 Don't sound very enthusiastic, but OK. So let's see. Last time, we talked a little bit about core black holes. I gave you the metric in excruciating detail in three different coordinate systems. And we talked about some properties, the metric, in particular, the ergo sphere and the horizon. And then we talked about geodesics, like throwing a little grad student into the black hole and just trying to measure his or her motion. And I had given you homework. How many of you did your homework? OK. So the homework was to show that you have some geodesic equations here for a homework solution. So we had something like d z, or I'll just write it as d x d tau square plus gamma mu, say alpha beta, u alpha, u beta was equal to 0. And I was saying that this is not great because it's a second order system of differential equations. And we would like to rewrite that as a system of first order equations to do a first order reduction. And one way to do that is to realize that the energy, which is defined as t alpha, u alpha, the angular momentum, I guess I put till this on this last time, which is phi alpha, u alpha, quarter constant, q, which is essentially the killing tensor k alpha beta, u alpha, u beta, are conserved in the sense that these are constants of the motion. So d by d tau of e is 0, d by d tau of l is 0, d by d tau of q is 0. And in addition to that, you have the normalization condition for time like geodesics that u alpha is minus 1. So the goal was to use these expressions to find a geodesic equation that looks like t dot equals something, r dot equals something, theta dot equals something, and phi dot equals something. So the way to actually solve this is just write it out. So you have that u alpha is some vector. I'm going to call it t dot, r dot, theta dot, phi dot, where the dot means the relative with respect to proper time. And so then e tilde, for example, or minus e tilde is going to be the metric, alpha beta, t alpha, u beta, and that is going to be g t t t dot plus g t phi phi dot, OK? And similarly with l, right? So if you go and you calculate l, you'll have a similar expression for q. You'll have a similar expression. And here you'll have minus 1 is equal to a similar expression with t dot squares and phi dot squares and r dot squares and all of that. So then all you do is you have four equations for, in principle, four unknowns, t dot, phi dot, r dot, and theta dot. And you just algebraically solve that. And ta-da, it's not that hard, right? You can find these expressions in just standard textbooks. I'm not going to write them again. But as a corollary of this extra credit, I guess, I called it homework was to show that these things are conserved. So let's see. Let's write over here. So the idea was that d by d tau acting on e, say, I want to show that this is equal to 0, question mark. So I can rewrite d by d tau on e as the four velocity u alpha times the gradient, the covariant derivative acting on e. But e was t alpha. I put a minus sign here. So it's t alpha u alpha. I have too many alphas. Ah! So when I teach this in class, I just annoy my students whenever they say something like that. I just put a little tiny dot here. Now it's a different variable. So let's call it beta beta. So you just use a chain rule here, and you get u alpha u beta d alpha t beta plus u alpha t beta here u alpha d alpha u beta. And then you recognize that u alpha d alpha u beta is the geodesic equation. So this vanishes. And then you recognize that here you have the covariant derivative of a killing vector, doubly contracted with two copies of the four velocity. These two copies of the four velocities is a symmetric matrix, which means I get the same thing if I symmetrize on those two indices. By the contraction I can then just symmetrize on these two indices. But the killing equation, remember, is that this vanishes, which then means that this has to be 0. That this is 0. And the same goes through if you calculate it for q. And obviously the norm of the four velocities is also conserved. So those are some of the tricks that you are to employ when you're dealing with geodesics. So now, we've all warmed up for the type of math. We're going to be, yes. Can you repeat that question? Because I have to repeat it to the thing. I couldn't hear you. Yes. So the question is, can you always show that the equations are separable in this sense of superability if your background is stationary and axisymmetric? And the answer is that that's not always the case. So for example, there's modified theories of gravity where the current metric is not a solution to your field equations. And in that case, the space and it's still stationary and axisymmetric in the sense that there is a conserved energy and there's a conserved angular momentum. And the norm of the four velocities is still the norm of the four velocity. But you can show that there is no second run killing tensor. So without the presence of this Q, you can't even start following this proof. So the issue of separability of the equations of motion is very tightly related to the petrope type of your spacetime. And proving that your spacetime is petrope type goes beyond just saying that it's stationary and axisymmetric. And it depends on the theory and the field equations. So talk to me about it later if you want because that's like a full lecture. OK, good. So now that we've warmed up, let's look at Emory's. So an Emory is defined as a definition. I'm going to do it the math way. It's a stellar mass compact object. So stellar mass compact object in my life is abbreviated as SCO because I don't want to write that a million times. In a generic, I'm going to say in spiral or orbit, around a supermassive black hole. So there are two acronyms for you. Stellar compact object or stellar mass compact object and supermassive black hole. So that's our definition. So even though a lot of what I do has to do with calculating these orbits and calculating the gravitational waves emitted by such systems, I think it's always important to remember like what sort of astrophysical scenarios could produce these binaries. So we're going to do like two minutes of astronomy. So just bear with me. So there's two, so formation scenarios. There's two classic formation scenarios that you should know of, and these are sort of fairly lengthy topics. So we're going to do it fairly briefly here. So the first scenario is called the capture scenario. And here's the idea that you have a galaxy. We're going to do formation scenarios with pictures only. So it got me my colors over here. So galaxies are orange. Everyone knows that. So here's my galaxy, and everyone knows that supermassive black holes are blue. So here's my supermassive black hole very much not to scale. But I was told that if I don't draw big, you can't see it. So imagine that you have some sort of binary that's over here, and it's going around this. Imagine like it's going around. Like that, some binary of two compact objects that are both small, like stellar mass, black holes, maybe something like that. But as you can imagine, galaxies have tons of stars, and there's tons of processes that occur in the galaxy that can affect the dynamics of my little two stars. See, unfortunately, we like to solve the two-body problem, but in real life, there are more than two bodies. And there's also more than three bodies. There's like a gazillion bodies. And so every now and then, it can occur that this binary will interact with a third object. The third in this chord is going to be painted purple. So here is the bad guy. And the bad guy interacts with this binary. And the end product of that interaction can be that the disturber here becomes bound to one of the components of my good binary. And now these two go around. And the other component of the binary is sort of ejected. And there is a probability that that ejected component will be headed toward a galactic center. And if that's the case, eventually it will get close enough to this supermassive black hole that it will sort of loop around. And this distance might still be very, very large, right? So this is not a small distance. But what happens is that in the process of coming with a small impact parameter to the supermassive black hole, this small object emits gravitational waves, which everyone knows are white. And when that happens, the binary formed by the small compact object and the emery, which where this binary in the orbit was very, very large, the seven major axis is very, very large, becomes much more bound because now that orbit has lost a lot of energy in this burst of radiation that it emitted. So what happens now that instead of this one going back to its initial trajectory that's over here, it becomes bound, bound to the supermassive black hole. And you might think, like, this is incredibly unlikely. But it turns out that if you take a probability that's really, really tiny and you multiply it by a gazillion, which is the technical word for the number of stars in a galaxy, then you get a number that's finite. And the particular value of what that number is, like how many events like this happen inside of some volume centered at the solar system per year, nobody knows. There's very complicated astrophysical processes and there's rates. So there's the rate of how often this would happen and they range from zero times per year to like a thousand times per year for systems that would be detectable with gravitational wave instruments in a decade. How are we going to be able to figure it out? Well, either someone really, really, really smart is going to put in all of the physics that are needed to solve this problem, maybe on supercomputers and calculate it and convince us this is the right number or much more likely we'll turn the instrument on and we'll measure and we'll see, like, ah, the answer was 11. Okay, so that's one formation scenario. Another formation scenario is, what did I call this one, disk formation. This one is less likely than that one but let's describe it anyways. Okay, so in the idea of disk formation, what you have, you have your supermassive black hole which, again, is blue, a lot of galaxies that hold supermassive black holes tend to have accretion disks around them. Those accretion disks need not always be perfectly thin disks like we like to imagine, but a good chunk of the time they are or they're expected to be. So let me draw a disk. And so imagine that here's the black hole, I'm gonna look at the black hole, like from Bert's eye view, like from the top. So here's the disk. Imagine that the disk is thin, like a thin disk. And there's a lot of gas here and maybe the disk stops some distance away from the horizon. That distance might be the innermost stable circular orbit, it might be a little bit closer in, doesn't really matter. Point is that this disk has an enormous amount of mass but in any particular region of the disk there's very little density. It's because the disk is gigantic. Okay, so it's very small density but then you integrate over a large volume, you get a large mass. And what can happen is that in the outskirts of the disk the gas that is present can fragment and form stars. Just like stars are supposed to form, say in the solar system, right? You have a giant molecular cloud and it's all collapsed and they're fragmented. Pretty as a little star. So this star, this sun-like object but more massive maybe that the sun-like object is gonna be moving around here, it's gonna take forever. Eventually, it's gonna migrate also, it's gonna migrate when we say migrate we mean it's gonna move in closer to the black hole. Slowly, very, very slowly because it's very far away. Not due to the emission of gravitational waves, has nothing to do with gravitational waves. It migrates in because it's immersed in a disk and as it's moving around there's viscous drag because of interactions with the material but also the object produces little ripples, little waves on the disk and those little waves on the disk carry away angular momentum which then back-reacts on the orbit by pushing it in a little bit. Eventually, if you wait a long, long, long time, this star can go supernova. Supernova is red, ka-poof, okay? And if the original object was massive enough after it goes supernova, it can live behind a black hole. So after this explosion, now we have a little black hole. Now, you're gonna ask, oh, but the thing went supernova so it blew up the disk. Then there's no, it didn't blow up the disk because there's not enough energy to blow up the entire disk and disrupt the disk. You might disrupt a little bit of the neighborhood of this disk but this will refill very, very fast. So if you wait a little bit longer, it's like nothing happened but now you have a black hole. And this black hole, guess what? What's gonna do what black holes do? Which is just go around. And eventually, just like the star was being dragged into the supermassive black hole, this object will slowly move closer and closer in until eventually it enters the gravitational wave-dominated regime. And now you have a memory. Okay, so I don't work on formation scenarios because I'm not an astrophysicist so please don't ask me questions. But I think it's important for all of us to know what the formation scenarios that people have in mind are for the models that we are trying to solve for. Like so that we know that this actually are supposed to happen in nature, how they're supposed to happen in nature. And if you wanna read more about this, there's thesai after thesai that have been written about it. Yep. No, is there no questions? Yeah, yeah, very good. So the question that Bolo asked is, it is important that the object be compact. That is, it did not be a star for the definition of an embryo. Why is that? That's a very good question. Here we're dealing with supermassive black holes. So certainly if the supermassive black hole has its masses like 10 to the six, 10 to the seven, the tidal force that it exerts on a companion, say when the companion is at the horizon or closer to the horizon, is actually very small. Why is that? Because the tidal force goes as the Riemann tensor and the Riemann tensor goes as the mass of the object divided by the separation, say cubed. Okay, so if your separation is roughly the horizon, so M, then M over M cubed is one over M squared. And if M is giant, then that force is very weak. So you can certainly have a star go right through the horizon. Don't even notice that there was a horizon there. I mean, there will be tidal deformations a little bit, but it won't get disrupted. So in principle, you can have an object like a neutron star or like a regular star falling into this black hole. The reason that we don't consider that is because the strength of the gravitational waves that they will emit is much, much smaller than the strength of the gravitational waves emitted by a small black hole going into a supermassive black hole. Because the gravitational waves' strength scales, if you wanted scales with a mass of the objects, but when they're black holes, but when they're stars, they scale essentially with the density of the object. So, okay. All right, so with that short explanation in mind, let's now talk a little bit about modeling. So, modeling. So I like to describe this modeling of, so what are we modeling? We're modeling the gravitational waves emitted by this emrys, okay? And so to understand the gravitational waves, you're going to understand the motion. Unfortunately, the problems are coupled, so we need to understand the motion and the gravitational waves at the same time. So the way I like to describe this is through this very nice diagram that I first saw right down, written down by Lior Barak. So Barak has a very nice review article. That you can all read. So Barak is a professor in Southampton, Lior Barak. And I think it's CQG 2609. There's others, Eric Poisson, the same Poisson that wrote the Poisson Toolkit book and the Poisson and Wheel book at all. He has a living reviews in relativity, LRR 14 from 2011 also. So these are your references if you want to read more about it. But in any case, let me plot, I don't want to plot this, okay. Let me plot on this axis, the orbital separation R divided by the, say, total mass of the system or the mass of the supermassive lacool. Let me plot here the mass ratio. So the mass of the small compact object divided by the mass of the big object or the total mass, it doesn't matter. And so the mass ratio ranges from zero to one. That's the domain. And the separation ranges from when the thing merges. So maybe not zero, but whatever you want to call it, the horizon, I'm going to call it zero. It doesn't really matter. To all the way to spatial infinity, right? And so in this domain, what I want to understand is what approximations can I use to solve for the trajectors and the gravitational waves? Or what methods can I use? So you see, if the mass ratio is very close to one, well, the first thing I'm going to say is like, I know. This is all nonsense. Just throw it all into a supercomputer. Let the computer deal with it. I'm going to have to worry about it. The problem is that if you have one of these extreme mass ratio orbits, the amount of time it takes for the orbit to adiabatically shrink by a little bit scales inversely with the mass ratio. Okay, so if your mass ratio is very, very tiny, you have to wait a very, very long time for the gravitational waves to react on the orbit and for the orbit to shrink and for the system to merge. On top of that, if you have an extreme mass ratio in this spiral, you have to model in your numerical grid not just the supermassive black hole, which is actually pretty large, but also the small compact object. So you have to resolve the compact object, which again, that resolution, the difference in scales is again of order to mass ratio. And on top of that, you have to then also model the gravitational waves that typically have a wavelength that is much larger than the size of the small compact object. And that needs to be extracted very far away from the binary system. So all of that means that if you actually try to push your numerical simulations to the embryo regime, which is over here, you are just out of luck. Okay, so there's some sort of limit as to how well numerical relativity, not how well, but some limit on what numerical relativity can do today. Okay, some bar here. On top of that, even if you're considering comparable mass binary, so like two black holes, equal mass, if you put them really, really far away from each other, then the amount of time they're gonna take to spiral in, it's gonna be enormous. And you only have as much time to run your codes as the lifetime of a grad student thesis. So that puts a bound on the maximum separation, which means you cannot really go all the way to spatial infinity, because then your grad students don't graduate. Okay, so then that defines some sort of regime here where NR works wonders if you actually work on it very carefully. And this goes back to Pewter's talk, where in order to do the numerical relativity problem, you have to take your space time and foliate it with space like hypersurfaces and do a three plus one decomposition and then ensure that your constraints are satisfied, blah, blah, it's all very complicated. It's all very cool, it's all stuff I'm not going to talk about. Then what else can you do? Ah, so Pewter talked about the post Newtonian approximation, right? He was actually talking about the post Minkowski and approximation. It's an expansion in M over R or in weak fields. If you want in capital G, the strength of the coupling constant of the gravitational interaction being very weak. But on top of that, you can also use an expansion in small velocities. That expansion of the small velocities is also sometimes called the post Newtonian approximation. I'm going to talk about that tomorrow on my birthday. I might add. You're welcome. That's okay, because that's like what I actually do. So it'll be easy. But of course, as the orbital separations become very, very small, the strength of the gravitational interaction becomes huge and the orbital velocities can become actually not much, much smaller than the speed of light. So clearly, these post Newtonian approximations cannot work when the orbital separation or the velocities, when the orbital separations are small or the velocities are large. So there must be some different color. Pn is blue. There is some minimum regime here, some line below which I cannot go with my post Newtonian approximation. So, but I can go all the way here. So you would think that I can extend this all the way to like tiny mass ratios. Okay, that the post Newtonian approximation, expansion in velocity will work even when you have a tiny, tiny object going around a supermassive black hole, as long as the separation is large. It turns out nobody fully understands this. So math people, you're welcome to work on this if you want. That the properties of the post Newtonian expansion that has actually been taken to what we would call in physics, three loop order. So that's an expansion that has been taken up to V over C to the seventh power relative to whatever the leading order term was in the expression that you're trying to calculate. Okay, that's a very high order perturbative expansion. So you can look at the convergence properties of this series. And you can try to study, well, you can try to understand whether that series converges to something as you add more and more terms or whether that series is asymptotic or divergent. And we don't know, we don't know for sure whether series asymptotic or not. What you do know is that as you look at more and more extreme mass ratios, the coefficients in the series become larger and larger. Which is suggestive that an approximation that might have been okay over here if you fix the order to like V over C to the seventh might not be as accurate if you increase the mass ratio, if you make it more extreme. So it turns out that for the purposes of observation and for modeling, the Poisson-Gener approximation is not really valid or accurate enough as the mass ratio goes to zero. Okay, great, so boop in. So what can we do now? Well, anyone wanna guess what's the obvious perturbative scheme that we can use? I love it when audiences are very participative. Argentinian guy, what you gonna do? Very good, I was not going to talk about that, but no problem. So yes, these are essentially Taylor expansions in V over C, okay? So you might ask, well, we know what to do to improve the convergence properties of a series. We resummit, okay? So there's ways in which you can resum these expansions and one way that's very, very clever and very effective is the so-called effective one body approach. You can think of it as a resummation. That tries to make post-Newtonian, which I forgot to label, more and more accurate if you want more and more valid in this regime and actually in this regime too. Like he wants to push this line to the left and this line down, okay? But no, what I meant is, what do we do here? How do we solve the problem here? Waltz didn't. We do black hole perturbation theory, right? We expand our problem in Q much less than one, okay? And that's a good parameter to expand because unlike the velocity which changes as your orbit evolves, Q doesn't change because, well, it changes because objects could accrete a little bit of mass energy. But for the most part, the mass is essentially constant so Q is essentially constant. It's a good expansion parameter. So here's where cell force calculations or black hole perturbation theory calculations are done. Now, do not go and tell your friends, oh, Nico told me that these are the lines and I can sort of infer that this is zero and this is one, this is like 0.25 and this must be 0.6. Like clearly where I drew this line, it's a cartoon, right? Like nobody knows where these lines are. Like at all. So it is entirely possible, but we do know that positonium seems to agree with numerical relativity in this regime from the comparisons that have been done. Especially when you resum them. We also do know that cell force calculations seem to work or seem to reduce to the post-Newtonian results if I take the cell force calculations that are just solutions in Q much less than one and then on top of that, I do a post-Newtonian expansion, a V over C much less than one expansion. Then if I do those two expansions here and then I take my post-Newtonian one and I do an expansion in Q much less than one, then those two bivariate expansions match. So that's good. And there have been comparisons between cell force calculations and numerical relativity simulations in some regimes, but it's not clear how well those two match because NR can't really go here. Okay, lecturers are not allowed to ask questions. I'm kidding. Yes, correct. Excellent. So let's explore that idea because that's exactly what we're gonna do. Thank you. It's like I planted that question. Okay, so historically there have been many, many ways in which you could try to model M-RES using a Q much less than one expansion. I am going to explain this or present just a few of the methods. There is no time to present all of the methods. All of the methods are approximate in one way or another. Some are way better than others in terms of like accuracy of your approximation. So let me first show you the methods that are least accurate, but easiest to understand. And hopefully before I'm done, I'm gonna be able to derive some of the equations that are used in cell force analysis. So now I need to cheat because otherwise I'm gonna get the equations wrong. So let me begin with the cladgiest of the cladgiest approximations. Sometimes called the semi relativistic approximation. I felt like I had to describe this because it was pioneered, well, because I'm in Italy and it was pioneered by Remo Ruffini. So I figured if I don't mention Ruffini at least once then people are gonna get mad at me. Bigger font. Yes, Ruffini and Sasaki. And they have a very, very beautiful paper in PTP progress in theoretical physics. It is a Japanese journal, 66 from 1981 that you can refer to, very readable. So what they said is, like Peter was saying, like if q is zero at that point everything is just a geodesic, right? So I can certainly take my metric to be the metric of my supermassive black hole plus a perturbation that's going to represent gravitational waves, just the gravitational waves on top of the supermassive black hole spacetime which are produced by the small compact object going around. And so the four velocity of the small compact object is gonna be represented by the background solution which is the four velocity of geodesic motion plus a small correction to that trajectory. And in principle you could go to higher and higher order. So the equations that you need to solve remind you are, guess I'll write them over here because they're always going to be the ones we are going to solve. g mu nu is equal to eight pi t mu nu, g and c are one. And obviously, conservations of stress energy tensor, okay? So in the words of Wheeler I suppose, at least as far as I know, matter is telling curvature or spacetime how to curve and then the curvature of spacetime is telling matter how to move but these equations are obviously coupled so you need to solve them sort of simultaneously. Now, if you neglect, if you do this expansion and you plug it into these equations then you get that too leading order. This background metric has to just satisfy the vacuum mines and equations without a source. And that's just the current metric. So yay. And in the absence of a small object then that source, that current metric is stationary, it's not moving anywhere. So to essentially first half order higher than that I'm gonna have a point particle which is how I'm going to describe this more compact object moving or a test particle moving in the background of the current metric. Which then gives me what Peter was saying the geodesic equation for the point particle. But then what I need to know is okay now that I have that point particle the test particle moving in a geodesic how does it back react on the metric to generate the gravitational waves? So I have to go back to here and now I have to put on the left hand side that the metric is the metric of a black hole plus a preservation H and I have to put in on the right hand side that now I have a stress energy tensor and that stress energy tensor is that of a test particle and that test particle is moving with a trajectory or warline given by the solution to the geodesic equations. And so if you do that you'll find so equation one one becomes box on H mu nu trace reversed in this case equals a minus 16 by T mu nu after we use a particular gauge condition which again, Peter was calling the wave condition wave coordinates turns out that that particular coordinate system the wave coordinates that satisfy box X equals zero imply when you do a perturbation about Minkowski the gauge condition that D mu on H mu nu equal to zero with a particular definition of H I can get into this later if you want I probably have to describe it when I talk about post-Newtonian tomorrow. So in their approximation, what they said is in order to solve this equation to make things even easier we're gonna say that this box operator which is supposed to be a G mu nu covariant mu covariant nu we're gonna replace for G mu nu and for the covariant derivative the back that them in koski metric so we're gonna make the approximation that just when I solve this equation just when I solve that equation that the background is Minkowski okay and if I do that then oh and I'm also going to say that T mu nu is that of a point particle or a test particle so you have some mass for the small object and then the four velocity mu mu nu and then a direct delta function for dimensional of the coordinates X row minus zero of tau theta if you do that there's also many questions okay yeah yeah so that's what I was trying to say I went through it a little bit fast so if you actually do this calculation properly you realize that this is not the only term you have if you do a perturbation about the car metric you also have a term that depends on the Riemann tensor and this will be the box of the car metric so the approximation that Sasaki and Rufini did again this is the clugiest of the clugiest models okay it's just say well when I have the linearized equation I'm going to replace what was the super massive black hole background with a flat Minkowski background which is totally wrong but we're gonna do it anyways okay so if you do that then the Riemann tensor is zero and then this is the box of Minkowski and that's good because I think if this is the box of Minkowski I can invert this because there's a well-known Green's function for it yeah so the matter field is that of a test particle that has a mass M and it's in motion with some four velocity like so that is localized as a distribution plus of matter hmm sure let's give it that name it's a test we call it a test particle in physics yeah so I didn't quite say that explicitly this is the trace reverse metric perturbation so it's H mu nu minus one half eta mu nu times the trace of H calculated with a Minkowski metric yeah so that trace reverse is computed from the eta trace okay so then if you do that turns out you can solve that you can you know the solution to that equation is very easy well G mu nu H mu nu is going to be minus 16 pi box to the minus one of this and then I'm done right supposed to be a joke and okay so you use your Green's function and you turn up you end up getting and it looks like minus four M times the distance to to your field point where you're calculating this times U mu U nu divided by a vector L alpha U alpha and all of this evaluated at retarded time so retarded time is t minus t minus r if you want a field point r leave it like this to retarded and L alpha here is the vector one times the n unit vector that points from the field point to the location of the small compact object or vice versa actually small compact object to the field point and U mu here is a four velocity which you write as gamma some Lorentz factor times one comma three velocity and presumably you have this because you've used your geodesic equation to solve for the velocity of the test particle so what is this solution what do we call that solution in physics yeah it's the linear feature solution for radiation so it's B card I don't know how you pronounce these things solution of electrodynamics for a particle that's moving on a flat background that's a good question so so in practice we tend to write these things in in coordinates that are adapted to the Minkowski background okay so because there's the box of Minkowski so we typically use for numerical reasons just Cartesian x y z coordinates if you can use spherical coordinates if you want but then numerically you get into issues sometimes when you hit the poles because you don't know how to evolve so you have to be a little bit careful about that but yes they have to be adapted to the background good nobody liked the solution but it's a solution from the 80s so you know now we're going to accelerate 10 years I'm going to show you a less cludgy waveform called the clutch waveform or Cluj depending on how you pronounce it I pronounce it clutch so this is an idea so the the idea here is like let's try to come up with some sort of approximation that solves these equations but a bit more accurately right like you don't want to be using you don't want to be using this cheat this incorrect step of throwing away the background completely it's going to be totally off so so how do we do it we do it as follows we go ahead and we again follow what Peter was saying earlier so at zero's order in Q leading order in Q when Q is zero then I should have a test particle moving in a geodesic now that geodesic is going to have some so to leading order I'm going to have a geodesic it's characterized by some energy I'm going to momentum and Carter constant okay so let me draw a two-dimensional this is a three-dimensional space right and Q let me just draw it in two dimensions let me suppress Q for now so E and L so I can pick a point here some initial condition for my geodesic I just pick an orbit with some energy and some angular momentum and that's just a geodesic if I run a computer it'll just go around forever and ever and ever if I pick E and L very very intelligently I can even make that orbit be a circle so but I could have picked another point this would be another geodesic and another geodesic and another geodesic and another geodesic these are all different geodesics look at my beautiful grid of geodesics so I've gridded this phase space okay so what happens in reality is that if I start with some geodesic I start my evolution here somehow that geodesic needs to evolve and spiral into the black hole why is it spiraling into the black hole because the geodesic is disturbing the space time and that generates gravitational waves the gravitational waves carry energy and angular momentum and current constant out of the system which then back reacts on your orbit forcing it to spiral in forcing the E, the L and the Q to change in response to the emission of gravitational waves so there has to somehow be if I'm here some way to compute how how E and L will change due to the emission of gravitational waves in a small neighborhood of that phase space point so if I can do that I can compute like a little arrow okay and if this grid is very very dense this is not trying to scale so let me just draw it like this it's an arrow I could then jump from here to here to this other point and then there I can go and I can solve my geodesic equation again and calculate my rate of change of E and L and Q due to the emission of gravitational waves and calculate another arrow so let's say this one takes me this way so I jump so I jump to some point here and I do it over and over again computing little arrows that will take me hopefully through phase space from geodesic to geodesic to geodesic okay and then you join the idea is that you join all of these geodesics together to get the full trajectory okay ask me again in five minutes so what we're doing here is something that it has been known in celestial mechanics for a very long long time it turns out it's called um oscillating orbits so the idea is that here I have an orbit that's a circle and then I am losing energy and an angular moment well that's just I'm losing energy such a way that after I lost enough amount of energy I turn into some sort of other circle there's some limit that you can take in which this geodesic sort of oscillates onto the next geodesic and then oscillates onto the next geodesic and so on and so forth sort of like shrinking slowly from one circle to the next circle to the next circle so this was an idea that was put forth for a variety of people but the reference that I typically follow for this is Hughes prd 2000s I think there's two papers or he explains this thing so the main bit that's missing here is how do I calculate e dot l dot and q dot right how do I calculate if I have a geodesic how do I calculate how much how much the energy and momentum of the current constant change due to the emission of gravitational waves so you need to prescribe somehow e dot l dot and q dot and there's a couple of ways in which you can do that one way in which you can do this is by saying I'm going to use the post Newtonian expressions for e dot l dot and q dot so expressions that are only valid in the small velocity limit again totally wrong because the geodesics are not necessarily slowly moving but you can certainly do it and see what happens so option one option is you can do p n a better option is you can use something called black hole preservation theory which I'm going to describe on Thursday to calculate something called psi 4 which is one of the Newman plan road scalars and that's psi 4 you can eventually show it can be used to calculate the amount of energy carried out by gravitational waves at spatial infinity okay psi zero would allow you to compute the amount of energy absorbed by the black holes as the waves go into the horizon so that's another way in which you can compute this okay so this is sometimes called the sort of frequency domain method for calculating because there's a frequency domain method to calculating this psi 4 the post Newtonian prescription of these variations of constants is much much more kluge but definitely it's something that people have done in the past so that's method two and if you want if you want something that's more complete what you can do is remember that you have expressions for t dot r dot you know theta dot and phi dot right into one from for geodesics that's how I started my entire lecture and these things were functions of the coordinates but there were also parameters e and l and q here which were supposed to be constant and the same thing here and the same thing here and the same thing here so what you could do is if you have some sort of analytic way to prescribe e dot and l dot and q dot say from post Newtonian theory then instead of just solving these equations numerically on a computer you can solve these equations enhanced by these equations which now means that your constants are no longer constant but they're varying on a different timescale but they're varying so if you solve if you do that then that's another way of implementing this idea of escalating orbits okay so in my last 15 minutes let me tell you why all of this is completely wrong and let me tell you what people have done to fix this problem or fix the problem with these approximations okay so who can tell me what's wrong okay who can tell me one thing that's wrong no need to list everything what have I not included when I do the calculation this way the cell force what is the cell force right so the small object is right so the small object is not a test particle and because it is not a test particle it curves space time and because it curves space time the curvature that it presents on space time actually back reacts onto its own trajectory and that effect needs to be taken into account sometimes that effect is called the conservative part of the cell force because it's symmetric and they're time reversal for example it's if you want the part of of the force that comes from the space time metric or from the curvature in the space time metric produced by the small object and this thing that I'm calculating here e dot and l dot and q dot they are essentially also producing a cell force but it's a dissipative cell force okay it's telling you how how your geodesic is not really a geodesic but it's the trajectory is forced if you want by some sort of radiation reaction force okay so this includes some part of the dissipative cell force but we want to include the whole thing so cell force and if you have questions think Boggs in the audience and he can tell you how he derived these equations a long long long long time ago better than I can um all right so so this is this is my derivation of this thing so a this is a non very much non rigorous derivation of this thing what is it that you called it Peter it was a very dirty proof of how you do this result right anyway so so what is the motion that you're trying to consider the motion you can write it down as the second derivative of the trajectory plus some gamma prime alpha mu nu d z mu d tau prime d z nu d tau prime equals zero where primes here denote the trajectory on the spacetime g mu nu which is equal to g mu nu of the supermassive black hole plus a correction h mu nu due to the small compact object so if you want if you knew how the small compact object is curving spacetime you could in principle then take that metric whatever that metric is and plug it into here into the Christophels and and calculate and calculate the geodesic equation on that background so this goes to what you were saying earlier so you can think of this as the geodesic or geodesic motion in a perturbed background that's one way to think about it another way to think about it you can reinterpret this as you know geodesic motion so no primes now which is now not equal to zero but it's actually equal to some force f alpha that I'm going to call the self force and for good measure to make the unit work let me multiply the whole thing by m the mass of the small compact object so you reinterpret this as a trajectory on a background that is just the background of the supermassive black hole but that is forced okay so these these two descriptions one of a trajectory on a perturbed background versus one of a trajectory in an aperture background that is forced can be made essentially equivalent so what you do now what you do now is we're here is you what we want to do is we want to calculate what this self force is so so what is the self force is f self force so we do some change of variables you know you know that d by d tau it's going to be equal to just d tau prime d tau d by d tau prime and similarly for the second derivative it's going to be two terms in this case okay and so we go back to this equation over here which I'm going to call equation two and so equation two can be written as let's say m times d2 c alpha d tau squared d tau prime over d tau squared plus d2 tau prime d tau squared d z alpha d tau prime plus the second term which I'm not going to touch m alpha mu nu d z mu d tau d z nu d tau okay so all I've done here is I have replaced in equation two d by d tau squared by something that depends on d by d tau prime this should have been a prime here sorry yeah so there's two terms here so if I apply another d by d tau there's going to be one that acts on here and it's going to give me d2 tau which is the last term and another term is going to oh I don't see what you're saying so this what I wrote is this here no no so this is correct so all I've done is I'm rewriting this equation so I am not going to rewrite that in terms of tau prime so that's correct okay so just doing it in steps so just give me a second so what I'm going to do now is I'm going to use that d2 z alpha d tau prime 2 is given here by equation 3 by this expression so I can rewrite this as minus m gamma prime alpha mu nu minus gamma alpha mu nu times d z mu d tau d z nu d tau plus this second term d2 tau prime d tau squared want d tau d tau prime d z alpha d tau so after you manipulate things a little bit you can see like for example this last term here the second term is appearing right here this term over here comes from from there and so this term over here is the second term in parentheses over there and so now you notice two things one thing is that this is the difference between two Christoffel symbols calculated in essentially two different spacetimes one is a spacetime that is not perturbed and another one is a spacetime that is perturbed so when I calculate that difference the difference is just going to depend on covariant derivatives of the perturbation because the covariant derivatives on the background are going to cancel the other thing you notice is that this term over here you know it depends on on d z alpha d tau but d z alpha d tau is a four velocity and you know that the equation two you know that equation two is saying that f s alpha written in terms of the four velocities u mu d mu u alpha and if I dot this whole thing with u alpha then this right hand side is zero because u alpha u alpha is normalized to minus one so u alpha times f cell force alpha is equal to zero which means that the only part of the cell force that we sort of care about for this derivation is the part that is perpendicular to alpha to u alpha okay so this piece here must be containing a part that's perpendicular to u alpha and a part that's parallel to u alpha and the part that's parallel to u alpha at the end of the day must be canceling with this term so since we only care about the part of u alpha that is perpendicular as a part of f alpha that's perpendicular to u alpha we can project which we're going to do now and so the projection of f alpha cell force is just g alpha lambda plus u alpha u lambda so we project perpendicular to u alpha and you end up getting minus one half m g alpha lambda plus u alpha u lambda times times covariant derivative nu h lambda mu plus covariant derivative mu h lambda nu minus covariant derivative lambda h mu nu where these covariant derivatives are associated with the supermassive black hole background so the kermetric okay and this whole thing is sometimes called m times covariant derivative this is defined to be m times covariant derivative alpha beta gamma acting on h beta gamma we're going to do a piece of the cell force which is what you care about so now now you're almost done because um well you are essentially done what you need to do to solve for the trajectory of the small compact object is to solve this equation over here this is a geodesic that is being forced by this term over here so this is to first order so it's going to be the supermassive black hole background remember that's also the covariant derivative associated with the background so you can calculate all of the terms here but of course this thing also depends on h so you need to have some sort of prescription for what h is so if you do properly an expansion of the Einstein equation about a curved background smbh plus h score what you get is box of the trace reversed metric perturbation plus two times the Riemann tensor mu alpha nu beta acting on h bar mu nu equal minus 16 pi some t alpha beta for your matter source and again here box is supposed to be box of the background and r the Riemann is supposed to be the Riemann of the background so now you have two equations that are coupled one for the metric perturbation and one for the trajectories this is not the equation for the trajectory the equation for the trajectory is equation two with f-cell force this thing and now we're done right easy peasy right so bob solved this in do you remember when this was on 70s 80s still in my thumb there bob okay did but do you remember the year this time okay so what I was going to what I was going to ask the students was like are we done like is this is this it and the answer which bob already gave you that no this is not it because that h mu nu is singular at the location of the point particle or has a piece that's singular at the location of the point particle so you need to figure out a way to resolve this this issue so problems just over here and this is this will conclude my my talk running almost over um so small in quotation mark problems so one your test particle has a divergent self-filled at the location of the small compact object other little problem delta functions their distributions are not strong field representations uh say point particles and so on and so forth so the reason is that if you actually take a bowl of matter and you compress it more and more and more and more and more to turn into a test particle then at some point the energy density that's contained inside of this body will become so so large that that particle will collapse into a black hole by something called uh the hoop conjecture which you can read about you want by thorn in the 70s so the resolution there's well there's different ways in which you can you can show this result but one way is to treat the small compact object as a black hole treat the skull as a black hole and use my favorite mathematical technique which is asymptotic matching so here what you're doing is you're solving for the problem in a neighborhood that's very very close to the small compact object and then in an area that's farther away from the compact object and then there's an overlapping region of validity where you just match the two approximations and if you do a calculation correctly you discover that the cell force really is not m times d alpha beta gamma acting on h beta gamma but rather it's m d alpha beta gamma acting on the regular part of h beta gamma where h mu nu has been decomposed into a singular part plus a regular part which by regular here we mean a part that doesn't diverge the location of the test particle so um this sort of expression with the regularization including is what Bob was referring to as the mi sa ta ku equation which stands for mino sasaki tanaka queen and world equation um so I leave it there there's a lot more work that uh I had to close over for obvious reasons that goes into exactly how to extract this regular part of the metric perturbation a lot of these calculations are numerically so subtracting a singular part is quite tricky there's gauge issues blah blah blah metric reconstruction um so this is a very hot activity of research and by the way this is just the first order in q and we need everything then just second order so good luck with that fortunately I am not working on that so I'll stop there thank you