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Jorge Pullin: Okay. So our speaker today is Stefan Weigel, who will speak about effective Ltb models.
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Stefan Weigl: Thank you very much for the introduction. And first of all. I would like to thank the organizers very much for inviting me to give this talk here.
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Stefan Weigl: As already said, I am going to talk about effective Ltp models, and you can already find 2 associate papers to this talk on Archive, and they will be published soon.
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Stefan Weigl: The my collaborators in this project were Christina Giesel, Hong, Guang, Leo, Eric Ruddett, and Paramp Singh.
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Stefan Weigl: So let me start with a brief introduction to the topic.
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Stefan Weigl: So the overall question, which is, of course, a monumental task is, how do quantum gravity effects influence leg holes?
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Stefan Weigl: And this is
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Stefan Weigl: and a a topic through many quantum gravity communities. And here I tried to list a few people who were working on this from our community, and this list is, of course, far from conclusive. So I already want to apologize in advance if I forgot here some persons
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Stefan Weigl: to be more concrete. one class of models so called, eternal black hole models.
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Stefan Weigl: and these are models which try to already which start from an already formed black hole and try to to include now quantum corrections motivated from the quantum cosmology.
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Stefan Weigl: and this is possible due to the isom isometries into setup. For example, in the interior we will have a Kantowski sucks space time, and therefore we one can use some resize from look quantum cosmology in order to do this, and I want to refer at this point to 2 recent reviews on this subject.
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Stefan Weigl: Here, however, we are, yeah in into interested in another perspective. Especially, we want to see how this dynamical formation process is is
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Stefan Weigl: is happening. So we want to see what happens in a gravitational collapse with some quantum gravity effects
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Stefan Weigl: the typical setup. To to reduce the the complexity of this question is to only and consider spherical symmetric models. And further, as matter, we will restrict ourselves to dust. So this is a perfect fluid, with no pressure.
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Stefan Weigl: and in classical tr.
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Stefan Weigl: If on classical classical general relativity, the solution of the Einstein equation is given by the so called le Metre Tolmann, Bondiemont as space times.
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Stefan Weigl: And further, we will do this in in this in in our approach in an effective description. So this means, we will work in a classical model which includes some correction functions. And these correction functions have the role of of implementing the quantum gravity corrections
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Stefan Weigl: add a quantum gravity effects
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Stefan Weigl: in the literature most models described the Oppenheimer Snyder scenario. So this means that here one has a homogeneous dust ball
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Stefan Weigl: embedded in a vacuum. And
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Stefan Weigl: this simplified setup allows, for example, certain for example, to use classical junction conditions in order to glue this interior space time, which is usually modeled by some Friedman, or, better to say, modified Friedman equations to some exterior
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Stefan Weigl: space time, which when us which one is usually imposing certain symmetry restriction on. So we have symmetry. And also we consider aesthetic space times.
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Stefan Weigl: And one interesting question that arose in this context is in these dust. Collapsed models is whether there is a discontinuity in the gravitational feet forming
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Stefan Weigl: after the bounce. So before the bounce, more or less, all models are coherent, but after the bounce they they they diverge a bit and related to this question, is also, the fact how these equations of motion look, and if we can have a decoupled equations of motion since coordinates similar, like in the classical case.
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Stefan Weigl: And this is where we want to try to to contribute to this discussion, and our approach is to construct effective Ltb models from an underlying effective spherical symmetric model. As a one plus one feeds theory
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Stefan Weigl: under certain assumptions. I will come to this later.
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Stefan Weigl: and this means we are especially interested.
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Stefan Weigl: Consider a general answer for these effective models, and we will not only restrict ourselves to
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Stefan Weigl: to a certain polymerization schemes like, for example, scheme which can be also motivated from a reduced quantization. But here, more more. Our perspective is to see which kind of restrictions on this general effective model. Such a dynamical and implementation of an Ltp. Sector constraints.
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Stefan Weigl: And one of the advantages of this approach is then, that since we are working in in Ltb, we can basically consider arbitrary dust mask profiles. So this means as a special case. This includes vacuum or polymerized vacuum and open habits, nider scenario, but also real smooth in homogeneous dust. Profiles, too.
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Stefan Weigl: And one advantage of knowing this spherical symmetric model is that, for example, there's no area gauge set on this radio coordinate. Yet. So in distance we have. There are some coordinate transformations.
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Stefan Weigl: possible.
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Stefan Weigl: And with that I want to show you the plan of the talk. So first of all, I am going to talk about classical LTB. And space times in the class and economical framework with connection variables.
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Stefan Weigl: Then we will. turn our focus to constructing effective Atp models. And we will do this by analyzing the dynamical stability of a so-called adb condition under the effective dynamics.
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Stefan Weigl: Then I will consider a concrete model which we will. which we will construct from from from from adapting to improved acc dynast solution in the marginally bound case.
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Stefan Weigl: and also see that extended mimetic gravity can give an underlying, coherent lagrangian of this canonical framework.
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Stefan Weigl: Then we will discuss further the polymerize vacuum solution and compare to other models in the literature.
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Stefan Weigl: And finally, I will conclude with a summary and an outlook.
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Stefan Weigl: Okay, so let us start now with classical LTB. Models.
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Stefan Weigl: So we work as usual in real bear variables where the Hamiltonian has the standard form. It's a combination of constraints, since we are in spherical symmetry, and we only have one nontrivial spatial direction, which means we only have one different morphism, constraint, and also only one Gauss constraint, and this is the scalar constraint.
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Stefan Weigl: We can impose the spherical symmetry on the triad and the connection components, and this will restrict the freedom of this components to only 3. And we can see that the spherical symmetric metric can be written in terms of these trial components, for example, in terms of EX and E. Phi, which is related to 2 components. And here we can see
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Stefan Weigl: they are not. they are not completely determined by by this this to try components only up to a rotation of thereof. And this we can fix when we when we use the Gauss constraint.
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Stefan Weigl: So fixing that goes constraint into setup, we are then in a phase, space which has only to them 2 degrees of freedom anymore. And this and as conjugated pairs, we can choose. Then this symmetry restricted triad components, and the components of the extinction curvatures.
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Stefan Weigl: Additionally, we will consider we will add dust to our gravitational system, and we call it T, and because we will use it as as a dust, time, reference, feed
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Stefan Weigl: so in this. Try it. Variables. We can now write the Ltb solution, which is now only a function of ex and the dynamical equation which has to hold. And here we have 2 free functions. So first of all, we have the Ltp function, which is this curly E, and it's only a variable on X has no phase, space, dependence. And this curly F, which is
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Stefan Weigl: basically a measure of the of the of the dust. Mass distribution. And this LTB. Function is a measure of the total energy of a shell at radio coordinate X
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Stefan Weigl: and discussed this effort by the total gravitation, gravitation, a mass inside of a shell at radius X
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Stefan Weigl: and also some, some, some, some some definition on over in the case that this Curly E is 0. We will call this the marginally bound case, and if it's non 0, this is the non marginally bound case, the marginally bound case is very closely related also to the vacuum case.
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Stefan Weigl: So now, comparing this, the Ltp solution
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Stefan Weigl: to the
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Stefan Weigl: the general form of the spherical symmetric metric we see we have to do a certain identification. First of all, the lapse function has to be one, and the shift vectors vanishing. And further, there has to hold a relation between the 2 triad components, and this condition we will call the Ltp condition.
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Stefan Weigl: and it turns out that we can reduce the spherical symmetric. metric! the spherical symmetric system to this l tb. Sector by implementing 2 gauge fixings.
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Stefan Weigl: So first of all, we can gauge, fix the the Hamiltonian constraint with respect to the dust, time, gauge. So this will set, then the the labs function to one, and further we can gauge. Fix the different constraint with respect to this ltp condition.
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AAipad2022: So after gauge fixing what are the degrees of freedom left? Is the
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AAipad2022: F and N. Energy density free now or
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Stefan Weigl: and no, no, there, there are no degrees of freedom, and we have 2 canonical variables left, which is will be one component of of these extensive curvature and one of the triads.
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Stefan Weigl: But what about the matching? The matter is is reduced as a because of the dust time. Gauge.
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AAipad2022: Okay, thank you.
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Stefan Weigl: Thank you for the question.
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Stefan Weigl: One interesting fact here is, one has to be careful a little bit with this gauge fixing when working in the marginally bound case, because here, when we already implemented this task, time, gauge turns out that the Ltb condition is not anymore conjugated to the, to the dipom constraint.
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Stefan Weigl: So this means, here they actually are first class, and in this sense we have to add the atp condition as another first class constraint to the system. And this will mean that in this setup no physical degrees of freedom are left anymore. But we will see this is consistent in the Hamiltonian framework, because here also the Hamiltonian will only be a boundary term.
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Stefan Weigl: All right. So let's turn our focus to the Fa. To effective Ltb models.
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Stefan Weigl: So as motivated from the fact, what's happening in the in the classical system?
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Stefan Weigl: We will directly go into a partially gauge fix effective system where we implement already the dust time gauge fixing because we are mostly interested whether we can make this Ltp reduction. So this means we will consider a Hamiltonian where we now have a gravitational contribution to the scalar constraint. And this data means that it's an effective one or polymerized one.
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Stefan Weigl: and
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Stefan Weigl: then we have the one diffomorphism constraint which takes in these variables this form.
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Stefan Weigl: Here's already one of the assumptions we make. We will not, consider an effective different constraint, but work with the classical one.
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Stefan Weigl: and the form of this gravitation! Contribution of the scalar. Constraint is given by this by this term here. So we have introduced here 3 polymerization functions H. One and H. 2 and an F function.
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Stefan Weigl: The 2 H functions they encode inverse. Try it inverse, try it corrections, and have a classical limit of one.
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Stefan Weigl: And this this generalized autonomy corrections. Because of how we introduce it has a classic element of 0. And this not only in encodes Holonomy corrections because it's it's allowed to depend on this extrinsic curvature components, but also encodes right corrections due to this ex component.
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Stefan Weigl: And in this way we define our general answers of effective models.
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Stefan Weigl: So are are these constraints, I mean, in which you polymerize one, but not the other.
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AAipad2022: is the yes. What is the constraint? Algebra
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Stefan Weigl: really come to this? So first of all. Here. One can see that because we don't want to change the density weight of the scalar constraint. This will mean that once on bracket, it has still the normal form between the diffum constraint and the scalar constraint.
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Stefan Weigl: So this is just a normal relation. And of course the X with the axis has also exactly the the same relation as in the classical case. The interesting question is now whether still the basically the bracket of C delta with theta closes. And in in this Hamiltonian context this means, if C. Delta is a conserved quantity in the different sector.
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Stefan Weigl: and it turns out if you want to have this already? This imposes severe. Restrictions on the polymerization functions, namely, no polymerization of Kx will be allowed anymore.
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Stefan Weigl: And further, there has to hold a relation between the inverse triad corrections and the Holonomy corrections in this way. And this is not only a restriction because of these derivatives, but also the left hand side is only a function on EX. So in these Holonomy corrections, now they only they are only allowed to depend on the K. Phi. X. 20 curvature. They also have to be in this combination only a function of the triad.
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Stefan Weigl: So here to to to show how we define this F. One F. 2 functions. It means that F, one is encoding the Kyle extension curvature and Phi direction squared term. And this is the linear K 5 term.
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Stefan Weigl: And one note is because we and I wouldn't call this constraint algebra here, since we are already in a partially gauge fixed Hamiltonian
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Stefan Weigl: and and the constrained algebra should be analyzed in the fully gauge unfixed system. So this means it depends also what kind of Meta you couple to the system in general. But it turns out this is very similar to what also Alloa Alonso Bada, considered when they couple dust to the Meta and analyze the closure of the constrained algebra. This relation here.
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Stefan Weigl: So now we want to come to the effective Ldp sector.
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Stefan Weigl: and this means we will have, we will introduce, first of all, an effective atp condition. So to remind you, the classical one has this form where we have the relation of the trial components. Just to this atp function E, and
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Stefan Weigl: now we also allow the phase, space functions to appear, and also arbitrary derivatives of thereof.
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Stefan Weigl: and what we will do now is to see when we can. When we make this Ltp condition dynamically stable. And this was already. Investigated in a smaller in in an early context. By by
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Stefan Weigl: and here we will extend and and generalize these results. We will call such dynamical, stable adp conditions compatible.
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Stefan Weigl: And our strategy to do this will be that we that we are going to work on the equations of motions. So this means the the dynamic stability is then equivalent to the questions for which effective ltp conditions due to 4 equations of motions reduced to only 2 in the sector where we have a vanishing shift. Vector the different constraint and the Ldp condition vanishing.
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Stefan Weigl: and the 2 variables one usually chooses, and which have the simplest equations is K, phi, and EX.
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Stefan Weigl: So first result in this investigation is already that the compatible atp conditions are not allow, are not allowed to actually depend on this arbitrary derivatives. But first of all, there's a contribution which which is basically done on margin Deli case. So there's a dependence on this Ltp function. Only K fine ex dependencies allowed in this G Delta function.
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Stefan Weigl: and only if we only considered a marginally bound case. Also there is an while this Kx Twitter Tilde dependents allowed.
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Stefan Weigl: So this is basically their contribution. This corresponds to the marginally bound case.
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Stefan Weigl: And now, you want to see? Which kind of restrictions do these these compatible idb conditions come impose on the polymerization functions which we have in our scalar constraint, effective scalar constraint.
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Stefan Weigl: First of all, the first observation is that there is an additional constraint when an KX. Polymerization is involved, and
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Stefan Weigl: when we are in an unmarkingly bound case. This means that actually also no Kx polymerization is allowed.
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Stefan Weigl: However, if we, if we in a much in a marginally bound case.
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Stefan Weigl: we have to basically restrict ourselves to to an Ltp condition which has only these 2 dependencies to also get the same result. So the marginally bound case one has a little bit more wiggle room, but when restricting the form of this Ltp condition, one gets the same result as in a non marginally bound case.
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Stefan Weigl: So secondly, when we separate this add function dependence on the to data. So this means we figured out this
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Stefan Weigl: this Ldp function in a non-marginally bound case.
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Stefan Weigl: then the compatible atp condition is only aligned to to only allow to depend on this invest on this trial component. And further.
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Stefan Weigl: there has to be a a relation fulfilled between this the atp condition and the Holonomy corrections. And if you remember from the slide before this term looks very similar to the conservation of the Delta case, and so, naturally, there's a close connection, then, between
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Stefan Weigl: having a compatible Ltp condition and a conservation of C. Delta. And this to also have the conservation of C. Delta. We can write in this context. And this relation.
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Stefan Weigl: Yeah, we can now also see if we want to go to the classic if you want to and have a classic condition in the effective framework. So this means that there's no trial dependence in this G twiddle.
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Stefan Weigl: This means that actually the we have this. This thing has to be one. So there is a is A. This relation on the Holonomy correction has to hold alone from the compatibility of and of the classical Ltp condition in this effective framework.
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Stefan Weigl: And further, if we want to have additionally the conservation of the data also, there is another condition on the inverse riot components.
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Stefan Weigl: One observation, for example, is here that we cannot only have in this, in this sector, the and non-trivial non-trivial. H. One correction infest right correction. Since then, this relation is violated.
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Stefan Weigl: Okay, so now let us concern. Let us go discuss the dynamics of these models.
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Stefan Weigl: So considering a system with compatible Adb conditions, we can very generally show that these equations of motions are actually decoupled if we look at at a
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Stefan Weigl: at also at the case, that the C. Delta is conserved. So this effective scalar constraint is a conserved quantity.
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Stefan Weigl: then the dynamics of these models in the Addb sector have the following form. So this looks a bit complicated. But the important thing is basically that no radial derivatives are appearing here on the right hand side.
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Stefan Weigl: And this condition, these equations of motions are applicable in the marginally and also in the non marginally bound case.
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Stefan Weigl: and they are not marginally bound case. We can get to the same result when we implement the gauge. Fixing with respect to this Ld effective Ltp condition and compute the
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Stefan Weigl: that the rug bracket associated to it.
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Stefan Weigl: And this result, for example, shows that the assumption of decoupled chess, which was which was assumed in some dust collapse models can be true.
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Stefan Weigl: and another effect. And and again, I want to say that in this Hamiltonian framework, so the solution is parametrized, first of all by this Ltb. Function, E. Which is not a phase-based function. It's just a radial function, basically a free parameter, and also the conserved quantity mass of a shell.
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Stefan Weigl: And this is especially intriguing. Now, this decoupled sector, since now we, we can build a concrete model. From from improved accuracy, dynamics. So and a general strategy to do this is basically
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Stefan Weigl: we can choose an an effective accuracy model as a starting point.
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Stefan Weigl: and use foregone corollary or the foregone green box to identify. The the form of the effective Fergusymatic model and the Ltp condition. So
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Stefan Weigl: I can read this basically off. And in this way
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Stefan Weigl: we can can first of all get the underlying spherical symmetric model that has no area gauge implemented yet. So we have this coordinate transformation, freedom on the radio coordinate.
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Stefan Weigl: And secondly, we have a dynamical stable reduction to the Ltb sector. With this effective Ltp condition so, and here dynamically stable with respect to the effective dynamics.
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Stefan Weigl: And third, these equations of motions are decoupled and and exactly coincide with the chosen Acc model.
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Stefan Weigl: and sometimes, which will be the case, and the model will. consider! Next one can even then relate this effective Ferguson model, the Sis effective Fergus symmetric model to an underlying cover and Lagrangian.
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Stefan Weigl: and in our model this will be extended magnetic gravity in the co-moving gauge, and in this sense we can then regain the coordinate transformation in the temporal coordinate. Since they then correspond to redefining the dependence on the on the the time dependence of this magnetic field which serves as a clock.
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Stefan Weigl: We will see this like later.
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Stefan Weigl: So we want to start now with the following, equations of motions. So in in B+B coordinates, we can write. This improved accuracy! Dynamics in in this way, where V is now related as the power of 3 half of ex, and b is k phi, divided by square root of EX, and this alpha parameter is related to the barbarian parameter and minimize area gap and accuracy.
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Stefan Weigl: And
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Stefan Weigl: Comparing this with the general form of the equations of motions. We can then deduce that and this spherical symmetric model is the underlying
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Stefan Weigl: model, which is already found by Tipuvala. Here, in this paper. due to the effect that we don't have any inverse try corrections. For example.
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Stefan Weigl: the compatible Ldp condition is exactly the classical one.
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Stefan Weigl: and this also means that we are, and in this C. Delta conserved sector.
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Stefan Weigl: and we can rewrite these equations of motions which we have under foreground page in terms of modified Friedmann equations, and in terms of this capital R. Coordinate, then the then, the metric takes this form here, where the the case, when now the radial derivative of R. Vanishes. This metric is degenerate, and this is a so-called shell crossing.
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Stefan Weigl: and sometimes the shell crossings can also give rise to singularities. It depends on how the mass function is chosen.
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Stefan Weigl: and we and we will come back to this matter. And due to this modified Friedman equations, we can for and, for example, see that we can also have bounces in these models for certain critical energy densities.
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AAipad2022: For the, for the upcoming talk. Just take over this model that was joining
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AAipad2022: 2,012 by Tibetwala. So what have you? What I mean?
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AAipad2022: What it? What more have you learned now? Like
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AAipad2022: more extended your work.
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AAipad2022: which is much more general.
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AAipad2022: If, in fact, we're going to get
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AAipad2022: the underlying spherical models symmetric model to be the same as before.
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AAipad2022: Why, I mean? Why, the the extra work you did. So what did we learn? New? What is a new thing that we learned?
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Stefan Weigl: So so basically, what we learned is in in in this setup. We can. So so, Tipper, while I did not consider any Ldb sector or something like this, and this is exactly the connection that we show, so that basically we can implement an effective adb reduction. Starting from this effective model and have decoupled equations of motions which exactly are the form of this improved accuracy dynamics.
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Stefan Weigl: This is basically our input. What tperwala came to this very symmetric? Lagrangian, if I'm correct by considering different polymerizations and and analyzing whether the constrained algebra closes.
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Stefan Weigl: and and this is also similar related in our language to the effect that the C. Tilde is conserved.
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Stefan Weigl: So in our. And so what is different, or what is our new endpoint is is basically that we that we show this relate this effective Ltp relation and the equations of motions there.
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Stefan Weigl: Okay.
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Stefan Weigl: Well, then, let me continue here. For the for the upcoming talk. We will now re restrict our analysis to this marginally bound case. So this means we will set this energy function to 0.
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Stefan Weigl: And then we can find the analytic analytic solution to these decoupled equations of motion, and this is given exactly by this expression here.
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Stefan Weigl: Where this Beta is an arbitrary function of X, and this curly F is again the master distribution we can choose, and if we have if we consider homogeneous dust, which means that this is just. X to the power of 3 and some constant, and the solution was already derived in these 2 papers here.
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Stefan Weigl: Setting this constant.
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Stefan Weigl: then we are in the vacuum solution.
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Stefan Weigl: And we can see, for example, due to this square here, that we have a type symmetry, and also since here we can choose without loss of generality. This Peter, to be just X. This metric is, of course, clearly stationary
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Stefan Weigl: and
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Stefan Weigl: from studying. Now, the solution for mass profiles of this vacuum, or Oppenheimer Snyder collapse? We. We can see that we actually will not have any shake crossing singularities.
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Stefan Weigl: but in the
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Stefan Weigl: general, in homogeneous case this is not true. And here I want to refer to one very new paper, which was just uploaded last week.
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Stefan Weigl: and we can then also investigate the formation of horizons by computing the expansion parameters related to the solutions. And what we will see is
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Stefan Weigl: that depending on the demands which depends on X has to be above a certain lower mass bound, and this is the same mass bound as derived by these 3 papers, and this will mean, in particular, that, for example, for for low, for small x there will be no horizons forming.
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Stefan Weigl: we will see this at the very end of the talk, also in a picture in A, on Home, in the, in homogeneous case.
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Stefan Weigl: So let me now talk about this underlying, covariant Lagrangian.
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Stefan Weigl: So because we, and as spherical symmetric models this is only a Tel. 2D. Action. But what what can be shown is that this primary Hamiltonian that we consider so in our effective spherical, symmetric model. This can also be derived from this Lagrangian. In the co-moving gauge.
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Stefan Weigl: So this ex. This extended memetic theory is an it's a conformal extension of Einstein gravity.
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Stefan Weigl: And we have 3 parts basically, in this action. First of all a curvature part, and secondly, here a mimatic, a mimetic condition on the mimic, the mimetic condition, or the mimetic fit phi, and a potential which encodes higher derivative coupling. Between the the mimetic field and the geometry in x and y
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Stefan Weigl: and
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Stefan Weigl: One thing to note here is that we not couple this with the typical determinant of the metric, but with respect to the triad such that we have a parity odd function, and this will then allow for us that, for example, one of the triad components in this setup, the if I is allowed to change the signature.
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Stefan Weigl: and this is necessary to to to
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Stefan Weigl: correctly reproduce the accuracy. Dynamics in this model
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Stefan Weigl: and this mimic model naturally defines a foliation, because the smooth memetic field due to the mimic relation
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Stefan Weigl: basically can can foliate the that that space time into space like slices where the file is constant. And this means that without loss of generality we can fix. We can say we can reduce to the case where the magnetic field is only allowed to depend on this time calling that in our case, because we go to the into the school moving gauge. This will be just T.
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Stefan Weigl: And, as I said, the higher derivative coupling in X and Y can be seen in the definition, a
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Stefan Weigl: yeah of these quantities where we have. The, with respect to this. With respect to the 2D metric, and here and this coupling, and in the co-moving gauge, actually, these quantities, X and y directly relate to the extrinsic curvature of on a on such a Phi even equal constant slice.
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Stefan Weigl: and the difference now to classically, gr can be seen. For example, when we take the variation of this Lagrangian and and and derived upon all of the Einstein equations.
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Stefan Weigl: because.
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Stefan Weigl: first of all, there is a non rotational dust field. Appearing source from the mimetic field. And additionally due to this mimic potential, we have an additional stress energy. Tensor appearing due to this higher derivative coupling. And and we can basically add this now to the classical Einstein tensor
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Stefan Weigl: have an effective Einstein tensor in this way, which now includes also quantum gravity effects or effects from this polymerization.
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Stefan Weigl: I think it's bit confusing for the audience. And
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AAipad2022: so this is a completely new chapter. This has nothing to do with what it's talked about before. Is that correct?
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Stefan Weigl: So this this relation one can do only for certain effective models for the mimic, for the effective models I showed you here. This is possible, but for general effective models one cannot do this relation to the Lagrange to display. But your model has no scalar field. Your model does not. So what is the relation between what was done before? And now what you're doing now?
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Stefan Weigl: So so yeah. The. So this is not a new reside. But we have. You had no scale of fee. And now, suddenly, there is a scale of feed. There are various other things happening.
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AAipad2022: so it would be helpful for us to understand the relation between what you spoke up to now, and what you are now talking about, that would be very good for us to understand.
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Stefan Weigl: Okay, so so wh. What is good with this link to this Lagrangian is 2 things. So, first of all, we have now a possibility to get a Lagrantian to our Hamiltonian.
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Stefan Weigl: and and therefore Einstein equations. And secondly, we we regain, in a certain sense, this temporal transformations, but which are then these redefinitions of the time, dependence of this mimic field.
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Stefan Weigl: And secondly, or thirdly, we can. And this, yeah, the effective system. So this polymerization contributions can be sourced by this mimic field. That's the thing. So so in this sense, one can interpret these polymers effects as coming from exotic meta, contributions to
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Stefan Weigl: to to to to our system.
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AAipad2022: You also said one stage, that if I change the signature I don't understand the meaning of that word. If I just a function, what does it mean if I change the signature so so when we now look at the dynamics, we will see that actually in vacuum at the bounce, for example. This, if I try. It will change the signature
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AAipad2022: signature in sign or signage sign
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AAipad2022: sign. Okay? Okay? Yeah.
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Stefan Weigl: So it will come, go from a positive value than to a negative values. Mostly. That's what I mean.
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Hongguang Liu: Okay?
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Hongguang Liu: And if you look at the if you look back the right hand side also, that's the after. I didn't think that's why it's related to the previous and license which we have a desk copy, and then everything can
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Hongguang Liu: rewritten the RGB. Covenant.
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Hongguang Liu: I think we have a.
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AAipad2022: So you are saying that this is
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AAipad2022: I mean, you can think of the Phi as it does. And I didn't understand the statement that you just made.
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Hongguang Liu: This is Rick.
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Hongguang Liu: Yes, yes, you can. You have seen you have seen everything at the desk at the desk.
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AAipad2022: Okay, thank you.
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AAipad2022: But dust. But previously does not null gravest right in the
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AAipad2022: in the previous transparencies. It was
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Stefan Weigl: so. So is this a completely different system. No, it was also notation. It does. But this, if this mimic feed is not only the mimic dust sourcing, but also the polymerization contributions to the Hamiltonian. So it has 2
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2 chops. Basically.
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Stefan Weigl: it's our reference field, but also sourcing our if our polymerization effects so quantum gravity effects in this setup as an effective model
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Hongguang Liu: and also previous analysts. Also. No version, no rotational. That's because we don't. we don't contribution to the
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Hongguang Liu: we only use to depend dependent time.
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Stefan Weigl: okay, I hope that? Answers that question.
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Stefan Weigl: So then let me continue here with a specializing on the polymerized vacuum case.
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Stefan Weigl: So now we want to and restrict ourselves to the vacuum, which means that this dust mass distribution function will be set to constant. This is already done, and just to papers, too.
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Stefan Weigl: and then the solution takes the following, from where now we can introduce this zet, which is just X minus t, and we can see that this metric is clearly stationary cost is the only dependence of the metric on this R function, as we saw.
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Stefan Weigl: And
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Stefan Weigl: now one just has to be careful a little bit, since this vacuum does not mean that we have a flat geometry. So so first of all, vacuum means that this dust contribution we saw in the Einstein equation, the energy density, lambda is 0, but the curvature is not vanishing, and this can be seen, for example, here. That even at the bounce, where it has the biggest value. It's still bounded.
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Stefan Weigl: But it's never vanishing. And this is due to this higher derivative coupling of the mimetic field in terms of this additional stress energy tensor appearing in the Einstein equations, and we would interpret them as a quantum gravity effects.
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Stefan Weigl: And, as I said already it's everything boundless. So this means also we don't have any share. Crossing singularity appearing in this vacuum case even at the bounds, which is the most quantum region.
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Stefan Weigl: And here, what I also already mentioned on the first slide. We will have to signature. Change any Phi since due to the relation due to the Adb condition, we have a relation between E 5 and the radio derivative of X
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Stefan Weigl: and plugging in our solution, we can see that the bounce which is at set equal 0 we have here different signatures and the derivative.
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Stefan Weigl: And this this is consistent with the fact that also here the metric is degenerate. So if I go trans. Going from a positive to a negative through 0, so at that bounce, it's actually degenerate. But one can check that, for example, also that geodesics can pass through this the bounce.
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Stefan Weigl: And so it's more a coordinate effect, and not a real physical singularity in this sense.
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Stefan Weigl: So it's not a, it's not a coordinate effect, right? I mean, if you just vision right then.
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AAipad2022: whether metric is digital. It or not is not a coordinate effect. Right? I mean.
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Stefan Weigl: yeah.
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AAipad2022: So okay.
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Stefan Weigl: it's it's not a similarity, but it is still a
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AAipad2022: I mean, it's not a metric at that point, if you like, because if it's degenerate by definition, it's not a metric. Okay?
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AAipad2022: But the dynamics is digit. The dynamics is deterministic. Nonetheless.
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Stefan Weigl: Thank you.
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Stefan Weigl: So what we now can what we now want to do is to to relate our solution via coordinate transformations now to other models in the literature, and also, for example, discussing the the appearance of shocks in this context. And
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Stefan Weigl: first, let let us focus on the poly polymerized vacuum case, and here we can
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Stefan Weigl: transform from our Ltp coordinates Tx to to watch it like coordinates Tau and R, where the metric takes the the. This typical form as a function of this AR, which is then now not only the classical result, but also has some quantum corrections proportional here to this Ipa square, and this is, exactly the same solution which was also derived in these 3 papers. For example.
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Stefan Weigl: however, one thing to note here, this coordinate transformation is is only well defined when when r of set is monotonic, and we saw that R is a a function of of set square. So this means that actually, before the bounds are after by the bounds, we can do this, but not at the bounds. And in this sense, the space times can have different global structure.
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Stefan Weigl: And
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Stefan Weigl: now, concerning the area gauge. Here, basically 1. One works in Gtp coordinates. And to to get to this coordinates, one gauge fixes the different morphism constraint. When we are in this very symmetric system. With respect to this area gauge fixing and not with respect to the Ltp gauge fixing.
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Stefan Weigl: And then the solution of the Si shift vector is given by by this sign of the of K. 5. And we can see that we can exactly reproduce the model of Hussein Kelly, Santa Cruz, and with new wing and 22.
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Stefan Weigl: And then, after gauge, fixing the different morphism, constraint, we will, which one can then reduce to the LTB. Sector. However, in this context, the equations of motions are not decoupled.
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Stefan Weigl: and to relate our solution directly to this choice of coordinates, we can for example, we can make a corner change to this Gtps and and and what we can see here is that then the shift vector, is just a tempered derivative of our solution are.
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Stefan Weigl: And this means that
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Stefan Weigl: our solution impose naturally, also a sign change on on of the shift vector at the bounds which is not observed when one directly works in in the ready. In this our coordinates
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Stefan Weigl: in the vacuum case. And this. Roughly speaking, also, is the is the is the reason for for discontinuities appearing so for for the appearance of shocks in the Oppenheimer, Snyder collapse scenario.
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Stefan Weigl: So to to see this global structures. In in more detail. One can, for example, consider
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Stefan Weigl: X equal constant space times in these other coordinates. And let us focus here and on this first picture on this first figure. So in this, watch it like coordinates, Tau and R plotting this X equal constant at your desk. We see when we check in which directions.
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Stefan Weigl: increasing. TS, that's the sign vector the sign arrow, and the Red arrow is pointing towards increasing X direction and and now, comparing these 2 arrows before the bounce to after the bounce we will see from the perspective of Ltp coordinates for T. And X. This orientation is changed.
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Stefan Weigl: and further, even more severely.
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Stefan Weigl: since this X. Equal constant geodesics are also the word lines of our clock field. Phi.
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Stefan Weigl: we see. They intersect after the bounce with word lines of of a greater X. And this means that we have to to introduce discontinuities in our clock field.
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Stefan Weigl: which, very similar to the observation in the paper of and this violates the smoothness of our so in this sense this tower coordinates are not a good description of the of the dynamics.
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Stefan Weigl: and what one has to do in order to to get the same global structure in R coordinates as as in the in the Tx coordinates is to glue, basically a different patch of r coordinates after the bounce with a different orientation, which is exactly described by Johannes Mitch in this paper.
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Stefan Weigl: and
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Stefan Weigl: technically, since we on the under level of solutions. There is a 2 to one correspondence between the set or X coordinates and the R. Coordinates we can identify 2 different geodesics. When concerning their coordinate transformations, and
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Stefan Weigl: actually when when we want to see if in our setup also, shock solutions are allowed. We can introduce effective junction conditions which are derived from this effective Einstein equations in our case, and see if there are a non-trivial surface matter. Contributions appearing.
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Stefan Weigl: and due to this 2 to one correspondence, not only we can have the usual gluing, but also a gluing where we change to the orange line, which is exactly the minus set
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Stefan Weigl: geodesic at the bounce. And what we see I don't want to go there into details is that
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Stefan Weigl: basically in in both ways of gluing, no surface and stress energy tensors appearing. So this means without violating the smoothness of our clock field of our mimic feed, and in this way we we don't see any any shocks appearing.
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Stefan Weigl: And here, on the right hand side I just want to show the conformal diagram of our space time. So this is very like the Reissner north stream. One.
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Stefan Weigl: The only difference is that due to our Orient, change our orientation change at a bounce. We have to glue of another tower of these
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Stefan Weigl: These Penrose diagrams which are identified at at this minima radius and direction, and in blue. One can see, for example, one of
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discussed lines
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Stefan Weigl: going through this through the bounce, and then ending up in another universe with apparently universe with a different orientation than the original one.
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Stefan Weigl: And these are in both directions here
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Stefan Weigl: infinitely extended.
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Stefan Weigl: So now I want to only very shortly speak about the inhomogeneous case.
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Stefan Weigl: since here one can see that now a single shake crossing singularities up here. Also observed, in in this in this paper.
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Stefan Weigl: so as we saw for very small X values there can be no horizons. Which I hear this white dashed lines, but when we have a bounce there is also shake crossing, appearing, and what the shade crossing singularity is doing. And it's it's basically protecting or separating the space time into 2
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different patches with different orientations.
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Stefan Weigl: So here the annotation is switched. With respect to this one. It looks like these singularities are weak singularities, because also to designing can pass through. But this needs further investigation, and this work in progress.
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Stefan Weigl: And with that I want to come to a summary. So first of all,
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Stefan Weigl: I showed that our framework allows the construction of effective Ltp models with Holonomy and inverse right corrections under certain assumptions. So, for example, that no polymerization of the different morphism constraint is, is
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Stefan Weigl: is implemented.
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Stefan Weigl: Then we can see that certain effective, effective Ltp models have decoupled dynamics, and this then allows us to to take an Aq. C model as a starting point and construct from their efficioratic model for an inhomogeneous dust collapse.
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Stefan Weigl: and sometimes as in in this concrete model, we can use the underlying limited model to to to provide a Lagrangian formulation and also have regain all the coordinate transformations.
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Stefan Weigl: And in future. One interesting question would be, for example, to study further phenomenological properties like whole evaporation, using these effective Einstein equations from our underlying covariant framework.
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Stefan Weigl: And
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Stefan Weigl: additionally, another perspective would be to extend the A the analysis to accuracy models which has an asymmetric bounce. And this is work in progress right now.
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Stefan Weigl: And lastly, another perspective would be to
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Stefan Weigl: look to study more closely the different polymerized we can have in this class of effective theories. And for example, see if there might be a pick of like a Crm in this context, this is also work in progress. And
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Stefan Weigl: with this I want to close the talk and thank you for your attention
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Jorge Pullin: questions. I believe.
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AAipad2022: Yeah, II have a couple of questions. One. If you can go back to your slide. 17, just for a second.
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Stefan Weigl: Now.
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AAipad2022: Beck.
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AAipad2022: so I mean II there, there is some confusion in the community, and I would like to
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AAipad2022: take this occasion to
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AAipad2022: understand the the current our our current understanding.
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AAipad2022: I mean the the paper that you referred to. I think if I remember correctly.
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AAipad2022: They said that it's really the pandemic coordinates
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AAipad2022: which was use a Parliament, or that was wrong in the sense that that gave rise to some coordinate singularity, whereas physically, nothing is happening.
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AAipad2022: and and certain
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AAipad2022: effects that were seen before.
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AAipad2022: By, for example, by the the group in Canada, where, where? Just coordinate artifacts, and that, in fact, there was no physical problem at all.
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AAipad2022: And you are saying something now from the point of view of the mimetic field, Phi.
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AAipad2022: and because they did not have mimetic fields at all.
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AAipad2022: But but is a statement that what you are saying in terms of mimitic field is exactly the same as what was in Taniya saying.
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AAipad2022: or is there something different?
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Stefan Weigl: Partly, yes. So so 1 one argument is,
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Stefan Weigl: because of the due to the different properties we post to our clock field, which, for example, is a discontinuity or a smoothness.
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Stefan Weigl: For example, this does not allow shocks, but so so in the sense, what what in this paper was done is that the show? If you use different coordinates, you can. You can have no singularities, and therefore no shocks have to be implemented. However, from our inspective we can extend this this work in the in the sense that
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Stefan Weigl: because we have now a a full model in the sense we don't use junction conditions.
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Stefan Weigl: we can actually, in the Oppenheimer Snyder case. See from our, for in this a Ltp coordinates there is no singularity appearing, and even if you try to use junction conditions at the boundary of the star.
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Stefan Weigl: there is still no surface energy, density. And this and this is in our case, the the analog on to shock solutions. We have to do it like this because we don't have coupled differential equations. So it doesn't make sense here to go into speak solutions. But we can go to the Einstein equations and and look for distribution, a solution which is a exactly function pond. Now in our effective framework.
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Stefan Weigl: So in this sense, we, we extend this a bit. Because of our underlying Lagrangian formulation, and say, even when one tries to, because of this, 2 to one correspondence of T and X, or Atp coordinates to this Gtp coordinates, we can. We can consider a different gluing than the normal one, which and and even then we don't find shocks.
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AAipad2022: Okay, good. And the second thing was about again. This the I would like to understand better understand the relation with limiting gravity
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AAipad2022: from what you know, what the other. So the way you presented it was also kind of starting with
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AAipad2022: certain answers for for this desk collapse right for the Ltb model, and the I mean, you know that you don't change it if you want some constraint, but you probably by the totally constraint, and so on, so forth.
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AAipad2022: And then you arrive at some point. And then you said, basically, Yeah, but these are the equations. But these equations could also be derived from the pneumatic gravity. If you make some identifications with the scalar field as being
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AAipad2022: dust, erotic, irrotational dust field. That's all right. But could I not also take the point of view that the fundamental theory just limited. And I just write it down the questions.
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AAipad2022: And then what you're done is a sector of that theory. Is that a consistent thing, or am I missing something? If I took that point of view?
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Stefan Weigl: yes, but but, as I already tried to say before, this cannot be done for arbitrary, effective models. So so only for certain polymerization functions. If you want, we can tune this and mimic potential here such that it can source this. This quantity effects.
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Stefan Weigl: But you're completely right in in this subsector. One can see this as the underlying theory in a certain co-moving gauge.
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Stefan Weigl: Yeah, which is only a might restriction on the geometry. As I pointed out here with this natural 48 point of view, that what you wrote down Lagrangian is a theory that you look at.
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AAipad2022: and it is not obvious that this theory contains, you know, TV. But but, on the other hand, you show that it does contain it to be. And this is explicit proof of that.
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AAipad2022: That's I mean it does it? It seems like many, many other components, right?
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AAipad2022: Ltbs, and so, okay, thank you. Thank you. That's that's very clarifying.
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AAipad2022: Yeah, I ask this question because you know, we've got experts. Hong Kong Wong was talking about before, and I think Vicar is an audience so that people may want to contribute so that we can connect various approaches.
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AAipad2022: Thank you very much
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Jorge Pullin: regard.
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Viqar Husain: Oh, yes, I just wanted to ask a clarifying question. I think the results, for, as I understand the results of no shocks both in of Hazini at all's paper, and your work relates only
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Viqar Husain: to the Oppenheimer Snyder model. Is that correct? And not to be in homogeneous profiles? Oh, yes, yes, yes, for sure. II only wanted to say this gluing is only possible in the Oppenheimer Snyder case, of course. Yes, so II just wanted to clarify that in general, when you do have
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Viqar Husain: the. And this was our recent work. With Ed and Francesco. In our recent work. What we show is that the decoupled odes that you obtain are actually the characteristic equations of a nonlinear Pde.
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Viqar Husain: And so when you actually look at that nonlinear, PDE. And look at the most general initial data profiles.
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Viqar Husain: Then shock waves are an essential part of that.
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Viqar Husain: So that's just a comment and encouragement for you the rest of you to look at that paper
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AAipad2022: right? But for for you're talking about yes, the the full. It's basically a one plus one field theory.
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Viqar Husain: and the one plus one field theory can be looked at in more than one way one way to look at it is to consider the equations that we just saw, which were the decoupled point odes.
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Viqar Husain: And what we show in that paper is that these decoupled point odes are exactly the characteristic equations of a nonlinear. PDE.
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Viqar Husain: And the of the nonlinear Pd can include shock waves.
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Viqar Husain: Yeah, I'm really happy that you know. P. Various people spoke up, because then one can get a global picture. So I think there are multiple aspects aspects of this, and I think it's as as you
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Viqar Husain: I mentioned Stefan. I think it's very important to study the inhomogeneous cases from multiple angles as well.
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Viqar Husain: yeah, for sure. Because the the Openheimer Snyder collapse, in a sense, is, of course, only an idealized picture. Yeah, it's basically a quantum mechanics model. Right? So.
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Stefan Weigl: yeah.
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Jorge Pullin: Ed.
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Edward Wilson-Ewing: yeah, thank you. The car actually said most of what I wanted to say. But let me just add one quick point.
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Edward Wilson-Ewing: If we look at the Oppenheimer Snyder case, what happens is that we find that these characteristic curves that require us discussing that they cross.
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Edward Wilson-Ewing: And so in Openheimer Snyder. It's a little confusing, because this could signal either hey!
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Edward Wilson-Ewing: Formation of a shock or the fact that the coordinates fail.
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Edward Wilson-Ewing: and it's not obvious a priori, which one is the case.
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Edward Wilson-Ewing: and this is what was pointed out by in these other papers. And also it's Stefan has been presenting here. But if you look at more general cases. So if you allow for in homogeneity and homogeneous profiles, for example, including profiles that are very, very close to Openheimer Snyder. Then you get a shell crossing singularity. And then it's clear
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Edward Wilson-Ewing: because of this singularity. What it's saying is that actually, these
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Edward Wilson-Ewing: these characteristic curves crossing is physical is not a Coronadic artifact, and at that point you should really be looking for shocks.
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Edward Wilson-Ewing: So let me stop there.
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AAipad2022: Yeah. But I don't understand why you say the open Amish side. The case is,
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AAipad2022: there's ambiguity, I mean, you know, if, in fact, there existed
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AAipad2022: I mean, it is a bit like saying that there's ambiguity in the case of the Schwarzschild horizon. If there is singularity, not because it's some coordinate, there is, and some other coordinates there isn't.
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AAipad2022: And the whole point is that if there is one coordinate system in which everything is regular, then there is no singularity.
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AAipad2022: So I so II fully appreciate what you said just now, but you know, moojing this case that I fully appreciate.
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AAipad2022: But I don't see why there is any room for ambiguity in the open. I'm Ashinder case, and I would really like to understand that because I may be missing something here.
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Edward Wilson-Ewing: well, I would. I would just say that you can take the inhomogeneous case
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Edward Wilson-Ewing: and take a profile which is close to Oppenheimer Snyder, and you can take the limit.
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Edward Wilson-Ewing: And even when it's arbitrarily close. There'll still be a shock. Cross. Sorry a shell crossing singularity that'll form. So in that case, if you view it as a limiting process, then you may expect there to be a shock to there too. But if if you take it separately, then then I agree that there exist these extensions without any singularities there. There's no question about that. So I mean, it's a bit like saying that
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AAipad2022: if I looked at Short Shield as a limiting case of pricing. Then there is also always a inner horizon.
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AAipad2022: And it's true. So so same thing here for open it is another one, I mean, if you like, if it's like for the basis or something.
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AAipad2022: it's it's a special.
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AAipad2022: III completely maybe more accessible.
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AAipad2022: may maybe we should just say, this is a non typical case
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Jorge Pullin: on one.
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Hongguang Liu: Yeah. So that's the further, because we have a general solution. And then we can see that the
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Hongguang Liu: probably weekly. That's why, every time
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Hongguang Liu: you'll see the tricks and the the classical video. So we go to the shop solution. Okay, cool.
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Hongguang Liu: we can, we can have these kinds of decisions.
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Hongguang Liu: And and also
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Hongguang Liu: back to the whole
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Hongguang Liu: Edward side. And then.
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Viqar Husain: okay, we go. I just wanted to add one more comment in. Even in a classical Ltb collapse with no quantum corrections of any kind. There is initial data that leads to a shock wave on the infall
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Viqar Husain: for certain sets of initial data, and that's a little known fact. But it already exists in the literature.
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Hongguang Liu: Yes, yes. Why.
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the compensate moving treachery.
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Hongguang Liu: And we see that with red line battery.
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So why hold the red line? Why on the
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Hongguang Liu: produced by classical
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Hongguang Liu: classical?
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It's another
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Hongguang Liu: of the graphic possible.
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Jorge Pullin: Hmm. Any other questions
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Jorge Pullin: we have noticed. Thank the speaker again.