0 00:00:02,330 --> 00:00:15,450 Jorge Pullin: Okay. S0 0ur speaker today is Eric Lewandowski, who's speak about conformally invariant approach to 1 00:00:15,710 --> 00:00:17,080 Elzbieta Lewandowska: Thank you. 2 00:00:17,300 --> 00:00:19,610 Elzbieta Lewandowska: Hello, everybody. 3 00:00:19,680 --> 00:00:22,590 I welcome you from war. So 4 00:00:23,410 --> 00:00:25,520 Elzbieta Lewandowska: perhaps some older. 5 00:00:25,720 --> 00:00:34,640 Elzbieta Lewandowska: The listeners remember the Conference G. R. 20, G. R. 20, and am all the 10 6 00:00:34,970 --> 00:00:47,440 Elzbieta Lewandowska: it that we held in war. S0 10 years ago the conference dinner for that conference was organized in the garden of our 7 00:00:47,450 --> 00:00:50,280 Elzbieta Lewandowska: a king's castle. 8 00:00:51,060 --> 00:01:06,710 Elzbieta Lewandowska: and today you may see the familiar garden familiar to you from from our Conference dinner, and in this garden Joe Biden will be delivering a speech to to the world. 9 00:01:07,080 --> 00:01:30,900 Elzbieta Lewandowska: So it shows that my choice of the place for the conference. Dinner was not so bad. I actually this I this today. I I I satisfy to buy. By the way, we organize that Conference dinner, although it was not good for delivering speeches. But I hope that Joe Biden has better 10 00:01:31,370 --> 00:01:36,800 Elzbieta Lewandowska: equipment and the support. Let me turn to the scientific part 11 00:01:37,330 --> 00:01:39,180 on my talk 12 00:01:39,600 --> 00:01:40,540 Elzbieta Lewandowska: A. 13 00:01:41,280 --> 00:01:58,490 Elzbieta Lewandowska: When we do general relativity, we usually consider like equations that are not conformally invariant. On the other hand, there are some aspects of space-time of general relativity. For instance, when we go to 14 00:01:58,610 --> 00:02:06,640 Elzbieta Lewandowska: a conformal completion to the conformal boundary of space, time. when a 15 00:02:06,860 --> 00:02:11,730 Elzbieta Lewandowska: when conformally invariant framework would be, would be useful. 16 00:02:11,940 --> 00:02:20,550 Elzbieta Lewandowska: Well, but since Einstein equations are not conformal, invariant, well, the the the the 2 things somehow are not compatible with. 17 00:02:20,580 --> 00:02:21,610 Elzbieta Lewandowska: He's out there. 18 00:02:22,150 --> 00:02:31,080 Elzbieta Lewandowska: but but it's not so bad. They can be partially made, partially compatible. And this is what my, what my talk is about. 19 00:02:31,360 --> 00:02:35,020 Elzbieta Lewandowska: I would like to. I will use 20 00:02:35,030 --> 00:02:38,760 Elzbieta Lewandowska: the mathematical notion which is called 21 00:02:38,860 --> 00:02:45,140 normal conformal cotton connection. 22 00:02:45,680 --> 00:03:05,120 Elzbieta Lewandowska: So let me begin with a introduction to to the to this cartoon construction. However, I should also emphasize that that this work was done, together with with my Adam Bots. 23 00:03:05,450 --> 00:03:07,260 Elzbieta Lewandowska: Yeah, our student. 24 00:03:15,430 --> 00:03:17,270 Elzbieta Lewandowska: Okay. So 25 00:03:17,680 --> 00:03:25,200 Elzbieta Lewandowska: the way current town introduced his various structures was quite. 26 00:03:25,530 --> 00:03:37,280 Elzbieta Lewandowska: quite physical in a sense, namely, he would always start with a model example when something is very simple and very obvious. And then he would 27 00:03:37,460 --> 00:03:39,620 Elzbieta Lewandowska: a generalize 28 00:03:39,670 --> 00:03:47,780 Elzbieta Lewandowska: this example by keeping some properties and and relaxing some some other. 29 00:03:48,540 --> 00:04:01,540 Elzbieta Lewandowska: Okay, so let me start with a model example of of Carton connection. So consider a lee group. G and the mower cut down. Form on on this lee group. 30 00:04:02,100 --> 00:04:12,320 Elzbieta Lewandowska: Consider Subgroup. H of the group G. And consider the question 31 00:04:12,590 --> 00:04:18,829 Elzbieta Lewandowska: bundle. This is the principal fiber bundle. The bundle space is the group, the bigger group. 32 00:04:18,910 --> 00:04:24,510 Elzbieta Lewandowska: the bundled base Space Space Manifold is the quotient g quotiented by H. 33 00:04:24,570 --> 00:04:28,780 Elzbieta Lewandowska: And this structure group is is the subgroup 34 00:04:29,010 --> 00:04:32,830 Elzbieta Lewandowska: H. So in this case. 35 00:04:33,080 --> 00:04:41,810 Elzbieta Lewandowska: what is the cartoon connection? It's just the them a motor carton, one form 36 00:04:41,970 --> 00:04:43,010 Elzbieta Lewandowska: itself. 37 00:04:44,830 --> 00:04:57,110 Elzbieta Lewandowska: So you can now sync what properties the I the maverick our town form. So so you can see this is not what we call a band, a connection on the on the bundle right? 38 00:04:57,190 --> 00:05:12,450 Elzbieta Lewandowska: Because connection on the bundle usually defines some horizontal space; that is, it has some kernel, and in this case a a a cartoon form actually takes values in the the algebra of this bigger group. 39 00:05:12,610 --> 00:05:21,390 Elzbieta Lewandowska: So so it means that it's in some sense defines and map between the space 40 00:05:21,400 --> 00:05:23,460 Elzbieta Lewandowska: tangent to the group 41 00:05:24,190 --> 00:05:32,080 Elzbieta Lewandowska: and it to the, to the yeah, I mean to the bundle and the the algebra of of the group. 42 00:05:32,380 --> 00:05:37,030 Elzbieta Lewandowska: Well, in in the case when the bundle is the group itself. This is quite, quite obvious. 43 00:05:37,400 --> 00:05:40,390 Elzbieta Lewandowska: Not a a map. and 44 00:05:40,430 --> 00:05:46,250 due to identity satisfied by my work. Our town form 45 00:05:46,430 --> 00:05:47,380 Elzbieta Lewandowska: a 46 00:05:47,800 --> 00:06:01,630 Elzbieta Lewandowska: This equality is satisfied. I I hope you can see my course, or if you want me to, may, if you want to make me happy. Please unmute yourself and say, Yes. 47 00:06:02,310 --> 00:06:03,140 Elzbieta Lewandowska: Yeah. 48 00:06:03,290 --> 00:06:04,090 Adam Bac: yes. 49 00:06:04,210 --> 00:06:15,370 Elzbieta Lewandowska: Thank you. Thank you. Okay. So what this equation means, which is the identity for for the our cartoon form. It means that the curvature 50 00:06:15,650 --> 00:06:18,950 Elzbieta Lewandowska: of this Carton connection is 0. 51 00:06:19,790 --> 00:06:21,960 Elzbieta Lewandowska: Okay, so here is the 52 00:06:22,220 --> 00:06:24,850 idea of generalization. Let us 53 00:06:24,930 --> 00:06:25,510 Elzbieta Lewandowska: Hmm. 54 00:06:26,180 --> 00:06:29,970 Generalize this example, such that this curvature is not 0, 55 00:06:30,600 --> 00:06:41,820 Elzbieta Lewandowska: so? And well, I'm not going to give you the precise definition, because, anyway, I W. What I will need, I will decide. Define precisely in in in due time. 56 00:06:42,190 --> 00:07:01,980 Elzbieta Lewandowska: But I want to give you an idea such that you can actually yourself go for a walk and come back after, but not now. Maybe after the talking, and after 5 min walk you can go back with your own definition of of of what should be carbon connection. S0 0nce again there is given a bigger group and the smaller group 57 00:07:02,010 --> 00:07:11,960 Elzbieta Lewandowska: as before. However, now we consider a principal, and and we consider a principal fiber bundle P. With the structure group H. 58 00:07:12,460 --> 00:07:20,760 Elzbieta Lewandowska: And we make sure we make Choose P. Such that the dimension of P. Equals the dimension of of G. Of this model group. 59 00:07:22,140 --> 00:07:32,630 Elzbieta Lewandowska: And now we consider a one, for on P. That's well that has as many properties of the 60 00:07:32,650 --> 00:07:39,410 Elzbieta Lewandowska: which means we want it to take values in the Lee algebra of G. 61 00:07:40,780 --> 00:07:43,450 Elzbieta Lewandowska: Secondly, if we 62 00:07:43,700 --> 00:07:47,090 Elzbieta Lewandowska: transform a mandal 63 00:07:47,140 --> 00:07:58,270 by using the the right action of the structure group, then we want this, the form to transform in the same way as the maverick. Our town form transforms on G. 64 00:08:00,130 --> 00:08:07,590 Elzbieta Lewandowska: And also the my cartoon form is a sort of identity map. So it maps 65 00:08:07,890 --> 00:08:09,130 Elzbieta Lewandowska: Victors 66 00:08:09,190 --> 00:08:27,580 Elzbieta Lewandowska: tangent to group into elements of the algebra. So in in a to to the corresponding, so so to in in such a way that the the generator is actually the the the vector the lady variant vector field on a group 67 00:08:27,610 --> 00:08:46,050 Elzbieta Lewandowska: is mapped into the generator that generates this, this, this one-dimensional group of transformations. So we require the same about a however, not for all the algebra elements of g but only for the algebra elements of for the algebra of the group H. 68 00:08:46,310 --> 00:08:52,990 Elzbieta Lewandowska: And that gives generalization of the marker down form. But in this case this 69 00:08:53,040 --> 00:08:59,820 Elzbieta Lewandowska: this form has a curvature, so so it can be, it can be curved. It's not flat 70 00:08:59,820 --> 00:09:14,160 Elzbieta Lewandowska: example anymore. So and every cartoon connection, every every E family of of of cases always comes from some model example like this, which is flat in the sense of this 71 00:09:14,270 --> 00:09:15,530 Elzbieta Lewandowska: identity. 72 00:09:17,170 --> 00:09:31,850 Elzbieta Lewandowska: A. There there is a known example in in even in general relativity, which is a fine Carton connection. So in this case G is the group of the 73 00:09:32,330 --> 00:09:44,320 Elzbieta Lewandowska: It's fine transformations of some vector space. H is the group of all the linear transformations of that vector of space. 74 00:09:44,910 --> 00:09:48,030 Elzbieta Lewandowska: And now we choose a 75 00:09:48,470 --> 00:09:51,570 Elzbieta Lewandowska: So we we start with the money fault 76 00:09:51,940 --> 00:09:54,960 Elzbieta Lewandowska: and the bundle of co-frames 77 00:09:56,020 --> 00:10:05,380 Elzbieta Lewandowska: such that the dimension of the tangent space is the same as the dimension of the vector space in which those groups are defined. 78 00:10:06,550 --> 00:10:13,710 Elzbieta Lewandowska: And so so the dimensions now feet, and the dimension of P is the dimension of G. 79 00:10:14,090 --> 00:10:22,520 Elzbieta Lewandowska: And now we can start with arbitrary linear connection, so with the with the standard 80 00:10:22,630 --> 00:10:27,650 linear connection defined on the to frame bundle. And then 81 00:10:27,860 --> 00:10:32,760 Elzbieta Lewandowska: for this, in our connection we can find an a fine connection. 82 00:10:33,450 --> 00:10:41,510 we which which, which, briefly speaking, because it has one more role in the one more column. 83 00:10:42,190 --> 00:10:47,790 Elzbieta Lewandowska: and it has this such that the curvature of this fine connection 84 00:10:48,140 --> 00:10:52,710 Elzbieta Lewandowska: encodes the curvature of the linear connection 85 00:10:53,240 --> 00:10:56,220 Elzbieta Lewandowska: and the torsion of the linear connection. 86 00:10:56,260 --> 00:11:06,190 Elzbieta Lewandowska: and then we are happy that the torsion, if we are, if we don't, kill it by just assuming that the the the connection instruction free. So if we admit torsion. 87 00:11:06,200 --> 00:11:12,580 Elzbieta Lewandowska: then distortion becomes a part of the curvature of the corresponding a fine connection. 88 00:11:12,850 --> 00:11:21,170 Elzbieta Lewandowska: and then those connections are. or a are pretty well known. 89 00:11:21,410 --> 00:11:34,100 Elzbieta Lewandowska: There is also another sort of special case, or maybe even exactly the same, which is called in by people doing particle physics is is doing, called the within connection. 90 00:11:34,670 --> 00:11:48,330 Elzbieta Lewandowska: So in that case the group G is the group of isomorphisms of 2, plus one dimensional Minkowski space. The group H. Is the group of Lawrence transformations which is clearly the subset. 91 00:11:49,320 --> 00:11:53,330 Elzbieta Lewandowska: and we consider the orthogonal co-frame bundle 92 00:11:53,950 --> 00:11:56,640 over to the 2 plus one dimensional 93 00:11:56,770 --> 00:12:10,930 Elzbieta Lewandowska: in space time. A. This spacetime doesn't have to be Minkowski. If it's it, it it it, it. It is in general, if it is a curved space to that to plus one dimensional space. And but but those groups correspond to 94 00:12:10,950 --> 00:12:16,860 Elzbieta Lewandowska: 2 symmetries of Minkowski spacetime. So then for Gamma, we choose the metric connection. 95 00:12:17,440 --> 00:12:29,760 Elzbieta Lewandowska: and for a we choose the a fine Carton connection, which is the restriction of the 96 00:12:31,160 --> 00:12:40,600 Elzbieta Lewandowska: And with this connection we can write the chair and Simon's action, and it turns out that it's equivalent to the Einstein equations, and I think that within. 97 00:12:40,610 --> 00:12:51,130 Elzbieta Lewandowska: in in eighties a use this to quantize, to to provide one more quantization of 2 plus one gravity. So that's why it is 98 00:12:54,460 --> 00:12:56,490 Elzbieta Lewandowska: theoretical physics 99 00:12:56,880 --> 00:13:01,570 Elzbieta Lewandowska: provided that 2 plus one gravity is very 100 00:13:03,030 --> 00:13:08,070 Elzbieta Lewandowska: hey? One technical remark. So 101 00:13:08,120 --> 00:13:11,100 Elzbieta Lewandowska: one technical remark. So 102 00:13:11,680 --> 00:13:12,770 Elzbieta Lewandowska: the 103 00:13:16,430 --> 00:13:18,300 Elzbieta Lewandowska: given a 104 00:13:19,490 --> 00:13:21,660 cartoon connection 105 00:13:21,990 --> 00:13:35,510 Elzbieta Lewandowska: and and the curvature of cartoon connection, we will do exactly the same what we usually do when we when we work with either with young meals, fields, or gauge fields, or or when we work 106 00:13:35,660 --> 00:13:53,350 Elzbieta Lewandowska: where we work with with with Levita connection. And so, instead, we not really work directly on a bundle. we will, mathematically speaking, we will choose some section of of locally defined on some open set 107 00:13:53,730 --> 00:13:59,680 Elzbieta Lewandowska: open subset of space-time. A section of this bundle, and we will pull back 108 00:13:59,870 --> 00:14:09,870 Elzbieta Lewandowska: connections and curvatures. But due to the the covariance of pullback and exterior, derivative actually. 109 00:14:09,950 --> 00:14:21,860 Elzbieta Lewandowska: most of the formulae still work after this pullback. So so we will end up, considering some a gauge, some section dependent connection which transforms 110 00:14:22,260 --> 00:14:24,950 Elzbieta Lewandowska: with respect to some 111 00:14:25,300 --> 00:14:26,190 Elzbieta Lewandowska: hey 112 00:14:28,850 --> 00:14:30,130 Elzbieta Lewandowska: field 113 00:14:30,490 --> 00:14:39,990 Elzbieta Lewandowska: and the corresponding curvature transforms a little simpler. That's why we we like curvatures 114 00:14:40,050 --> 00:14:42,350 Elzbieta Lewandowska: of what will you find connections. 115 00:14:42,910 --> 00:14:54,180 Elzbieta Lewandowska: and we will even drop this symbol of the pull back of section. So we will just identify connection with the pull back of this connections to make life easier and 116 00:14:54,400 --> 00:15:05,850 Elzbieta Lewandowska: a Anyway, we do it for calculations, so it it gives us a framework which is, which is easy to use in every time, any time when we want it to be very, very 117 00:15:06,370 --> 00:15:13,210 Elzbieta Lewandowska: precise. We could go back to to Bundles and you. We could formulate everything in terms of of the bundles. 118 00:15:14,360 --> 00:15:26,230 Elzbieta Lewandowska: Okay, so this is a general general introduction to cartoon connections, and not really very exact and complete. 119 00:15:27,050 --> 00:15:29,700 Elzbieta Lewandowska: So let us now turn 120 00:15:29,890 --> 00:15:41,860 Elzbieta Lewandowska: to the normal conformal carton connection, and I will define it exactly in this sense, in some gauge. So so we you actually don't see the bundle at all. 121 00:15:43,080 --> 00:15:45,990 Elzbieta Lewandowska: Okay. So this is a working definition. 122 00:15:46,370 --> 00:16:01,470 Elzbieta Lewandowska: So consider 4 dimensional spacetime. Actually, today we will be strictly 4 dimensional, a a lot of definitions. I mean this definition of the of this K Carton and normal conformal connection. It is 123 00:16:01,590 --> 00:16:12,140 Elzbieta Lewandowska: the same for arbitrary dimension and arbitrary signature, however, dimensions for will be important from the point of view of equations, which will be, we'll consider 124 00:16:12,150 --> 00:16:20,150 Elzbieta Lewandowska: so cause here four-dimensional space time. Consider a metric tensor on this spacetime Here i'm using a normalized 125 00:16:20,320 --> 00:16:44,340 Elzbieta Lewandowska: a a normalized frame. However. it may be the null frame or normalize, to be to t0 0ne timeline, to to be orthogonal, s0 0rthonormal. So so, Ned, let us not specify exactly. It's just normalized, and and it means that Etta equals constant, and this is certain matrix that has signature minus plus plus. 126 00:16:44,910 --> 00:16:47,960 Elzbieta Lewandowska: We will also use the volume of this 127 00:16:48,010 --> 00:16:50,670 defined by this frame. 128 00:16:51,010 --> 00:17:05,130 Elzbieta Lewandowska: and I I I. Here we we we it a consider 2 different phones for epsilon. So this font is for epsilon, which is just a symbol plus minus one. 129 00:17:05,270 --> 00:17:16,359 Elzbieta Lewandowska: while this font will be for the volume element, so the difference is in the determinant. But but still the determinant is a constant number, so the difference is not so. 130 00:17:16,700 --> 00:17:17,540 Elzbieta Lewandowska: Be 131 00:17:17,770 --> 00:17:31,960 Elzbieta Lewandowska: so in okay. So let us now turn to the construction of the conformal of the of the connection. So in this case the group for this connection will be so t0 4. 132 00:17:32,250 --> 00:17:33,740 Elzbieta Lewandowska: So here we have. 133 00:17:33,930 --> 00:17:47,100 Elzbieta Lewandowska: We have a S. 0 1 3 is our group of Lawrence rotations. However, the group for the connection will be S. 0 2 for Thely Algebra. 134 00:17:48,350 --> 00:17:54,060 Elzbieta Lewandowska: and you can think of this as more more more precisely as the group 135 00:17:54,260 --> 00:17:56,570 preserving 136 00:17:56,770 --> 00:18:02,720 Elzbieta Lewandowska: this form queue. In the middle of this form we have the form, Etta. 137 00:18:02,830 --> 00:18:16,660 Elzbieta Lewandowska: but there are 2 more directions. and the form looks like this in in in those a additional direction. So this form is defined in 6 dimensional space. 138 00:18:17,130 --> 00:18:22,370 Elzbieta Lewandowska: and the matrices that preserve this form are 139 00:18:22,410 --> 00:18:25,020 Elzbieta Lewandowska: set our our 140 00:18:25,050 --> 00:18:43,100 Elzbieta Lewandowska: group G. No, they A. A normal conformal Carton Connection may be defined as follows: this is not the general definition. This is the which you read in in in in, in, for instance, in Kobayashi. 141 00:18:43,170 --> 00:18:48,960 Elzbieta Lewandowska: or in some, or in cartoon. But but it is a working definition which is, which is equivalent. 142 00:18:48,990 --> 00:19:00,520 Elzbieta Lewandowska: So from the data which is given from this orthonormal frame, consider the following matrix of one forms. So here we put the the elements of frame 143 00:19:00,730 --> 00:19:15,990 Elzbieta Lewandowska: it, that one at the 2 with the 3 that 4 0r or we come from 0 t0 3. Here we lower the index by using a so it's just lower it index. It's, it's not. It's not that different different object? 144 00:19:16,110 --> 00:19:19,380 Elzbieta Lewandowska: Gumma is the metric. 145 00:19:19,600 --> 00:19:21,520 Elzbieta Lewandowska: and a 146 00:19:22,010 --> 00:19:27,010 Elzbieta Lewandowska: a, not a, a. a a twisting connection. 147 00:19:29,610 --> 00:19:33,840 Elzbieta Lewandowska: and a a P. 148 00:19:33,940 --> 00:19:41,870 Elzbieta Lewandowska: Is so, so, so, so P. Is constructed from the Ricci tensor 149 00:19:42,350 --> 00:19:52,970 Elzbieta Lewandowska: and from the Ritchie Scalar. So so these are these are the elements of the Riemann tensor and a 150 00:19:53,570 --> 00:20:05,650 Elzbieta Lewandowska: and well, and from them we construct the the, the, the the Ricci. This. 151 00:20:05,860 --> 00:20:11,440 Elzbieta Lewandowska: So this we turn it t0 0ne form, and we put put here. 152 00:20:13,130 --> 00:20:24,750 Elzbieta Lewandowska: So. given theta. We, we can calculate the the metric and and a a rotation free connection. and then 153 00:20:25,110 --> 00:20:35,960 Elzbieta Lewandowska: we have a We have this put here the minus scout and tensor, and this set our matrix. So what is so important about this matrix? We could 154 00:20:36,310 --> 00:20:41,690 Elzbieta Lewandowska: we we could define something else. But what is so special about this matrix 155 00:20:42,990 --> 00:20:53,200 Elzbieta Lewandowska: is that it has a very nice transformation property. So, namely. if we rescale our frame by a function F. 156 00:20:54,070 --> 00:20:58,660 Elzbieta Lewandowska: And for this function we construct the following matrix. 157 00:20:58,940 --> 00:21:05,090 Elzbieta Lewandowska: So here this matrix has zeros. This is the direct hit, the chronic or Delta. 158 00:21:06,340 --> 00:21:10,930 Elzbieta Lewandowska: A. However, here this on this lower 159 00:21:11,450 --> 00:21:20,530 Elzbieta Lewandowska: it we have the conformal factor and the gradient of the conformal factor, and even the square of the gradient of the conformal factor. 160 00:21:21,690 --> 00:21:28,720 Elzbieta Lewandowska: So this matrix itself is a bit complicated. It contains derivatives of F. 161 00:21:29,080 --> 00:21:33,420 Elzbieta Lewandowska: However, that transformation law is very nice. 162 00:21:34,860 --> 00:21:40,900 Elzbieta Lewandowska: A. Of course we can also apply Lawrence transformation to the frame. 163 00:21:41,070 --> 00:21:54,220 Elzbieta Lewandowska: and then H. Is much simpler. It is just the the the, the, the, the the the Lawrence transformation in there a central block, and and one and one in the corners. And this this is it 164 00:21:56,290 --> 00:21:59,880 Elzbieta Lewandowska: now a. To tell you all the truth. 165 00:22:01,050 --> 00:22:09,920 Elzbieta Lewandowska: In this theory we also applied the following gauge transformations where B is arbitrary one, I mean, these are 166 00:22:11,770 --> 00:22:22,890 Elzbieta Lewandowska: be with upper index, is arbitrary. Vector Field and B, with low lowered index, is low lowered by by by, by by metric, is a 167 00:22:23,450 --> 00:22:28,750 Elzbieta Lewandowska: the arbitrary one form. So so these are components of the one form. 168 00:22:28,840 --> 00:22:30,610 Elzbieta Lewandowska: and this is B squared. 169 00:22:30,870 --> 00:22:35,240 Elzbieta Lewandowska: However, this those gauge transformations. They 170 00:22:35,340 --> 00:22:36,940 Elzbieta Lewandowska: we're used 171 00:22:37,110 --> 00:22:45,530 Elzbieta Lewandowska: actually to to make this and this term 0. So so in the in my definition, I gauge fixed already 172 00:22:45,710 --> 00:22:51,850 Elzbieta Lewandowska: a little disconnection. And and so those transformations are not allowed. 173 00:22:52,020 --> 00:23:10,410 Elzbieta Lewandowska: Actually, if I allowed those transformations, then then those transformations here would be also a little hmm different. So I could split this, those transformations, this transformation int0 2 0ne, which is just the scaling and the other one, which is of this. For 174 00:23:11,190 --> 00:23:15,150 so so here we we have this gauge fixed 175 00:23:15,230 --> 00:23:22,860 Elzbieta Lewandowska: the version, and and we don't apply. However, if somebody wanted to have a general version then and then. 176 00:23:22,900 --> 00:23:26,130 and then one can just just 177 00:23:26,250 --> 00:23:40,230 Elzbieta Lewandowska: apply those gauge transformations and obtain the most general form of the connection. However, every connection, Kb. Can be. It turned into this, written in this gauge in this in this way. So that's why I do it, because I find it convenient. 178 00:23:40,440 --> 00:23:42,780 Elzbieta Lewandowska: A question. 179 00:23:43,120 --> 00:23:45,070 Erlangen: Hi! Hi! You! 180 00:23:45,090 --> 00:23:59,480 Erlangen: So these these other these. 181 00:23:59,860 --> 00:24:01,830 Erlangen: How how can I understand them? 182 00:24:02,730 --> 00:24:09,300 Elzbieta Lewandowska: I don't understand it, either. So that's why I I 183 00:24:09,410 --> 00:24:19,210 Elzbieta Lewandowska: gate. But do do we know how they act on Teeta? 184 00:24:19,340 --> 00:24:22,240 Elzbieta Lewandowska: They are somehow related 185 00:24:22,370 --> 00:24:29,420 Elzbieta Lewandowska: with, okay, so I can. So so I can give you. I can give you interpretation 186 00:24:29,480 --> 00:24:40,740 Elzbieta Lewandowska: which will be familiar to to to many of you, namely. 187 00:24:41,040 --> 00:24:48,150 Elzbieta Lewandowska: is the following: that we first extend our 4 dimensional space time 188 00:24:48,970 --> 00:24:55,530 Elzbieta Lewandowska: to be a section of a of a five-dimensional null call. 189 00:24:57,430 --> 00:25:07,780 Elzbieta Lewandowska: and then the conformal trans it's not arbitrary. Now surface it's a null cone, which means it. It has 0 sheer and only non-zero expansion. 190 00:25:08,660 --> 00:25:19,310 Elzbieta Lewandowska: and then on this now con this now cone is embedded in some 6 dimensional space time. and those transformations correspond to this. 191 00:25:19,370 --> 00:25:24,000 Elzbieta Lewandowska: to this additional vector that is transverse t0 0ur t0 0ur 192 00:25:24,190 --> 00:25:27,990 Elzbieta Lewandowska: now surface, and to the choice of this vector 193 00:25:30,610 --> 00:25:31,850 Elzbieta Lewandowska: so 194 00:25:31,910 --> 00:25:41,600 Elzbieta Lewandowska: so they are not from our space that they are. 195 00:25:42,900 --> 00:25:58,700 Elzbieta Lewandowska: and I think they may be also relevant for something, but in I just first of all switch them off to to see what I can do without them. But I also think that maybe they they they are related to them some some interesting charges or or 196 00:25:58,910 --> 00:26:02,450 Elzbieta Lewandowska: no currents. Thank you. Thank you for for this question. 197 00:26:04,120 --> 00:26:11,530 Elzbieta Lewandowska: Okay. So now, what about the curvature of our connection. So let me remind you that the connection transforms in this way. 198 00:26:11,830 --> 00:26:18,760 Elzbieta Lewandowska: so the curvature will transform in a in there nicer way. 199 00:26:19,820 --> 00:26:24,630 Elzbieta Lewandowska: Now i'm sorry. But here, this this guy here I 200 00:26:24,870 --> 00:26:33,280 Elzbieta Lewandowska: it it should be on some other transparency. So just ignore, ignore this. What is here. Just Just ignore this thing 201 00:26:33,580 --> 00:26:50,250 Elzbieta Lewandowska: it it will be later. I I i'm by by mistake. I I I also put it here. Okay. So the connection transforms in this way the curvature transforms nicer. So let us see what what is the curvature? So the curvature is defined as follows: but if we 202 00:26:50,330 --> 00:26:55,070 Elzbieta Lewandowska: do the calculation, then we find that curvature is actually quite 203 00:26:55,120 --> 00:27:02,180 Elzbieta Lewandowska: nice. So it has zeros here, and zeros here the vial tense, or in the middle. 204 00:27:04,010 --> 00:27:18,290 Elzbieta Lewandowska: and the covariant derivative of the scout and the the minus scout and tensor with respect to Gamma. So by this d I denote the exterior, covariant derivative with respect to the 205 00:27:20,040 --> 00:27:26,310 Elzbieta Lewandowska: So so you can see that. For instance, if Einstein equations are satisfied, then we will have 0 here and 0 here. 206 00:27:26,620 --> 00:27:36,740 Elzbieta Lewandowska: So there is some nice interplay with the Einstein equations, because they mean some sort of reduction, at least for them on the level of the 207 00:27:37,350 --> 00:27:38,420 Elzbieta Lewandowska: Hmm. 208 00:27:38,540 --> 00:27:45,160 Elzbieta Lewandowska: on the level of the connection, on the curvature. Well, maybe also, if if a 209 00:27:45,190 --> 00:27:56,060 Elzbieta Lewandowska: in the case of in Lambda is 0, then they also mean a reduction of connection. So so, so, so there is some. They, they, they, they they! There is some interpretation for. 210 00:27:56,860 --> 00:28:02,130 Elzbieta Lewandowska: for, for for for for for understand equations, if they happen to be satisfied by the metric test. 211 00:28:03,500 --> 00:28:22,440 Elzbieta Lewandowska: And so now, if we write bunky identity, so this bank identity comes for free. It is just a just, identically satisfied, very simple algebra. But now, if we use this F. Here, then we will obtain some some differential identity satisfied by the Vi tensor and discount. 212 00:28:24,100 --> 00:28:30,610 Elzbieta Lewandowska: Okay, and and please ignore this. I will go back to this later. Okay. 213 00:28:30,660 --> 00:28:34,520 Elzbieta Lewandowska: now, we can it sync of our 214 00:28:34,920 --> 00:28:35,790 Elzbieta Lewandowska: a 215 00:28:36,570 --> 00:28:40,890 Elzbieta Lewandowska: part. Time connection is of a young meals field. So we can 216 00:28:41,220 --> 00:28:43,760 Elzbieta Lewandowska: act on this with 217 00:28:45,050 --> 00:29:03,520 Elzbieta Lewandowska: and right the left hand side of the young music way, so which is int in if we use differential forms, it is just the covariant, the river, the exterior derivative acting on the Hodge, dwell on F. So this is this: now we can do a calculation. 218 00:29:03,690 --> 00:29:06,680 Elzbieta Lewandowska: and after the calculation we find 219 00:29:06,880 --> 00:29:10,500 Elzbieta Lewandowska: that the result that we have a lot of zeros. 220 00:29:11,520 --> 00:29:15,470 Elzbieta Lewandowska: The only non-zero are the column 221 00:29:15,810 --> 00:29:18,650 Elzbieta Lewandowska: left column and bottom row. 222 00:29:19,390 --> 00:29:28,160 Elzbieta Lewandowska: They are 3 forms. So we can write them in terms of hold of hosts. Dual applied to the, to the one from Frame I: 223 00:29:28,260 --> 00:29:29,470 Okay. 224 00:29:29,750 --> 00:29:42,010 Elzbieta Lewandowska: And and the coefficients are certain tensors. This tensor looks like this. and it is known as the So this back tensor 225 00:29:42,040 --> 00:29:45,290 Elzbieta Lewandowska: you can. Now you can see that if 226 00:29:45,350 --> 00:29:49,940 Elzbieta Lewandowska: space-time satisfies the Einstein equations. So if 227 00:29:50,710 --> 00:30:15,170 Elzbieta Lewandowska: If a Ricci is 0, then this is als0 0. This is als0 0. But even if Ricci is proportional to gab. then this is als0 0, because the covariant derivative. Of of Lambda times. G is 0, and here we contract with by tensor and by tensor doesn't have to any non 0 traces. So the the back tensor again vanishes. 228 00:30:15,220 --> 00:30:18,550 Elzbieta Lewandowska: So this is exactly one of 229 00:30:18,730 --> 00:30:22,410 Elzbieta Lewandowska: a useful in this whole call 230 00:30:22,450 --> 00:30:25,090 Elzbieta Lewandowska: game elements, namely. 231 00:30:25,270 --> 00:30:40,230 Elzbieta Lewandowska: the integrability condition for the Einstein equations that is conformally invariant. So if Spi Metric tensor satisfies the Einstein equations with cosmological, constant vacuum, and in equations, then 232 00:30:40,320 --> 00:30:46,170 Elzbieta Lewandowska: the back tensor vanishes. So this is a. And and this vanishing is 233 00:30:46,490 --> 00:30:50,140 Elzbieta Lewandowska: a a conformally invariant property. So we can take 234 00:30:50,400 --> 00:30:54,960 Elzbieta Lewandowska: any representative of the conformal class of 235 00:30:55,630 --> 00:31:06,590 Elzbieta Lewandowska: metrics, and if the back tensor is 0 is not 0, and then there is no chance that there is a solution to the vacuum machine equations among in this conformal class. 236 00:31:07,210 --> 00:31:12,960 Elzbieta Lewandowska: If it is 0, then? Not necessarily. And I will turn to this 237 00:31:13,210 --> 00:31:17,810 Elzbieta Lewandowska: later. No. in terms of our 238 00:31:18,250 --> 00:31:26,670 Elzbieta Lewandowska: our very cartoon connection, the vanishing of the back 10. So it's like like the young meals equations. 239 00:31:26,730 --> 00:31:28,860 So it takes this this for 240 00:31:29,660 --> 00:31:32,340 Elzbieta Lewandowska: which suggests to us some 241 00:31:32,850 --> 00:31:37,410 Elzbieta Lewandowska: some, a, a. some considerations which will be presented 242 00:31:37,960 --> 00:31:39,820 Elzbieta Lewandowska: presented below. 243 00:31:40,620 --> 00:31:44,460 Elzbieta Lewandowska: And now let me make one more comment. 244 00:31:45,440 --> 00:31:46,570 Elzbieta Lewandowska: So 245 00:31:47,290 --> 00:31:54,120 Elzbieta Lewandowska: we can consider this Spino representation of the in a normal conformal carton connection. 246 00:31:55,150 --> 00:32:01,270 Elzbieta Lewandowska: which means we can we know how to represent 247 00:32:01,770 --> 00:32:06,540 one form in terms of spinner, so so we can also extend it to this. 248 00:32:07,330 --> 00:32:24,140 Elzbieta Lewandowska: And then what we obtain is actually known as local twist or connection. So if you have a book by Penrose Rindler, and you find section on local twist or connection, then, even though Cartel is not mentioned there, probably. 249 00:32:24,290 --> 00:32:26,970 Elzbieta Lewandowska: unless there is some newer addition. 250 00:32:27,090 --> 00:32:33,930 Elzbieta Lewandowska: This is exactly equivalent to the to the normal conformal Carton collection. 251 00:32:34,120 --> 00:32:40,480 Elzbieta Lewandowska: It was observed for the first time by by by a guy whose name was Merkulov. 252 00:32:40,820 --> 00:32:45,790 Elzbieta Lewandowska: You. It was rational region, I think, working somewhere in the Uk. 253 00:32:46,950 --> 00:33:00,520 Elzbieta Lewandowska: Okay. But now let me also mention the reducibility. So we have already seen that Einstein equations, they lead to some reducibility. There are some other reducible cases. So if 254 00:33:00,660 --> 00:33:04,250 Elzbieta Lewandowska: in particular, if this connection. 255 00:33:04,590 --> 00:33:14,950 Elzbieta Lewandowska: in fact, takes values in the algebra as you one comma 2 embedded in S. O. T0 4, 256 00:33:15,480 --> 00:33:19,260 Elzbieta Lewandowska: which means, if it it can be gauge to this form. 257 00:33:19,510 --> 00:33:27,070 Elzbieta Lewandowska: then something very unusual happens then the space-time conformal geometry admits solution to the twister. Equation 258 00:33:29,330 --> 00:33:37,080 Elzbieta Lewandowska: and Such Metric tensors are are not trivial. They are either Minkowski or they are. 259 00:33:37,110 --> 00:33:47,820 Elzbieta Lewandowska: or or this solutions are not twisting. Then there is there. There are some sort of 260 00:33:47,940 --> 00:33:59,650 Elzbieta Lewandowska: a a metrics, they they said. The Featherman family of space, of metric tensors or conformometric tensors which were introduced by a featherman. For for some different 261 00:34:00,160 --> 00:34:04,320 Elzbieta Lewandowska: reason, and among those space times you can find 262 00:34:04,460 --> 00:34:13,560 Elzbieta Lewandowska: examples. There were found examples, explicit examples of metrics, such that the back me tensor vanishes. 263 00:34:13,690 --> 00:34:16,370 However, they are not conformal to Einstein. 264 00:34:17,130 --> 00:34:22,020 Elzbieta Lewandowska: just to make sure that that this condition is really is only necessary, not 265 00:34:22,830 --> 00:34:23,800 Elzbieta Lewandowska: sufficient. 266 00:34:25,230 --> 00:34:28,270 Elzbieta Lewandowska: Okay. So here was my 267 00:34:28,560 --> 00:34:34,690 Elzbieta Lewandowska: So I completed here the introduction to the a normal conformal part time connection. 268 00:34:36,170 --> 00:34:42,260 Elzbieta Lewandowska: And now let me turn to applications which we which we would like to. 269 00:34:42,350 --> 00:34:45,960 Elzbieta Lewandowska: to which we propose in in our work. 270 00:34:46,139 --> 00:34:48,929 Elzbieta Lewandowska: Okay, so let us 271 00:34:49,940 --> 00:34:56,710 right for the a normal conformal carton connection. 272 00:34:57,220 --> 00:35:09,800 Elzbieta Lewandowska: the L Lagrangian 4 4, which looks exactly like the young meals lag around. Yeah. Well, except that we that we remember that our variable now is not 273 00:35:09,960 --> 00:35:15,490 Elzbieta Lewandowska: the connection. it is the the the frame. 274 00:35:16,140 --> 00:35:22,430 Elzbieta Lewandowska: and and actually and and and and here this star is also 275 00:35:22,460 --> 00:35:36,950 Elzbieta Lewandowska: defined by this frame. In in a sense, I I mean, or it is defined by by metric tensor. So so this is a different different theory than just Jos. Yang Mills theory. 276 00:35:37,230 --> 00:35:47,400 Elzbieta Lewandowska: Well, however we may, we may, we may use the L lagrangian of of young Men's theory, and and insert in this Lagrange and the curvature of the 277 00:35:47,430 --> 00:35:52,780 Elzbieta Lewandowska: so. So there is given some some orthonormal frame. 278 00:35:53,040 --> 00:36:02,810 teta, and from this frame we calculate the the the this, this carbon connection, and then the curvature, and and and then it defines this 279 00:36:03,050 --> 00:36:06,700 Elzbieta Lewandowska: a a young meals looking 280 00:36:06,810 --> 00:36:09,450 Elzbieta Lewandowska: for differential 4, for 281 00:36:09,900 --> 00:36:16,060 Elzbieta Lewandowska: well we know very well that in 4 dimensions. This 4 form is conformally invariant. 282 00:36:16,120 --> 00:36:21,640 Elzbieta Lewandowska: So if we rescale if we rescale a metric tensor, then 283 00:36:21,770 --> 00:36:41,580 Elzbieta Lewandowska: well, now we have t0 2 t0 2 transformations, and Well, I mean when we rescale that then we rescale this background matrix, I mean this metric which is uses background. In fact, it's not background, is dynamical, and at the same time we rotate F. But the the rotation of F. But this object is 284 00:36:41,580 --> 00:36:50,270 Elzbieta Lewandowska: invariant with respect to those gauge transformations of F. So at the end of the day. This is invariant with respect to conformal transformations. 285 00:36:50,380 --> 00:37:00,580 Elzbieta Lewandowska: so actually it even has a familiar form. So if we work out what it is, then we find this is the square of the by tensor. 286 00:37:00,600 --> 00:37:05,460 Elzbieta Lewandowska: and and i'm sure that many people considered such such lagrangian 287 00:37:05,790 --> 00:37:13,680 Elzbieta Lewandowska: A. Then the new element which we introduce is that we want to 288 00:37:13,720 --> 00:37:21,370 Elzbieta Lewandowska: treated by using this this a cartoon connection. So so, then we think we, we. 289 00:37:22,170 --> 00:37:24,940 Elzbieta Lewandowska: this is proper, because of 290 00:37:25,280 --> 00:37:28,700 Elzbieta Lewandowska: of this nice properties of that connection. 291 00:37:29,400 --> 00:37:33,250 Elzbieta Lewandowska: Okay. So now let us consider the variation of this lagrangian. 292 00:37:33,260 --> 00:37:35,080 Elzbieta Lewandowska: So let's go slowly. 293 00:37:35,150 --> 00:37:45,780 Elzbieta Lewandowska: So this is variation with respect to the Orthonormal frame. So even when I write, a variation of a a is not independent by 294 00:37:46,650 --> 00:37:49,180 Elzbieta Lewandowska: it, it is. 295 00:37:49,560 --> 00:37:55,970 Elzbieta Lewandowska: it is we, i'm, varying a with respect to, to, to, to this call free. 296 00:37:56,850 --> 00:38:03,570 Elzbieta Lewandowska: plus the term when when we there is a variation of holes dual and 297 00:38:06,140 --> 00:38:09,090 Elzbieta Lewandowska: plus a 298 00:38:10,740 --> 00:38:13,400 Elzbieta Lewandowska: plus the term which can be 299 00:38:15,710 --> 00:38:17,480 Elzbieta Lewandowska: integrated 300 00:38:18,480 --> 00:38:20,360 Elzbieta Lewandowska: integrated by 301 00:38:22,340 --> 00:38:23,190 Elzbieta Lewandowska: here. 302 00:38:24,550 --> 00:38:27,420 Elzbieta Lewandowska: So this is this: this boundary term. 303 00:38:28,060 --> 00:38:43,800 Elzbieta Lewandowska: However, as far as this term is concerned, we notice that this whole dual actually comes with the violence I mean in this, in in this, in this expression it only comes with the violent answer, and the by, tensor has this nice property 304 00:38:43,860 --> 00:38:46,040 Elzbieta Lewandowska: that the Hodge dual 305 00:38:46,660 --> 00:38:57,890 Elzbieta Lewandowska: acting on the E is a second pair of indices, is the same as the Hodge duel. Acting on the first pair of indices, the very non-trivial property. 306 00:38:57,940 --> 00:39:10,150 Elzbieta Lewandowska: but due to this property this amounts to to to to the acting. With this fine with this fixed tensor on the on the of the first pair of indices. 307 00:39:10,320 --> 00:39:24,080 Elzbieta Lewandowska: and this tensor is independent of the of the frame. So for this reason the variation of this of star is is just is just 0. So it means that we from this variational principle we 308 00:39:24,550 --> 00:39:26,380 obtain 309 00:39:26,960 --> 00:39:39,030 Elzbieta Lewandowska: the the local equations and those local equations is this young mill's equation, or, if you want, it, is the the vanishing of this, which is the same as vanishing of the back. Tensor 310 00:39:39,220 --> 00:39:41,740 Elzbieta Lewandowska: plus the boundaries. 311 00:39:44,540 --> 00:39:47,810 Elzbieta Lewandowska: and this boundary term 312 00:39:47,820 --> 00:39:49,950 Elzbieta Lewandowska: can be promoted to the. 313 00:39:50,160 --> 00:39:51,890 Elzbieta Lewandowska: to the simplectic 314 00:39:52,570 --> 00:40:04,810 Elzbieta Lewandowska: potential density, because it it becomes simplic, potential only when we integrated along some cosy service. But but here. It is just just a 3 form, so I' that's why I call it density. I'm not sure if it's 315 00:40:05,800 --> 00:40:15,590 Elzbieta Lewandowska: and the the proper name. But but I need some name to to explain the name of the integrating this. So in this way. So this way we obtain the 316 00:40:15,630 --> 00:40:16,500 Elzbieta Lewandowska: hey? 317 00:40:16,580 --> 00:40:21,680 Elzbieta Lewandowska: A nice proposal for simplicity, density 318 00:40:23,300 --> 00:40:24,330 Elzbieta Lewandowska: A, 319 00:40:24,480 --> 00:40:37,710 Elzbieta Lewandowska: which is variation of a which star, if we can we. And now but from this form we can immediately see that this simplectic, the potential is, is a conformally invariant. 320 00:40:37,880 --> 00:40:50,520 Elzbieta Lewandowska: because conformal transformations they amount to such transformations. And then here is we use, trace. So they are killed by the trace. So they are. and this expression is invariant with respect to 321 00:40:50,930 --> 00:40:53,920 Elzbieta Lewandowska: it it. With respect to those transformations. 322 00:40:54,150 --> 00:41:06,600 Elzbieta Lewandowska: on the other hand, we can work out what it is, and then we find that in terms of the space-time metric tensor, it is variation of of the orthonormal frame which 323 00:41:06,840 --> 00:41:14,270 Elzbieta Lewandowska: the Hodge dual of the derivative of the minus scout intens, or plus variation of gamma, which 324 00:41:14,370 --> 00:41:19,600 Elzbieta Lewandowska: how to do all of the of the 2 form obtained from the vial. Answer. 325 00:41:20,530 --> 00:41:31,650 Elzbieta Lewandowska: So this, this. So the conclusion is that the the potential which is written in this way is so nice that it is conformally invariant 326 00:41:31,900 --> 00:41:34,270 Elzbieta Lewandowska: that conformal invariance will be used 327 00:41:34,730 --> 00:41:37,510 Elzbieta Lewandowska: in in a few slides below. 328 00:41:39,420 --> 00:41:44,010 In order to understand what actually we are doing. Where we are. 329 00:41:44,970 --> 00:41:53,460 Elzbieta Lewandowska: we decompose the the young muse lagrangian with the we we call it 330 00:41:53,730 --> 00:42:04,100 Elzbieta Lewandowska: int0 2 terms. The first term is the euler. a density. so after integrating it becomes the euler 331 00:42:04,140 --> 00:42:16,030 Elzbieta Lewandowska: invariant. But but here we this is before integrating. and we're where this currently are. It is just the the curvature of the Levy Chivita connection. 332 00:42:16,250 --> 00:42:17,800 Elzbieta Lewandowska: the curvature, 2 4, 333 00:42:19,530 --> 00:42:23,540 Elzbieta Lewandowska: and the variation of this euler part. 334 00:42:23,730 --> 00:42:35,050 Elzbieta Lewandowska: Not surprisingly, it doesn't give any local equations. It only gives the so it only gives the simplicity. Again, the simplectic potential which we can 335 00:42:36,250 --> 00:42:41,720 Elzbieta Lewandowska: choose to be this on that and the second term L one. 336 00:42:42,100 --> 00:42:47,000 Elzbieta Lewandowska: It can be written in this way in terms of this scout and tensor. 337 00:42:47,990 --> 00:43:00,270 Elzbieta Lewandowska: and this is the variation of the second term which produces the the the equation, the condition for Bach that it has to vanish, and also defines another symbolic 338 00:43:00,460 --> 00:43:06,420 Elzbieta Lewandowska: potential which can be calculated. and so to be to be 339 00:43:06,720 --> 00:43:19,650 Elzbieta Lewandowska: this. So in this way we have decomposed the a symbolic potential density defined for the Carton Young meals theory into the euler 340 00:43:20,400 --> 00:43:22,590 Elzbieta Lewandowska: simplectic potential. 341 00:43:23,740 --> 00:43:26,450 Elzbieta Lewandowska: plus there 342 00:43:27,310 --> 00:43:29,830 Elzbieta Lewandowska: a simplic potential 343 00:43:30,020 --> 00:43:32,590 Elzbieta Lewandowska: of this and 344 00:43:33,110 --> 00:43:39,800 Elzbieta Lewandowska: of this lagrangian of this other Lagrangian which which could be used to to give them the back 345 00:43:40,070 --> 00:43:46,410 Elzbieta Lewandowska: it the the of the back T. So, however, the Lagrangian itself is not conformally invariant. 346 00:43:46,720 --> 00:43:55,550 Elzbieta Lewandowska: So s0 0ur choice of Lagrangian makes it conformally environment. Okay. So let us see what we can learn 347 00:43:55,660 --> 00:44:00,460 Elzbieta Lewandowska: from this. Okay. S0 0ne of applications of this framework 348 00:44:00,610 --> 00:44:09,880 Elzbieta Lewandowska: is that actually we go back to the Einstein equations. Right? Okay. So if the Einstein equations are are satisfied. 349 00:44:10,360 --> 00:44:11,250 Elzbieta Lewandowska: then 350 00:44:11,500 --> 00:44:18,770 Elzbieta Lewandowska: the back tensor vanishes, and then the scout and 10 minus scout and tensor just becomes this. 351 00:44:20,020 --> 00:44:23,620 Elzbieta Lewandowska: So it is just constant times, times, times this. 352 00:44:24,220 --> 00:44:32,320 Elzbieta Lewandowska: and then our our simplic potential, takes this simple form. So this is variation of gamma, which 353 00:44:32,400 --> 00:44:35,220 Elzbieta Lewandowska: how to do all of the bio 354 00:44:35,790 --> 00:44:36,890 Elzbieta Lewandowska: to form 355 00:44:38,770 --> 00:44:39,880 Elzbieta Lewandowska: A. 356 00:44:41,720 --> 00:44:50,860 Elzbieta Lewandowska: On the other hand, we can also calculate what is this Theta one which was defined on the previous transparency, and it takes 357 00:44:51,060 --> 00:45:00,170 Elzbieta Lewandowska: this for. and this form should be pretty familiar to you, namely, if we consider a 358 00:45:00,320 --> 00:45:11,130 Elzbieta Lewandowska: Lagrangian, which is just the the Einstein Hebrew lagrangian, however, written in terms of the of the of the orthonormal frame. 359 00:45:12,370 --> 00:45:24,680 Elzbieta Lewandowska: then it takes has this form. and if we now calculate variation of this Lagrangian that will give us the Einstein equations and the boundary term. 360 00:45:25,230 --> 00:45:43,650 Elzbieta Lewandowska: Then we can see that this boundary term which we promote to the this is the one could call it Palatini also. But but here we our variable is is Teta. So connection is not the independent variable, so that's why I don't call it 361 00:45:43,910 --> 00:45:50,410 Elzbieta Lewandowska: so. So then this Lagrangian gives the I shine here. He gives them a simplic potential of this, for 362 00:45:50,610 --> 00:45:53,690 Elzbieta Lewandowska: So now, if we compare this 363 00:45:54,420 --> 00:45:57,730 Elzbieta Lewandowska: With this. then we can 364 00:45:58,070 --> 00:46:04,780 Elzbieta Lewandowska: a a conclude the following: that the simplic potential density 365 00:46:04,800 --> 00:46:17,540 Elzbieta Lewandowska: defined by the is the a simplic potential density of the theory that would be defined by the Euler invariant 366 00:46:18,610 --> 00:46:19,890 Elzbieta Lewandowska: minus 367 00:46:20,150 --> 00:46:26,890 Elzbieta Lewandowska: this number times the simplic potential density of Einstein Hilbert theory. 368 00:46:26,940 --> 00:46:31,920 Elzbieta Lewandowska: However, this is all true only if we restrict ourselves 369 00:46:32,810 --> 00:46:35,400 Elzbieta Lewandowska: to the solutions of the Einstein equations. 370 00:46:36,130 --> 00:46:53,380 Elzbieta Lewandowska: but we can do it because every solution of Einstein equations is a solution to the back way. It will be, has vanishing back. So we solution to this young music equation, and it's just a subspace on which this simplectic format is, is works in this, and defines some 371 00:46:53,540 --> 00:46:55,890 Elzbieta Lewandowska: currents and etc. 372 00:46:57,980 --> 00:47:02,670 Elzbieta Lewandowska: Okay, so let me show you how how we can apply this to 373 00:47:02,800 --> 00:47:11,470 to this cry of asymptotically. So here I will assume the spacetime is the sitter, but but for us in but for anti, the sitter, it it it it works. 374 00:47:11,550 --> 00:47:17,990 Elzbieta Lewandowska: it works the same. So let us consider them pepper man Graham coordinates. 375 00:47:19,240 --> 00:47:31,100 Elzbieta Lewandowska: So we consider a solution to Einstein equations with with the cosmological constant bigger than 0 in a If a firm and Graham coordinates the metric tensor can be written in this way 376 00:47:31,540 --> 00:47:38,280 Elzbieta Lewandowska: and describe the conformal infinity corresponds to Rho equals 0. So here we have 377 00:47:38,750 --> 00:47:49,710 Elzbieta Lewandowska: infinity. However, if we look at this metric tensor inside the parentheses. It is finite on this cry. So this is this non physical, rescaled 378 00:47:50,110 --> 00:47:52,950 Elzbieta Lewandowska: rescale metric. And now 379 00:47:54,030 --> 00:48:01,890 Elzbieta Lewandowska: notice the following. Since our theory is conformally invariant. 380 00:48:02,080 --> 00:48:09,010 Elzbieta Lewandowska: And actually we don't require, I mean, conformal to rescaling already 381 00:48:09,120 --> 00:48:20,090 Elzbieta Lewandowska: destroys the the the Einstein equations. However, we don't require Einstein equations only require vanishing of the back tensor. We can well apply our framework to this metric. 382 00:48:20,510 --> 00:48:30,640 Elzbieta Lewandowska: but if we apply it to this metric, then certainly the simplic potential density will be finite on the on the sky. 383 00:48:31,150 --> 00:48:43,110 Elzbieta Lewandowska: But our physical. the coming from the physical metric simplic potential density equals due to the conformal invariance, this one. 384 00:48:43,170 --> 00:48:45,770 Elzbieta Lewandowska: So it means that our 385 00:48:47,170 --> 00:48:56,950 Elzbieta Lewandowska: original simplic potential density which we've defined for this this Cartonian meals potential will be finite 386 00:48:57,220 --> 00:49:00,990 Elzbieta Lewandowska: all the way to this cry, and on the screen, too. 387 00:49:01,470 --> 00:49:07,010 Elzbieta Lewandowska: A. And this is sort of surprising, because if instead, we take Einstein. 388 00:49:07,060 --> 00:49:27,620 Elzbieta Lewandowska: the just Einstein lagrangian, then that the corresponding simplic potential is not finite. We have to do some renormalization. So here we is already. We. We don't know what is the result. But even without starting any calculation we from just from the properties we can see that our result will be finite. 389 00:49:29,100 --> 00:49:40,720 Elzbieta Lewandowska: and probably non-trivial because this relation between the Lagrangians for cosmological, constant, not 0, is is not trivial. So let us see what it is. 390 00:49:41,760 --> 00:49:49,020 Elzbieta Lewandowska: So what there is this, this familiar analysis of the expansion of the metric tensor 391 00:49:49,300 --> 00:50:07,720 Elzbieta Lewandowska: on on the scri. We know that so we have this G 0 which gives the the background 392 00:50:07,720 --> 00:50:14,680 Elzbieta Lewandowska: constraints. It is traceless and due to Einstein equations, traceless and divergence less and otherwise. It's free. 393 00:50:15,870 --> 00:50:20,650 Elzbieta Lewandowska: And we want to check this formula 394 00:50:22,360 --> 00:50:30,670 Elzbieta Lewandowska: in this, for this metric tensor and calculate what it is. So we do it, we do the calculation. And now we find the following: that this guy 395 00:50:30,710 --> 00:50:45,020 Elzbieta Lewandowska: actually vanish, that this guy is not vanishes in diverges on sky like one open row. However, the by tensor vanishes on this cry, so when they meet together they produce something which is not 0. 396 00:50:45,680 --> 00:50:53,920 Elzbieta Lewandowska: S0 0n the sky. We introduced the volume. Hmm. By contracting. by contracting. 397 00:50:54,020 --> 00:50:59,750 Elzbieta Lewandowska: Our four-dimensional volume is the vector field defined by the 398 00:51:00,000 --> 00:51:03,010 the the by the vector for the D. 399 00:51:03,860 --> 00:51:16,840 Elzbieta Lewandowska: So this is our choice of orientation. Then we consider the pull back of this 3 form to subscribe. and indeed, it defines a finite 400 00:51:16,950 --> 00:51:18,260 Elzbieta Lewandowska: finite result. 401 00:51:20,330 --> 00:51:24,610 Elzbieta Lewandowska: which is a variation of. So if we apply this 402 00:51:24,710 --> 00:51:27,820 Elzbieta Lewandowska: to variation of our frame. 403 00:51:27,950 --> 00:51:44,150 Elzbieta Lewandowska: it's some on and in the background of of of some frame. So then, if we apply this variation to here, and we replace the variations of of at the by variations of the Eta defining this 404 00:51:44,440 --> 00:51:46,640 Elzbieta Lewandowska: of this non- physical metric. 405 00:51:47,560 --> 00:51:52,340 Elzbieta Lewandowska: then this is what we obtain. So this is variation of this G. 0 406 00:51:52,360 --> 00:51:57,610 Elzbieta Lewandowska: in this expansion. Times T. 0, where T. 0 is, in fact, G 3. 407 00:51:59,130 --> 00:52:02,760 Elzbieta Lewandowska: Okay. let us compare it with them 408 00:52:03,840 --> 00:52:06,090 Elzbieta Lewandowska: with the known 409 00:52:06,580 --> 00:52:08,230 methods. 410 00:52:08,970 --> 00:52:10,200 Elzbieta Lewandowska: So 411 00:52:10,230 --> 00:52:13,250 once again I re-wrote what we would we 412 00:52:13,270 --> 00:52:16,770 Elzbieta Lewandowska: obtained was was 413 00:52:16,780 --> 00:52:18,530 this. 414 00:52:19,230 --> 00:52:21,080 Elzbieta Lewandowska: A. On the other hand. 415 00:52:21,160 --> 00:52:34,410 Elzbieta Lewandowska: If we now go to standard definitions, then in standard definitions, the holographic stress energy tensor is defined by our T. 0 as follows: so in terms of the holographic stress energy tensor. It will be this. 416 00:52:35,930 --> 00:52:47,200 Elzbieta Lewandowska: And now, if we consider the Einstein original Einstein Herbert action, then we have to add some boundary terms, and we have to add such term. and 417 00:52:47,290 --> 00:52:57,920 Elzbieta Lewandowska: it then we obtain from such action we obtain the pullback of the of the 418 00:52:57,940 --> 00:53:10,360 Elzbieta Lewandowska: a simpletic potential of this war, so by comparison, shows that our Lagrangian gave the same pull back Modulo, the the the the coefficient, which 419 00:53:10,460 --> 00:53:17,030 Elzbieta Lewandowska: which agrees also with with what I, with this decomposition which was made 420 00:53:17,390 --> 00:53:18,340 Elzbieta Lewandowska: out here. 421 00:53:19,280 --> 00:53:31,340 Elzbieta Lewandowska: Well, a second question is, what is on the scri of a simpletically flight space time. But here we are not actually expecting anything that would be good for the degrees of freedom, because 422 00:53:31,450 --> 00:53:37,070 Elzbieta Lewandowska: in this decomposition, maybe I I will write this decomposition again. A. 423 00:53:37,110 --> 00:53:50,900 Elzbieta Lewandowska: Because this the composition is the follows. So conformal is so this cartoon young meals. The simplic potential is the simplic potential of the topological theore minus lambda 424 00:53:51,360 --> 00:53:54,790 Elzbieta Lewandowska: times, the simplic potential of the 425 00:53:55,080 --> 00:54:11,330 Elzbieta Lewandowska: of the Einstein Hilbert. So if I, lambda is 0 for a simpletically fled case, then then this term will drop. So we just we just deal with the simplic potential of the euler theory. So not surprisingly what we calculate here 426 00:54:11,350 --> 00:54:23,150 Elzbieta Lewandowska: defines a a vanishing the currents. And however this is this: is there the result? 427 00:54:23,440 --> 00:54:31,780 Elzbieta Lewandowska: Talking about currents? We could go back to the, to t0 0ur theories, so so, s0 0nce again to the beginning. 428 00:54:31,850 --> 00:54:36,320 Elzbieta Lewandowska: and we can ask, what are the current? So there is the 429 00:54:36,330 --> 00:54:41,440 Elzbieta Lewandowska: I consider that the few more reason this theories the 430 00:54:41,660 --> 00:54:54,050 Elzbieta Lewandowska: this theory given by this by this, a. A a cartel young meals, and as well as general relativity. So we can ask, what is the 431 00:54:54,460 --> 00:54:58,550 Elzbieta Lewandowska: the a current for neither current Northern Ireland for the 432 00:55:11,300 --> 00:55:22,870 Elzbieta Lewandowska: 2 form. minus the term proportional to the equations, to the which vanish. If we, if we assume that we are on the shell, so it means that the 433 00:55:23,170 --> 00:55:27,450 Elzbieta Lewandowska: charge corresponding to the theomorphism is this. 434 00:55:27,750 --> 00:55:44,500 Elzbieta Lewandowska: We can also consider a symmetry defined by generator of the Lawrence rotations or conformal rescaling. So if I call it just L. Then the corresponding charge, then the corresponding current is, is this 435 00:55:44,700 --> 00:55:54,110 Elzbieta Lewandowska: so? It is exterior derivative of the corresponding charge. So here this charge looks like valid talents, or con contracted with with. 436 00:55:54,510 --> 00:56:02,020 Elzbieta Lewandowska: If this is Lawrence rotation, so it will be contracted with generator of Lawrence rotation. If this is conformal. 437 00:56:02,180 --> 00:56:10,300 Elzbieta Lewandowska: a a rescating. Then then I think that at least in case of hmm, I see an equations. And this is just 0. 438 00:56:13,410 --> 00:56:17,200 Elzbieta Lewandowska: Okay. So to summarize. 439 00:56:19,080 --> 00:56:20,480 Elzbieta Lewandowska: to summarize. 440 00:56:21,860 --> 00:56:34,050 Elzbieta Lewandowska: we defined here a an approach to the A, a. to to to space time, which is conformally invariant. 441 00:56:34,540 --> 00:56:35,510 Elzbieta Lewandowska: Hmm. 442 00:56:35,650 --> 00:56:52,950 Elzbieta Lewandowska: It has this property that if we consider solutions to Einstein equations, then the necessary conditions which are so. So. In other words, we we, we consider here a conformally invariant theory, such that it contains the anish time 443 00:56:53,330 --> 00:56:54,280 Elzbieta Lewandowska: theory. 444 00:56:54,330 --> 00:57:01,890 Elzbieta Lewandowska: So whatever currents are defined in terms of this conformal theory, they also 445 00:57:01,900 --> 00:57:03,300 Elzbieta Lewandowska: have them. 446 00:57:03,430 --> 00:57:11,760 The the the satisfy the I didn't the same identities. If it for Einstein's space time is because Einstein's a special case of that theory. 447 00:57:12,520 --> 00:57:18,120 Elzbieta Lewandowska: but the advantage our formula we have is that they are conformally invariant. So 448 00:57:18,250 --> 00:57:25,810 Elzbieta Lewandowska: they, for instance, we can go with them to the conformal completion, and we we don't 449 00:57:25,930 --> 00:57:27,180 Elzbieta Lewandowska: a a 450 00:57:27,270 --> 00:57:30,400 Elzbieta Lewandowska: face, any any infinities there. 451 00:57:31,710 --> 00:57:34,730 Elzbieta Lewandowska: So this is it. Thank you. 452 00:57:35,180 --> 00:57:36,500 Elzbieta Lewandowska: Thank you very much. 453 00:57:47,690 --> 00:57:48,760 Elzbieta Lewandowska: Hello, Hello. 454 00:57:49,270 --> 00:57:51,970 Hey, You A question in the chat. 455 00:57:52,460 --> 00:57:54,120 Elzbieta Lewandowska: Okay, let me 456 00:57:54,340 --> 00:57:56,250 let me see. 457 00:58:08,570 --> 00:58:20,720 Elzbieta Lewandowska: Okay. So in my talk I considered a lambda bigger than 0. So so it in here I gave some 458 00:58:20,810 --> 00:58:23,140 specific, exact formulae. 459 00:58:23,590 --> 00:58:25,980 Elzbieta Lewandowska: I is the 460 00:58:27,180 --> 00:58:48,140 Elzbieta Lewandowska: perhaps some. Some details are different for the ads, because then this this boundary is time Like, however, from the point of view of of Cartine connection and and all this framework, this is not obstacles. So if we start with. So so 461 00:58:48,410 --> 00:58:49,700 Elzbieta Lewandowska: all the 462 00:58:50,430 --> 00:59:06,020 Elzbieta Lewandowska: carpent connection part is unsensitive completely on the value of cosmological, constant, and actually is valid even for space times that don't satisfy Einstein equations at all, but that that's satisfy only the the back equation. I mean the 463 00:59:06,310 --> 00:59:14,140 Elzbieta Lewandowska: what is sensitive on this, whether this is ads as anotic ads or asymptotically. Ds. 464 00:59:14,240 --> 00:59:17,940 Elzbieta Lewandowska: Is this a a 465 00:59:18,250 --> 00:59:21,000 Elzbieta Lewandowska: form of metric tensor? And then some 466 00:59:21,190 --> 00:59:30,600 details come, and but but I think that this going to the limit would be similar, and we would obtain a similar. 467 00:59:30,700 --> 00:59:34,200 Elzbieta Lewandowska: a similar solution. 468 00:59:36,850 --> 00:59:37,900 Elzbieta Lewandowska: A. 469 00:59:39,890 --> 00:59:41,530 Elzbieta Lewandowska: So 470 00:59:42,340 --> 00:59:43,280 questions for you. 471 00:59:44,120 --> 00:59:45,630 Elzbieta Lewandowska: Thank you for asking. 472 00:59:46,170 --> 00:59:58,910 Simone: Hey? How you like. Maybe do you think you pointed out about the conformal gravity processing all the solutions of a Gr. That is is already true for vile, square, right 473 01:00:03,050 --> 01:00:09,050 Simone: internal frame and Carton connection construction. Maybe. Can you give us some 474 01:00:11,510 --> 01:00:16,150 Simone: some broad comment on that? Apart from the they can. You got aspects? 475 01:00:16,700 --> 01:00:27,140 Elzbieta Lewandowska: Okay. So vile. Tensor by itself, indeed, is conformally invariant. however. 476 01:00:27,440 --> 01:00:32,480 Elzbieta Lewandowska: the the the proper, but but it's not really by itself 477 01:00:32,740 --> 01:00:36,550 Elzbieta Lewandowska: curvature of of anything except 478 01:00:37,210 --> 01:00:43,190 Elzbieta Lewandowska: the normal conformal cartime connection. So this is like 479 01:00:43,220 --> 01:00:58,530 Elzbieta Lewandowska: I could compare it to well, when somebody would be dealing with some components of of of remontense, or but would not define Riemann Tensor, but just played with the components I could come and and show. Oh, by the way, all those 480 01:00:58,530 --> 01:01:05,310 Elzbieta Lewandowska: components they can be collected into a single conformally invariant object. So so we just 481 01:01:05,380 --> 01:01:08,430 Elzbieta Lewandowska: restore, restore some 482 01:01:08,830 --> 01:01:12,050 Elzbieta Lewandowska: geometric a 483 01:01:12,050 --> 01:01:36,250 Elzbieta Lewandowska: structure which is behind the the file tensor 484 01:01:36,550 --> 01:01:38,290 Elzbieta Lewandowska: various identities. 485 01:01:38,890 --> 01:01:56,980 Elzbieta Lewandowska: However, there is this question which we actually didn't answer yet, and and I i'm sure you are. You are sensitive on this, because you you in your papers, you often discuss this, that here we use a frame rather than metric tensor. So 486 01:01:57,070 --> 01:02:00,700 Elzbieta Lewandowska: so there is a question if we could 487 01:02:01,100 --> 01:02:02,930 somehow. 488 01:02:03,340 --> 01:02:18,760 Elzbieta Lewandowska: Well, what is the difference between? So it seems that that that that would we obtain is is just what you would obtain from this frame formulation of of of general relativity. So we, which is a little different than what we obtain from the metric 489 01:02:19,170 --> 01:02:30,130 Simone: formulation. But due to your work we know we know what is the the difference. So I 490 01:02:30,560 --> 01:02:31,460 Elzbieta Lewandowska: thank you. 491 01:02:33,910 --> 01:02:34,800 What? The 492 01:02:34,820 --> 01:02:38,030 Wojciech Kaminski: So it's another, a comment 493 01:02:38,160 --> 01:02:40,940 Wojciech Kaminski: so so to to 494 01:02:41,270 --> 01:03:00,560 Wojciech Kaminski: so so I think to in addition to what you like, that we are gaining something, because if we using a frame in in this the normal cartoon connection, then we also have to very easily this simplic potential that is conformal in via. 495 01:03:01,000 --> 01:03:07,040 Wojciech Kaminski: And this is very important in our work. So if we just use by square. 496 01:03:07,330 --> 01:03:17,300 Wojciech Kaminski: then in principle, them okay, maybe we obtain some. So I hope one can obtain the a conformal invariance, selective potential, but it's not 497 01:03:17,380 --> 01:03:27,130 Wojciech Kaminski: of the month. So in this a a carton approach, we just getting it for free. So this was a common. 498 01:03:28,140 --> 01:03:29,310 Simone: I see. Thank you. 499 01:03:35,820 --> 01:03:39,110 Erlangen: Can I ask another question 500 01:03:46,350 --> 01:03:55,520 Erlangen: in this normal Qatar connection framework, the symbolic structure actually differs by the variation of the euler term? 501 01:03:55,580 --> 01:03:56,360 Elzbieta Lewandowska: Yes. 502 01:03:56,640 --> 01:04:03,480 Erlangen: so how does this affect? This, then, also shows up in the definition of the charges, I suppose. 503 01:04:03,670 --> 01:04:04,680 Erlangen: Yes. 504 01:04:04,740 --> 01:04:21,860 Erlangen: give you more, for for instance. So I have 2 questions. So how do the charges now? This defer from the usual charges in like ATM framework, or just? 505 01:04:23,100 --> 01:04:24,230 Erlangen: And 506 01:04:26,120 --> 01:04:28,620 Erlangen: and then whether this difference 507 01:04:28,720 --> 01:04:37,250 Erlangen: is only. And how this difference also change better. This difference changes the fluxes, like the change of 508 01:04:37,270 --> 01:04:40,760 Erlangen: charges between different cross-sections, for instance. 509 01:04:40,770 --> 01:04:50,050 Erlangen: What would be very nice, for instance, is, if the difference is so. If this change of the simplic structure does not affect the 510 01:04:50,100 --> 01:04:51,280 Erlangen: the flux. 511 01:04:51,620 --> 01:04:55,570 Erlangen: the difference between charges between different cross section. 512 01:04:55,610 --> 01:04:59,300 Erlangen: Is there something like that, or or is it? Is it not? 513 01:04:59,530 --> 01:05:07,580 Elzbieta Lewandowska: Yes, it it seems to be true; for for the the few more reasons that that this. 514 01:05:08,080 --> 01:05:14,240 Elzbieta Lewandowska: that this euler term it doesn't change the the currents. 515 01:05:14,690 --> 01:05:16,860 Elzbieta Lewandowska: so it provides the 516 01:05:16,910 --> 01:05:28,310 Elzbieta Lewandowska: finishing currents. However, it changes our notion of of charges. It it suggests some different charges, but 517 01:05:29,080 --> 01:05:32,320 Elzbieta Lewandowska: but it doesn't change the evolution of the charges. 518 01:05:35,050 --> 01:05:39,610 Elzbieta Lewandowska: and it restores that conformity. 519 01:05:39,870 --> 01:05:41,950 invariance, or covariance. 520 01:05:42,680 --> 01:05:45,180 Erlangen: Thanks for this clarification. Thanks 521 01:05:48,470 --> 01:05:50,290 Okay, any other questions for you. 522 01:05:54,550 --> 01:05:58,900 If not, let us. Thank you. Thank you, Eric, very much for the Okay. 523 01:05:59,310 --> 01:06:00,750 Elzbieta Lewandowska: Thank you for coming. 524 01:06:05,910 --> 01:06:07,420 Simone: bye. Thank you.