 I was a late addition to the conference speaker list. Apparently it was because Tiibo wanted to hear about the latest and greatest in gravitational self-force, so I'm hoping that he will not be disappointed. We'll see by the end. Okay, so the general context is something we've heard a lot about already at the conference, which is just the gravitational two-body problem in the relativistic regime. It's a very complicated problem with a lot of parameters in general. Here's a plot of the binary parameter space, just two of the dimensions in the parameter space, which is the orbital separation, how far apart the two bodies are, and the mass ratio. And in different regions, we use different approximations or different methods. So if the bodies are widely separated, then their mutual gravity is weak, and we can use Post-Mtonian theory or Post-Minkowski in theory. If the two bodies are very close together and roughly equal-sized, then the most accurate method is full numerical relativity solving the fully nonlinear Einstein equations. Now what I want to focus on is this corner down here, which is gravitational self-force, where one of the bodies is much smaller than the other one, and we can use perturbation theory in the limit of the mass of this one being much smaller than this one. And so I think this is a very important limit for multiple reasons. So one is that currently most modeling has really focused on the comparable mass regime. So you imagine a binary in spiraling down over here. We start off in Post-Mtonian regime, end up in the numerical relativity regime, and you combine them together with effective one-body theory, and you have your current models that are used in LIGO. But as detectors get more precise and as we see broader frequency bands with detectors, we will see much greater variety of binaries, and in particular we will see higher mass ratios. So we've already seen mass ratios of order 1 to 10, and we'll see more and more extreme mass ratios as we go. It's important to look in this limit to get accurate models in that limit. There's also more fundamental reasons why this limit is very interesting. So as Alessandra told us, the limit is partly already built into EOB. You can kind of think of EOB as mapping this problem onto that problem. So it kind of provides a way to tie together this whole binary parameter space. Now the follow-up talk by Leo Barak will really focus on that idea of how gravitational self-force can really inform EOB and other things. I'm going to be talking about another aspect of why this is interesting. So not only can it model binaries with very extreme mass ratios, but typically we find it's actually remarkably accurate even as we move towards the comparable mass ratio regime. It's much more accurate than you might expect. Okay, back up. But the original motivation for looking at this small mass ratio regime really came from a particular kind of astrophysical system called extreme mass ratio inspirals. So this is really taking the limit of small mass ratio where we're imagining a stellar mass object, a neutron star of black hole of say a few tens of solar masses orbiting around a massive or supermassive black hole in a galactic core. So here you typically have mass ratios of order 10 to the 5, so very extreme. And these are going to be important signals for Lisa and other space-based detectors because they emit in the Miller Hertz regime. And this small guy does many, many thousands, tens of thousands, hundreds of thousands of orbits very close to the big black hole. And the orbits are typically very complicated. So you get a very precise probe of the regime of strong gravity near the large black hole. And so you get a lot of information about fundamental physics, about astrophysics, a lot of unique information that's typically much more precise than other systems. Okay, but I don't want to focus on Emory's in this talk. I want to emphasize that the self-force theory is not just a way of modeling Emory's. It's a general method of tackling the Einstein equations. So we imagine having an exact spacetime that consists of a background spacetime plus perturbations due to a small body in the spacetime. So this background can be anything. It doesn't have to be the background of a big black hole, but in a binary this will be the spacetime of the larger body as if that body were isolated, unaffected by the small body. And then you add perturbations due to the small body. So this quantity epsilon is proportional to the mass of the small body. Now these perturbations deform the geometry of spacetime and that deformation affects the motion of the small body itself and it exerts what we call a self-force. So this is the covariant acceleration in the background spacetime. If you ignore these perturbations to the geometry you just have zero on the right-hand side, just a Geodesic equation for a test body in the background. But once you add these perturbations you get an H1 contributing to a first-order self-acceleration or self-force H2 then leads to a second-order and so forth. So this is a general method and the first part of this talk is going to be talking about how this method works independent of what you're modeling. It doesn't have to be the two-body problem. Then we will get into specifically modeling the two-body problem. So I'll split it into three pieces. First is self-force theory, the local problem. So dealing with the behavior near the small body in a way that's essentially independent of what the external spacetime is. Then go to the global problem which is solving the Einstein equations in the whole spacetime and then it becomes essential what is the external geometry and in the binary case the external geometry will be the spacetime of the larger black hole. Then I'll present the newest results which are at second order in perturbation theory and what we call post-adiabatic waveforms. Okay, let's jump in a few slides here. Okay, so the local problem. So here I'm illustrating a binary but the point is the local problem you zoom in on the region near the smaller body and it doesn't really matter what's out here. It could be anything. And most kind of foundational self-force derivations are based on this idea of matched asymptotic expansions. This idea goes back many decades in applied mathematics but in dealing with the two-body problem it goes back mostly to the 60s and 70s. Tebo actually used this in the contribution to the 1982 conference proceedings that we've heard about from multiple speakers already at this conference. So it's a well-developed method. So the idea is outside in the external universe use a perturbation theory where you're treating the small guy as a perturbation on top of whatever is out here. So that's what I just grabbed on the previous slide. But as you get close to the small one its gravity becomes dominant over the external gravity and so then you adapt a different approximation where you're approximating the space-time here as perturbations on top of the space-time on the small guy. And then in this buffer region in between the two regions you can feed information back and forth. And from my perspective the interesting thing and the important thing is to feed information from the inner region the inner expansion out into the expansion of the outside world and essentially integrate out this small region. Okay, so the name of the game here is just solving the Einstein equations in that buffer region and you get quite a bit of information just from the Einstein equations. So first off you find that the local solution to the Einstein equations in that region splits quite neatly into two pieces. So here's a space-time diagram with time running up into the right. Here's the small body moving through the space-time. This is represented as if it's a material body so the curvature is finite but it could be a black hole. It doesn't matter. It can be anything. It could be something even more exotic than a black hole. No matter what it is you get this nice split into two pieces. You get a self-field which is directly determined by the object's multiple moments. A leading order of the mass, sub-leading order of the spin, sub-subleading quadruple moment and so forth. And then you have a remainder that's very slowly varying in the neighborhood of the small body or on the scale of the small body. And this defines an effective metric which is the external background plus this other field. And this effective metric is a smooth vacuum metric that's determined by global boundary conditions. So the self-field is determined by the multiple moments locally. The effective metric, the effective external metric you can only determine it once you know the global boundary conditions and you solve the full problem. Okay, the other thing you get just from the Einstein equations in the buffer region is an equation of motion for the object's effective center of mass. So this has been derived over many years starting at zero-th order test body motion going up to first order and then second order most recently. So here I'm presenting the results that I derived about 10 years ago which is that at second order so up to order epsilon cubed terms what you get is just geodesic motion in this particular effective metric. So this is now the covariant derivative of the effective metric. This is the proper time in the effective metric. And again this is derived directly from the Einstein equations outside of the object as we heard from one of the speakers earlier in the conference. Just the vacuum equations in the vacuum region already tell you the motion of the object in the non-vacuum region. Okay, so also this is all done outside the object where everything's perfectly smooth. There's no regularization of infinities. There's no assumptions about HR. Everything is just derived from the Einstein equations. Okay, but it's actually convenient to introduce singularities into the system and that's what I was meaning by saying that we integrate out the local behavior, the local region. So we don't really care much about the fine details of what's going on very, very close. So we just cross that region out. We then take the analytical form of the metric outside of the body and just extend it down to a representative interior curve. The curve actually lives on this space down over here. And then it becomes singular just like if you extend a Coulomb field from outside of a charge distribution. It will become singular as 1 over r as you approach the center of the charge. Okay, so the reason this is convenient even though it introduces singularities is that you don't have to care about the fine details of what's going on in here. You just deal with the multiple moments that define this field outside and you can show that this is actually equivalent or a way of deriving a point particle representation of the small object. So here we define the stress energy of the object to be just the curvature of the metric. We find what this metric is. We calculate the curvature and we call that the stress energy. There's quite a bit of mathematical subtlety in properly defining this thing because this doesn't have a natural definition as a distribution because it's too singular. But you can find what we call a canonical definition in this recent paper with one of my students and find the stress energy that we called the Depweiler stress energy because he was the first one to write it down although he didn't derive it, which is the stress energy of a point mass in the effective metric. So we have test body motion in the effective metric through order epsilon squared and we have an effective stress energy tensor for what we can now consider a particle that is just a point mass in the effective metric. If you add spin to the object you'll have a spin term. If you add a quadruple moment you'll have a quadruple moment term. Okay, so that's all I wanted to say about the local problem near the small body. So now let's switch to the global problem which is solving the Einstein equations. Given that local representation solve the Einstein equations in the spacetime of a big black hole. So at 0th order in the external spacetime you're starting off with just a test mass moving on a geodesic in the Kerr geometry of the big black hole. That geodesic is characterized by three constants of motion, the energy, the azimuthal angular momentum and what's called the Carter constant to the orbital inclination. So you have three constants of motion and you have three associated phases of motion. So you have radial oscillations in and out. You're also circling around azimuthally and you have polar oscillations up and down and so you can associate a phase with each of those motions that runs from 0 to 2 pi in each of the in each of the periods and construct these phases so they have constant frequencies. So the rate of change with time is just a constant frequency constructed from these three constants of motion. So that's the geodesic case. Now, once you're taking into account the metric perturbation and the local self-force then these constants are no longer constant. They start evolving slowly with time and we have two distinct time scales in the system. So this is similar to what Bella was talking about either yesterday or the day before. The separation of time scales. We have a radiation reaction time which is roughly the time over which these constants change. That's of order 1 over my small parameter so it's a long time and then we have just the orbital periods associated with those frequencies those slowly evolving frequencies. And what Hinder and Flanagan showed 13 years ago is that on the long time scale this radiation reaction time those orbital phases have a very neat form or a neat asymptotic approximation. So you start off with a 1 over epsilon because the phases accumulate roughly linear with time so after a long time you end up with a lot of accumulated phase. And then you have subleading terms that's independent of epsilon plus corrections but each of these coefficients are functions of a slowly changing time not t itself but epsilon times t. Now if you've correctly calculated this and correctly calculated that then you have the phase accurate up to a small correction that goes to zero when the limit is epsilon goes to zero and so that should be precise enough to do very accurate parameter extraction from an observed signal of course depending on the mass ratio but for small enough mass ratio you're guaranteed to have very small areas here. Okay so what do you need for each of these things? So this first term is called the adiabatic order phasing and that's determined just by the average dissipative piece of the first order cell force so it's basically just the slow dissipative change in the system a leading order then this is called the first post adiabatic order and you'd have second post adiabatic there and so forth you can keep going up. The first post adiabatic order is determined by the average dissipative piece of the second order force and the rest of the first order force so stuff that wasn't already included here so this is why we have to go to second order this is why I talked about the second order metric early on all of this has to be done up to second order to correctly get to this post adiabatic phasing and if you omit this you omit something potentially quite large because it's independent of mass ratio this could be 10 radians it could be even one radian would be not good enough for us so it's something we need to include okay so this is a typical calculation at first order where you think okay the cell force is having a small effect so let's approximate the motion as a geodesic in the black hole spacetime you can then use either time domain methods or frequency domain methods and because it's not a geodesic it's orbiting around the black hole forever emitting waves forever with basically fixed frequencies so this approximation breaks down after some time which is called the defacing time oops back up so you can imagine this is accurate in some band of time here but not over the full in spiral so what do we do instead to get accuracy on a long time scale we use this multi-scale expansion which builds on work by hinder and Flanagan for the motion but now we're dealing with the full Einstein equation not just the motion so we have a set of slowly varying parameters both the parameters of the orbit but also the parameters of the big black hole all of these are changing with time so we'll just call this list curly j here and the rate of change of time rate of change of those orbital parameters and system parameters is slow with time so you start off with the first order driving force second order driving force and these coefficients are only constructed from these system parameters then you have still these phases of motion that are varying on the rapid time scale and the orbital time scale and you can build this up such that these are still actually exactly the geodesic frequencies as functions of these slowly evolving system parameters now we adopt this multi-scale expansion for the metric where we have amplitudes that the only time dependence is in these slowly varying parameters so you have slowly varying amplitudes and then rapidly oscillating phase factors that are coming from these orbital phases so you can substitute this into the Einstein equation and then solve order by order for this guy here then solve these ODE's all through the system so you can imagine at each point in this parameter space you're solving the Einstein equations to calculate these amplitudes and to calculate these driving forces then once you've got that you've got the rate of change of these guys which then moves you through the parameter space so all the hard work is setting up the equations in the right way such that you can solve this in advance and then rapidly zoom through the parameter space and we now know we can do this we can generate waveforms natively in milliseconds using tools developed by these people in combination with this two-time scale expansion okay so results so far the only case we've concretely been solving things at second order is quasi-circular orbits around a non-spinning schwarzschild black hole so here our system parameters are just the orbital frequency itself the mass of the big black hole and a small slowly evolving correction to the the spin of the black hole so we start off with zero spin correction but then it fluxes going to the black hole and the spin can evolve and we have the only phase now is just the azimuthal phase of the orbit so here we're never neglecting dissipation we're never neglecting the non-zero radial velocity but the structure of the calculation is still essentially that we have a sequence of circular orbits that we then evolve through so we solve the Einstein equation for some fixed frequency and then evolve evolve the frequency to get the waveform so this was supposed to be the simple test case that we started working on in 2013 I've been working on with Niels Warburton and Barry Wardell at UCD in Ireland but it's taken us eight years to work out all the details and develop the analytical and numerical infrastructure to make this work so here's the first result we got which is a calculation of the binding energy of the system so we go to infinity we measure the bond mass of the system we then subtract the rest masses of the two objects so one is the rest mass of the small guy the other is something we actually measure from the numerics which is the mass of the central black hole and then the binding energy is just a difference now I don't want to get into what is being compared against what here that's a kind of worms all I want to say is we've calculated this thing from the bond mass at infinity and done some checks on it to make sure it's behaving properly so we've got the bond mass as a function of frequency the next thing we calculated more recently is the energy flux out to infinity so this is just a rate of change of the bond mass with the retarded time along scriplus so here we're comparing against actually full numerical relativity and post Newtonian theory this is for mass ratio 1 to 10 these, this green curve and the green squares is what we call one GSF so just leading order in self-force theory the blue curve here, this wiggly curve is full numerical relativity this orange curve is 3.5 post Newtonian fluxes and then this red curve that cuts through the numerical relativity curve that is our calculated fluxes at second order in the mass ratio so you see we agree pretty remarkably well with fully non-linear relativity over a big frequency range even though the mass ratio isn't particularly small here so this is the flux basically as a function of frequency for the 2-2 mode but our agreement is also good for the other modes okay, this is mass ratio 1 2 equal sized objects so you would think mass ratio 1 our approximation would be total garbage, why would it work at all we're expanding in powers of the mass ratio but no, it's good so it's important to see the scale here obviously we can see a difference between these curves but our second order self-force calculation in red is off from numerical relativity by I think less less than 1% even here at the end where it starts to go bad the reason we get this turnover and things start really going bad is the two-time scale expansion itself breaks down at the innermost stable circular orbit so we haven't yet tackled the problem of extending into the plunge that's going to be a separate problem that we have started working on we don't actually have it working yet okay, so we have pretty remarkable accuracy there now in the two-time scale expansion that I laid out there are several expressions for the rate of change of the orbital frequency in terms of the local self-force we still have not successfully calculated the second order local self-force but we are able to generate waveforms at first post-adiabatic order just using the Bondi mass-loss formula or the balance law at infinity so we start off with the Bondi mass it's by definition the binding energy plus the two rest masses the flux is by definition the time derivative of this and we have the binding energy as a function of frequency and as a function of the parameters of the big black hole now we can invert this to get an evolution equation for the frequency if we approximate the mass of the big black hole and the spin of the big black hole as constant so that's not actually consistent at the order we're working with but numerically these things are so small that we have very negligible impact on the gravitational wave so we actually get good accuracy as you'll see even neglecting these things so then we can just rearrange this equation to get the rate of change of the orbital frequency from the flux and from the binding energy and then we expand this out in powers of the mass ratio and get this waveform so this is again mass ratio 1 to 10 blue, the blue dotted curve is full non-linear numerical relativity the orange curve is our waveform generated with a two timescale expansion and we see even though we're neglecting these rates of change of the black hole parameters we have pretty amazing accuracy so if you look just by eye here it looks like we have 100% accuracy if you zoom in near the end you do see we're starting to accumulate some defacing towards the end this is only 4 cycles before merger or actually 3 cycles before merger and here we have about 0.1 radian of error so pretty amazing accuracy even though we're very far from the extreme mass ratio regime this is only 1 to 10 mass ratio okay I think I'm out of time so maybe I'll just leave this up big message accurate, much more accurate than you'd expect so some questions or comments everybody's happy this please sorry this rate of change of angle velocity I thought, I mean naively I thought this should be related to the flux of angle or momentum can you also compute the flux of angle? we have not computed flux of angular momentum because of the structure of the two-time scale expansion you should be able to get it from either one for quasi-circular orbits okay so I think there are no questions from remote people so let's always picture again, thank you thank you