varying on the scales which are shorter than the scale of the wavelength in space, not in time. In time, they're gonna scale with the same time scale because everything in time changes with V. But in space, they're gonna be separated and this is gonna scale like one over. That's a mom, you know. What am I doing this? I'm doing this for this audience. And now, because you know this, we solve for those guys and plug it back in the action which formally means we integrate out these guys using the background field method in which this H is treated as a background. So we gauge fix the same way that I was doing everything but with covariant derivatives in which you keep this H inside the covariant derivatives. That's the standard background field method. To construct you T minu which is conserved and it's now this guy. That couples linearly to the radiation field which is your background. So you look at the linear piece and that pseudo tensor now includes all the potentials. And this guy is conserved. This guy is indeed conserved. And now includes both the source and also the part that is pure gravity like masses. So this is how we incorporate all the self gravity. We have to do the exact same thing that I just did, except that the leading order term, this would be the leading order term. Now we're gonna have a quadruple that's gonna have a T00. And this T00 we computed from the source but also from all the self gravity parts that are gonna come from here. Okay, and from the binary system diagrammatically what this means, you have two sources. I have this potential guys. I have my radiation guys. I need to compute things like this where I integrate out my potential modes and they all dress this guy. We'll see this a little bit more in detail when I do the modern view. So I'm trying to do the parallelism the way we basically rephrase many of these calculations in terms of five hundred. Exactly, exactly. It is the final size of the binary, yes. So now the binary became a point. In this expression, this is the multiple moment of the whole binary system. So the whole binary system or whatever distribution of matter you have in there, we map into something that only has, it's a zero plus one theory. It only has time dependence. And it's coupled to this radiation field. So this expression will stay exactly the same except that now the time dependence of this guy will include all the self gravity. Okay. So now there are also high multiples and you might ask, well, how are we gonna get those? And let me just, before I derive the quadruple formula, let me just make one quick comment because this will help us to also make a connection with the way we do things. Which rather what we do things rather than solving equations of motion, we go into the action because the action captures all the information that we need for the problem at hand, which is compute the radiated power and the binding energy of the system. Okay. So I wanna make a little connection with that, which you already see over here because we are writing an action because this is the action that you vary with respect to H to get the equation of motion. Right. And this gets dressed once you include all the self gravity. Okay. And if you're doing this at the level of the equation of motion, you get a bunch of guys here to the end power. Some of those we call potentials. We solve for them, we plug them back and we get something linear in the radiation. Okay. But instead of plugging back in the equation of motion, we do it at the level of the action by computing those diagrams because it's nothing but integrating out in the saddle point approximation, the potential modes. This is the way we rephrase the calculation. But I want to make one connection here which will be a fairly straightforward and to see how we reproduce these manipulations that I just told you at the level of the equations from the point of view of an action. This is my action. So what I do, which is similar to how this cube came out is do a multiple expansion of this field. So here is a T and an X because this is localized and the source and these variants are much longer scales and we multiple expand this field H menu. So we take the H menu and we multiple expand it around the center of mass that I'm gonna put at zero just for simplicity and we just doing this in space. Okay. This is a multiple expansion. And now we plug this into the action and then you see immediately what just happened. The same thing that just happened at the level of the equations. If we look at the radiation field only, the first step is this and therefore we get Tij. We get Tx, d3x and we get dT, H, T, zero. And this is the guy we just had a second ago. This is the guy we just had a second ago that we rewrote as a half d zero square, t zero zero, XR, XJ, d3x. Why this is great? This is great because now this action becomes localized. I integrate out the R scale. There is no more R here. It's only a theory that lives in a warline. It's only time dependent. So I match my whole thing into a point and I produce the multiple that I was looking for with the full team menu that I also includes the potentials. So you get a half integral dT. Those two derivatives I can now integrate by parts and I'm gonna hit this guy. If you include the other components you will see that you're gonna build up the Riemann tensor and it has to be because we do everything in a different variant way. So what we're going to get is a Qij, which is this guy. And those two derivatives on the other guy with the factor of a half you're gonna build up the electric component of the Riemann tensor. In fact, you build the vile tensor and we'll see why the traces are not gonna matter, why the traces don't matter and also this means if this is the vile tensor this is symmetric trace free. And this is my action. Then the action match into this with a Wilson coefficient that is time dependent. So I map into a warline theory, zero plus one. And now if I go from here and try to compute the one point function just compute the one point function with a source, localized source and I go to the TT gauge, I get this expression. Now from here, if I was doing a field theory I could derive a power and immediate power just from the graviton production or I can do the following trick which is I do the optical theorem. I take the imaginary part of the self energy there's Qij, Qkl and here there is an Eij, Ekl propagator and I do the trick of using Feynman boundary conditions which by the way the optical theorem is not quantum it's completely classical too. It could have been discovered after Maxwell and I use the optical theorem. That relates the imaginary part when you integrate out the radiation field using the Feynman boundary conditions the optical theorem tells you that this which basically puts the guy on shell is equal to twice the emission power which is what you are after. So you can integrate out this guy, do this calculation and then you will see that you're gonna get many, many derivatives that are gonna land in Q and then you're gonna get the quantum performance. I will do this properly later because we will see how this recovers all the orders. Okay, this is the filter way of doing the optical theorem or computing the amplitude of gravity to emission. That's the way we kind of slick way to find the total power but the way it was originally done or what people do is from here they go to the point in vector and they just compute the power by integrating over the surface but the answer obviously is the same. Why am I doing this? Because this is the way we start. I start here. I construct a theory which leaves already here which is a zero plus one theory. By this invariance I can construct the form of these terms. It has to couple to this guy and it's an object that I will call quadruple. I don't know where it is yet but I can match this object by doing these manipulations backwards and then I read what this Qij is by looking at this and doing these manipulations. And therefore I learned that the Qij of my zero plus one theory is this moment with the T00 that includes also the self-gravity. I go to higher orders, how? Well by keeping these guys by now looking at this guy and the higher multiples. And then you're gonna get here things like this and this is group theory. This is symmetric and then you have, okay I'll do this otherwise I get too distracted. So you'll see that you can decompose these guys in moments and then you start dressing up higher order corrections to this guy which go beyond this. Where there is a two here, okay? So this is the way we're gonna do the calculation. Now this is not the way that most people do the calculation but for us it seems very systematic to do it at the level of the action. So this is multiple moments at the level of the action matching into our world and theory. And once you know who this Qij is either you do this trick or what I'm about to do to get the power. And once you know the power and you know the binding energy which I haven't told you yet how to compute we can try to get the information about that picture. How much time I have? 20 minutes or shoot. Okay, so quickly. All right so this is the pseudotensile guy not for the short most for this guy. The point in vector. Okay I was gonna do more derivation blah blah blah. Okay this is this guy then. For the TT gauge it gets like this. This just follows is this conserved pseudotensile lambda or he has a million different names. That basically tells you that there's the conserved guy that tells you about this was a big discussion, right? People whether this gravitational waves carry energy or not and so on and so forth. This was not resolved maybe until much later. Even Einstein did not believe in gravitational waves. He thought the full nonlinear equations of GR did not allow for gravitational waves solutions. He didn't believe in gravitational waves. He didn't believe in the expansion of the universe. He didn't believe in black holes. So you see you can be a genius and still screw up. So what happens here is that the T, the flux we have the TCOI component but because we are in retarded time all the derivatives with respect to space and time essentially are the same. Where is my, did I erase the H? I erased H. Well because the retarded time the things are evaluated in T minus D so it will take the R, the X position. When you take that derivative with respect to X it's the same as the derivative with respect to time. So this is essentially the same as the TCOI. And that's why I'm gonna use COI. So the E dot that I'm gonna get from the flux is gonna be, now the keeping track of everything is gonna pay out because this is the natural scale for luminosity. And this is integral of this zero. I don't know where I'm pink now. H, I, J, T, T. H, I, J, T, T. And the surface integral that we do has some distance T and then this and here I put the four pi of the 32, okay? So what I have to do now is just plug the expression that I had and I erased for H, I, J, E, T here. And sorry, here's T, T, I had the projection. And this is a bunch of delta and N so what's gonna happen is very simple. These T's are gonna cancel. This is gonna give me an extra derivative. So we're going to have something like this like, well the C's and the G's, you see the G's, there are these zeros here. So one of each is gonna give you C5, C5. So there's gonna be C10 downstairs and G's up. So you're gonna get a G over C to the five. So it's gonna be reversed. And we're gonna get QI, J, triple dot. QI, J, triple dot. Symmetric trace free, symmetric trace free. And the coefficient will be a bunch of integrals of this guy that has the P's, which has this guy. So there's gonna be integrals of moments of the unit vector on the sphere. And then you know, for example, NISJ over four pi is delta J over three. And things like this, there's one over 15 with a bunch of deltas with the high ends. If I'm not gonna do this, you get a five. But this is E dot. And this is a quadruple formula. Now for us, this derivation will be compute this guy, project this into symmetric trace free tensors, which will be the same as doing this integral, and then getting the imaginary part. The imaginary part will put those guys on shell and it will give you sort of like the phase space part. And that will give you extra time derivatives because from here you get two only. So there's gonna be a d3k over k and that d3k over k gives you a k square that gives you an extra integer. I will do this calculation. And then you reproduce this. The nice thing about doing the calculation this way is that when I tell you here what comes next by constructing this theory at the beginning, I will start from this theory. And I can do this generically. I can get the generic formula and then I do matching to tell you who discuss here are by undoing what I just did this way. So instead of going this way, I'm gonna go this way. Very good. So do I have more comments here? Blah, blah, blah, blah, blah, blah. No, so okay, good. We're done. Calculation now, we're gonna try to get this data. Now we're gonna do the binary. And we're gonna compute the radiation of a binary system to compute the E dot for the binding energy which is gonna be crucial to get this evolution. We're gonna just use the Newtonian part. So this is like the Newtonian waveform approach. They said that Newton doesn't radiate. This is Einstein, but it's weak field like linearized gravity. So it's linearized radiation plus the Newtonian energy. The Newtonian energy we know we need because this is what it's gonna complete the dots. This is an expression for the radiation and who's E? Well E is gonna be the binding energy. We're gonna take a dot and then we're gonna see how the orbital frequency changes with that. And this is related to the frequency of the gravitational waves which is what we observed. Okay, so that's basically the idea. Now this is formally we're doing some kind of a diabetic expansion because we're saying that the radiation is small as you will see. And therefore we're treating as if we take a concept quantity and we basically equate the concept quantity over orbit so we integrate over periods to get an average quantity and that change of that quantity in time is small and therefore we can go orbit by orbit adiabatically, right? If I were to re-derive instantaneously this equation by not doing this adiabatic trick, I would have to do something else. Which we do to compute the radiation reaction force which by the way we'll see which order this radiation reaction force will enter but I'm not gonna do that here because of time obviously. But it's very interesting because it's related to the cell force problem which is very important for Lisa for the stream mass ration spiral which we cannot do this adiabatic expansion because the velocities are large. Okay, so now very quickly, hopefully. Hopefully I can get to this. Okay, so now Y and X. We have, this is one and two. One, M one, center mass, X two, M two and we have a frequency, this thing is moving and this is just Kepler. So this is moving with some orbital frequency, omega. It's given by Newtonian force and this is R. So a few things that I'm gonna need. I need the mass ratio. I'm gonna use the symmetric mass ratio which is mu over M and M is the total M. This is useful because in principle this is a stranger between one and two which the post Newtonian case and like the cell force cases, comparable masses we have invariance with respect to one, two. Okay, so if I take the density now, it's very simple. There is no self-gravity at this point. So we just have a bunch of masses moving in a spacetime. This is our mass density. So we can compute the quadruple. We can take derivatives, so I'm gonna simplify. I'm gonna just jump into the answer. With two derivatives, which is what we need. Two, you're gonna get a bunch of omegas mu R square. Well, it has to scale like mu R square and the two derivatives are the omega square and there is a matrix here that goes like this, cosine of two omega t, x, y, c plane. This is an x, y plane, so this is x, y, and c. Sine two omega t, sine of two omega t minus. This we will need for the amplitude, but if you wanna get the power, you take an extra dot. By the way, here you see why the frequency is gonna come out twice, the orbital frequency. There's obviously the group theory argument or the spin-to-fill argument, but it's also the actual calculation, which often helps. So now we compute the e dot and this is the famous 32 over five, g newton over c to the five. New square omega six R four m square. Do that as an exercise. And this can be rewritten as 32 over five, new square, this ratio to the two-third, everything to the fifth times the plane luminosity I was telling you before. This, by the way, I don't know if you guys know, it's about 10 to the 52 watts. So gravity is weak, so whenever g newton is huge. So the whole point is what is this, right? Because otherwise this would be a humongous luminosity. How you don't, it's humongous because the sun, if you compare with the sun, the sun is about 10 to the 24 watts, I think. And there are about stars, there are about 10 to the 26 visible stars in the universe. This, as much as I remember, someone saying this, so you see this combined is still less than this, right? And that's why people told you when you saw colloquiums about the LIGO detection that within fractures of a second, the amount of energy emitted was more or the power is more than the entire visible universe, right? Why would we have to see what this number is, okay? Because this, for normal matter, as an earth's sun is tiny. Who is this guy? Well, this guy is my ex-parameter that I had in my first lecture, which scales, as you can see by using Kepler's law, scales like V square. So this is V to the 10. So this effect here for comparable masses, this is about a quarter, this is V over C to the 10. So depends how fast you're going with respect to the speed of light, you will see how much you can radiate, okay? I will do some estimate to try to feed some of the data here to know how big this coefficient is, okay? But the 10th power kills you unless you have a lot of mass moving relativistically, okay? And getting very close because that's where you increase the velocity, right? Okay, so from this guy, we're almost done, now we need the E. So this is the E dot as a function of omega for a circular orbit. Now we need the E as a function of omega. This is trivial because I'm using the Newtonian approximation. This is simple X which I can write. Let me just write it. G, sorry, this way, and GM omega over CQ. It's just the binding energy, right? I'm using the potential, the virial field, right? The potential kinetic energy are comparable. So your binding energy in order is just this, right? In Newton, right? And I rewrite this as a function of omega and I get this. And this is 1 half MB square. I mean, it doesn't matter. Okay, so now I take a derivative of this expression as a function of time. I get omega dot here and I equate to the power and I get an equation for omega dot. How the frequency changes with time. So the expression that you get at the end of the day, I'm trying to solve an equation for omega dot that's gonna be related to the frequency of the gravitational wave up to a factor of two. So how do I get this? So essentially you're taking the omega dt. So I can take E of omega and do the omega dt. And this is E dot. So if I take this down here, I get the omega dt as this. And then you see this is flux. This is derivative with respect to omega or binding energy as a function of omega. So you see where this is going now. If I get higher corrections to the flux and the binding energy, I get higher all the corrections to this expression. How much better I can feed the data, okay? This is the leading order calculation. So from here, I get an expression that often is written this way for the rate of change of the orbital frequency that goes like this, and it's new. And again my beautiful parameter coming back to the factor. And this is B to the five. So it's a very small effect. So the shrinking of the orbit is a small effect so they justify this adiabatic expansion. This adiabatic approximation which I compute the binding energy as if it was conservative and I equate to the flux because I'm doing an integral over an orbit. Over an orbit, there's a very small change in the energy. That's also the way you got the hostile impulse, right? How they change in the period and they change in the orbit with time, circular orbit into circular orbit, even though that's not exactly what is happening, right? If you're gonna really track the motion, you have to do something else. And if you push me at the end of the lecture, I might tell you how to do that. So now, this is the way I write things that make manifest the expansion pattern. So the radiation reaction force is a 2.5 PN effect that you see from here, it's V to the five. So if you wanna include the effects of the back reaction of the emission, this is a 2.5 PN effect where PN is a V C square. And it's non-conservative, that's why it scales with fraction of five. It's V to the five. It's not time reversal anymore. Okay, so here we're almost done now. Now it comes to that famous cheer mass. The cheer mass comes because I rewrite this because you see there is a total M here, there's a nu here. So what people do, very smartly, is to rewrite this in terms of something called the cheer mass, I cannot do that called the M. So I'm gonna call it MC in the book. It's the same, but it's defined as nu to the 3.5 times the total mass, which is that combination up there. And why this is important because that's the mass scale that enters in this equation, which is what I will get from the data because that tells me how to solve how the change of frequency in time for the gravitational waves. And then however time I have, I will try to get the data. Tell me how much time I have. Five minutes, okay. All right, I'll do this very quickly. And then next time or during the discussion, ask me again, okay? Let's see how much and then I wanna explain. Okay, so number one, the frequency because the factor of two is omega over pi for the orbital. This is the orbital motion. This is the frequency of the gravitational wave. I'm not gonna keep gravitational wave there anymore. So I'm gonna solve, then they will give me, this will give me an equation for f dot over f square that can integrate as a function of time because there you see how it evolves with time. And therefore I solved that equation. I'm gonna skip a few steps. I'm gonna give you the answer. So between frequency one and frequency two or time t one and time t two, this is how the frequency evolves with time using the leading order waveform or w omega dot over square at leading order. This equals to a fifth g chirp pi third per c cube times the time that goes between one and two. Which means I can solve for the chirp mass if I know the time between two frequencies. So we can look at the data. So as we see here, the gravitational wave is seen in the lower, in the lower panel. Here at about 40 hertz, but see what's measured here, I don't know why, but okay. So around here, 40 hertz, you get like an amplitude of about 40. So it's more like 30 earlier, but here's around 40. And then it goes very high, about 300 is the kind of frequency around here. So because this is a high power of the frequency to a high power, as the frequency goes high and let's say it's infinity here, we can drop this term between about 30 and what, 42 between maybe 35 say and 43, like 0.008 say. So we drop this term and we say this is about 40 hertz and this is about 0.8 seconds. And now we put the Gs and the Cs and all that and we get a chirp mass, which is about 30 sermas. This is a very rough estimate using the leading order. This is not how this is done, right? This is just educational guess of leading order matching of my waveform, my prediction for the change in the frequency in time will give me for this information, this chirp mass. Now remember the definition of the chirp mass and this is m1 and 2 over m1 plus m2 squared. If I parameterize as m1 like this and m2 like one minus Y then this function is Y, one minus Y. That's kind of like this, a quarter. It's always less than a quarter, which means that we can put a lower bound on m from the chirp mass that tells us that the mass is bigger or equal to m chirp and this is about 70. We still don't know what the individual masses are but we got the chirp mass extremely well from the measurement. This we can get extremely well and we get even better and we have a lower bound on the total mass. So how are we gonna get, and I haven't even said anything about this thing, how are we gonna get the m1 and m2? So we need to break the degeneracy. So this is done in different ways but from the analytic side, from the perturbative side which you can actually compute, this is done by including higher order corrections. If I go to this expression and I tell you what comes next then it won't come out exactly as a function of m chirp anymore. It will depend on the X parameter which is what counts as B over C. So X is gonna come in as a function of m. It's gonna, if you factor out the m chirp it's gonna have the mass ratio and by measuring that you're gonna get the mass ratio. Now because it's done by B over C, the accuracy by which you're gonna get the masses is not as good as the accuracy by which you get the chirp mass but that's life. But what we're gonna do is something kind of like quick and I'm wrapping now with this. Unfortunately I had a few more things I can tell you. Maybe next time I'll do very quickly the other things because they're very nice because we can get how far the source was, how much energy is released all from the leading order waveforms. So the only thing I wanna tell you is let's look at the cutoff frequency. So the cutoff frequency is around 300 Hertz. So if we naively think that that happened when the two holes are on top of each other so we can estimate what the typical cutoff frequency is by just Kepler, if we put here the Schwarzschild radius of the total mass and if we do that we get a mass that goes something like this and we go from an orbital frequency to an actual frequency CQ over G newton and then from here using that the cutoff frequency is about 300 Hertz which you see over there which by the way really suggests that these guys are very compact that you could have any other star which is bigger it would have shut off much earlier with these masses. So this is really really compact because you get more or less the right mass which you could have inverted and say how close did they get and they get almost as close as the Schwarzschild radius with this cutoff frequency. So clearly these objects are having 30 sort of mass energy mass inside almost a Schwarzschild radius. So this is a proof that things like really, really almost look like black holes are out there in nature with masses well about what we had seen before. So if we do that then we get the total mass is about 70 which is comparable where we said I had lower mass this is very naive, you get that M1 and M2 from this analysis you get something like 40, 30 not quite but similar versus this was 36 and 29. This is what was done numerically the best fit. So it's not so bad. Next time I'll start I'll give you how we get from the waveform amplitude the luminosity the instance how far the thing was which was around 400 megaparsec how much energy was released which was two, three sort of masses and we'll see a little bit more about the amplitude and how we get things like this which are how they look like in the detector band. Okay, thank you very much.