Jorge Pullin: Our speaker. Today, Simonis Paciale will speak about localized energy density of gravitational waves.
Simone SPEZIALE: Thank you very much. So I will.
Simone SPEZIALE: In this talk, I will present some results. Some of the results that appeared in a recent paper with a buy that was followed, but there was anticipated by a short version, and they're both on the archive and then some other results with Antoine, my Phd. Student, on his third year that should appear soon.
Simone SPEZIALE: and I should start with the an apology. The fact that in this talk I will not talk about loop quantum gravity. But actually my motivations for doing these things started very much. We look onto gravity.
Simone SPEZIALE: And it was quite some time ago when I was asking this question about what happens to this nice interpretation we have of quantum space. If, instead of using a space like upper surface, we use an Oliver surface and the interested in doing that
Simone SPEZIALE: is, then there's there's access to constraint, free data that have been identified on our library surfaces so that maybe one can get a a an idea what the physical quanta are instead of having the problem of solving that constraint on the contour space networks.
Simone SPEZIALE: Now, asking this type of questions actually ended up opening, being equivalent of taking a red pill and going down a very long rabbit hole, out of which I still haven't gotten out, because it has been. very, very interesting. Research directions, with lots of interesting questions to discuss, and lots of excellent colleagues. With whom I collaborators with whom I've been writing papers on this topic in the last few years.
Simone SPEZIALE: So with that apology, let me tell you that today's talk would be on some of the most basic questions that one has in a classical general activity, and which is the meaning of energy and a gravitational theory. And how do we establish the gravitational waves of physical and carry energy? And
Simone SPEZIALE: because I like to be broad, and at least at the beginning be as broad as possible. I've prepared that an introduction to the problem with some very basic stuff before I will move on on more technical new results that concern these 2 papers I have mentioned.
Simone SPEZIALE: and so please interrupt for any questions at any time, on any of these parts.
Simone SPEZIALE: So first part, so introduction to the problem. I would like to talk about what happens first of all to Hamiltonian generators in the presence of radiation, and then the special case of the additional subtlety that occur in general activity in the second time.
Simone SPEZIALE: So now, just to refresh ourselves, one approach to the discussion of energy and classical mechanics, there is one very elegant and simple way to talk about the energy which is to identify it as the generator of time. Translation. Symmetry!
Simone SPEZIALE: So this can be done by looking at the time, flow the over Dt. And hooking it with the Symplet. 2 form omega, and that gives us the Amazonian, which in simple system is kinetic term plus potential term.
Simone SPEZIALE: Now, this we can do straightforward in a finite the system with a fine number of degrees of freedom in 50, or you don't want us to dig into account boundary conditions, because, for instance, consider a finite region given by a time like cylinder with Sigma 0 and Sigma one initial and final slice.
Simone SPEZIALE: and t the time like boundary, then one can have waves. So, for instance, wave solutions that go out or come in, and therefore the system is open, and in this case
Simone SPEZIALE: one does not have a simple amygdalion description. So what can be said in that case
Simone SPEZIALE: well, the result that one has from just analysis of field ofigations of motion is that this implanted 2 form is closed on shell, and therefore, by Stokes theorem, we know that it's integral on Sigma would be the same as the integral on Sigma one plus the integral on the boundary.
Simone SPEZIALE: Now typically in in field theory. You would assume, for instance, that this boundary is very far away. A spatial infinity feels a a vanish go down to 0 at some speed, and therefore there is no flux of Omega. Let me go with simple flux, namely, the integral of Omega, and T vanishes at the boundary.
Simone SPEZIALE: In this case the simplest 2 form is conserved, namely, it's integral. On Sigma. 0 is the same as the integral on Sigma one, and then one gets an Amazonian, which, for the time, time, translation vector field would be given in terms of some integral of space, of an Amazonian density, which, for, let's say a scalar field would be like something like v dot square plus anything else.
Simone SPEZIALE: And I'm mentioning the fact that there is a notion of local energy density, because, of course, this is something that will be lost in the gravitational case the notation I'm using with this capital I and this big delta is the infinite dimensional version of Little I, the interior product
Simone SPEZIALE: and little D, the exterior derivative.
Simone SPEZIALE: So then, on the other hand, if there is flux allowed, let's say, outgoing or incoming radiation. Then this Intel on the boundary of Omega is no longer 0, so we cannot really say. Of course it's still closed, not shell this in 32 form. We cannot say that this, conserved in time in the sense of it, is independent of the cushy slice we use in this cylinder here.
Simone SPEZIALE: and because it's not conserved, these means that also you might end up having a an issue with the usual construction of a Mithonian vector fields. Because if you have some vector fields that deform the corner of Sigma by moving it upwards or downwards. For instance, these these transformations will actually see the energy flowing out or coming in. And therefore, these transformations in phase, space cannot be a Mithonian
Simone SPEZIALE: and of course there are very interesting examples of such transformations. Precisely. Time translations at this boundary would be of the sort we move in the future. This corner of Sigma 0, we see fract coming in or or out, and therefore that transformation it seems intuitive that it cannot be just Hamiltonian transformation, because doesn't preserve the sympathetic structure.
Simone SPEZIALE: So to make the description simpler and the this entangle incoming from outgoing is convenient to switch from a time like boundary to a null boundary. Then I can take it this way or the river this way, and I will only have to be concerned without going flex or a opposite way, and then it will be only incoming flux that I have to deal with, and to make the discussion even more precise. Let's push this boundary all the way to infinity.
Simone SPEZIALE: So it it is actually a section, a region of scribe. Then I can think of Sigma 0 and Sigma one as 2 partial cushi hyper surfaces like some hyperbolic slices that intersect sky at 2 given cuts.
Simone SPEZIALE: And now I can talk about different face spaces. For instance, the phase space associated with Sigma 0 would contain associated with Sigma 0. So it's like a partial cause. She slice because it misses. Some of the information that is associated with a complete kushi slice, which is, for example, the green line here.
Simone SPEZIALE: so an upper surface that extends all the way to spatial infinity 0. This would be a complete cauchy slice. Then I can also talk about the radiative data
Simone SPEZIALE: and the radiative phase space which is associated with this region. N. Of scribe. and so the information contained there together.
Simone SPEZIALE: with the I wrote these the wrong way around. I'm sorry
Simone SPEZIALE: the last sentence here should have corrected it. So the information contained on the radio, this piece on N and on the partial coaches license one is equivalent to the information contained on Sigma 0.
Simone SPEZIALE: So here should have been Omega Sigma. 0 also contains the cushion data on Sigma, one union end.
Simone SPEZIALE: and
Simone SPEZIALE: it's interesting and useful to introduce also currents for this in practice. 2 form, I'm gonna call it little Omega to distinguish it from Big Omega, so that by integrating little Omega that I derived from the field equations or the Lagrangian. If you want
Simone SPEZIALE: for a given theory that I integrated over the different surfaces, and I get these sympathetic forms associated with these face spaces I've been talking about.
Simone SPEZIALE: Now, how about the Amazonians that one can construct in these different face spaces?
Simone SPEZIALE: Now, in general, if you have a symmetry. Xi. Would be mainly interested in the case of gravity and dipomorphisms, but some of these formulas actually are more general than that.
Simone SPEZIALE: And so one thing that one can prove is that the equivalent of the final dimensional hooking of the sympathetic to form with the symmetry generator
Simone SPEZIALE: is not necessarily just the delta of
Simone SPEZIALE: the Amazonian. But there might be an extra piece. Let me forget about that extra piece for a second. Let me just focus on what one gets for the Miltonian. If size is the symmetry that we expect from another. Tm, that the Amytonian should be associated with another current.
Simone SPEZIALE: and in the case of diffomorphisms, this is the form that one can write for the nether current where Tita is the boundary term that appears when you do the variation of the Lagrangian. And this interpretation of a simple potential.
Simone SPEZIALE: So, for instance, if we take just a scalar field in Minkowski, then size any of the healing vectors.
Simone SPEZIALE: And, for instance, if by Xi, we take the translation killing, vector then another current would be the energy, momentum tensor.
Simone SPEZIALE: And then these are the currents. If we integrate over Sigma 0, we will get the total energy momentum on Sigma 0. Whereas if we integrated over n this piece of of scribe. Then we will get actually the energy momentum flux.
Simone SPEZIALE: a gross cry.
Simone SPEZIALE: So you see, these different phase spaces, omega sigma 0 or omega N. May come with the their own different generators. One of them is interpretation as total energy, momentum, and the other one as total energy, momentum flux.
Simone SPEZIALE: and this you can do for any size, so it could be. and excited is a symmetry.
Simone SPEZIALE: For these example, of of scalafili, minkowski.
Simone SPEZIALE: Now, of course, there is this a little extra piece I didn't talk about yet these diix ita.
Simone SPEZIALE: And so, in order to include that this total energy, momentum, and energy flux are really the Amazonians. One would have to discard this term.
Simone SPEZIALE: because otherwise it spoils. I mean the I mean tone, and if you want is non integrable, or is not an exact one form in field space.
Simone SPEZIALE: So when or how we can get rid of that term. Well, in the example of the scalar field the simplest 2 form is
Simone SPEZIALE: delta pi delta phi, or delta phi delta pi.
Simone SPEZIALE: and then one can see explicitly that this term has this form, pi delta phi, and then the boundary. 3. Volume 3 form. So let's say, we are on N scalar product with psi
Simone SPEZIALE: so examples in which these terms vanishes, and then we have integrability is, if size tangent to the corner that I'm taking the pull back on, because then this term vanishes.
Simone SPEZIALE: or alternatively, if at that corner pi is equal to 0 or delta Phi is equal to 0
Simone SPEZIALE: now by equal to 0. You can think of it as some stationary condition. So I'm looking for some special solutions, such that the velocity is 0, because pi is just phi.so in that sense these solutions are stationary.
Simone SPEZIALE: whereas delta frequency, we can think of it as be imposing some conservative boundary conditions.
Simone SPEZIALE: Now notice that, for instance, these
Simone SPEZIALE: stationary consideration for integrability defines actually some in some sense, a dance up a dense subset, namely, solutions that at this corner have pi got 0, but not at the rest of N. For instance.
Simone SPEZIALE: And actually one thing that we discussed in the last paper with a buy, and that we come back to. and that actually, you can make this intuition rigorous by introducing a topology in the radiative phase space.
Simone SPEZIALE: and that allows you to define the Amazonian generator, using only the integral piece and then extending it, using the continuity defined by this topology to the full face space. So we'll come back to this. But let me just mention it in this simple example, where
Simone SPEZIALE: there are any other gravitational theory.
Abhay Vasant Ashtekar: and and also for the audience. It's always confusing by your your U use of word stationality, because for a stationary means, time independent.
Abhay Vasant Ashtekar: And these are not time independent solutions. Right? I mean, it's just 5, I mean, I can have a coach's license on which pi vanishes somewhere, and that doesn't make it stationary solutions confusing
Simone SPEZIALE: absolutely. So if you put right. So pi is phi dot equals 0. So if you post this everywhere on N,
Simone SPEZIALE: they would be talking about some stationarity everywhere. But if we just impose it up one slice of one instead of time. As you just said, this is not imposing stationarity, so in this sense this is not. This is a dense subset. Intuitively, because you're not imposing stationary, we're still allowing for arbitrary time dependence.
Simone SPEZIALE: Yes.
Simone SPEZIALE: Now this discussion of detail media just gave with the example of the scanner field is actually much more general than that.
Simone SPEZIALE: So in using these generous formulas in principle, the between exist, if these abstraction is a total variation, and in general this can occur if size tangent to the cross sections.
Simone SPEZIALE: and intuitively, that means that these are transformations that don't move the corner. So whether there is flux or not.
Simone SPEZIALE: It doesn't matter, because these transformations don't see the dissipations. And so in that sense they are naturally a Miltonian.
Simone SPEZIALE: and the alternative is.
Simone SPEZIALE: if Theta vanishes now in the world zup as construction, which I will come back to, and that I've already mentioned. This Theta vanishes if one looks only a perturbations around stationary solutions now, stationary solutions in the sense that I just mentioned, namely, there is no flux nowhere across of N,
Simone SPEZIALE: and therefore these are very special solutions that are measure 0 subset in the face space.
Simone SPEZIALE: So in that sense. These these Amazonians are
generators in a very big sense. So only for perturbations around this measure, 0 substance
Simone SPEZIALE: subset. Sorry.
Simone SPEZIALE: But again, one can apply this more general procedure. Not just in the case of the Scalar field number in general, in the sense, then, if it is possible to endure this ready face space without topology.
Simone SPEZIALE: then one can look for the Tita. That vanishes on a then subset identified by the topology.
Simone SPEZIALE: and then do the same procedure of extending it by continuity to define the Newtonian generator everywhere in the phase space. Now, the interesting thing of this procedure is that in the case that in the cases that we started it.
Simone SPEZIALE: it reproduces the same result. The one obtains with the world zoom as procedure, so the construction of charges and flexes is consistent.
Simone SPEZIALE: but the advantage of doing so is that instead of talking about Amazonians. surface charges on a given cross section, Sigma, one talks about a Newtonian, says the fluxes on N,
Simone SPEZIALE: namely, if you go back to the previous discussion I had, we can think of different face spaces, face, and sigma, or face base on end. If you talk about the face base and Sigma, then the natural interpretation for would you like to call Metonian is a surface charges. But these don't exist in general because of the presence of flux, so they only exist in a measure 0 subset
Simone SPEZIALE: around the stationary perturbations.
Simone SPEZIALE: But if we look at the face space associated with the end, so a portion of scribe, then we can define an Amazonian that is valid in the all ready to face using this construction.
IEM-CSIC: But these interpretive but these are Newtonian, does not have the interpretation of a surface charge, but rather of the flux of the symmetry going through Nope. Is that a question? Maybe.
IEM-CSIC: Well, could you comment on ambiguities in detail, like, for instance.
IEM-CSIC: So canonical transformations?
Simone SPEZIALE: Right? Very good. Yes, that's a very nice question a little bit maybe I'll come back to it. But let me answer it right away. So in fact, this split that I'm using here. In the first line in which I say that generator is a total variation plus an obstruction, is actually ambiguous as well can get just pointed out.
Simone SPEZIALE: because the sympathetic potential is not unique. And another current also is not unique. In particular, there are 2 types of ambiguities. There are ambiguities that come from adding boundary terms to the Lagrangian
Simone SPEZIALE: and ambiguities that come from modifying the sympathetic to form by a corner term.
Simone SPEZIALE: Let me remove the second. Let me say that this in practice 2 form is given once and for all, so I don't touch it. So the left hand side is unique of this equation. I'll come back to those at the at the end of the talk, but let me ignore them for the moment. So the left hand side is unique. I still have the ambiguity about being a boundary term to this. To the Lagrangian we changes.
Antoine RIGNON-BRET: but so you think you can mute yourself.
Simone SPEZIALE: Thank you.
Simone SPEZIALE: So if you change did so, if you change the Lagrangian by boundary term, you change, did in such a way that Xiao omega does not change.
Simone SPEZIALE: but the split between delta and nonintegrable changes. In a sense, you are always free to take an integrable term and put it into non-integrable term.
Simone SPEZIALE: So this is where the Worldsupas procedure is important, because it comes in. Sorry, I should have said, because it fixes this ambiguity by giving a list of rules that you have to use in order to select the good Tita, and therefore get rid of this ambiguity.
Simone SPEZIALE: The beautiful thing of this new procedure well, which is basically rooted in the old work of a buy and and struggle of 81 that we've been looking at is that it is the choice of topology is that when you do the step of endowing the phase space with a topology that step
Simone SPEZIALE: not only it allows you to think of the integrable term
Simone SPEZIALE: as the Admin alone, because you extended by continuity, but also it selects what is the right integral term.
Simone SPEZIALE: So this is why I've written. I've written here that the topology is such that a certain theta is 0 on a 10 subset. So if you want to give you an example, a simple ambiguity in Theta is that you could go from Pdq. To Qdp.
Simone SPEZIALE: But when you talk about the topology, for instance, you will choose, for instance, the way we constructed in the paper is that we use, we define topology. We have momentum norm.
Simone SPEZIALE: So, for instance, norm 0 means pi, or P. The momentum equals 0, and so then naturally selects the polarization Pdq. As opposed to Qdp.
Simone SPEZIALE: So, in a sense, this construction kills 2 birds with one stone
Simone SPEZIALE: selects for you. It gets rid for you of the ambiguities and gives you a procedure to think of them. If Tonian, as the flux over the whole face space in spite of the existence of this obstruction.
Simone SPEZIALE: If there are more questions about the meeting I'd be happy to come back to, but I hope that this is enough for now. Wolfgang.
IEM-CSIC: yeah, for sure.
Simone SPEZIALE: Okay, thanks. Okay. So I've already mentioned the various aspects of generativity. But now let me go the deeper into the question of specific question of energy
Simone SPEZIALE: for the sake of the broad community. I would like to start from very basic stuff. because it's always a nice discussion. I hope it won't take too much time.
Simone SPEZIALE: And remember that the reason why Ngr is difficult to talk about A notion of gravitational energy is because of the femorphism invariance that tells us that cannot be a local energy density, as the reason for Scalar, Fielding, Minkowski, phi dot square plus blah blah blah. The terms that have written at the beginning, and the nice intuitive argument that one gives in class is that by the equivalence principle we can always locally set to 0 the gravitational field
Simone SPEZIALE: that means the metric is flat. First, derivatives of the metric are 0. And if you think about a standard energy momentum tensor like for a scalar field, depends on first derivatives of the field. And so
Simone SPEZIALE: an equivalent object like that would be 0.
Simone SPEZIALE: But then, if it is a tensor, it has to be 0 in any coordinate system, if it is 0 in one coordinate system. And therefore we get this argument that there cannot be some energy momentum tensor for gr.
Simone SPEZIALE: Now.
Simone SPEZIALE: this subtle situation actually, had a very strong impact in the development of the field because people were even worried about whether gravitational waves really existed then. And one of the key questions was, Well, they exist if you actually carry energy. But what do they mean by energy? What do we mean by energy of a gravitational field? Since it is so slippery.
Simone SPEZIALE: and I don't know if it was bonded that says this, but in a sense one can be very concrete, and say, if gravitational ways are real well, you should be able to heat water with them. So we have a bucket of water, and we would like to know when gravitational ways pass by whether they actually hit the water.
Simone SPEZIALE: So we have Newton's bucket. Now this is Bondi's bucket is not quite the same, but it's again a bucket full of water that we'll be talking about again later.
Simone SPEZIALE: And so there are various way out. So the most commonly used one is to give up the use of a tensor and use few tensors. This is a Lando, if sheets approach which is championed by the post. Newtonian post-mico. Minkoskiens approaches, and so people in that case usually are happy with setting up a preferred coordinate system for a preferred background. So a lot of covariance is lost in this approach, and this gives rise to subtleties.
Simone SPEZIALE: but it is very common. A less common approach is to well face the fact that the first non-trivial quantities, then, are second derivatives, and then work with the bill. Infant. Answer. But it doesn't even have the right dimensions for being an energy. So this is kind of like a non mainstream approach. I would like to say that the most beautiful approach, and that for me is the
Simone SPEZIALE: deepest one is to use asymptotic symmetries when they exist, or any way observables using these Hamiltonian and other methods.
Simone SPEZIALE: In a sense.
Simone SPEZIALE: the bar can, doesn't allow me to see what's written here. Yes.
Simone SPEZIALE: in a sense. In this case we're solving the problem. What do we mean by energy? But looking at the certain class of quasi local, observable, sometimes called, which are not integrals over the old space in this case, but integrals over the boundary of space. So they're surface integrals.
Simone SPEZIALE: This is how the nether theorem and the Newtonian methods allow us to identify an ocean of energy.
Simone SPEZIALE: and in particular, the way the problem of energy traditional ways was initially settled in the sixties was with this famous bondi formula, the energy loss that tells us that there is a certain quantity e that monotonically decreases as gravitational ways are emitted
Simone SPEZIALE: from a system. So N. Here is called the new sensor, and it's a way of characterizing gravitational radiation in a coordinate, independent way. And this E is precisely a surface integral. So is an example. So what do we mean by energy? Here? We actually mean the total energy of the system, because this is an integral over the sphere at a given cut of scribe.
Simone SPEZIALE: And now the beauty of this simple thing that approach is that both left inside and right inside it can actually be identified as amithonians in these 2 different senses.
Simone SPEZIALE: The left hand side is the surface charge. So it's the Amazonian in the weak sense, on the face space associated with Sigma, whereas the right hand side is the Amazonian in the in the full sense on the radiative phase space.
Simone SPEZIALE: If we are in between these 2 cuts. Okay, so let me tell you how these formulas are derived. How do we get from nether theorem to this formula.
Simone SPEZIALE: So the nova. So these are the formulas I've written before for the Scalar, Fielding, Minkowski. the node where the first one is the nether current associated with, let's say, an isometry.
Simone SPEZIALE: And the second line is the Amazonian generator.
Simone SPEZIALE: Now, the novelty of when the global invariants associated with killing vectors becomes diffumorphism, invariance in general activistic context, or even for gauge theories in which we have local gauge symmetries is that actually, this nether current is exact on shared.
Simone SPEZIALE: So we can write this the D of something that we call the surface charge.
Simone SPEZIALE: For example, if we do these with the choice of split choice of sympathy potential that is usually taken from from the protection, the standard or obvious or naive sympathy potential. Then the surface charge is given by the comma 2 form.
Simone SPEZIALE: and if we do it for a gauge theory, for instance, is given by the electric field the momentum.
Simone SPEZIALE: So then, for this reason, the Amazonian generator also becomes a surface term, because, you see, the obstruction was already a surface term, and the integrable term now also becomes a surface term. So everything is just a surface term, and that's actually pretty, I'm gonna say, consistent with our intuition of gauge symmetry, as being just a redundancy of the description, and not really a symmetry, because what this means is that if there are no boundaries.
Simone SPEZIALE: then they simply 2 form, as the J is the generate along the different morphines, transformations.
Simone SPEZIALE: and so we recover our intuition that diffomorphism are just gauge because they correspond to the general directions of the sympathetic to form. But if you have a boundary, this in practice to form as non-trivial support on boundary, on the thermorphisms that are non-trivial at the boundary, and so we recover a notion of actual symmetries for the diffomorphisms. Now, whether these are symmetries or not. Of course, it depends on the boundary conditions and the solution space that one is looking at.
Simone SPEZIALE: because this might prevent, the some of the diplomorphism, or even all of them
Simone SPEZIALE: to fix ideas. Think of the simple example when we send the boundary to spatial infinity, and then, full of conditions, say that the metrics should be flat down there, so that we say asymptotically, flat matrix, a spatial infinity.
Simone SPEZIALE: But then there's not a unique notion of asymptotically flat metric. We could fix it to be in the form, you know, minus 1 1 one, the standard form. But even that is not unique, because we have the freedom of doing Poincare transformations, and so boundary conditions that say that the metric is the Cartesian flat metric at spatial infinity come with an infinite residual pitch. Symmetry of Poincare transformations.
Simone SPEZIALE: And that's precisely where the non-trivial gauge direct non-trivial directions in the ziplty 2 forms will reside.
Simone SPEZIALE: And so, if one computes the electrical engineer. In this case one gets surface charges, and one can show that the surface charges are precisely
Simone SPEZIALE: the adm surface charges that they were previously constructed. So, in other words, these construction of the Amazon generator automatically gives you what Regent it Boeing used to call the improve the generators. They were adding by hand the boundary terms, so that they so the Demotonian would be differentiable when boundary terms are included.
Simone SPEZIALE: namely, Poisson Bragg. It would generate the expected gauge transformations.
Simone SPEZIALE: And and this is what you automatically get with this construction.
Simone SPEZIALE: That's a a a special infinity. That is simple, because in this case the full of conditions of a synthetic flatness impose that the standard synthetic potential is actually exact, a special infinity, and therefore integrability follows trivially.
Simone SPEZIALE: This also means
Simone SPEZIALE: that the sympathetic to form when we push it to this time and activity that we push it to spatial infinity is pullback is 0. So there's no flux. So everything is consistent. All the vector fields are a Newtonian, because, anyway, there's no dissipation that can be seen as spatial infinity.
Simone SPEZIALE: Now, if you do the limited scribe. On the other hand, there is flux, so we don't expect the to be in exact one form.
Simone SPEZIALE: and, in fact, you do the calculation, and you find out that it is not so. These flux and the system is decipherative.
Simone SPEZIALE: Now, ideally, we would like to identify the no radiative subset of the face space as the set of solutions for which data bar equals 0, so that we could say when there is no radiation.
Simone SPEZIALE: Well, then there is no flux. And then my symmetries are Hamiltonian vector. Fields. The problem is exactly what Wolf already pointed out that Tita Bar is ambiguous because I can take a piece of L and put it in Tita bar.
Simone SPEZIALE: and therefore, by doing this I change the solutions for which Theta bar is 0, and therefore I change the notion of
Simone SPEZIALE: stationarity. So I would like to avoid the disambiguity.
Simone SPEZIALE: And there are 2 procedures so avoiding disability means identifying the fluxes. What should be the correct fluxes, and here have written the explicit formulas that when you do this change or split, you go from Tita to Tita bar. Then accordingly also Q. As to change any changes from queue to queue bar.
Simone SPEZIALE: and the sum of the 2 is still the same, because, like, say, Omega is independent of this change of split.
Simone SPEZIALE: Now, what the Worldsupas procedure did was to narrow down a preferred data bar with a list of requirements and effectively in many cases this gives you a unique data bar. And so the procedure works. And one identifies faxes and charges.
Simone SPEZIALE: The alternative that we're pushing is to actually use this argument of adding a topology, a choice of topology to the face space, so that actually, one can talk about Amazonians in full generality on the radiative phase space and not just around measure, 0 subsets.
Simone SPEZIALE: And actually, the 2 procedures. We check them in 3 different cases. They give the same result. So that's actually quite nice and surprising. They're very different approaches give consistent results.
Simone SPEZIALE: And when you apply to Pms, these consistent results. Select as flux the formula to Dubai and Michael Strobel, add in 81,
Simone SPEZIALE: in which the flux is given
Simone SPEZIALE: portion of scribe between a cat called S. One and another. Cut call S. 2 of the News Times. A certain transformations on the sheer which can be explicitly written like this here F contains super translations and the boosts. With respect to a choice of full issue given by U. Equal, constant.
Simone SPEZIALE: And why are the boosts and rotations.
Simone SPEZIALE: So that's a result that one gets with the willsupus procedure or with the procedure.
Simone SPEZIALE: And then the question is, can we now also identify charges. So these would be so now the flexes will be the Amazonians for the radiative phase space do we have, also charges that we can understand as weak, and for the Sigma face spaces. That's a little bit more that's required as an extra step.
Simone SPEZIALE: It's not so automatic. And again, there are 2 alternative procedures, the world Zoomus. One is basically based on bootstrapping the charges from the fact that we know already an original queue which is common, and we know L, namely, the difference between the bar and the standard simplicity potential. So you can compute the charges like this. But actually, it's a little bit subtle to do it like this.
Simone SPEZIALE: because the problem is that Comar
Simone SPEZIALE: depends on higher order extensions of the similarity vector fields which are non-canonical. They have to be fixed in some ways typically are fixed in a field dependent way. And so this introduces quite some noise in the calculation, and sometimes this calculation is presented in a bit of a sloppy way in the literature.
Simone SPEZIALE: And so one is to be actually careful about doing it like this.
Simone SPEZIALE: All of this is to say that it looks simple to do the calculation like this, but it's not exactly super simple. One has to be careful.
Simone SPEZIALE: The alternative way of doing. The calculation, which is the old way in which they and Scroll
Simone SPEZIALE: did it was to look for so for integrating the fluxes, namely, just look for potentials, using the E instance equations such that their time derivative would give the flux.
Simone SPEZIALE: And also this procedure is a little bit cumbersome. You have to use the Einstein equations. And you're actually integrating. So it's also a little bit cumbersome. But it has the advantage that you never need to talk about extensions of the symmetry vector fields. You never need to talk about choices of Sigma. So it is more intrinsic in some sense.
Simone SPEZIALE: So the 2 procedures have difficult aspects and compelling aspects. And again, in the examples that we started, they actually give the same answer at the end of the day. So that's also quite nice. And in the case of Bms, the answer you get at the end of the day is this expression for the charges
Simone SPEZIALE: which coincides with very old formulas that have been established already long time ago, Garo's super momentum and the destroy bilangular momentum
Simone SPEZIALE: And now I can finally go back to
Simone SPEZIALE: the bonding formula.
Simone SPEZIALE: because so this is the general flux balance load that one has for every bms symmetry that's cry. And we can now specialize in specialize it to Y got 0. So this Y refers to a choice of creation. As I was saying, as you can see from this expression for the symmetry vector, field.
Simone SPEZIALE: Then, if we specialize it to this choice, it reduces to this formula, here the second formula, and for T equal one.
Simone SPEZIALE: What happens for table. One is that we can integrate by parts the derivatives in the second term, the soft flux Ddn, and so we only get this N squared, and so we recover the bondi formula
Simone SPEZIALE: for arbitrary T. We get a generalization of the bondi formula to an arbitrary flux of super momentum.
Simone SPEZIALE: These formulas also contain the flux balance for angular momentum and center of mass, which are contained in the Y part, and I will not be talking about
Simone SPEZIALE: so let me say a little bit more about this.
Simone SPEZIALE: super translational part of the flux.
Simone SPEZIALE: and actually no, before saying that. So let me summarize where we are. So far.
Simone SPEZIALE: So basically, we have, I'm like to say, as beautiful synergy of results.
Simone SPEZIALE: because there is this world zoom as procedure for describing and obtaining charges and fluxes, Asians or weak Amytonians, if you want associated with Sigma.
Simone SPEZIALE: and what we've been pushing with a buy is this old idea of the 81 paper that was associated with the old scribe, also finite regions of Scribe that one should think of of the Amytonians as being the fluxes on this planet region.
Simone SPEZIALE: because the flux and definite region can be understood as a Mithonians. For
Simone SPEZIALE: they all radio phase space in a strong sense, not in a weak sense.
Simone SPEZIALE: And so these are 2 complementary features. And what is interesting is that the the Amazonian that is, the flux on a final region of scribe is on shell exact. And so it's a difference of 2 corner charges, and we could prove that the difference of these 2 corner charges is coincides with the weak onions in
Simone SPEZIALE: the Worldzupa sense.
Simone SPEZIALE: Now, this is also where the Bms boundary conditions are important.
Simone SPEZIALE: because it's important that all the terms that are time independent are also universal. So in some of the recent literature people have been investigating extensions of the submitting groups from Bms to larger groups, like extended bms, generalize, Bms, Bmsw, and these are situations in which.
Simone SPEZIALE: this rigidity, that everything that is time independent is also universally lost, typically and so additional subtleties arise. In that case.
Simone SPEZIALE: but in the standard context of vms, the solution is extremely consistent between the 2 different viewpoints. And now the papers with a bike
Simone SPEZIALE: contained also completely unrelated discussion on the geometry of Scri, regardless of the construction of alternative and fluxes
Simone SPEZIALE: which had a very nice
Simone SPEZIALE: description in terms of unified the framework that can describe both sky and the weekly isolated horizons. And so I apologize. I didn't have time to talk about this by gave 2 weeks ago a very nice presentation of this here a perimeter.
Simone SPEZIALE: Hopefully, I don't know if that one is available online. I think it might be on Pira, because this is actually also very elegant and very nice, and very helpful for the understanding also of Miltonians and charges and flexes. The fact that there is a common genetic framework that describes both settings which may be surprising because black hole and cosmological horizons
Simone SPEZIALE: are typically isolated or weekly isolated surfaces in the sense that there's no flux that goes across them. On the other hand, when one thing's about Scribe, one thing's about an arbitrary large flux of gravitational radiation that is being registered there. So their physics is completely different, and the beauty of the description we point out with a buy is that there's actually a unique equation
Simone SPEZIALE: that describes when specialized to both settings that describes this very completely different physics.
Simone SPEZIALE: And this beautiful equation is that
Simone SPEZIALE: all of these? Both cry and background cosmological horizons are non-expanding horizons. And in that case there's a unique connection that is induced by the space time connection. And this is what I'm calling D, forget about the bars here.
Simone SPEZIALE: and then J is just an arbitrary one form on the number surface. And so the time derivative of these
Simone SPEZIALE: unique connection on the developer surface can be split in this way that involves the the rotational one form while tensor and reaches and reach a tensor.
Simone SPEZIALE: Forget about Alpha. It doesn't matter here. And that. And the beauty is that different pieces come into play
Simone SPEZIALE: when the horizon is actually a physical horizon in space-time, or when it's scrying, namely, and horizon in the conformal completion. because basically, in one case you use the Einsteins equations, whereas in the other case, use the conformalized tense equations.
Simone SPEZIALE: and the solution of these is the different pieces of these equations. Actually complementary pieces of these equations are set to 0. And that's why that's how very completely different physics emerges.
Simone SPEZIALE: And so in that domain is thorny as that of observable Cgr. I find it quite remarkable that not only we have very different procedures like this one, based on sympathetic structure, and they will zoom us one based on extensions and the sympathetic potential, they give consistent results. But also this procedure can be applied to very different settings, like charges and fluxes at sky or at physical black core and cosmological horizons.
Simone SPEZIALE: And I know that, for instance, buy a new record, extending also some of these ideas to asymptotically de sitter space times. So it's a very fertile mathematical and physical framework that one can work with.
Simone SPEZIALE: So now like to. Also, in the last time that I have with talk about something a little bit more speculative, and I would like to come back today bond this packet for a second.
Simone SPEZIALE: So let's look again at the super translation, fax balance law.
Simone SPEZIALE: and let me give you slightly more details. So when T equal one, we recover the total energy, as I mentioned now, T is an arbitrary function. On the sphere is the
Simone SPEZIALE: is the symmetry parameter, and so, if we choose a boundary frame, namely, a round sphere, then we can use spherical harmonics to decompose the scalar function and talk about the L. Modes of this spherical harmonics, then L. Equals 0 and L equals. One corresponds to the global translations of the Poincare group. So L equals, 0 is like T equal one, that those are the time translations. And that's where the bondi energy is.
Simone SPEZIALE: The one modes will be associated to the momentum, the global momentum of the
Simone SPEZIALE: Bms group.
Simone SPEZIALE: On the other hand, all other modes from L equal to onwards. These will correspond to super translations which in flat space times we don't usually use them because we have a preferred pointer group. But in asymptotically flat space times these emerge naturally, because there's no in general, there's no unique selection of a preferred sub group in the presence of radiation. And so these are part of the symmetries that you cannot get rid of.
Simone SPEZIALE: And the important argument that bond used to uniquely identify gravitational energy was the fact that the flux is strictly negative. So there's no ambiguity on the fact that energy is being carried away when there is some gravitational interactions, like in the picture of the coalescing black holes I put at the beginning of the talk. But these occurs only for the L equals 0 nepon one modes.
Simone SPEZIALE: because the point is that this Dd. Operator here you integrate it by parts twice, and then it acts on d on T, and it's just a property of the Laplace of the
Simone SPEZIALE: spherical harmonics on the sphere. That yeah, I should say that N is traceless. And so it's a special property of spherical harmonics on the sphere that the L equals 0 and one modes are eigen modes of the operator. Dd, traceless.
Simone SPEZIALE: So that's why you get a strictly negative flux for the global translations, but not for the super translations.
Simone SPEZIALE: Then consider now is, if, instead of talking about total energy like in one. You would like to talk about some localized energy like how much energy actually will hit the bucket that is here. Then you would like to take a profile. 40. That is not Evil one, but is like picked on some region of the 2 sphere, name, namely, let's say, the region where the packet is.
Simone SPEZIALE: but these profile some Tp profiles on some point will necessarily include
Simone SPEZIALE: higher modes than l. One, and therefore the flux will not be strictly negative. And so this charge, associated with this higher modes, generic super translations, which is some type of mass multiple, if you want does not have a negative, definite flux.
Simone SPEZIALE: so it is only possible to characterize a strictly negative, the total loss of energy.
Simone SPEZIALE: If you look at the the flux of a mass multiples. Is not the strictly negative. So one could say that localizing energy, density still remains somewhat elusive even at sky, even with the additional energy. Sorry, even if with additional universal structure, the one has there.
Simone SPEZIALE: then somehow, we can only talk about. You know, these negative, strictly negative flux of the total energy, and so on. This bucket.
Simone SPEZIALE: we'll have to look at mass multiples, and the flags will be either positive or negative. but can do. Can one do something? Can one do something about this?
Simone SPEZIALE: So this is where I would like to be a little bit more speculative.
Simone SPEZIALE: Now notice that the soft term so the one that was flux is not strictly negative is actually a total time derivative, because, n, we can think of it as the time derivative of the shear.
Simone SPEZIALE: and therefore we can just write it as time derivative of the shear and move it to the left hand side.
Simone SPEZIALE: namely, we could rename renomate the charge to be, not the standard mass aspect, but the standard mass aspect plus Dd. Sigma. Now the numerical factor is not one, but I realize my numerical factors. I was copying them wrong. So please don't take me seriously about this one. It's some number there. It doesn't matter for our considerations. Now.
Simone SPEZIALE: with this redefinition you will have a modified super momentum charge, which is no longer. It denies garlic, super momentum. But the advantage is that you will have negative energy flux for every mode.
Simone SPEZIALE: Then you could say that the energy loss is unambiguously negative for any multiple.
Simone SPEZIALE: Now, a similar mode. These modification actually, as we consider in the past, we found the paper by Dane and Moriski, in which they were looking at this definition of super momentum in some construction that they were calling nice cuts that I'm actually not able to talk about, because I don't remember the details. But they do consider this definition of the charge.
Simone SPEZIALE: Now, what is the obvious problem with the definition of the charge is that it would not be an Amytonian. It will not be an Hamiltonian.
Simone SPEZIALE: charge, because we have just spent half an hour arguing that I can uniquely identify charges and flexes by requiring that they are covariant and requiring that they are, and I don't get this, I get something else.
Simone SPEZIALE: so
Simone SPEZIALE: that should be the death. The the death of this alternative construction.
Simone SPEZIALE: however.
Simone SPEZIALE: charges that we constructed are indeed unique. Once the sympathetic structure is given. So let me go back to Samo. To the one ambiguity the wolf mentioned, which I I asked him. Let me not talk about that one yet, because I wanted to talk about it. Now
Simone SPEZIALE: there is an ambiguity that in finite dimensional system it doesn't exist. The only ambiguity in the synthetic structure is adding a boundary term to the Lagrangian. But in field theory you have an additional ambiguity in that the field equations
Simone SPEZIALE: only determined is implanted to form up to corner terms. namely, structures that that doubly doubly exact the buy exact both in space, time, and field space.
Simone SPEZIALE: This corresponds to modifying the synthetic potential, not by adding a boundary Lagrangian, but by adding a corner term.
Simone SPEZIALE: Now, the problem with this is that a priori? You have no control over this addition.
Simone SPEZIALE: So you could say, Well, what are you doing here? You're just adding by hand, if you don't, modification like this, something that you don't control. So it's completely artificial.
Simone SPEZIALE: But actually, the the question is more. The other way around is that when you take naive or simple or or standard simplity potentials, you're already making choices on these, alphas! And let me give examples in which non-trivial, alphas occur because they occur all the time.
Simone SPEZIALE: A very simple example is actually the difference between the simplest potential. The one gets within the Coverian Facebook from Einstein, Eilbert.
Simone SPEZIALE: and the sympathy potential the one uses for the Adm phase space.
Simone SPEZIALE: So they Saybert, one can be written as G delta gamma, and if you take the pull back on, let's say, a space like or time like ever surface. You can write this as Qdp, and that's the standard admetic potential. Which is the reason also why we say that they stay in Bertha
Simone SPEZIALE: like Grand Jan, is compatible with Neumann boundary conditions as opposed to Dirichlet. Because you fix the momentum to be constant for the variational principle. But this identification is only valid up to a corner term.
Simone SPEZIALE: Now this corner term it so happens that you can think of it as just a a pure gauge description of whether your time, like and space like boundaries meet at an auto corner or a non-organic corner, namely, you can think of it as a description of the radial shift. Vector
Simone SPEZIALE: and so you could fix coordinates so that you set it to 0. But
Simone SPEZIALE: if you talk about arbitrary coordinates, then a priori, you have a term like this, and I can tell you where it was relevant in the literature. This discussion, another example that in our community is very nice is going from metrics to, because there, too, you have a corner term that gives the difference when you read the naive, sympathetic potentials or obvious potential from both Lagrangians. And so in the past few years, there's been an intense discussion over these corner terms differences
Simone SPEZIALE: in various papers. If you were at the last Loop Conference, an analogous corner term was invoked by Miguel in his talk because it is needed to talk about the super rotations. So some of these extensions of the Bms symmetry at infinity.
Simone SPEZIALE: So then, we could ask, Is there a corner of the information of the sympathetic 2 form that actually allows us to turn on the standard bms charges into these ones with strictly negative flux.
Simone SPEZIALE: I claim that the answer is Yes, this will be found without one to answer the question, though in the last I have like 7 min
Simone SPEZIALE: I would like to tell you a little bit more details about advise radiative phase space at Scribe, which is actually extremely rich and non-trivial.
Simone SPEZIALE: And there is this 81. There's this 2 81 papers, the Prl. And the J. Mat Feece.
Simone SPEZIALE: I would highly recommend the team at fees. There's so many details and so much physics in there that has been rediscovered in the last few years. It's quite surprising. So the point is that
Simone SPEZIALE: so? As I mentioned briefly, at some point By's insight was that the radiative phase space at sky can be described using connections.
Simone SPEZIALE: but connections no more than just the the gravitational radiation gravitational radiation is given by the news, which is some components of vile if you want.
Simone SPEZIALE: But the connection also knows about the gravitational shear on any given cross section of sky. And this is more information.
Simone SPEZIALE: then the news for 2 reasons, first of all, because the news in some sense, is the time derivative of the
Simone SPEZIALE: of the shear, and therefore there's like an integration constant if you want, in going from the news to the shear, but more importantly, because this time independent information that is contained in the shear includes the effect of super translations which are time, independent deformations of cross sections.
Simone SPEZIALE: And so this information that one can read from the connection if one takes an origin as reference and cannot be read from the news.
Simone SPEZIALE: So just to make this more concrete.
Simone SPEZIALE: you know, first of all, concerning the fact that this year has more information than does just the radiation. Let's not forget it. Even in the flat space time. You have infinitely many, not just congruencies that are sharing.
Simone SPEZIALE: even though there's no radiation anywhere. And as I just mentioned, at scry, even in the present. Whether this radiation or not, we can create additional shear that has nothing to do with the radiation by just doing a super translation.
Simone SPEZIALE: So
Simone SPEZIALE: the direction in the space of connections that they're sensible to. Changing this year without changing the news. We're called vacuum directions in a biased paper. Because you can think of if you want to vial as the curvature of this connection, and so their vacuum, in a sense analogous to
Simone SPEZIALE: those special connection in a young mill series that are vanishing curvature. Now.
Simone SPEZIALE: it's interesting to parameterize this vacuum directions in a in a vectorial way, in a tensorial way. And this we can do if we choose an origin, say 0 share for a chosen cross section, then all vacuum connections can be parameterized with respect to this choice of origin with a certain operator acting on a choice of
Simone SPEZIALE: super translated initial super translated slice. With respect to this chosen cross section which we can call a bad cat, because, you see, even though there is no radiation, we would see shears. Historically this type of
Simone SPEZIALE: cat subscribe called bad cats.
Simone SPEZIALE: And now, if you do this this parametization of sorry. And this part of way of parameterizing back in directions in the recent literature has been called super translation field. In this very well known paper of compare Future Rusigone
Simone SPEZIALE: and Storming, there is also a way of thinking about it as the super translation Goldstone. So there's lots of research these days on various aspects of gravitational and partial physics in which sometimes you hear this name. Super transition field super transition, goldstone. Well, would they really mean? It's just this choice, at least, as far as I understand, is this choice of parameterizing a vacuum direction in a by space.
Simone SPEZIALE: Now using this parametrization.
Simone SPEZIALE: we can now define something that, compared future with Zeoni called the covariance here, namely, the difference between the shear of in the presence of radiation
Simone SPEZIALE: and the sheer on a vacuum direction with the same. Sorry. Just so. You have a connection, you just remove the radio part, and you get one of these vacuum connections. You subtract them. And so you get some relative. Sorry. Yeah, some relative shear
Simone SPEZIALE: the difference between the 2 that was called covariant share. But compare the reason why was covariant called covariant
Simone SPEZIALE: is that, if you know something a little bit dms. Your intuition should tell you that this quantity is invariant under super translations, because super translations will act on the sharing the same way regardless if there is radiation or not.
Simone SPEZIALE: And so the key property of these
Simone SPEZIALE: covariance here is that under super translations it transforms homogeneously without the inhomogeneous term associated with the standard shear.
Simone SPEZIALE: and this is due to the fact. It was forms under vms differently than the shear, because the vacuum directions also transforms non trivially.
Simone SPEZIALE: so they actually destroyed. Flex cannot be written in terms of what cannot replace this share that appears there with this covid share, because it is additional transformation here.
Simone SPEZIALE: But if you could.
Simone SPEZIALE: then you see immediately that the flex will have only the hard part, because the soft part comes from the non-trivial, inhomogeneous transformation of the sheer. So if one could do that, replace. This implanted flux with this one, then it will have only hard part.
Simone SPEZIALE: And now what we claim is that actually, if you modify the symplectic to form by corner term. that is actually possible. If you allow for this modification.
Simone SPEZIALE: you don't need to add initially, I thought one needs edge modes. No, you don't need it. You just need to use this vacuum direction treated differently than the the dynamical directions, and constructed these corner term out of it.
Simone SPEZIALE: and then, furthermore, the synthetic flux is also unique. If you further require that the charges are fully covariant and vanish
Simone SPEZIALE: so.
Simone SPEZIALE: And, by the way, even though the flex is, they wanna propose that? Or consider by then, Maurisky, the actual charge is not what they had, because what they had would not be covariant. It will not transform consistently
Simone SPEZIALE: So there will be an extra term associated with this choice of vacuum. Basically.
Simone SPEZIALE: Okay. So that's what I wanted. And you will read about this, you getting some of this stuff in the short paper we put up with Antoine earlier on last week, and the specifics of this new choice of sympathetic form will appear hopefully very shortly.
Simone SPEZIALE: And so, to conclude, because I've run out of time the
Simone SPEZIALE: so I've tried to. Hopefully give a nice introduction to the topic. It's very technical. There are a lot of subtleties. Some have typically to do with general activity, some have nothing to do with general activity. They just have to do with the presence of open systems. And so it's hard to the temptation to talk about. Everything is strong, and I hope I didn't make a mess in trying to talk about too many things
Simone SPEZIALE: but then, after this introduction, I specialize that the discussion today
Simone SPEZIALE: results that we obtained first with a buy and then we didn't one and so summarize in the paper with a buy. The main point was that we showed that the geometry of sky can be put on the same footing as the geometry of background cosmological horizons, because they're all examples of weekly isolated horizons in a purely geometrical sense, but purely geometrical. I mean without invoking field equations.
Simone SPEZIALE: and they're dramatically different. Physics can be abus understood precisely from the different field degrations that one imposes on them, namely, iceense equations in one case, and the conformal license equations in the other case, and this allows also unified framework to describe fluxes and charges which remarkably ends up giving results that are consistent
Simone SPEZIALE: with the Walden Zoo Pass procedure, and in the paper we highlight the pros and cons of the different procedures, and how they should be compared to one another.
Simone SPEZIALE: And then, while these construction is very unique, once the sympathetic to form is given, I've tried to argue that there might be, mathematical or physical situations in which is interesting. To modify this in print, to form by a corner term.
Simone SPEZIALE: And if you do this.
Simone SPEZIALE: you can actually consider a new construction for the Bms charges in which you get a negative, definite flux for all mass multiples, and not only for the first 4,
Simone SPEZIALE: and that
Simone SPEZIALE: the construction is based on the modifying by a corner term that is based on vacuum directions only.
Simone SPEZIALE: and the could be used to describe this bond debugger and some notion of localized energy density.
Simone SPEZIALE: and II think I should stop here and thank you for your attention.
Jorge Pullin: So any questions
Jerzy Lewandowski: I have a technical question.
Jorge Pullin: Go ahead.
Jerzy Lewandowski: A about that analogy with, Between sky and non expanding horizon.
Jerzy Lewandowski: So on, it seems to me that on Scribe this rotation, one for Omega is 0 by definition, by construction of Scribe, isn't, isn't it?
Simone SPEZIALE: Let me? Yes, let me get the formula.
Simone SPEZIALE: So the answer is, Yes, but let me.
Jerzy Lewandowski: I think it.
Jerzy Lewandowski: It was feature engineer for me.
Jerzy Lewandowski: I'm
Simone SPEZIALE: I don't know if you're I'm sorry.
Simone SPEZIALE: It might be
Simone SPEZIALE: so. I want to do
Simone SPEZIALE: stop sharing so that you could see my face. But maybe I should not have
Jerzy Lewandowski: if I can see. Okay. Okay.
Simone SPEZIALE: okay. So
Simone SPEZIALE: where was it? No. After gr, yes, of the intermediate somebody
Simone SPEZIALE: here. This formula here? Right?
Yes.
Simone SPEZIALE: yes. So so in this form. Yes, now so this rotational one form that many people in the literature called also, How do you check one form?
Simone SPEZIALE: So this is what captures the angular momentum, or the on a finite
Simone SPEZIALE: on and horizon at a finite distance if you want to in physical space-time.
Simone SPEZIALE: And so these is just in the case of physical horizon. This is just
Simone SPEZIALE: it's if you want, it's time. Dependence is entirely determined by the flux of gravitational waves coming through. And so, if it's weekly isolated, there is no such flux. And so it's time dependencies, in a sense trivial.
Simone SPEZIALE: And so it these remains a corner degree of freedom.
Jerzy Lewandowski: Yeah. But my my question. My question is that.
Jerzy Lewandowski: as far as I understand
Jerzy Lewandowski: the this analogy on spry but
Jerzy Lewandowski: horizon, which Omega is 0.
Jerzy Lewandowski: Yes, who or no?
Simone SPEZIALE: Yes, yes, I think so. Yes.
Simone SPEZIALE: not just omega so okay, so, Omega, if I remember correctly, definition, Omega contains both. I just check and the in affinity. Right? So, because oh, yes, right? Because omega dot the normal gives you the affinity. So Scribe is actually not only so the all of Omega is 0, so Sky can take it as extremeal, and then also the rotational one for me. Zoom. Yes, right.
Jerzy Lewandowski: So I want to make a remark now
Jerzy Lewandowski: most sophisticated. So if you consider a a a space time of not right.
Jerzy Lewandowski: A new man, Umpi Cabarino like like, then.
Jerzy Lewandowski: This Omega will be no 0. But so I'm wondering how
Jerzy Lewandowski: what difference does it make?
Abhay Vasant Ashtekar: Yeah. Well, interesting work to be done? I would say, No, no, I mean in that case not equal to 0. But the topology has 3. There are no 2 sphere cross sections, and so we're talking about it.
Abhay Vasant Ashtekar: Or you can allow Omega to have
Abhay Vasant Ashtekar: wide singularities.
Abhay Vasant Ashtekar: and I think long time ago Amitabasen and I have shown that in that case you can define both
Abhay Vasant Ashtekar: for momentum as well as
the net. 4 momentum, and then that 4 momentum is absolutely conserved.
Abhay Vasant Ashtekar: So I mean, it is different.
Abhay Vasant Ashtekar: Because you've got wide singularities.
Abhay Vasant Ashtekar: But I don't think that there is any.
Simone SPEZIALE: well, one subtlety that might maybe appear if you're allowing for singularities, or any way, if you change the spherical cross section of the sorry the spatial topology of the cross sections
Simone SPEZIALE: one. So this is also what happens when you do extended bms, so these super rotations are stronging. Miguel was talking about, and one thing that we have noticed to be done to one, and that we've written about it is that if you do these.
Simone SPEZIALE: you lose the uniqueness of garage tensor, because that one heavily relies on the compact topology of the spheres.
Simone SPEZIALE: And so something's become So in particular, the issue of covariance becomes trickier.
Abhay Vasant Ashtekar: But in this case, because because of the structure.
Simone SPEZIALE: but still you, when you check conformal invariance.
Abhay Vasant Ashtekar: not not solutions. You do not. You do have the roast answer that you can just get up.
Abhay Vasant Ashtekar: But maybe this is too technical for general.
Jerzy Lewandowski: Thank you
Jorge Pullin: other questions?
Abhay Vasant Ashtekar: Yeah, II have a question. If you can go back, go to your slide number. I think it is 40, something 43 or something in which you add the
Abhay Vasant Ashtekar: 43? Yes, exactly. Yeah.
Abhay Vasant Ashtekar: So that 2 points, I mean your motivation here was really coming from, you know. is there localized energy.
Abhay Vasant Ashtekar: But, as you just pointed out.
Abhay Vasant Ashtekar: if, in fact, the topology is not as is 2, then we don't have Goro stencil. In fact, if the topologies I mean locally, you cannot talk about body news.
Abhay Vasant Ashtekar: And so even your first term an A BNAB is not meaningful
Abhay Vasant Ashtekar: unless you have got a global topology. Right?
Simone SPEZIALE: Yes, I'm assuming here that the topology is still the one of the sphere. In this discussion. I didn't say that T should become. I'm just saying, what if T. Is localized
Abhay Vasant Ashtekar: another? You probably didn't mean it. But you kept saying that. Well, for the first 4 4 y. Lms, you get this positive. That's not true. Right? Because if I have a pure space translation.
Abhay Vasant Ashtekar: you know cos theta sine theta cos phi for TII can get anything I want right? Because sine Theta cos Phi, and so on are negative somewhere. You probably meant something like.
Abhay Vasant Ashtekar: it's a time like super translation or something, and you ought to first of all, say what what you mean by
Abhay Vasant Ashtekar: by that. I mean, it's not an arbitrary super translation, because it just won't be the case, that
Simone SPEZIALE: all I meant was that the news appears square, the switch. They are the part of the fax, that's really all I was referring to.
Jerzy Lewandowski: Yeah. But the function P can be negative.
Simone SPEZIALE: I'm sorry they
Abhay Vasant Ashtekar: multiplied by capital. T is the energy density.
Abhay Vasant Ashtekar: And as you reckon, I was was saying that this tea
Abhay Vasant Ashtekar: it can be negative for a space translation, and certainly, for if you just had L equal to 2 and equal to 3, etcetera, then she's positive somewhere and negative somewhere, and I can have Nab Nab
Abhay Vasant Ashtekar: to have completely localized as you wanted in the part where T is negative.
Abhay Vasant Ashtekar: and therefore your, your, your, your total answer would be, would not be positive.
Simone SPEZIALE: So yeah, I agree so let me rephrase what I was saying by requiring that right? You're right. The flux will not be strictly negative, but it would be hard, purely hard, as in depending. Only so, it would be independent of the sign of the news if you want.
Simone SPEZIALE: Yeah, maybe that's absolutely yes.
Abhay Vasant Ashtekar: yes, because it would only depend quadratically on the news.
Okay.
Abhay Vasant Ashtekar: no. I meant that if you move, that's very different from localizing energy. Right? Sorry.
Abhay Vasant Ashtekar: That's very different from localizing energy.
Abhay Vasant Ashtekar: Okay? But second question was that I mean, I think you kind of tangentially refer to it. But if I look at the second equation on your slide. right?
Abhay Vasant Ashtekar: That charge is not necessarily
Simone SPEZIALE: Minkowski, and they're not given by this formula.
Simone SPEZIALE: There's an additional term. Yes, but you're not telling us what the formula is. So that's fine. Okay. But when when it's basically that one, plus the additional term that you require
Simone SPEZIALE: contains. You can write it like this. So this s is this difference between the actual share and the vacuum shear and Delta XI of U. 0 is the transformation of the vacuum. Shear.
Abhay Vasant Ashtekar: Okay, thank you.
Jorge Pullin: More questions, Antoine.
Antoine RIGNON-BRET: Yeah. Sorry. I just wanted to come back to. Comment about the negativity here. What?
Antoine RIGNON-BRET: Wh? What I want to just wanted to say, that's any. It's true that of course it depends on the sign of tea. But if you take, for instance, the test function on the solicit sphere, so that is positive. So you can think about the
Antoine RIGNON-BRET: as a very general notion that I mean, of course, the sign of energy depends on the sign, on the time translation. Right? So if you just take a notion that is very picked
Antoine RIGNON-BRET: at some point on the celestial spheres. It's what we mean by localized visibility. I sphere such that the the the see, the parameter is positive. In some sense you would say that
Antoine RIGNON-BRET: you know the flux is negative, but of course, if you have a super translation, and you have a variation of the sign of T on the solutions here. You cannot give any. you don't give any information of the sign of this what I wanted to.
Antoine RIGNON-BRET: because I can always take a T which it's true for time. Locally time future time directed super translation.
Abhay Vasant Ashtekar: Yeah, that's that's the site. Okay.
Abhay Vasant Ashtekar: But but your final expression, what charge is also linear in T or.
Simone SPEZIALE: yes.
Simone SPEZIALE: the
Antoine RIGNON-BRET: yeah. Yes. So you have.
Antoine RIGNON-BRET: So in fact, it's he only enters in this Delta XI of you.
Antoine RIGNON-BRET: In fact, it's like a covariant version of the Moorishi charge that. And because, as you said before, the Morrishi charge before does not vanish in Minkowski space time. That this one does
Abhay Vasant Ashtekar: is also no covariant, in fact.
Antoine RIGNON-BRET: But yes.
Jorge Pullin: any other questions.
Jorge Pullin: Okay, let's thank the speaker again.
Simone SPEZIALE: Thank you. Bye.