 And then I'll let you know when we start. And I think Roberto can help me to check if we are actually going live. I think we are right. OK, perfect. Yes. So welcome, everyone. Thank you for joining us for today's low physics webinar. My name is Alejandro, and I'm going to be your host. Today we are presenting the large mass ratio limit of the relativistic two-body program, bias code hugues. Professor hugues earned a bachelor in physics from Cornell University. And he's PhD from Caltech, California Institute of Technology, where he worked with Professor Kip Thorne. So we also have a low physics talk by Kip Thorne. So you might want to revisit that also. And then he did postdocs at the University of Illinois, Caltech, and the Cavali Institute for theoretical physics before joining the MIT physics faculty as faculty in January 2003. He's currently the Margaret McVicar Faculty Fellow and the division head for astrophysics at MIT. Professor hugues works on various topics related to astrophysical applications of Einstein General Theory of Relativity, with a particular focus on modeling of gravitational wave sources and how we can use them as a tool for observational astronomy. We are very happy to have him in our webinar series today. So people remember you can ask questions over email through our YouTube channel or Twitter. And then the questions will be read at the end of the talk. So without further ado, we will turn in time to Scott. Thank you for joining us. I think you can share your screen now. We'll start sharing my screen. Thank you very much for hosting me. And I've been talking with you over email for months now. I don't think we have ever actually met in person. So I'm looking forward to the day when that can happen. So it is a nice side effect of this pandemic that doing things remotely has become somewhat easier. So I'm enjoying that. And I love the opportunity to do things like this. So as Alejandro described, I'm going to tell you a bit about how we study the two-body problem in general relativity, the relativistic two-body problem, in particular focusing on a limit that allows us to both solve this problem with very high precision, which as I'll describe in the talk is of particular interest, but is of direct interest for upcoming future gravitational wave measurements. Now, I like to set the context for why we are interested in this problem by sort of reminding everyone that if we go back to Newtonian physics, the theory of gravity that describes how we study the motion of bodies due to their mutual gravitational attraction for a couple of centuries before Einstein came along, the two-body problem is not that difficult. If you've studied university-level physics, even high school-level physics, you've learned already a lot about this. And in fact, when you dig in a little bit, you find that the complete solution to the motion of two bodies under the mutual gravity according to Newton is quite simple. So suppose I have a body with mass m1. I have a second body with mass m2. And they are attracted to each other by their mutual gravitational attraction. Well, I just remap them to a total mass, capital M, indicated here on this slide, a reduced mass, which I've written the Greek letter mu indicated on this side. I assign to this binary. I can now, I'm going to call it a binary from now on. I can assign it an energy and an angular momentum. I can remap the energy and the angular momentum to two different parameters. I'll call p, which is technically it's known as a semi-latus rectum, but it sort of gives me an idea of the mutual separation of these bodies. In orbital eccentricity, which I'll write epsilon, and then the complete solution for the motion of these two bodies is given by these two functions I've written down here, r of theta and t of theta, which sort of parametrically describes the motion of the two bodies about one another. If gravity, if the study of gravity had ended with Newton, I would now end my talk and say, are there any questions? That would sort of not really be worth your money. So fortunately, additional sciences come along since then. So how do we describe the motion of two bodies according to Einstein? So what we teach our students in our general relativity classes is you begin by building the spacetime of a large gravitating object. I'll say a little bit more about this process later on in the talk. But the basic idea is that the geometry of spacetime, g mu nu, the metric of spacetime, is determined by the distribution of mass and energy and momentum in some kind of a large gravitating object. It's a complicated procedure to build that spacetime, but let's say we've done it. Then freely falling objects will respond to that body's gravity by following geodesics, trajectories that give us extremal time as measured by clocks that are carried by those objects themselves. And so there's an equation that describes that, where in this equation I've written down here, these gamma coefficients are essentially derivatives of the metric. This reproduces Newtonian gravity and the proper limits and gives us a lot of new features as well. A lot of fun stuff. I'm teaching our general relativity class right now and we're just getting into this. We're getting into all the fun you can do with it. Seeing how Einstein is different from Newton. This is actually, though, very oversimplified. It assumes that those freely falling objects are just what we call test bodies. These are objects that respond to gravity, but they don't actually generate gravity. We're sort of assuming that these little bodies fall through spacetime without bending spacetime themselves. And that is totally wrong. Particularly when the freely falling body is not a test mass, if it's actually a planet or a black hole or a neutron star, in fact, both bodies are going to affect spacetime. And part of what makes this problem complicated is that actually defining what constitutes each body might itself be ambiguous. Fundamentally, part of the challenge here is that in general relativity, we don't actually have really a two-body problem. We have a one spacetime problem. When you study the motion of objects in general relativity, it's all about computing the spacetime in which your two objects live. And the truth is, we can only unambiguously describe that spacetime as describing two-body motions in certain limits. So let's consider what the challenge is for two black holes. Our goal is to solve the Einstein field equations, a set of coupled, nonlinear, partial differential equations with the following boundary conditions applied to them. We're going to require that the spacetime be purely vacuum, that there be no matter or energy other than things related to gravity in there. There's no baryons. There's no fields, no electric fields, no magnetic fields. Purely empty space. That actually is the one thing that simplifies this analysis. We're going to require that at some notion of an initial time, the spacetime unambiguously contain two black holes. And I'm going to require that nothing come out of those holes event horizons. By the way, we don't actually know where those event horizons are. So I'm going to need to set an event. I'm going to need to set a boundary condition on a surface whose location is not known. Now it's now well known that we can routinely solve this problem using numerical relativity. But as Alejandro's introduction made clear, I'm an old man. And so I saw the process of getting to these videos, which are now routinely shown by my colleagues and I at various conferences. This was a really long, difficult, bruising journey. And even today, although we can solve this problem for many, many interesting situations, it is very resource intensive. This takes a colleague of mine who shall remain nameless. I remember he once was refereeing a paper where someone was presenting many calculations of gravitational waveforms from binary black hole collisions. And he said to me, I think I'm going to require them to plant a tree for every waveform they put together when you take into account the amount of carbon dioxide that they must have generated making this catalog. It's a very resource intensive process and it's difficult. So there are cases where a really clean notion of it being a two-body problem makes better conceptual sense. And what I'm going to do is focus on one of these in particular in this talk and talk about how we can use the insights in this limit to guide our understanding of the general problem and to really get the physics out of this resource intensive, difficult calculation. So one case where you unambiguously have two bodies is when these are not vacuum situations like two black holes, but they are bodies with matter. Then they have surfaces. They have a place where the fluid of neutrons or whatever comes to an end and the vacuum of space begins. That defines a body in a sensible way. You still have the challenge of computing the spacetime that goes through both of those bodies. And it should be born in mind that the bodies can disrupt. You can have a well-defined isolated body that kind of explodes or disrupts and throws matter all over the place, vastly complicating the calculation. I can have objects that are clearly widely separated. And when that's the case, then what is going on is they will be interacting weekly with each other. So I can treat each body as an isolated one moving in the spacetime of the other body. And to some extent, what is known as the post-neutronian approximation of general relativity is based on this notion, or at least it builds this notion into it. And it's also been hugely successful. The one that I am going to focus on in this talk is the large mass ratio. The idea of the large mass ratio is that I have one big body and then a smaller one that has a relatively weak influence on the spacetime of the large body. I will treat the total as the sum of these two, but I'm going to be able to treat the contribution from the smaller body as a perturbation to that of the larger one. So let me tell you a little bit about part of what motivates this is that, in fact, that setup of one big body and one smaller one exactly describes one of the most interesting astrophysical sources of gravitational waves that people like me are thinking about, which we call them the extreme mass ratio in spirals. The idea here is that if I look at a galaxy, basically every galaxy that we see that's of a certain mass, larger than a certain mass, has a large black hole at its core. And surrounding that large black hole is a cluster of stars. Some of those stars evolve and form black holes and neutron stars. And so there's a little cluster of compact objects down there. Multibody scattering will occasionally put one of those massive compact objects onto a strong field, gravitational wave-driven orbit of the big black hole. And when that happens, it becomes a gravitational wave source. Events like that are going to be rare per galaxy. It requires that you not just have the right multi-body scattering events, but you have to do it and you have to scatter it in the right direction so it gets captured by the big black hole and forms this kind of a binary. So although it's rare per galaxy, it probably happens on the order of once every million years or so per galaxy. The gravitational waves that they generate will actually be audible to space-based gravitational wave detectors. In particular, the plan detector, LISA, the laser interferometer space antenna, out to cosmological distances of order of redshift of 1, which is a couple thousand gigaparsecs. When you fold in the fact that it's maybe only once per million years per galaxy, but you have hundreds of millions of galaxies that will actually be radiating, you reasonably expect there to be dozens to hundreds of events like this per year in a detector like LISA. And if we want to be able to measure the waves from these things, we need to be able to make models, particularly if we want to measure them and then interpret what the measurements tell us. So because these systems have very large mass ratios, the properties of the in spiral and the gravitational waves that they generate depend more than anything else on the properties of the large body spacetime. So by measuring these waves, it offers us an opportunity to probe and to test the properties of the massive compact objects that live in the cores of many galaxies. Now those objects are presumably black holes. They are presumably the core solution of general relativity describes those black holes. But we'd like to test that. Even if that, in fact, is exactly what describes these objects, we want to be able to measure their properties with very high precision. And indeed, when people do so far, people have done kind of quick analyses of these things. I shouldn't say quick. Barack and Cutler spent quite a long time doing this analysis. But we now have somewhat more sophisticated tools and so we're due to revisit some of these analyses. But we find that we can do things like measure the mass of the black holes in the cores of galaxy using these extreme mass ratio events to 0.01% accuracy, incredible precision. We can measure the spin in the mass ratio of the binary that forms to similar degrees of accuracy and a lot of other interesting properties of the big black hole. This would provide a wealth of information about the properties of very high precision information about the properties of black holes in galaxies and also on the scattering and the properties of the star clusters that surround these black holes. So this should be candy for astronomers who are interested in black holes in the cores of galaxies. From a more foundational physics perspective, it also offers us a really interesting opportunity to test the spacetime nature of those objects at the cores of galaxies. So what we have been able to formulate is the idea that you can essentially measure the multiple structure of the black hole, the presumed black hole at the core of the galaxy. So to give sort of a motivation for this, I am showing here the multipolar structure of the Earth's gravitational field as inferred from measurements by the GRACE experiment. GRACE stands for Gravity Recovery and Climate Experiment. What this is, it's a set of spacecraft that orbit the Earth and by very precisely mapping out the properties of the orbit, they can infer what the distribution of mass is within the Earth that produced it. And then they kind of exaggerate the bumps in this. Of course, it's mostly spherical. But some areas of the Earth are heavier than others. And this helps to show that. So I was going to show this movie again. So you can see things like a region here in the South Pacific where there's a little bit of extra mass. India, there's a little bit of lower mass than normal. And so by making precision measurements of orbits, we can learn about the properties of the body that generates the gravitational field of those orbits. So when we do this for black holes, we expect that we can likewise measure the multipolar structure of the spacetime of those black holes. And there's an expectation that if this is in fact a Kerr spacetime, the spacetime that was discovered by Roy Kerr that is expected to describe astrophysical black holes, the multiple moments have a very simple structure that only depends on the object's mass and spin. We want to test that. So far, we do not have data that allows us to do that. But if we can measure these extreme mass ratio inspirals that are described by the large mass ratio limit of the two-body problem, that should give us a perfect tool for doing this. Finally, it just should be noted, as I've said, that it's just simply a clean limit of the relativistic two-body problem. And password has actually shown that the large mass ratio limit works surprisingly well even when the mass ratio is not that large. So I'm showing now an old paper that compares the output from various approximation methods, including those based on the large mass ratio limit with exact calculations using numerical relativity. And so they take the large mass ratio limit and then by a few clever tricks, they actually take it to a mass ratio of one, which is not a large mass ratio. And yet nonetheless, the predictions from those calculations agree with the exact calculations from the numerical relativity simulation extremely well and are much quicker and easier to calculate. Now this was done so far for sort of the simplest cases and a lot of us are working on what we need to do to improve the large mass ratio calculations to try to extend this to a broader class of problems. Let's say a little bit more about that later in the talk. So let's talk now about how I'm actually going to do my large mass ratio analysis. The concept behind this is very simple. I take the metric of spacetime, I write it as the exact solution describing a black hole, my G alpha beta BH, and then I add a perturbation to it that describes my small object that is perturbing that spacetime. In principle, I now just have to take the tools that we learn about in the general relativity class to generate the Einstein field equations. And then I just kind of introduce this sort of perturbative expansion that many of us learned initially in a quantum mechanics class. We just go through this thing and just say, I'm going to keep all my terms to linear order in the perturbation H, enforce the field equation and see what I get. In practice, this doesn't actually work that well. Okay, so if you try to do what I just described for a general black hole, you can get some equations that they're unmatched and we don't find that they're actually really easy to solve except in a special case where the black hole is non-rotating where it's a non-rotating Schwarzschild solution. They are because of spherical symmetry. There's some wonderful cancellations and it works out okay. Nonetheless, a person who was, so if I am Alejandro's academic grandfather, his academic great-grandfather, who was my undergraduate research supervisor, Saul Tkulski, developed a way of describing black hole perturbation theory that works for all black holes. And the clever trick that Tkulski came up with is instead of working with the metric, start out with the tensors that give us spacetime curvature, perturb them and then use a particular framework that allows us to very nicely pick out outgoing and ingoing radiative degrees of freedom in those curvature perturbations and see what that does for us. So there's a relatively few, if you study some general activity, there's a simple way to understand Tkulski's more general approach. If you start with what's known as the Bianchi identity that governs the curvature tensors of general relativity, okay, so my derivative operator here is a covariant derivative in my spacetime. If I start with this Bianchi identity and then I take one more derivative, I can turn this into a wave equation for the curvature tensor. That's literally all Tkulski did. You just turn that into a wave equation. Then what you do is you perturb that curvature tensor. You manipulate this with a couple of tensor identities. You apply the field equation to get rid of a couple of terms, change your curvatures into stress energy tensors. And then we use what's known as the Newman-Penrose framework is a way to allow us to pick out certain components of the tensors that correspond to outgoing and ingoing radiation. A lot of math happens. This is what Tkulski earned his PhD for. So I'm doing this in one slide, but this is a lot of work. But once you're done with all that work, you have an equation. So it turns out to be a simple partial differential equation for a scalar quantity. That scalar quantity is actually a complex thing. And so it's got two independent functions in that. That ends up being something that describes the radiative degrees of freedom in the curvature of a black hole. This is actually kind of an amazing equation if you sort of wanna just nerd out on the mathematical beauty of it. There's a parameter in here, which I've labeled S. And what you find is that if you set S equal to either plus or minus two, this describes the perturbations due to a gravitational influence on the black hole. If S equals plus or minus one, it describes electromagnetic perturbations to a black hole. If S equals zero, you can think of it as arising from some kind of a scalar particle. So this ends up being a beautiful master equation. Now, let me back up for just one second. Over here on the right-hand side of this equation, I have this T. That T is the source that is actually perturbing my black hole. That's where a lot of the work in this subject comes in. How do I describe the object that is perturbing my black hole? What we are going to do is we're interested in binaries. We're interested in the two-body problem. And so what we do is we take it to be a small body that has a given multi-polar structure associated with it. So it's gonna be dominated by the body's mass. If you think of it as just a pure monopole, then it's a little mass M, and I can characterize its stress energy tensor by what I've highlighted in blue here. Now bodies are not completely described just by their mass. They have other properties. In particular, if it is spinning, then there will be some mass currents. And that generates a dipole term to this stress energy tensor. And the dipole is related to the spin annual momentum of that body. We could go further. We could include terms for the quadrupole moment of this body, octopole moment, et cetera. You can sort of almost imagine, I showed you a little movie of the multi-polar structure of the Earth. You could include terms for all the complex lumps that make up the distribution of an object like the Earth. But when we're talking about the kind of things that are likely to be astrophysical sources of gravitational waves, two is probably enough. Some people are thinking about whether we need to include the third one right now. But the truth is the vast majority of efforts so far has focused on this monopole term that's over here and work that's related to what happens when this smaller body is spinning and there's a dipole moment, it's just now becoming very active. I'm going to focus largely on results relating to this monopole, to the body's mass. And toward the end, I'll begin describing some recent results and some efforts describing what we do when we include the body's secondary spin. So now we've got this equation. We know something about what it's telling us, how do I solve it? What do I need to do to solve Tkolsky's equation? So there are three major ingredients on this. The first one is I need a very accurate, complete description of the body's motion, the thing that is actually producing the T on the right-hand side of this equation. How do I describe that? Then I need to be able to solve for the field psi, which will then tell me about the curvature perturbation that arises from my source. Then I need to find a way of computing the back reaction that this change to the body's space time exerts back on the orbit and talk about how that changes the body's trajectory on very long time scales. So let's begin by just talking about the small body's motion. Let's do the first step of this. So a beautiful framework describing this comes from my colleague, Anna Flanagan, who's someone who, he was a graduate. He had just defended his PhD when I started graduate school. So Alejandro, he's like your great uncle or something. And he and his former student, Tanya Hinderer, developed this way of describing the motion of the characteristics of black hole orbits using an action angle variable framework. So in this sort of system I've written down here, lambda, just think of that as a time variable. I can answer technical questions about that later, but just think of it as a time variable that's well-adapted describing orbits of black holes. The parameter epsilon is mass ratio. And so the action and angle variables, basically the angle variable is you can just think of as things that relate to the motion in the four coordinates of space time. So we're gonna use T, R, theta and phi. And the angles are conserved integrals of the motion that describe the properties of orbits in a black hole space time. These Gs, the lowercase G and the uppercase G, they represent self-force effects. They are sort of effects on the motion of the orbit that arise from how the small body changes the space time itself. So to understand what's gonna happen, what I wanna do is look at this equation order by order. So let's just, first of all, throw away everything that's scaled with the mass ratio. I'll just do order epsilon to the zero. So what you find in that case is your integrals of the motion are constant. So there is no forcing term that changes those J's and the angle variables just increase at a rate that's set by sort of a generalized notion of a frequency. This, at least for when the small body is just a monopolar mass, this is just a funny way of rewriting the equations that describe GUDs and orbits. If the body has some dipolar structure, it's a funny way of describing the motion if it's a spinning bottom. That's it. Now when I include effects at linear order in the mass, I get these forcing terms. So one of the things that we have learned is that to understand what's going on with these forcing terms, it's useful to describe them in a Fourier expansion. So both lowercase G and uppercase G, just think of them as some functions that depend on a couple of these angle variables. I can expand them in a Fourier expansion, okay, defined as follows. And when I do that, that allows me to sort of separate out parts that accumulate over many orbits, sort of the bit that has an average value for these things and other parts that just oscillate. So if the orbit did not evolve, they would in fact average to exactly zero. The orbit's actually gonna evolve slightly, so they don't perfectly average away. So these Delta F terms give a small but non-zero contribution. There is one exception. I'll be happy to answer questions about this at the end, but there will turn out to be some interesting moments that can change the story a little bit. It's a sidebar, so I'm not gonna talk about this too much, but in case anyone has heard of this, we are aware of the, there's some interesting issues happening at what are called resonances. So when I do this, the equations of motion look like this. So I've now separated out the average of these terms and the oscillating pieces. The average of the lowercase G function basically tells me about how the frequencies describing orbits shift thanks to the way the small body changes that big body's space time. Okay, so you can sort of think of this as almost like your, the Newtonian analog of this would be a slight shift to Kepler's law because the body's own gravitational field is changing the binary's interaction. The average of the capital G is describing how these integrals of the motion, the energy and the angular momentum, which I'll talk about in a little bit more detail later, how they change due to the self interaction. This in fact ends up describing the way that energy and angular momentum are lost in the system due to gravitational wave emission. So let me just quickly go through. We then find when we're studying these kind of things that there is a term that describes the motion on very short time scales. There is a leading dissipation that changes the orbit on long time scales. And then there are corrections that come in at higher order and are all things that we are slowly learning how to take into account. So what I'm gonna focus on for the next couple of minutes is an approximation we've developed that is at the moment the most mature and complete description of these things. We're gonna take the motion to look like the black hole orbits you learn about in textbooks on short time scales. And I'm gonna use this orbit average self force, this orbit average dissipation to describe the motion on long time scales. We call this the adiabatic approximation because it sort of looks like the system is adiabatically evolving through a sequence of simple black hole orbits. You can in fact think of the system as essentially flowing through a sequence of the kind of orbits you learn about in textbooks with the rate of flow simply set by the rate at which gravitational wave emission changes these integrals of the motion. Okay, and as I said a moment ago, this is currently the most mature and well-developed approximation of modeling these things. And so those who are interested, this reference here is a paper that I led, got published last year that describes where we stand in modeling these things. So let's talk a little bit about these orbits. So these, you take the GD's equation that describes motion in any space time, focus on black holes and those equations of motion very famously turn out to separate so that I get separate equations governing the radial, the polar, the theta angle, and both the time and the axial motion in respect to this coordinate system. You also find when you do this that these orbits are characterized by three integrals of motion. There's an energy. This arises from the fact that black hole spacetime is my background big body spacetime in this problem is time stationary. It's not dynamical, but it's the same from moment to moment. They are also axially symmetric. And so there is a notion of an angular momentum associated with this. And there's a third constant, usually just called Carter's constant that exists due to a more complicated symmetry that I can't describe as quickly or easily as this. But it's worth noting that when the larger black hole is a non-spinning one, this is nothing more than the square of the other two components of the angular momentum. Once I've selected these three quantities, I can now look at the geometry of a black hole of it. And you get this sort of really cool structure where they end up sort of filling a torus in the spacetime near a black hole. So once I've selected energy, angular momentum, and this Q, that tells me the minimum radius, the maximum radius, and sort of a range of angle over which the orbits varies. Once I understand this motion, I can also identify three distinct orbital periods that characterize the body's orbit. So there is a T sub R, which tells me about the time it takes for an orbit to move from minimum radius to maximum radius and back. There's a T sub theta that tells me about the time to move through its range of the theta angle. And there's a T sub phi that tells me about the time to revolve in the actual direction. Each of these periods corresponds to a frequency. Once I have a complete understanding of this, that gives me a tool for characterizing in sort of a Fourier transform sense, all the functions that are important for studying these orbits and quantities related to these orbits. These orbits is worth talking about, they're pretty cool. So when you look at these frequencies, one thing you find is that they differ significantly from each other. They all asymptote to Kepler's law as you go into the weak field. But the most interesting stuff is when you go into the strong field, when you're talking about reason to gravity close to the black hole. So I'm plotting here one example which shows the frequencies, modulo two pi associated with some orbit, associated with the radial motion, the theta motion and the phi motion. So in this plot where I've sort of chosen particular values of eccentricity, orbital eccentricity and orbital inclination. And I've plotted it versus the minimum radius in the way to the point at which they get closest to the black hole. Notice they all asymptote, they come together as our men gets large. So this is for an orbit that's very far away from the black hole, all three of these frequencies are the same. But as you move into the strong field, they begin to differ significantly from one another. I like to think of this as almost being like the Zeeman splitting of features in the spectrum of a hydrogen atom when you put it into a strong magnetic field. It's not a strong magnetic field that's breaking this line, it's actually strong gravity that's breaking this line. So this is a way in which the spectral characteristics of orbit is giving me, of orbits, is giving me information about the properties of strong gravity. So because these frequencies are all different from one another, when I actually look at an orbit, what I'm seeing is that it's gonna move through radius and the theta angle and the phi angle at different rates and their mismatch leads to all sorts of interesting procession effects. You get periastron procession as things moving through the strong field. You'll notice in this particular orbit I've got plotted here at whirls around the black hole a few times before it comes back out. The orbital plane is continually wobbling. It's a very complicated kind of a figure. So I've understood the orbit. That gives me, that means I now fully understand the T that's on the right-hand side of this perturbation equation. Now I just need to solve for psi. So part of why Tcholski earned his PhD thesis was that he showed that this equation can actually be turned from a partial differential equation into an ordinary differential equation provided he sort of picks the right basis of functions to describe the solution. So if you expand it in a particular harmonic and multipolar expansion, this turns into something that's not that difficult to solve. You know, it's hard enough. I spent a lot of time writing code to do it, but it turns into something that can be done using ordinary differential equations. You can solve it to very high precision. In particular, if the source is a black hole orbit, the only frequencies that contribute to the psi through our perturbation to the spacetime are gonna be harmonics of the orbital frequencies. And so that gives us a very closed set of frequencies that we need to solve over and really expedites solving these equations. Once we've got this whole thing solved, we can build a complete solution for that psi. Once we've got the psi, we can describe gravitational waves very far away from the body. We can describe what the tidal impact that the tidal stresses exerted on the black hole by the orbiting body are. And if anyone's interested, there's, I can point you to papers where there's some of the technical details of this, but the key thing which I wanna emphasize is once we solve that equation, we can get everything that's of interest for describing the distant radiation and how the black hole interacts with the orbit. Once we have those, we can extract from them how the body's properties evolve due to radiation being emitted and due to a tidal interaction with that small black hole, or sorry, with the large black hole. And again, actually, okay, I forgot to put this in here and here are a few papers I've been involved in that summarize how we do this and give a lot more details. The key thing I wanna emphasize is it's now basically a solved problem. All I need to do is put all the pieces together. I examine a lot of binaries. I densely cover the parameter space that describes orbital kinematics and I allow these orbits to evolve due to gravitational wave emission. I basically compute rates of change for each orbit, use those to evolve from orbit to orbit, and then stitch together the result to assemble the gravitational waves I get from this. So here's an example of how I go through this process. So here's a set of orbits that start out with zero eccentricity. And so what I do is I just compute on a grid, a dense grid of orbits, what the gravitation waves look like from each orbit and how gravitational waves tend to sort of push the system, the direction in which the system tends to flow. And so in this particular case, I'm thinking about a system that has zero eccentricity and so the orbit is completely described by a point in one axis is orbital radius, the other axis is sort of cosine of a particular notion of inclination. And you can see two examples of gravitational waves that arise from orbits in this parameter space and this plot shows the direction in which I evolved from orbit to orbit. One quick technical note, I've exaggerated the angle here somewhat, best to show that it's a non-zero angle. It's actually very small and so to a good approximation, we find that this inclination angle remains practically constant over an in spiral. Another one is think about an orbit that lives in the equatorial plane. It varies between an R min and an R max that I've characterized with these parameters P and E, but it lives in the plane with theta equals pi over two and a similar kind of thing here. Here are two examples of gravitational waves in this from this parameter space and this plane shows the arrows that describe how radiation tends to push you from orbits to orbit. Once you stitch, I've filled several disc drives with data describing these kinds of things. Then once you do that, you just need some tools to describe how you step from orbit to orbit and you assemble the gravitational wave. So here's an example. This is actually based on a slightly cruder approximation of things, I should update it at some point, but it still does a very good job describing this. So I'm gonna show you now what you get when you look at a 10 solar mass black hole spiraling into a million solar mass black hole. This signal would actually last a year, probably a little bit more than you would be willing to wait for me to play a video for you. So it's been sped up by about a factor of 100,000. So what you're gonna see in this video is the trajectory followed by the small body as in orbit's the big black hole. You're gonna see the gravitational wave form it produces at the bottom. And just because it's fun, I've turned it into a sound so that you can actually, your ears can allow you to sort of track what the information the gravitational wave sounds like. So let's go. Hopefully the sound is coming through clearly. You can see it's a very complicated shape. These are all those processions you saw in that previous movie just rapidly moving from one configuration to another. The buzzing that you are hearing is because the black hole, the orbiting body rather, moves more quickly near the black hole than when it's farther away. So it's kind of going fast slow, fast slow, fast slow. The neck comes together and makes this sort of buzz as it all, as it all grows together. As in spiral proceeds, the body moves faster. The orbit becomes more circular. At this point, I stopped it about one day. It would very quickly plunge into the big black hole and the signal would end. But that complicated pattern on the bottom there, that is what we hope to model very precisely and what we hope to measure. And this is a difficult problem. Fortunately, although the waveform is itself complicated, it can actually be thought of as the sum of many, many thousands of relatively simple elements. So in a system like what I just played for you, there's actually approximately sort of a thousand of what I like to call voices that contribute to this. And each voice has an amplitude that evolves in a very smooth and simple way. Because every one of the voices that contribute to this is so smooth and simple, we can exploit that to compute these waveforms very quickly. We have to do a lot of relatively expensive pre-computation and then storing of all our data somewhere. But because everything is so smooth and well behaved, you can interpolate the data very, very accurately and recover that waveform with significantly less calculation. In fact, making a waveform like that by taking advantage of pre-computing various things and storing them, I can make waveforms like that in approximately a second. So that roughly one year long waveform requires about one second's worth of calculation using a bunch of, I have some colleagues who know the computational ends of some of these things. They use graphics processing units and a few neural network tricks to get that down to about a second. That's not my expertise. So thank God for co-authors. And it gets it down to something that works incredibly efficiently. My colleagues and I, in fact, I have a telecom to discuss this shortly after this talk is done. We're working on extending this to the case. It will be slightly slower because they're just a more computationally complicated problem, but we still think it'll be on the order of a second or so. So I wanna emphasize one cool bit of physics that is in this. So when I calculated that in spiral, I was computing the rate at which energy and annual momentum flowed to infinity due to gravitational waves. And that flowed to infinity, it's always positive. When gravitational waves fly away from the system, some of it's eventually measured, we hope by our detectors, but it always takes energy away from the binary. But there is also some radiation that's absorbed by the black hole. And the radiation absorbed by the black hole is a little weird. Whether it is positive or negative depends on the relative frequency of the orbit and the black hole spin. What we find is that if the orbit's axial motion is faster than the black hole spin, then this sort of energy going, sorry, the radiation going into the horizon takes energy away from the orbit. But if the orbit is slower than the black hole spin, we actually find that that contribution adds energy to the orbit. So when we initially began seeing these in our calculations, it was surprising. But it was a substantial effect. We actually find that for large mass ratios, especially if the larger black hole is rapidly spinning, there are messing around with the rate at which you absorb energy changes things by several thousand radians, several thousand orbits over the course of an in spiral, many times larger than you would need to see this effect of the detector noise. It's huge. So it's safe to say my reaction was, huh, this is really confusing. You know, a horizon is supposed to be a perfect sync of anything. How can energy or angular momentum come out of it? Well, it turns out there is an exactly analogous phenomenon that we see in Newtonian gravity called Pytocouple. So imagine I have a moon orbiting a fluid planet. So the moon raises a tide on the planet and the planet bulges. Fluids are viscous though. And so the bulge lags in time via stresses of the applied time. And when you take into account that both the moon is orbiting the planet and the planet is spinning, what we find is that if the planet is spinning slower than the frequency associated with the orbit's velocity, then thanks to that viscosity, the bulge will lag the position of the orbit. If the, oh, and when that happens, the bulge exerts a torque that takes angular momentum and also energy out of the orbit. The orbit slows down, but at the cost of the planet spin getting a little bit faster. If the planet is spinning faster than the orbit's frequency, the opposite happens. The bulge will lead the orbit's position. There is a torque that adds angular momentum to the orbit. The orbit goes a little bit faster and the planet spin slows down. This in fact is exactly what happens in the Earth's, the Earth-Moon system, okay? So the moon is actually spinning it up a little bit misleading. It's gaining angular momentum due to the tidal coupling of the oceans and the crust and the atmosphere to the moon's gravity. And as a consequence, the Earth's spin is getting a little bit slower. So the net effect is the orbit gains energy if the orbital frequency is greater than the spin frequency of the planet. It loses energy in that case. It gains energy if the orbit is slower than the spin frequency of the planet. It is exactly the situation we see in the black hole analysis. So does this intuition, I mean, come on, we're talking about a fluid planet versus a black hole. Is there any validity to this analogy? Well, it turns out that an event horizon behaves remarkably like a viscous fluid when a tide acts on it. So if I have fluids, there's an image that I took from a Wikipedia article. And I imagine I have fluids flowing in some kind of a flow and a tide acts on them. Well, what happens is that tide shears the fluid elements that are moving in that flow and that generates heat, okay? So that generates heat, that generates entropy. And so the rate of heat generation is actually governed by the viscosity. And so if you're interested in understanding how much entropy is generated in this fluid, there's a simple equation from fluid mechanics that describes this, which relates the rate of entropy generation to the temperature of the fluid, the shear viscosity of the fluid, and the shear of the fluid due to your tide. Turns out there's quantities known as horizon generators which you can kind of think of as null Gds, light beams that live in the event horizon and help to trace out the event horizon structure in space time. They are also, they are sheared by a tide in a way that looks just like how a fluid element is sheared by a tide. The entropy of a black hole is its area. Okay, that is a famous result due to Jacob Beckinschein and Stephen Hawking. And when you work through the math associated with this, let me go back to what the equation looked like for fluid mechanics. The rate at which the entropy of a black hole is increased, it's governed by an equation that looks just like the fluid equation. Indeed, you can actually identify a term that looks just like of this cost of it. So we can actually look at the generation of entropy, the rate at which a black hole's area is increased due to the tidal influence in gravitational waves, is what this is telling us. And indeed, when we actually compute how strong this effect is, it is huge. So we actually find that the rule of thumb is that over a measurement, if some effect changed its phase by a rating or so, you should be able to pick it out of the noise. So we actually expect for the measurements of stream mass ratio in spirals where a small body sprung into the big black hole, it will change the phase by dozens, to hundreds, to thousands, possibly up to tens of thousands of radians, way above the noise. So this is an example of the kind of cool physics that we hope to do with these kinds of systems. So as I wrap up this talk, I've kind of given you the impression that there is, you know, I hope I'm getting the impression that there's a lot that we've understood. We have a great set of tools that we can use, but I wanna emphasize that there is a ton of work to be done. So in terms of what I talked about today, I have neglected several effects. The first one is various other things in the self-interaction that I did not include in these models so far. When I include the sort of what we call the conservative part of the small body's self-force, that also that changes the body's orbital frequencies and work by former postdoc of my Niels Warburton working with my colleague Chuck Evans and his student Tommy Osborne. They've shown that if you, you know, fold in what that thing does, you get dozens of, again, dozens of radians need to be taken into account. That is also a huge and important effect. And we're just starting to think about how we fold those into our models, okay? A lot of work to be done. There's an oscillating contribution to the dissipation, which I think we actually have an idea of how to begin looking at this, but we haven't done quite as much work on that. We do know that to order of magnitude, it'll be similar to this other effects, and we need to do a lot of work to just make sure we're doing it properly. Neglected effects too. Everything I've talked about so far is at first order in the mass ratio. That's an infinite series. What about second order? So order of magnitude analysis show us that the second order is going to measure. It's going to matter and we need to model it. But to get those terms properly, you have to know the first order terms to incredible precision. The source governing the perturbation equations at second order in the mass ratio depends on sort of the solution to the first order of perturbation squared. So if you have any error in your first order solution, you're going to completely screw up your second order solution. This has been a focus of a tremendous number of my colleagues in what we call the Capra community. So this is based on a series of meetings that we have for about 10 years now. And very recently, they have computed a situation where they can now sort of do with the full second order self-force in spiral through a sequence of circular orbits for a Schwarzschild black hole. So it took about literally 10 years to write this paper that I've got referenced here. But what's striking is that now that they've done the large mass ratio limit to second order in the mass ratio, they nail the results that can be produced using full three-dimensional numerical relativity and making the waveforms that now that they've got the tools together, they can make each waveform in about a millisecond. It is incredible. They can sort of take hundreds of hours, thousands of CPU hours on a supercomputer and reproduce it with a milliseconds on a laptop. A lot of pre-computing went into that, but it's striking. Neglected effect three, that small body that's spiraling into the big one, it's not just a point mass. So this is something I've been thinking about a bunch lately and so I'm gonna take a few minutes towards my conclusion to sort of sketch a few of the effects that happen. When the small body itself has some structure, it's not point like. And so you can think of it as an extended body and as an extended body, it's sort of, for lack of a better word, it tastes the space time in which it's sitting. And what we find is that there are sort of forces and processional effects that arise due to the small body structure coupling to the curvature tensor of the background space time. So it's not just falling in geodesic, it's feeling a force that pushes away from the geodesic and it's trajectory processes due to the fact that the spin sort of processes that moves along its orbit. Fortunately, we can treat the secondary spin itself as a small parameter and that allows us to set up a whole new set of perturbative expansions. And there has been a ton of work in recent years in developing the framework to solve these equations and look at motion in this framework. And so I'm highlighting here a couple that I personally had found to be really useful and valuable as I began to learn this field. Let me give you an example of one thing that happens. So if I have a small spinning body in an equatorial orbit and it's just going around a non-spinning black hole which I forgot to write about. So this is an orbit of a short shield black hole and in a particular time parameterization, the frequency is associated with the radial motion and the axial motion. So if it's time to go around the axis of the black hole, there's a simple form that's not that hard to derive. So this is what it looks like to linear order in the small bodies orbital x and trisving. Now I allow that small body to have some spin and I start picking up additional complicating effects that tell me that this additional force it's feeling shifts all these frequencies. These are significant. That small body spin is at the same order as the mass ratio of the small bodies itself a black hole. And so this will influence the observables. We need to take this into account. And indeed one of our goals is to use this to get better understanding of the two body problem in general. So we have to understand this not just for the simple case I've laid out here but for general orbits, general small body spin orientations. I will advertise work by my student Lisa Drummond who although I'm an author, I will give her the line share of the credit for this work. We've recently done a very systematic analysis of how to apply this to general black holes and general orbits. And so a couple of papers in the archive going through that that Lisa should be very proud of. Also, okay. So one other neglected effect. What about the environment that these binaries live in? As general relativity theorists, we often live in this sort of platonic ideal of the universe where I have my black hole, I have my body orbiting the black hole and nothing else. That's false. Especially as I'm thinking about extreme mass ratio binaries, this is the thing that lives in a galaxy. Okay, so there's gonna be other stars and other black holes that are near the system that are gonna gravitationally perturb it. So I'm involved in one of these papers working with my colleague Beatriz Bunga. We showed that if there are additional sort of reasonably strong gravitating bodies near that system, especially when certain frequencies combine on resonance, they kind of can hammer the binary and change the evolution in a way that I've not described to you today. You also have to worry about the fact that there can be matter accreting on the black hole which will exert drag on the orbit. And it introduces additional dissipational effects that are gonna change the N-spiral. And so Alejandro's PhD supervisor, Nico Yunus, was a co-author on this paper with Vence Coxess that I think laid out a really nice framework for understanding this. There's been more modern work looking at numerical models of accretion flows that I will also advertise for this. I'm not involved in either of these things, but I began thinking about how to fold this and some of the framework I have been developing. Punchline I would say, so this is my last slide, is that the two-body problem in general relativity is basically solved. These computational breakthroughs in numerical relativity basically tell us that we can solve, throw enough CPUs at a problem and burn enough natural gas to power the computers that solve them. You can basically model any binary system that you care to. But we wanna also understand the physics of those things. And the large mass ratio limit, we have found to be a really powerful tool for studying this problem and developing insight into the physics of these black holes. And especially for students who might be listening to this thing, I will advertise that there is a lot of work to be doing. And one of the other versions of this talk I give, I have a little picture of a one day when my daughter was two, she just made a little sketch that looked remarkably like this extreme mass ratio N-spiral. And so I often show this and say, we start training our students early when I would show her picture with that thing there. But all joking aside, there's a lot of room and a lot of need for people to help us out with this. So I will end there. Thank you very much. Thank you, Scott for this fascinating talk. I see I receive some questions, maybe the coordinators will have. I don't see one in the YouTube channel. So let me start with this one there received. There is a little bit long. So let me see, it's a question where it has several questions. So it says, what sets the limits of convergence in this problem? How do we know it actually converges? Is it because of numerical relativity comparisons? And how do we know it's not fairly close to the black hole? And how do we know is the correct answer? Quote unquote, I guess. No, that's a great question. And it's an important question. And part of the answer indeed comes back to the fact that we have multiple ways of attacking this problem. And we can look at, so the truth is if we only had one technique, you should very much be worried about that. But by using multiple ways of studying this problem, looking at the numerical solutions, numerical relativity solutions, by looking at this sort of wide separation expansion, we can see that all the different techniques converge onto a similar behavior. In terms of how do I know whether it's working well in different domains near the black hole, that's kind of a question that depends on the properties of the solutions to these differential equations. And I've not gone into the details of what some of them are, but they in fact, all of these things have certain boundary conditions that they must satisfy. So for instance, you require, things can only go into the event horizons, things can only go out. If you sort of think about this, you make a causal diagram describing this. So at null infinity, you only have outgoing fields. On the event horizon, you only have in going fields. And when you satisfy that, the event horizon, assuming the differential equations end up admitting well-behaved solutions. But then the question is, do I know whether this solution is convergent or not? We have only recently really begun to understand how to solve the second order equations. So far I can tell you that it's, we're physicists, not mathematicians. We don't have a proof of convergence, but we kind of have an empirical evidence from the calculations we have done so far that nothing is blowing up in our faces. That may change at some point in the future if we discover that there are some issues here. Actually, we give one example that does sort of point in that. We actually find that the equations of motion in this sort of multipolar description of these bodies, it could be subject to some pathologies. And that's just related to the fact that whenever you treat a body rather than as an extended object, but as a system of pure multiples, you're gonna have a few kind of artificial features associated with that. And we have to be careful and understand that. And to make sure that any pathologies associated with that do not pollute our solution. Thank you. We'll see if there's a follow-up question or a comment. Yeah, hand up. There's another one. Oh, yes, I'll go through that in a second. There's another question here. How do we treat the intermediate range? I guess the imbri case. That's an outstanding question. So, strictly speaking, this framework, we know it works, our sense, which I will again, I'm relating to that first question, our empirical sense is that the sort of first order and perturbation theory works fine when the mass ratio is sort of 10,000 to one and more extreme than that, 10,000, 100,000, a million to one or higher. The numerical relativity in principle can do any problem, but it gets computationally intractable when the mass ratio gets too large. And so there's been a lot of work going up to mass ratios about 15 to one, about 21. There's an intermediate regime here, sort of 50 to one, 100 to one, where neither technique really works that well. The hope is that these second order techniques will bridge that gap. And I again, emphasize that to hope we don't actually have completely established that that's going to be the case, but early indications, at least for the simple situations that we've been able to study so far, suggest that that is the case. And let me just comment that astrophysically that regime is of huge importance. So one of the most important sources for future space-based detectors is likely to be black hole, the assembly of black holes at high redshift, sort of early cosmological assembly of black holes. And those early black holes are, according to all the wisdom we have so far, they sort of assembled hierarchically when the galaxies that they're embedded in also assembled hierarchically. And when you go through the statistics associated with the simulations, which, you know, take them with a grain of salt, but nonetheless, it's at least an indicator, systems with mass ratios of 20 to one or 50 to one or 100 to one are pretty common. So it's quite likely that to build good models for the intermediate mass ratio regime will be important for the astrophysics of future measurements. Thank you. Okay, I see here, yeah. John, Alex, you can unmute yourself, yes. Yes, thank you, Alejandro. Okay, first of all, thank you for the great talk. And I just have one question and it's referred to your last slide, what you say the two-body problem is basically solve it. That may be a bit of an overstatement. Yeah, I just want to know because the two-body problem is just a set of partial differential equations. And in a mathematical context is if you say, this is solved because you have the function that put into the equations that satisfy the partial differential equations. And this is not like I have the functions. So when you say basically, because you have a set of observations and that, so I just want that you emphasize what you mean in basically solve it. Right, so that's a great, thank you for the request for this clarification. So what I would say about this is that for any astrophysical system that we observe today and that we are likely to observe in the next 10 years, I think we can do a numerical relativity simulation that accurately describes it. Some of them will be very challenging. So systems of very large mass ratio, systems with high eccentricity, systems where the members of the binary are very rapidly spinning. Those will take significant CPU resources and it's quite possible that the codes will encounter instabilities that will make them a little bit more challenging. But the vast majority of systems that we're likely to encounter in the next 10 years and I would say all that we've observed today, we can model those. And the key thing which I want to emphasize though is that being able to run a numerical relativity simulation that allows me to sort of describe it, that's important and that's necessary. But I think that the kind of quasi-analytic sort of mixture of analytic and numerical techniques, large mass ratio limit being one of them, that we use to understand these binaries gives us a lot of insight into the physics of what is happening with this. So you know, akin to the sort of analysis we did where we looked at how the tidal coupling can change the rate of in spiral. Things like that can be handled much more simply in a perturbative calculation than in a numerical relativity calculation. Okay, and last one. In some part you say that one of the things that is for further work is think about the environment. And what if, because I don't know how difficult could be if you start just by adding a new body and you talk about a three-body problem, for example, is too hard or is like the next clear step or is not the case? So I'm gonna advertise a little bit. One of my former postdocs working with one of my graduate students, so that's the paper by Carl Rodriguez, did some work looking at this problem. It's the three-body problem is tractable. If you can treat it hierarchically, okay? So what I mean by that is imagine you have a binary and you have two bodies that are strongly interacting with one another. Their orbits are really rapid. And then you have a third body that's sort of distance from it. So that its influence is, you know, it's non-negligible, but it's not evolving very rapidly. So then you have what you call a hierarchical binary that we are developing some really powerful tools for modeling. If you have three bodies that are really close to one another so that there's no hierarchy of scales between them, that's hard. I don't think we really necessarily know how to do that. Fortunately, from an astrophysical perspective, the hierarchical problem is probably the more relevant one, okay? Because what you're going to get is when you have a really tight binary, there'll be other stuff near it, but near as a relative term, right? It'll be kind of, you know, be sufficiently far away that it's not significantly dynamical, but it does exert an influence. So, you know, there's, again, I'm gonna come back to one of the points I made. There's a lot of room for people to think about problems in this venue. And so we have early indications and some great first steps in some of the directions for these things, but this is not the end of the story by any means. But thank you. Thank you so much. Sure. Thank you. Do we have another question here from Roberto Nicolas or maybe I can read the last one. We are a little bit past the hour. Let's see here. Okay. I'll ask the last one. Scott, thank you again for this nice talk. I would like just to see your vision about how modeling this, like for the detection purposes. So let's say we have Lisa. I'm assuming we need to have a model that will have the resonances, these environmental effects. Like, so do you think it's going to be something like a sort of clutch where we start doing it and then now we put the resonance where it's important and then we keep doing that. So it's going to be like a Frankenstein model or do you think at the end of the day we might be able to capture all of these events, it affects with a great- And the end of one's self-consistent model. I think initially, you know, you sort of, you're going to start with Frankenstein, okay? So you'll start with Frankenstein because we know how to do many of these elements sort of on their own. And you have to sort of figure out a way to sort of to piece them in there. But one thing that's worth emphasizing is that let me talk about, you know, some of these neglected effects. So hopefully you can still see my screen. When we started including things like some of these, some of these self-force effects that are not yet included in the adiabatic models I talk about. So like for instance, this oscillating contribution to the dissipated part of the self-force, that actually incorporates the resonances. To give an example, it's similar to what you just used, right? And so if we have a way of folding that in in a computationally effective self-consistent way, the resonances would come along for free essentially. So I think some of the effects will be kind of like that, including for example, the conservative part of the body's self-force. I actually think doing that as sort of like a separate Frankenstein addition to the adiabatic models might be surprisingly accurate and effective. We'll need to test that by having a self-consistent model that does everything all at once. What I suspect will happen is that when we make a model like that, we'll have sort of like our beautiful gold standard that probably takes 1,000 CPU hours to compute. And then we'll do these Frankensteins might take seconds to compute. And we'll just check and see how good does this Frankenstein do compared to the gold standard? And I think that's where a lot of the activity is going to be over the next five years, especially as plans for studying the data and doing the science with space-based detectors that it's really kind of solidifying now. That's when people are going to begin thinking about these things. Awesome. Thank you. Thank you very much. And thank you everyone for attending to our low physics webinar today. Remember, you can see the schedule for the next talk. So Scott, thank you for your time and thank you for contributing to low physics webinar. Let me just emphasize again, it was a really, I have found in the pandemic, the lack of being able to travel, part to me hate Zoom, right? So there's so many things that I want to get together with someone and have a conversation and then go out for a beer afterwards. Doing this though is a wonderful side effect of all this. So I really appreciate the opportunity to be able to give this presentation. This is fun. And I hope that this was something people found interesting. I had fun doing this. So I very much appreciate the invitation. Thank you. Thank you very much. We are very happy for this excellent contribution. So see you next time. So next time. All right. Take care everyone. Adios. Take care. Adios. Thank you very much, Scott. Okay.