 Donc, j'ai voulu finir la partie de gravité, donc c'était un peu rapide, donc je vais traiter l'expansion multiple de l'électroënité, qui est techniquement plus simple. Il a les mêmes problèmes que la gravité, il y a une transformation de gage, il y a plus de multiples que ce que vous voulez. À la fin, vous avez deux sets de multiples moments. Donc, let's me expliquer ce qu'il y a de scratch. Donc, la expansion multiple de Maxwell's equation with sources. En, donc, nous sommes comme la gravité, nous imposons la condition de transversalité, la condition de l'électroënité de l'électroënité, c'est 0. Ensuite, Maxwell's equation devient cette, minus 4 pi par c, j mu. Ok? La seule différence, donc, pour Einstein, cette A devient h bar mu nu, qui était là avant ma kinéassistance, il a évoqué tout. Et la source à deux indices, c'est t mu nu. Ok? Et, et, let's remember also something, which is true in both cases. There is a compatibility condition, evidently, if you impose this condition, and with this thing, it's because the source was conserved, because if I now take the divergence of the left hand side, it has to be 0, but we know it was 0. So, before I did anything, because of the structure of Maxwell's equations, that at the structure d nu f mu nu equal, now, I don't know the sign for sure, at this thing, there is a Bianchi identity, which says that if I take d mu nu of this, the source had to be 0, which is well known, and it's also true for Einstein's equation, except it's the contracted Bianchi identity for the Einstein case. Now, what I said is that if you decompose the source J mu, let's say, contravariant in the charge density, with the conventions you need a velocity of light here, and the ordinary spatial current. Similarly, A mu contravariant, if you decompose it in a scalar potential and a vector potential. Ok? Then, as I said, this thing you can solve by using this general formula. So, it gives that the scalar potential phi is a sum with this decorating coefficient sum over all l from 0 to infinity of special derivative. So, I remind you that this is a repeated special derivative l times with indices. And then, here, you have f l of u over r, where u here is t minus r over c. So, what we are doing here, let me repeat, is that we have a source, which here is J mu. This source is generating waves, ok? You take a point, a field point here, and you decompose here in this form, in terms of symmetric trace-free tensors, this is a symmetric trace-free tensor, which is a retarded wave, also, because it's a function of t minus r over r. With this derivative, it's a solution of this equation outside the source, ok? And A i is 1 over c, the sum of minus l over factorial l, d l of, here, you can use the formula above, but you say the index i is like an extra index, 1, 2, 3, and, but the multipolar decomposition is a decomposition which is symmetric trace-free with respect to these l indices, ok? And at this stage, the point, the formula that I did not remember this morning, but I have it here now, is to say, this object is irreducible. It is a symmetric trace-free tensor, so it's a nearestable representation of the rotation group. Here, I have an object, which has symmetric trace-free with respect to these l indices, but not with respect to this extra thing, but I can decompose it in three parts, u i l plus, you put just some numerical coefficients. I will comment about numerical coefficients in a moment. The Levy-Chivita tensor, ok, so let me put hats over these objects, plus 2 l minus 1 over 2 l plus 1 delta i i l d l minus 1. Let me make the remark I wanted to make. One advantage of working with symmetric trace-free tensors is that everything is rational. When you work with YLMs, all the coefficients have square roots of l l plus 1 and things like that because the normalization condition makes square roots appear everywhere, ok? Here, it is algebra among tensors with l or l plus 1 indices et you just have factors that come from the dimension of space or 2 l plus 1 that comes from some dimension of d l spaces. Now, what are these? Here, I have written the explicit formula, which says that these objects belong to the tensor product of the space of vectors and the space of irreducible tensors with l indices. Et ça a été décomposé. C'est une sémi-trique trace-free tensor avec l plus 1 indices. Donc, ce sont les l plus 1. Ici, il y a un objet qui est une sémi-trique trace-free tensor qui a cette notation, je ne l'ai pas défendue, l minus 1. Je dis que l est un multieindex. L signifie i1, i2, i l. L minus 1 est une notation condensée Juste pour dire, i1, i2, i index l minus 1. Ça signifie l minus 1 indice, mais aussi un multieindex. Donc, ceci signifie l minus 1 indice ici et un extra index ici. La notation, quand les indices comme i, j, k sont mises dans ces brackets, ça signifie que c'est une projection qui fait une sémi-trique trace-free. C'est-à-dire que, premièrement, on fait une sémi-trique trace-free, puis on fait tous les traces. Je vous rappelle que, si j'ai, pour exemple, une sémi-trique avec juste 2 indices, ce objet signifie que, premièrement, je fais une sémi-trique trace-free, donc je prends la sémi-trique partie, et ensuite, je prends une troisième de delta ij, la sémi-trique trace-free. C'est un exemple d'une sémi-trique trace-free tensor qui n'est pas sémi-trique, et qui fait une sémi-trique trace-free, qui est sémi-trique sans trace. Donc, cet objet, comme l'indice, est sémi-trique trace-free, donc il s'agit de DL. Et cet objet s'agit de la représentation DL minus 1, parce que c'est sémi-trique trace-free avec seulement l minus 1. Et c'est une identité, si les objets qui sont dénotés ici sont les suivants. Ce objet, sémi-trique trace-free, c'est, comme je l'ai dit, je prends l'objet avec l'indice plus 1 et ensuite je le projette pour être sémi-trique trace-free, c'est un opérateur de projection. Le C, qui n'a que l'indice plus 1, est obtenu par prendre mon objectif G, parce que je décompose ça. Donc, ça a l'indice plus 1, donc je peux l'écrire comme G, A, B et puis l'indice minus 1 qui reste. Epsilon levitivita ici j'ai I1 up to L minus 1 donc je peux ajouter DIL et c'est un extra-indice A, B et puis je prends la sémi-trique trace-free projection sur ces l'indices donc cet objectif est sémi-trique trace-free et ces extra-indices sont contractées. Et le dernier objectif c'est le plus simple qui a l'indice minus 1 c'est simplement obtenu par prendre ce truc et prendre une trace mais évidemment je ne peux pas prendre une trace sur ces indices parce que c'est trace-free donc le seul trace que je peux prendre est entre un index à la fin et un index là-bas parce que c'est sémi-trique de toute façon donc ça veut dire que j'ai l'indice minus 1 qui reste. Donc, c'est un très trivial vous avez une identité où epsilon apparait deux fois il doit apparait deux fois parce que tout est en variant dans ce que je fais donc si j'ai changé le signe de epsilon cette convention change un signe et j'ai utilisé la identité que le produit de deux epsilon est le summe de delta donc c'est juste une identité algebraique qui est la translation dans le langage d'un tensor de ce fameux truc que la représentation de la vector de spin 1 times spin L c'est spin L plus 1, spin L et spin L minus 1 ok, juste 2 mais maintenant je peux plugue ce ici mais maintenant je dois utiliser deux faxes différentes je suis dans un gauge de la Lorentz je peux ajouter à MU je peux faire une transformation de gauge ici je suis dans l'électromagnétisme donc la transformation de gauge est un gradient d'un scaler pourquoi en gravité c'est le gradient de la vector avec deux termes symmétriques et maintenant pour rester dans la gauge de Lorentz si je suis dans la gauge de Lorentz ça signifie qu'au MU il est 0 mais ça signifie que j'ai besoin de blocs de z equals 0 donc la transformation de gauge d'expansion est à dire xI, une solution générale sera un summe de L dL xL de T minus R sur R donc je peux jouer donc j'ai une séquence infinie d'électromagnétisme qui est la décomposition de la transformation de gauge donc si je ajoute ce terme ici je vais avoir une formule qui dépend de FL les variantes donc maintenant je regarde les objets irréduciables donc cimétriques trace retensors donc c'était par cette formule c'est donné et souvenez-vous, ce FL là c'est... ces objets sont obtenus par la solution de cette équation comme je l'ai dit cette équation est de la même type la équation de scale d'exemple, il y a un extra index qui ne plaît pas... qui est un index numérique donc je peux utiliser ces formules pour compter FL et cette GIL comme fonction de la source donc pour exemple, FL sera un intégre sera ce intégre intégre de la source la source DFI est la source RON j0 en ce sens et puis XL donc elle ressemble FL à cette étape ressemble à un moment de charge mais ce n'est pas le début parce que je avons d'autres choses qui sont ul cl et dl Il y a plus d'indices, ok, mais pour l'élément générique, il y a des tensors symmétriques, qui sont aussi mis en ordinateur, et sorry, il y a le delta L de Z, qui, non, j'ai mis la barbe rouge, c'est-à-dire cette chose, avec cet ordinateur sur le delta L, ok. Mais maintenant, j'ai ces objets, qui sont en ordinateur sur la source, et ensuite, j'ai ce XIL, qui peut faire tout ce que je veux, pour simplifier la formule, ok. Et j'ai quelque chose d'autre, qui est que la source doit être conservée, ok. Et la conservation de la source, qui implique ce truc, si j'applique cette formule ici, je vais avoir des relations entre ça, parce que vous voyez, si je prends le temps dérivertif de phi et le espace dérivertif de ça, je vais avoir des expansions multiples de ce type. Si je réplique ça, je vais avoir des expansions multiples en termes irréduisibles, si vous avez quelque chose dans les termes irréduisibles, quelque chose doit s'améliorer, je veux dire, chaque chose irréduisible doit s'améliorer. Donc, cette condition, ou cette condition, me donne des relations entre le temps dérivertif de phi et le espace dérivertif, parce que, en faisant ça, j'ai aussi une identité, qui je peux même écrire, du papier avec un... oui. Je vais l'écrire. Donc, ça sera plus clair, pour exemple. J'ai une identité, qui est le temps dérivertif de phi et le espace dérivertif est equal à quelque chose comme integral over x, xl, integral dz, delta l, z. Et ensuite, j'ai minus d by dx i j i plus z over c n i d by d le temps dérivertif d j tilde i. Ok? Oui. Et puis, cette chose, je peux intégrer par part et puis, j'ai une relation entre des moments ici, les uns qui apparaissent ici. Et puis, on doit travailler toutes ces identités et utiliser xl, pour que, à la fin, vous trouverez... Oui, ici, le u signifie t. Oui. J'ai changé les papiers et des choses comme ça. Oui. Oui, c'est viable. Je vais essayer de donner la logique du truc. Alors que, après une transformation de gage, dont vous computez, vous computez ce que vous voulez ajouter. Vous computez le xl par utiliser ces identités entre le fl et les autres multiples moments. Alors que, à la fin, dans un nouveau gage, c'est le prime de tnx, ce qui est le summe de... Alors, je vais faire le cas d'électroënité. Factoriel l dl ql de t-r-c-r et ai prime dans ce gage tnx est-1-c le summe d'A-L cette fois, c'est un détail de spécifiquement car il n'y a que l-1 et ce qui est le time de ces objectifs c'est Bastille d-r-c-r D-R d-r-c et plus 1 over 2L plus 1, 2L plus 3, ok, sorry, I'm reading the wrong line. Where did the thing disappear? Here, it is plus L over L plus 1, epsilon i a b d a L minus 1. Sorry, it's not the final formula. Ok, no no, it is, sorry, it is. I forgot all this notation. Of T minus R over C divided by R. So, let us look at the result. After a gauge transformation and using the conservation laws linked to the fact that I am in a, the source is conserved. So modulo is this gauge transformation. The general electromagnetic field is expressed in two sets of multiple moments. A QL, a Q for charge, so it's a charge multiple moments. It is the analog of the multiple moments of the charge distribution, but in a relativistic context. And M is for magnetic. It is a magnetic multiple moment. And the explicit expression of these objects are that QL is what is the good formula here. Do we have a better formula here? So that means that out of this F UCL, you kill two of them. One by the conservation law. Exactly. Exactly. The counting is that there were four sets of multiple moments. Because you can count an infinite sequence of multiple moments like a block. One of these infinite sequences is killed by a gauge transformation. And the other one is killed by the consequences of this relation, which is scalar. This relation has no indices. So it gives, when I expand it, an infinite set of relations between the derivatives of these four things. And indeed, thank you Natalie. The counting is that I have four, minus one for gauge transformation, minus one from the conservation law. And the same thing in gravity will give four times more gauge transformation and four times more conservation law identities. And then this eliminates from ten multiple moments, minus four, minus four. It gives at the end two sets of multiple moments. That's why electromagnetism like gravity at the end has two degrees of freedom. Because another way of saying this is this number of big sequences of multiple moments counts like one scalar field. Because remember, one scalar field is an infinite sequence of multiple moments. So one sequence of STF multiple moments is like one scalar field, one relativistic scalar field. We know that gravity has two degrees of freedom like electromagnetism. That's why at the end, each degree of freedom scalar is like an infinite set of multiple moments. And that's why you expect, by the same counting, that there will be only two sets of multiple moments. For electromagnetism, the explicit expression is that there is an integral over X, there is the integral over Z between minus one and plus one. And then there is this delta L, and rho bar in this sense, where... Sorry, yes, there is no bar. What I mean is this thing. So let me put a tilde, where rho tilde means that the time argument is shifted from T to T plus Z X over C. You don't shift the space argument, you shift the time. So you introduce Z inside, it's an operation. Once you do that, so Z has appeared, then you need to integrate over Z. C'est la fonction de Z. C'est la première terme. Donc, cette terme ressemble à la densité de charge avec une fluctuation dans l'argument de la time. Il y a plusieurs moments, comme un moment quadruple de la densité de charge. Mais ce n'est pas la vraie réponse, parce que c'est ce que vous obtenez pour FL. La formule pour FL, c'est ce que j'ai écrit. FL, c'est ça. Mais ce n'est pas encore le moment physique, parce que j'ai besoin de faire une terme extra de la transformation. Et en ce cas, vous trouverez que la terme extra est minus 1 par C squared, 2L plus 1 par L plus 1, 2L plus 3. Il y a le delta, qui a maintenant un index L plus 1 comme la fonction de Z. Et ici, vous avez X hat A L D of J tilde. Donc, si la time argument est T, ce sera D by DT. C'est la time derivative de la currente, qui est une vector. Donc ici, vous avez une vector qui est contractée, donc c'est STF parce que de cette chose. Pourquoi ici, j'ai un scaler et c'est STF parce que j'ai cette chose. Donc, c'est comme un type électrique. Donc c'est un type électrique multiple moment, STF multiple moments. Et vous avez un type magnétique, type magnétique, type ML d'un argument. Il dépend de l'une variable. C'est un formula très similaire, plus simple pour le cas d'électromagnétisme. Delta L of Z. Et ici, vous avez X hat L minus 1 M tilde of I L hat, où M est la densité de magnétisation. La densité de magnétisation signifie que c'est une vector donc c'est X cos J. Vous avez des currentes. On parle d'électromagnétisme. Il y a une densité de charge et il y a une densité de current. Vous avez un type magnétique comme il s'appelle en pairs de currents. Vous dites que c'est un petit loop qui est comme un petit moment magnétique. C'est le moment magnétique de l'infinité. C'est la densité de magnétisation qui est un fil de vector à l'intérieur de la source. Et cet fil de vector vous avez un multiple moment et vous vous faites une trace de frein. Ce formulaire est très simple. La seule subtile c'est que l'argument de temps est shifté par quelque chose. La relation est ici à chaque stage. La propagation de la vitesse de la lumière est là. Pour l'électromagnétisme à la fin, la transformation modulaire est transformée. C'est pourquoi il y a un prime. Il y a des dérives de ce moment ou des dérives de temps du même moment et des dérives de space de cette densité magnétique. Ok ? Et et ces formuleurs si vous regardez en Jackson, si vous ouvrez le book de Jackson, vous ne trouvez pas ces formuleurs parce que tout est fait si vous assumez l'exponential i omega t qu'il y a des fonctions harmoniques et puis vous utilisez des fonctions baisselles et des choses comme ça. Donc, ça a l'air un peu plus compliqué que les fonctions baisselles. Pourquoi tout est algebraique et la seule subtile c'est d'avoir cette fonction ici. Mais là-bas, il y a la transformation qui donne la fonction baisselle. Pourquoi est-ce qu'il vient de ce delta L ? Parce que c'est une sorte de fonction. Oui, il vient, oui, je vais expliquer. Il vient, oui, c'est mystérieux mais c'est aussi une violation de causalité dans le sens que c'est dit, oui, je vais discuter. Ce formulaire est dit que si j'ai la source ici et si je compute le fil MU à ce point, dans le temps spatial ce truc ne dépend seulement de la source en retardant le temps. Parce que je fluctue pas par T minus R mais par T plus R vers C ou minus R vers C. Donc, j'ai la fluctuation qui ressemble à un causal. Et la raison est la suivante. C'est que la composition multiple dit que si je suis ici je prends une sphère à la même radieuse R et puis sur cette sphère je décompose ce fil dans des choses réduisables avec respect aux rotations sur cette sphère. Et donc, ce que je fais c'est que je combine le fil tous tous ces coups de la source dans la définition de la composition multiple. Cette composition multiple n'est pas locale ici. Et donc, elle utilise une avantage de la source qui a cette structure double de l'icone. J'ai besoin de prendre la densité de source ici. Donc, j'ai besoin de tous ces avantageurs. Et ce truc est cette Z. Et ce qui se passe c'est que pour les hauts valeurs cela devient plus fort et plus fort et vous êtes plus près de l'icone. Et c'est ce qui est l'answer. Et pour la gravité je ne vais pas rewrite la formule que mon assistante m'a évoquée. Mais l'important point c'est qu'il y a seulement deux types de moments multiples un type mass à l'intérieur d'un type électrique ce qu'on appelle le type mass et un type spin l'analogue de magnétisme est spin en gravité. Et ces moments, je vais juste récolter leurs notations. Elles étaient appelées I, L et J, L. Actuellement, les lettres J sont choisies pour évoquer le spin. Donc, ce sont les moments de spin. Et puis, vous regardez pour une lettre plus près de J. Et I est mieux que K. Donc, c'est pourquoi cela doit être avant. Ça veut dire qu'il y a une combinaison de ceux qui sont associés avec une polarisation ou une LCT et une autre combinaison. Non, maintenant, on va à la polarisation. Ok. Ceci que je dois rewrite dans la gravité j'ai vraiment besoin de la gravité. Donc, je vais rewrite la gravité dans une corner. Je ne vais pas rewrite la formule avec toutes les coefficients, mais juste pour avoir la structure. Je ne suis pas dans un gage. Le point que je veux expliquer c'est que je ne suis pas dans un gage où je vois les deux polarisations de la gravité. Je suis dans un gage harmonique et je peux toujours faire une transformation gage pour protéger sur ce que l'Igo et Virgo peuvent voir. Et c'est le next thing que je veux expliquer. Ok. Je vais rewrite seulement celui-ci. Ok. C'est le summe de l'Igo. C'est juste pour le fun. Et vous avez ce truc. I-L-T-R-C-R. Donc, c'est la simple chose. Plus le termes de gage. Maintenant, H-0-I il contient quelque chose où je n'ai pas essayé les coefficients. Il y a d'I-L-1. C'est la prime dérivertif de celui-ci avec l'indice I-L-1 avec l'argument T-R divisé par R. Et ensuite, il y a un troisième terme qui est H-I-A-B d'A-L-1 J-B-L-1 de T-R-C par R. Donc, au niveau de l'I-0 de la métrique, j'ai la prime dérivertif de l'argument et j'ai la prime dérivertif de l'argument T-R-C-R et au niveau de la métrique dans l'argument vous avez quelque chose avec l'argument T-R-C-R d'actuellement vous pevez c'est une très simple chose. Ici, il y a l'indice L mais ils sont tous contractés et cela donne un scaler. Ici, je veux un vector de l'I-1 mais sur la métrique 3-3 et comme vous voulez le dérivertif de la fonction de T-R-C-R, vous contractez l'autre dérivertif. Ici, vous avez deux indices mais vous avez besoin de deux dérivertifs pour compenser et ici, vous utilisez R et vous utilisez la loi de conservation. Dans ce cas, vous devez absolument être en condition qu'il soit satisfait parce que vous utilisez cela pour relater les moments multiples qui étaient 10 moments multiples minus 4, minus 4 qui ont fait 2 moments multiples donc ici, vous avez deux dérivertifs et ensuite, il y a un autre termen avec coefficients et puis vous avez le delta non, non, sorry, ce n'est pas cette formule et puis vous avez D, ici vous avez J epsilon A B I J dot index J B L-2 comme fonction de T-R-C sur R et vous avez ces dérivertifs donc, tous ces objets je le répète, tous les objets qui appuient ici, ils sont de type un certain nombre de dérivertifs et puis des fonctions avec des indices de T-R-R donc, tous les blocs de construction sont en train d'établir les dérivertifs avec des dérivertifs spéciaux c'est la solution de l'équation de D'Alembert et les objets qui arrivent sont des représentations irréduciables de la rotation des groupes et vous avez deux types le même IL appuie de temps différencié et donc, c'est un moment de type un moment multiple et c'est un moment de type spin un moment multiple ok et les formules c'est ce que j'ai écrit avant, qui était ici-dessous, je pense que c'était ici vous pouvez encore le voir je vais faire juste le premier terme comme IL de son argument c'est un intégre d'un espace ou c'est un T d'un taux d'or de Z delta L, XL et sigma à l'argument du shift et ainsi des autres termes où sigma c'est ce que j'avais écrit avant c'était T00 plus TSS sur c2 donc c'est un moment de type mass et le autre là qui a de la dense momentum touche sain Je ne sais pas si c'est juste que je peux recommencer, c'est un peu merveilleux. Donc, ici on est dans l'homony-gauge. Et, effectivement, j'ai choisi un gage. Ok. Mais maintenant, pour répondre à la question de Pierre, si je veux... Donc, cela exprime, selon la source, cela exprime le fil que cela émite. Mais maintenant, je suis intéressé, je suis là. J'ai un interferomètre ici. Et je veux savoir, dans le sens que je l'ai expliqué avant, farf de la source, il devrait être une main-gauge, ok? D'abord, pour répondre à la question, j'ai oublié un film de Hugo. Quand j'ai... Alors, let's... let's look again at this, to have this clearly. When... let me do even an example, when I have one special derivative of something which is f of t minus r over r, just to be clear. What is this? This is minus... if the space derivative acts on this, I get minus ni over r square f of t minus r, ok? But the special derivative can act on the r which is here. And therefore, I have an extra term which is minus f dot at the argument of t minus r of the gradient of r itself, which is ni, ok? The unit vector divided by r. So, and if I have more derivatives, I have more time derivatives of f. But I will always have a term in one over r. So, when I go very far, actually, I will... only this term will survive. And therefore, if I go in the wave zone very far, I will have something which looks like a plane wave, which decreases like one over r. The coefficient of one over r here will be obtained, for instance, by taking the derivative acting only on this guy. Everybody acts on this, here and here. Everybody should act on this, and then I will have this plane wave. I have part of the thing which is a plane wave one over r. Now, what is observed by a gravitational wave detector? As we said before, on suppose, in a different gauge, this thing should be equivalent to an H ij tt, because we have proven this morning that given a general solution of the linearized Einstein equation very far, there exists a transformation of gauge where I have only spatial components where 0, 0 is... I want to go in a gauge where H 0, 0 is 0, where, or H bar is the same thing for this thing, vector part is 0 and only the special part ij is non 0. But still, I am in a Lorentz gauge, I am in a harmonic gauge, which means I have the transversality condition that this thing will be 0. As we have proven this morning, and the trace of H will be 0. So, staying within the harmonic gauge, I can now refine the gauge. The problem is the harmonic gauge is absolutely not unique. I can change harmonic gauge a lot. And here I want to change it in order to extract from these formulas what will be observed by LIGO because this is what you want to compute. The point, the reason why I explain this is that these formulas allows to compute the multiple moments that are emitted. But now the next question, and this is what you compute analytically, is what do you observe when you have LIGO-VLGO? And now I will answer this question. I need to go to this gauge. How do you do that? The way you do that, is that first, and that's clear here, if I go very far from the source, I will have 1 over r wave, but locally this 1 over r wave is like a plane wave, as we know locally it's a plane wave. And what did we prove for a plane wave? We proved that a plane wave in general looks locally something like that. There is a phase factor and then there is an amplitude. And in the TT gauge, in the TT gauge you project everything in let me draw something. Now the drawing is not in space time, but in space. I have the source here. This is emitting waves. These waves are multi-polar decomposed. Now I look in some direction here and I say locally these waves here will be plane, because these are spherical waves, but spherical waves very far from the source looks like a plane. What is the direction of propagation? If I am at a point x here the direction of propagation is n, actually. Is the unit vector from the source to where I am. This is the wave I am here, so in this direction. And therefore what I need to do is to project on the plane what we know is that the wave the wave is obtained is only transverse to the direction of propagation. So there is a tensor which is symmetric trace free in the local plane on this sphere tangent to this sphere where I am orthogonal to n. So I take n as a unit vector I take the two plane orthogonal to n and then I need to project this wave on this plane in a form which is a symmetric tensor without trace. And this is the answer in the sense that in direction n I can define a projection operator pij of n which is delta ij minus ni nj. This is a projection operator in the sense that if it acts on a vector if I take a vector v pij vj the projection of v is what? It is this it is the projection v can be projected in a long n and then perpendicular to n and what I have written is the part where I have subtracted what is perpendicular. So this thing projects a vector of arbitrary direction in a vector orthogonal to the direction of the wave. Now hij tt is obtained by doing this projection twice because you prove this you go from this gauge to a tt gauge at the end of the day you find that if you want to compute what LIGO detects here or at least what is the wave the wave comes from some direction I can take it as something existing in the plane perpendicular to the direction of the wave then the LIGO detector is also at an angle so I will have also cosines of projection on LIGO but this is a separate calculation I compute the 2 polarisation of the gravitational waves but I have a tensor so what I should do is that I take h in any gauge for instance in this gauge I take it's 1 over r so I take the 1 over r amplitude of the wave let's say a kl t minus r over r I take this, this is the plane wave from these formulas and now I want to project it as a tensor in this plane so what you do is you take this projection operator for the direction n jl for n this project the tensor in the plane but it has still a trace so you need to subtract the trace and therefore the projection operator is this minus one half of p ij of n dkl you just prove this I give the final result this is an explicit algebraic formula which says given actually in any gauge where but which satisfies Einstein equation and which is wave like I take its special components and then I do this algebraic projections where n is the direction from the source to where I am but this formula allows one to compute explicitly the ok so now I can combine all formulas that is to say I take these formulas which express the wave around the source in multiple moments and then I ask what do I see here what and then you get you get really the decomposition of what you see in the LIGO detector as a function of the source because what I have solved by these formulas is that I can compute the wave emitted to all orders in relativistic I made no slow motion approximation ok the source can be ultra relativistic it has to be I am in linearized gravity so the next difficulty a go beyond linearized gravity and we will see how to do that yes yes yes yes yes yes yes yes yes yes yes yes nobody I have I have the quadrupole, the octupole the magnetic one yeah tourist order we will get this yes the idea is to get the full thing yes so let me first define if I have let me write it this way in linearized gravity let me call this operator here yes this is a projection operator which contains 4 indices ok so let me call this thing p with 4 indices so p ijkl it's an algebraic operation which takes a tensor and projects it in something simpler which has only components in the plane orthogonal to the direction n that I chose the formula which is transfer stress less to polarization of the gravitational waves there is a factor 4 which comes from just the normalization there is ok so what you do you take these formulas and as I said you keep only the part of the wave which is in 1 over r ok so for instance actually and only h ij because finally all the other components they are not important for the projection they must be related in some way that's why actually it's the same object that appears here so immediately I see I will have at least 2 derivatives of this guy and at least 1 derivative of this but as I said if I take the 1 over r part I will have l minus 2 more derivatives because I need to act on the argument t minus r so actually if I am at a multiple l I will have l derivatives ok because I have 2 derivatives plus l minus 2 means that the multiple l will enter with l derivatives so the quadruple will enter with 2 derivatives the octuple with 3 derivatives etc in the waveform ok but at the end you find that you have this projection operators i prime j prime ij in sorry I am changing the notation each time so let me keep the notation the unit vector now is just denoted capital N instead of small n but it's the same as before it's to distinguish the field point from the source point ok and you have a sum over l now when you do the sum you find that the multiple moments for l equals 0 and 1 they disappear why ? because for instance the multiple moments for l equals 0 is the total mass of the system which appears like a schwarzschild m over r ok but it is non-radiative ok so so this term when you do the tt expansion this one disappears ok similarly for the the angular momentum of the source which is a constant quantity and the center of mass thing so the physical degrees of freedom that you record at infinity they start with the quadruple magnetic mass type or spin type then you have an infinite sum where also for the moment I have kept the velocity of light ok it looks like a nuisance to keep the velocity of light but actually what is important and this is in a sense what Einstein already had in mind is that the multiple expansion is at the same time an expansion in powers of 1 over c if you take higher multiples they contribute less and less in the sense that they contain more powers of 1 over c that's why the quadruple always dominates it's the leading order term and then the octuple is a small correction 1 over c square formally compared to the quadruple etc so you get this thing then here you have objects that I will denote this way l minus 2 so t minus r now it's just capital letters nl minus 2 so minus 2l over c l plus 1 l plus 1 factorial epsilon a b i v j b l minus 2 n a l minus 2 ok so the mass multiple moments I will define what is this object ok at this stage I am defining the following thing I am saying that the two polarization of the gravitational waves can be written as an infinite sequence of multiple moments of the TT waves these objects are symmetric trace pretensors and they parametrize the multiple moments of a general wave ok so there are two types of multiple moments but the work we have done before has expressed these things in terms of i l and j l which themselves are expressed in terms of the source so actually I will be able like Einstein did at the lowest level as an integral over the source but now there is no approximation as been made of the small v over c let me for the mass type moments because these are mass type moments the power of 1 over c is l why there is an extra power of 1 over c l plus 1 for the magnetic thing because always magnetic spin effects in gravity as an extra v over c so there is less emission there is always for instance the magnetic type quadrupole is less important than the mass type quadrupole that's why Einstein wrote a formula for mass type quadrupole he was fully right it is the first term if I write this explicitly just to see how it looks there is so it goes like 1 over r ok sorry I changed the notation several times but what is small r here becomes capital r it's just the distance away from the source ok and here I am taking only the coefficient of 1 over r by using these formulas and I write the formula explicitly the 4 has become 2 because there is a 1 half here so I have this projection sorry i prime j prime ij which depends on the direction from the source and now let me write explicitly the beginning of this thing yes sorry the result of linearized gravity is that these objects ul so symmetric transfer we fell in this is finally equal to the elf time derivative of the mass multiple moments where this notation means as a function of one variable the elf time derivative so it's dl over dt to the power l of il so it's just a notation we have a function of time because you can write 1 dot when you have 1 time derivative 2 dots when you have 2 time derivatives but when you have l dots it's not convenient so you need to know how many time derivatives so in linearized gravity this u here is just the elf time derivative and I explain why the reason is here it appears with 2 time derivatives I have l minus 2 space derivative but they act on this thing here on the plane waste thing this is finally l derivative of the mass moment same thing for the vl as a function of t in linearized gravity this is the spin type multiple moments differentiated again l times you can see that it's the same number of time differentiation together with the formulas which gives that il is some integral over the source so now I have an explicit formula in linearized gravity which says if I have a source with some t mu nu in the source from t mu nu I compute it's just a notation the sum t00 plus tss over c squared I call it sigma because it comes in naturally by the way it is the combination which enters all over cosmology in cosmology you have rho plus 3p ok this thing is the energy density plus 3 times the pressure ok and the fact that in cosmology in some formulas this is important is again important here for inflation for instance it's very important because it changes the sign it's just saying that Einstein's equation yes actually the reason why this quantity enters is that if you look at r00 from Einstein equations the source here is t00 plus tss ok which appears on the right hand side of Ritchie not of the Einstein tensor of Ritchie that's why it comes in here so let me I was writing explicitly to see what it means so this term using this formula and summing over L it gives iij with two derivatives this is the Einstein term two derivatives of the quadrupole moment except that evidently Einstein had said that the quadrupole moment is given just by the first term here the usual quadrupole moment while here we have a relativistic quadrupole moment with many relativistic corrections so even if you keep only this term you have more the next term is 1 over 3 c the third derivative of the octupole moment contracted by the vector n from the source which enters here ok plus 4 over 3 c epsilon I will stop after this term ij anb ok with two derivatives so the in the waveform the quadrupole moment of spin type enters at a coefficient 1 over c smaller than this that's why Einstein had said that lowest approximation I have only the quadrupole but here we see quadrupole octupole spin type quadrupole and then you have infinitely many term like that c squared for instance the next term which is the multiple moment is something like that you just have more and more n vectors, more and more time derivatives and things like that ok so this is the result of linearized gravity and then from this result you can already compute for instance what is the energy flux and the angular momentum flux when you do that at this stage it's not clear why it is useful to have a decomposition in symmetric trace free objects the reason is that when you compute the energy flux for instance the computation of the energy flux is essentially will be energy flux I mean all over the sphere that is to say what is the total energy flux in gravitational waves emitted from the source ok, essentially this formula is you have an integral over the two sphere so you have the surface of the two sphere of h i j t t dot h i j t t dot the square modulo coefficient you have the wave optimization of the waves and they are like the energy is proportional to the time derivative of h why, because as we know for a scalar field for instance the t mu nu of a scalar field is d mu phi d nu phi so it can equal minus one half of a thing but it's always time derivative of the field that come in the energy ok, so similarly here although I don't want here to enter you define energy flux in these waves by the way in the book which was here this morning of madame choquet this is also discussed in the high frequency thing you define the t mu nu of the gravitational waves as the right hand side of Einstein equations when you write Einstein equations and you say on the right hand side I put non-linear terms coming from the gravitational wave itself they contribute like an additional term to t mu nu and this give this Einstein had computed even before getting generativity what is the stress tensor the pseudo stress tensor of gravity he had conservation law so he used that and he had the correct formula ok, he could not compute the angular momentum but he could compute the energy flux anyway, so imagine now doing this calculation what do you do you take this object you take a time derivative so immediately each derivative here will augment by one unit the quadruple moment will have 3 times derivative this one will have 4 and things like that and now you take this square and now the fact that these are symmetric, these are irreducible things means that the cross products all vanish because when you have irreducible representations they are orthogonal with respect to the natural norm that you have and therefore when you compute this square you will have only the square of this the square of this that add linearly this is the nice thing so that for instance for the energy loss you will get I don't know if I have the coefficient here but I will write the essence of the thing like the energy flux let's say flux in energy is sum over L with some numerical coefficients of if I take this object this would be UL with one derivative square plus VL with one derivative this thing square with some numerical coefficients and if I replace this formula UL is the plus derivative of the L-type mass multiple moments this means I have a sum over L starting with 2 where I have the mass multiple moments with L indices differentiated so let's get it right L plus 1 times square with some numerical coefficients plus the JL differentiated L plus 1 times square with numerical coefficients and the old Einstein quadrupole formula yes and indeed now if I keep the 1 over c square I have I had 1 over c so now I get 1 over c square and actually if I put everything I have this 1 over cl ok so it means so let me let me get them ah this grows like 1 over c 2L and this grows like 1 over c 2L plus 2 ok ah yes modulo and overall factor so this is saying for each value of L the magnetic thing the spin thing is less important by v square over c square but it also means that the octupole is 1 over c square smaller than the quadrupole and the exact decapole is 1 over c square even smaller and this is very important because this is saying that when I have a binary when I have a source with a characteristic velocity v over c if v over c is a small parameter the multiple expansion is also directly an expansion in v over c I know that if I want to compute things up to v over c 6 or something like that in the energy flux I only need to compute up to the moment or whatever and I can neglect everything of high multiple moments and this is the way post Newtonian method can compute things nearly to the end by including more and more things although there is the non perturbative aspect that we will come back to so now I have talked for one hour so let us stop for a break of 15 minutes and I will continue with the non linear things so now I will sketch on, because everything I said was to give was to give some tools because these tools will be useful for the non linear theory because evidently we are not interested in linearized gravity linearized gravity is never a good approximation in astrophysics ok as soon as there is a system with internal gravitational forces and we are talking about a binary system since the non linear effects of gravity are the ones that create the force between the two objects and therefore for binary systems I am never in linearized gravity so even at the lowest approximation that's why actually the paper of Einstein with the quadrupole formula cannot be applied to a binary system it can be applied to an oscillating ball of fluid something like that but not to a system which is held together by gravitational forces you absolutely need to include non linear effects of gravity even for the lowest order quadrupole formula for a binary system ok and this was realized first realized by Landau in their treaties in 1944 they explicitly say we need to include non linear terms and at the end when you include non linear terms you recover the same linear we are talking quadrupole formula of Einstein and I want to go beyond that because this is what we need to do to higher approximations ok let me just keep that somewhere I will add one formula here which is just to make the connection with what is done in explicit in LIGO type things is that when you have ok what in practical terms what what people call the multiple expansion of the gravitational waves is measured by some quantities which are called which are projection along tensors spherical harmonics ok so there is a theory of spherical harmonics which generalize the scalar spherical harmonics and I must say it always looks complicated but the actually the result is very simple to understand because when you decompose the tensor h i j t t along tensors spherical harmonics which are labeled by spherical harmonics with m going from minus l to plus l ok I won't I'm just saying what is the relation because what you find in textbooks and the quantities I have introduced here the relation is very simple is there is Newton's constant there is the distance to the so I'm in the wave zone there is a square root of 2 for some historical reason like in the Fermi constant something like that is the only square root which appears there is a 1 over c l plus 2 ok then there is a 16 pi over ok this is not important ok you just have to know this and then you have two numerical coefficients that because now we are in the realm of normalized spherical harmonics was integral over the sphere you will have square roots for the first time because each time you have ylms you have square root of something so here you start having square roots is just a convention before we did not have square roots 2l l minus 1 but what do you have here you have the ul which is this ul ok which itself does not contain square root plus 2i the mass type you see the notation is always to have the first letter is the more important mass type and then you go in the alphabet a little bit later like ij and uv to have something that you can understand and then there is a 1 over c at the level of the amplitude there is a square root of l plus 2 over 2l l plus 1 l minus 1 you have vl and then here so here these are symmetric trace free tensor with l indices why this is a complexe number it's a projection now you project symmetric trace free object on a basis so you compute a number like the components of something because yeah maybe let me note this m goes from minus l to plus l as we all know ok this means m takes 2l plus 1 values ok plus l plus 1 is precisely the dimension of the space we talk about and therefore any object which in symmetric trace free form is a symmetric trace free tensor with l indices is a member of the irreducible representation of dimension 2l plus 1 of tensors of euclidean tensor and therefore there existes many bases on which it contains as useful information only 2l plus 1 components real components for ul so you can indeed there is the same information in ul and something with just lm indices ok but now for the waveform the convention is to compute this in terms of this by having here this where I I had introduced this this morning remember that the usual ylm so you take the textbook with ylm and you say ylm has a function varying on the unit sphere with angle theta phi I can parameterize it in terms of a unit vector n with n squared equal 1 and actually as I explained this morning it is a polynomial in n and this thing is ylm nl simply so it is a polynomial of health order in the components of nl the components n let me remind you is nx ny nz with nz cos theta and here this is sin theta cos phi and sin theta sin phi ok so if I replace this in here I will have I will have exponential i m phi ok the usual thing and I will have polynomials in sin theta and cos sin theta of the logendre type that I have already explained this morning anyway this defines this thing and as I explained this morning this object for a given lm is a symmetric trace-free tensor for l indices and the set of these 2l plus 1 tensors is a basis of the set of symmetric trace-free tensors you use this basis to transform this symmetric trace-free tensor in a number which is the components and this number with this coefficient gives you hlm and this is hlm is what is often plotted in theoretical computation for instance the main quadrupole wave is h22 this is even in the most relativistic case by the way yes let me mention this here I was saying when the source is moving slowly the multiple expansion is an expansion in powers of v over c ok what is surprising in a sense is that when you do this decomposition for coalescing black hole even near the end essentially h22 is the most important thing and the other things are smaller and they become comparable to h22 only really very near the coalescence and their contribution to the energy flux remains small so in that sense this ultra relativistic thing of the coalescence of these two black holes has the good taste of being not too relativistic nearly up to merger so this type of formula I will use later but now I want to explain what is the connection between this hlm and the formula which connects this hlm to htt yes so the good thing is that my assistant answer now because everything is on the blackboard so htt is this so let me indeed let me now summarize the thing so I don't need this let me put this up so something very simple happens in linearized gravity which does not happen at all in linear gravity but we will still use part of the thing in linearized gravity yes in linearized gravity they were three types of multiple moments in the sense that we said H bar mu nu the gravitational waves this was a formula written somewhere here is expressed in terms of source multiple moments it is a functional of two set of multiple moments where I just sketch is an integral over the source of the mass density xl with some decoration and things like that so this was something I compute from the source like if I have a binary system I take the quadrupole moment of the mass of the neutron star the mass of the neutron star from the source the wave in multiple decomposition but now I had also the wave at infinity I transformed to a TT gauge I have the wave that LIGO is detecting this wave was expressed in terms of radiative multiple moments so HTT which is computed from this so it's not used is expressed in two multiple moments but even if I am in a non-linear theory and define our ul and vl not because the formulas will not be all correct so I can say if I am in a non-linear gravity I take the wave at infinity it has this form it's just a parametrisation of the wave I decompose it in multiples and therefore it has some radiative multiple moments now in linearized gravity we have found for the linearized case ul is simply the time derivative of jl so this is a relation which is valid only in linearized gravity and it will be modified ok yes so I had those three things I can integrate this over the source I can parametrize the solution outside the source in terms of these multiple moments and then I have radiative multiple moments which are time derivative of this so I know everything now what happens in the in the non-linear case is much more intricate but one can use still things so I will sketch the technique we use so this is what we call the MPM multipolar post-mincoscan so now so multipolar post-mincoscan by the way nmincoscan means not post-Newtonian we are going to do expansion which will use and the idea of this expansion was first proposed by Bill Bonner in a restricted setting and keep on emphasise maybe it's usefulness and then we have taken up the thing and pushed it as a tool for computing things in a non-linear way so what do you do you are going, you have a source which for instance is made of two now it's a space-time picture so the source I'm really interested in is the motion of two black holes or two neutron stars in space-time so they make like this DNA thing ok they move in space-time and you want very far from this so one billion light years away here you want to know in this direction what what is the h t t i j wave that you get here ok the way you do it is you can still use these formulas in the sense that if you imagine that I take a big sphere one billion light years around the source which is maybe a bit difficult but you decompose in this symmetric trace rate tensor and you just parameterise the wave here by this sum it's just a parameterisation but now you want to know what is the relation so you have this UL and VL these are radiative moments at infinity that you can observe and to answer the question of Natalie they are all gauge invariant I have fixed things so that is gauge invariance and these are like the two degrees of freedom of the wave but in a multipolar sense now you want to know not in linearized gravity I need to take non-linear effects many non-linear effects in gravity so how do you do things you do things in successive approximation first you are going to decompose you decompose the spacetime in various zones ok around the source and then you are going to use different approximation methods in each zone there is a zone the big approximation method I mean the one with a big domain of validity will be this multipolar post-minkoskin or Blanchéd amour higher approximation method which will be in a big zone which goes essentially to infinity and the reason why it does not go fully to infinity is that there are some logarithmic terms and you want to modify because you work in an harmonic gauge yes this will be done in an harmonic gauge now the harmonic gauge is bad when you go in a radiative zone very far because there are logarithmic terms that pop up but these logarithmic terms they are not very difficult to handle and there is a paper of 1987 by Luc Blanchéd to show how to deal to introduce the bondy-sax type coordinate system so that you have a good expansion in powers here you will have some coordinates t, r, theta, phi in this region you will need to change the coordinate r to another coordinate to absorb some logarithmic things ok and to have a good expansion of the bondy-sax type ok so it is part of the scheme but most of the calculations they will be done here and here in the zone near the source you will use post-Newtonian theory but later because this is a convenient way to compute the field generated near the source the main the meaty thing in the MPM is what happens in this thing so the scheme is the following what you do in this exterior zone so this is exterior zone first you are outside the source so you solve Einstein equations in vacuum so you solve r mu nu equals 0 but how do you do it you do it by your post-Minkowski expansion which means in general it's good to work now with what was called in the old days the Gothic metric which means g mu nu inverse ok et you write the Gothic metric as eta mu nu euh let me yes so there is if I want to keep the same if I was writing here this way I think if I put the minus h bar this h bar in linearized theory is the same as the h bar before because the perturbation away from the Minkowski metric this is actually the h mu nu minus minus one half h actually in the calculations you don't put the minus sign here you put it somewhere else ok let's not worry about just the sign and you solve h equals 0 in terms of this the advantage is that when you are in harmonic gauge this object so let me now forget about signs and notations so I put plus h mu nu it is this thing or if you want you can put a Gothic h mu nu ok now the harmonic condition says let's put a Gothic h mu nu that this is your exactly so the gauge condition is exactly that the divergence of this thing is 0 because the divergence of eta mu nu is equal to 0 and now when you write Einstein equations you will so when you write amu nu equals 0 you will have equations which you write formally as d'Alembert of this h mu nu ok is a sum of terms which contain 2 derivatives and then I have terms quadratic in H 2 derivatives I have terms cubic in H and then 2 derivatives and then quartic in H ok I want just write Einstein equations ok we know it's not going to be always very good but each time there will be a problem we will locate it and then take into account the physics of what happens because this d'Alembert is a flat spacetime d'Alembert this way you can solve really explicitly that's why it's called postmenkowski ok now why is it called multipolar postmenkowski because at each level you are going to solve h mu nu as a successive thing saying is g h1 mu nu plus second order in g h2 mu nu plus third order h cube mu nu etc you push the calculation up to the needle where what h1 mu nu will be the general solution of d'Alembert of h1 mu nu equals 0 but this means h1 mu nu is like a linearized gravity wave outside the source we want to get the field which is obtained using boundary conditions of no incoming gradation so we look for a general solution of Einstein equation outside the source which is retarded type waves but we have solved this problem it is given by this there is the general formula for a general h mu nu that as we are non linear we need to write that h1 mu nu is so let's call this thing this formula the canonical solution of Einstein's linearized gravity expressed in terms of two sets of multiple moments except that the multiple moments that we are going to put here are not related to the source because we are we are somewhere here we are just parameterizing the thing so the two sets of multiple moments so h canonical mu nu we call them to distinguish them ML and SL but just to remember that M is mass type and S is spin type like this and you need to take into account a gauge transformation because the general solution is in a general gauge you need actually to take into account later some changes of gauge so you need to put back the various now I forgot how many there there are some gauge multiple moments the ones I said we can eliminate sorry there is W yes there are four that's what I said before there are four coordinate that we can make so there are four multiple gauge transformations which are there you put them but they play a minor role but it's important for the scheme to keep there and now now you iterate Einstein equations explicitly and the power of this method is because you will be able to solve Einstein equations explicitly because you combine post-Minkowski expansion nothing at this stage but you inject an explicit multipolar expansion for the solution and now you look at the solution so let me write here at the solution for the second order you have this thing and in the right hand side what do you have you have two derivatives and you have something quadratic in H1 but H1 is the sum of function of T-R over R space derivative each one is of this type you take this, you add more derivative ok, if it is a time derivative puts a dot here if it is a space derivative it augments the number of ends so it has still the same structure but it is non linear so to compute explicitly this non linear thing one way is to expand everything because as we explained before this thing is like F with the L derivatives plus 1 over R square N some unit vector F with L-1 derivatives and things like that it is at the end what you get is something fully explicit and you need to do I mean it's not that you argue it will be of this type, you compute things this thing is finally will contain powers of 1 over R in general of the 1 over R type there will be function of T-R because you have products of function of T-R and then there will be indices ok so depending on this you will have new new indices also floating around and then ok after doing this thing here yes, this function of T-R is the product so here it is the product for instance of ML with some derivatives T-R times a different set ML2 the T-R with a certain number derivative with some contraction of indices so now these were irreducible representation of the rotation group but now you make the products of two irreducible representation like the L1 times the L2 ok, depending on... and then you need to re-decomposite in irreducible parts to have something simple actually what is most important is that the angular part you can always when you have angular parts you can re-decomposite in n hat L because there is just a formula for instance if I have n i n j I can say this is n hat i j plus 1 third of delta i j so ok I just use this to all orders so at the second level therefore you need to solve things like that ok but now the difficulty is the following this is a function of T-R so it is a kind of wave which goes like that and it contains inverse powers of 1 over r but it contains very high inverse powers of 1 over r because if you think just of the quadrupole for instance the quadrupole contains 1 over r cubed i j for instance of T-R if you take a time derivative and you take the square you will have 1 over r6 things like that and then higher multiples they go like 1 over r6 and what do you want you want to solve this equation now if you try to solve this equation I compute the green function which is the green function acting on the source but what is this source this source is something which has an angular dependence but it has a radial dependence ok and the radial dependence is divergent when you go formally to i equals 0 which is ok because I am outside the source so I should not go inside i equals 0 I should solve this problem keeping outside the source but the power of the method is that you don't do that you formally now forget about the source you say let's because the idea is the following I have replaced the source which is physically there by a fictitious word line which has delta function singularities and derivative of delta function singularities in the sense that it is emitting multiples so multiple things you know if I take d'Alembert of dL of ft minus r over r d'Alembert of ft minus r over r is minus 4pi f of t delta 3x it is a delta function at the origin ok it is a distribution theory formula if I am in distribution theory if I take L derivative I will have L special derivative of the delta function so it means the multiple decomposition means I have replaced the physical source which was regular which existed by a formal set of derivative a skeleton of derivative of delta function something very singular when I compute the products of the singular things that increase towards go to infinity so all these integrals are infinite meaningless but this is where you start doing some technological thing which was inherited by what we had been doing before for the 2pn problem is you use analytic continuation you say, ah yes Adamard does not work here so it is more Schwartz like or Gelfand type analytic continuation you say a power of R because the this thing diverge because at the origin by the way this green function what is this green function the green function that one is using which is this thing this thing is a convolution operator with it is a green function between the field point tx the source point let's say t'x and this thing is what I had written before it is delta of t-t'-x-x' over c divided by x-x' and you integrate over x' so you have a Poisson integral like 1 over x-x' and if you multiply this by something which goes like 1 over r' to something 3 plus plus plus plus it diverges because the integral at r' equals 0 becomes integral of d3x' over r'3 plus something and this thing does not exist it is a divergent integral so what you do but you are not interested in this integral you are interested in solving this equation which is a different problem so you have the right to do whatever you want so that at the end you get a solution which is r to the r0 to the power b where b is a complexe number that you are going to choose and if the real part of b is large enough this gives this says that the integral now becomes convergent because you see if I have an integral let me give an example if I have d3x over r'3 ok I go to polar coordinates I have let's put r to the power n and let me change x' to x this thing is r squared d r d omega d angles divided by r to dn ok so I have an integral over the angles and then I have integral of d r over r to dn minus 2 ok if n is equal to 3 or larger this is log ethnically divergent or worse but if I multiply this by r over r0 to the power b this becomes a convergent integral at i equal 0 if the real part of b is large enough and therefore I can compute this thing as a function of the parameter b now you prove that this integral is a meromorphic so it exists in the b plane so in the plane of the variable b which is a complexe variable you go sufficiently far on the right so that you can compute the integral here and then now you ask is this an analytic function of b the answer is yes because it's trivial to look at it and the only singularities will be poles so actually by analytic continuation it will define a meromorphic function of b but at the end of the day what you want is to go back to b equal 0 this way you will have a solution of the problem you wanted if you stay away from b equal 0 you don't have a solution of the problem but the worst that can happen when so you continue b up to b equal 0 is that the worst that can happen is you can have poles at b equal 0 but it's clear I mean when you look at it you find immediately that the how is it called the residue of the pole I mean the singular part will be actually at the first order you get only simple poles 1 over b they are not always poles but if there is a pole it will be 1 over b and the coefficient of 1 over b immediately you prove is a homogeneous solution of the problem so you can subtract it and if you subtract the singular part you have what is called the finite part so now you add but not in the sense of adama of the finite part because now this is valid you put a finite part for b equal 0 which when you take the Laurent expansion so this thing is a function of b which has a coefficient like b to the k plus a coefficient b to the k minus 1 plus c1 over b and then c0 and then the parts of order b the finite part is this you just subtract everything which is a pole or multiple and then what you prove is that c0 is a solution of the problem you wanted so it's just a mathematical trick and the reason why it is useful is that you can compute all these integrals explicitly and therefore this way given any at any order of nonlinearity you have integrals which you can compute you have products of STF tensors you have function of r because the reason why this thing simplifies a lot is that all those things are just functions of t minus r and then there is a power of r and then there are angular dependence it's not like a complicated function that you don't know it is something polynomial essentially modulo some function of t minus r and with Luc we could develop formulas for computing this thing at the nonlinearity level which means that given any order you can, if you want, compute things you need somebody very energetic like Luc to pursue these two high orders and thanks to this work and energy one got results at the nonlinearity levels but the logical point is the following so I start from this I give unknown seed multiple moments so at the level h1 you put some unknown multiple moments then you compute the solution at the next order which is a function only of these multiple moments but now with nonlinearity and then you can continue so this way but at this stage you have now to worry we should be in harmonic gauge so we have to check is the harmonic gauge condition satisfied it is modulo something where you can correct so you can generate with some tweeting something which is at each level a solution of the harmonic gauge condition and a solution of Einstein equations therefore at each stage so you do that so this way you generate an h at any multiple order because formally you can continue this evidently it gets very very complicated so you push things only to some level but which is a function of unknown multiple moments and the next step is to say ah but how do I determine these multiple moments because now I have computed the exterior so one computes the exterior metric here outside the source as a formal functional we are also when you do that you never do a post-Newtonian expansion because a post-Newtonian expansion is valid only in the near zone but it is crucial here to have something which is valid in the wave zone also you want the validity of this expansion to go from outside the source up to infinity essentially modulo these logarithmic terms that you deal with separately and the next step is to say how do I now compute the value of these things especially these two moments in terms of the source variables before you go there yes because you have this all these multiples ms which are for the moment arbitrary they are arbitrary, they parametrize the solution then when you regularize you insisted on this one over r power but all that is multiplied by this ffm ffm yes and for the moment you have no idea of the dependence in t minus r of these things they are related then get the results so don't you have to make assumptions about the behavior you compute things what you mean the behavior they how did I verge at the origin of some ffm the coefficients of the one over r what you have to prove is you have to prove that to all orders because what happens is that when you go to for instance if you keep the number of multiples fix you say I go up from l equal 0 to l equal l max to make things clear when you do that the order of divergence at the origin will increase at each order of the post-minkowski approximation ok and to be able to prove that there was an analytic and then logarithmic, logarithm pop up and you have to prove an analytic continuation in presence of lawyers with Luke spaces that we called ln which had a special signification for me but anyway that was just a name which contain logarithm up to powers n of the logarithm and you prove that ln goes to ln plus 1 and then you can do we prove to all orders in expansion in G that you can always do the analytic continuation and you can always solve the problem formally ok so it generates a solution at the end of the Einstein equation we are not saying it's the it generates the general solution in the sense that this solution contains the number of arbitrary multiple moments that exist that you expect to exist in a physical solution but at this stage you construct a solution and the next step is to relate it to the source c'est que you solve now a different problem is that you solve a post-Newtonian problem now you reintroduce the source so you say ah but here I have matter ok which moves around and now I am going to solve Einstein equations by a different perturbation scheme which is a post-Newtonian scheme so a post-Newtonian scheme instead of using d'Alembert thing you write d'Alembert as delta minus d defend d d d l s die et you treat this as a small term that you put on the right hand side and you solve poisson type equations the problem of... the post-Newtonian scheme is very good for computing explicitly things up to some order but it gets bad when you go in the wave zone, that is to say it has a domain of validity formal domain of validity which is limited to be at a distance from the source d'un wavelength émitté par le système. Mais la compétition extérieure est valide dans une zone qui est formelée à une radio qui est plus grande que quelque chose qui est lié à un wavelength parce que vous n'avez pas fait de l'approximation de la zone. Vous êtes post-Minkowski. Donc la seule chose est que vous devez être beaucoup plus grand que la taille de la source. C'est-à-dire S. Et si maintenant la taille de la source est beaucoup plus petite que la taille de la wavelength émittée par la source, mais précisément, c'est une approximation post-nutrienne qui dit que les velocités sont plus petites que la unité et beaucoup plus que la unité, vous avez une zone intermédiaire entre deux chelons, deux tubes. Vous savez, vous avez un tube ici et un autre tube. Et dans ce tube, les deux formules, donc, ce sont toutes les formules de la série. Je veux dire, les formules d'expansion, l'expansion à l'extérieur est une expansion en pouvoirs de G et l'expansion à l'intérieur, c'est une expansion en pouvoirs de 1 à la c². Et puis vous utilisez la méthode de la expansion asymptotique qui est à dire, dans une zone intermédiaire, les deux expansions doivent être d'expansion du même objet, le même métier, la transformation modulale, parce qu'on ne sait pas si le gage harmonique qu'on utilise à l'extérieur est le même gage harmonique qu'on utilise à l'intérieur. C'est pourquoi vous devez mettre le gage général, parce que sinon, il y aurait un mismatch même dans le régime de gages harmoniques. Et cette information, vous pouvez transmettre une computation explicite pour compter cette chose comme fonction de la source, parce que la approximation post-Newton compute tout en termes de la source. Ici, vous compute tout en termes de ces objets qui sont connus par dire que les deux choses sont les mêmes, vous compute ces choses en termes de la source. Pour finir, je vais illustrer la première chose non triviale qui est la approximation de l'approche 1 pn. Ok, je peux vous demander une question. Oui. Une petite question. Dans le cas d'étro-dynamiques, vous pouvez faire la même chose, mais dans ce cas... C'est une théorie linéaire. C'est une théorie linéaire, donc c'est beaucoup plus facile. L'électro-dynamique? La réponse était exacte. Je n'ai pas besoin de... C'est parce que j'ai toutes les équations de l'électro-dynamique. Dans le cas d'étro-dynamiques, cela dit que la source émette quelque chose qui est combinée avec quelque chose d'autre de la source. Et ensuite, j'ai des vertices cubiques. Donc, je dois prendre l'interaction des deux multiples de la source qui génère quelque chose à l'extérieur. Et j'ai beaucoup de renormalisation des choses à prendre en compte. D'abord, je dois dire, parce que la première chose qu'on a faite avec Luc était la première approche de post-Newton. Et il y avait quelque chose remarquablement simple qui n'a pas été appuyé à d'autres autres. Luc, en plus, quand l'un a ajouté l'approche de l'électro-dynamique, a trouvé un bon truc, qui est la suivante. Oui, laissez-moi... Laissez-moi écrire, peut-être, le type final des formuleurs. Oui. On verra. Donc, ici, vous devez prendre vos mains. Les formuleurs finales sont que vous voulez compter les multiples. Donc, on appelle ça les multiples radiatives. C'est vraiment ce qu'il y a à l'infinité. Il paramétrisera ce que vous voyez dans un détecteur à l'infinité. Mais le problème est quand je n'ai pas dans l'électro-dynamique, je pouvais rappeler ceci à l'électro-dynamique de IEL. Et IEL était un intégral sur la source. Donc, tout a été résolu. Maintenant, je vais avoir plus de termes non-linear. Je vais avoir... Je suis en train d'une relation où l'UL est relative à un intégral sur la source mais qui prend un effet non-linear dans une façon systématique. Et l'MPM method donne ça. Et je vais montrer les types de formuleurs que vous avez. Parce que vous devez distinguer deux types de non-linearité qui viennent. Mais je vais faire les types de formuleurs que vous avez. Vous avez les choses que vous êtes intéressés dans, qui sont les multiples moments à l'infinité. Donc, il y a l'UL et le VEL. Je vais le faire premièrement, l'UL est equal à... Donc, il y a plusieurs multiples moments dans le jeu. Pour l'instant, j'ai dit qu'il y a ces moments qui peuvent être appelés les algorithmes multiples moments que vous introduisiez dans la solution avant que vous les relatez à la source. Puis ils disparaissent. Après un moment, ces moments sont répliqués par quelque chose en termes de la source et je vais le faire les formuleurs à ce stage. Vous trouvez que l'UL est donné par une formule qui ressemble comme l'autre c'est le dérivatif de quelque chose qui sera expéré en termes de la source mais je vais... je vais le faire mais j'ai d'autres corrections qui viennent de quelque chose que Yvonne connaît très bien qui est le fait que quand vous êtes dans le temps espace l'UL ne se propague à l'infinité mais il y a une telle que l'UL a une partie à l'infinité et ces choses quand vous les computez expérimentalement signifie que ce que vous avez observé à l'infinité ce que vous avez observé à l'infinité ne sera pas seulement une fonction de la source à l'un des retards du temps comme going straight to the source to the velocity of light mais elle contient an integral over the infinite past c'est à dire what the source did a billion years before we saw it before the light came already a billion years after that let's say ten billion years before still influences the gravitational wave at infinity these are the so-called hereditary effects as we call them with look so you have integrals over the past and these can be computed explicitly the first at lowest order you get this integral 0 to plus infinity detail of the same object so remember L means L indices ok L plus 2 derivatives ok at a time U the same U is a retarded time ok it's actually I should have kept the thing that it's not small U it's capital U the reason why it is capital U is that this U is not is like t minus r minus 2g the total mass over c squared log of r like you know in the correction there is a logarithmic correction the retarded time you have to take into account the fact that the light cones are not flat they are bent by logarithmic terms by the total mass of the system and other things so you have this type of logarithmic corrections that you normalize this way but this is still in a sense the real retarded I mean it's the retarded time ok in a curved space time but you depend on this something before with a logarithmic kernel log ln means napier log natural log ok c tau over 2 r0 plus numerical coefficient don't try and vl of U is given by the same type of formula except it is the other multiple moments gl l of U plus the same coefficient 2 gm over c cube and this m actually should be the better it's the ADM mass of the system ok 0 to plus infinity total energy of the system of gl l plus 2 times u minus tau log of c tau over 2 r0 r0 is the scale to do these computations you need to to regularize divergent integrals and there are logarithmes coming in so you need to introduce a scale r0 this scale does not enter in physical effects you prove at the end that the scale is arbitrary ok but there is so here tau is a time c tau is a length you divide by a length it is a dimension less thing in the log and then there is a different numerical constant here pi l and there are extra terms this is the first tail effects Luke has studied the tails of the tails the non-linear tails there are terms beyond that you can compute but but here I have introduced il and gl what are these these objects are obtained by the finite part with respect to so to compute these you need again to introduce a finite part and an analytic continuation with a factor r over r0 to the power b ok but it has a different meaning now because the finite part you needed here was because of a UV divergence that is to say a divergence at the origin ok while here you will have an infrared divergence which is a divergence at large spatial distances but it's just the same thing that you see in a different way because once you are inside the source you have divergence there outside you see them differently anyway the formalism tells you you have this and this thing the nice thing is that here you put the il as computed in linearized theory that is to say these formulas and the other formulas that my assistant has erased and but remember this is an explicit integral ok in terms of these integrals were integrals over T mu nu of the source this was in linearized gravity so these things were integral of something with some kernels T mu nu what you do is you take the same functionals but you replace T mu nu by tau mu nu and you have the same formula so you use the formulas of linearized gravity that's why I said the formulas of linearized gravity are useful because jL linearized T mu nu goes to tau mu nu where tau mu nu is the right hand side of Einstein equations when you write it in the form T mu nu equal so now I am not sure of the sign but there is it's like what I was for each mu nu you have finally something like minus or plus I'm not sure of 16 pi over c4 times T mu nu when you write Einstein equations you have this plus all the non-linear terms that you can put here these are non-linear terms and you call the sum of the T mu nu sorry yeah it's there is the G also there is the determinant which appears here ok so everything here the determinant G T mu nu plus the non-linear terms in Einstein equations which are D H D H plus H H H all the terms you give it a name you say ok it's the right hand side of Einstein equations I call it tau mu nu ok but the nice thing is that formally replacing in these formulas T mu nu by tau mu nu gives this part of the multiple moments to which you must still add another non-linear term which is the hereditary tail terms but at the end this you can compute because you compute it now this integral is is dominated by what happens in the source but it extends outside the source but the finite part with the R over R0 to the B allows you to define the thing uniquely and this gives the good answer that you can compute to any order you want in post-Newtonian theory now so this scheme which has used post-Minkowski and post-Newtonian at the end gives something practical something which is not easy when you go beyond the leading order but which has been pushed to quite high orders and which is the basis of what will be used to compute the radiation of binary black holes up to merger because you will use this post-Newtonian these post-Newtonian things allows you to compute the waveform so the waveforms here would be this HLM which was somewhere now it disappeared it's up there HLM yes it is there ok so you want to compute the the multiple things emitted really by a binary black hole ok and what you do you have this scheme which you combine post-Minkowski and post-Newtonian thing but which allows to compute the waveform as an expansion in powers of v over c this expansion will lose its validity when you go near merger and we will discuss another method which is the resumation methods which are used in the effective one-body formalism to go up to merger starting from things which should not be allowed to be pushed up to merger did I want to say something else or just to actually we can probably stop here everybody must be tired yes I won't ah yes let me just mention one thing that if you do this at the 1 pn approximation because this is something at least very simple in this scheme if you do this at the 1 pn approximation that we have done with Luke before Luke found this formula you find and this is remarkable that the formulas of linearized gravity if you write them in terms of the contravariant components of t which is not the same as writing them in terms of the covariant because I am now in non-linear theory are exact that is to say that they give the correct 1 pn mass type multiple moment emitted by the source so to compute the emission of a binary system from the mass type multiple moments at the first post-neutral approximation which means going beyond Einstein the remarkable thing is that they are given by the same formulas as linearized gravity ok it's an accident at the 2 pn order they are non-linear terms that are more complicated to compute that you need to tackle with ok peut-être que ça pille vers nos Landau-les-Chitstem good question actually what it means is that when you compute the sigma which was like so when you compute let me write this because I noted it so the technical thing is that yes I have written it here it is because it is because this thing which is the so it's the sum of this and this and all the non-linear terms at the 1 pn happens to be the thing with the contravariant components like that there is an extra term because still there are non-linearities like the determinant here it's not the same but the extra term is 1 over 2 pi g la place of the square of the Newtonian potential where d'Alembert of V is minus 4 pi times g sigma so it's a relativistic but it is it's a Laplace of a thing and when you replace the Laplace here you find it does not contribute by integrating by parts because you compute when you compute the quadruple moment you or anything because the Laplace goes on xi, xj and gives 0 because it is harmonic function so at the 1 pn there is something nice okay the non-linearities are like non existants but at the second post-Newtonian and third post-Newtonian you need to use complicated calculations you need to use dimensional regularisation but that will be discussed next time okay thank you