 your life. Okay, I think we are live. Welcome, everyone. Thank you for joining us for today's low physics winner. My name is Alejandro, and I'm going to be your host. Today, we are presenting Ultra Compact Swarshield Star and Exalted Eco by Camilo Posada. Camilo obtained his PhD in physics from the University of South Carolina in 2017, before becoming a postdoctoral researcher at the research center for theoretical physics and astrophysics in OPAWA. He was at the group of Relativistic Astrophysics at the University of the Federal Dua BC. His research interests are on relativistic astrophysics, compact objects, medical relativity, and gravitational wave sources. We are very happy to have you, Camilo, and for everyone listening to us now live, remember that you can ask questions or email through our YouTube channel or Twitter, and then the questions will be read at the end of Camilo's talk. So now, without further ado, we will turn the time over to Camilo. Thanks for joining. Thank you very much. So thank you, Alejandro, for the introduction and also for the invitation to this webinar. So I'm very happy to be here. So I'm going to share my screen now. So you guys should see this right now. Okay. So let me move this a little bit further here. Okay. So I will talk about this as I have been calling it Ultra Compact Schwarzschild-Star, an exotic eco, but I will go into some arguments why I'm calling this like that. So, okay. I'm sorry, I have some issues here. Okay. So I would like to give first some motivation for all of this work in principle. So as we know, black holes, they have this reputation like in the scientific community and there is a lot of consensus on the existence of black holes. However, they have several paradoxes which remain unsolved. For instance, the information loss, which is crucial for black hole physics. Also the huge black hole entropy implied by the Hawking effect. Also like the well-known curvature singularities in the geometries of the interior of mathematical black holes. So these are still open questions that are waiting for satisfactory answers. So that's why several authors have proposed in the last decades some alternatives for black holes. So I'm probably, if you are familiar with the literature on this, so some people call it non-singular black holes or black holes, mimickers also, or echoes, acronym for exotic compound objects. So you can actually see the this review from 2019 by Carlos Sampanio on the subject if you want to learn more about it. So just to name a few of those echoes, so you can find things like boson stars, grava stars, falseballs, black stars, anisotropic stars, and the list continues. So at the present, this is an important point. So there is no direct observational evidence of existence of the black hole even horizon. So even with the data that we have from like Virgo gravitational waves and including the picture of M87 from the even horizon telescope. So these things we know are sensitive to the light rain at the east coast. So the inner most stable circular orbit. So which actually is actually quite outside beyond the even horizon. So this is important because sometimes I can see still people believing that we are actually seeing a black hole and it's not true. So you are not seeing an even horizon there. So that people have to be careful with those things. So on the other hand, if we are proposing something different to a black hole because some exotic compound object or non-singular black hole with a surface instead of an even horizon. So any physical phenomena near the surface or near the will be horizon and the possible structures if you have some tension model like a grava star and the old one. So these things could be proved in principle after the maybe a ring down in this in the well-known phenomena of echoes. I can see this is small like a reflection that between the east coast and the surface of the between the light ring and the surface of the object. So eventually these modes will leak out and we could observe those things. So this could be implications of a surface instead of an even horizon. So however, the existence of these echoes are still not conclusive. So there's actually a lot of debate whether or not LIGO will go observe those things. So okay, so this is okay. So again, as I said before, so the black hole has a lot of reputation and actually now it's part of the pop culture nowadays and you can see actually from movies, the most recent one, Interstellar, nice movie actually. So actually, probably you might know that Kip Turn, sorry, I didn't start my watch, so still in time here. Sorry about that. So Kip Turn actually was advising Christopher Nolander in the film in order to depict this very nice image of a black hole. So and also a bunch of memes you can find nowadays with the M87 picture. So I picked this one in particular with the Lord of the Rings thing and it's also even in music. Like you can see this record cover with a Christian disc near the black hole, like this some black metal bands. Actually, the music is not quite good, but it was nice to cover. So it's now part of the pop culture, so it's really something like it's getting a lot of attention. So okay, so I'm going to be now a little bit technical, probably not much. So hopefully this isn't going to be so painful for all of us. So just to describe a little bit, putting context, more ideas. So we go back again to the social solution. So which corresponds to the first exact solution to Einstein's equations from Boris Bach in 1916 few months after Einstein's equations were published. So and actually describes basically the vacuum exterior gravitational field of a spherically symmetric mass here. So there is one parameter only the total mass M as measured at infinity, of course. So and there are two peculiar things on the solution. So first of all, so an article zero. So we can see there are in principle, infinite title forces. So there is a singularity here. And actually, when you go back to calculate things like a variance, a scalar invariance. So actually, you also find out that these things are divergent at that point. So this actually is telling you that this is an essential singularity. So there is a breakdown of the theory that at that point. So so there's nothing to be there. So and there is a second point, which is quite peculiar, which is this R equal to M, which is called the spatial radius. When this is goes, this goes to zero, and this is actually divergent. So in principle, it's an infinite red sheet hypersurface. So moreover, there is a reversal of the roles of T and R. So this time coordinate and here are the radial coordinate. So they change the role once you close that surface as time like an spatial coordinate. So that has implications actually to the definition, as we know it so far of an even horizon, which is this marginally trapped surface from which matter and information cannot escape classically. So the idea is, initially, as we know, that anything once closes this surface, and the spatial radius will be totally lost from the exterior universe, let's say. So here we are actually showing, I'm showing here this embedding diagram, so in two dimensions, so we are fixing two coordinates here. So this is an upper surface, two dimensions upper surface. So showing the spatial geometry, the exterior spatial geometry here. So moreover, this is like asymptotically flat solution, meaning that when you go far from the source of an argos to infinity, we should record any constant space time. So you could say like, okay, so if I go back to look at this as what she's reduced, so when you go back again and do that size to see what happens with these colors in variance, in variance of a cube, a cube, and so on. So these things are actually finite. So it seems like nothing really dramatic happens at the horizon in the space time geometry. So in principle, you could say, okay, I could change coordinates so that I could remove the singularity. So that's been, that has been done. And in principle, you can see like at the Kruskal extension, adding some coordinates and so on. However, you have to be careful when you do those things, because you can do that in principle, the analytic extension, for example, when you do Kruskal extension. So only if the time you knew the energy momentum tensor is zero in the horizon, otherwise you cannot do that. So that's an important point that also it's crucial to discuss. Moreover, the analytic extension involves the inclusion of complex coordinates, which sometimes seems a little bit strange, given that we have physical coordinates here. Okay, so it took almost 50 years for, in order to find like an extension of the solution, now for the case of a rotating black hole. So it actually was found by Roica in 1963, which corresponds to the exact solution for a stationary axis symmetric field outside a rotating black hole. So now we have two parameters in this solution, the total mass and the angular momentum to J. Sometimes it's convenient to define this parameter J over M here. So as you can see, this is a more complex solution. Of course, it has more substance than the Schwarzschild metric. So in principle, for the static case, when you turn off the angular momentum, so stop the rotation, so we go back to Schwarzschild as expected. So also important point, this is asymptotically flat and invariant under simultaneous inversion of Young-Phi as you can see from here. So this is crucial because there were many attempts before Kerr to find a rotating metric, which failed in the sense that they were not asymptotically flat as expected. So it was really the correct one, let's say. So there are two killing vectors associated to this metric, which are the t and d phi. So once again, this killing vector is actually to go a bit technical, not much. So I'll just tell you about the isometrics in the spacetime. So the symmetries of the metric basically. So here in this picture, I'm showing some pictorial diorama, which you want of the Schwarzschild, the Kerr spacetime. So now there are two horizons here, so we actually come from the solution when this is equal to zero you see here. So this is a quadratic function. So now a symmetries are ring, so I will not go into more detail on this. So and one particular characteristic which is quite interesting of this solution is that it predicts the existence of the so-called Ergosphere. So an Ergosphere is this region, where we cannot find any static observer here. So in other words, any observer in this inside the Ergosphere will be dragged together with the rotation of the black hole and will move along with it. So you can find a static observer here, where once you cross this static limit, you will be dragged also together with the rotation in this black hole. So this is a very important region and actually it has implications, more implications for example in the Penrose process, which actually is a consequence of the existence of this Ergosphere and more things. So I will not go into more detail on this, this is just to mention that. So also there is an important point which is that there is no Birkhoff's theorem for the Kerr metric and Birkhoff's theorem just to recall is this theorem proved by Birkhoff's course that says that any solution to answer this question, which is static and a spherically symmetric has to be its bashing, what a patch of its bashing. So you cannot do the same for Kerr basically. So the idea is that it is not true that the exterior space time of any rotating object is described by Kerr. You could find that quadruple moments are a combination of those from Kerr, but not necessarily Kerr. Now, when you have a black hole, you can actually prove that this is the state of this gravitational field. Okay, so that's after improved by Israel, so which I will mention now later, perhaps, I think. So okay, yeah, okay. So and once again, this will be connected now with the idea that we have these so-called uniqueness theorems that were like a lot of work on that back in the 60s, 70s, the Golden Era of the year, let's say. And a good way to summarize this is that a black hole has no hair. Basically that this is telling you that external gravitational field and electromagnetic fields of any stationary black hole with charge. So in this case, it would be the Kerr-Newman metric black hole. So are determined uniquely by three parameters, mass, charge and intrinsic angular momentum. So this is very important as again, as I said, so there was a lot of theorems back in the 70s regarding this result. So okay, so on the other hand, we could say that black holes are inevitable if they are in principle two conditions and goes back to the theorems, singularity theorems proved by Hawking and also terms in principle. So first of all, you have a trap surface. Okay, so once again, like once it's there, so no information can go out from there. And one of the energy conditions is satisfied. So you have the weak energy condition, which this is, let's say like a mathematically something like this, but you know, there was telling you that the energy density has to be non-negative. Okay, so we are assuming like common matter, as we know. And also the strain energy condition will be satisfied, meaning that the energy density to the sum of the three main pressures is non-negative too. Okay, and suppose it's bigger than equal to zero. So of course, you could say, okay, I can find some compound, some substance, some mechanism, something that violates one of these conditions. One good example, for instance, is the cosmological dark energy for the so-called lambda term in the astro-secretion. So we know that these dark energies, we believe now is this unknown substance, that's why we call it dark. So anything we don't know in physics, we call it dark. So it is a known thing that in principle is making the universe expand faster and faster. So the question of a state for that, for that energy is p equals minus rho. So in principle, it violates this from any condition. So that happens, so what happens, therefore, with the singularity theorems? Okay, so that will be the starting point, because I will go back to this later. So we could provide hints if we are trying to find something that is not singular for the outcome of gravitational collapse. So all right, so this is just to summarize some of the paradoxes of this, of the idea of black hole again. So the crucial one is information paradox, and there's been a lot of work on that so far. But it seems to me, I mean, as far as I know that there is still no consensus on the solution of this. She's crucial because we are talking about fundamental issues with quantum mechanics in the sense of loss of information. So also, the car family of black holes, they have singularities and close time cures in the interior, which are problematic, of course, went back to the formula of Hawking for the temperature of a black hole. So you remember that the expression is one goes like a one hour total mass and it's proportional to H bar. So in the classical limit, that temperature goes to zero, but the entropy goes to infinity, you go back to the formula for entropy for black holes. So it's something like you have something super cold, but infinite entropy. So that's peculiar in the sense that going back to the conceptual level, what is the interpretation of the entropy of a black hole in this case? And we go back to Boltzmann. So we learn from Boltzmann that the entropy is actually associated to the law of energy of accessible energy of states. So in this case, what will be the interpretation here when we go back to the when we go to the context of black holes? As also as has been observed by some authors, it's possible that you could have potentially large unbounded semi-classical effects of stress tensors produced by certain mechanisms, for instance, vacuum polarization. So therefore, you have that. So the idea of extending a vacuum global solution to the interior is not possible, which is basically the idea of analytic extension when you do cross extension for instance. So that's a crucial point also that, moreover, probably some of you might be aware of this paper in 2014 by Hawking. Well, he actually wrote, there are no black holes. So you can see actually the point of debate I can imagine and the controversy on this. So in principle, the resolution of these issues is expected from some, let's say, accepted theory of quantum gravity, which still we still don't have that. Therefore, it will be important and it's worthwhile to find alternative models which we could somehow try to get some hints to solve these paradoxes. So, okay, here is some catalog of some of the horizonless compact objects that have been proposed during the years. So I'm really stealing this from the review by Cardo Sompani from 2019. So I will just probably discuss briefly some few of them, like the fluid stars going back to Shafidon Tekowski and has been studied quite in terms of stability and gravitational wave signatures and isotropic stars also, which formation remains unknown, but there's some work on stability, electromagnetic signatures, gravitational waves and so on. Probably the most popular ones, or I would say in the literature is the boson stars. Osiratons, which it seems in principle that the formation mechanism is the most, is the better, let's say, compared to the other ones, or at least the less controversial, let's say, and there's been a lot of work in terms of stability, signatures and gravitational waves, of course, and electromagnetic. The Grava stars, which I will talk more about it later, which still the formation remains a mystery. So what has been some work on stability, electromagnetic signatures, gravitational waves, like a Quasi-Normal Morse and so on. So I will have included some of my work here, but they didn't, so okay. And anyway, so the list is long. So you have ABS bubbles, antideciter bubbles, wormholes, which are quite interesting and also mystic. So there's some work by Bronikov, most of it on this subject, force balls, super spinners, collage polymers. Well, you see bunch of fancy names. So it's quite a big list. So, okay. So I will go to talk about this Grava star, so which I've been concentrating some of my part of my research. So if nowadays, you go to Google and you type Grava star, probably the first thing you see will be something like this. Okay. So this is a nice looking Bluetooth speaker. I think it's 40 or 50 bucks. I'm not sure what that means. So probably you go a bit into the underground music like me, perhaps you can find also that Grava star is the name of death metal band from Greece, I believe. I think that is still active, but they are not very famous. So, okay, hopefully they are not going to be offended if they see my video, but what it is. And you go to physics now. So you can actually find a Grava star was this alternative solution to black holes proposed 20 years ago now by Masuramotola. So the idea was that during the collapse, there is certain phase transition near the would be horizon in the original model 2001, where the final outcome will be some interior region, which is basically described by a P equals minus equation of states like a negative pressure fluid, so kind of like a dark energy bubble, which you want. Okay. So of course, you have to match that with the exterior like which is bashing. Okay, because there is uniqueness there as we know. So in order to do that surgery, let's say you have to introduce, so they introduce this thin shell of ultra stiff matter. So, which is because of the flow. Okay, and then that's the main composition, like the five layers composition of this, this object. Okay, so again, so this negative pressure. So actually has the effect of this effective repulsion, okay, which avoids in principle, the appearance of a central singularity. Okay, so this actually was not new, to be honest, and actually back in the sixties, there was a proposal based on that by Liner and Sakharov, actually they were trying to find a way in order to avoid the appearance of the central singularity in the collapse. So the only way would be to have something with this effective repulsion, like a P equals minus negative pressure fluid, which again, by only just on any condition, therefore you can evade the singularity problems. Okay, so this Gravastan was found to be stable back in 2001. So once again, there is no even horizon. Okay, here, so we don't have any issues with loss of information like in black holes. It's not singular at the center, like black holes. This is a regular interior described by the Ceter metric, kind of like the Ceter type metric. Okay, there is no entropy is very small, but that is more compared to black holes, which is problematic. And time is global. Okay, so global in the sense that we know like given Ceter metric, you can actually predict the whole evolution from there, from initial data. So in 2015, there was some interesting observation for the, in terms of this model, and was found out, was found out that the Gravastan actually can emerge naturally, emerge naturally. When you go back to study carefully, the solution, the second paper by Esbarchil with the constant density interior solution, we call it Esbarchil starvation. Okay, so I will mention a little bit about this in the next. So we go back to this second paper by Esbarchil, again, 1916, where we actually solved Einstein's equation for a fluid with constant density, or what is called the incompressible limit. Actually, he called it like that in his paper, back in the day. So, and again, it's interesting because it's an exact analytic solution for Einstein's equation. So then actually, you can check like most, I mean, most of the e-app textbooks nowadays, this solution, like Schott's, Karol, Weiber, gravitation, of course, and so on. Okay. So, and again, so the assumption is here, the density is constant for all values of pressure. So the pressure actually decreases with radius from the center to the surface, but the density remains constant. So let's discuss the positive side of this conflict of the solution. So first of all, it provides a simple analytic toy model, really, for interior of a relativistic mass. And in principle, I mean, you could think like, okay, probably constant density is a bit unrealistic, probably it's too restrictive, but you could think something like having a fluid with certain composition which varies with the distance with the radial coordinates. So for instance, like the regions with higher pressure without me at the center, you could change the composition such that the density remains constant compared to the points on the surface where the pressure is lower. Okay. So kind of like a hand-time payload, as gravitation put it, or you can, you don't want to worry about it. So you just simply imagine some global water of limited size. Of course, you cannot expand, go extend this example too far because this is gonna not gonna work along the scale, like for a big thing. But in principle, it's not so crazy to have this situation. So in that sense, this interior bashing solution provides a nice example of hydrostatic equilibrium in general relativity. So the negative side or the contrast of this is that again, it describes a perfect fluid in the incompressible limit. So meaning that the speed of sound goes to infinity, it's not well defined. And so that's problematic. And also, there is a discontinuity in the density at the surface. So we really don't believe that, for instance, astrophysical objects of interest, like a Newton-Ester behaves like that. So a Newton-Ester, which would expect like the density decreases with the radius and goes to zero near the surface. So something more like, for instance, the Tolman-Seven solution, or the Buchdar solution provides probably a much better description from the interior of Newton-Ester compared to the constant density. Nevertheless, the solution is still interesting as a tree model and we'll have some repercussions for the graph as far as we will see in the next, okay? So first of all, let's observe that this solution is actually non-singular as long as the pressure remains finite. So you just study the behavior of the central pressure, so which is the point of the the Huyger pressure you can find. Actually, it's the Ebergen when the radius reaches these nine-fourths, the total mass. Here I'm putting, I'm sorry, I didn't say from the beginning, I apologize. I'm putting this gravitational constant and the speed of light equal to one, so this is natural unit. So we don't worry about that now. So there are some GEC here somewhere here. So we will forget about it now, okay? So this is just out of where M equal nine-fourths. So actually, it's varsity in his paper, and actually, so for those who are interested in this, you should actually read this paper, this something that you should read at least once in your life, very nice. So he actually noticed this limit is his solution. So later on, Moodal, 1959, actually proved that this is a maximum limit for any configuration with any equation of a state, okay, given certain conditions. So first of all, we assume that the general relativity is the correct theory of gravity. So we use science as equations. Second, we assume a perfect fluid and a spherical is symmetric. We also assume that the pressure is isotropic, meaning that the radial components of pressure are nearly equal to the transverse pressures and the tangent pressures. Density and pressure remain non-negative, okay, important, and the density decreases monotonically with the radial coordinate R. So under these five assumptions, Moodal showed that any configuration should be less, the component should be less than this, or R over N should be bigger than this nine-fourths, okay? So in principle, educacity is an inequality which actually is telling you that the constant density saturates this limit. It's put you in the equal case here, and you can see here, okay? So this is the maximum case you could consider, the most extreme case you could consider. So this toy model is giving us like the top there, so from down or from there to below, so we are okay with any other solution. So this will be important because it will be the starting point for what comes next. So now, so you could show, and actually was shown by, recently by Masoud and Motto, in 2015 that if you look at it a bit more carefully about the solution, you can actually show that you can evade the Moodal bound for the constant density solution if you allow the existence of an anisotropic pressure. So once again, a theorem is as good as the assumptions used to prove it. So if I violate one of those, I could evade this Moodal limit. So for instance, if I allow the existence of an anisotropic pressure at some surface in the star, I can evade the Moodal bound, and actually that's what happens here. So you can notice that you start to squeeze the star from above, so you start to make it more compact, okay? So I squeeze this star so the radius starts to approach the Moodal bound. So I go to Moodal, so if I introduce an anisotropic stress on that surface, I can actually go beyond that, so I can integrate that divergence, and there is some you want to know about this when the paper is with the details on that. So you can actually integrate that divergence and can go beyond Moodal. What is interesting is that once you cross Moodal, there appears another bubble here from the interior, first at r equals zero, which has a negative pressure in the interior. So I just keep, I just keep squeezing from above as I'm showing here in these different snapshots, and you keep squeezing those, so the radius of the star from above starts to approach this by shear radius. This negative pressure bubble keeps growing from inside. So kind of like a as above, so below, somehow, so to speak, such that when you approach this by shear radius completely, so the whole, this whole bubble of negative pressure fills the whole interior of the star with the, and becomes essentially a Grava star, okay? So that idea, so that was a very interesting observation of that solution. So what is interesting about this point is that this alternative was already there in this by shear solution, so it's inherited only from an exact solution to Einstein's equations, okay? So that's what is interesting. We just have to be a bit more careful about this Boudin limit and how to kind of manage to obey it. So, okay, so that was the observation that led to the idea that by this limiting adiabatic process as I was showing before, as the radius approaches this by shear radius from above. So this is by shear star, this constant interior solution essentially turned out to be the Grava star, basically, okay? So this negative pressure, like a bubble of that energy surrounded by this now a boundary layer of anisotropic stresses. So now we are getting rid of this thin shell that was put back in 2001. Here we don't need to introduce that thing here, as you can see. So, and basically the idea what you are left with this metric for the interior, which is the metric for static Grava star interior. So notice here that the one over by shear radius squared is somehow like the zeter horizon. This had like a zeter form, okay? So now this constant that was left undetermined in 2001 was actually now is fixed because this actually corresponds to the exact value that gives you the correct matching with the Sbashian exterior at side, okay? As we can see here. So again, this metric is just consequence of the Sbashian interior solution when you approach the Sbashian period, okay? So it's not put at home, we're just coming from there, okay? So, okay. So once again, also this pressure becomes essentially minus, so the pressure because basically minus rho, and again, it's just consequence when you go back to check the relation for pressure from the constant Sbashian interior solution, you approach the Sbashian radius, you ended up here. So you're just coming from there, okay? So it's not being put at home in any way. Okay. So that was an interesting realization. And once again, so the idea here is instead of this tissue now is that in this gravel study, you have this surface located at this Sbashian radius where there is a discontinuity in the surface gravity. So the discontinuity in the surface gravity actually gives you this contribution as a delta function form of anisotropic stress. So you have like some anisotropic stress on the surface. So that's why we could evade the Brudel bound, which again provides a surface tension given by this. Notice that this is twice the smart value for the smart formula for black holes, which is 1 over 8 pi m. So, but here is, I want to stress this out because here you have a surface tension associated to a physical surface, which is given by this anisotropic stress, okay? In black holes, you have a surface tension associated to the horizon, but it's vacuumed there. So it's not clear to me that surface tension in the case of black hole is, how do you interpret those things? Okay. As I said before, so this ultra-compacted splash, it starts as I've been calling it or the gravel study, you want to put the name. So it has maximum components of one half, like a black hole, okay? You could put up to some possible plank scale correction, we will talk about that now. And we can evade the Brudel bound once again because of the existence of this anisotropic stress, okay? In principle, you could think as this gravel started in 2015, I would like to call it, so people start to kind of make the differentiation with the 2001 model. So it could be seen like the universal limit of this old gravitational connoisseur star proposed back 20 years ago, with this ultra-stiff shift, the one I discussed before with this thin shell of matter and so on, okay? So this is really like a little bit different in that sense. So another important point here is this, when you see these papers on all gravastars, of course, there is a time like other boundary because you have an ultra-steam, ultra-keying, you know, super thin shell of matter there. So it's a time like other boundary there. So it's a time like surface there. So in contrast with this one, here, we are locating exactly, you can see this delta function is big that is bashing radius, so this is an old surface now, okay? It's an old surface, so it's infinite redshift there, okay? So this is important because this will be connected with something I will discuss in the next. So, okay, so, okay, I started my clock a little bit late, but okay, I think I'm fine, okay? So you have still like 15 minutes? Oh, great, okay, excellent, okay, thank you very much. So, okay, so I can, so this is bashing star, so yeah, the star is so far. So, so together with Cecilia Pidentin, in my time in Brazil, so we started the stability of these things on the radial oscillation using the well known Chandrasekhar theory of pulsations. So it's, in principle, is seen that these things are stable against radial perturbations. Also, it's on colleagues here, you know, power, we actually constructed an anisotropic version of these ultracompatis bashing star or the grava star. So using the so-called minima geometric deformation and GD method. So, no, no, it's, it's a, it's like it's a more, with more substance, let's say, it's, it's another with more substance compared to this, to this other model. Also, we, together with Roman Konopli and Sasha Sidenko, he had, you know, power, we also started this, the, the actual perturbations, the gravitational wave signatures of these objects. So this, in principle, are stable against these actual perturbations. I will show that later. However, we did not find any echoes, and I will talk about that a little bit later, connected with this surface, the key part is that this is a null surface again here. We were talking about this null surface. So that's something that people sometimes skip when they, they read this. Okay. Also, most, most recently, together also with Cecilia again, her student, Victor Guedes, we studied the tidal reformability of these objects. And actually we also found that they don't have any love, like similar to the case of black holes. So this will have important consequences to, to realize whether or not this is a good black hole miracle. So I will talk more about this a little bit more in the next. And also this year, there was a, appeared this, now an extension to slow rotation of this new Brava star. The important point here is that quantity, like the moment of inertia, in mass-quadruple moment, approach to the corresponding curve values, at least at second order in the angular velocity. So the consequence of this is that there is no way to distinguish this Grava star from a curved black hole. Okay. So moreover, this paper actually, it's like an improvement of an old proposal I had for a slow, slowly rotating Grava star. So the approach here was different because they considered the whole configuration already, like the final solution, let's say, the final Grava star. So I didn't use the Grava star and start to squeeze it, but there were some issues with that. So this could be seen like an improved in many ways of that old model I had. So, okay. So in principle, also this, in terms of the rotation, this Grava star is a very good black hole miracle. So I will discuss briefly now some of my recent work on, on the signatures of action and gravitational waves and the type of deformability. So the question was, okay, are these things stable against action perturbations? So the idea here is going back to the problem of actual gravitational perturbation. So we simply stop this equation. So now we have some potential here, which of course will be determined by the spatial star solution and so on. So as usual, so we introduce the R star coordinate or the total coordinate, which when R goes to infinity, this R star goes to infinity and this is a crucial point here. So when R goes to R0, let's say called this R0 corresponds here in the solution to this region R0. So in this bubble that is growing from the interior. So once again, this R0 is a null surface. It's a null surface. It's not a time like surface. It's a null surface, which eventually when you complete, when you approach completely the slushy radius, it is placed on the surface. But again, it's a null surface. So we place the boundary conditions at that point, but it's null. So at the end, the boundary conditions that you place there are exactly the same as in the case of black hole because it has infinity reaction. So I'm emphasizing this because I have some discussion that some people will not get in this point. Anyway, so in the vacuum exterior, so we go back to the standard regular equation, as we already know very well. So and then you can solve the problem. So we found some time domain profiles for the perturbations, as you can see here. So this is showing the initial thing down. Okay, we are considering the case, the illegal two case, which is the relevant one for the problem. So and we compare with the case for black hole. So the black hole line is actually black. It's below these lines. So it's probably, no one's in there, but trust me that is there. So believe me. So you can also check the paper here and put in the reference just in case. So, so here we are showing the profile for three different spatial stars. So the problem was the same. So the philosophy is the same. So we take this partial star, we start to squeeze it. Okay, again, so different components. And then we see what happens. Okay, and then we start to approach this partial radius. Okay, so as you can see for a big spatial star. So now we are beyond moodle here. For instance, for 1.1 is by she read you, which is the magenta line here. So the initial freedom is the same as black hole is usually suspected for an echo. Okay, but then of course the timeline, then the power low tail is a bit different from black holes. However, as you keep the squeeze in this star, as you can see here from the red line, which is one point of spatial radius, or this one, which is the closer one, which is the blue line, we are very close now to spatial radius. Notice how the initial ring down the posterior, the exponential decay and the power low tails actually approach exactly almost exactly like the case for a black hole. Okay, so this is crucial because in principle, this is telling you that in terms of gravitational wave signature for actual modes, we have exactly the same behavior as black holes. There are no echoes, you can see echoes here. Okay, and again, because once you place the boundary conditions on the surface of zero, which is a null surface, so this will be the same boundary conditions on the case for black holes. Moreover, as you can see the perturbations are actually decreasing with time, so this is stable, otherwise it will be going up. Okay, so this is in principle an important result and it's showing that in terms of now gravitational wave signature for actual gravity perturbations, this thing behaves as a black hole. Okay, in the, of course, in the final configuration, then you have completely these bashing radius. Okay, so now the last part of the talk will be regarding the tidal deformability and log numbers. So what is log? So in principle, the tidal deformation of compact objects will actually place a very important role in the mission of gravitational waves during the spinal, binary spiral. Okay, and also actually provides also important information on the internal structure. Okay, so you can imagine like you have two objects here, so like this, like a typical binary. Okay, so the gravitational force, the fate of this one on the other one, will induce quadruple moments on this object. Okay, so this, look at this external disturbance of this object on this one is what we call tidal effort. This tidal effect, so the response of this object due to these external fields is what we call tidal deformability. Okay, and these things are actually determined by the so-called log numbers, and the log just comes from Augustus' log, who actually was a mathematician and geophysicist who actually introduced this formalism back in 1909, when he was studying the tidal deformations and all these pieces behind this for the earth. Okay, so eventually we learned that we could actually extend this for any component of course. Important that these log numbers, of course, depend on the question of state, and we will see also that it depends strongly on the components. So once you can learn more about this, you can actually tell what's going on in the tidal result with the question of state. So there's been a lot of work on that by the Munagar, being an internal person also, in connection with black holes and Newton stars and so on. Okay, so for instance, from the event 1707-17 from LIGO and AMBIRGO, so which is this binary of Newton star binary system, so this actually put important constraints on the tidal deformality of a Newton star. So therefore, from there, we can start to, once we know that, we can go back and start to see, to start to put constraints on the possible equations of state to see which one is the best. Trying to find out what is the question of a state for a Newton star. Okay, so okay, so going back to now for black holes, there's something interesting because black holes actually has been shown that they have zero tidal deformability for rotating and not rotating cases. There was important work by Golibeck, Poisson, Cheyenne, the Taken, the Taken-Casals recently. So what it means is that of course, if you have some object near the black hole, like a Newton star, of course, you will see some, there will be some deformation. So the result what is telling you is actually an observable infinity, we don't see any deviation from the typical asymptotically gravitational multiple moments of the black hole. Okay, so and again, so the result actually that all these love numbers banished for black holes is still very fascinating and actually it's now motivating a lot of work for a lot of people actually this year. Very interesting paper by Poisson and some other people and trying to find like deeper, that's the significance of deeper explanation to this result that these love numbers are zero for black holes. Okay, so on the other hand, when we go to exotic compared objects or echoes, it's been shown that the tidal love number for these things in principle has been suggested that they scale like one over log of theta where this theta is this parameter tells you how far you are from the, how far is this update from the, from this partial case, from this partial horizon. So in principle, as you can see compared to Newton star, so the tidal number for echoes actually with certain Planckian corrections, of course, near the horizon, are smaller four to five orders of manage. So in principle, well, now nowadays this is a bit beyond the current light sensitivity. So, okay, so then again with Cecilia again, and with Victor Gheres, so we actually started what happens now we can see that this is partial star, so do the exercise again, so take this partial star, the constant density solution, start to squeeze it, go beyond Buddha and see what happens with the tidal informality as we approach this partial radius. Okay, so actually the result is quite interesting because, okay, here first of all, I'm showing some results, so this is the tidal informality, this is the components here, so we can see the result here like actually components you can see is, excuse me, the tidal informality is highly dependent on the components, first of all. So again, this is the book, the limit here, so this configuration has been studied up to here by the Muran Agar in 2009, and then what we did was just fill it up and go beyond that to see what happens. So we can actually see this monotonically decrease behavior of K2 as a function of the components here and actually approach to zero as the components approaches one half. So one particular point here, and this is an observation that is important to stress out, is that notice the value at the Buddha bound, so we actually found that K2 at Buddha is around this value, so notice the striking question of K2 going from 0.75 for almost zero components to this value at the Buddha limit, so this is interesting, and if you do now a 15 line for this, which actually was left undone, but I just recently did it in some paper that will be with a period of proceedings, so actually notice that this is more like an exponential decay, this is the best 15 line for this tidal informality as a functional component, so notice that it has a better exponential behavior, so actually this is not a lock scale behavior I was proposed for echoed by Cardo and collaborators. So naturally, even without this, you cannot notice up to here, which was already known, that this decreases quite faster than the lock scale behavior. So once again, at least for this assumption, back in this paper, the idea was that the lock is, they assume three different, I think three or four types of echoes, the old gravastar workhorse, I believe, and some unisotopic stars, so from there they say like, okay, it seems like all echoes probably will have this lock scale behavior, so it seems to me that that conclusion was a little bit hasty in the thing that we actually found here, that this echo, which has been shown to be a good black hole mimiker doesn't follow that, okay? So moreover, this approaches to zero again, in the spatial limits, so it's like a black hole again, okay, in that sense. Okay, so in summary, so as I said at the beginning, so the several paradoxes posed by classical black hole has actually motivated the emergence of alternative models, mainly in the form of these echoes, or exotic compared objects. So now this gravastar proposed 20 years ago, nowadays we realize it as emerging naturally when you go back to the exact solutions to Einstein's equation for that constant density fluid of this partial star, so it's already there for more than 100 years, let's say. So finally, this gravastar, the 2015 model as a limiting ultra compass partial star, let's say, in terms of things like moment of inertia, mass particle moment, tidal deformability, gravitational wave signature as we saw, behaves as a black hole, basically. Okay, so in other words, this gravastar 2015 could be, in principle, a very good black hole mimiker. I will just leave some probably news on gravastar, which were quite on the hot topic, let's say, still probably not so much now, but back in the gravitational waves came out first, so there was a lot of debate whether or not they would observe in black holes gravastars, so there was some news on that. So, okay, so I think that's all. Thank you. Thank you very much, Camilla. Let me see if there are questions over one of these. I think we still have five minutes before the hour. Okay, let me start with something. So just to make sure if I understood these objects, and so typically the motivation comes from all the problems you mentioned at the beginning, and then there are several things we don't know, but these objects in some sense are still classical and they satisfy Einstein equations, right? So could you please comment again, so what problems they might be solving or is for the interest of having black hole mimickers per se? Yeah, yeah, yeah, exactly. So let's recall that, for instance, the Venor Ison has particular issues in terms of the information paradox, right? Okay. The idea that you have some pure quantum state and eventually becomes a mixed state, so you have no conservation of information there and has problematic issues in terms of unitary evolution in quantum mechanics. So these objects, in principle, when you remove the Venor Ison, you are avoiding that, okay? At least in this object here, the graph after the time is global, so you don't have this analytic extension, okay? So you don't, the GTT component inside as outside remains time-like, okay? So there is no reversal like in black holes, so you don't have any problem with information loss. So you can do quantum mechanics there without any problem. This is a regular interior, again, so it's not singular, basically the center, so it's well-behaved at the singularity of radical zero in contrast to black holes, which are singular. So you could say, of course, we cannot see that singularity because it's hidden behind the Venor Ison, but again, it still remains the question like, okay, we have something that is not correct. Of course, people claim, well, general relativity is not probably the final theory that we are looking for, okay? So at that point, we should have things like correction from quantum mechanics, and those things should be seen near this radical zero. So, but again, we still don't have, there's no consensus on a possible quantum theory of gravity, so therefore, so another point should be, okay, let's see if we can propose something different and then try to see what we can learn from that, okay? So that's the main idea behind all of these objects. So now it's some people claim like, actually, there's no need to remove that, just like you can actually have all these possible scenarios together, like a living together, like you could have gravastars, forceballs, black holes, and so on. So it's like a whole menu of things without the need to one overcome the other. So still, I'm not, we're not discussing that right now, let's say, so, but that's kind of like the main idea. Thank you. Walter? Yeah, you could just continue with Alejandro's question. Thank you, Camilo, for the stop. So, right, so this proposal of the ultra compact stars, so gravastars is not really, it's not that if we, for example, ever detect one, we are then concluding that black holes are not present, right? This is not, this doesn't exclude at all the existence of black holes, right? This is just something that could exist, right? Well, the point is like, as I was showing here, like, in terms of gravitational waves and things like tidally formability, this behaves as a black hole, basically, okay? So once again, if now we talk about observation, for example, like the LIGO-Virgo data or the even horizon telescope, those things are sensitive to the light ring and are equal to 3M for its bashing. So beyond that, you cannot say anything. That's the point. So the geometry exterior for these things are actually exactly the same as a black hole, as a bashing, right? So of course, what people propose, like, if you have some horizon less compact object, like with some surface, some tin shell, for example, you could have echoes, right? Like, as you have, you have these trap modes between the light ring and the surface. So these modes trap there eventually leak out, and you could observe that in some signal of LIGO-Virgo. So here for this one, again, this is a no surface. So even though there is a hard surface, it's not. So the infinite, there is infinite ratio there, okay? So as you see in your analysis, we didn't find any echoes, okay? So it seems like this thing is really behaving like a black hole. So if you see a black hole, there's no way right now to say, okay, this is a black hole or a graph, because those things behave the same in principle, according to our results. For this graph, I start in particular, the one I'm talking about the 2015. So the good thing of this one is we are solving the problem of singularity of the interior that is not even horizon. So you are getting something out. Right. Well, you are displacing the question or you're basically avoiding the, having to answer the question about what happens around an event horizon, right? But there are other solutions to answer any questions that also have horizons and that we might have like the receded horizon. And then we still have to answer for the questions of what happens with quantum theory around those horizons. But also, and then what happens with other types of black holes, like primordial black holes that are not so much seen as stars, but something that was formed from other densities very early in the universe, would that also be, I mean, it would be also acceptable to interpret if we ever detect primordial black holes as primordial gravity stars or? In principle, yes. Because again, the only parameter handy solution is the mass. So you can imagine like a super massive black hole 10 to the five, six solar masses or 10 solar masses or something. It's no problem here in the solution because it's the only parameter, free parameter in the solution plus the, plus the spin of course. Okay, so in that sense, like in the in the same range of masses that you can find black holes, you also can cool find this gravity star. So there is no limitation in the mass, in that sense. Okay, so going back, let's say to, as you, as you point out, like to primordial black holes or so on. So, so the point here is just to kind of emphasize the idea is, so you have this object that in, in terms of signature of gravitational waves, tidal deformability and rotation, quadruple moments behave exactly as what we expect for a curve black hole, essentially without the problems of black holes, meaning there is no horizon, there is no similarity at radical zero. So of course you could ask the, the real question here, the main question would be, okay, so how do you explain this, that you have something with positive pressure because you started with the constant density, and then something became negative pressure. What happened there? That's the main problem, actually, in this model, but we don't know that. And that would be more like, let's say the microscopic physics in that model, like what's going on there, what kind of phase transition that you have something with positive pressure matter, and suddenly become something with negative pressure. That's the real issue there. So that's something that we don't know. Okay, so, but if you assume that, and you allow this thing at the end to form, as I show here, all the possible signatures for observation and indications of this thing is actually the same like a black hole. So you have a good black hole mimicry in that sense. So that's the main idea. Yes, first of all, Camilo, very interesting, the talk. So I was one, I mean, correctly, if I, if I'm wrong. So let's say that in the Gravastar, the main ingredient is the kind of what is helped supporting the avoiding the collapse of the, of the object is the kind of the amount of dark energy in the interior of the grass. Yeah, so the repulsive effect, the negative effect. Yeah, the repulsive effect, the kind of the Gravastar not to disappear. So is there is a way or kind of, like the question that Walter was saying that how to produce these Gravastar in the, in the universe kind of, because in, in our time, kind of, it's very hard to have a kind of a bubble of dark energy or is it, it might be related with inflatone fluctuation that produce these bubbles. And that's the, that's the million dollar question. And yeah, that's, I really don't know to be honest, because it's, as you see, like, as, as we show how this emerges naturally, it's purely using Einstein's equation. So it's purely like a microscopic approach. So now how really you can produce that mechanism that you have something with positive pressure becomes negative pressure. And it's, it's something that actually we cannot learn from only from Einstein's equation. Now we might need to go to the microscopic there and probably with some field theory to really to see, oh yeah, that situation is possible under these circumstances such that you can produce this and we have observed that. So, so yeah, so I think that's the main drawback of this model. That said, the mechanism of formation and I think most of these echoes have this issue, as you can see, like from postbones and so on, probably in the list I was showing here at some point. So the mechanism of these things is, as you can see, the formation is, we don't know really. So, so we can in principle learn something about the phenomenology of those things in terms of stability and signatures, rotational weights and so on. But the mechanism of formation remains still, except for probably for boson stars, which is probably the most, let's say, studied so far. But other than that remains kind of like a mystery. So, so I think that's the main drawback here. But, but it still is interesting to see that there is an alternative there that probably you could, you could learn something from it. Once we have something more like, you know, image or something like some quantum theory gravity that probably could predict these things. Say, okay, yeah, actually these things could emerge from that big theory, but we are still far from that, I think. So, so that question remains, I cannot answer that. So, yeah, it's still open. No, but it's, yeah, super valid. Yeah. And I think there's a lot of people trying to find out how. So, because one of the novelty of the Gravastar, as far as I understood, is the fact that it's a solution, it's a new solution of the Einstein equation. Exactly. It's not any object. Exactly. Now, you're not putting something there and, okay, this is not all, it's really, and actually this one in particular, as I showed here, is really emerging from the Sbashi interview solution with a well-known exact solution to answer the question. So, it's really, it's already there. It was already predicted by Sbashi. So, 100 years ago. So, probably, he had to go beyond this to realize that, you know, but of course, he had to wait a little bit for some mathematical tools to come to know that. So, yeah, so that's the idea. Okay, thank you. Okay. I think I just have one last question, Camilo. And it's, so you said you can, at some point in your talk, you said, okay, it's because we have to do it carefully, how to build them, and then just to like start analyzing them, because as you said, they were like in the initial solution, but then this was not seen, let's say, or explored at that point. And then you say, okay, we can evade this book out, limit. But then my question is how delicate it is, or in the sense how easy or like how fine-tuned you have to do your staff to get the audience. And my question, and then like the big question I wanted to ask here is like, in the sense that you can show, okay, the quasi-normal modes can be, I don't know, somehow mimic, there is no title, love, or etc. And then you can prove this, oh, it's a stable and they're not radial. Now let's go to Axial, and perhaps later Polar. And that's, my question is, it's always like you can just do it with the same object, or you have to fine-tune the new condition for the other test or this other test. Because I remember when these mimicers appear, and there was this paper by Pani at all, they said, oh, because the ringing part could be anything. But then people say, okay, but then you would have to consider that the whole gravitational wave form from the spiral to the merger to that, because otherwise you cannot stick. So my question is related to these sort of thoughts, like is it the same object that you can prove all these nice mimicers, or for each test you just have to fine-tune? Surely it's the same object. As you can see here, for instance, for the for the accept perturbations, we use the same object. So probably I didn't go into the details here because I wanted to put it a little bit less technical, but it's the same object. So many, this one here. So for the tidal informality, it's exactly the same. So this is the same object. So we are simply taking the basher start, squeeze it, and then see what happens when we crossbook that and apply this basher area. So the only parameter in this solution, again, is the compactness, basically. So there is no fine-tuning, like, okay, for certain parameters, you'll find this, but for some other parameters, you don't find that. So what happens, for instance, when you see these papers by Cecilia, Retzola, and also Cardozo, Pani, and so on, with the old Gravastar, you can actually start to modify these things. Yeah, yeah. Put in different widths, different composition inside, like some anisotopic fluid. So you can start to fine-tune these things. Here, no, actually. This is just one parameter solution. It's very clean in that sense. Okay. So that's why I think it's, for me at least, well, I mean, I was proposed this problem first by Masud and I was working with him, Dr. Laina. So from there, I said, okay, it's quite interesting. So let's see how far we can take this. And the solution even is so simple, what allows all these analyses with one simple parameter. I think that's the beauty of it. So I think in contrast with other objects, with fine-tuning and so on. So yeah, that's it. Great. Thank you, Camilo. I don't see more questions. I don't see them checked. I'll check if we have something. So thank you. And I believe Roberto can correct me. This is the last webinar of this year. I guess so, yes. Yes, maybe. Yes. Stay tuned. We might forget. We're going for holidays now. Okay, very nice. Now we go for holidays. Yes. So stay tuned for the next season. We will still be discussing these nice signs and we will also keep continuing with great speakers. So thank you very much, Camilo. And thank you everyone for joining us. Thank you very much for the invitation. It was good to see you. So thank you. I hope that we will see you in the next session also. So thank you very much. Bye-bye. Thank you. Bye, everyone.