Categories en Physique — Marc Lachieze-Rey Talk Part 1 (contd.)

0:00 I'm a bit counter-intuitive, because if we talk about the one after the other, it's not necessarily that it's one who is the cause of the other. the cause it's to say even if there is a force and here it's confonding chronology and causality different chronology it's a very wrong word but it's the word two elements chronologically it's a curve of the genre of time between the two. The genre of time is not the genre of light. It's to say there is an interaction, for example, there is a particle which is propagated from this event to the other, or an ohm sound, well, it's the same thing. D'accord? The relation causal, it's more large, it's to say, temporel or light. And it's true that in all cases that we know but it's equivalent. In fact, we define a structure like this, and if I take... Well, topologically, I have here the future chronological that I call... I don't know if it's J or I, I don't remember. I think chronological... Chronological, it's I. So I have... So there are notions topologiques, obviously, liées to all that, and it's why, in the paper that we studied, obviously, we tried to express with the categories both the two structures, namely the structure causal and the structures of inclusion, which, finally, rapport to the topology, because, obviously, at the beginning, On a une géométrie différentielle, c'est donc construit déjà sur une variété topologique, avec une topologie qui est en gros, en tout cas localement, celle de la Terre.

2:30 Alors, donc... Dis-moi, quand tu dis qu'il y a quelque chose qui s'est propagé de la Terre, that means exactly that there is an element of tangent at the beginning which by transport parallel to the curve in question will arrive on an element assigned to the curve. Yes, an element of tangent which is the same causal, which is the norm null or negative, which, in parallel, according to the curve, since it means geodesic, It will be found here. But you have also the geodesic of Jean-Respas. And in this case, it means that you have a vector which will be transported parallel to him-même in remaining one of Jean-Respas. Well, she is the one of Jean-Respas, because, obviously, we don't have a geodesic. We can never have a particle, or we can never travel in the space. We can travel in the time. Well, in the time and in the space at the same time. So it doesn't exist. It doesn't exist. What do you mean? It doesn't exist. It doesn't exist. It can exist. It's called a tachyon. It's something that goes faster than the light. Because you have understood that when I say T and C, and X, it's CT. So when I say x2 égale 0, pour un rayon lumineux, et la même chose que de dire, c'est dans cette unité-là, c2 dt2 moins dx2 égale 0, donc c2 dt2 égale dx2, que je pourrais écrire, dx sur dt en module égale c, c'est-à-dire, ça veut bien dire, ça se propage à la vitesse de la lumière, It's another way to say that. So, in Mikoski, we see that when I decide a right like this, the pente of this right, this right, if it is a short time, it represents the line of the universe of a particle. The pente of this right, it's the weight of the particle in this reference that I chose. And the relative relative,

5:00 l'angle entre les deux et faire un changement de vitesse comme je vous l'ai dit, c'est faire une rotation hyperbolique si j'étais ici ça voudrait dire quoi ? si j'interprétais ça comme la ligne d'univers d'une particule ça voudrait dire qu'elle va à une vitesse plus grande que la vitesse de la lumière bon, en relativité a priori ça n'est pas interdit c'est ce qu'on appellerait un tachyon sauf que vous voyez que aucune rotation hyperbolique ne peut me faire sortir du cône de lumière. Donc, ça veut dire que si j'ai une particule normale, j'ai beau l'accélérer autant que je veux, je ne pourrai jamais la faire passer de l'autre côté. C'est-à-dire qu'elle ne pourra jamais avoir une vitesse supérieure à cette lumière, ni même une égale. En revanche, les tachyons pourraient se transformer entre eux. Donc, les tachyons, ça a été étudié de manière théorique. Alors, moi, je n'ai jamais regardé ça, but I believe that it could not exist in our universe, but from the point of view of the theory of relativity, we are obligated to add as a supplementary, for example, that the tachyons does not exist. For example, in the first theory, in the first version of the theory of the cord, these first versions predicted the existence of the tachyons. and in fact the physicians at the time considered that it was a bit embêtant but it was not really redimitoire it was to say that we could accommodate the physics where the tachyons exist so you don't ask what happens if there are tachyons who pass here I don't know but it's not totally exclusive we can talk about tachyons we can when I have a variety Well, it can be very complicated in its topology and in its metrics, and I can study it from the point of view of its structure causal. C'est-à-dire, si j'étudie sa structure causale, je n'aurai pas toute l'information sur la variété, puisque je ne connaîtrai au mieux que la partie conformément invariante de sa métrique, mais j'aurai déjà beaucoup d'informations. Et, alors c'est là qu'il y a des travaux extrêmement subtils dont je ne vais pas vous donner tous les détails,

7:30 mais on peut définir des conditions sur la structure causale d'une variété. I'm not going to give you all because there are a lot of people. So what we need to know is what we want to call causality. It's to say, what is it that we want to say that an space-temps, well, an space-temps, an space-temps, if you want, has the good properties causals? In general, it has a good proprietor if an event is in the future of another, it is not in his past. In other words, for example, I am not in my own future or I am not in my own past. We have to exclude things like that. Or, in relativity, if I don't add these conditions, it is possible. Autrement dit, en relativité, il est tout à fait possible d'avoir des solutions qui obéissent donc aux équations d'Einstein, toutes les prescriptions de la théorie, dans lesquelles on a des courbes de temps fermées. Close Time-like Curse. Qu'est-ce que c'est ? C'est simplement une courbe comme ça, fermée dans l'espace-temps, telle que le vecteur tangent est toujours de genre temps. Alors, évidemment, ça paraît assez bizarre, c'est contraire à notre incution. Ça n'existe évidemment pas dans les espaces-temps les plus simples qu'on peut construire comme Minkowski ou autre, Mais il existe des solutions des écosystèmes d'Einstein avec des trucs comme ça, la plus connue étant le modèle de Ptl, mais il y en a plein d'autres. Et si je veux en construire un, il y en a un qui est tout bête, c'est que je vais simplement prendre un cylindre. Un cylindre comme ça, où voilà la dimension temporelle, et voilà la dimension spatiale. Si c'est un cylindre, mettons qu'il soit à deux dimensions, il est plat. So it's a thing that describes an universe completely wide of matter. So it's also resembling, in this sense, that the space-time Minkowski which also describes an universe wide of matter. So if you want, it's Minkowski on which I made a changement topological, which I cut and I recollect. That's exactly right to Einstein

10:00 with, just a tenseur of energy and impulsion, no part of it. And you can see that there's a curve of time, and the beach, all of that are closed. So this is a solution to the general relativity. If we lived in a thing like that, what does that mean? for example in the universe, I want to draw my future, my future, I want to draw my mirror like this, obviously there it continues, and it continues in it, so I am in my own future, and I am also in my own past. It means that when our time proposes are cool, it should be a periodic function. Well, it's not easy to interpret, and so we have to say, something like that, we don't want to. So, is it reasonable or not to reject something like that? There is no response there, there is no response there, there is no response definitive. Some indications, it is that we need to distinguish... Evidently, if we say that, all of a sudden, we think about the famous paradoxes I'm going to kill my grandmother, and so my grandmother will not have children, and so I don't exist, so it's not the same paradoxes. Now, when we look at things in a little fine and we try to ask what is really the time and its history that we have lived, On se dit que, évidemment, pour un système très complexe, parce que chacun de nous est un système très complexe, surtout Jean-Jacques, le temps, rappelez-vous, n'est pas défini. on peut le considérer du point de vue de la relativité

12:30 comme finalement un ensemble de lignes d'univers un très grand nombre de lignes d'univers on va avoir quelque chose si on regarde les lignes d'univers des particules 10 puissance 23 si on voulait une situation qui permette ce fameux paradoxe ça voudrait dire It's not only one line of time, but it would demand a very large number of lines of time fermées, and in addition, they would be fermées in a way of coherence. I don't know if it's clear, but it would demand... The situation in which we would have the temporal paradoxes that I just described would demand something much more cher, if you want, that one single line of time fermées, or even several lines of time fermées. And what many physicists think, and personally, but we will talk about it also, is that there is absolutely no paradox to imagine that in the space-temps, there is plenty of time fermes for a particle. Because to say that a particle has a line of time closed, it doesn't make absolutely no problem. Because if you want, a particle, well, I don't know how to say it, but it's something that it would be better to think about it, but imagine that a particle has a line of time closed. There is absolutely no paradoxes that we can do. Autrement dit, par exemple, je ne peux faire aucune expérience physique, c'est le même temps que je peux faire, ce n'est pas une question technologique, mais je ne peux concevoir aucune expérience, même une expérience de pensée, qui puisse permettre de répondre à la question, telle particule que j'observe, par exemple dans un accélérateur, est-ce qu'elle a une ligne de temps fermée ou pas ? Maybe we can exclure that the circumference of a line of time is smaller than that. Well, obviously it is quite difficult to think about these questions. And if it interests you, if you do it, you should know that there are plenty of errors in the literature. It is to say that, as it is a subject quite subtle, when we talk about time, we tend to, in a certain hypothesis,

15:00 to use the current language and forget that we have made this hypothesis. So, I just want to look at that in the literature, there are a number of different things that are said there-dessus. Heureusement, there are also a number of different things just. But it is simply that it is extremely vigilant when we talk about these things. So... So the question of the fact that a single particle is not produced, there is a lot for that. Yes, there is more than a region of space-stand. Because in theory, there is an explanation on the seuil of paradoxicality, which is in-dessous of a certain quantity... Yes, there is no paradoxes, because there is no constitution. Well, just now... It's quantified, that? Yes, just now, it's what I will absolutely announce now. Well, we ask the question, is it an universe where there are lines of time closed, which is solution to the relativity general, is admissible as a solution with a physical sense. And then if the answer is yes or no, we can define the conditions that are more strict or less strict. And, just like this, there is a hierarchy of conditions that have been established in this genre. Alors, je vais vous les énoncer et je ne vais pas vous les définir parce que les définitions sont quelquefois très techniques. Je vais vous donner les références où vous pouvez les trouver. Alors, il y a ce qu'on appelle... Alors, la hiérarchie va dans ce sens-là. Il y a... Excusez-moi. Excusez-moi. Ah oui, voilà. It starts with a condition called the hyperbolicity global, I will describe it, then the stability causal, then the causality forte, then the distinguishing space time, Ensuite, la causalité tout court En espace-temps, peut être ou non causal Ensuite, il peut être ou non chronologique Et puis, dernière condition Non totalement vicieux

17:30 Alors, je crois que vicieux, ça veut dire That would mean that in each point, there is at least a temporal curve in this case. The most interesting thing is, I think it's the hyperbolicity global. Dire qu'un espace-temps possède une hyperbolicité globale, c'est simplement dire qu'il existe une surface de Cauchy. Il existe au moins une surface de Cauchy dans cet espace-temps, une hyper-surface de Cauchy. Est-ce qu'on demande quel est le genre d'espace ? Oui, il existe une surface de Cauchy de genre espace. What is a surface of Cauchy? It's a hyper surface, so dimension 3, in this case-là, which we demand that it is differentiable, that it is even, yes, there are some conditions that vary according to the authors. I don't think that we should demand that it is lisse, but I don't know. So, two genres espaces, telles qu'elles sont intersectées exactement une fois par toute courbe causale, qui sont complètes dans la variété, bien entendu. C'est-à-dire qu'on ne va pas avoir une courbe causale qui commence et qui suit ici. Alors, ça veut dire quoi ? C'est une position très forte, en fait. that means that I have my variety so there exists a surface like that tell that whatever is the causal I mean, so the genre of time so the genre of space that I dessine there is a force a point where it will be d'accord that means what puisque la relativité est déterministe et que toutes les théories physiques qu'on connaît aujourd'hui sont déterministes, eh bien, il y a toujours, si vous voulez, toutes les courbes qui sont dans le passé d'un événement arrivent ici,

20:00 donc je peux décider de décrire l'état du monde sur cette hypersurface. super-surface, comme je vais définir une fonction temporelle et je vais dire ça, c'est t égale t0, un temps originel, si vous voulez, peu importe, c'est pas un temps, c'est une fonction temporelle, So I define all the conditions here and in virtue of the propriety that I have announced and of the determinism, that means that what happens here is completely determined by everything that happens in the intersection of its passé causal with this surface of Cauchy. Do you always talk about the geodesic? No, no, no. It's about all the curves. But all the curves, how do you do it? Causes. What do you mean? How do you do it? How do you do it? How do you do it? How do you do it? N'importe quelle curve of the sort of time. Ah, but I don't do it. If it was a geodesic, it would be perfectly defined by its vector here. But here, I take n'importe quelle curve causale. Well, it's just property like that, it's to say that it can coincide in a certain interval and then... If you want to... Well, it's not determinable, it's not like you said earlier. It's to say that it's the same until a certain point and then you can have more information. Yes or no? I don't understand the question. What do you call it? The surface of Cauchy has an influence in the past and in the future, but in the space. Well, in this case-là, for that space-time is global hyperbolic, we demand that there is an hyper-surface of Cauchy that there is no one in every space, etc. It's very important, and it's all. It's very important that the space-temps can be created like this. It's very important. It's very important. What? No, no, not at all. There are a lot of varieties that don't belong to this. Let's start by the Schwarzschild's model.

22:30 We don't belong to this. Well, let's say these are the most simple ones that we can imagine. that it is globally hyperbolic in a region of space-time. It's to say that we can have a space-time in a variety of complicated and say that a part of the space-time is globally hyperbolic. There is always a question. Yes, of course. It's a notion that is quite a global. But what I mean is that, for example, when you have a singularity, rather than when you have a horizon, you will, for example, ask that there is a surface of Cauchy dans toute la partie de l'espace-temps qui est en dehors de l'horizon. Ou des choses comme ça. Mais si tu veux, il y a des tas... Bon, ça, c'est très utilisé parce que il y a des tas de... Tu peux faire beaucoup de choses dans ces cas-là. Par exemple, il y a une approche pour quantifier la gravitation qui s'appelle l'approche canonique, tout simplement. Donc, je ne vais pas en faire des détails. Mais ce qui est intéressant, c'est que quand tu as une variété comme ça, You can do a follow-up of your variety, it's-to-dire that you can have a family of hyper-surfaces, it's just simply to describe that, a family of hyper-surfaces, which are all... Well, it's a follow-up, so you write your variety like a reunion of hyper-surfaces which are all in correspondence with the hyper-surfaces of Cauchy, and so if you can do a follow-up like this, which allows you to define something that you will call time and space. It's very nice because if you know each point you can define the time and space, it means that the geometry of the space-temps, which I call a chrono-géométrie, you will be able to consider it as the variation in the time of a spatial geometry. geométro, geométro-dynamique, geométro-dynamique. Dans l'espace-temps, il n'y a pas de dynamique, si vous voulez. On ne sait pas comment s'écoule le temps, etc. C'est une variété qui a une certaine géométrie. Si dans cette variété, j'arrive à définir quelque chose que je considère comme un temps,

25:00 la variation dans cette direction, de ce qui est transverse, c'est-à-dire de l'espace, This will be equivalent to the geometry, but now I see it from a point of view dynamic. And when I want to do what we call a canonical canonical, this canonical canonical, we call it canonical because it is done on the model of the first quantum theory which has been built. Of course, the first quantum theory which has been built, they have been built with the notion of time. because it was either the quantum physics the most simple which is in the end of the time newtonian or the time of the relativity restreint which is defined in a way rather simple so to be able to apply a procedure of quantification canonic we have to have something which represents the time we have to have something to determine the dynamic what do we have to change Well, it's just simply, in this case, compared to the time. Obviously, after, we can define things a lot more general, a lot more general than the quantification canon. We don't want to let the decoupage in the space and the time. But if you want, there is a lot of work done with this hypothesis of hyperbolicity global and which allows us to make this feuilletage. Ah, oui ? Oui ? Ta chrono-géométrie, là, telle qu'elle est définie, la géométrie dynamique, elle se raccorde généalogiquement à un métier de Wheeler, tel qu'il définit son géométro-dynamique. Ben, si tu veux... Wheeler, effectivement, a prononcé ses mots, parce que, si tu veux, quand tu veux quantifier, l'idée, c'est de passer d'une géométro-dynamique classique La relativité, c'est la chronogéométrie, c'est-à-dire la variété d'un espace-là. Dans les cas sains, qui sont ceux-là, et puis évidemment, heureusement, il y en a d'autres, on a envie de revenir à notre formalisme intuitif, donc on interprète la chronogéométrie comme une géométrodynamique, c'est-à-dire une variation dans le temps d'une géométrie spatiale. Et ensuite, on a envie de quantifier, c'est-à-dire on a envie de construire une géométrodynamique quantique. It's what said Wheeler, he said that we pass from the relativity interpreted as a geométrodynamic classique to a geométrodynamic pointy. In fact, a question, just at the beginning of your exposé, I don't understand, but, let's say, the particle of Oselia?

27:30 Yes, well, I'm in relativity, for me, a particle, it's an universe that I understand. N'importe who? Yeah, it's a little bit, you know, because each of us has a particle, we have a universe of the genre of time. Yes, but after the question, maybe you could say, but if there are two lines that intersect, you know, it's not a problem? Well, yes, there is a problem between densities, but I don't know if it's a problem. So, no, there can be an interaction. Well, an interaction, these two lines intersect. If you align it like this, it is like this. At this point, the two particles become the same. Yes, but wait, I'm okay. You think it's not really pertinent? No, but your question is pertinent, if you want, but it obliges to take off from the relativity. Because the relativity, if you want, considers the particles as objects, or idéal, a little like Newton. and we know today, for example with the quantum physics, that you can't consider a particle as an object punctual ideal, you don't want to localize it, you have to associate at least an extension which is its length of debris, etc. So, already at this stage there, there is a contradiction between relativity and quantum physics. So, the relativity renounce to say on an echel microscopy. But there are contradictions, I'm sure. For example, if you suppose that she has really an extension nulle, it means that she will engendrer a curve of the space-temps which would be infinite. Well, all of that is not the same thing. So in general, when you use a particle, you place in the approximation that we call test particle, C'est-à-dire que tu supposes que la géométrie de l'espace-temps n'est pas modifiée par la particule elle-même. C'est-à-dire que la particule se promène dans un espace-temps dont la géométrie est donnée par les équations d'Einstein, tenseur d'Einstein proportionnel au tenseur d'énergie et à impulsion de la matière. Mais dans ce tenseur d'impulsion de la matière, tu n'inclues pas la particule elle-même, la particule test.

30:00 Tu regardes, sinon tu es obligé de te demander qu'est-ce qui se passe... interesting, but which is not resolved. But it doesn't change with the dynamic geometry? Because if I look at the wheeler, it was actually a little... But when you talk about the changement of the geometry, of the space, it's the same type of change that the movement of particles? And what is these two changes? If you don't know about it, I don't use it anymore. I don't use it anymore. I don't use it anymore. I don't use it anymore to explore the geometry of the space-temps. When I have an space-temps, it's something abstract. Now, what I'm interested in is that if I... For example, in this space-temps, I'm going to ask what are the rayons lumineux in this space-temps. I dessine tous les rayons lumineux possibles ça va me dessiner une espèce de tissu et en regardant l'ensemble des rayons lumineux ça va me donner une ça va me permettre de reconstituer la géométrie qu'est-ce qu'on fait finalement en cosmologie ou en astrophysique ? l'espace-temps, je ne peux jamais observer l'espace-temps directement en revanche, qu'est-ce qui se passe ? on a des objets lointains qui sont des galaxies, des quasars, des trucs comme ça which are in space-temps, which envoie the rayons lumineux which are propagated in space-temps. And I, I receive these rayons lumineux on the maritime or on the telescope or on the radio-télescope. And if I make some reasonable hypothesis on the objects lointain, to say that these objects have the same proprieties as the objects here or with a certain evolution, they are parted in a relatively regular way, it means that with these hypothesis plus the information that I receive on my telescope in general, I reconstitute the tissue of the geodesic and it's exactly that the principle of the test cosmology it's in analyzing all these geodesics of longitude nulles that I arrive I reconstruct more or less the geometry of space I can't reconstruct it completely, but I can have some information on it. For example, the median curve. And today we know that the median curve,

32:30 it has a matter of lambda that we call the constant cosmological, which is not what we expected. And we reconstitute it like that. So, it's finally the spaces-temps the most simple, very caricature, so that, obviously, the space-temps of Minkowski is like this, all the spaces-temps of Robertson, or the models of Friedman-Lemaître who describe what we call the Big Bang are like this our cosmology is described in this way and we have no reason to think that our space-temps is more complicated that our universe is more complicated at all at the level of global it is to say maybe there are here these little these little irregularities that we don't get into account. If you want, it's like when we talk about the surface of the Earth, we'll say, in gros, it's a sphere. Well, it's true, there can be a mountain, a cavern, and all that, it's quite complicated, even from the point of view topology. But in gros, we make a lissage and we say that it's a sphere. When we do the cosmology, it's the same. We can have an space-time which, localement, has a bizarre propriety. Even there are noirs, there are noirs, maybe there are singularities. but when we do a sort of lissage, we don't forget all this we don't forget the form global and we don't describe it by a simple model called the model Friedman-Lomert because it's them who gave us the equations in the years 1920 and these models entrent in this class there this is a model very simple on can in these models define a sort of function temporal which we call the cosmic why? because there is a of the space-temps by the hyper-surfaces of the genre of space. And so at each hyper-surfaces of the genre of space, we can associate the value of a certain temporal function that we call temporal temporal, because in addition, we can impose that this temporal function coincide with the temporal temporal of certain observators, that is to say the observators of certain observators en chute libre là-dedans. Donc cette fonction est bien définie, même si elle n'a pas toutes les propriétés d'un temps. Là aussi, on lit beaucoup de bêtises, parce que ce temps cosmique,

35:00 quand on parle de l'univers, de l'âge de l'univers, etc., c'est bien défini, mais ça n'a pas les propriétés d'un temps, dans le sens, comme on parle du temps newtonien, ce n'est pas un temps qui s'écoule de la même manière pour tout le monde, etc., il faut faire plein attention à tout ça, mais ces modèles sont quand même relativement simples It is defined in a way unique. The cosmos? And it depends on the conditions. Because if we can... Like our space-temps is very simple, in fact, it obeys to what we call the principle cosmological. What does the principle cosmological mean? In general, we can say that L'espace-temps admet des sections spatiales à symétrie maximale. Les sections spatiales à symétrie maximale, on les connaît, c'est forcément R3, S3 ou H3. and we can impose, in this case-là, that the feuilletage is done just by these sections spatiales to the maximum symmetry. In this case-là, it is unique. Because, of course, we can feuilleter by n'importe what. C'est mon principe optimale, est-ce que moi j'ai un principe optimale ? Ah oui euh... Non, si tu veux, ce principe cosmologique c'est un principe, c'est-à-dire ça veut dire que dans un premier temps on va supposer les choses le plus symétriques possible, alors on pourrait dire aussi que l'espace-temps est à symétrie maximale, space-temps in symmetry maximale, there are three possible, which are Minkowski, where the curve is null, decyther and anti-decyther, where the curve is positive or negative, it depends also on the signature that we have chosen, but decyther is an hyperboloid 4 dimensions and anti-decyther is an hyperboloid like that, today we think that our space-temps it is pretty close to Citer, as you know. Why do I say that? Ah, yes, it is a problem with the concept cosmological perfect. So, obviously, we are going to build the models, first

37:30 the most simple and then the most complicated, with the idea that the first simple model simple that we will describe our universe, we will choose, we will not complicate things. So, first, we would ask a sort of concept cosmological perfect, which would say In this case, we know that it is not Mikovsky because it is wide, there is no material, so it remains docitaires or anti-docitaires. In all rigueur, it is impossible because docitaires or anti-docitaires are universes vides. For an universe that is rigoureusement docitaires, there is no material at all. So we are not really docitaires, but we are perhaps not far away. So, then, we're going a little bit further. Since our universe is not conformed to the concept cosmological perfect, it's a little more complicated. The concept cosmological is very short. The space-temps is not at the maximum maximum, but it admits the sections spatiales at the maximum maximum. And there, we construct... On peut montrer que si on admet ce principe et si on est dans le cadre de la gravitation, de la relativité générale ou même dans un cadre plus large qui est suivi de n'importe quelle théorie de la gravitation de type métrique, à ce moment-là, on a une contrainte très forte sur l'espace-temps et on peut le décrire par ce qu'on appelle la métrique de Robinson-Walker. Well, it doesn't matter what you want, but it's the most general metric that allows us to describe an space-temps. It's the equivalent of the equivalent of the space-temps that we call Friedman or Lemaitre. And we build models with this, and among these models, there is a whole family that works very well for describing our level. That's what we call the Big Bang models. Well, these models of Big Bang, they are not completely determined. There are a whole family because... Well, we can very well see how this family... ...se parametric, if you want. Because in an space-temps, like this, we can always show that there is a system... We can always find a system of coordonnées where this metric will be written like this. dt2 moins une certaine fonction comme ça, dsigma2

40:00 où dsigma, c'est la métrique justement d'une section à symétrie maximale donc c'est la métrique soit de R3, soit de S3 soit de H3 donc finalement, on voit que la simple hypothèse du principe cosmologique avec une théorie métrique nous dit que la géométrie de l'espace-temps de quoi ? Que d'une certaine fonction, r de t, qu'on va appeler le facteur d'échelle, parce qu'il nous indique comment, en gros, la longueur de quelque chose va augmenter avec le temps, et puis du caractère sphérique, hyperbolique ou plat des sections spatiales, dont on sait qu'elles sont à courbure constante. Voilà. Et donc, à ce moment-là, ça c'est a function arbitraire, and then there is finally the sign of the curve, if you want, which we call k. So, these are the two parameters of these models, such that R, obviously, is a function. So, this function, we will, at his turn, we will parameter by the current value R0, and we can always choose the coordinates of the way that R0 is the rayon of curve of the space space, and R0 will be the rayon of the space space, suppose that the space space is not null. Otherwise, R0 would be a paramètre arbitraire. The first derivative of this function, logarithmic, R point over R, today. We have a development of Taylor, if you want, to reconstruct today. This is called the constant of Hubble. The second derivative, R second over R, today. so we are going to parameter it like this, for that it is without dimensions, and we call it the parameter of deceleration, this is measured at a better than 10% près, this is measured at a better than 10% près, and then there is something that is defined by the third derivative, we call it the parameter W, but I would say it is measured at, I don't know, maybe 100% près, but it's already not so bad. And what's interesting is that it would be maybe minus 1, and if it would be minus 1, it corresponds to what we call the constant cosmological. Well, I didn't want to enter into all these details here, it was simply to talk about the concept cosmological and the concept cosmological parfaits,

42:30 which was to respond to the question. Just, je vais peut-être simplement terminer, donc il y a toute cette hiérarchie de conditions de causalité sur une variété, donc qui commence avec l'hyperbolicité globale. causal, s'il n'existe aucune courbe causale fermée en aucun point. On va dire qu'il est chronologique s'ils admettent aucune courbe fermée de genre temps nulle part. La causalité est forte, c'est une condition un peu plus technique pour la définir, mais en gros, ça va revenir à dire que, non seulement il n'admet pas de courbes causalement fermées, mais il n'admet pas de courbes causales qui soient presque fermées. C'est-à-dire qu'il n'admet pas de courbes causales qui reviennent dans un voie d'usinage très petit du point initial. Bon, tout ça s'exprime de manière technique, évidemment. On définit aussi la causalité stable, etc. Bon, mais ce n'est pas ça, finalement, qui est intéressant. Donc, ce que vous avez bien vu, je pense, c'est que, tel que je définis là, la structure causale d'un espace-temps est équivalente à sa structure conforme. It's to say that it's completely determined by the structure conforme. So we're going to say that if two spaces-temps are transformed conforms to each other, that means what? That means that in a row, I can represent them on the same variety, with two metrics that are related by a function. At this moment, these two spaces-temps have the same structure causally. But, precisely, the paper that we saw the other day, it is a paper, I didn't know that, which suggests to use a notion a little more large of equivalence causal, which is the following, which is

45:00 And so, we can define a kind of causal structure more large. It is to say that we could have an espace-temps where we can say that they have an identical structure even if they don't have the same structure conforme. an identity causal more strong than the identity of the structures conformes. And in general, the thing is the following, I will try to say it in a way intuitive, C'est que, en gros, supposons que j'ai deux variétés, M1-G1 et M2-G2. Bon, la première chose qu'il faut aussi se rappeler, que je n'ai pas redite mais qui va de soi, j'espère, c'est qu'un espace-temps, c'est une variété modulo tous les difféomorphismes. un difféomorphisme phi qui transforme une variété différenciable M en une variété M', c'est une carte inversible qui, à chaque point M, associe un point M', vous savez qu'un difféomorphisme agit canoniquement sur les vecteurs, les fonceurs, les formes, etc. Donc, il va agir canoniquement sur la métrique G, puisque c'est un tenseur, et bien la variété M, G et la variété, disons, phi M, phi étoile G, du point de vue de la relativité, sont la même. I would say even, from the point of view of the variety of Riemannian, it's the same thing. If there is a diphéomorphism and we transport the champ to any other champs, to be more exact, it would be ajouted, since there are all other sensors

47:30 that represent the material, here, it's the same, we transport it as we need with the diphéomorphism.