 I mainly discussed what is a gravitational wave and what is the analytical theory of the generation of gravitational waves by general sources. So I won't come back to that within a mixture of post-Minkowskiian and post-Newtonian theory, but at the end everything is expressed in post-Newtonian expansions. In the first part of the lectures today I will discuss the dynamics of compact binaries. Compact means objects which are small radius compared to mass, I will explain that. Then I will explain why the post-Newtonian theory here are not fully satisfactory. I mean they are necessary ingredients, but they lose their validity at the most important moment which is for the strong signal of the merger of two black holes. And for this one developed resumption methods which takes perturbation expansions and allows to predict things when the expansion is no longer valid. And in particular one combination of resumption method which is the effective one-body method which was introduced here at HES in 1999 and 2000 and this led in work with Alessandra Buonano to the first analytical prediction of the signal of coalescence of two merging black holes, complete signal from the minus infinity in time to plus infinity in time. This prediction as we will see is already very close to what has been observed, but it has been improved when five years later, numerical relativity, so numerical relativity started only in 2005, this was in 2000, but numerical relativity helped to improve each ingredient there like the dynamics, the generation of gravitational waves and this led to numerical relativity improved version of the effective one-body templates for gravitational wave signals which have been developed in parallel here at HES with Alessandro Naga and in the states in the group of Alessandra Buonano and in the last hour where I will use I think transparencies to be able to show nice pictures, I will show the comparison between the predictions from EOB-NA, EOB and EOB-NA and the data. Now there is a notation that maybe you can get used to immediately that I will use all over. We will be talking about a binary system of two masses M1 and M2 and it is convenient to denote the sum of the two masses by capital M and the so-called reduced mass of Newtonian mechanics which is the product of the masses divided by the sum you call it mu which is a traditional notation and then in a paper with Natalie in 1986 we add in this paper many formulas where there were pure numbers like 3 minus this ratio mu over M which is a dimensionless number the ratio of the effective mass divided by reduced mass by the total mass and after a while we realized that it would be good to introduce the notation for this dimensionless ratio and as the next Greek letter in the alphabet was mu we said let's call mu this dimensionless ratio which is finally the product of the two masses divided by the square of the sum of the two masses this is a dimensionless number which is which goes from 0 to 1 fourth in the equal mass case it is equal to 1 fourth as you can easily find and if one mass is much smaller than the other one it is a very small number okay and so it interpolates between the extreme mass ratio limit and the equal mass limit where this parameter is equal to 1 fourth we will also need later this important conversion that given a solar mass given Newton's constant and the velocity of light the combination g times solar mass divided by the cube of the velocity of light is a time which is equal to approximately 5 microseconds and more precisely 4.9255 microseconds and more precisely it is known to 10 digits but we will not need 10 digits today and maybe in the future gravitational experiments will need the 10 digits although I doubt it now the first point is dynamics of strongly self-gravitating bodies so the traditional perturbation theory in generativity is to say okay Einstein's equations which have written explicitly last time are non-linear PDEs for the metric and since Einstein who introduced this method in 1913 actually 1912 you can do perturbation theory saying the metric is equal to the Minkowski metric plus a perturbation called H mu nu and when this H is small you can hope to solve Einstein equations but the problem is that if you consider a body with a certain mass m let's say a body a a is a label with a mass m a and a radius r a and if you solve these two first approximation you find that let's say one one of the components the time time components but the other one are similar if you evaluate it at the radius of the object this thing is 2 g ma divided by r the distance from this but if you evaluate this at r equal the radius you have now you need okay if this object is a neutron star a typical mass ma is 1.5 times a solar mass now the there is also an analog of this that if you take g times m sun divided by c square this is now a length instead of being a time and this length is like something like 1 1.5 let's say kilometer if you use round numbers so if you take a neutron star with radius of order of 10 kilometers although it's not well known so you compute this 2 times 1.5 divided by 10 okay 1.5 solar mass times 1.5 finally with a factor 2 you find that h zero zero at the surface is equal to 0.45 in other words the metric at the metric at the surface is minus one i'm using the minus plus plus signature plus 0.45 so instead of being a small deviation you see that the perturbation away from minkowski is enormous okay it's it's half of the value and if you take a black hole by definition a black hole the surface of a black hole is such that r is 2 g m because the surface of a black hole is such that g zero zero is zero so at the surface of a black hole perturbation theory is even more wrong than from a neutron star so the weak the usual weak field expansion is unjustified and therefore one needs a new approach to be able to deal even in post-neutrality with the motion of two black holes or two neutron stars and and and there exists such a method uh this method is what can be called the matched multi chart approach this method was first used in a sense by manas in 1963 then for the motion of black hole by peter deas okay in 75 then there were contributions by ron kates in 80 and then by myself for for what we will discuss uh in 82 a full full account full account was given in the in my lisouche lectures of 1982 published in a in a book edited by uh natalie der rel and vpr so what is the idea the idea is you cannot use a coordinate system of this type where you have this uh deviation so when you have two bodies like two black holes so this is now a spacetime picture this is a space picture okay now time goes vertically space goes horizontally this is the world line i mean this is the motion in spacetime let's say of one black hole or one neutron star this is the world tube of another black hole or another neutron star and the idea is that you should use several coordinate system actually the idea is linked to the mathematical definition of a manifold the manifold is defined by an atlas of of maps of charts okay and and you just do that you say that near each object you should have a coordinate system so near object a you have a coordinate system let's call it x a alpha and uh same thing for the other okay here it is xb and then in between and outside the two objects you use a coordinate x mu of this type and then when you and and this coordinate chart is a global coordinate chart where you exclude what happens near the two bodies here you can use an expansion of this type eta mu nu plus h mu nu at order one plus h mu nu at order two you can use an expansion because if you stay far away from the two objects h mu nu is small and then you can use an expansion of this type but then when you are close to the objects you use a different expansion you don't use you don't say that the metric is equal to flat space plus small perturbation you say that the metric here and you use a coordinate system uh let's say it's convenient to use indices from the first part of the greek alphabet so alpha beta here as a function of capital x gamma is equal to the metric of an undisturbed uh object that you talk about like an undisturbed black hole or an undisturbed uh neutron star so you have the full non-linear solution of anstein's equation of uh you know for a neutron star of the tolman openheimer volkov equations for instance or for the schwarzschild solution if you have a non-rotating black holes so you put this as a zero th order uh solution g zero uh of x okay and then you look for first order perturbation of this but the small parameter now is something linked with the the far away presence of the companion okay so the small parameter will be the mass of the companion divided by the distance away between the two objects which is a small parameter because when you sit on this neutron star uh essentially if the other companion was very far you would be an isolated neutron star so there must exist a perturbation theory of this type and then um then you use a matching method okay matching of asymptotic expansions you have two different expansions two different perturbation theories one here one here and then you look for an intermediate zone which is uh some overlap between this uh yellow coordinate system and the white one where the two expansions would be valid and then you match the two things there are techniques for doing that and using this you can propagate information from what is the source if it is a black hole a neutron star in the external uh thing and and then this allows to compute everything you want the metric and the motion uh of some word line defined by this object here when doing that uh you can prove something which i had called the uh effacement of internal structure which was the following thing that you can distinguish uh whether you have a neutron star or a black hole by introducing uh something which is the so-called uh love number of uh a neutron star k2 there are actually two uh love numbers uh the love number of a neutron star uh which i explicitly introduced in 1982 uh is when you look at the perturbation of a neutron star by an external field is the the coefficient which measures the fact that the neutron star will be deformed and that that this deformed because there is a companion so it's a tidal deformation and this deformation will induce a quadruple moment in this neutron star and this quadruple moment will have a field which generates a field okay by the way i remember that uh after introducing i mean love number love is the name of an english scientist we had introduced love numbers for the tidal deformations of ordinary bodies but i remember when i gave a talk uh in the Schloss ring ringberg Schloss uh the metrius christodulu was there and then he said ah tibo i thought you had introduced love number because this is your name you know of love as i said no no this has nothing to do with me but uh it is there and so what what you find is that for a neutron star k2 is non-zero so it distinguishes a neutron star and a black hole and you find the effect here in this in this scheme so it's the effect of the deformation of this star by the presence of this companion gives a change in the metric so it's the structure dependent change of the metric which is proportional to k2 there is a numerical coefficient then there is uh there is actually the in lowest approximation something which is the the tidal the tidal field of the companion the second derivative of the newtonian potential and then this creates a quadruple moment which therefore generates a quadruple uh field uh here due to the deformation of this uh times a numerical coefficients which is actually the the it which is the radius to the fifth power but you can rewrite the radius to the fifth power just this way by a trivial identity but for a neutron star a black hole uh by definition as we said the radius is of order of gm over c2 this is it was one half for a neutron star and it is one for for a black hole and therefore formally this effect when you say that this is of order one is of order you know one over c2 to the five one over c2 to the five or one over c to the 10 is something which when you compute the equations of motion uh yes because I have put also the one over c2 here it's called the five pn approximation okay this is something which corresponds to a term in the equations of motion which is v over c to the 10 smaller than the leading newtonian order okay and this is very small even as of today okay so small that it it's too small to be computed today and too complicated so the idea was that this is negligible and therefore uh up to yeah no this was obtained within yes this was obtained within the relativity calculation this was what was new you can if you forget everything about anstein you can also find it within a newtonian approximation of saying reaction of the deformation yes yes but to justify it one needed this game but but at the end it's indeed scaling its dimensional analysis okay yes there is an this is the quadruple yeah the index two here means for l equal to okay there is a full theory of relativistic love numbers that we have developed with alessandro nagar in recent times so but this was the beginning now but what I want to say is that once you understand that this is saying that although we are talking about the motion of extended bodies neutron stars or black holes so a priori you say it should be very complicated I have at each moment to take into account the fact that I have this surface and what happens to the surface once you realize that this body the internal structure is as a very very small effect if you can neglect this thing then you say the bodies should be equivalent to point masses I can skeletonize them seen from outside by point masses because they have no structure and therefore the conclusion is that one should have a scheme where you can skeletonize that is to say skeletonize extended bodies like black holes if they are compact extended bodies and therefore solve a dynamics where you have only two world lines okay which means an action and then and then you look at a mu nu plus h mu nu okay plus h mu nu two that is to say you can forget about the scheme of the internal structure and say I will only solve this problem where the action is okay so there is the integral of the Einstein-Bert action okay r with a coefficient c4 over 16 pi g and then this is the integral of a spacetime square root of g and then you you skeletonize you represent the bodies by world by point masses which means that you put an action which is minus g mu nu of z d lambda d z d mu d z d nu okay you have world lines and along the world lines you write the metric the the proper time then this is the action and then you solve this in harmonic coordinates yeah this is later yes yes spins is dealt with by an extension of this which which is much more complicated actually spin but it can be done okay and it is done I mean eob is developed with spin to very high order I want to give the main idea so now what kind of equation so at the end it gives Einstein's equations for instance arm mu nu equal a pi g over c4 sum over the the two bodies of the stress energy tensor of each body minus the trace term in dimension d where this thing is the variational derivative the action with respect to g mu nu so the stress energy tensor of a point mass is an integral m a c square the world of the proper time along the world line g z d d z d nu d z d nu times a delta function of x minus z but now we have a problem because you have delta functions but by the way to complete last time I explained how you define gravitational waves by solving Einstein equations and in particular by using so-called harmonic coordinates it's convenient also although we'll see that there are problems with this to use harmonic coordinates in in a general sense so harmonic coordinates are defined by the condition that the crystal fold symbols contracted over this is zero which is equivalent of saying modulo a sign that the gothic metric which is the square root of the determinant is equal to zero and by the way Einstein already used this condition in 1912 okay some people say ah let's call it the don't dare or whatever Einstein in 1912 knew about harmonic coordinates and worked for years trying to reuse the Ritchie tensor using harmonic coordinates and then he used them for gravitational waves also in 1960 when you do that the advantage is that the Ritchie tensor here becomes a diagonal pd operator in the sense that yes if I put minus two then this gives g alpha beta d alpha beta g mu nu plus terms of d g d g first derivative quadratic contracted with the inverse metric here you have the metric with two indices down so you have all these indices are contracted in some ways and all possible waves exist so so the advantage is you see this operator becomes a block operator but in a curved space time but when you solve this equation by saying I will look for a solution of the type g mu nu equal eta mu nu plus h mu nu this operator becomes the d'Alembert operator in flat space time and then you can use perturbation theory saying the type of equation you have at lowest order for instance is d'Alembert of h mu nu at first order is minus 16 pi g over c4 t mu nu at first order minus 1 over d minus 2 eta mu nu t1 where t mu nu at first order is this quantity in which you replace the metric by eta mu nu okay and then this is fine this is well defined but now let us see what it means first I am computing the metric at point x I have two were blind I have a point x a field point and this thing I can say I can represent by a diagram saying that h mu nu at point x is equal to the inverse of this operator which is the green function for the Klein Gordon equation so so so this this thing and then it's okay so this wiggly line means eta mu nu sorry eta mu alpha eta mu beta minus 1 over d minus 2 eta mu nu eta alpha beta this is a projection operator for tensor indices times g the green function which g0 which satisfies equal the delta function of x minus y okay which is explicitly known in four space time dimension you take the retarded green function so you can solve by successive approximation this diagram is just saying this is linearized gravity and and this dot here means I have t mu nu 1 this t mu nu which is which is written by a delta function integrated of this world line so the source is something which vanishes everywhere in space time except on one or two world lines and the metric can be written at lowest order as this plus the other diagram which is this so this is the the thing generated by mass a this is the thing generated by mass b at the next approximation when you solve now this equation to the next approximation you have something like the long bear of h-menu second approximation in the right hand side contains various types of terms like h1 with two derivative of h1 due to to this term okay put on the right hand side plus all those terms which are first derivative quadratic in the first derivative plus a term which is which comes from the source if you expand the source the source depends on g mu nu here here so if you expand you have also h1 times t okay and now this gives extra diagrams in perturbation theory these diagrams are sorry let me out of this type okay i compute at point x i have something generated by mass a minus b and then they combine at this point and this cubic vertex means that i'm solving angstein equations with terms quadratic there yes yes but the the nice way yes let me explain this now if you relax angstein equations by harmonic condition you don't have equations of motion you can solve angstein equations for any motion of the sources okay you don't have constraints on the motion of the sources at the end once you have solved for the metric you want to impose that the metric satisfy this condition your solution must satisfy this condition and writing this imposes a constraint on the sources and you find that this constraint is equivalent to saying that this is the equations of motion of the world line in the metric in which they are embedded okay that's what you you prove formally and technically in this thing so but it's convenient to solve the thing without imposing the equations of motion and then write the equations of motion and then at the end you solve angstein equations when everything is satisfied the point i want to make here is that they are not only this diagram but they are diagrams of this type okay and now sorry at point x and now so if you so you have diagrams which are like Feynman diagrams okay actually those diagrams are equivalent to what Fokker Schwarzschild and people in the 1920s and 1916 had introduced before Feynman and Feynman had studied the papers of these people and then he understood that in them it was convenient to draw a diagram so Feynman was interested in the quantum problem here we are interested in the classical problem we are just saying here that perturbation theory is perturbation theory we have greens function concatenation of vertices and green function the full the full proof that you have indeed Feynman rules for this was given in a paper by me and Gilles Esposito-Fares in 96 so it's convenient it's a language okay but what is not a language is that you see here things which from the quantum field theory you would call a loop okay a loop is a diagram topologically you have this thing now what they do mean loops classical loops exist okay and and they mean self-energy effects if you compute for instance what means this term okay in lowest approximation these terms means that I must compute like the inverse that inversion which in lowest approximation in the inverse Laplacian of the the gradient of the first-order metric h1 but the first-order metric is ma over r okay is is the is the Coulomb potential 1 over r the Newtonian potential if I take a derivative I have ma over r square if I take the square I have ma square over r4 and now so this thing means this at lowest order but this is an integral in space d3x with the x field minus x this x where I am of ma square over x to the 4 but this integral is divergent because I am in three space dimension and I have a function which is singular like 1 over the distance to the fourth to point in space okay so this thing is infinite okay and and this thing is what this thing means that I have a correction to the source delta x minus z z a and then this multiplies in space it becomes this and this is ma divided by the Newton potential x minus z times this but this is this delta function is saying that I should compute things at x equals z okay but at x equals z this is infinite okay so this thing is also infinite okay so so here we have a price to pay it was very convenient to say I mean there was a proof that you can skeletonize extended bodies by point masses but once you do that in non-linear theory like Einstein's theory you have infinity to cope with okay and I will before discussing how you deal with that let me say that here I was discussing how you compute the the field once you compute the field you can compute the equations of motion and you can also prove and that there is a reduced action which is called in the in the gravity literature Focker action because Focker was the first one to do that for electromagnetism okay for electromagnetism you have only one diagram which is this one which is the one photon exchange between two charges and Focker in 1920 wrote and actually Schwarzschild in 1915 already knew that and Focker wrote in 1920 what is the action of exchange of one what we call one photon now between two world lines which is a classical relativistic version of the the Coulomb interaction you can do the same thing in gravity but in gravity the the action of two world lines as the one graviton exchange we are now in a sense you close you see you close this thing by putting it here so you compute the interaction energy of the two world lines I don't want to I mean two years ago I have given a full set of lectures on on this the Focker action so there are videotapes of this so people interested can see on IHS website the full discussions in French but the blackboard drawings are in any language so you have you have now really loop effects and then at at higher order you have things like that but you have also things like that okay you have things like that you have things like that like that and with Natalie we computed this thing up to two loops okay so we've I will give the the results now but what I want to discuss now is how you deal with so this means as it is clear here and is clearly here means self-gravity effects it means that when I have a point mass this point mass generates a one over our field around this one over our field as an energy there is an infinite energy concentrated at r equals zero and it is part in gravity any type of energy gravitate so this has to be part of the business you cannot delete it completely without knowing what it does okay so how you do that a good method the first good method which was used because when you do that at for instance in the book of Landau leaf sheets where this is treated at one pn this is easy enough that you can say okay this term let me delete it and let's not worry about it when you add to pn this is more complicated there are things there are divergences that depend on external viable so you cannot just throw them and say this is zero you have to deal and a convenient way is analytic continuation which in the brand introduced by Marcel Ries okay and I want now to explain this quickly and by the way this was the early method used in quantum field theory in the in the 49 and things like this was introduced by Marcel Ries and was used for some years in quantum field theory let me explain this method which is the one I had used in 1982 to regularize all the dangerous things that happened at the second post-neutronian approximation the basic idea of Marcel Ries is that the problem the problem comes from what the problem comes from the what that you had point particles which were therefore a delta function on a word line and the idea was let's replace the delta function by something which is regular or in some sense and and then do the computation with this regularized delta function and the regularized version of the delta function in d dimension is what Marcel Ries called z with an index a in dimension d and it is the following object it is minus the Minkowski squared interval to a power a minus d over two divided by some function of a that I will not define explicitly times a step function so a step function is this you know 0 and 1 after x0 times a second step function these two step functions means that this thing is non-zero only in the future light cone of x okay so it's one here it's zero outside but essentially this thing is is the interval square this distance square with a minus sign to some power but but the trick of Marcel Ries was that this thing satisfied the following identity plus two and z a convoluted with z b is z a plus b okay remember that yes a what is the first equation delta of t sorry this this here z below below yes sorry this is d'Alembert the d'Alembert the wave operator acting on z a plus two is minus z a plus two with the same a so what is a a is a complex number okay and for any complex number h here is a product of gamma functions it's a numerical coefficient we see such that when you do the convolution of these things in spacetime okay it is a somewhat complicated integral you fall back on this with the property that it's the same function but the index a is replaced by the sum of two things now as you can see immediately z0 from this thing is a unit for the convolution z'A times z0 is z'A okay so it is the unit convolution thing but what is the unit convolution it is the delta function okay so for convolution in spacetime okay delta in dimension d so z0 is the delta function so the idea of risk is you replace delta which is z0 by z'A where a is epsilon is a small number and then you are going to take this small number to zero but instead of saying it is small you say I do analytic continuation I take a in the complex plane so I have the complex a plane and then I replace in Einstein's equation the right hand side the delta function by this and then I do the computation for any value of a and then when you do that you don't have the singularities you can do all computations when a is is in a certain regime domain of the complex plane why is it so because if you look at what happens for the the three dimensional I mean the integral along the world line was a point particle in space was like a delta function this is replaced by a times the distance so I have a point in space okay z and then I have delta function is something which is only here now I have the let me call r the distance between x and z the point the risk thing is of this type okay in first approximation it is this when a goes to zero this goes to zero so it goes to zero except that there is a singularity like one over r cube and then you find that this thing is indeed a representation of the delta function when a is small but the idea is now the integrals that were infinite for instance was the integral over d3x and and when you solve now what is the the Coulomb potential the Newtonian potential is Laplace minus one of this but this is Laplace of minus one of this and then you find that this is r to the power a minus one that is to say what before was one over r now becomes r to a different power and when I compute for instance this times the gradient of this square I have something which is r to the 2a minus 4 integrated over space but this thing if the real value the real part of a is large enough okay so when you are somewhere here this is a convergent integral and this convergent integral is also an analytic function of a and then what you use is the property of continuation of I mean analytic continuation of identities and for instance the basic idea of this is that given any function f if I compute this integral zero to plus infinity x to the a f of x dx all the integrals you have finally when you use polar coordinates they will be of this type what was before diverging will be something like that where a is a or 2a minus 4 it's some linear combination of a some integer but this thing first you decompose it in a integral from zero to some number plus an integral from c to plus infinity you decompose the integral into now you look at each part the problem here is that this was diverging at x equals zero but if you take the real part of a large enough you can integrate at x equals zero and then when you do that you find so that it converges here and that it is a metamorphic function in the complex plane that is to say you can extend it uniquely in the full complex plane a but the price you have to pay is maybe at the value you are interested which is a equals zero maybe you will have a pole okay if you have a pole you have to deal with the pole if you have no poles this is a way of defining the thing there and for instance if you compute this way this integral this integral is when I use polar coordinates I have a 4 pi then I have dr r square r to the 2a minus 4 okay but this integral I can write as integral from 0 to r0 of dr r to the 2a minus 2 plus integral from r0 to infinity of dr r of 2a minus 2 each one of these integral is convergent in a part of the complex plane okay if you compute these two integrals and if you add them you will find that this is zero okay surprising but true and and same thing with the the delta function then becomes zero so at the second post-Newtonian approximation now let me summarize the work at the second post-Newtonian approximation which means when you go up to two loops okay it was shown that the analytic continuation where a by analytic continuation goes to zero gives a finite result so this is a way to compute something you compute a metric you compute equations of motion evidently at the end you have to prove that what you compute is what you want you have to prove which is proven in my lectures of lisouche that it is a solution of Einstein equations and that it satisfies all what you want and that it satisfies actually the proof was to say that this is this construct a solution of Einstein equations as a function of two world lines okay with some structure near the world lines that near these world lines it looks like a schwarzschild solution a non-linear schwarzschild solution because you you you find this way all the expansion of the schwarzschild solution up to 2pn in that case actually you need 3pn for this and then I had proven that if there exists a solution of Einstein equation of this type which looks like schwarzschild in a technical sense near each world line it is equal to the unique to this solution we were talking about before okay so that was the proof that this was giving the correct equations of motion now what was the result the result was the the 2pn dynamics and 2.5pn with natalie so I've written here the the new the post-neutral expansion of the amiltonian okay here we were in harmonic coordinates actually there are subtleties in harmonic coordinates there does not exist lagrangian there exists a generalized ostrogatsky lagrangian you know that you can go to different change of coordinates so that you have an ordinary amiltonian okay all that has been discussed at length I don't want to discuss it again at the newtonian approximation the amiltonian of two bodies is p square over 2 minus 1 over r here I am using scaled things where the gm is the radius is divided by gm and the mu also the relative the mass is also scaled away to have something nice okay otherwise it makes it a new will be so this is newton kinetic term minus potential energy minus one over r at the one pn approximation which was first worked out by lorenz dross in 1917 and anstein fell off man in 1938 the lagrangian was first written down by fixed handholds in 1950 you have this which is already you know there is a p4 p4 means by the way means the vector p square square etc okay p4 you have p square one over r you have n is the unit vector between the two bodies that we are talking about okay here you have n dot p which is the radial momentum square and then you have a one over two r square interaction okay yes the one pn at the one pn there is an ordinary lagrangian as discussed in landau lift sheets the surprise with natalie I remember was that there were theorems saying that does not exist a lagrangian okay and then indeed when we found the equations of motion we looked for a lagrangian and we found that they existed the lagrangian but it had to depend on higher derivatives okay at 2 pn already at 2 pn but you can also eliminate at any pn order actually you can also eliminate higher derivatives and have an ordinary lagrangian and an ordinary amiltonian at 2 pn this was the answer we got so you see it starts getting now messy okay more terms there are actually 11 different terms no not 11 here we'll get the counting and you see that the coefficients start depending also of of the coefficient nu which is the symmetric mass ratio so remember nu is mu over m it goes between zero and one fourth and you have terms linear and nu terms quadratic in nu at this level okay now at the time in 1982 in the proof that at 2 pn this was finite I had also realized and published the fact that there is a pole at 3 pn okay that is to say at 3 pn the next order there will be a pole 1 over a and then to me it looks surprising because this proof was saying if there is real problems of the fact that the point particles are not points but extended objects this should be at 5 pn not 3 pn the resolution of the the thing came when one explicitly computed the 3 pn thing but in order to compute the 3 pn thing which was first done by a Yaronovsky Schaeffer and myself in in 2000 one needed a better regularization methods because actually to go to to 3 pn you need a refinement on this thing but this refinement was invented after Ries it was invented in 1972 independently by Bolini and John Biaggi and by Herat Hofth and and Weltmann and this is dimensional regularization okay dimensional regularization is also dimreg is also a method of regularization based on an 80 continuation but instead of smoothing delta function in an artificial way what you do is in fact you look again at this formula of Ries this formula of Ries I have written it in any spacetime dimension d okay and then you find that the critical thing is that there is a power of the interval which is a minus d and here we were using the fact that I do an 80 continuation in the power because I am saying I kept d equal 4 but I do an 80 continuation in a dimensional regularization says no the says the delta function is z0 so I put a equals 0 but I play with d if I write the green function explicitly as a function of d and I take the dimension as a complex parameter now so if I am able to compute this perturbation theory where the green function is at each stage computed in dimension d and I have an explicit analytic function of d I can now do an 80 continuation of d and this method is practical in classical gravity I mean it works and it gives and then it generates a solution of Einstein equations which is what you want let me stupid question because it is clear that it works for the divergence you are talking about on the blackboard in the middle about the one over R to the fourth for example yes but when you have terms which transform the stress energy tensor delta times one over R yes let me explain this indeed to explain this actually it is different because when I do for instance when I do this computation so when I do what was the integral over space now it's convenient to capital D is the spacetime dimension okay it's convenient to call the space dimension small d this way you remember what is space time it's the big one and space is the small one so small d is 3 plus epsilon okay where epsilon is a complex parameter when you have the integral over space of the derivative of h the Newtonian potential square okay what happens is that h is the solution of the Coulomb of the Einstein yeah I mean Newton equation in space dimension d so it is m over r small d to the minus 2 the Newtonian potential you know is linked to the dimension of space it's one over r because d equals 3 but when d is not 3 you have this but here you have also the d which is here so this integral is d dr times r to the d minus 1 times r to d so it's times r to d d minus 1 square so there is a 2 so but this does not cancel the there is still a d so you can play with d so you get the same result as before for the delta function what happens is that you now have a delta function but you compute r to the d minus 2 at i equals 0 but if d is large enough this is 0 simply because it's regular at this thing so this thing is 0 in in dimension d okay at at the 3 pn approximation there are really three loops in the sense that if you project the diagrams the diagrams here were in spacetime okay but actually the green function which comes in the calculation is always the green function linked to the propagation in space the Laplace minus 1 and one of the important diagram both of this type okay where this thing means that I have sourced by the mass m1 I have three fields okay I have three lines so this means it is a term of order m1 cube why here at mass m2 this is of order m2 square because I have two lines and then this diagram means that here I have a scalar degree of which propagates which is the longitudinal type of gravity term why here I have the gravity wave which is the transverse trace less as I explain gravity term propagating which is sourced by the derivative of this okay so this is this is an integral and which is as you can see really three loops because technically although these are classic it's a classical theory and in quantum field theory you would open the lines and not call them loop diagrams when you compute them not in the Fourier domain but in X space they are really as difficult as loop diagrams to compute and the number of loops is proportional to the difficulty okay now when doing the computation in adm coordinates which is Arnovid-Deser-Misner Hamiltonian formalism from general activity you find that the answer is finite after analytic continuation in space time except that there are poles but the poles cancel so it's delicate because when you have one over epsilon minus one over epsilon you want to be sure that the and then to to get the calculation right you really need dim reg and this was the calculation done with Jarnowski and Schaefer in 2000 where dim reg was essential and then you see now at 3 p.m. the result is much more complicated and also for the first time there are transcendental numbers coming in okay here there are rational numbers only while here you have zeta 2 pi square over 6 that come in and when you do the calculation I will stop for the break now actually I've essentially finished yeah let me finish then we will have a 15 minutes break when you do the 3 p.m. calculation in harmonic coordinates you find that indeed there is a pole now so it looks strange because there is no pole if you do this in a certain coordinate system and there is a pole if you do it in harmonic coordinates which is so this confirms what I had said in advance in 1982 there will be a pole there is a pole but the new information is that this pole can be renormalized away in something which is the position of the world lines because when you do the computation it's a computation which depends on the choice of a world line in spacetime okay and this world line represents what it represents a fictitious center of mass of the body but if you have a black hole or a neutron star that does not exist the center of mass okay it's not black hole it's certainly not defined so it is something which is an intermediate technical object these world lines and you can shift it you can say ah I have a bare world line I have a physical world line in spacetime which is like a bare world line plus a shift of the world line which can be singular like 1 over d minus 3 and this is what you find that is you say if you shift the world line mathematically by something singular but which is small because it is g cube over c6 or something like that you know 3pn order it's a small term at 3pn order and then it shifts away all the poles there and then you find that the result is the same equivalent to that okay so this calculation in harmonic coordinates was first done by Blanchet, myself and Gilles Esposito-Fares and with Yannoski and Schaefer we have shown the equivalence between the two calculations okay so this is 3pn let me mention briefly some technical tools just because these calculations you know they can be done because I mean there are hard integrals to compute okay so just to give you a flavor to do these computations you need some old results and new results I will only mention here the old result like the old result is the following is this diagram this diagram means that I have a mass point at point z1 a mass point at point z2 and then I compute the gravitational field in generativity due to the non-linear interaction between the two actually this diagram means I have Laplace minus 1 Laplace minus 1 Laplace minus 1 which means that this thing is Laplace minus 1 of 1 over x minus z1 x minus z2 okay these are the the green function for the Laplacian in three spacetime dimension 1 over r 1 over r for two points and then you compute again the Poisson integral of this this integral is already non-trivial to compute it was first computed by Fox in 1939 and actually it's a very simple answer and I remember when I was a kid I did not know at all evidently the work of Fox and I rediscovered the correct result by using some technique so actually this thing is r1 plus r2 plus r12 so it is the log of the perimeter of the free point so you have three points you have the distance r12 you have this distance r1 r2 you take the perimeter of the triangle the log of this and this satisfies this thing it's okay this is Fox and and then there is a very nice formula due to Marcel Ries which I will just write which is when you want to compute an integral of this type which is you have two points in space you take the distance to the point r1 to some power you take the distance to the second point to some power where these are arbitrary complex numbers and then you can do this explicit computation this is pi over two over five and then you have the the distance between the two points to the power b plus c plus three and then here you have the product I will only write of three gamma functions so you have three gamma up and you have three gamma down I only write a few of them to indicate the structure so it is like a generalized beta function with three gammas and three gammas and there are generalizations of this so you need these x-space integrals to compute the thing to be able to compute the 3 pn and we will briefly discuss the at 4 pn the calculation at 4 pn has been done only I mean by us Yaranoski Scheffer and myself in 2015 14 okay and there there are 100 terms in the Hamiltonian okay it's a very complicated thing and there is a subtlety which is a non-local in time new thing but that I will discuss when needed okay let's have a 15 minutes break why I mean erasing I just want to mention that there is something very interesting and subtle happening at the fourth post-neutral approximation which is that up to this level one could approximate the interaction between two world lines the interaction between two world lines goes through retarded greens function but at lowest order this retarded greens function says that the gravitational interaction propagates at the velocity of light and between two closed objects this is a small effect compared to the the space they move in the meantime okay but at the 4 pn approximation there is something which is non-local in time the green function contains something which is an integral over the infinite past and formally the dynamics depends on something which is a functional of the infinite past and for the conservative dynamics of the infinite future also this is linked to something we had first found with Luc Blanchet in 1988 which is a breakdown of the pn approximation the notion of near zone gets completely I mean gets very subtle at 4 pn you need to take into account the infinite past hereditary effect of what happened in the source billion years ago in the dynamics of the source now okay but this is a very small effect but it must be taken into account explicitly okay now coalescing binary black holes about that point so you said that the 4 pn amintonium had 200 terms something like that yeah 109 okay so the one you wrote is just no I've written so first there are 109 terms plus the non-local terms which is an integral in I have not written the non-local term here and then when you go to the center of mass thing this reduces the number of terms I don't know the exact number I've written only the first one and the last one here it would need the full blackboard to to write the thing now so the motion of two we are now discussing the motion of two black holes okay yeah that's gets more complicated than needed okay now these two black holes they go around each other because of the retarded interaction there is they are damping forces in the system which we which were first directly computed by natalie and myself in 1982 so part of the equations of motion as a damping term which makes that the two objects gets closer and closer and then this means this system go faster and faster and they emit gravitational waves of increasing intensity and increasing frequency now one can distinguish various periods so this is a spacetime diagram time is minus infinity in the past for for for most of the motion of these two black holes in spacetime one can say that this in spiral is adiabatic okay oh yes let me also adiabatic means that you can consider that the two objects at each moment are on a Keplerian orbit and you take into account like a circular change of the parameters of the Keplerian orbit as a function of time as a slow thing as an adiabatic change okay but when you get near merger and before merger it becomes non adiabatic the things start falling okay so there is a non adiabat the late in spiral becomes non adiabatic then there is something we shall describe which is called the plunge and after the plunge there is the merger somewhere here and then the ring down okay and now how do you compute this first there is a nice simplification that I had explained last time which was found by peter's in 1964 using the lowest order emission of angular momentum from the system and energy from so you you compute the you use the quadrupole approximation of anstein 19 6 1918 which gives that there is a quadrupole wave from this quadrupole wave you compute the energy sent to infinity and the angular momentum also or you use what we found with natalie which is directly the radiation reaction force which takes away from the system indeed energy in angular momentum in agreement with these quadrupole type things and then the first thing you find that peter's found is that the eccentricity decreases faster than the semi major axis of the orbit so more precisely in lowest approximation the eccentricity square divided by the period to the 19 half modulo terms which tends to think is constant and therefore as the period gets shorter and shorter the eccentricity gets smaller and smaller which means that the system even if it started you know being very elliptic on a wide orbit will become circular on a wide orbit before this wide orbit shrinks to a smaller and smaller orbit it's a it's a rather fast process so it simplifies a lot because then you can discuss the in spiral along quasi circular orbits so you can do computations to high order in post-Newtonian thing but assuming the two objects go on circular orbits and this simplifies the computations a lot okay now what is i have sketched in the first lecture what is the logic of doing that the logic is that you need equations of motion which includes both conservative effects and radiative damping effects so if you are for instance in arnovid desert mizner coordinates the conservative dynamics is written by amilton's equations okay for the relative motion also you eliminate you go to the center of mass and you eliminate the motion of the two objects and you replace it by the motion the relative motion of x1 minus x2 okay you go to the center of mass where p1 plus p2 is equal to zero and then in the center of mass you find that the action depends only on r where r is the is x1 minus x2 is the relative thing and that the derivative of the action with respect to r which is the momentum conjugates to the relative thing is equal to p1 and minus p2 in the center of mass so the two things are zero and the first one is the the the relative momentum p okay and this piece satisfies equations of motion the usual amilton equations of motion where the amiltonian if you use post-newtonian theory is given by this at newtonian approximation this one pn this two pn this three pn and now we know since last year the four pn thing okay but if you if you write this the two objects they go around in orbit eternally okay they don't lose energy it's a conservative system so they would never sink we know that there is a radiation reaction force that you have to add how do you compute the radiation reaction force so as I said in post-newtonian theory h is the sum the full h actually is mc2 the total mass of the system plus h newtonian plus 1 over c square h 1 pn plus etc so you have an expansion of the dynamics in powers of v2 over c2 on one side now when you compute this you compute it by also by computing the the waves emitted at infinity by the method the blanche d'amour higher method I have recalled in the first lecture and this also which is quite complicated and you need to compute integrals tail integrals you can compute especially for circular orbits where things simplify a lot what is to high post-newtonian approximation 3 pn and 3.5 pn not yet 4 pn that's because there are problems there the the the losses of energy uh angular momentum and actually you compute essentially the loss of energy because it's enough for computing what you want for circular orbit so I don't write again the formulas that I had written on the first thing coming from the various work there okay and now the usual approximation that you find in many papers on this but which is not enough that's the point I will say is to say okay I have a system which has some energy of the system and I want to write that this system loses energy so I will write a kind of balance equation saying that the loss the the change of the energy of the system is equal to minus the energy loss to infinity so minus the flux so this f is the flux of energy and for circular orbits in quasi circular orbits the system is described by only one degree of freedom you just need to know the radius okay at each moment or the more invariantly the orbital frequency omega so omega capital omega is 2 pi over the period of the system and if I were in newtonian approximation I would say I would have Kepler's law which says that gm is omega 2 a3 this is the so called uh John Wheeler calls it the one two three law you know because you have this to the power one this to the power two and this to the power three you just need to remember where omega is first or a is second okay and you remember it because the velocity uh and then you can introduce a velocity associated to omega that you define as the usual formula omega time say okay now you can express if you want by using this formula this formula if you were using also at the newton approximation you have the energy is gm u over 2 a so you can express everything in terms of of these objects okay you have this equation to eliminate a in terms of omega you eliminate a in terms of v so you find that the energy of the system is minus one half of mu v square where v is this v which looks familiar but there is a minus sign because there is the that's the variable theorem the minus one over r of the attraction energy is compensating the plus one half of the mu v square and you have this thing if if these objects are function of v you can say it's a differential equation in v if i know this function this thing becomes partial derivative of v with respect to v times or total derivative times v dot dv over dt equal minus f of v so i could say that this equation tells me dv over dt is minus the flux divided by d by dv so i can compute v as a function of v i mean v dot as a function of v this is so if i know this to like four or five pn approximation i can compute this to some approximation and that gives me a differential equation to this approximation let's see what it gives at the just for orientation okay at at the lowest approximation which is the Einstein quadrupole formula the flux is g over okay as we said this is g over five c to the five this five is two times the spin plus one that's why there is a five i mean Einstein knew it implicitly then you have the third derivative of the quadrupole moment square when you compute this for two point masses you get 32 over five g new square that's a new times v over c to the 10 okay now you can solve i have written on the blackboard all the equations you need you can solve v as a function of time at the lowest approximation and you find that v over c is one half of tau a constant integration constant minus tau to the one eight where i have introduced tau which is the real time there is again a coefficient one fifth for the reasons i explained there is new which is defined there and then this is a time in seconds i need a dimensionless equation yeah v v is a velocity here in the usual units v over c is a pure number okay therefore here i should have a pure number so i need to divide t by something which makes it dimensionless and then it happens that this is gm over c cube okay so anyway you define something proportional to time and then you get that the solution from at the lowest really lowest order angsteinian quadrupole formula is given by this which which means what which means as a function of time t there is a critical time tc where this thing is infinite when t goes to minus infinity v over c tends to zero and then it has a singularity which is a weak pole you know minus one to the eight so it takes a very long time for the velocity which is slow slow slow and at the end the velocity in that sense v over c shoots up to infinity infinity at this stage just for a pedagogical reason also the in the quadrupole approximation the gravitational wave frequency omega is twice the orbital frequency the reason is that as i explained in the first lecture the electricity of the graviton is two and this electricity in other words there is a quadrupole emission means the frequency that you observe is twice the orbital frequency okay and you can compute using this formula to lowest order that omega dot over omega square so gravitational wave omega is a frequency omega dot is as the dimension of a frequency square it's a inverse time square so if i divide by omega square it's a dimensionless number this dimensionless number when you compute things is a certain number 48 over 5 times a certain mass gm omega over 2 c cube to the 5 3rd now this object here is something with the dimension of mass and this object is called the chirp mass and it is equal to nu which where nu is the symmetric mass ratio defined there times the total mass m so this is the so-called chirp mass why is it called the chirp mass because this is saying that at each moment the frequency of the gravitational wave is increasing okay but i can compute that if i compute this number the frequency dot how it increases the derivative divided by square of the frequency and i express it as the 5 3rd power of the frequency itself i need to put in front something with a dimension now so if i can observe this chirping if i can observe the the value that omega you know if i have omega as a function of time and i compute at a certain value omega what is omega dot what is the slope i can compute the chirp mass because this is the first thing that could be done for this event you observe something it is chirping you can compute a mass from it and then in the case of this you find that this mass is something like 70 solar masses okay actually less okay actually it would be better to define the chirp mass but with 4 nu here uh because then for all the systems the 4 nu would be essentially 1 because nu is 1 4 for the equal mass ratio and as it is a maximum it is equal to 1 around the maximum for many mass ratios then we will know that the chirp mass is essentially the sum of the two masses then it would be 65 so i must modulo the redshift cosmological redshift but now even if i am at this approximation and i say okay now let me go beyond yes now exactly this is lambda olive sheets now if i view each of this function f of v is the lowest order so which is 32 over 5 nu nu square v to the 10 times corrections now we are going to see in powers of v over c which contains both odd and even starting at three odd things and the energy e of v is the newtonian minus one half of nu v square times also v over c square which is one p n correction v over c four which is two p n v over c six which is three p n sorry two four six and now one even knows the v over c to the eight okay and then the id would be ah i will use the better formulas and i need them now let us see if we first really need them what is the last time there was a question by you no saying what does it mean v over c what is the v over c at merger okay this the answer to this depends on how you define v over c okay but let us define v over c observationally uh by what is uh yeah if i combine these formulas here and here i find that v is equal to g m omega to the one third at the newtonian approximation but now i can define v given uh given a gravitational wave with frequency omega i can define an associated orbital frequency which is omega over two it's a conventional thing then i plug it here it defines something with dimensions of velocity to see that it has a good dimension you can just divide by c here then it becomes c cube inside the one third but now we know g m over c cube is precisely your time so this is indeed dimensionless okay and now we can ask uh from the effective one body uh method that i'm going to describe now one can compute in the formalism what is the gravitational frequency at merger in a precise sense okay now when you and we will also show analytically uh how you do that now in the eub formalism the gravitational wave frequency for the two two mode l equal to m equal to mode at merger okay over c cube you find that this is 0.377 okay or roughly 0.38 now this means that the g m omega formally associated to this is half of this so it's something like 0.19 okay one eight eight eight five yes for an equal mass case yes sorry it's uh it's for an equal mass case but this is the case relevant because uh the correction due to you are negligible in this case okay now uh if i compute this is at merger now i can use this formula to compute v over c at merger for an equal mass case it's just a convention just to know where we are in the expansion we want to know at merger this expansion what are the terms there okay and then you find that this is so you can do the calculation yourself this is 0.191 third and this is 0.573 this is why one says you know that this is roughly half the velocity of it's even more than half the velocity of light at merger but now here i have expansions okay but but in front there are coefficients okay the coefficients are not one okay actually the coefficients grow fast and we understand why they grow fast and if you compute the term at v over c six for instance the term at v over c six the 3 p n term in the flux as a coefficient which is 123 minus 16 log of 2 v over c okay times this if now you put this number and you put this you find that at merger the 3 p n correction here is equal to 4.37 compared to one so you have an expansion which is 1 plus plus 1 plus 2 plus 3 plus 4 you have something which is not at all small compared to one so in that sense v over c one half is not one half okay it's more like one the the terms in the expansion are comparable to the lowest order term so this shows that you cannot at all use post uterine expansion at mergers okay now you have to stop before to use them now and also another way of saying that is that here i have the ratio of two expansions so when i have the ratio of two expansions one plus v square plus v divided by one plus i can do two things i can leave this ratio as is or i can re-expand everything or i can push it on the right hand side when i so i have many ways of solving this equation which we had introduced with uh higher and satyaprakash we call them t2 t3 f2 f3 okay you can uh the same balance equation you can solve in post uterine theory in many ways and you can compare whether the solution gives the same prediction at the end and then you find that they diverge a lot that near merger the predictions are totally different so it shows intrinsically that you cannot priori trust the post uterine expansions up to merger okay just the fact but there is worse than that which is the equations they contain the conservative force and the radiation reaction force here i was saying the radiation reaction force are corrections due to higher post uterine that become larger than the leading order term okay formally actually they are not we will see that when you re-sum them they are not larger that the in a sense the angstania and quadruple thing is a good approximation but you need to re-sum thing but the Hamiltonian is worse because for the following reason here i have i have this Hamiltonian here if i look at the the circular orbits okay how do i get circular orbits from this from this Hamiltonian i take h and i say circular orbits is the effective potential when i say that i have a fixed angular momentum l equal p phi which means that p square is the radial momentum square plus p phi square over r square so i write that the i write that h is the radial momentum square plus the centrifugal energy which is the energy linked to the angular momentum minus the potential energy okay that's well known it's a solschrodinger equation okay for instance but this means that the effective potential is the sum of one half l square over r square which is positive and goes to plus infinity and the minus one over r here and the sum over two give something like that i have near i equals zero a centrifugal barrier i have at a very large the attraction of the coulomb i mean Newton force and therefore a circular orbit is when you sit at the minimum okay but from this diagram you see that if i use the Newtonian approximation here i will always have a circular orbit even if the radius even if the two objects are as close as you want they can go on a circular orbit but now if i start adding these terms here you see that i have minus one over but here i have plus one over two r square which has the opposite sign so it gives it gives a term which and then you have more and more terms when you do that you find that it becomes totally unreliable in the sense that if i use 2 pn or 3 pn i get completely different answers at some radius okay when the radius is of the order of 10 gm i find that depending on the approximation i use i get nonsense or whatever but now we can remember also something that there is a limit where we know exactly what happens it is by using the parameter on u okay u is the mass ratio the symmetric mass ratio when this symmetric mass ratio is small which is not the case but let's consider formally the case where it is small i have a small particle around a big mass the big mass is a schwarzschild black hole and therefore i know what is the effective potential the effective potential is the effective potential for a particle moving in a schwarzschild black hole and this effective potential has not this shape it is quite simple the effective potential for schwarzschild is the following it is it's for an energy square and it is equal when you solve a geodesic in schwarzschild you find 1 minus 2 gm i have put back gm now here gm was 1 okay 1 plus angular momentum square instead of having a sum of the kinetic the centrifugal terms s square over r square minus gm over r i have a product and because of the product and because this thing vanishes at i equal to the potential the effective potential has a qualitatively different shape it is okay let me redo it because it vanishes at i equal to m so it means this potential also it's normalized to 1 at infinity so it goes to 1 it is like that it vanishes when the radius is 2 gm and then the circular orbit is this one okay but i see also that i have new circular orbits which is the maximum of the potential these are unstable circular orbits so we learn that and then so in this limit or small new i know the potential should be like that what does it mean i know that the system is losing angular momentum so if the angular momentum diminishes this thing becomes like that and at the end these these two maximum and minimum they merge i have an inflection point which means that when a test particle is around the schwarzschild black hole there is a radius below which there are no circular orbits at all okay stable circular orbit this radius is when you compute it is 6 gm is three times the schwarzschild radius so this is saying i expect that the the exact two problem thing might be of this type okay at the time when we work on that there was no agreement on what you would expect but you would expect that qualitatively the thing should be of this type and which is another reason of saying why post-Newtonian you cannot use because below like 10 m the thing gives different 6 m they give total nonsense okay in general so this the orbit where i have the inflection point is called i mean there are several names but i call it the last stable orbit lso now all this was understood in the in the 1990s and many people try to go beyond this i mean there was a problem post-Newtonian was not good enough one of the attempts was done by bala ire satyaprakash and myself in 1998 and there were two ideas initially we said okay let's assume adiabaticity we will find it's not a good approximation till the end but let's assume that the problem is to transform these expressions in better expressions that incorporate the physics you know like for instance there is a last stable orbit so let's incorporate it in some way and to do that with ire and satyaprakash we i need this so but i can use this so this is dis by the way i already said it but let me repeat most of the references to these papers you will find in as as comp as links actually on the web page of ihs there is a on ihs web page there is a gravitational wave web page and they are clickable things so when you see names in blue click on it and then you have the paper or at least the access to the paper so with ire and satyaprakash we said part of the problem is that this function e of v which is directly linked to this is not good and we should introduce a better function and we said the better function should be the square of a modified energy and this modified energy we said for some reasons should be this was a guess okay let me put the c4 over 2 m1 m2 c4 so e as dimension of mc square and all that so it's a dimension less energy okay sorry no maybe i should not have put the c let me take away the c's i take c equal one now this is an energy it's an energy sorry it's a dimension less energy okay and we said let's replace this let's take the post-neutral expansion of this function instead of the original function and because we know that this function has a linear pole in the schwarzschild limit let's take a padé approximation of this function for the for the following reason when i have all these are expansion in powers of of some variables when i have a function f of x and the only thing i know is the expansion at the origin okay what can i do to improve the the question is a foreign if i have a post-neutral expansion which is a power low expansion how can i improve the function why are the coefficients increasing so fast why is the thing so bad at the merger the reason is when you think about it is the exact function seen in the complex plane as singularities okay and now if i consider a function let's say of a variable v and in the complex v plane i know that this function has a singularity somewhere okay singularity can be a pole a branch cut thing like that but we know from complex function theory that the first singularity in the complex plane determines the radius of convergence and therefore if i know where the first singularity is i know what is the radius of convergence and i know what how the coefficient grows because if the radius of convergence is somewhere let's say v singularity this means that the coefficient they are this means the series is like one over v to the power n it's of this type and then this means the coefficient cn grows like one over v the first singularity to the power n okay and and if this thing is closer than one then it means the coefficient grow exponentially okay and and this was saying we understand that there is indeed on some function there is the lso singularity another function there is the light ring singularity because there is a second singularity which is that the unstable orbit circular orbits they also exist only up to 3m below 3m there are no circular orbits at all and when you think about it it proves that the function the flux function for instance will have a singularity at the light ring in the limit when you go and therefore you expect it and therefore at the time we said okay so let us replace and now how do you really eliminate these singularities to have a better function what you can do if you know there is a pole you can say let's factor the pole but factoring the pole mr. pade in france told us how it is to say let's replace a post Newton power low expansion by the ratio of now let's assume the first coefficient is one one plus n1v plus n2v square divided by one plus d1v plus d2v square that is to say if you replace a power low expansion by a rational fraction you can here inject the information or you can guess where is the pole and once you do that that is to say if you do a pade approximate it's factors the pole and therefore the function is better represented that was the first idea to use pade expansion for both for the energy and the flux okay and and actually it worked to some extent okay but then in 1990 so that was in 1998 okay in 1999 another idea came how to re-sum the dynamics better and actually this idea came the weekend while I was reading the french version of the book of itsyxon-zubair and I've checked that it's not in the english version okay that is in the french preprint version there is a chapter on the hydrogen atom energy levels and they say there is a work by brezin itsyxon and zin justin amplified by further work by Todorov which allows to re-sum ladder diagrams for this thing and at the end they say a good function energy function is this thing so when I saw that what was working for including two body effects or the hydrogen atom including this function that we had introduced for another reason the year before I said ah looks interesting and then with alessandra bow nano we pushed the id further and then the eob method which was invented then so this is now uh bow nano uh 99 the what is this effective one body method the method which will incorporate when needed previous re-summation is the following you have a real two-body problem which is this thing okay and as I explained before uh this means I have two masses m1 and m2 that move around and they interact at lowest order they interact by linearized gravity but at higher order they interact you know what in qft you would say there is a blob of all the non-linearities okay angstein equations are non-linear so the interaction between these two things contain a lot of sub interactions okay and this you know how to compute by perturbation theory so it's one so this is equivalent to the post-Newtonian expanded amiltonian okay but the idea is to say we want to incorporate something which is a deformation of the exact case we know which is when u is zero I know I have a test particle going around a schwarzschild black hole so in that limit I know the exact dynamics is a geodesic motion in the schwarzschild matrix it's something non-linear it's something non-perturbative so the idea is to say let's imagine that there is some metric g mu nu effective which is not the schwarzschild metric but which is a metric which depends on the parameter nu and which is a continuous deformation of the schwarzschild metric in the sense that when u is zero it is the schwarzschild metric of mass capital M but when u is not zero it's something else that you need to determine such that a particle mu of mass mu where mu is the usual Newtonian effective mass reduced mass having a geodesic dynamics which means the action is integral of mu the proper time in this effective metric that you don't know and you want to say I want this thing an effective particle moving in an effective metric to be in some sense equivalent to the two-body dynamics that I have computed here okay so this is the basic idea of this effective one-body formalism for the dynamics will come to the so it will be a way of resuming in a sense the Hamiltonian okay this part but we will need also to resum the radiation reaction separately okay one needs to do several things how to how to I said something but these are words okay how how do you compute also here I have unknown functions okay how do I compute these functions okay the idea is something I learned from John Wheeler which is this is a purely classical problem okay although as before we have seen five man diagrams for a classical thing but it is useful to think quantum mechanically that is to say if you think of this classical problem as a quantum problem it simplifies okay why because when I have a two-body problem like that quantum mechanically it means I have energy levels energy levels you know etc spd etc quantum energy levels they are classified in quantum mechanics by two quantum numbers two integer quantum numbers one which is the the angular momentum quantum number l lh bar is the angular momentum this is s state which would be l equal zero p state is l equal one and then there is a second quantum number which is the the number which counts these things like this is the ground state n equal one n equal two n equal three it's the radial quantum number and then you can define actually the the the principal quantum number of the Coulomb problem is l plus nr plus one where nr is the radial quantum number it's the number of nodes in the radial quantum wave function you add n and you add one in order to avoid the zero thing okay and as you all know and if you forgot I remind you that for instance in the Coulomb problem the energy levels are like one minus two n square okay they only depend on on at lowest approximation on the total principal quantum number okay but now this is for the real this is for the two-body problem each real but if I think of this problem the dynamics of a particle in some metric first you realize that because there is an SO3 symmetry look at this okay here I have only p to the fourth p square and p square this is explicitly invariant under the group of rotations in three dimensions okay the relative dynamic and therefore this problem should be also spherically symmetric okay therefore I'm looking for a spherically symmetric metric here therefore I have a spherically symmetric Hamiltonian also on the other side so I will have also levels I have a different problem and now once you say I think in terms of these levels immediately you say ah but the correspondence if I want this to be the same as this I want this thing to correspond to this this thing to correspond to this this thing to this clear quantum mechanically I have integer quantum numbers here therefore it means that the effective angular momentum should be equal to the real angular momentum in the center of mass frame and the the the radial quantum number effective should be equal to the radial quantum number of the other side but now we are interested in a classical problem okay so let me take the large quantum number limits h bar going to zero l going to infinity what is what determines these quantum numbers this is the Borsam Eiffel rule and the Borsam Eiffel rule is saying that ni plus one half h bar is equal to the action variable for the degree of freedom i I have a separated dynamics and the action variable integral of pdq on the cycle divided by 2 pi after if you don't divide by 2 pi you have h here but if h bar you have 2 pi okay and in the large integer number this is saying that these quantities which are the action variables I should be identified between the two problems so at the end you are saying I want the real Hamiltonian so let me see as a function of the action variable of the real problem to be related but related means what you realize also when you do a little computation that you cannot equate this energy to this energy you need to allow for an arbitrary map between the two energy scales so let's be generous and say I have I have put already unknown functions here so let's say I put a new unknown function which is the energy map f so there is an unknown function f such that this should be equal to this effective energy as a function of the action variable of the effective thing but I identify the arguments of the effective things to the effective thing so now it gives me an equation of this type by the way in celestial mechanics the Hamiltonian expressed as a function of the action variables for a separate problem is called the Dolone Hamiltonian okay so sometimes we call it the Dolone Hamiltonian and technically the the reason why with Alessandra we could proceed fast with this idea is that with Gerard Schaefer in 1988 to 10 years before we had computed the 2pn Dolone Hamiltonian which is a non-trivial calculation let me describe very quickly so if you take the 2pn Hamiltonian you have this beast okay now you want to say let me solve this problem exactly as an integrable problem by by doing what I take the 2pn adm Hamiltonian I go in the center of mass I'm already in the center of mass and then I write that the action is minus the energy t plus s0 of the of all the variables I separate time okay I have the Hamilton Jacobi equation which tells me that e real is equal to the Hamiltonian the real Hamiltonian has a function of r and ds0 by dr that's the usual now time independent Hamilton Jacobi equation I write that s0 okay this is a spherically symmetric problem so you can solve the problem in a plane first saying okay let's restrict to because theta phi you know you take just phi and then you write that s0 is l times phi whereby definition l is p phi is the angular momentum full angular momentum of the system plus a radial action but the radial action will be of of this integral of pr where pr is obtained by plugging this in this which gives an explicit equation so you plug this you get pr square plus pr 4 plus pr 6 equals something you can solve this equation perturbatively as a function in pr square by expanding in 1 over c square you can eliminate the higher power so it gives that pr square is something which is that pr is the square root of something when you compute this something you find that it is the Newtonian approximation would be pr square you know is energy plus 2m over r so you get a plus 2m over r plus sorry plus the angular momentum over r square but you get relativistic correction this coefficient modify and then you have extra term like d1 over r cube plus d2 over r4 plus d3 over r5 okay you get something like that now you need to integrate this over a cycle and this is where that we've get out we discovered that you can do this in the naive way of what in physics we are told today or you can go back to good authors and in the book of sommerfeld at on bow they were techniques for computing these things because this is what sommerfeld had to compute okay he was solving the problem with relativistic correction he was the way it's he found the fine tractor constant you know it was the first correction to the hydrogen levels in the borsom feld thing and at the time they knew how to compute these things and they said what is this integral this integral evidently in modern times we could say it's a it's a conceivably zaggy period but we don't need to say that it's a period of an hyper elliptic curve okay so this integral is something i have a square root so it's a square root of a polynomial in one over r so i have some algebraic geometry here kistoff but but what you want is you have this thing vanishes at the point r1 and point r2 it depends on the energy in angular momentum of the system for a given energy angular momentum and one over c square there is this is zero here and here and i want the integral between the two things i want this integral the real integral from this point to this point this period but the trick of sommerfeld because it's very bad if you do that you realize you cannot do the computation because the borders depend on one over c if you expand you have infinite integrals you need the adama party finish here and there it's very complicated why there is a very nice trick you say this i can take a contour integral the contour integral in the complex plane will actually give me what i want because i can deform the contour back on this thing so it is the same because i have no there are singularities but they are somewhere else so if the contour is close enough i can do that but once i am outside i can now expand everything formally in powers of one over c square because i don't have singularities you see the problem is that this thing has singularities here and here so if i expand the singularities get worse if i am here i have no singularities so so we use formulas of this type and then we computed the answer and then let me write and by the way and this is needed for the effective one body thing so what you get for instance explicitly i will write the answer to the one p n thing we computed it at two p n and then with janowski and shaffer when we computed the three p n we computed the delone amiltonian at three p n in 2000 2000 the classical delone amiltonian is a function of two quantum numbers i mean classical thing where l is what i define and i define the classical analog of n which is simply uh the sum this is called the delone first variable is the sum of l and i are and as you recognize this is the sum of this and this okay modulo h bar and what you find is this the real is mc square when i add it minus one half of mu alpha square over n square so this is the Coulomb thing minus one half of one over n square the famous Coulomb thing where alpha alpha is g m mu okay i hope i did not forget some scaling is there a signal i think it's just g mu i did not put it there uh and uh is it okay uh yes i think it is okay and then so this is the Newtonian answer very easy to get now at the one p n you have a correction alpha square over c square one over c square thing and then you get two coefficients six over nl minus one fourth of fifteen minus nu over n square okay and then this is one p n then at at two p n you have more complicated things like depend which are polynomials in one over n and one over l and things like that but this is an explicit expression and now how do you get the effective one body yes it's a transformation of the two p n amiltonian in the lone action variables now uh now let's consider this problem so it is the deloné version of this now let's do the deloné version of this but for this i need now to say what is this effective metric how can i compute something without knowing the function here you make something which is as i said we know that this metric has to be spherically symmetric therefore if it is spherically symmetric i can write it as a general spherically symmetric metric as can change the gauge and go to schwarzschild coordinates therefore this ds square so ds square effective g mu nu dx mu dx nu i write it as minus a over c square dt square plus b over dr square plus r square d omega square where this is the metric on the sphere d theta square plus sin theta square d phi square okay as usual notation and i go to schwarzschild coordinates where i have r square therefore a general spherically symmetric metric is parameterized by two functions a of r yes it's clear it's time independent no no no it's look at this this amiltonian is autonomous time you look at the dynamics the symmetries of indeed because it's that's what is confusing this thing is not the real metric that each body sees where it's more complicated it's an effective thing at the dynamical level but now what we know as we said what this function a of r should read it should be a function of nu and should reduce to the schwarzschild uh so you know two things okay first it should reduce to schwarzschild in some limit and also as i am doing everything as post-neutering expanded it should admit a post-neutering expansion and therefore i can just say this thing will be one plus a unknown coefficient a1 that i have to determine gm over c square r plus another coefficient a2 nu gm over c square r square plus etc you continue a3 b of r nu should be also one plus b till day one of nu gm over c square r plus b till day two etc this square plus b till day three a till day three this cube etc now i have reduced the problem to computing coefficients a1 a2 b1 b2 r so it looks better similarly for this function f because this function f uh let me parameterize it immediately i can write in reverse i can write this thing saying that the effective energy divided by mu c square which is a dimensionless thing should be equal to one plus e real binding divided by mu c square times one plus alpha one e real binding over mu c square plus alpha two this square plus etc where i just define the binding energy is just the energy minus mc square actually it is the energy i've written here okay usually you forget about the mc square here we are relativistic so we have to remember there is an mc square somewhere so if i put it i have to subtract it so here i've just written the most general function which reduces the newton approximation saying it's the same energy modulo the mu versus m and it has relativistic correction with unknown parameters alpha one alpha two but now the game is i want to determine all those parameters by some equation what is the equation the equation is this so i need now to compute the deloné amiltonian of this effective dynamics with this so what what i do is i write the amilton jacobi equation so one writes mu square plus g mu nu effective d mu you know p mu p mu where this is the the action for the effective problem equals zero it's the matchshell condition like lambda olive sheets and it's the simplest way to solve the geodesic dynamics by amilton jacobi but but if i plug all these here i have the same type of equations are here i have the same sommerfeld type integrals but except that now they depend on arbitrary coefficients a1 a2 b1 b2 but i can do the calculation and then when you do the calculation you find that the effective energy has a functional of g mu nu effective that is to say as a functional of these unknown coefficients e is equal as a function of the two radial numbers i mean the two numbers of the effective problem but i identify them to the e is equal to because it is a test particle of mass mu the first term is not mc square is mu c square okay that's it that's life which shows that i cannot identify this to this okay i need some transformation but the rest i have at the newtonian problem i know the answer actually the answer will be the same because let's say if i put a priori let's say this is the quantum the law never involves of the effective problem let me put primes but i will identify the primes to the unprimes the first term is just the coulomb it's the newtonian thing it says there is a one over r potential in both problems it comes from this term okay as we know this is the newtonian approximation that gives the one over r thing okay and then yes and now but then i get correction one plus alpha square over c square and here i get instead of numbers i get combinations of the unknown parameters like for instance the first term where i have n prime l prime i get a till day one square minus a till day two minus one half of a till day one b till day one okay which is instead of this coefficient and the second coefficient is minus b till day one minus seven over eight eight till day one over n prime square and actually in writing that i think that i have already assumed to get this coefficient one here that that this coefficient is is minus two which is just okay the newtonian approximation of this is is just given by this term so i need to identify this coefficient by the usual newtonian approximation and then you find that a till day one must be equal to minus two it's a convention actually it's just a way of i mean it's not totally a convention because i have put the sum of the two masses okay it's the the the newtonian interaction is proportional to mu times the sum of the two masses the product of m1 m2 which is equal to this anyway a till day one from the newtonian approximation must be minus two so the unknowns are this one at at the for instance at the one pn approximation uh this is unknown this is unknown and this is unknown so i have three unknowns already at the one pn approximation and how many equations do i have i can identify this and this modulo these equations so it gives me an equation which introduces a further unknown which is alpha one so actually at the one pn approximation i have three unknowns which are alpha one a till day two and b one and i get two equations and these two equations they are this so i will not repeat it's this equal to this no i will repeat because it's modified by alpha one so one equation is a one square minus a two minus one half of a till day one bay till day one equal six so this is the effective problem this is the real problem then i have this coefficient here b till day one minus seven over eight eight till day one but when i use the transformation f i have an extra correction coming from alpha one and when you do the computation you get minus one half of alpha one and this is must be equal to this coefficient 15 over four minus new over four where 80 day one is minus two okay so because of this i have two equations for three unknowns at this stage although we relaxed this thing and we looked for more general solution we have a long discussion with alessandra let's make the further assumption that not only the the coefficient of g zero zero in this effective metric is schwarzschild like but also the coefficient of the of the radial thing that is to say let's assume that the linearized gravity of the effective metric is equal to the schwarzschild gravity in fact there is justification for this one can prove it is a necessity finally but let's assume this so if i assume this now it says i am assuming that b till day one is the the the coefficient of schwarzschild of the term one over r which is plus two okay now with these two assumptions is that like fixing the post Newton parameter gamma to one uh for the effective metric yes yes let's say that so under these two assumptions now so i know b one so you see this is an equation for a two now that i can determine and now i can determine alpha one okay and the result is you find and that's where things get simple that a two is zero and alpha one is new over two and now let me comment and that yes and let's stop that's that's the end theorem theorem uh what does it mean this thing this thing means that at the at the one pn approximation uh this term is zero therefore and and and and these two terms are like schwarzschild so this is saying that at the one pn approximation i have a test particle moving in a schwarzschild background and a test particle moving in a schwarzschild background under the only condition that the the transformation contains here a coefficient which is new over two so a simple energy map is equivalent so sorry a geodesic in schwarzschild is equivalent to the one pn angstein infield of man dynamics so this amiltonian which when you have to work with it is always more complicated than it looks okay which are four different coefficients is equivalent to schwarzschild so it's the first simplification of the eob thing okay something which does not look at all like schwarzschild because uh this thing is equivalent in disguise to a test particle mu moving in a schwarzschild thing to one pn okay and then after lunch we will discuss beyond one pn okay let's stop here the mass is mu always the mass of the particle to all orders it's m it's the sum of the two masses yes yeah it's m capital m the sum it's very simple it's like in the newtonian case it's a sum but it's at the one pn level now okay