Jorge Pullin: Okay. S0 0ur speaker today is Eric Lewandowski, who's speak about conformally invariant approach to
Elzbieta Lewandowska: Thank you.
Elzbieta Lewandowska: Hello, everybody.
I welcome you from war. So
Elzbieta Lewandowska: perhaps some older.
Elzbieta Lewandowska: The listeners remember the Conference G. R. 20, G. R. 20, and am all the 10
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
Elzbieta Lewandowska: a king's castle.
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.
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
Elzbieta Lewandowska: equipment and the support. Let me turn to the scientific part
on my talk
Elzbieta Lewandowska: A.
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
Elzbieta Lewandowska: a conformal completion to the conformal boundary of space, time. when a
Elzbieta Lewandowska: when conformally invariant framework would be, would be useful.
Elzbieta Lewandowska: Well, but since Einstein equations are not conformal, invariant, well, the the the the 2 things somehow are not compatible with.
Elzbieta Lewandowska: He's out there.
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.
Elzbieta Lewandowska: I would like to. I will use
Elzbieta Lewandowska: the mathematical notion which is called
normal conformal cotton connection.
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.
Elzbieta Lewandowska: Yeah, our student.
Elzbieta Lewandowska: Okay. So
Elzbieta Lewandowska: the way current town introduced his various structures was quite.
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
Elzbieta Lewandowska: a generalize
Elzbieta Lewandowska: this example by keeping some properties and and relaxing some some other.
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.
Elzbieta Lewandowska: Consider Subgroup. H of the group G. And consider the question
Elzbieta Lewandowska: bundle. This is the principal fiber bundle. The bundle space is the group, the bigger group.
Elzbieta Lewandowska: the bundled base Space Space Manifold is the quotient g quotiented by H.
Elzbieta Lewandowska: And this structure group is is the subgroup
Elzbieta Lewandowska: H. So in this case.
Elzbieta Lewandowska: what is the cartoon connection? It's just the them a motor carton, one form
Elzbieta Lewandowska: itself.
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?
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.
Elzbieta Lewandowska: So so it means that it's in some sense defines and map between the space
Elzbieta Lewandowska: tangent to the group
Elzbieta Lewandowska: and it to the, to the yeah, I mean to the bundle and the the algebra of of the group.
Elzbieta Lewandowska: Well, in in the case when the bundle is the group itself. This is quite, quite obvious.
Elzbieta Lewandowska: Not a a map. and
due to identity satisfied by my work. Our town form
Elzbieta Lewandowska: a
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.
Elzbieta Lewandowska: Yeah.
Adam Bac: yes.
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
Elzbieta Lewandowska: of this Carton connection is 0.
Elzbieta Lewandowska: Okay, so here is the
idea of generalization. Let us
Elzbieta Lewandowska: Hmm.
Generalize this example, such that this curvature is not 0,
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.
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
Elzbieta Lewandowska: as before. However, now we consider a principal, and and we consider a principal fiber bundle P. With the structure group H.
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.
Elzbieta Lewandowska: And now we consider a one, for on P. That's well that has as many properties of the
Elzbieta Lewandowska: which means we want it to take values in the Lee algebra of G.
Elzbieta Lewandowska: Secondly, if we
Elzbieta Lewandowska: transform a mandal
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.
Elzbieta Lewandowska: And also the my cartoon form is a sort of identity map. So it maps
Elzbieta Lewandowska: Victors
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
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.
Elzbieta Lewandowska: And that gives generalization of the marker down form. But in this case this
Elzbieta Lewandowska: this form has a curvature, so so it can be, it can be curved. It's not flat
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
Elzbieta Lewandowska: identity.
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
Elzbieta Lewandowska: It's fine transformations of some vector space. H is the group of all the linear transformations of that vector of space.
Elzbieta Lewandowska: And now we choose a
Elzbieta Lewandowska: So we we start with the money fault
Elzbieta Lewandowska: and the bundle of co-frames
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.
Elzbieta Lewandowska: And so so the dimensions now feet, and the dimension of P is the dimension of G.
Elzbieta Lewandowska: And now we can start with arbitrary linear connection, so with the with the standard
linear connection defined on the to frame bundle. And then
Elzbieta Lewandowska: for this, in our connection we can find an a fine connection.
we which which, which, briefly speaking, because it has one more role in the one more column.
Elzbieta Lewandowska: and it has this such that the curvature of this fine connection
Elzbieta Lewandowska: encodes the curvature of the linear connection
Elzbieta Lewandowska: and the torsion of the linear connection.
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.
Elzbieta Lewandowska: then distortion becomes a part of the curvature of the corresponding a fine connection.
Elzbieta Lewandowska: and then those connections are. or a are pretty well known.
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.
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.
Elzbieta Lewandowska: and we consider the orthogonal co-frame bundle
over to the 2 plus one dimensional
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
Elzbieta Lewandowska: 2 symmetries of Minkowski spacetime. So then for Gamma, we choose the metric connection.
Elzbieta Lewandowska: and for a we choose the a fine Carton connection, which is the restriction of the
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.
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
Elzbieta Lewandowska: theoretical physics
Elzbieta Lewandowska: provided that 2 plus one gravity is very
Elzbieta Lewandowska: hey? One technical remark. So
Elzbieta Lewandowska: one technical remark. So
Elzbieta Lewandowska: the
Elzbieta Lewandowska: given a
cartoon connection
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
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
Elzbieta Lewandowska: open subset of space-time. A section of this bundle, and we will pull back
Elzbieta Lewandowska: connections and curvatures. But due to the the covariance of pullback and exterior, derivative actually.
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
Elzbieta Lewandowska: with respect to some
Elzbieta Lewandowska: hey
Elzbieta Lewandowska: field
Elzbieta Lewandowska: and the corresponding curvature transforms a little simpler. That's why we we like curvatures
Elzbieta Lewandowska: of what will you find connections.
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
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
Elzbieta Lewandowska: precise. We could go back to to Bundles and you. We could formulate everything in terms of of the bundles.
Elzbieta Lewandowska: Okay, so this is a general general introduction to cartoon connections, and not really very exact and complete.
Elzbieta Lewandowska: So let us now turn
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.
Elzbieta Lewandowska: Okay. So this is a working definition.
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
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
Elzbieta Lewandowska: so cause here four-dimensional space time. Consider a metric tensor on this spacetime Here i'm using a normalized
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.
Elzbieta Lewandowska: We will also use the volume of this
defined by this frame.
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.
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.
Elzbieta Lewandowska: Be
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.
Elzbieta Lewandowska: So here we have.
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.
Elzbieta Lewandowska: and you can think of this as more more more precisely as the group
preserving
Elzbieta Lewandowska: this form queue. In the middle of this form we have the form, Etta.
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.
Elzbieta Lewandowska: and the matrices that preserve this form are
Elzbieta Lewandowska: set our our
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.
Elzbieta Lewandowska: or in some, or in cartoon. But but it is a working definition which is, which is equivalent.
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
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?
Elzbieta Lewandowska: Gumma is the metric.
Elzbieta Lewandowska: and a
Elzbieta Lewandowska: a, not a, a. a a twisting connection.
Elzbieta Lewandowska: and a a P.
Elzbieta Lewandowska: Is so, so, so, so P. Is constructed from the Ricci tensor
Elzbieta Lewandowska: and from the Ritchie Scalar. So so these are these are the elements of the Riemann tensor and a
Elzbieta Lewandowska: and well, and from them we construct the the, the, the the Ricci. This.
Elzbieta Lewandowska: So this we turn it t0 0ne form, and we put put here.
Elzbieta Lewandowska: So. given theta. We, we can calculate the the metric and and a a rotation free connection. and then
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
Elzbieta Lewandowska: we we could define something else. But what is so special about this matrix
Elzbieta Lewandowska: is that it has a very nice transformation property. So, namely. if we rescale our frame by a function F.
Elzbieta Lewandowska: And for this function we construct the following matrix.
Elzbieta Lewandowska: So here this matrix has zeros. This is the direct hit, the chronic or Delta.
Elzbieta Lewandowska: A. However, here this on this lower
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.
Elzbieta Lewandowska: So this matrix itself is a bit complicated. It contains derivatives of F.
Elzbieta Lewandowska: However, that transformation law is very nice.
Elzbieta Lewandowska: A. Of course we can also apply Lawrence transformation to the frame.
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
Elzbieta Lewandowska: now a. To tell you all the truth.
Elzbieta Lewandowska: In this theory we also applied the following gauge transformations where B is arbitrary one, I mean, these are
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
Elzbieta Lewandowska: the arbitrary one form. So so these are components of the one form.
Elzbieta Lewandowska: and this is B squared.
Elzbieta Lewandowska: However, this those gauge transformations. They
Elzbieta Lewandowska: we're used
Elzbieta Lewandowska: actually to to make this and this term 0. So so in the in my definition, I gauge fixed already
Elzbieta Lewandowska: a little disconnection. And and so those transformations are not allowed.
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
so so here we we have this gauge fixed
Elzbieta Lewandowska: the version, and and we don't apply. However, if somebody wanted to have a general version then and then.
and then one can just just
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.
Elzbieta Lewandowska: A question.
Erlangen: Hi! Hi! You!
Erlangen: So these these other these.
Erlangen: How how can I understand them?
Elzbieta Lewandowska: I don't understand it, either. So that's why I I
Elzbieta Lewandowska: gate. But do do we know how they act on Teeta?
Elzbieta Lewandowska: They are somehow related
Elzbieta Lewandowska: with, okay, so I can. So so I can give you. I can give you interpretation
Elzbieta Lewandowska: which will be familiar to to to many of you, namely.
Elzbieta Lewandowska: is the following: that we first extend our 4 dimensional space time
Elzbieta Lewandowska: to be a section of a of a five-dimensional null call.
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.
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.
Elzbieta Lewandowska: to this additional vector that is transverse t0 0ur t0 0ur
Elzbieta Lewandowska: now surface, and to the choice of this vector
Elzbieta Lewandowska: so
Elzbieta Lewandowska: so they are not from our space that they are.
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
Elzbieta Lewandowska: no currents. Thank you. Thank you for for this question.
Elzbieta Lewandowska: Okay. So now, what about the curvature of our connection. So let me remind you that the connection transforms in this way.
Elzbieta Lewandowska: so the curvature will transform in a in there nicer way.
Elzbieta Lewandowska: Now i'm sorry. But here, this this guy here I
Elzbieta Lewandowska: it it should be on some other transparency. So just ignore, ignore this. What is here. Just Just ignore this thing
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
Elzbieta Lewandowska: do the calculation, then we find that curvature is actually quite
Elzbieta Lewandowska: nice. So it has zeros here, and zeros here the vial tense, or in the middle.
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
Elzbieta Lewandowska: So so you can see that. For instance, if Einstein equations are satisfied, then we will have 0 here and 0 here.
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
Elzbieta Lewandowska: Hmm.
Elzbieta Lewandowska: on the level of the connection, on the curvature. Well, maybe also, if if a
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.
Elzbieta Lewandowska: for, for for for for for understand equations, if they happen to be satisfied by the metric test.
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.
Elzbieta Lewandowska: Okay, and and please ignore this. I will go back to this later. Okay.
Elzbieta Lewandowska: now, we can it sync of our
Elzbieta Lewandowska: a
Elzbieta Lewandowska: part. Time connection is of a young meals field. So we can
Elzbieta Lewandowska: act on this with
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.
Elzbieta Lewandowska: and after the calculation we find
Elzbieta Lewandowska: that the result that we have a lot of zeros.
Elzbieta Lewandowska: The only non-zero are the column
Elzbieta Lewandowska: left column and bottom row.
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:
Okay.
Elzbieta Lewandowska: And and the coefficients are certain tensors. This tensor looks like this. and it is known as the So this back tensor
Elzbieta Lewandowska: you can. Now you can see that if
Elzbieta Lewandowska: space-time satisfies the Einstein equations. So if
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.
Elzbieta Lewandowska: So this is exactly one of
Elzbieta Lewandowska: a useful in this whole call
Elzbieta Lewandowska: game elements, namely.
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
Elzbieta Lewandowska: the back tensor vanishes. So this is a. And and this vanishing is
Elzbieta Lewandowska: a a conformally invariant property. So we can take
Elzbieta Lewandowska: any representative of the conformal class of
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.
Elzbieta Lewandowska: If it is 0, then? Not necessarily. And I will turn to this
Elzbieta Lewandowska: later. No. in terms of our
Elzbieta Lewandowska: our very cartoon connection, the vanishing of the back 10. So it's like like the young meals equations.
So it takes this this for
Elzbieta Lewandowska: which suggests to us some
Elzbieta Lewandowska: some, a, a. some considerations which will be presented
Elzbieta Lewandowska: presented below.
Elzbieta Lewandowska: And now let me make one more comment.
Elzbieta Lewandowska: So
Elzbieta Lewandowska: we can consider this Spino representation of the in a normal conformal carton connection.
Elzbieta Lewandowska: which means we can we know how to represent
one form in terms of spinner, so so we can also extend it to this.
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.
Elzbieta Lewandowska: unless there is some newer addition.
Elzbieta Lewandowska: This is exactly equivalent to the to the normal conformal Carton collection.
Elzbieta Lewandowska: It was observed for the first time by by by a guy whose name was Merkulov.
Elzbieta Lewandowska: You. It was rational region, I think, working somewhere in the Uk.
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
Elzbieta Lewandowska: in particular, if this connection.
Elzbieta Lewandowska: in fact, takes values in the algebra as you one comma 2 embedded in S. O. T0 4,
Elzbieta Lewandowska: which means, if it it can be gauge to this form.
Elzbieta Lewandowska: then something very unusual happens then the space-time conformal geometry admits solution to the twister. Equation
Elzbieta Lewandowska: and Such Metric tensors are are not trivial. They are either Minkowski or they are.
Elzbieta Lewandowska: or or this solutions are not twisting. Then there is there. There are some sort of
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
Elzbieta Lewandowska: reason, and among those space times you can find
Elzbieta Lewandowska: examples. There were found examples, explicit examples of metrics, such that the back me tensor vanishes.
However, they are not conformal to Einstein.
Elzbieta Lewandowska: just to make sure that that this condition is really is only necessary, not
Elzbieta Lewandowska: sufficient.
Elzbieta Lewandowska: Okay. So here was my
Elzbieta Lewandowska: So I completed here the introduction to the a normal conformal part time connection.
Elzbieta Lewandowska: And now let me turn to applications which we which we would like to.
Elzbieta Lewandowska: to which we propose in in our work.
Elzbieta Lewandowska: Okay, so let us
right for the a normal conformal carton connection.
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
Elzbieta Lewandowska: the connection. it is the the the frame.
Elzbieta Lewandowska: and and actually and and and and here this star is also
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.
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
Elzbieta Lewandowska: so. So there is given some some orthonormal frame.
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
Elzbieta Lewandowska: a a young meals looking
Elzbieta Lewandowska: for differential 4, for
Elzbieta Lewandowska: well we know very well that in 4 dimensions. This 4 form is conformally invariant.
Elzbieta Lewandowska: So if we rescale if we rescale a metric tensor, then
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
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.
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.
Elzbieta Lewandowska: and and i'm sure that many people considered such such lagrangian
Elzbieta Lewandowska: A. Then the new element which we introduce is that we want to
Elzbieta Lewandowska: treated by using this this a cartoon connection. So so, then we think we, we.
Elzbieta Lewandowska: this is proper, because of
Elzbieta Lewandowska: of this nice properties of that connection.
Elzbieta Lewandowska: Okay. So now let us consider the variation of this lagrangian.
Elzbieta Lewandowska: So let's go slowly.
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
Elzbieta Lewandowska: it, it is.
Elzbieta Lewandowska: it is we, i'm, varying a with respect to, to, to, to this call free.
Elzbieta Lewandowska: plus the term when when we there is a variation of holes dual and
Elzbieta Lewandowska: plus a
Elzbieta Lewandowska: plus the term which can be
Elzbieta Lewandowska: integrated
Elzbieta Lewandowska: integrated by
Elzbieta Lewandowska: here.
Elzbieta Lewandowska: So this is this: this boundary term.
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
Elzbieta Lewandowska: that the Hodge dual
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.
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.
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
obtain
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
Elzbieta Lewandowska: plus the boundaries.
Elzbieta Lewandowska: and this boundary term
Elzbieta Lewandowska: can be promoted to the.
Elzbieta Lewandowska: to the simplectic
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
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
Elzbieta Lewandowska: hey?
Elzbieta Lewandowska: A nice proposal for simplicity, density
Elzbieta Lewandowska: A,
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.
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
Elzbieta Lewandowska: it it. With respect to those transformations.
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
Elzbieta Lewandowska: the Hodge dual of the derivative of the minus scout intens, or plus variation of gamma, which
Elzbieta Lewandowska: how to do all of the of the 2 form obtained from the vial. Answer.
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
Elzbieta Lewandowska: that conformal invariance will be used
Elzbieta Lewandowska: in in a few slides below.
In order to understand what actually we are doing. Where we are.
Elzbieta Lewandowska: we decompose the the young muse lagrangian with the we we call it
Elzbieta Lewandowska: int0 2 terms. The first term is the euler. a density. so after integrating it becomes the euler
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.
Elzbieta Lewandowska: the curvature, 2 4,
Elzbieta Lewandowska: and the variation of this euler part.
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
Elzbieta Lewandowska: choose to be this on that and the second term L one.
Elzbieta Lewandowska: It can be written in this way in terms of this scout and tensor.
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
Elzbieta Lewandowska: potential which can be calculated. and so to be to be
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
Elzbieta Lewandowska: simplectic potential.
Elzbieta Lewandowska: plus there
Elzbieta Lewandowska: a simplic potential
Elzbieta Lewandowska: of this and
Elzbieta Lewandowska: of this lagrangian of this other Lagrangian which which could be used to to give them the back
Elzbieta Lewandowska: it the the of the back T. So, however, the Lagrangian itself is not conformally invariant.
Elzbieta Lewandowska: So s0 0ur choice of Lagrangian makes it conformally environment. Okay. So let us see what we can learn
Elzbieta Lewandowska: from this. Okay. S0 0ne of applications of this framework
Elzbieta Lewandowska: is that actually we go back to the Einstein equations. Right? Okay. So if the Einstein equations are are satisfied.
Elzbieta Lewandowska: then
Elzbieta Lewandowska: the back tensor vanishes, and then the scout and 10 minus scout and tensor just becomes this.
Elzbieta Lewandowska: So it is just constant times, times, times this.
Elzbieta Lewandowska: and then our our simplic potential, takes this simple form. So this is variation of gamma, which
Elzbieta Lewandowska: how to do all of the bio
Elzbieta Lewandowska: to form
Elzbieta Lewandowska: A.
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
Elzbieta Lewandowska: this for. and this form should be pretty familiar to you, namely, if we consider a
Elzbieta Lewandowska: Lagrangian, which is just the the Einstein Hebrew lagrangian, however, written in terms of the of the of the orthonormal frame.
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.
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
Elzbieta Lewandowska: so. So then this Lagrangian gives the I shine here. He gives them a simplic potential of this, for
Elzbieta Lewandowska: So now, if we compare this
Elzbieta Lewandowska: With this. then we can
Elzbieta Lewandowska: a a conclude the following: that the simplic potential density
Elzbieta Lewandowska: defined by the is the a simplic potential density of the theory that would be defined by the Euler invariant
Elzbieta Lewandowska: minus
Elzbieta Lewandowska: this number times the simplic potential density of Einstein Hilbert theory.
Elzbieta Lewandowska: However, this is all true only if we restrict ourselves
Elzbieta Lewandowska: to the solutions of the Einstein equations.
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
Elzbieta Lewandowska: currents and etc.
Elzbieta Lewandowska: Okay, so let me show you how how we can apply this to
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.
Elzbieta Lewandowska: it works the same. So let us consider them pepper man Graham coordinates.
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
Elzbieta Lewandowska: and describe the conformal infinity corresponds to Rho equals 0. So here we have
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
Elzbieta Lewandowska: rescale metric. And now
Elzbieta Lewandowska: notice the following. Since our theory is conformally invariant.
Elzbieta Lewandowska: And actually we don't require, I mean, conformal to rescaling already
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.
Elzbieta Lewandowska: but if we apply it to this metric, then certainly the simplic potential density will be finite on the on the sky.
Elzbieta Lewandowska: But our physical. the coming from the physical metric simplic potential density equals due to the conformal invariance, this one.
Elzbieta Lewandowska: So it means that our
Elzbieta Lewandowska: original simplic potential density which we've defined for this this Cartonian meals potential will be finite
Elzbieta Lewandowska: all the way to this cry, and on the screen, too.
Elzbieta Lewandowska: A. And this is sort of surprising, because if instead, we take Einstein.
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.
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.
Elzbieta Lewandowska: So what there is this, this familiar analysis of the expansion of the metric tensor
Elzbieta Lewandowska: on on the scri. We know that so we have this G 0 which gives the the background
Elzbieta Lewandowska: constraints. It is traceless and due to Einstein equations, traceless and divergence less and otherwise. It's free.
Elzbieta Lewandowska: And we want to check this formula
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
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.
Elzbieta Lewandowska: S0 0n the sky. We introduced the volume. Hmm. By contracting. by contracting.
Elzbieta Lewandowska: Our four-dimensional volume is the vector field defined by the
the the by the vector for the D.
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
Elzbieta Lewandowska: finite result.
Elzbieta Lewandowska: which is a variation of. So if we apply this
Elzbieta Lewandowska: to variation of our frame.
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
Elzbieta Lewandowska: of this non- physical metric.
Elzbieta Lewandowska: then this is what we obtain. So this is variation of this G. 0
Elzbieta Lewandowska: in this expansion. Times T. 0, where T. 0 is, in fact, G 3.
Elzbieta Lewandowska: Okay. let us compare it with them
Elzbieta Lewandowska: with the known
methods.
Elzbieta Lewandowska: So
once again I re-wrote what we would we
Elzbieta Lewandowska: obtained was was
this.
Elzbieta Lewandowska: A. On the other hand.
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.
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
Elzbieta Lewandowska: it then we obtain from such action we obtain the pullback of the of the
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
Elzbieta Lewandowska: which agrees also with with what I, with this decomposition which was made
Elzbieta Lewandowska: out here.
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
Elzbieta Lewandowska: in this decomposition, maybe I I will write this decomposition again. A.
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
Elzbieta Lewandowska: times, the simplic potential of the
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
Elzbieta Lewandowska: defines a a vanishing the currents. And however this is this: is there the result?
Elzbieta Lewandowska: Talking about currents? We could go back to the, to t0 0ur theories, so so, s0 0nce again to the beginning.
Elzbieta Lewandowska: and we can ask, what are the current? So there is the
Elzbieta Lewandowska: I consider that the few more reason this theories the
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
Elzbieta Lewandowska: the a current for neither current Northern Ireland for the
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
Elzbieta Lewandowska: charge corresponding to the theomorphism is this.
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
Elzbieta Lewandowska: so? It is exterior derivative of the corresponding charge. So here this charge looks like valid talents, or con contracted with with.
Elzbieta Lewandowska: If this is Lawrence rotation, so it will be contracted with generator of Lawrence rotation. If this is conformal.
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.
Elzbieta Lewandowska: Okay. So to summarize.
Elzbieta Lewandowska: to summarize.
Elzbieta Lewandowska: we defined here a an approach to the A, a. to to to space time, which is conformally invariant.
Elzbieta Lewandowska: Hmm.
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
Elzbieta Lewandowska: theory.
Elzbieta Lewandowska: So whatever currents are defined in terms of this conformal theory, they also
Elzbieta Lewandowska: have them.
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.
Elzbieta Lewandowska: but the advantage our formula we have is that they are conformally invariant. So
Elzbieta Lewandowska: they, for instance, we can go with them to the conformal completion, and we we don't
Elzbieta Lewandowska: a a
Elzbieta Lewandowska: face, any any infinities there.
Elzbieta Lewandowska: So this is it. Thank you.
Elzbieta Lewandowska: Thank you very much.
Elzbieta Lewandowska: Hello, Hello.
Hey, You A question in the chat.
Elzbieta Lewandowska: Okay, let me
let me see.
Elzbieta Lewandowska: Okay. So in my talk I considered a lambda bigger than 0. So so it in here I gave some
specific, exact formulae.
Elzbieta Lewandowska: I is the
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
Elzbieta Lewandowska: all the
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
Elzbieta Lewandowska: what is sensitive on this, whether this is ads as anotic ads or asymptotically. Ds.
Elzbieta Lewandowska: Is this a a
Elzbieta Lewandowska: form of metric tensor? And then some
details come, and but but I think that this going to the limit would be similar, and we would obtain a similar.
Elzbieta Lewandowska: a similar solution.
Elzbieta Lewandowska: A.
Elzbieta Lewandowska: So
questions for you.
Elzbieta Lewandowska: Thank you for asking.
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
Simone: internal frame and Carton connection construction. Maybe. Can you give us some
Simone: some broad comment on that? Apart from the they can. You got aspects?
Elzbieta Lewandowska: Okay. So vile. Tensor by itself, indeed, is conformally invariant. however.
Elzbieta Lewandowska: the the the proper, but but it's not really by itself
Elzbieta Lewandowska: curvature of of anything except
Elzbieta Lewandowska: the normal conformal cartime connection. So this is like
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
Elzbieta Lewandowska: components they can be collected into a single conformally invariant object. So so we just
Elzbieta Lewandowska: restore, restore some
Elzbieta Lewandowska: geometric a
Elzbieta Lewandowska: structure which is behind the the file tensor
Elzbieta Lewandowska: various identities.
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
Elzbieta Lewandowska: so there is a question if we could
somehow.
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
Simone: formulation. But due to your work we know we know what is the the difference. So I
Elzbieta Lewandowska: thank you.
What? The
Wojciech Kaminski: So it's another, a comment
Wojciech Kaminski: so so to to
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.
Wojciech Kaminski: And this is very important in our work. So if we just use by square.
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
Wojciech Kaminski: of the month. So in this a a carton approach, we just getting it for free. So this was a common.
Simone: I see. Thank you.
Erlangen: Can I ask another question
Erlangen: in this normal Qatar connection framework, the symbolic structure actually differs by the variation of the euler term?
Elzbieta Lewandowska: Yes.
Erlangen: so how does this affect? This, then, also shows up in the definition of the charges, I suppose.
Erlangen: Yes.
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?
Erlangen: And
Erlangen: and then whether this difference
Erlangen: is only. And how this difference also change better. This difference changes the fluxes, like the change of
Erlangen: charges between different cross-sections, for instance.
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
Erlangen: the flux.
Erlangen: the difference between charges between different cross section.
Erlangen: Is there something like that, or or is it? Is it not?
Elzbieta Lewandowska: Yes, it it seems to be true; for for the the few more reasons that that this.
Elzbieta Lewandowska: that this euler term it doesn't change the the currents.
Elzbieta Lewandowska: so it provides the
Elzbieta Lewandowska: finishing currents. However, it changes our notion of of charges. It it suggests some different charges, but
Elzbieta Lewandowska: but it doesn't change the evolution of the charges.
Elzbieta Lewandowska: and it restores that conformity.
invariance, or covariance.
Erlangen: Thanks for this clarification. Thanks
Okay, any other questions for you.
If not, let us. Thank you. Thank you, Eric, very much for the Okay.
Elzbieta Lewandowska: Thank you for coming.
Simone: bye. Thank you.