Dinner conversations

0:00 And so, the main object is the topos. The topos is underlined, so I'm using the fact that we can use the theoretical language under the condition that we can use some actions like I've shown before, not abstract actions. And there we will have an object where the lines are unlimited. We can construct from R the equivalence x square is equal to 0, and then we have the axiom here that says that it is in small and incurred straight lines, and in terms of formulas, it says that whenever we have a couple, they give us reals, and then we look at the function that every decimal has the same a plus bt. Then this function is there. And as an example of that, we have the definition of derivative, because if I take that point x0, so we look at the function around x0, f of x0, and then by this axiom, we are given a couple of three b's such as f of x0 plus a plus dd for every dd. If we do d equals 0, then we have, of course, the formula f of x0 plus d is equal to x0 plus b times d, where it would be b, and b is what is called derivative. So this is the slope of the straight line, which is equally small. And so the usual groups can be divided. I remark that every function is smooth. Secondly, the arc is not filled, and the classical logic is compatible with this axiom.

2:30 But, as I say, there are toposes which are, therefore, really consistent in this view. Now, as I say, art is not a field, but it is a field sometimes, a field in the sense of a cog, which means that if not n elements are all equal to zero, then at least one of them is a unit. An important axiom will develop some of the linear algebra in the topos, and as I said, there are topos models with revisionistic logics. Now, this is a very, it's a formulation that we will talk about that with the existence of this topos model. Let me mention some infinitesimal objects. And of course here we have, well this is over-optimistic because there are topos models. I really don't know whether there are topos models, but this would be precisely problems of analysis for what I'm going to do. Of course there are topos models, but for what I'm going to do, and for the time being, there are no, I cannot prove that there are topos models for all the actions I need. Now, some infinitesimal objects, V is one of them. And V2, which I kind of also defined it. So these are the steps of V1, V2 in these as a V1. So I'm sorry, this is V2. That's a V1, V2, 0 to 0. And another example is the Liebhaber. Okay, a very fundamental notion in synthetic differential geometry is the notion of a microlinear space corresponding to the notion of generalized manifolds in some way. And in order to define that notion, let me start. With the diagram that you see here, 1, 2, and 2, and these are I1 and I2 are the inclusions of B to the first and second. And then this is not a commitment, but our beliefs that this is a commitment in the sense that if you put formal definitions of commitment...

5:00 How do you show that? Well, f is of the form a plus cd by the box of your axiom, and g is of the form a plus cd because remember that f of 0 is equal to g of 0. Now, such diagrams for which r believes that it's a collinear, we call that an r collinear. And then we define m in a collinear if m believes that all collinears are collinears. All r collinears are collinears. So as an example, let's show that m is a collinear of the power of this. Then m to the e is just evaluated at 0. It's a vector bundle, a bundle that will show that addition is correct. So, m is the unique limit. If I put in this axiom, in this diagram, I put m is equal to r, then this is also true. And this, if you think for a moment, this says the map from m to the b cross over m, so the fiber product into m to the b2 isomorphism. Now this, we take the diagram to this.

7:30 We add that portion to this diagram, so m to the diagonal, m to the d, and then the addition as follows is this, or the same fiber, and we can add them and we get the point on the same fiber. So, in the particular case in which M is R, for instance, then we simplify this addition, because M plus, F plus G over D, by this definition is H, the H that we have obtained over there, C, E comma D. So this is A plus B plus C. Now the properties of microlinears, that R is microlinear, and limits of microlinears are microlinear. And if f is a micrometer object and x is arbitrary, then 10 to the x is micrometer. And to the subject of Newtonian gravitation. Here I'm going to make a parenthesis. So to say the following, say that, you see, what Einstein does when he tries to derive his Laplace equation of field theory, And so Einstein started from that and he proceeded then in a way that I would consider rather formal because he looked at this operator in the left hand and he said what should take the place of this? What should take the place of this?

10:00 He deduces this by very, by formal consideration. For instance, he said that only second derivatives appear in the definition if we identify five with a metric, because he introduces a metric, and then he says, well, then everything should be a tensor, and therefore only second derivatives at most of this tensor. I was always surprised about that, and I was wondering whether there was something that was more physical, gravitational equation. And then I found the book by Sachs and Wu called General Relativity for Mathematicians. Nice physical interpretation of this. And the physical interpretation is the following. You see some Gedanken experiments. You imagine that you have a famous example of a lift that is freely falling. So the lift is considered to be a laboratory in which you can measure things. Then you imagine that you are sitting here, a lift freely falling, and then you have...

12:30 So suppose that you have, here you are, and here you have an apple, right here, but look, this apple is closer to the center of the earth than you, therefore the traction is greater, and therefore with time this apple would go this way, similarly if an apple is on top of you then the apple would go in another way. Now what they do is they look at the The gravitation of every apple situated at a given distance, let's say, the relative gravitation with respect to you, you look at all these gravitations, you make an average of those, and the average of those, all the gravitation of the apple with respect to you, and these are now to be equivalent. So I like this formulation because it uses well understood notions by the notion of... I'm sorry, like the notion of acceleration. And then what these people tried to do is from here to proceed now to obtain the actions of general relativity. To tell you the truth, I did not understand their derivations. And then I took the way out. So to become very, very simple-minded and then to say, well, I don't care about that. I'm going to see my way. How are we going to proceed to consider to replace R3 by space-time rather than this complicated definition that we have, and then proceed accordingly? I hope that everything I did is correct. If there are physicists in the audience, something's wrong.

15:00 The only thing I can say in my paper is that at the end, I got the equation of Einstein. So I imagine that, of course, this doesn't say anything. Everybody knows Deodorus Cronus among others, right? The stoic, the stoic school. So anyway, we proceed. So what I'm going to do is to give an mathematical formulation of the apples. Or rather, I mean, I'm going to say what the apples are, and then what you are doing there. And you see these people say the apples should be very close to you. So I take apples as being infinitely close. So, then there is a picture. So, I imagine that here you are, and then I'm going to take a round U-sphere, and I'm going to take all the apples which are at the distance h, and so, by definition, then this means the set of h U, such as U-sphere, and h-ish. So, these are the points at the distance h from 0, from U. I see the trajectory that you are describing, gamma, you describe, and so I imagine that stars here, I have an ax here, and then I imagine that gamma zero, your position is x, and then the velocity is u. So I can imagine u, so u being about s2, so that the velocity would be unique, and so this could be in all directions. We should start at this level, let's say, and then let's start, therefore, at the point x plus h of u with the velocity u.

17:30 And recall now that the velocity in the trajectory, I'm sorry, acceleration is equal to minus gamma of phi, so of gamma of u. So phi is the potential, okay? I just look at that, I apply this to this guy, to the apple, and then I take the difference to see what is there, and so if I take the difference with a little Taylor series, you can show that minus delta, I call that the difference, minus delta u h is equal to h times phi of u, and where phi of u is the linear transformation All of this is very simple and so now let me state the proposition. It's a proposition that can also be shown easily, is that if you take the mean relative acceleration, so you have to define what is the mean relative acceleration and the way you do it is... The sphere has, of course, a surface of 4 or 5, so that in order to take the mean or relative acceleration, you divide that 4 or 5, and then you add all this. Sigma is the surface form. And then you can show that this is equal to h3 of nabla squared. So, this means that if this average is 0 for every h, then this is...

20:00 Because if something is universal, you can't solve it, you get this. So this is the interpretation, this is the formulation. The distance h from 0 is 0 for all h in this right here. It's pretty simple, but notice that another fair part is the trace of psi, where psi was the linear transformation that I defined before. And look, here we are taking all the h's, therefore it seems more natural. Rather than taking only ages, take families, that gamma, age, and t equal age values of this family, and therefore we can consider there is a one-to-one connection relation between this family and the maps that I call values of u from r into e to the d. We satisfy the property that we give u of t by definition of age equal to gamma. Okay, so then, why I do that? Because this will be something that will play a crucial role in the PhD. Okay, so, in some way, this was E, I forgot. Ah, E was the space. Ah, okay. And then, this plays, as I say, plays a role in the... You see, some of you can imagine as being just this bunch of families, so this is a bunch of trajectories. So these are, so to speak, all these trajectories, you can think of these as being the neighbors of dual trajectories.

22:30 Unfortunately, there is no good name for that, and both names I can't really call that any more. Notice that this, from the property, is that the W of U makes this diagonal computer. Because, why? Because gamma zero is gamma zero of U, and you can check that gamma zero of U is just dual. Objects like that are very well known in classical differential geometry, and they are called vector field along. And for that we said you have here a map, and then you have here the vector field along. Not a vector field, because a vector field should be defined on the whole thing, but it's along a map. This is called a neighbor, and so we're going to use neighbor in the general case. So Einstein's theory of gravitation, I'm going to distinguish as it was said. Here, postulates I wrote in the sense that the postulates that Einstein postulated about the space-time. And this is something, of course, cannot be proven on top of the error now. On the other hand, I will give some axioms which are of the nature of mathematical axioms and these can be tested. Okay, so the postulates are the following. Space-time...

25:00 Let me say that Einstein assumed that was a four-dimensional Riemannian or semi-Riemannian manifold. I think that one of the main contributions that I have here is that I'm going to deduce the Einstein equation for the vacuum case without the Riemannian metric, just using parallel transport. So the second is the free-falling particles, so this is very clear, everybody knows what the geodesic is, and so if you just free-fall them, it describes the geodesic, and then the third, we have curved manifolds, interact according to Einstein's filter, and these three equations, let's say, as Wheeler has put it, matter tells space-time how to curve, and curved space tells matter how to move. A fundamental idea is that gravitation, for instance, there is no action at a distance, that was one of the troubles that Newton had, but then he said, whoever believes in action at a distance, anybody believes in that, and that's what he used. By the way, this seems to me one of the reasons, once I heard that theory expressed that this was one of the reasons why Newton was planning to write on Newton's...

27:30 Mathematical principles of natural philosophy, and apparently what he wanted to write was 60s of natural philosophy, but then he could never explain this satisfactorily to his mind, and then he said, well, let's be happy with the mathematical principles only. That I heard, I don't know whether this is known. Okay, now let's proceed to the mathematical formulation. So in the context of SPG, I'm going to assume that spacetime is microlinear space with a symmetric connection. Well, symmetric connection, the best way of thinking of that is the easiest way of thinking of it. If you want to be a part of the curve, then you can put, you can transport this vector parallel to itself, so this means it's just a transportation, parallel to itself, it's just to be alive, it's just to imagine what you're doing, and satisfy some conditions in reality and so on. So this is, it's just a fact that you can put. You know, for instance, this is interesting because this appears even in very elementary treatment, for instance, in the notion of acceleration, when you study, because it's moving, and then you can define the velocity by just taking the difference between this and this, but this is the same, it's at the same point. But when you want to define the acceleration, then you are in trouble.

30:00 Because the accretion are two vectors which are in different starting points, and so even there you are assuming that without saying that that is, of course, in the obvious, but I'll stop. And then I assume that this is n-dimensional in the sense that the fibers contribute an n-dimensional vector space. Here I'm using vector space in the sense of... It should be really called field, but remember I call that, I'm assuming that we have a field in the sense of we can, and so this is n-dimensional, things which are n-dimensional, which are not monocles, coming from monocles, but instead the infinity, right, is n-dimensional and doesn't come. Now, the law of motion can be formulated as follows. You can define, for every gamma, you can define for every mass, therefore, for an eyeball, for every... You can define the derivative as being a map from r into m to the d, and this is the finance of all that. So this, what we know now is, this is a vector field of a gamma, right? We satisfy the gamma d of b by definition of velocity, and now what we require is that this satisfies this angiodesis. And geodesics can be described in either two ways, it's defined in terms of. So what this says is that if you take a vector field and then you make a parallel transport, if you take a vector tangent in the manifold on a curve and then if you make a parallel transport and if you arrive to another tangent then.

32:30 You will say that this, I'm writing this, all this is real technical stuff, but because this will appear later, it makes you, for the time being, it's not important, it's just technicality, it's just for the job. So once again, maybe this will be jealously, I'll have a transport, I'll transport this, I want to convert, then I always say something which is a tangent to a version, so I start with something. Okay, three. This is the real main objective of this talk. So this is how to get Einstein's equation. Now people use physicians for deductions. They don't use it in the sense of a magician. It's not a deduction, but it is. So this, all this goes... Some deductions are better than others because less terminology, less choice and so on. So then this is a deduction I'm going to obtain in this sense. So we're going to start by defining the logical geodesic reference.

35:00 So the logical geodesic reference, friends, will be the analog of this set of, remember, of these trajectories that I started with in the beginning. So you remember these trajectories, well, we are assumed, we are in the sense of Einstein, these trajectories should be geodesic, you see. And then here I write my first axiom that should be called axiom four, but for some reason, and it says that there are enough local, and what this means is the following, if I take u in the form of zero, remember that I have here a field in the sense of cochlear for different from zero can be interpreted as saying that at least one of the components is. All of these are such that, so suppose that we start a tangent vector at x, zero, then there is a vocabularies that we explain, which starts at that, which for zero gives you, and then the fundamental notion of neighbor. The answer we had before, now there is a little, is that, suppose I have an integral curve of q.

37:30 Then I'm going to find a neighbor that I'm going to define now in Q. So this will be relative. And then it says that if along gamma, such as the elitist, the beta, you know, invoke our integral curves, I think they call W, H. So then this is the axiom 3. And now we define this notion of the neighbor of gamma. And similarly... In the axiom 3 we have the axiom 4 that says that there are enough neighbors. So what this means is that suppose that q is a locality and that r is an m, is an integral curve of q, and u is a vector of gamma 0. Then locally there is a neighbor of gamma n. And here we are, you can show not only... These are the different nations. In some ways, more technical, in the sense that it simplifies what we are going to do. Because you know the notion of vector field along gamma is something very, very restrictive.

40:00 Vector field can be defined all over, or at least on an open beam connecting that. But here we have vector field only along gamma, so it's natural to ask ourselves whether we could extend it. And this can be done, and this is, I mean, this can be applied to the state, that if you have a vector field and gamma and integral curve of Q, and for every neighbor, so every neighbor, this is locally, a vector field, a real vector field, if you compose this with that, so if you restrict this to gamma, then you obtain the regional. If it's a covalent derivative, some number squared, this is going to take the place of the acceleration of five bullets per second. This is mx is the tangent space, mx is the tangent space. So, I'm sorry, the analogy to Newton and case, we postulate Einstein's vacuum as being that the trace is equal to zero.

42:30 So, here we have the first formulation of the vacuum, which corresponds to this. Now, I hear that this doesn't help us too much because, you see, the trouble with this formulation is that it depends on too many things, and I don't want to explain that, but you see that the worst thing is it's uninformative because what we want to do, we want to connect your memorabilia with Carpenter, as occurred in my paper, so we would like to connect, to connect what we, this... And so on. This is the main theorem, theory that can be found now in Sachs and Moore, because at this point I made contact with Sachs. This theorem says that if K is a locally gelated electron frame, and gamma is an integral part of Q, and W is a neighbor of gamma and Q, then this acceleration, right, at the point zero, this generalized acceleration, it's called variant acceleration, can be described in terms of the rhythm of... I am really sorry that I cannot say too much about the Riemann-Chrysler-Thompson tensor, but it's a tensor that measures the curvature at the given point of the variety and...

45:00 The tensor depended on connection on Riemann-Chrysler-Thompson? Yes, it depends on the connection. Because you can formulate this into different ways, so... Yes, yes, this depends on the connection. But because this is the only thing I have... So, okay, so this theorem, I'm going to give a proof of that theorem. I can't give the proof because it's very tricky in some way, but I have to use, here I use a book by Lavantin, called in French, in English it's called Basic Notions of the Superlative Principle. Then he proves, he defines the notion of tensor rules. And then he proves the usual properties of that. Now, to prove that is very interesting because it's an application of all this law. There is an application of lambda calculus. So it's really an application of all these truths that have been in logic. Well, for some people, these are good news. For others, these are horrible news. But anyway, this goes into... And I would say that, unfortunately, the book in English was not careful, and then it made the whole proof be nonsense, because this is the problem that happens when one devises one's own notation. People will not respect that especially. And so he used something like, instead of quantifier, he used something like U according to the scope. The scope was immediate, but if the scope was... It was a part of that double. And then in the English text, they put something that correspond to say, this is equal to you, and then you can show from there that all are molecular to zero and space.

47:30 That showed how the small change that he introduced was rather unfortunate because of this problem. So, if somebody wants to try to get the... Okay, so I'm going to, I recall this relation that we're, so that, of course, this is a classic that everyone knows, but, x, y, x, y, x, y, and then r, x, y, x, y, x, y, x, y, x, y, x, y, and then, not the same thing, however, But this is the minor problem, and it can be solved, is that in the preceding, I had, I wrote this in which, what is gamma here? Gamma, point, and W, C, and so on, whereas now I'm applying that to vector fields. Okay? So then there is, there is a difference. However, we can, so we have to connect of number of vector fields with number of vectors. But it's not very difficult because, you see, if you take a vector, then you can generate a vector field, and all of this has to be done.

50:00 Okay, and then we're ready to the proof of the theorem by axiom 5. Remember that showed that vector fields along the curve were just restrictions. And then take the definition, the number of numbers. And then you use now these formulas to write this by switching these two by adding this extra number one that you can switch by then adding a new bracket. And then you create a new bracket where there is only this one left. And now you write this. You get that immediately because it's a formula. And look, here we have not only Q is 0, because Q was locally written in crime, and this of course is 0, because it's 0, so all that you get is this. And now you restrict to gamma. So gamma, and then you obtain pre-composed, pre-composed or post-composed, I leave it to you. I'm going to define, to show how these things tie it down with Einstein. In this lecture I'm going to define the Grischi tensor.

52:30 Okay, so the Grischi tensor is defined for two vectors tangent at the point x. Then I do the following. Look at m of x. So this was the tangent space m at the point x. There you remember that I assumed that it was a thin finite dimension. So I take a basis for that. And I take a few of these. And then you define which tensor you mean as this expression, what hat are you. So you switch here, you mean, right, and then you take one hat. If you look at the classical textbooks, you don't see that. You just see one hat here. Why is that? Because you can show, classically, that this is symmetric. But of course then you obtain the same thing. I cannot do that because I was not able to prove that that was symmetric. So because of that I have to take the symmetrization. And now I'm going to compute the trace. The psi was, therefore the trace is equal to the sum of wi, dual base. But then, by definition, this is reaching 0. And so then we have a better formulation, Einstein-Biden-Fried equation, which is ratio of gamma 0 to gamma 0. Remember that gamma was, gamma, I'm sorry, was, and then at that point, so all of the key, all the trajectories of the speed is just a point. So we are still far from union of x, but then a little underground, so I will not.

55:00 The algebra is a very simple algebra, and I use heavily the fact that R is the worst about the models, a very similar talk from a month ago, and I'm sure at the time that we have let me tell you that I didn't write as axioms some things that, some questions of analysis, I thought that had been solved once and for all, but it's the main uniqueness of... This, as I said before yesterday, was true throughout the book. Unfortunately, last month I found a mistake in that paper, and I tried to prove that in simpler cases, in the simplest topos, the topos, what is called the Cahiers topos, and I did not succeed. On the other hand, I refused to believe so clearly.

57:30 So let me finish up by saying that he's in honor of Anders and that, as I said before, I have worked with him since 75 and it has always been a pleasure. I've learned quite a lot from him and I hope that he will continue, I know that he will continue working. Questions? Comments? Thank you for the nice words. We certainly have to figure out a better way of describing Ritchie tensor rather than going into coordinates. I don't have any coordinates. You are quite right. I don't have any other way of describing this. There is only one place where I had to assume that the spacetime was finite dimensional, and that was to define the notion of trace. I don't know whether somebody has defined general trace, trace independent of finite dimensionality. And I think that these two questions that you just mentioned and these others are somewhat connected. I have no idea about, but in functional analysis there are some operators where you can define these things that usually only are defined for finite dimensional.

1:00:00 Aha, Fretholm operator, well I know, there are some names around that, where you can do a little of this finite dimensional. Well, for the time being, I don't care too much about just mathematical questions, but for instance, why I left that indefinite, for instance, I said that I work with an n-dimensional thing, is that now n can become anything. And this, you can do everything like this. But then this has something interesting. We can work on string theory, say, you know, how can we have this? This is a very good example in which things can be used in this way. The other is that it is possible to derive also the character milfs. I have no idea. I mean, you know, I just started with that, and I really like all suggestions. I wouldn't be very happy about that, I don't know. That's all. That's the extent of my knowledge, what I feel right now.

1:02:30 About this connection, they were writing equations on the . Actually, in physics, it was quite, well, almost at the beginning. I mean, there was so-called quadratic formulation and later Einstein-Cartan theory. The connection problem actually which appears in physics is that we need certain compatibility between methods to have this what we call, I mean, physics and kind of concrete solutions, we need to have some certain compatibility. At the end of the day, we need a relationship in the context of this what we have here, despite striking what was one of the last works of Anders, that there appears certain compatibility between metrics in terms of, well, in the context of energy and therapies. To some extent for free, yes. I mean, it appears in a purely geometrical way, yes, when one thinks in terms of the first and second neighbors. There might be some kind of new perspective that somehow might need the compatibility to stabilize our solutions and to have the physical meaning for them are connected to the question of

1:05:00 The embarrassment is that, for instance, Einstein, in his book, he talks about, he says, the fundamental notion is not metric, you know, in his 54, you know, just before he died, in this, he called it non-symmetric field theory, right, that appeared as an appendix to the meaning of relativity. And then he says, as, who was the man that is... The fundamental notion is a notion of parallel transport. But then I tried to see how just from parallel transport, you know, he obtained that, and I couldn't see that. At one point he introduced a chi that looked suspicious, like a magic wand. It was maybe not unique, but then he did something with the chi. So it was not clear to me that just from the notion that he considered to be the only thing that were really fundamental, he obtained... So he was not supposing that a connection is symmetric? Yes, he has one. He has a theory of non-symmetric connection. Yeah, okay. The connection is not symmetric. I got a very horrible equation. Of course, the classical relation between the connections in the metric is that the metric is invariant on the parallel transport by the connection associated to it, and therefore you can get back the metric just by transporting a given metric at one point around, so there is a very close relation between the two.

1:07:30 By the way, for instance, This is considered as an alternative approach to... Yes, but it's very curious because, as you say, he takes a notion of metric at one point, transport, but somehow I found myself that it depends on many factors, and he never showed that it's independent of factors, and the only way I was able to show that when I was taking that approach was to introduce the geometry. So, for me, it's not clear that these approaches do not... Maybe we can carry the conversation on over coffee, and let's say we start again at 11.15, and let's thank the speaker again.