Roger Penrose, G Garas'ko, D Pavlov & S Siparov video conference

0:00 Thank you for your attention. Right, can I just introduce these gentlemen to you? The gentleman on my right is Dr. Pavlov, who you did meet briefly at the Newton Institute a couple of years ago, and then this is Dr. Gerasco, the author of the paper, and the gentleman on the right who is going to act as their translator is Sergei Siparov from the University of St. Petersburg, who lectures on variational principles in astrophysics. And he's going to do all the translation, I'll just simply hand over to him, and thanks very much indeed for coming in like this at such short notice. I'm sorry we kept you waiting. Okay, far ahead. Is there any way you can turn the sound up? Can we turn the sound up? This is the halo. I see. Can you hear us okay? I can hear you fine. Okay, we can only just hear you. I don't know why. How about that? Still, still a bit low, but we can just hear you. It's okay. That's exactly right, not bad. Yes, that's better. Okay? Okay. Thank you for letting us talk to you and we would like to ask you several questions and to talk about the things that would probably be of interest, not only to us, to you too. And I think that Dr. Pavlov will start and tell us something about what he's working on. We'd like to give you the essence in very brief words.

2:30 The group of physicists, consisting of about ten people, is investigating the two... The first algebra is the direct sum of four complex algebras and the second algebra is the direct sum of four real algebras. These algebras look alike algebras of quaternions and bipaternions, but do not coincide with them. The main difference is that the group of holomorphic functions on these algebras is an infinite parametric and not a paraharmatic as in for tenors. The geometry that stands behind this algebra is not the remaining one, but it's learning. При этом, в геометрии, стоящей за алгеброй, являющейся прямой суммой четырех действительных алгебр, and the geometry that supports the algebra, which is the direct sum of four real algebras, имеется предельный переход к геометрии Галилея, то есть к геометрии классической доревитивистской физики. Case and that is classical physics. In geometry, which is a direct sum of four complex algebras, there is a subgroup which presents Lorentz rule. But this subgroup is not a subgroup of isometric transformations of this algebra, but a conformal one. We started to look at the properties of these algebras in order to find the correlation between these algebras and space-time.

5:00 And one of the amazing properties turned out to be the fact that the similar transformations of Lorentz in the Minkovsky space look very similar. One of the surprising properties is that the transformations, which are similar to Lorentz transformations in the regular case, look very much alike, but differs, but still differs from the regular case. Shall I hold it down a bit, Roger? Is that better? Oh dear, no it's still not visible. No, it's a blur. He won't, he can't read it unfortunately. We'll just have to talk, just have to talk through it. They haven't given us one, we asked for one but they couldn't find one. Absolutely. Yes, four coordinates, either real or complex. And contrary to quaternion algebras, these algebras are commutative and associative. And it's tempting to infer these divisors, the isotropic vectors, on the right cone.

7:30 The only difference is that the cone is not like a regular cone, but it has facets. Connes is not a quadratic function. It is governed by a function of the fourth order, a quartic, yes, a quartic. Transformations which can be performed according to these rules leads not to the dipole anisotropy but to quadruple. But in the first approximation, we can see dipole anisotropy. For the first case, we don't see any differences for Doppler effect in every case. You can see quadrupole and octupole modes. In the anisotropy of the microwave background, which also could be correlated to some quadrupole We can also see not only dipole modes, but quadrupole and octopole as well. I would like to get more information about the possible anisotropy which is connected with this quadrupole and octupole anisotropy of microwave background.

10:00 If it turns out that there are not only dipole variations, but quadrupole too, maybe this could be explained not only by Doppler effect, but by geometrical effect dealing with Finsler geometry. Have you heard about the cosmological effect, which is called avel axis? This effect looks like it. Now Dr. Pavlov would like to ask Dr. Gorasko to tell us a few words about what he is doing. It is known that Finsler produced such a geometry which is mostly close to classical mechanics. We can say that Finsler in a sense geometrized classical mechanics. But the field theory was not geometrized up to now. It seems that I managed to find an approach and with the help of which I tried to geometrize the field theory. The action for this field is a volume in the field space.

12:30 Similarly to the case of an implicit geometry, action is the length of a node or the length of a trajectory. The results appear to be amazing. First of all, we've got a field theory which does not contradict the Einstein theory. But in which you have a general momentum tensor, connected by the conservation laws. Moreover, it seems possible to unify Electromagnetic and gravitational fields. It seems that it's partly possible, not in general case. But this is possible not only in Finsler space H4, which Dr. Pavlov mentioned. But it is also possible in the pseudo-reminant space of general relativity. Einstein's equations start to be the connection between the energy momentum tensor and the metrical tensor. This is what I'm working on. There is no curvature tensor. It doesn't appear.

15:00 It's not an area of tensile space that we can introduce a curvature tensor. Because a metric tensor has four indices. But! But! There is always an element of volume. Oh, sorry. But there is always an volume element. Sorry. There is always an element of volume. But there is always an element of volume. Sorry. But there is always an element of volume. Sorry. But there is always an element of volume. Sorry. But there is always an element of volume. But there is always an element of volume. In case I work with pseudo-rheumatic spaces, of course, they are necessarily non-available. But when I work with a space where the metrical tensor has four indices, then it's difficult to introduce a curvature tensor there. But the geometric algebra still works. Thank you.

17:30 As far as I understood, he is engaged in some spaces that have a light cone, in which there are vectors that form a sphere, a four-dimensional sphere, and where there is a group, it is formed. It is necessary to perform conformal programming there. In the case when our space turns out to be finster, do we have the same programming and how to use them? And the question arises about the choice of the direction of the time axis. Yes, yes. So, you are absolutely right, these are very important questions. And what happens? First, I will start with the time axis. If we have hypercomplex numbers that we compare events with, in this case, time,

20:00 In this case, time is associated with the unity of the general algebra. But it is not enough. We must demand some properties for this basis. It is very important that hypercomplex numbers have exponential presentation in this basis. This demand is correlated with the norm of the number and the matrix of the space. But this is also not enough. The basis must be such that the time is, so to say, right. To obtain the non-relativistic mechanics in the Galilean space when using the non-relativistic limit transition as the conformal transformation. We regard a space of associative commutative hypercomplex numbers. For example, the space H4. These hypercomplex numbers are isomorphic to the algebra. So this is a direct sum of real quadric diagonal matrix 4 by 4. So this is a direct sum of four real sets.

22:30 And this corresponds to the space with a metric with a more or more metric. The metric function is defined by the polynomial of the fourth order of differentials. And this space has an infinite dimensional group of conformal transformation. The power of these transformations is similar to that of conformal transformations on the complex plane, but this set is a little bit similar. When we make a transition, C turns to infinity, light speed turns to infinity, we transit from this Berg-Mohr geometry to classical Galilean mechanics, classical Galilean geometry. If we make a transition, C turns to infinity, we get the regular geometry.

25:00 And we can say that Minkowski geometry and four-dimensional Berwald-Moore geometry, if we neglect the ratio V to C in the third power, Then these geometries do not differ. This is first. Secondly, it is absolutely true. The important thing is that the conformal group of this Bohr-Mohr space is infinite-dimensional and contains as a subgroup the Lorentz group. But this is a conformal group and not an isometric group. There are still some problems, but we are just dealing with them. The group of conformal group of Berger-Mohr space is isomorphic to Lorentz group.

27:30 It means conformal in general mathematical sense. If the element of length During the coordinate transformation, the coordinate transformation remains the same element of length multiplied by a positive scalar function depending only on the point. Then this transformation is conformed. If you have the fourth order, do you mean that you multiply the fourth order by the arbitrary function? No. Transformation... Is that visible?

30:00 ds is an element of length. You can't go there. I'm so sorry. Is that visible, Roger? You take your meter and multiply it by the electric field. No, no. I make a coordinate transformation. A metric function transforms in such a way that it remains the same only multiplied by a scalar function. In this case, whatever... So these transformations are called conformal. Sorry, could you say that again, Roger? Let's take a Minkowski space. If we try to find such transformation they transform the element of length of Minkowski into Minkowski element of length in the other coordinate system but multiplied by some coefficient.

32:30 These transformations are found and they have a 15 parametric group. These transformations are found and they have a 15 parametric group. They produce an infinite dimensional magical space. I think Otto wants to respond. The question is this. Conformity can be understood in two senses. We can talk about conformity of transformation in space, and we can talk about conformity of transformation of space itself. In one case you have a 15-dimensional group, in the other you have an infinite one. This is what you mean. Do you want to understand what you are talking about? I will try it in another way. Conformity of transformation can be considered as a transformation that preserves angles. Let us put it in another way. Conformal transformation can be regarded as a transformation which conserves angles. We can also introduce an angle in Finnsley geometry, and a transformation, which is called conformal, which preserves the result or acts as an achievement.

35:00 The very simple question is, is it active or passive? It's the same thing. And so you can define the concept of conformal transformations, and so they coincide, these two definitions. He thinks that these definitions are coincide, both of them are mean the same, meaning this scaling and this conserving angles are mean the same. If we make a scale transformation in every point, Then all the definitions of angles, then all the angles, then all the angles, but in Riemannian space there are very few such transformations, they make only a The only exception is the Euclidean plane. In one case there is an infinite space, in the other there is an infinite space. In the infinite space there may be an infinitely parametric conformal group, just as it takes place on the complex plane, on the Euclidean plane. On the five-dimensional space, there could be an infinite parametric group in the similar way we have it on a complex plane.

37:30 We can do it this way and we can also regard it as a coordinate transformation. There is no difference. Is this an active transformation? Or is it just a reshaping? There are conformal transformations of coordinates and there is a concept of a conformal transformation of space. Passive and active. I agree. He is talking about this. Yes, absolutely. I agree. What are you talking about? I am talking about both. He speaks about both conformal transformations, active and passive. Sometimes we regard the conformal transformation of coordinates, and sometimes we regard an active transformation. In a psychological sense, they do differ. But if they have an infinite parametric Then there exists an infinite parametric group of conformal transformation of a region of space into the region of space.

40:00 And vice versa. In the article that we sent you, you can look at the formula. He doesn't know what to say right now. I just expect the problem in the paper to be more clear. What does he mean? I don't know much about geometry. Yes, you do. Do you have an idea of what geometry should be?

42:30 We are at the beginning of our journey. We have not reached the spinors yet. Right now I haven't come up to spinners or twisters, but we're just trying to make correlations between this geometry and the geometry of normal space. Since we have very little time left, if I could just suggest, since we only have about five minutes, could you put to Roger the general point you were making to me this morning about the way that you think of holomorphic? In this space we are guarding, as well as in the complex space, there are holomorphic functions that are defined in this space and they are...

45:00 They are connected to conformal transformations. And that is why the study of this holomorphic function on the set of hypercomplex numbers is the problem of the same interest as the theory of functions of complex variables. In particular, this provides the possibility to construct fractals in such sets as Mandebrot and Julia sets, only in three and four dimensions. Dr. Powell would like to show pictures which are also some... These are rectangles constructed in this new geometry. Can you see that one? Oh, you can? This is just visualization. We will not distract Professor Penrose anymore. We thank him for finding time to talk to us. We are very happy and hope to meet again at the conference in other cases. Thank you very much for coming and talking to us. We are very thankful that you have found time to just listen to what we said, and we hope that we will meet and discuss it more deeply in some other conferences or some other places. Thank you very much. Thank you very much, Roger, and thanks for coming at such short notice. I'm really sorry for disrupting your day, and I'll be back in touch. Thanks again very much indeed. Oh, you wanted to make that a historical point, didn't you, about the wrong turning of the film.

47:30 It's okay, I've got notes. Okay, cheerio, Roger. Thanks again very much indeed. Okay, cheers. You're going to be immortalized on camera. Cheers. Thanks again. Just on a practical point, you didn't have any problems with the people there in the room in Oxford? Okay, excellent. Thank you. Thanks again very much indeed.