Hal Haggard: Welcome. It's my pleasure to introduce Idris Belfaki, who will be telling us about effective loop, quantum gravity, black holes, covariant, improved, dynamic scheme.
Hal Haggard: Thank you.
Hal Haggard: Address.
Idrus Belfaqih: Thank you all.
Idrus Belfaqih: I also would like to start by thanking the Ilqg Committee for this wonderful opportunity. It is a great honor for me to be here today.
Idrus Belfaqih: truly excited to share my work. But I also must admit that I'm feeling a bit nervous a bit quite terrified for the last couple of weeks.
Idrus Belfaqih: and but hopefully, I'll survive through the end of my talk.
Idrus Belfaqih: So I will be presenting my recent research here at the University of Edinburgh with my supervisor, Sudhasatwa, Brahma.
Idrus Belfaqih: and also with our collaborators, Martin Bojewald and Ivan Duke at the Penn State University. So I will be talking about our recent development in the solutions of effective covariance loop quantum gravity
Idrus Belfaqih: in particular for discussing about the vacuum solution and its properties.
Idrus Belfaqih: Think the slide is not working. Okay.
Idrus Belfaqih: let me start with a couple of motivations. As we all know, one of the fundamental questions in physics is actually about the quantum nature of gravity itself of how to quantizing general theory of relativity. It's not that we don't have any ideas about how to do it. We have a couple of models in order to quantize gravity
Idrus Belfaqih: loop. Quantum gravity has been so far one of the prominent model to quantize gravity or to study about the discrete geometry of the spacetime.
Idrus Belfaqih: And not only that, Luke quantum gravity also offered a fruitful and also phenomenological predictions
Idrus Belfaqih: to use that to use this model and try to see the smoking gun of a quantum aspect of gravity.
Idrus Belfaqih: There are a couple of laboratories or a couple of probes that we can use to study it.
Idrus Belfaqih: Among them is the initial state of the universe.
Idrus Belfaqih: and also the study about black holes itself.
Idrus Belfaqih: The effective model of loop quantum gravity applied to homogeneous spacetime has been very useful. In particular, it's already giving us so much through the effective loop quantum cosmology at which we see that it could resolve the singularity at the Big Bang.
Idrus Belfaqih: and also provide us with so much more information. And therefore many people actually try to figure out and apply the same game. They try to play the same game, to study about the black hole physics.
Idrus Belfaqih: however, the extension from loop quantum cosmology to loop quantum black hole. It's not going to be that simple, because in loop quantum cosmology we have a trivial defomorphism, constraint, while in loop quantum gravity in application to black hole physics in particular, because we have the spherical symmetric. Then there will be a remnant of diffiomorphism in the radial directions.
Idrus Belfaqih: and that must be treated with a lot of care.
Idrus Belfaqih: Now that we know that Black Hole is also one of the laboratory, then we see that there's a couple of interesting questions regarding the Black Hole physics, namely, one of them is related to the singularity issue, which is similar to the singularity issue that appear
Idrus Belfaqih: in the study of loop, quantum cosmology, or in in the general relativity
Idrus Belfaqih: in the Black Hole physics we also see there is a singularity at the center of a black hole, and this issue. Many believe that the issue of this singularity at the center of a black hole is actually an artifact of taking Gr. 2 seriously down to the Planckian scale.
Idrus Belfaqih: where, as I mentioned before, that many scholars believe that it should be that near the center of a black hole or near the Planckian scale the quantum effect must be dominant.
Idrus Belfaqih: non-negligible. And that's why
Idrus Belfaqih: we hope that this quantum effect will somehow resolve this singularity issue as well at the center of a black hole.
Idrus Belfaqih: And the second interesting questions is about
Idrus Belfaqih: the nonunitary evolutions of a quantum fields in the exterior of a black hole. This is the
Idrus Belfaqih: problem or paradox that introduced by hawking at the seventies, where a black hole turns out to be non stationary or non static, at least
Idrus Belfaqih: it will emit its energy, and therefore it will be decreasing its size, and the temperature will be rising, and it will be emitting more and more energy. And that raises the questions about the fate of the matter that, initially generating this black hole itself, where does it goes? And therefore it's been famously known as the Information Loss paradox.
Idrus Belfaqih: and some of the people believe that quantum effect near the Planckian scale must be resolving this issue as well.
Idrus Belfaqih: And the 3rd interesting questions is about, we believe that black Hole is a stationary stage of a collapsing star, and it's also interesting to see the dynamical process generating this black hole itself. So adding matter, starting with some matter distributions. And then this matter distributions collapse under its own gravity. And then what happened? Whether it's going to go down to the center of a black holes, or whether it's going to
Idrus Belfaqih: bounce near the Blanken scale. That will be another venue at which quantum gravity will be important.
Idrus Belfaqih: So, as as I mentioned before, the success of loop quantum cosmology
Idrus Belfaqih: inspired many people to study about the applications to the spherical asymmetric system.
Idrus Belfaqih: and some of the people tried to search for the covarian picture out of these formulations.
Idrus Belfaqih: and there's already a couple of work recently by some group
Idrus Belfaqih: at which they could capture the Holonomy or non-perturbative corrections from loop quantum gravity in the case of a spherical asymmetric system.
Idrus Belfaqih: And
Idrus Belfaqih: and also now that we have this particular model, now that we have this model that satisfied the covariance conditions. Then one might ask questions about what is the prediction coming out of this new model or constructions.
Idrus Belfaqih: whether it's going to give us a new venue.
Idrus Belfaqih: One particular model of interest is to use the mubar seam
Idrus Belfaqih: using the loop quantum cosmology terminology.
Idrus Belfaqih: and this is in particular interesting, because we know that in the Mu Bar case, usually one
Idrus Belfaqih: attach or connected the area of the discretized spacetime
Idrus Belfaqih: to the area gap of loop quantum gravity. So heuristically, this could be one of the motivations to say that this is the prediction coming out of the effective model of holonomic corrections from loop quantum gravity. And another point where this is quite important is because
Idrus Belfaqih: coming out, coming out of the phenomenological aspect.
Idrus Belfaqih: such that the extrinsic curvature, which is one of the phase space variable of our fundamental variables that people usually use in quantum gravity
Idrus Belfaqih: could grow unboundedly even in the region at which the gravitational field is weak, and therefore we need some corrections that will suppress this
Idrus Belfaqih: unbounded grows at the classical scale, such that we will eventually obtain the the proper classical limit when the gravity is weak, and therefore we need
Idrus Belfaqih: a corrections, as I previously mentioned, to suppress this inside the trigonometric functions, or inside this holonomic correction to suppress
Idrus Belfaqih: these unbounded corrections.
Idrus Belfaqih: But one might also then be worried regarding the Hollande functions. Now, if one turns to the quantizations now, we have
Idrus Belfaqih: hallenomy, flux, hybrid operator.
Idrus Belfaqih: But then it turns out that the covariance quantizations provide us a picture at which one could relate the the solutions.
Idrus Belfaqih: the solutions in the where Lambda or the Holonomy functions is a is a functions of density
Idrus Belfaqih: to another phase, space at which this will no longer be true, at which the argument inside the trigonometric or inside the Holonomy functions will be a parameter related to the fiducial length.
Idrus Belfaqih: and therefore the canonical transformations is pretty much useful when one would like to consider quantizations. For instance, one could use the formulations at which the parameter inside the Holonomy functions is independent of the phase, space, or background metric.
Idrus Belfaqih: So now, with this motivations, I will provide you with some takeaways in case you don't make it through the end of this talk.
Idrus Belfaqih: One of them is that we figure out a way to solve this effective model. We regain the vacuum solutions.
Idrus Belfaqih: and the solution turns out to be robust in the sense that we gain a non singular black to white hole transition solutions connected by a wormhole-like structure.
Idrus Belfaqih: And this structure as well, turns out to be generic for arbitrary holonomy parameter. So the solutions that we provided in here later on will be, we're not fixing particular lambda. So the lambda or the holynomy functions remain arbitrary.
Idrus Belfaqih: and the conclusions that we get this also could be applied to any lambda with only a couple of restrictions that will be mentioned in detail later on.
Idrus Belfaqih: The second important
Idrus Belfaqih: things that we gain is that we also because we also studied about the propagations of quantum field
Idrus Belfaqih: in the exterior background, and we consistently regain the hawking thermal distributions.
Idrus Belfaqih: and by consistent in here, I mean, we try to derive these distributions through some root.
Idrus Belfaqih: and it providing us with the same answer. So in that sense, we could say that this computations is robust, and the conclusions generated by this can be.
Idrus Belfaqih: It's a robust predictions coming out of this prediction.
Idrus Belfaqih: I should also mention that later on we will see that the hawking temperature
Idrus Belfaqih: is recovered only when the holonomy function is scale dependent. So when the holonomy functions is related to the area placket of the area gap in loop quantum gravity, then you will get the thermal distributions, the temperature coming out of these thermal distributions corresponding to the
Idrus Belfaqih: hawking temperature. But the difference is that now not only we have this temperature, but because the spacetime itself has the inherent
Idrus Belfaqih: area gap or or length cut off.
Idrus Belfaqih: Then there will be an upper bound in the hawking temperature due to this cut of length.
Idrus Belfaqih: And lastly, another interesting feature is that it turns out that the geometry, if you take the limit of M goes to 0, or the mass of the black hole goes to 0. It turns out that you don't get a trivial
Idrus Belfaqih: spacetime flat space-time, but there will be a remnant in the spatial directions, and this remnant of non-triphal or non-euclidean three-dimensional space could be acceptable because
Idrus Belfaqih: one would expect that loop quantum gravity in Planckian scale can be expected to leave traces of this non-smooth geometry, even in an effective descriptions.
Idrus Belfaqih: So here's the online of the talk today. So as a starter, I will be starting with the framework of how do we start with some ingredients
Idrus Belfaqih: to gain a proper way of incorporating non-perturbative corrections? When
Idrus Belfaqih: we have a non-vanishing diffiomorphism constraint
Idrus Belfaqih: trivially vanishing deformorphism constraints, so I'll be skimming a bit about the framework.
Idrus Belfaqih: and then I will be discussing a bit more in more detail about the general vacuum solutions. What is the geometry coming out of these solutions that we have.
Idrus Belfaqih: and what is the properties of these solutions?
Idrus Belfaqih: And then, once we have the solutions, we're going to play a bit with it and see the thermodynamics and semi-classical analysis coming out of these solutions as well, and I'll be closing the talk with a couple of conclusions.
Idrus Belfaqih: So let's start with the framework.
Idrus Belfaqih: To give a bit of a context.
Idrus Belfaqih: I will start with the classical structure of canonical general relativity.
Idrus Belfaqih: So the canonical formulations of Gr itself, initially initiated by Dirac and Berkman.
Idrus Belfaqih: and once usually started by choosing particular time directions in the spacetime. So you're choosing hypersurfaces.
Idrus Belfaqih: In this kind of setup one will see that the structure of Gr turns out to be similar to a Gish theory of a 1st class constraint.
Idrus Belfaqih: and the Hamiltonian is composed of a Hamiltonian constraint which act as the generator of the normal direction, so it will be perpendicular to the spatial hypersurfaces that we choose at the beginning.
Idrus Belfaqih: I should also mention here that the notations that I adapted in here is using.
Idrus Belfaqih: not metrics, the fundamental variables, but using instead densitous, trite, and extrinsic curvature as our fundamental variables or phase space variables.
Idrus Belfaqih: and also in the spherical symmetric there will be a diffeomorphism, constraint which is not trivial. In the full theory you will have 3 components of this diffeomorphism constraint, but because in a spherical symmetric you have the angular symmetry, then the angular part of the diffeomorphism, constraint will be trivially vanishes, and you leave with
Idrus Belfaqih: a single diffiomorphism, constraint in the, in the, in the radial directions, and this will gives you transformations along the spatial slice in the spaceline.
Idrus Belfaqih: and the specific brackets found in gr is usually known as the hypersurface deformations algebra, and, as I previously mentioned that the theory is 1st class constraint, and how to see it is actually through this hypersurface deformations algebra, because now the algebra between the generators of this normal and tangential direction turns out to be close.
Idrus Belfaqih: As you can see, the most important bit that will be useful in our discussion later on is the last part of this bracket which is between 2 Hamiltonian constraint. As you can see, the Hamiltonian constraint between 2, the brackets between 2 Hamiltonian constraint will generate tangential deformations just as depicted in this picture. So if you take n. 1 and N. 2
Idrus Belfaqih: or N. 2 and n. 1, then it leads you to transformations in the spatial directions.
Idrus Belfaqih: And the important part as well is that the space timeline element?
Idrus Belfaqih: If we, if we so this is the classical line element. If we see here
Idrus Belfaqih: the red color, which is the radial directions radial component of the metric, turns out to be equal or the same as the inverse of the structure functions appear
Idrus Belfaqih: in the Hamiltonian, in the brackets between 2 Hamiltonian constraint in here, and that will be one of the main important bits that I will be also mentioned later on, when I formulate the ingredients or recipe to gain covariance.
Idrus Belfaqih: Hamiltonian.
Idrus Belfaqih: So let's start with
Idrus Belfaqih: some answers. So let's start with the recipe, I mean, we now will. We will now formulate, the ingredients on how to deduce not only the not only the dynamics of this not only the dynamics of the phase space variables.
Idrus Belfaqih: but also how to deduce the spacetime geometry
Idrus Belfaqih: by requiring the system to be fully covariant
Idrus Belfaqih: in what follows. Again, as I previously also mentioned, that I will be adapted the variables here to be the variables that people usually use in spherical metric loop, quantum gravity.
Idrus Belfaqih: so the metric will be, will not be. The fundamental. Variables
Idrus Belfaqih: will not be the configuration space, but will be something which is constructed out of these phase space variables. So if you start with some, if you start with some answers, Hamiltonian, if you consider generic Hamiltonians, for instance, Hamiltonian, coming from regularizations, procedure of loop quantum gravity.
Idrus Belfaqih: then one must ensure that the constraint bracket between this constraint remain 1st class, and not only it must be remained 1st class, but it must be in the form of hypersurface deformations, algebra, and the reason for that is because we would like to deduce what will be the
Idrus Belfaqih: via the component of the metric later on in our geometry.
Idrus Belfaqih: So let's say, we start with some modified Hamiltonian constraint.
Idrus Belfaqih: We get some general modified structure functions as F over here, which is a function in general, it will be functions of density, stride, and extrinsic curvature, and I should also mention that the function need not to be equal to the classical one.
Idrus Belfaqih: I give you a glimpse of what will be the
Idrus Belfaqih: structure functions later on. That will appear if you take the Hamiltonian constraint to be gaining some of the non-perturbative corrections coming out of the Holonomy corrections. So, as you see, if we take that the modified structure functions will be given by these expressions, which I will later on mention again in the next slide.
Idrus Belfaqih: and, as you can see, this is not only a functions of densita stride, but it will be also a functions of K Phi or the configuration space.
Idrus Belfaqih: And there is also a global factor appearance in this expression which will be again later on mentions about it.
Idrus Belfaqih: So this 1st condition is usually known as the anomaly freedom conditions.
Idrus Belfaqih: And again.
Idrus Belfaqih: it turns out that the hypersurface deformations, algebra by itself is not sufficient to give you the ingredients for the theory to be covariance, or at least it will not provide you with the Covariance geometry.
Idrus Belfaqih: And if you want this theory to be providing you a covariance theory in the geometry picture as well, then you have to require that the gauge transformations generated
Idrus Belfaqih: by these constraints must be corresponding to the infinitesimal change in the coordinates or the infinitesimal transformations law for the metric, the spacetime metric. So this will be a stronger covariance conditions.
Idrus Belfaqih: And therefore, if, if, for instance, if one used these conditions for
Idrus Belfaqih: the off diagonal component of the metric.
Idrus Belfaqih: for instance, in the radial. Take, for instance, as an example, a sphereticles metric system, and then you take this condition for the off diagonal component of the metric. Then you'll see that on the left hand side, if you compute the bracket between the Gta metric. With respect to these constraints.
Idrus Belfaqih: you will see the appearance of the structure functions on the left-hand side.
Idrus Belfaqih: However, if you perform infinitesimal coordinate transformations in the spacetime metric in the off diagonal component, then you'll see that on the right hand side in here you will see a component
Idrus Belfaqih: that will be proportional linearly to the radial component of the metric.
Idrus Belfaqih: and therefore for these 2 to be compatible to each other, then it means that the structure functions on the left hand side must be equal to the inverse of the component of the radial metric, and therefore the desired line element instead now will be given
Idrus Belfaqih: by these expressions so essentially in here, in this ingredients that being proposed.
Idrus Belfaqih: There's a couple of people proposing this, such as Asir and David Brizuela, and also Martin and Eric.
Idrus Belfaqih: In this model, then not only
Idrus Belfaqih: the dynamics is being defined by this new corrections, but also the geometry must be something which is derived out of the requirement of invariant
Idrus Belfaqih: Next, we are going to construct genetic, unsat Hamiltonian.
Idrus Belfaqih: and at which we maintain, or we preserve the spatial derivative
Idrus Belfaqih: terms to be remained classical, namely, containing up to second order, derivative and quadratic and 1st order derivative
Idrus Belfaqih: of the phase space variables.
Idrus Belfaqih: and leaving the divio morphism constraint untouched. But, as you can see, there is this couple, there's many of new components in here, a naught ax, Xn, etc. And all of these new coefficients
Idrus Belfaqih: will be a phase, space function. So it will be a phase, space function, but not their derivative.
Idrus Belfaqih: And eventually, so you can see the detail in their paper in the Asir and the paper, and also in Martin and Eric paper.
Idrus Belfaqih: you'll see that this coefficients can be deduced
Idrus Belfaqih: once we use the ingredients coming from the previous slide, so by requiring that the model to generate a hypersurface deformations, algebra, and also the gauge transformation must corresponding to the coordinate transformations. Then these coefficients will be fixed up to certain arbitrary functions.
Idrus Belfaqih: and then, as I previously mentioned, so following.
Idrus Belfaqih: Yeah. So in the final expressions which I'm not providing here.
Idrus Belfaqih: there are still a canonical transformation, residual canonical transformations that one can still exhaust.
Idrus Belfaqih: And by this canonical transformations one can go from to either using non-periodic phase space, and by non-periodic I mean setting the holonomy parameter lambda in here to be a functions of ex. So you can then associate this to the area gap of loop quantum gravity.
Idrus Belfaqih: but the price that one will pay with the plutonium will be non-periodic in Kfi.
Idrus Belfaqih: But then one could also perform, or one could also use different phase. Space coordinates such that lambda is.
Idrus Belfaqih: We are using constant lambda, and in this particular case
Idrus Belfaqih: the phase space will be periodic. So the Hamiltonian will be periodic in K Phi.
Idrus Belfaqih: But there is, but there will be the the non constant lambda. Corrections.
Idrus Belfaqih: instead of appear within the trigonometric or holonomy functions, it will appear as a coefficient in some of the Hamiltonian constraints terms, so you still could get the same predictions. You can still get the suppressions that I mentioned at the beginning of the talk, but that suppressions will not be appearing within the trigonometric functions, but it will appear instead
Idrus Belfaqih: as the component of the Hamiltonian constraint.
Idrus Belfaqih: and this turns out to be quite useful.
Idrus Belfaqih: because then one could formulate the dynamics of this theory, either using a holonomy scale dependent
Idrus Belfaqih: or using scale independent parameter, but at the same time one allowed for scale dependent fx. Because non-triphal lambda or non-trivial Holonomy parameters appear as the coefficients in the Hamiltonian constraint, and this will provide you some tools. For instance, if you would like to perform
Idrus Belfaqih: a quantizations out of this method, for instance.
Idrus Belfaqih: we don't do that. But, for instance, if one would like to do quantizations out of this.
Idrus Belfaqih: then, of course, one would like to instead using
Idrus Belfaqih: this periodic phase space. Because then in here the argument inside the sine functions will be very much close in definitions with the
Idrus Belfaqih: Holonomy that people usually define in quantum gravity, at which the argument inside it is a length
Idrus Belfaqih: connected to the fiducial coordinates.
Idrus Belfaqih: The next thing is that we're also defining or computing the observable of the theory.
Idrus Belfaqih: And the reason for that is because, using this observable relations, one could simplify it, or one could eliminate the K. Phi that appeared inside the structure functions such that now your structure functions will be purely functions of density, strides or momentum space in your phase, space variables.
Idrus Belfaqih: And by observable we mean that it will be scalar functions. Out of these phase space variables, where the gauge transformations generated by the constraint on these variables can be written as
Idrus Belfaqih: linear combinations of the constraints.
Idrus Belfaqih: And if you use this observable relations. As I mentioned before, then, the structure functions now can be fully written in terms of density. Stride variables
Idrus Belfaqih: in here. Cf. Is arbitrary functions, but it's only a functions of ex and ex is the spacetime scalar in this theory.
Idrus Belfaqih: and to regain the classical structure, then one usually choose Cf to be one, and that will be the chose of Cf that we'll be choosing later on.
Idrus Belfaqih: There's a couple of lambda in here that I should
Idrus Belfaqih: defined to be clear. The lambda tilde is the constant lambda. So this is a lambda computed at the reference at some particular surface.
Idrus Belfaqih: and there is also another lambda, and this lambda, which is this one, is the one that I mentioned as the Holonomy parameter.
Idrus Belfaqih: At this stage Lambda remain arbitrary, so it could be any kind of functions.
Idrus Belfaqih: and it could be related to mu bar seam. For instance, if we relate the length of the physical length to the area gap of loop quantum gravity.
Idrus Belfaqih: And there is also another Chi in here. This is known as the global factor. And this is the global factor that attached to the Hamiltonian constraint, and it preserved.
Idrus Belfaqih: And it also shows in the expressions of the structure functions.
Idrus Belfaqih: And we will see later on about this
Idrus Belfaqih: structure. Or, Yeah, how how does it behave in the in the dynamical picture?
Idrus Belfaqih: if you are using? If you are seeing the expressions of the structure functions earlier, not this one but the earlier, where it still depend on the K. Phi.
Idrus Belfaqih: then the the structure function turns out to be symmetric under the reflections of the surface. So if you reflect
Idrus Belfaqih: that structure functions, and the Hamiltonian constraint with respect to
Idrus Belfaqih: K. Phi, equal to pi over 2 and Kx equal to 0, it turns out that the structure functions and the dynamic are generated by the Hamiltonian constraint symmetric under the reflections.
Idrus Belfaqih: And we will also going to see later on that dynamically. This surface, this surface in here will be related to the transition surface or the bouncing point in the pictures of loop, quantum gravity
Idrus Belfaqih: and also because the observable is invariant quantity.
Idrus Belfaqih: So you can compute the right hand side of these expressions at any point, and it will still spit out the same number. So if you, for instance, computed this reflection, symmetry surface, then it's going to give you these relations, and these relations later on will be shows a couple of times in the dynamical picture.
Idrus Belfaqih: where in here it shows you the locations of the reflection, symmetry surface, and later on it will be translated as the locations of
Idrus Belfaqih: and a translation surface.
Idrus Belfaqih: So let's that that will be the framework that we are going to use in our
Idrus Belfaqih: pictures. So let's start now with the let's use that new ingredients and New Hamiltonian constraint
Idrus Belfaqih: and the dynamical generated by that, and see what is the geometry predicted by that model.
Idrus Belfaqih: So let's start with stationary gauge. There's a couple of gauge that we can
Idrus Belfaqih: computed. But let's start with the diagonal gauge
Idrus Belfaqih: by setting the shift factor in the radial directions to be 0, and also choosing the angular component of the metric or the ex variables in the phase space variables to be X squared, or corresponding to the area of a sphere
Idrus Belfaqih: using the stationary conditions, namely, that the aerial gauge to be preserved.
Idrus Belfaqih: You will eventually get to these equations in here.
Idrus Belfaqih: and the only way at which this equation is 0 is to set this sign K phi, equal to 0, because the term inside the parenthesis is a square number, so it's always be bigger than 0.
Idrus Belfaqih: And of course there's a couple of choices of sine, K Phi, or sine, 2 K. Phi, equal to 0, and the choice that we are going to choose the principal branch, because with that choice that just will be providing us with a proper classical limit
Idrus Belfaqih: subsequently. So if you solve the diffiomorphism constraint, using the aerial gauge.
Idrus Belfaqih: the solutions, you will eventually get Kx also equal to 0. So now you almost solve all of the component in the phase space variables, namely, you already get ex
Idrus Belfaqih: k phi and kx.
Idrus Belfaqih: Now the remaining part is, if I
Idrus Belfaqih: one way to solve. If I is by solving the Hamiltonian constraint.
Idrus Belfaqih: and by solving the Hamiltonian constraint.
Idrus Belfaqih: you will be end up with
Idrus Belfaqih: differential equations. First, st order, differential equations of E. Phi.
Idrus Belfaqih: And it. It can be solved analytically, and the solutions will be given by these expressions over here.
Idrus Belfaqih: I should also highlight it in here. I might be forgetting to highlight in here that in deriving this
Idrus Belfaqih: equations over here.
Idrus Belfaqih: we are not restricted. The explicit form of lambda. So the Holonomy functions remain arbitrary up to this moment.
Idrus Belfaqih: and so the solutions in E phi is given by these expressions up to an integration constant, C. Phi.
Idrus Belfaqih: And one way to fix this integrations. Constant is by using the observable relations, because the observable relation is containing all of the phase, space, variables. And now we have all of the solutions of the phase space variables in this gauge.
Idrus Belfaqih: Then one can use that relations, and eventually you will going to end up with C. Phi. Turns out to be proportional
Idrus Belfaqih: to the observables, we still have one more ingredient to go, which is the lapse functions.
Idrus Belfaqih: and one way to solve the lapse functions is by exhausting
Idrus Belfaqih: the K Phi equations of motions. So because, K phi, before we already get the solutions is 0, so we can preserve that conditions.
Idrus Belfaqih: and that leaves you to an equations of the lapse functions.
Idrus Belfaqih: 1st order, differential equations of lapse functions.
Idrus Belfaqih: But this lapse functions turns out to be always attached to the global factors. Chi in here. So Chi is the global factor that I mentioned before.
Idrus Belfaqih: So that's why there is the Chi appearing in the denominator in these expressions.
Idrus Belfaqih: and Alpha is a constant of integration. So because we integrate this and the differential equations is 1st order, then you get the solutions up to this integration constant.
Idrus Belfaqih: and the functions J in here is defined as 2 m. Over X. Plus cosmological contributions. If we didn't set it to to be 0.
Idrus Belfaqih: Now that we have all of these ingredients to construct the geometry, we can now deduce the line element. We can get the line element
Idrus Belfaqih: and the line element will be given by these expressions.
Idrus Belfaqih: and there's a couple of locations at which the modifications appear. But the most prominent one is the one in the blue color, so the one in the blue color is something that later on will have a large contributions in defining the singularity resolutions in our picture.
Idrus Belfaqih: And but there is also a Chi factor. There's a global factor which is still appearance in both radial and temporal and temporal components.
Idrus Belfaqih: And that's still arbitrary up to this point. So Chi Alpha, and Lambda, it's still arbitrary, so we haven't fixed particular functions yet to it, and later we will see we will later on set
Idrus Belfaqih: the proper, the proper Chi to to get or to maintain a proper asymptotic limit as we go to to a large scale.
Idrus Belfaqih: If we take a look at the line element in the previous slide, then we see that there is a couple point. There's a couple places at which
Idrus Belfaqih: the metric be becoming singular. The 1st point is when the purple color, which appear as the denominator in the radial component.
Idrus Belfaqih: If you set this equal to 0, and then you solve the equations. Then you get the solutions to be corresponding to the black hole horizon.
Idrus Belfaqih: If you didn't set the cosmological constant to be 0. Then you're also going to get a cosmological horizon in this locations.
Idrus Belfaqih: But there is also non-classical singularities that appear in this metric, namely, given in the blue color. So if you set this blue color equal to 0, then the metric is being divergent.
Idrus Belfaqih: And if you take a look at this equation. It turns out to be exactly equal to the equations that I mentioned before. When we discussed the kinematical phase, space regarding the reflection, symmetry, surface. So by solving these equations you get the reflection, symmetry surface of our spacetime.
Idrus Belfaqih: But we haven't fully arguing at this point that this corresponding to the transition surface. We will see later that this is going to be corresponding to the transition surface. So we can't directly solve this equation. So you have to specify the functions. Lambda. First, st once you fix the lambda, then you can solve these equations, and then you will get the locations of the reflection, symmetry, surface.
Idrus Belfaqih: But one other thing that I should also highlight it in here is that the argument
Idrus Belfaqih: after lambda squared, which is one minus gx, is actually the one appears in this purple color.
Idrus Belfaqih: and therefore in the exterior region it will be positive, and this term will be negative in the interior region. So we can argue that this
Idrus Belfaqih: conditions, such that this equal to 0 must be happening
Idrus Belfaqih: within the horizon or in the interior region.
Idrus Belfaqih: The purple, as I mentioned the purple color is corresponding to the black hole horizon, and one can argue, and also shows explicitly that this will not be a physical similarities. This will be just an artifact of using.
Idrus Belfaqih: quote unquote, bad, coordinate system.
Idrus Belfaqih: So then, one could find a different code in the system.
Idrus Belfaqih: such that in this coordinate system
Idrus Belfaqih: the metric will be smoothly, will penetrate the horizon smoothly, so there will be defergencies or singularities. Appearance in this in the horizon.
Idrus Belfaqih: and this is usually known as the Pg Gauge, and you can solve this by starting with the Pg gauge in the phase space variable. So you restart your computations by again proposing the gauge and then solve the phase space variables out of this gauge and see the dynamics.
Idrus Belfaqih: Or you could also start with the Schwarzschild diagonal solutions and then perform a coordinate transformation such that you will eventually end up with this metric in the in the Pg cage.
Idrus Belfaqih: remember that as I so, of course, the the singularity at the horizon turns out to be coordinate singularity, and not true singularity. But what happened at the
Idrus Belfaqih: reflection, symmetry surface, so we still have to figure out the expressions of the Ricci scalar at this surface. So if you compute the Ricci scalar, then it turns out that it will be finite for any finite solutions and non-vanishing solutions, or at least non-negative solutions. If you get negative solutions, then the Ricci scalar remain finite at that point.
Idrus Belfaqih: but in in a way to get there, then it means that you have to pass through the classical singularity, then the solutions, then the singularity in that picture will not be resolved.
Idrus Belfaqih: So Richie Skeller computed specifically in that reflection, symmetry surface will be
Idrus Belfaqih: regular, provided that the lambda is a monotonically decreasing function, so it will not work for polynomial lambda.
Idrus Belfaqih: but also because the X lambda could giving you a couple of solutions.
Idrus Belfaqih: And if, let's say, it only gives you
Idrus Belfaqih: a minimum position, it means that there is no bound in the in the
Idrus Belfaqih: and the maximum point, so there's no maximum surface you still need to check whether the Ricci scalar will be bounded as well or not at the asymptotic limit, and it turns out to be
Idrus Belfaqih: true that the Ricci scalar is also finite at a large scale.
Idrus Belfaqih: again provided with the same argument as in the reflection, symmetry surface, namely, that the Holonomy functions monotonically decreasing.
Idrus Belfaqih: As you go to asymptotic limit.
Idrus Belfaqih: Now that we know that the Pgkh covered the region
Idrus Belfaqih: inside the horizon, but we see if we see carefully the metric will remain singular at the reflection symmetry surface.
Idrus Belfaqih: However, as we see before.
Idrus Belfaqih: The reflection symmetry surfaced is itself a finite surface. So it's not singular surface.
Idrus Belfaqih: So it means that one could find a specific code in a system such that it will cover this region smoothly.
Idrus Belfaqih: and since, as I mentioned previously, this will be located within the the horizons of a black hole.
Idrus Belfaqih: Then one might as well using a homogeneous gauge to solve or to see this, to solve this coordinate, or to look for this coordinate system I started by mentioning. There is also another homogeneous gauge corresponding to diagonal gauge, and this is similar to statics. Forces gauge that we defined earlier.
Idrus Belfaqih: just by swapping temporal and radial coordinates
Idrus Belfaqih: in the expressions of the metric. But yet this metric remain undefined at the reflection symmetry surface.
Idrus Belfaqih: So but one could also define. As I previously also already said, one can set up a coordinate system, such that the reflection, symmetry, surface
Idrus Belfaqih: can be penetrated smoothly, and this is done by using one of the phase space variables, namely, the angular component of the extrinsic curvature
Idrus Belfaqih: as the time coordinate, and using that, you will get this horrendous form, metric in this, in this gauge.
Idrus Belfaqih: horrendous as it seems, but this metric will be, penetrate your a reflection, symmetry, surface smoothly.
Idrus Belfaqih: and it will connect at your black to Whitehall interior, so to speak.
Idrus Belfaqih: But yet there is one remaining question still unanswered, namely, whether this reflection, symmetry surface, corresponding to the bouncing point.
Idrus Belfaqih: or whether we can penetrate this down to a smaller scale.
Idrus Belfaqih: And one way to answer it is by using the observable relations, or one could also using another set of equations of motions. But in here we're using the observable relations.
Idrus Belfaqih: So if you rewrite the observable relations, you can isolate the sine square of K. Phi, to be to be written to something like this.
Idrus Belfaqih: where B is actually defined to be. This functions
Idrus Belfaqih: and this functions, if you remember, is exactly the functions that appear when this equal to 0. You remember, this is actually the locations of the reflection symmetry surface.
Idrus Belfaqih: And if this is this, B will be bigger than 0 or positive definite in the exterior region. And it's still positive, definite in the interior region, because, of course, this term becoming negative, but still there is the lambda square that make this term be becoming less than one. But as you getting closer and closer the reflection, symmetry surface, then this term will be eventually goes to 0,
Idrus Belfaqih: and at that point the sine square will be saturated or hitting the maximum point. So it means that you cannot penetrate down below this reflection, symmetry surface.
Idrus Belfaqih: because then the argument for the sense were becoming undefined.
Idrus Belfaqih: Yeah.
Hal Haggard: Idris. I just wanted to quickly signal you the 10 min warning.
Idrus Belfaqih: Oh, okay, how far do I go? Okay, they still
Idrus Belfaqih: okay. I can probably a bit faster now.
Idrus Belfaqih: So I should. Now that there is this singularity being resolved, we could also ask questions about whether the singularity issue is also resolved
Idrus Belfaqih: using different rules, and it turns out to be true. You can also prove that the singularity is a result in this picture by using an argument of geometric conditions, or using a bundle of congruence. And in this bundle of congruence you will see
Idrus Belfaqih: that the reciro equations remain satisfied.
Idrus Belfaqih: and this term over here in the rectal will be behaving as like the energy conditions in the standard pictures of Einstein field equations.
Idrus Belfaqih: Indeed, in this case it will be violated, so the non geometric conditions will be violated in the case at which
Idrus Belfaqih: Lambda is monotonically decreasing.
Idrus Belfaqih: So I will be skipping a bit about this because most of this part
Idrus Belfaqih: actually already discussed during the talk in
Idrus Belfaqih: in Florida. So I'll be skipping ahead to the global structure. So the global structures that we gain is that given by this picture. So you see that instead of having a classical singularity at the center.
Idrus Belfaqih: the singularity at the center is replaced by a reflection, symmetry surface, which behave as a transition surface.
Idrus Belfaqih: and the previous slide is actually telling you that at some point gravity is becoming repulsive, and then you'll be eventually getting out to the anti-trap region.
Idrus Belfaqih: The couple point about asymptotic limit is, as I mentioned before, the global factor remain unrestricted.
Idrus Belfaqih: And one could restrict this global factor by requiring that the solutions to be asymptotically flat.
Idrus Belfaqih: But another interesting thing that coming out of this model is that in the 0 mass limit. So if you check the metric expressions previously.
Idrus Belfaqih: and then you study it. And then you take the mass goes to 0. It turns out that in order to recover flat time that you need to choose the global factor to be constant. And since this is constant, then there will be a remaining one plus lambda squared in here. So even in this case, the effective description providing you a non-trivial
Idrus Belfaqih: solutions, or there is a trace of discretized spacetime even in the limit of M goes to 0.
Idrus Belfaqih: So I think I will be a bit faster in here.
Idrus Belfaqih: and instead go to the part that I haven't discussed during the talk in Florida. So I will be talking now about recent progress that I gained in the Black Hole thermodynamics.
Idrus Belfaqih: As as so, we also studied about the solutions we studied the semiclassical analysis.
Idrus Belfaqih: quantum scalar field propagating in this spacetime, and we try to study about this through a couple of routes.
Idrus Belfaqih: and the 1st fruit that I will be mentioning in here is that we perform the near horizons expansions of the metric.
Idrus Belfaqih: and it turns out that the metric can be cast into a rindler form
Idrus Belfaqih: where the zeta factor in here is the coordinate near the horizon.
Idrus Belfaqih: and by using these expansions the metric looks very familiar, which is the rindler metric, and so one can deduce that the near horizon temperature will be given by these expressions
Idrus Belfaqih: and by incorporating the redship factor. Then one could get the local notions of temperature.
Idrus Belfaqih: and the observer at infinity will eventually measures the temperatures of the black hole to be given by the hawking temperature up to a global factor in here, which is Chi naught.
Idrus Belfaqih: I should also mention in here
Idrus Belfaqih: that there's a discrepancies with the hawking temperature, but this discrepancies actually only appear if we take the lambda or the Holonomy functions to be a constant number. If you take it to be scale dependent, then this lambda will be suppressed as you goes to infinity, and the chinot will be goes to one, and therefore you recovered the hawking temperature. In this limit
Idrus Belfaqih: we also tried to use different method in order to compute these thermal distributions.
Idrus Belfaqih: namely, using the tunneling formalism proposed by Parrick and Wilshek in the 1999. In this picture one could argue that the particle. The radiation is coming out of the interior of the black hole. So there's a pair creations just inside the horizon.
Idrus Belfaqih: and one of this pair is materialized to the exterior region, and it will bring some of the black hole's energy away.
Idrus Belfaqih: The important note in here that I should mention is that in their paper they highlighted. And this is also going to be important in our calculations. Later on.
Idrus Belfaqih: the Adm mass or the energy computed at infinity must held constant throughout the process.
Idrus Belfaqih: and therefore, as the black hole carries away the energy, Omega, then the black hole mass parameter, will be reducing by exactly M. Minus Omega as well. But this is not going to be the same in our effective descriptions.
Idrus Belfaqih: The tunneling amplitude then defined by computing the imaginary part of the actions.
Idrus Belfaqih: For the geodesic trajectories, and that will be given by this
Idrus Belfaqih: expressions. And the way one way to solve this is to use Feynman, i. Epsilon prescriptions where the pools of the integration it turns out to be located at the horizon, located at the horizons
Idrus Belfaqih: by performing epsilon prescriptions. One can then interpret this tunneling rate as the Boltzmann vector
Idrus Belfaqih: then one can deduce the black hole or the black hole as a thermal bath with temperature given by exactly the hawking temperature. Classically. So this is, I recalculate the classical expressions.
Idrus Belfaqih: and another interpretation is to interpret the argument inside this tunneling rate, or exponential to be the change of the entropy of the system.
Idrus Belfaqih: If you use that interpretation, turns out that the entropy must be proportional to the area of a black hole divided by 4 exactly the Bekenstein-hawking entropy.
Idrus Belfaqih: So let's play the same game for effective background that we have.
Idrus Belfaqih: Again, there's a particle, an antiparticle, and the particle is materialized to the exterior region. But keep in mind in here the Adm mass is not the same as the black hole mass parameter. There is a vector computed, and infinity that relate both of these quantities.
Idrus Belfaqih: and therefore you can see that the reducing black hole mass parameter will be given given by the blue color in here instead.
Idrus Belfaqih: So that will be the quantity that you have to use later on, when you compute the upper bound and lower bound of your action integrations.
Idrus Belfaqih: and that this will be the integral that eventually you'll get if you're using our effective metric.
Idrus Belfaqih: But the important thing is that it turns out this integral is not as messy as it looks, because the pole of the integration
Idrus Belfaqih: in the still in in the in located still in the horizon.
Idrus Belfaqih: and the component attached to the lambda. It's also located at this residue of the integral. So this component will be eventually
Idrus Belfaqih: not contributing to our computation.
Idrus Belfaqih: So using the same interpretation computed the same relations. Then you get that the temperatures computed by observer at infinity will be given
Idrus Belfaqih: by hawking radiations up to global factor similar to our computations near the horizon expansions. And I should mention again, that
Idrus Belfaqih: if we are using monotonically decreasing function, then the temperature saturates the hawking temperature.
Idrus Belfaqih: Not only that, but also the hawking temperature or the temperature at infinity will be bounded because there is an inherent cutoff in our model.
Idrus Belfaqih: and the black hole. Entropy measured by observer at infinity, will be also given by Baconson hawking up to this global factor which saturates to one. If Lambda
Idrus Belfaqih: also separates at infinity.
Idrus Belfaqih: So this is another way of doing the computations which we at which we are also computing, using Brown-ear quasi local energy and compute thermodynamic.
Idrus Belfaqih: It turns out that the entropy of the system is again turns out to be local notions. But the important bit that I would like to mention in here is that asymptotically it will be giving us these expressions, which is exactly similar to what we have, using the tunneling approach or using the near horizon expansions.
Idrus Belfaqih: Another way to compute the Black Hole thermodynamics is by studying the semiclassical analysis, namely, using the quantum field scalar field minimally coupled to these gravitational degrees of freedom
Idrus Belfaqih: by expanding, by using tortoise coordinates to expand this mode
Idrus Belfaqih: and also using crustical coordinates. One could relate one of these expansion to another, and keep in mind that the Tortua expansions is Tortua coordinates is only covered partial spacetime, which is the exterior region, but the crucial is covered. The whole region of the spacetime.
Idrus Belfaqih: and therefore, by relating them through the Bogouli books, transformations.
Idrus Belfaqih: and associating the term of these substitutions to Fourier transformations. Then one will get again the hawking distributions.
Idrus Belfaqih: Coming out of this effective descriptions.
Idrus Belfaqih: The final descriptions that I'm using is using the hawkings approach.
Idrus Belfaqih: namely, the geometric optic approximations, where, at the seventies. Hawking tried to
Idrus Belfaqih: rewrite the late modes at the scry plus written in terms of the early modes and the scry minus. So again, we are using the same method by tracing back this solution in this wave packet back in time.
Idrus Belfaqih: and of course there will be some blue shifted
Idrus Belfaqih: happening near the horizon. Then one can use the Wkb approximations.
Idrus Belfaqih: such that the phase of this will be surface of the surface of constant phase are null, and therefore we can, following the trajectory of this race back in time.
Idrus Belfaqih: So because I'm not sure that I think I'm already out of time. So I'll be skipping ahead about this I'm just mentioning.
Idrus Belfaqih: The result that I gain is that again I recovered the similar distributions. But now the interesting bit is that I also get the grey body factor coming out of the because now we considered also the potential
Idrus Belfaqih: appearance in this formulations, then some of the modes will be entering the black hole, and some of them will be reflected back. And therefore there is
Idrus Belfaqih: transmissions, coefficients known as the Greabody factor.
Idrus Belfaqih: and computing a gray body factor is not an easy task, and, in fact, there is no way of computing it in an exact way.
Idrus Belfaqih: But there is a way at which you can get this expression
Idrus Belfaqih: at least analytically, for low frequency or low energy mode.
Idrus Belfaqih: and by performing the calculations, by defining the region in the exterior of a black hole to be 3 region.
Idrus Belfaqih: so region one is the near horizon.
Idrus Belfaqih: The second region is the region near the peak of this potential.
Idrus Belfaqih: and the last region is the asymptotic region, and then you match the coefficients
Idrus Belfaqih: of the Schrodinger equations, Schrodinger solutions in each of the boundary interceptions of the solutions. Then you can get
Idrus Belfaqih: the coefficients out of these solutions.
Idrus Belfaqih: then you can compute the grey body factor by computing
Idrus Belfaqih: the flux of currents that goes to scri plus in comparison to the incident wave coming out of the past event horizon.
Idrus Belfaqih: And by computing this using our effective metric background.
Idrus Belfaqih: it turns out that the transmissions coefficients just get a little bit of modifications if you set the Lambda to be purely constant parameter.
Idrus Belfaqih: But in the case of
Idrus Belfaqih: non-constant Holonomy parameter, namely, if it relates to area gap of loop quantum gravity. Then there will be suppressions, as you can see from this red factor over here, and this is going to be quite important later on.
Idrus Belfaqih: if you, considering the emissions of energy
Idrus Belfaqih: out of this black hole or the Boltzmann factor coming out of this black hole.
Idrus Belfaqih: but the most 1. 1 of the important
Idrus Belfaqih: conclusions that we gain is that we get that the absorption rates
Idrus Belfaqih: of the black holes in this effective picture remain proportional to the area of the black holes. And this is actually a result. This result is actually
Idrus Belfaqih: concluded back in the seventies by Unruh, and also by Das Gibbons and Mathur.
Idrus Belfaqih: namely, that if you take any d-dimensional Schwarzschild black holes.
Idrus Belfaqih: either it's asymptotically flat or asymptotically deceiter or antideciter you'll get that absorption rates will be proportional to the area of the horizon, but they don't consider it effective model. So we came to the conclusions that even if we considered our effective descriptions, then the solutions remain preserved. So we still get the universality
Idrus Belfaqih: as they are using the Schwarzschild metric.
Idrus Belfaqih: So this is the final slide. I promise. So about the emission rates. If you compute the emission rates, it turns out that if in a constant holonomic case, then you will get exactly the same amount of energy emissions just differ by this overall factor.
Idrus Belfaqih: and that is not enough to stop the explosions of the black hole.
Idrus Belfaqih: But, interestingly, what we get is that if you take seriously the scale dependent holonomy, and then you perform the geometric expansions, then what will you get? Is that the energy emissions.
Idrus Belfaqih: of course, regain the classical expressions.
Idrus Belfaqih: But there will be another term that will try to halt this process. So there's a term that try to break
Idrus Belfaqih: the black hole from exploding. So essentially we could say that the inherent.
Idrus Belfaqih: discretized structure of the space-time itself eventually will stop or rescue the black hole from exploding.
Idrus Belfaqih: So I will stop here, I guess, with the slide about the conclusions, and thank you for for listening.
Hal Haggard: Thank you very much, Idris.
Idrus Belfaqih: I'm so sorry for using the whole time.
Hal Haggard: Kong, please go ahead.
Idrus Belfaqih: So thank you. Thank you for the talk. I have several comments on your result.
Cong Zhang: So the 1st one is is is what you mentioned about this black hole remnant. So what I want to see is actually so in our work like like 2020 or some, we. We started the quantum dynamics of this Aos model, and then we also found something like like the existence of a black hole remnant in this, in this evaporation.
Cong Zhang: Okay? And and then there is some other comments on the 1st part of your of your talk so. So can you go to this slide with this where you 1st introduce this structure function.
Idrus Belfaqih: Yes, let me actually
Idrus Belfaqih: here.
Cong Zhang: Yeah, here. So so here. So here you, you introduce this, you introduce this, this specific form of this modified structure function. And also you mentioned later that this can be written as a function of this Dirac of durable and and yeah, so actually in in our recent, in our recent paper with with Professor Landowski, Professor Ma and Yang. So so this one you also you also you also cited in your in your slice. So we prove that this.
Cong Zhang: this factor, this this modified structure function, can be arbitrary function of of this ex and direct, observable. And then this theory will be covariant.
Idrus Belfaqih: Hmm.
Cong Zhang: And and can you go to the next slide?
Cong Zhang: Yeah. And also here. So you assume this, this H. 2. There to be to be some quadratic polynomial of this, of of this, this, this E phi ex, and actually in our paper. So what we do is just just like we just consider an arbitrary function of those ex E. Phi and Kx. K phi. So we finally get some equation for the Hamiltonian constraint by solving which
Cong Zhang: you can get different solutions, and each solution will give you a covariant theory. So actually, we, we so far, we found like 4 solutions. And the 1st one is this classical? Is this classical expression of this Hamiltonic constraint?
Cong Zhang: And the second one is, is this one you show today.
Cong Zhang: and the 3rd one is the the 3rd one is another one, and we also find the space time resulting from this 3rd one and the and and the spacetime structure of this 3rd one is actually the same as this. This, like like like, is this one with like 2
Cong Zhang: horizons? Yeah. So actually, the space time resulting from that one is also a solution, a solution to this. This model, proposed by Edward and and his collaborators.
Idrus Belfaqih: So so you're saying that there's a couple of similarities essentially in our
Idrus Belfaqih: qualitatively, there's a couple of similarities in the solutions that we have.
Cong Zhang: Yeah, yeah, so, so, so, so, so the solution here is like one of the solution of our of our equation.
Cong Zhang: And we also have some other models.
Cong Zhang: which is also covariant, and and we also have the 4th solution, but it so. The 4th solution in this 4th solution it is not a polynomial of those ex. E. 5 anymore. And so in that solution. So we found the space time. The singularity of the space-time is actually resolved by anti theater, Schwarzschel spacetime with negative mass.
Cong Zhang: and there is no black hole to Black Hole transition in that model. But the advantage of that spacetime is is that there is no cauchy horizon, and that spacetime may be stable. But the point is like, it seems like that model is not related to loop quantum gravity model. So we still don't know what could be the underlying quantum material under that.
Hal Haggard: All right, let's move on to Eugenio, and we can come back to this if there's more discussion to be had.
UP: Whitmore 320: Thank you. This was very nice. This was very interesting. I have a question about your slide on the public will check formalism.
Idrus Belfaqih: Yeah, let me go to that.
Idrus Belfaqih: Actually, I think it's easier to use this.
Idrus Belfaqih: Do you want the classical or the.
UP: Whitmore 320: Nice pockets.
Idrus Belfaqih: Isn't it?
Idrus Belfaqih: Okay?
UP: Whitmore 320: Yeah. So I was intrigued by this remark that you have. Can you comment more about that? The distinction between the Adm mass
UP: Whitmore 320: and parameter, M. Data, piercing formula.
Idrus Belfaqih: Right? So yeah, at the beginning of the calculation, actually, we kind of confused, because I remember that at first, st when I compute this, I didn't get so so we initially computed using the
Idrus Belfaqih: brown-ier quasi local energy. And then we compute the solutions at infinity asymptotically. It will give us some expressions of the entropy. But then we are using this formalism, and this is the only formalism at which we gain different result among the others.
Idrus Belfaqih: And it turns out that I tried to reread again their paper. And exactly they're highlighting a point which turns out to be crucial.
Idrus Belfaqih: at which the Adms is the quantity, that one must be started as the constant quantity when they compute this quantity, when they want to compute this amplitude.
Idrus Belfaqih: and exactly in our model. Initially, when we compute about this, Okadms is different with M. But still they are both content.
Idrus Belfaqih: But then, if you're using that notions of Dirac observable in your computations. Then you're not going to get the same expressions as in the quasi local energy. And therefore in here, instead of starting by using the black hole mass parameter one. You should start with the Adm mass which is given by these expressions in here.
Idrus Belfaqih: and therefore by that, you can deduce what will be the change in the black hole mass parameter, and that will be given by these expressions, and only then you can redefine this effective this term as the effective Omega.
Idrus Belfaqih: and there will be the omega that appear in the upper bound of your integrations.
Idrus Belfaqih: and with that the conclusions will be coherence one to another.
UP: Whitmore 320: This is really interesting. The the public wheelchair formalism is the only one that I know that actually takes into account the back reaction to the emission of irrigation. So it's really nice that you can see the effect. Yeah, the modified metric.
Idrus Belfaqih: Right. Thank you.
Suddhasattwa Brahma: Yeah. And exactly
Suddhasattwa Brahma: a quick comment here would be that, of course, there are also like corrections, as you know, in the particle check, which is omega squared, and here also there would be like corrections which are slightly different from particle check. Those those would be like this, red Omega Tilde times blue omega. So that red Omega has, like small corrections in terms of this lambda infinity.
Suddhasattwa Brahma: So yeah, that's the back reaction that you that you mentioned exactly.
UP: Whitmore 320: Good, really nice. Thank you. Sudan. Yeah. And one more comment later in this slide that you have a remark on the temperature being bounded. Is this a bound of the smallest mass, or is it something else.
Idrus Belfaqih: Yeah, yeah, that's actually the same thing. Because so in here, I think I didn't provide the expressions. But so in the expressions, there will be a minimal radius. For instance, in the non-constant lambda there will be a minimum radius, and that minimal radius will be
Idrus Belfaqih: parameterized by the black hole mass.
Idrus Belfaqih: But so in the in the case at which lambda is constant.
Idrus Belfaqih: the smallest you can get from this value is actually 0. So you can go to 0 when the mass of the black hole goes to 0. But in the non-constant lambda there is an inherent cutoff coming from the loop quantum gravity area gap, such that as the black hole evaporating more and more energy, or it's emitting more and more energy, the size will be decreasing, but it cannot decreasing.
Idrus Belfaqih: and
Idrus Belfaqih: down to below the area gap of a black hole. So essentially, the area gap the inherent area cuts off
Idrus Belfaqih: that will stop this process for the black hole to be emitting more and more energy.
UP: Whitmore 320: Yeah, thank, you.
Hal Haggard: There other questions.
Hal Haggard: Kong? I wasn't sure whether you had a question also, or was it just the comments.
Cong Zhang: Yeah, so I just give the comment. So so, so, yeah, actually, what I want to say. So the last sentence I want to say is like, it seems like. According to to those work, it seems like this, general covariance could significantly narrow down our choice of the of the theory. So I think it's, it's yeah.
Cong Zhang: It should be considered.
Cong Zhang: Okay. Thank you. Thank you.
Hal Haggard: Well, I think with that we should thank you, Idris. One more time. Thank you for the talk.
Idrus Belfaqih: Thank you.