Einstein's field equations & SDG / discussion

0:00 So I am very, very particular to that. I believe at the end of the day, I will be able to tell you what kind of infinitesimals the real real numbers have. And we understand enough about quantum gravity, we'll understand, sure, there are infinitesimal numbers, they're the ones that are at the bound, at the limit of the information that can be transferred, and someday I'll even be able to tell you which kind we have. And I think the kind of calculus you get in a simplicial complex will at the end of the day. But in a way, we don't even need to even think of that question. You just go and find an elliptical algorithm, plug it in, refine the mesh, and the error goes to zero. Okay. That's it. Yes, and I agree with you, but I have to say that the way I actually ended up constructing these quantum field theories of simplicial complexes was by turning diagrams into oscillatory integrals and then using good old fashioned analysis, very classically showing that they actually turned out to be finite. And I felt you know, you have to murder the frog to dissect it. Something like that. at least you should I agree with you but at the present time the tools I have use delicate theorems about oscillatory integrals I can tell you where that comes from in your life unfortunately at this point your proposal is purely metaphorical I'm sympathetic with its philosophy, but I can't make matter of it. The point is, as the oscillation, you don't have to worry about the ultraviolet. There's an ultraviolet in this thing. But basically, you can control the errors as you go down. You're somehow enlarging your space of functions as you refine the mesh. But there's an error theory. I am very sympathetic to that point of view, but I honestly have to say at this point I don't know how to do it. And I have to say that

2:30 the thing of figuring out how to put the simplicial complexes together in a physically meaningful way is still in its influence. And I think that would be sort of surprises. Alright, so, what about the non-locality of one mechanic and the antagonist? it'll still be I mean look I'm doing a quantum superposition I mean all the quantum stuff I mean it's splitting in the structure because it's got a non-distributive lattice it'll all come out it'll come out that way it'll still get correlations but coming back to Einstein the greatest problem of quantum mechanics is the quantum mechanics actually I disagree I think really his greatest problem of quantum mechanics was his objection to it being based on a continuum I think unfortunately never wrote an essay about it, but I think... Sorry, I'm quibbling. But okay, that was certainly one of his great contributions, his best known contributions. So E.P.R. The main department was the role of the... Yeah, the question is, how do you... I mean, that was his great argument that in some ways that you have this non-mocallum he called went against... Okay, so what I would say that the same thing will come back. I mean, all right, there's an argument due to Carlo Rovelli, who attributed it to somebody else who never said anything about it, because he didn't want to say it was his own argument, but it's really his. Which he says that if you had a quantum manifold, it's enough to do classical dynamics on it to get the quantum theory of a particle. Because general relativity, the dynamics of a particle are determined by the metric. So if you quantize the metric, you should quantize the dynamics. So in other words, you'll just have the thing move along a geodesic, and because you're taking a quantum superposition of manifolds, you'll get a quantum superposition of paths, and you'll reproduce the... So we're still doing quantum superposition of sort of histories of the process, discrete descriptions of the space type. So if you ask where the thing comes up, you've still got a Hilbert space, and it's coming from the space that the particle is in, but it'll still have the same logic. So the Einstein-Piralski-Rosen will come out the same way

5:00 as before. I mean, we really do have to preserve it, because we've done all these experiments, we know it's right. We have to save the phenomenon. But that would be how it would come out. You'll have the quantum superposition occurring at the level of geometries, and the particle is embedded in that. And I think, ultimately, the particle is part of the geometry. Well, no, I mean, Einstein eventually, I think, gave in on that one. Maybe, you know, he didn't live to see the results. I didn't experiment, yes. I think you had to change, which is true. Yeah, I think so, too. Very quickly, what did John Bias need in any category for you to do? Or what did he do? Or which direction did he do? Well, he and I, well, he's more of a mathematician. And I'm more, he's a mathematician who's interested in physics. And I'm really a quantum superpositioner. So, there's a ladder of categorical structures. If you want three-dimensional topological field theory, you need a category. If you want four-dimensional theory, you need a two-category. But you can settle for a tensor category because it's a kind of two-category. I believe the world is four-dimensional. I refuse to be seduced by the blandishments of the straining era. over dimension, you're thinking metaphors. No, no, that's no, come on, it's a much so complex instead of dimension. Oh, dimension theory is much more general than metaphors. Okay. I mean, but, uh, okay, okay. Um, but, I mean, I think, see, I mean, they're, the most beautiful dimension in mathematics is four. Four is where the middle dimension knots. Four is where the smooth manifold theory is most complicated. Four is the critical for quantum field theories because the generic Hausdorff dimension of a Brownian path is two. So this is worth the limit of where they self-interest at. For us, where all the most interesting problems are, and it looks like we live in four dimensions. I think the whole impulse of trying to go to higher dimensions, which had such an enormous hold over people, it's almost impossible to get them to stop. But it's failed so magnificently for 25 years No, no, no. I'm talking about physics. I'm talking about calistic climb theories and string theories and all the rest of that. I think the discussion

7:30 has happened with plenty of time. Professor Reyes has to leave at least talk, so we have to thank you now. We love you. Thank you. I recall the guy whose name I was trying to think of that we were speaking at at lunch yesterday was John Corbett. John Corbett, yes. He has written a number of papers presenting C-O-R-B-E-T-T. He's an Australian mathematician and he's written a number of papers that you were presented today about quantum measurement, the relativization of the reals to a construction involving a quantum measurement. Yeah, I mean, it's almost exactly word to word as you were presenting today, so I'll certainly send you the references. But I don't see at all a connection with synthetic differential geometry. The whole point of synthetic differential geometry is to keep with the intuition that in the infinitely small functions are smooth and continuous. So where was the connection that you spoke of with synthetic differential geometry? I mean, this way of thinking of the real seems to destroy it completely. Well, it seems certainly to conflict with the motivation one has in the topos of smooth spaces. I think you have to do it in some sort of totals. Yes, but you do synthetic differential geometry in a very specific totals, the topos of smooth spaces. I agree. Well, but there are notions of smooth structures that come out of discrete spaces. I mean, there's a notion of something like a differential structure on a simple shakama. So, I mean, I don't think I can directly take synthetic differential geometry as she is spoke and important. But I think something in the same spirit. That's all I meant to say. So I deliberately try to bring myself out. You see it, and you accept that for 300 years there's nothing else. Yes, I spent the whole semester teaching your work in the graduate science. There still has to be calculus.

10:00 Yes, I've got them for you. Oh, God. Right. Give me two minutes and I'll give them. The only thing I need, is there any way, hang on, let's go over it, yeah, I agree with you, I agree with you, I mean there could be such, two categories are the work of God, the rest is the work of God. What you do is, or you put representations on You have a way of decorating the simplicial complex. You put represent... Or you can say, that's a fancy reason. Let me just say, let's do three dimensions first. Okay, it's easy. So how would I construct a topological field in three dimensions? I take a simplicial complex, and on each edge I would put a representation. And then on each face, I would put a tensor out there in between them. So I don't need a label on each face otherwise it might be a little bit too much. And then around each tetrahedron I would calculate the element of my Celsius, you know, tracing. And that gives me sort of my Lagrangian. And then it would form a state circle. Now with some overall labelings I take the product of this evaluation. I have to put it in normalization because life is rough. And then that expression is what's called the state of the sun. And just like just like a Lagrangian continuum field theory gives you a path integral so you can construct the whole theory. You can construct the whole quantum field theory

12:30 from this one. So that's the construction of a topological quantum field. And now it's more tricky in further pitches, but it's the same idea. But this is all done by the early 90s, right? Well, some of it is done. I mean, if you want to read it, it's all on GR-QC or quantum algebra. Okay, okay, if you want to read it. And what I'm saying, this is like bald head. So what's the progress of the last 15 years with regard to that? Can I... Oh, okay. Sorry. You can get some rest. You can get some rest. Yeah. So you'd be very good to answer that. Yeah, sure. No problem. No problem, sir. And, yeah, because it'll just be... It'll just be short. I don't mind asking you to be short. Um, well... Okay. Do you learn that? Do you learn that? Yeah. So, two points will be different types of... Thank God. So, this is... It's a sort of group or something like that. Okay, this is just that it's not the answer to all the problems. But it's not the answer to all the problems. You're saying that all the problems should take people to the world.

15:00 But again, it's the least convincing that this happened to be a problem. Okay, but it does seem to be a natural problem with the problem of limited information and a surrogate. And we also have points. There's also an equation about when it's equal to itself. Well, it's not equal to itself, it doesn't even exist. So points, whether they exist or not, depends on the observer. And then also there's this way of looking at a metric space where you say that the distance between the two points that helps you for which values of the the size of positive numbers is equal, so it's a point where people together are equal, and some field is there, and it's logical, so that is how they fulfilled by the former topic. Okay, can we resume, please? Friends, can we come back, please, to order? Our next speaker, unfortunately for me, has a website which has a very beautiful autobiographical note, which I advise you all to read, but I shall just read a few lines from it. I was born, if one can believe my parents and the competent authorities, on October 31st, 1937, in Santiago, Chile. And this reminds us all how very much dependent we are for outside information on the most crucial events of our life, our work. He went to an engineering school, some years of studying in engineering at the School of Engineering in Chile convinced me that my road led elsewhere. and he discovered the claims of mathematics, and he speaks up after the purgatory of the Esquirela de Infernalia, after the purgatory of the interview in school.

17:30 He finally entered the heavenly realm of present study and discussion, and of bohemian, right, as the applicatory of the intelligentsia of the country. After obtaining a full-ride scholarship, he went to the University of California in Brooklyn in 1951, where I studied mathematics and philosophy, taking a PhD in logic and methodical science in 1967. And we want to read two lines of this. I think it's critical descriptions. Put it to Americans in our... I must confess that in spite of the horror that this university, that's Berkeley, had obtained as the mecca of logic at the time and the richness and confusion of the 60s that marked me for the rest of my life, spent at Berkeley rather painful. Compared to Santiago, that's Chile, in that period, San Francisco cut a figure of a poor relative. I had met Allen Ginsberg in Santiago, a famous American poet, and I had made myself a grandiose idea of the cultural and Brazilian life that I would enjoy in that city. My stay in Berkeley left me with quite one certitude. I did not want to remain in the States. Fortunately, he received an invitation to go to the University of Montréal, where he has spent the rest of his academic career. All my career took place at the University of Montréal, assistant professor in 6873, associate professor in 79, full professor in 79, 2002, the year of my retirement. I regarded the year of freedom for myself, I'd probably share that feeling. thankful to the university for the intellectual freedom I enjoyed allowed me to follow a variety of subjects. That's the page of publications if you'll find on the website I chose. And rather than try to list all of his publications and many books and articles on various branches mathematics, category theory, applications of category theory to various subjects between the semantics and linguistics, other branches of linguistics. I'll just mention one recent book, written in collaboration with his wife, and the fortunate to have with us today, Marie Lapan Reyes, and another person, Pumaan Zofadai, is that the pronouncing? I don't know if it is. At any rate, wrote a book, Generic Figures and Their Gluings,

20:00 A Constructive Approach to Functor Adverbs. At present, this concludes, I continue my studies in mathematics, especially differential geometry, and I started some studies in physics, relativity, thus renewing some of the first loves of my youth, and indeed we're fortunate today that he's going to tell us about some of these recent studies. His topic is An Axiomatic Approach to Einstein's Vacuum Field Inversions. Professor Ains. Usually when he stands person who introduced you for the thing he said now I have to stand myself because he wrote my autobiography. Here, here. So after that, well, thanks anyway for inviting us, you know, to this meeting. And so, this subject is, since I was looking at that and trying to understand general relativity, I knew that that was hopeless from the beginning. and that I think that one can get some glimpse into that and I try to shoot that through synthetic differential geometry of which I knew more and so that's what I will try to do. So in order to start, I would like to say some words about synthetic differential geometry because I don't know to which extent that theory is known. So, anyway, I will make, I will talk about some basic methods. So, we have to start here with this theory, as we call it, that tries to do, as the word said, synthetic, differential geometry. which are very synthetic, means that we don't mean coordinates to the normal theory. Coordinates appear much later, so to speak, when we try to do computations for instance,

22:30 sometimes we introduce coordinates because we don't have to do that. And it's a theory that was introduced in 67 by Cyril Loebier and has been developed since then. And in some ways, it captures some of the intuitions of mathematicians in the 18th and 17th centuries. For instance, one of the intuitions was that in the implicit mode, any curve is a straight line. So, this theory has therefore a line that I call R. And then there are the objects, infinitesimals of first order, they are the objects of square zero. So, of course, R is not a field. And I think that already Gradendik have introduced this idea of infinitesimals in algebraic geometry. And so I will give you an example of what these infinitesimals do. So, how can we express that intuition that the infinite is small and it curves a straight line? Well, we have a map, which is a canonical map from R to R into R to the D, so that this, you should think of that as the maps from D to R, so infinitesimal curves. And so this canonical map associates the couple AB, the curve, which is straight line A plus BB, so it's an infinitesimal straight line. And then the axiom says that this is a direction, so that it says that any curve in R to the D is a line. so just with that you can define the derivative because you have a curve of function y equal f of x and then you have a point x0 then you look around x0 so the infinitesimals are around x0 so you write x0 plus d is equal to this by the axon is uniquely a decimal straight line of the form a flat B D leaving always an element of capital D for a unique because this is uniquely a

25:00 straight line and well you see immediately what is equal zero that a has to be a perfect zero and this B in uniquely unique you can define the derivative at t and from there the usual rules can be derived now one of the consequences of this is that every function is smooth as I said before the R cannot be filled and classical logic isn't compatible with this action for instance the point being that if you define a function that is 0 for 0 and 1 Obviously, that is not a straight line, it seems to be. And so, on the other hand, R is a field in a restricted sense. Sometimes it's called, in the sense of coq, which means that if not all the fi's are equal to zero, at least one of them is a unit. Now, this axiomal thing suffices for most of our purposes. and the fact that classical logic is incompatible with the axon doesn't mean that something portable has happened, no just that there are there may be other models and in fact there are topos models for this axon as somebody already mentioned in this topos model which are therefore the logic is efficient and so some infinitesimal that we have in this theory they are d that i already mentioned d2 which are the couple of d1 d2 and d such that the product of d1 and d2 is sort of equal to zero not only the product of d1 with itself and d2 with itself is zero but the product two of one zero another infinitesimal objects are the new problems of r now the fundamental notion which take the place of a generalized manifold in this theory is the notion of microlinear space and so to understand that I'm going to give a simple example so look at that diagram there with one

27:30 dd and d in d2 remember the two were the elements one d2 the product of the zero so there is an inclusion in the first variable there is inclusion the second variable and so that diagram although it's confusing it is not a coordinate one is a point i'm sorry one is a point what what you yes it's a terminal object right at one element yeah is the terminal object which is the same as r equals zero right here and r however r believes that that is the limit in the sense that if you take maps with target r then the f and g are such as f of zero equal to g of zero then there is a unique map B2 from B2 into R which makes diagram so let's look at the proof of that is very easy because it's equal you know should be according to the axiom so the problem a plus B B G should be the left on a plus C B and then the H the obvious way there is no other way of defining this has to be a plus b1 d1 i'm sorry plus bd1 plus cd2 and so then this diagram which is not a collimit but r believes in one in that sense is called an r collimit now we call an object m being micro linear if m believes that all the arco limits arco limits and so the first application of this which is very interesting is that if M is micro linear then how do you form vector bubble you think here you see an example of the synthetic nature of the theory you just take M to the D but nothing else it's just the infinitesimal lapse from B to and then there is an evaluation of zero is called pi and that is a vector mantle how can we see that

30:00 well let's try to see one of the examples this means that for instance we can add two vectors on the same fiber and so from the previous diagram you see if you look at at that diagram over here and if you put m instead of r then you see that this couple of maps which are equal to zero is an element here and therefore there is this is isomorphic to m to the d2 and therefore if we extend that diagram we enter the delta where delta is just the diagonal and then we compose all that and we have the addition let's check for instance in the case of R and then we make F plus G of B remember is equal to H of B B but H plus B B is a plus v plus ct and you see that we get the addition so that's the way that the properties now of micro linear spaces are very nice because r itself is micro linear but then limits of my micro linear are micro linear and the other very important one is if m is micro linear and x is It's just arbitrary, they tend to be neutral in it. So here, for instance, we have kinds of a manifold to another manifold, so to speak, it is a manifold. And so it has vector bundles, it has all properties that you want. Okay, so let me talk about the Newtonian gravitation. Newtonian gravitation can be presented in several ways and so there is first of all Newton law in the birth of gravitation but there is also a field theory that was discovered by Laplace and we said that it is the Laplacian of gravitational potential equal to zero this is of course I'm saying out vacuum matter and however there is another interpretation of those which is due to

32:30 Sachs and Wu as far as I know Sachs and Wu they are two mathematicians one is a mathematician and the other is a physical of anyway they work both on relativity and they wrote a book called general relativity for mathematicians in that book to obtain my rationalization. Although what I do is quite different from what they do. But the starting point is the same and the starting point is different. So they imagine that you are in a free falling lift. You know, it's a very famous experiment, an Einstein experiment, and then you are free-falling, and around you there are a lot of apples. The apples could be moving in different ways, right? Now, what you do is you take the average with respect to you of all the accelerations of the apples. So you take the acceleration of the apples, as with respect to you. It's a relative acceleration. And then you take the average of all of them. And then the average is zero. But the weight of the apple is proportional to its mass. It's not its mass. And that's it. You see, let's try to prove that. So, I'm going to... you see i have to take all the the apples which are at the sum very small distance so i interpret very small distance as an infinitesimal distance distance h since we have infinitesimal so i have to first of all describe the the sphere of radius between h of course i cannot define in the usual way, because h is equal to zero. So h squared is equal to zero, so it's like a block. However, what you can do is to take elements of this, the unitary sphere, and each one of them you multiply by h.

35:00 And so then that will be the sphere of radius h. So these are the points which are at the distance a from zero. And then I look at the trajectory. gamma is for u the trajectory and it starts at some point x and with the velocity u and then i look at the upper to simplify the thing i well i look at the the position with the starting point on x plus h u and the velocity i assume that is u and so this is what i call the trajectory of the gamma u h as well recall that from newton's law we have the acceleration is equal to minus gravitational the minus gradient gradient of the potential gamma of b and therefore i can take the difference between the acceleration of the apple and by acceleration and it becomes a force a difference between two gradients and I will call this respect of Delta u h and then you can look at the alpha coordinates of these vectors by just developing by Taylor series and you remember Taylor series ends at one point and then what would we have we have that we don't go to the fine starting from here in a linear map this is the fiber over X fiber over X and then this is the dinner map defined by this matrix. And if you look at this computation, so what we have proved is that minus delta u h, so this was minus this vector, is equal to h times phi u over phi u over u, that linear transformation. And the proposition is that the mean relative acceleration of all particles

37:30 separated by distance h from zero is one divided by four pi and an integral here i forgot to say sometimes two of course of minus delta uh times u because this would be the relative sigma and this is allowed to be h over three times the population and therefore and then as i say that was the study that was the formulation of second who the mean relative acceleration of all parties with a distance h from zero is zero for h so in virtue of the previous equation this means that h over three of the laplacian is equal to zero for H, and this means that this is by once another application of coclomir, this means the section KL, this means that the Laplacian itself is zero. And we notice that the Laplacian is a trace of this matrix. From that point of view, since we have to take, you know, all the H and then simplify by H, would be more natural to consider all these different trajectories as just one family would be a deep family so index by p which is here and it could be considered therefore as one map from r into p to the d defined in the obvious way w u of applied to t applied to h because you see one time we are here in w of u of t we are in a map we have a map from d into e and we have to to define it we apply to h and this will be gamma u h of t and we have a reformulation now that just the previous equation so that the acceleration of the particle of effect to myself is equal to phi of u at the point 0. And therefore, we can formulate Newtonian computational equation as a trace of phi equals 0. Of course, that turned out to be exactly the Laplace equation.

40:00 Now, why did I go through all of that? Well, because this gives me a very interesting notion. of neighbors along gamma you see is where you remember these were the neighbor trajectories around gamma and but appears now as a map from r into b to the d so this appears therefore as a vector field along gamma and define it this way so so this we call by abuse of language I don't have a better word that's what it sucks and who use I'm not very happy about that they called a neighbor don't know variation yeah because there's a lot of these are a lot of neighbors so But if you put an S here, you wouldn't solve the problem. So, in the theory of the dictation, there are some oscillated space-time and before many knows here but anyway we'll go quickly it's a four-dimensional curved manifold three falling particles describe geodesics straight as line in this manifold matter and curvature interact according to our equation in the words of quitter says matter tells space how to curve and curved space tells matter how to move so this is the interaction the main idea is gravitation has been incorporated into the geometry so for instance we do not have problems connected with actions at the distance that play and others now let's go into the mathematical formulation i'm going to uh distinguish i think as euclidian and others oscillates and actions so oscillates i will say oscillates in the physical sense axioms are mathematical, axioms, this is that this can be proved. So purely mathematical. So the structure of space-time, this is space-time, so is a microlinear space. But this was a

42:30 fundamental notion that in synthetic differential geometry that takes the place of a the micro-linear space with symmetric connection on parallel transport. Here, well, I think that will be, we have a map from M into D cross Z, which is a canonical map to K, and now economic and connection is a section of this map that has some properties properties of the reality I will not go into that that is the connection equivalently you can define the connection in this context in a very a geometric simplified way maybe you have a curve here and you have in this curve you have a vector which is tangent to the micro linear space and then a connection is a way of transporting that vector in such a way that some properties are preserved. So, for every, if here is gamma of zero, let's say, and here is gamma of H, then there is a projection between the fiber at this point and the fiber at this point. And this preserves some properties, but if the speed is double, then the speed is double, and so on and so on. Parallel transor, infinitesimal. Infinitesimal, yeah, that's the whole point. So every infinitesimal. And now this notion, you think, this turns out to be a group of infinitesimals. But now we can define these things infinitesimal because we have infinitesimals. That's more or less what they get from the infinitesimals. I think, yeah, I think he is, in some cases, for instance, we have been able to write down exactly this thing, our argument that we kept on, as that.

45:00 Okay, so then, and there is a law of motion, so space, the structure of space-time, space-time is microlinear space, with a symmetric connection, or parallel transport, which is n-dimensional, in the sense that the fiber over n is an n-dimensional vector space recall that vector space is really means the r module right and dimensional three r module that r works what we call field because it has this nice property and the law of motion means the following that we can define the velocity and the velocity is and then we can define the acceleration of being the double, I mean, just you iterate that. And then the gamma satisfies the equation for being a geodesic. This can be written in this way, so that gamma is nabla, number was the synergy connection gamma dot gamma dot or number if you look now at this it was called derivative by covariant derivative that is equal to zero so then we have formulated that all right now what is the main objective of this talk ah this is and the part three really the connection between uh matter and curvature so So, this is the part which is really the main object to be stuck and where some work has to be done. So, I start with the definition first. The locally geodesic reference frame is this vector field such that you can write this in two ways. This in terms of invariant and in covariate derivatives.