Einstein's field equations & SDG / discussion (contd.)

0:00 But it usually means an open set, or it means the infinity, and so on. By the way, there is an intrinsic topology that was discovered by Penrose, and this topology says, when something is open, it's just between, really between any arbitrariness of science. And then the topology says this is open if and only if you are given two elements, two elements U and X, either U is different from X, or X is in Q. So, this is remarkable because it's nothing. This usually, you see, usually one sees intuitionistic logic as something just very heavy in the world. The reason is something that, you know, limit us. In this case, this doesn't limit us but allow us to define the notion of intrinsic topology and which you can show that has very nice properties and so on. discovered by you know so then we have whenever I say locally I refer to and in the case of well-known the top all of other processes I will mention some of them at the end this coincide for the particular case of manifold and common coincide with this usual notion that we have oh i don't know how i'm acting I imagine that there are some actions missing. Don't say there is one. Well... This is exactly one of the bottoms, the last ones,

2:30 for the last few things. Yeah? Here? And please, please, Madame, please, and the last thing for it, of the fall. Yes, I, this, I, this happened when I was, yes, so this is six, this is seven, this is seven, okay. I have to go to that, I have to go. Locally geodesic, oh yes, action four. Okay, so action four says that there are, oh, then I need action three says that there are enough of this gadget, mainly this locality reference. geodesic reference frame. So in the sense that for every u, which is in the fiber of an electron, such as is different from zero, then there is one of these locally which is starting at u is equal to, and starting at zero is equal to u. So, of course the notion of gamma being integral curve, it is over there, namely, if you compose Q with gamma, then you obtain the velocity of gamma, at least it's just a notion of integral curve of the vector-fifth Q. Axion 4 is that it depends on the notion of neighbor, so I have to define a neighbor. so we usually we saw what the neighbor was in the case of Newtonian physics so I take gamma and this was a neighbor was a vector field along gamma but here we have one further property is that the lead derivative of this vector with respect to Q is equal to 0 so we define this this is a relative notion of neighbor in Q so the derivative derivative of omega of W is 0 or this

5:00 is the more intuitive notion that we can formulate here is that all the W of H are integral curves of Q, where the W of H of T, there is another way of defining, is W of T applied to H. And so this is the notion of neighbor now that we have. And axiom 4 is that there are enough neighbors so that if Q is one of these vector fields which are locally geoletic frame then r is an integral curve of q u is an element with the fiber then locally there is a neighbor of gamma and q furthermore this is unique now i hope that we are yes then very good so action five is different measure also like the others it's not a physical nature it's mathematical so that if you have a vector field and you have an integral curve of this vector field and a a neighbor w then this neighbor can be represented as the composite of a local vector field w tilde which is there with a composite gamma and furthermore we have here that they need bracket q w filter is equal to 0 so you remember the neighbor was from there the lead derivative is equal to 0 and here we have the bracket is equal to 0 and now we define size which is a linear transformation from M of X into M of X you remember how we can find that in the case of Newtonian is it so it's quite similar because we have to replace here the derivative by the covalent derivative second covalent derivative

7:30 of gamma dot of B of B, where B of B is a unique neighbor of gamma in Q of zero, at zero. And this is, therefore, the analog of the linear transformation that we had before. the linear transformation that we had before was the linear transformation that showed the relative acceleration of the particles or the apples or the take to myself here this is therefore the the acceleration of the neighbor of gamma with respect to gamma dot so by analogy to Newtonian's case we equation and the trace of that is even closer. So that will be the Aniston-Blocking-Feed equation. Now, the trouble with that, so this is a very simple formulation which in some way is really obvious what has been done in the Newtonian case, but there are some troubles. for instance it depends on too many things the main trouble is not that one but it is because what we want to tie we want to tie the curvature with the with this equation of in other words we want to see the trace of these connected to curvature, because up to now the curvature have not entered into, just in a complicated way, but you'd like to see a simple thing. And here it is the main theorem. The main theorem is that if Q is a local geodesic reference frame, and if this is an integral curve of q and w being a neighbor of q then expression phi of e which is explained in this way here, it is equal to the Riemann-Christophel tensor, RMS-0, RMS-0, RMS-0.

10:00 So now we connected this formation, so to speak, with the Riemann-Christophel tensor, this linear transformation with Riemann-Christophel tensor. Now, the proof of this is extremely simple, because it is a simple computation starting from the following thing, starting from properties that the derivatives have, and the definition or a property according to how you start of the Riemann-Christophel tensor. So, these two are the ones that we are going to use. So, by axiom 5, we can find W tilde, such as W in the synthetic table. The big bracket is equal to 0, which is an extension of W. And then, starting from that, we do the form of computation. This is, by definition, nablaq of nablaq of this. And this is nablaq of that expression. But that expression, by using the first property, is equal to this. But remember now that this is zero. And so this is equal to this. But then, using the second property, this can be written in this way, unfortunately. Now, we have here a number of q, q, you remember that was zero, because q was vocally genetic reference frame, and this is equal to zero because the bracket, the bracket is equal to zero. Therefore, we obtain that this second derivative is just r, q, w, to the q, and then you restrict this to gamma, and you obtain what I had said before. Now I need a definition, the definition of a Rishi tensor, so up to now, you see, there have been no indices whatsoever. Now I define the Rishi tensor as follows.

12:30 Take u, v, two elements in the fiber of an x, and I take the basis of m of x, that, you remember that was a vector field and I think it's dual basis and I define the rigid tensor as being one half of the sum of Wi this vector this tensor in same with the Rieman-Gristoffel-Dentos plus the W of that, the same but in which U and V have interchanged. You know why I do that? Because in this context, I cannot prove that Rieman-Gristoffel-Dentos if I define it in the classical way, which is just this. Classical way is that. It's symmetrical. But anyway, in the classical case, then they coincide. And if I look now what is a trace, sometimes the trace is recited in, but the trace is just that. So that shows that what I had. I guess that was true. So that finishes the proof, and therefore what How can we write that, just Rishi? It's equal to zero. That is the result of the computation. And from here, with some algebra, you obtain that Rishi is equal. And this depends on purely algebraic markers. So, now, since I talk about synthetic differential geometry, let me mention something about the models of synthetic differential geometry. So, the models of synthetic differential geometry are topos models, and maybe I will explain a little better what I'm doing there because it's not very clear.

15:00 So, let me start with the notion of the Infinity Ring. so see infinity ring to think an ordinary ring is a ring such that we can can interpret the polynomials and variables right forever again and what infinity rings is a ring in which we can interpret the C infinity function in an variable from part N so this is there is a theory of the sense of here of rings infinity ring examples of these are the compels that I mentioned there for instance C infinity of rn that is the free C infinity ring in n generators and then you can divide C infinity of n by i this turns out to be also C infinity rings the morphism in this particular case become equivalent classes of math, smooth math, which have the property that if F is in I, then F composed with 5 is in J. And two elements are equivalent if and only if. When you project that and they are removed from every time now the coverings are covered of this nature you see whenever u of ultra is a countable cover so i'm using here the notion l to say that the the sides are not the c infinity rings but the sides are the dual of c infinity rings and so then L of A would be the dual of that, and of course we know by Johnnet that this can be written as H to the A. So then here will be the dual of C infinity of Arendt, and this will be the dual of this algebra.

17:30 C infinity of Arendt divided by I, and so here it is . and so whenever so these are countable covers and so in the sense that whatever this is a comfortable cover of this we put that as a cover and L of A here is a 12 equal to 12. Now the the model for synthetic differential geometry proceed by specifying what kind of ideals are there you know we consider to be there and then F, which is one of the simplest to work, is a topos of closed ideals, it's closed in the sense of Whitney topology. And the other G is another, the topos of germ-determined ideals. And so that in order for a function to be there, I mean, the germ of the function should be the germ of the functions of the ideals. And so you have the germ-determined ideals. and then the proposition is that both of us have properties that represent the archivist so let me look at some of the example that we already use in our theory and r for instance is the line so this will be the dual of the infinity of r and as a representable cluster is h to the infinity of r and this is the line d will be the dual of the infinity of r divided by x squared this is what they're usually called the dual numbers and is h to infinity of r divided by x squared, and that, for instance, in these are our functions, r of a, this is h to infinity of r of a, and since this is the three-in-one generator, this is just simply a, and d of a are the element of square zero, as we wanted it to be. As an example, let's see how the integration axioms can be proved in this model. So this is not the best form of integration axiom, I'm just giving that as an example.

20:00 So, suppose we have an F. An F is what? F should be an element of R to the R, but at the same stage, that's a whole point, right, but at the top of the degree we have different stages. So, at the stage L of A, well, what is that? Well, by Gionera, this is the same L of A into R to the bar. But then by adjoinance and exponentiation means that this is the same as R times L to the A into R. And so remember that R was H to the infinity of R, and L of A then is actually infinity of R, N divided by I. And then here the product transforms I think that include co-products at that level and this means that this is the same as the infinity of R into that and once again in one generator this means just another here so another f of t can be written as f of t x model of i star now the star is just i but in this bigger field right in more variable so here you will see exactly what i mean because i was given here the proof appeared so i'm going to define the integral of zero to one of f of pvp How do I define that? The size is 0 to 1 of f of t dx dt modulo I star. Now, I have to prove, I mean, this is the obvious definition, right? So you have to look at, you have to think of this as a function, being as a function in parameters, modulo some idea. So then there are two difficult problems because you have to first notice that whether the things in parameters work and secondly you have to see whether this cogitivizes by the ideal everything goes okay. And so here I am not finished because I didn't show that this was all defined because I could have shown other things here. So suppose that I take two things here, F1 and F2, but then by the condition here I mean the difference between the two of them isn't the ideal.

22:30 I start so I wrote I wrote this explicitly this means of the form a I TX F I X where F I is an I and now if I integrate this then I integrate this with a straight to 0 1 so I call this capital I which is just a function of X and so I have a function of X and here that I of X get outside so integrate And so this, I is an I, and this means precisely that this element is an I star, so that that shows that at the same time that this exists and is unique. Then you can check that all the other actions, for instance, something which, things like equation that first of all the differential equation have unique solutions this it was proved some years ago and and then I can prove all the actions that I needed for my proof of Einstein vacuum field equations in this model for instance in the model G or in the model F or more generally in the model When all legs are closed and so on, there are several other baguines. And I think that this is a good point to smile. Thank you very much for this question. Any questions? First of all, a small instance of port. Around 1950, Andre Ray wrote a long draft for Bobaki, where he intended to give a description of differential geometry in a spirit with, well, the technique was slightly different, but the spirit was exactly the same as the one which has been developed in synthetic differential geometry. and he wanted to take seriously the idea that you can have to

25:00 intelligibly close points and the idea is that, he considered the well, in your setup he considered instead of having this infinitary quotient, he considered only finite dimensional algebra, which is just I mean, something localized at one point. for instance what is called very algebra because I know about that it was in 1953 it was published well just a short account it was published that was published yeah and I asked him when he came to Montreal whether he had published or written something else and then he told me the story that he had written that to Bourbaki you're so wrong because they were writing on Bourbaki on the differential geometry and then he thought this would be a nice way but then Bobaki didn't like it well there were a long discussion there were a long discussion for some years I see but the project came down and well also I mentioned that one when people who like this there might say what I mean we're standing with the definition of chief of skills in the beginning. I mean, I came in one proposal which was very much inspired by which is to introduce a scheme as a phantom from the rings to the set. An Anchebag group as a phantom from ring to group. And this has been developed to some extent in the book by Thomas Hume and Gabriel. And so these things come together. But I have to say we were inspired by the other guy and in some way you see puts more or less in a more geometric cut but he had done more or less as a break yes right because he had an tense or something whereas the tensor of course here becomes exponential and of course our intuitions are really exponential much easier than with tensors and so on so but this certainly played a very important role and and in fact the first model beside i think if i understand correctly there was one model by you that of all the infinity uh rings with the canonical topology

27:30 yeah the problem with that was very was very difficult to work no one knows the canonical nobody knows what could happen so then i suggested to somebody i was working with the book that looked at this paper of of andre dale and so it was looking at this paper of andre dale that he has he arrived to some of these ideas for the construction of the model because the first model was just a model So, the slide was things like this, right? N times L of W, I think, right? So that the word of W were a veil-algebar. So veil-algebar are autographs of the form, one way of saying is the same infinity of R-N divided by an idea of I, which means every element is new important there so well there are several definitions so by the way i i forgot to mention so there are two books that i use extensively were very useful to me the one was by renee I'm sorry, from Louvain Leneuve, yeah, and it's called, so there are two versions of this, there. In French, there is and in English, there are basic notions of synthetic differential geometry. So then, one is a translation of the other. However, you have to be aware there are horrible mistakes here because Lavandon obviously didn't like lambda calculus you see and in lambda calculus we can't say exactly where the scope of things are so but Lavandon that was in addition credit he said well instead of putting this where x was you know outside the scope and

30:00 so on he has wrote X and with a bar I mean what further up the scope see two bars and so on and then different when this was printed as it that was translated then the person just put one bar everywhere he thought it may have been a mistake in the text for the French text so then there are some And for instance, in this relation that I talk about, you know, the curvature and so on, then this appears there with mistake and make a whole proof of nonsense. And then the other book is the book by Gunetek and Reyes. and this is especially works on models of intuitive differential general so there's an enormous amount of models and so on and there are for instance there are even models in which there are invertible infinitesimals and not only nil-potent infinitesimals but even invertible infinitesimals and so on So that these are the books that I use in this context. So these are called Models of Smooth Infinite Malanity. But we chose a couple, as you had mentioned yourself. It's important that there was also a major work by Taylor. along in the 50s, I think, and even there were other attempts in history to use the new potens, which was clearly being used in practice by engineers and scientists to try to make this into a rigorous theory. For example, around 1900, when the Conti d'articolo matematico di Palermo, there are attempts to describe mechanical problems in this way. And recently, actually, finally, the five mathematicians have noticed that they asked you to review for the Cyan, another book on the subject.

32:30 Of Anna Kock, yeah. So I recommend people look at this review just for the sake of realizing that finally... Yeah. Oh, that's a great fact. Well, Anna Koch was his first book, and by this you are talking about the second edition. Second edition. Second edition to the book, yeah. Speaking of engineering at this point here. Whitney wrote a book called Geometrics Inclusion Theory, which Mark Rothenford has never read. The consequences of the including form has been Dennis Sullivan's rational homosophage theory, Joseph Dujo's sort of the street match theory, and later used by Winnemieu at a proof of re-finger conjecture, and various places, Federer and minimal surfaces. But the formula, this book's horrible, and he'll try to clean it up. coche problem and all that. But the formula, the key one that we love in finite elements in engineering, goes back to Andre B's 1952 paper, Sir Vizio and Isara. So that is a different paper, you're saying? Yes. No, it's a different paper. I also, I know that, yes, only I know that paper and even, I think, if I'm not mistaken, we put that into our book because we have uh the ram you know we would prove the ram theorem yeah in this context and and then well i'm not very happy about what it came out but then we we put upon point this this proof by the veil and this proof of veil is it's a really beautiful proof and it's very it's extremely constructive, you see, contrary to other proofs and so on, so that this could be carried out immediately, you know, into synthetic differential. Well, some people say it's the birth of a spectral sequence in all this, but it's wonderful, it serves me and me very well, and basically we use it for finite elements, and we engineers don't understand what I just said, but this is the end, the basis of it. Basically when you have something like back to the equation, it's stupid to say a differential form is like a scalar function, you really get into trouble.

35:00 If you say it's a differential form, it's a co-chain, and you use the Whitney or Vape formula, and you find out which works wonderfully. And so going back to this history, I wanted to go back because it seems like the problem started with me. So, from what I understand, everybody and their brother had a cohomology period named after them. So, and then one thing, as I understand, the Andre Day wasn't happy about, groups, but you don't have a correspondence on the co-chains. And you want to see more things, more on the co-chains. So going back earlier, so you see that I interpret this paper as a reaction to Albert Steenrod axiom. But the whole thing would not be a problem at all. I think the whole thing got interesting with Durand's theorem. And Durand has this beautiful paper, he wrote 1975, he presented in Romania, on how the, so Elie Kartal proved, worked out the concrete polynomials for the class of Lee groups, the proof was correct, modulo the isomorphism between homology and co-homology. And Durand proved this theorem basically to make that isomorphism. He went to Elie Kartal, he said, off the edge. And Leveig, Durand's advisor said, no, this is mathematically proofing here. To me, it's astounding that Stokes' theorem on medicals got proved. And then the question was, how general is this? Because we have triangulations, what is the role of this? So, to me, this is back to the history of Stokes' theorem. Yes, and I think I should mention also, I'm sorry that I forgot to mention, another source that was Edithman. Edithman, Jett. And then I, you know, when I was really at the very beginning working on that in 75 or so, on syntactic differential geometry, then Edesman invited me to Amiens to talk. And then he told me that this is the way he had thought of that, because I thought at that moment that we're generalizing Fermat, you see. And so I had called my title, you know, Calcul Inflictimale à la Fermat.

37:30 And then Erison told me that that's the way he thought himself that he was doing with the jets. Now the trouble with the jets is what André Veil pointed out. The jets are like the bricks of something, but you have to put them together. Like the prime numbers, but you need numbers. And so they're very loud about what you get when you put together. There was a lot of interaction between Eversman and André Veil. That was the time when Erosman was still part of Bobaki. He left Bobaki in 1948, and that was about that time. So there was a lot of interaction between Andrei and Erosman. Well, I hope that this could be documented better, because in some not-so-distant fashion, the archive of Bobaki will be able to work. I'm breathing on that, but it's difficult. And the only thing I found is that I will donate my own archive, although I have no permission. I donate my own archive, and then it contains a lot of drafts, and I don't have to have any permission. a final footnote sort of a footnote is that the great Sophie Armand Manifold he feels it was really in a paper by Volterra in 1889 and nobody knows this paper I don't know about it I don't think you know about it and the Dresden Dickers pulled it out in the 70s and he got inspired through various things and I'd like to understand that So Grethly was inspired by taking the United paper of Volterra, and, uh, I just wanted to know something about that, but it was pointed out to, by Diram, to Hiroshima, in fact, that there is a paper of Volterra, which anticipated many of the results, which they thought you were doing, you know, they anticipated, but there isn't a notion of manifold in it, how important it starts to work out. well I'm sorry not really done because that was a vacuum right the Einstein

40:00 vacuum field equation I the truth with equation of Einstein I don't know how to do that without metrics. Because it depends. You see, the full field equation of Einstein has, on one side, has this sensor, which is the Einstein sensor, and this is defined in terms of the metrics. I have a question, which is related. It's a very naive question from your point of view, because I don't know synthetic difference between geometry, but from physics point of view, we have two space-time structures, chrono-geometry, chronology and geometry, inertia gravitational one is represented well there's gravitation fields every by connection let's say symmetric connection to make it simple that's what I want yeah you're right so we think that's a symmetric connection the chrono geometry could be either the Newtonian chrono chronology with the absolute time and plus Euclidean geometry on the spatial hypersersion or it could be Lorentzian or it could be general authority and signature but very one can start with the symmetric connection. One can define the equation of geodesic deviation and interpret this physically as tidal forces without saying anything about the chronology write down the Ricci equals zero for the empty space case we can treat the matter cases without making any decision as to what the chronology is. One can then you probably know Elie Carton and Friedrich's impendent gave four-dimensional formulations of the Tonian theory, in which 3 g times to equal 0 are the field equations. So, up to this point, one has made no decision about the chronogeometry. So it's not clear to me, one could go with, in other words, one could use to develop the Tonian theory or the general logistic theory. Everything on the inertia the inertial gravitational field side, the connection side, would be the same. It's not saying to me where in your formalism the choice is made, or is it still open? You see what I mean? I think I could choose the component of the geometry and still say everything you said

42:30 about the inertial gravitational connection, or I could choose the general optimistic metric where the chronometry and geometry are inextricably united, and still everything on the right-hand where or whether you have made the choice in your formulation. I really couldn't stay because I... Okay, don't... If you were not leaving, I would have raised in the general discussion. But since you're leaving, I have to ask you that. But I think it's interesting to investigate whether you really are forced to make a choice. With most of your formalism, it seems to be not. only at some later stage you have to say well if it goes in a Newtonian way or it goes in general but this is an interesting point not just theoretically but practically as well well practicing sort of calculations because as I showed in a paper a few years ago one can use this Newtonian background theory as a starting point for a approximation procedure one does not have to start the approximation from Mikovsky's based on we'll start with this as you know the total gravitation to be in the fourth dimension formation is a zeroth order approximation so you already have a non-flat Riemann tensor even though the space is still flat and then you can you have to now expand the metric and the connection independently and then relate them to relations which give you the compatibility and this procedure has many advantages what has in the zeroth order approximately already has full military gravitational period. So it doesn't have to assume the gravitation period is weak, as one does in the usual approximation procedures. I will be interested in... Yeah, I'll send you that. Well, let's thank our speaker again. Thank you. Bon voyage.