 number three in the afternoon So I'll cover everything now before I go on to discuss a little bit about the the evidence for black holes let me complete what we were seeing yesterday and Because some of you were concerned about some of the things that I was doing. Let me point out number one So we were reducing the problem to a single ODE Okay for the fields for some master function It could be a scalar it could be a vector or could be a tensor the gravitational waves in the case of gravity What you should do. So I told you how will you compose? metric fluctuations in Tensorial Cerecal harmonics and The way you do so if you have initial data if you prescribe just initial conditions everything I said goes through if instead you want to Focus on some non vanishing stress energy tensor. So you you do exactly the same, right? So you take the field equations You linearize them around some background solution in So suppose for instance you have a small buckle or a neutron star Falling into a supermassive guy. Okay, then the background is vacuum so You're going to expand your metric around the background which is Schwarzschild and then you're taking linearizations of that. So your correct metric is going to be the background plus some fluctuation Right, you're going to expand Jimmy new up to linear order in H Okay, and you're assuming that the neutron star or small buckle contributes Something that scales like age. So so at the end you're going to write something like this Okay, and this would be the stress tensor of your point particle. Okay We know how to do this now, right? We decompose this in tensor spherical harmonics We project this on those harmonics and you get a set of Couple of equations the notebook that I sent you basically shows how to separate and decouple all the equations So at the end you're still going to get a single OD for Evolving the problem. Okay, so with some source term Okay, so this is note number one note number two is that even So suppose we do this for a point particle. Okay, so I take a small buckle of mass mu You could think of this for instance as a small buckle of a solar mass falling to a supermassive buckle we tend to the six solar masses these systems exist and We know we're going to see them in the future Okay, and this is fine because now we have a guy which is a million times smaller and less massive Than the other so I can apply I can use this procedure Okay, now let me tell you that this works astonishingly well even when you extrapolate To equal mass systems so for example if you do the calculations they're in the notebook the radiated energy When you throw a small buckle into a more massive Guy is something like this Mu squared over m Okay So of course smaller masses The less the energy you get If you extrapolate this to the equal mass case by promoting Mu to be the reduced mass of the system Then you're going to find that the radiated energy gives you something like this 6.5 times 10 to the minus 4m This is purely Protervation theory, okay, we should not be even allowed to do this Okay, just a while extra violation if you do this using numerical relativity So evolving non-linearly the full system then you find that you're going to get 5.7 times 10 to the minus 4 So you see Even in this while extrapolation you're within 20% of the answer So this is what you should if you have your preferred theory and you want to see how gravitation waves Are emitted how they work? This is what you should do before plugging everything in the computer try to understand the problem at linearized level first Okay, I don't think we know why by the way why this happens Why all the non-linearities that should occur in truly equal mass systems are washed away one of the possible reasons Is connected to the light ring it kind of prevents Radiation from escaping but but this is a no okay so yesterday we we were looking at scattering of waves of vocals and The collision of particles with black holes, okay, and we saw that Generically the late-time signal is something like that It's a ring down and we try to understand the ring down in terms of a superposition of complex exponentials Let me give you an example Which is a very clean and I think neat example, so I'm going to replace my effective potential Okay By a simple delta function, so I'll be working in minkowski when I mentioned just a delta function for the potential And see what what what comes out so I'll be working with this So I so remember this was basically what we showed yesterday Here we had an effective potential. I'm just making sure it's localized Okay, and then there was some source term Which was basically related to the initial data I'm going to assume that the time the momentum is zero. I only have the wave function at t equals zero So the source term looks something like that, okay To be able to solve this nicely I will also assume that the initial data in the initial configuration is localized. So this guy is A delta so I have a delta function the potential at x zero And I'm going to release a delta function at x equals x naught very trivial example Now all the wave functions we define the nice wave function on the left Which is outgoing the nice wave function on the right we can write them down and they're like this so L is e to the minus i Omega x on the left It's a in e to the minus i omega x plus a out e to the plus i omega x when x goes to infinity I Can get these coefficients now right all I need to do is so remember we have this problem X zero there's a delta function at x naught. I release my data The only thing I need to do now is require Well continuity of the wave function here and use this equation of motion to get the jump Okay, so I find If you demand continuity then you find that one is equal to a in was a out This is basically well because at x zero this has to be equal and the jump So you integrate across the delta function at x zero gives you Psi prime between plus and minus epsilon some small distance around zero has to be to be not Psi at zero Okay, so you find One is equal to a in plus a out and a out Minus a in plus one is equal to to be not over Omega okay The wrong scan is very easy to compute now the runs can between the two solutions is really just to i omega minus to be not So immediately you'll find quasi normal frequencies remember from yesterday the quasi normal frequencies are the poles Basically the zeros of the runs can so this tells you immediately that Omega q and m is Minus I you know So it happens that there's a single quasi normal frequency for this system, but but but it's there And if we know what is positive the system is stable Okay, it's minus i so As time goes by the fluctuation dams away very good the solution we also saw how to write down the time domain solution so It was something like this So first I started to come I start by computing the Solution in free space, which was just an integral with the source at large distances. We found this the X Psi L I omega so this is the initial data over the runs can okay But this is a delta function. So the integral is trivial so this is e to the i omega x I omega over the runs can a in e to the minus i omega x naught plus a out e to the i omega x not So this is done So in free space we're done We can invert now Right, and if you invert We need to do an integral in the omega space in frequency space. So we find psi So this is the time domain function is 1 over 2 pi integral of e to the minus i omega t times this guy the omega and Now we substitute right and you see immediately that these intervals are relatively easy, right? So these are 1 over 2 pi integral of e to the i omega x minus x naught minus t over 2 plus 1 over 2 pi Integral in the omega e to the i omega x plus x naught minus t a out over 2 In okay, but I don't know if this is basically just a delta function, right? This is the Fourier representation of a delta function delta of Of x minus a Is basically this Okay So the first term this term is going to give me a delta function And this is what I was referring to as the the prompt contribution So this is kind of the minkowski analog of the online cone propagation So this would be then I find psi is this guy. It's one half Delta of x minus x naught minus t Plus 1 over 2 pi integral in the omega of Minus i v naught over omega Over 2 so I agree. I'm rearranging some terms over omega e to the i omega x plus x naught Minus t how do we do this? Well, we close the contour There's a single pole. The pole is exactly at the question of frequency So the only thing we need to evaluate is the residue and We're going to find we're going to find that psi is one half of Delta x minus x naught minus t Plus let me rearrange again that term 1 over 4 pi in the wrong the omega minus i v naught over omega plus i v naught e to the i omega x plus x naught minus t Okay So the pole is here Minus i v naught at small time. So if t So this is minus okay if t is smaller than x was x naught You can close the contour on the upper half plane There's no poles that so you can close it because this guy goes to zero When you close it at large omega right if you close it with a large semi circle This term goes to zero. So you get nothing. There's no poles here. So so for t smaller than x was x naught psi is only one half of Delta x minus x naught minus t What is this? Well, we started with this problem okay Initial so where somebody is there plucks the field and I'm observing here Okay The first term tells me simply that a half of the initial data is traveling at the speed of light towards me So that's fine Now what I would expect is that the other half is traveling to the left right so there's another half Traveling this way it interacts with the potential barrier and some time after that Actually, I know precisely the time right it's the time it takes to get from here to here plus twice this time I should get something and you do get something because you find that for t larger than x was x naught The field is psi equals on half of Delta x minus x not minus t plus the residue contribution Which is v naught over two e to v naught x plus x naught minus t at these times, okay? So you see this is Very nice because this is Precisely the QNM contribution It's decaying Away in time precisely at the quasi normal frequency rate, okay that that's controlling the decay of the fluctuation Okay, of course, it's a very special example in the sense that you are able to compute very precisely The amplitude with which you excite the mode of the barrier okay But for Michael's basically the same thing happens or any other compact object instead of a delta function Potentially we had some effective thing But the picture is always the same you got a prompt contribution From the initial data then it goes the other half tries to go into the buckle It excites the modes of the buckle which seem to be localized at the light ring We saw it yesterday, right and after a time after and actually the time is two x naught Plus this one you can check Okay, you get something else and the something else is the ring down of the field very good. There are open issues Well, there are some important things. We don't understand yet Others we do understand For example, it was thought or it was kind of hope that that it would be possible To excite resonantly the modes of a buckle, right? For instance, you can you can do resonant excitation of the modes of a glass Right or the modes of a guitar It turns out that you know, of course, it's hard to excite resonantly The modes of a buckle just because they are localized at the light ring And if you think about the problem the light ring is basically the innermost Thing structure you have in the space time, which also means it's the highest frequency you have in the problem Right. So if you want to excite resonantly The modes of a buckle you'd need to find something that travels at the speed of light. So you need You need basically motion The equalcy and this Is very hard to do it's very hard to do because matter should basically Stop existing at the innermost stable circular orbit that has a lower frequency than the light ring Okay, we do not know well We have done a very poor job at understanding the amplitude of The amplitude to which we excite quasi-normal modes in buckle space times. Okay, we don't even know how much a Small in falling buckle To each extent it's going to excite the modes of a large buckle Theoretically I mean of course we can compute the time domain waveform, but at the theoretical level computing excitation factors Of buckle space times has not been done properly So this is one item to do again. Nothing basically is known about way time tails because you see This guy is going to get a prompt contribution is Going to get also a contribution from the potential barrier, but after that. So let me draw my observer again X X naught where the initial that I was this is the barrier So that's prompt contribution from initial data a quasi-normal contribution from the barrier some of the fluctuations travel on Right, but there are space time curvature. So they're going to get back scattered So some of them are going to fly back to him and this gives you a power law T to the minus well, it depends on the multipolar index in this way Okay, this has not been seen in any numerical simulation of real Initial data Okay, most likely because of numerical issues, but it's something we don't understand yet We can use this to test any alternative theory of gravity you want Use this means we can take your favorite theory of gravity. Suppose you don't believe GR is correct. So you add some Some other degree of freedom. This is going of course to change the way the black holes ring down the way the black holes vibrate So the idea well the optimal scenario would be to have a way Would be to find how the impact of changes in the theory in the question all frequencies because if you have this Well, I give you a black hole. I'll give you a gravitational detection. You measure two or three modes You can rule out GR and you can constrain the parameters of that theory, right? So so far this has been done on a on a case-by-case Analysis and there's a reason for that even though the problem is linearized. I mean it looks very simple Okay It's really really tough to make very generic predictions about how the frequencies are going to change in fact We know and I'm going to discuss in the afternoon of examples where the change is even non-perturbative You seem to add a small parameter, but the changes are well are dramatic and The same thing happens when you change boundary conditions. Okay, so let me go before I Start on the boundary conditions. Let me first tell you why We should even think about changing boundary conditions and this goes to the topic of lecture four Which is about testing the local paradigm. It's a bit funny that so I discovered not too long ago that Chandrasekhar was giving is a talk on the shvarshil in honor of shvarshil in Germany and There he said that a confirm so Of course, we believe the current metric should be the only stationary state of the field equations in vacuum. Okay, and There should be ways to test it even back then there were well people were starting to think about ways to test But Shander Zakhar made a remarkable claim that a confirmation of the metric of the current space-time or Or some aspect of it cannot even be contemplated in the foreseeable future I think he was he had in mind most likely, you know Tests using gravitational fluctuations or gravitational waves. He knew we were still Sometime to go before we saw something, but it's kind of interesting that three decades later. We at least Have something to work on. Okay, and there's something to work on is the first detection here This is the first detection is most interesting for me here and for you because it's so far The the event where the final stage is more clearly seen. Okay These are 30 solar mass wakals or objects colliding with one another and this gives you a signal Which is more clearly seen in the very late stages, right? So the question is this is fine. This is all compatible with two wakals Can we use for instance our ring down analysis to disprove any other object or not and There's two types of claims The first claim is well, this has to be a buckle It has to be a buckle because we know nothing else in the universe Which is able to dump a signal so quickly because you see this is the noisy data, right? The signal is excited. These two guys are going around each other But then all of a sudden everything dies so quickly And the argument is there's nothing I know that has such a huge viscosity That will dump the signal so fast. I mean that you see two cycles there or something like that, right? the other argument is well Maybe it's not a black hole, but let me check how compact it is. It's quite easy to do a calculation stance told you about The quadruple formula just using quadruple formula and energy balance So you know the energy flux you connect you equate that energy lost by the system to the binding energy of the system Okay, and you get basically the the evolution of the orbital frequency Of your of your binary. Okay, it looks something like this So the orbital frequencies related to the gravitational frequencies so gravitational waves are emitted at twice the orbital frequency So the expression looks something like this. Okay, not very illuminating, but anyway It seems it depends only on this church mass cal M and It depends on Merger time t naught. Okay, which we have no access to but given this data Ligo and you I mean I said down and I did calculation by I just using estimating roughly Frequency at two different points here before merger. You can so you have frequency at two instance Okay, you can go here. You have two later points. You estimate t naught and column Okay, it's quite simple. Of course, you will have some uncertainty I got the difference of 20% with the label results But anyway a factor 5 would be okay for me, but it's a factor 0.2 So you will find that the total mass of That system is of order 65 solar masses. Okay, if there's some uncertainty Maybe you'll find a hundred. Maybe you'll find 40, but that's irrelevant. So, okay, you have 60 solar masses But how much space is this mass occupying? Well, that's easy to do because you have Kepler's law So you can relate frequency to separation or there's even an easier way, right? You go here These are 0.05 seconds and you see something that's repeating Roughly five to ten times on 0.05 seconds, right? So you have something doing this and In 0.05 seconds. I see like five cycles. Okay If I am very very conservative and I assume that whatever is doing this is traveling at the speed of light Even if it's traveling at the speed of light You get something like 500 kilometers of separation. Okay, so it's a very quick estimate to see how big the object is So you're able to squeeze 60 solar masses in 500 kilometers Okay, so this is what we saw it's a very compact object indeed and so so the idea is well Then there's really nothing else we know of it has to be a buckle Okay, it's dark We didn't see this guy in the electromagnetic window. It's very massive and it's very compact Okay, and the argument then goes well, we know nothing more massive than free solar masses That's compact and dark. There's nothing else we know like this in the universe Therefore it has to be a buckle. I Mean of course there is an extra ingredient and the extra ingredient is That we know black holes form in realistic Situations for example, this is an example done five or six years ago You take two neutron stars those we've seen Okay, roughly 1.4 solar mass each of them And you try to merge them. This is a simulation done in the AI six years ago or so And this is nice because I'm going to use a fraction of it if it plays. Let me see So densities are huge here Okay, these guys exist, but they are so if you take a spoon of this material a teaspoon of this material it weighs As much as a mountain. Okay, so it's really dense Units are 10 to the 14 grams per cubic centimeter But it's interesting so as as as these objects start to in spiral you will see that there's tidal deformations Right, so the gravity of one of the objects is deforming the other Right, we can compute these numbers. We can we know how much they deform one another They're pulling the other material. They're losing energy via gravitational waves. So at some point they have to merge There's no question about this. They could form a single neutron star okay, and They do this if you're lucky enough if the stars are light enough most of the times You see huge tidal deformations Most of the times they're going to merge and do this they are going to form it looks white, but it's a buckle when Okay at the center You now form a buckle and all the collisions. We've done so far eventually you may even form a Metastable state which is a very massive neutron star At intermediate times, but eventually there are all these all collapses of locals So we know locals form We have no other alternative. Okay so So the common law is well, it has to be a buckle. So I'd like now to indoctrinate you in why We should keep an open mind as to this as to what the object must be actually Yeah, actually well the reason is a bit more refined than that, but I'll go there in a minute Now there's reasons why you should doubt you should at least keep an open mind about why There could be something else and the reasons are several for example We saw that the interior of a buckle has a diverging crutch when scalar. Okay, so there's a singularity there Gravitational fields are huge. We have no idea how to handle diverging gravitational fields We need something like quantum gravity or a theory that somehow Doesn't produce singularities in the interior. Okay, and of course you can ask why it's a huge coincidence That each time I form a singularity. I also form an horizon Right. This is basically the essence of the censorship conjecture But there's also another tweak to the story, which is the following you're you have to assume that even With the classical horizon there this classical horizon is able to shield any possible quantum effects that are Ineherent to the singularity. So you have to assume that any quantum effect Produced by this point like singularity is confined to it in the horizon. Okay So this is a point number one The point number two is that most of the objects we know Come with spin. I showed you an example. You throw two buckles at one another. We actually saw this happening in the universe The final state is spinning Now if you have a block of which is spinning or a block which has some minute charge They come with an extra ingredient. It's called a cushy horizon. I don't have time to go through the details But of course your eyes and means that if you try to do this kind of analysis fluctuations of the geometry You basically can't evolve you can't do time evolutions past that horizon. Okay, so in practice In practice if you throw yourself in one of these guys I'm not going to do it. I'm fine on the outside But if you're if you care enough about physics, I just said, okay, I'm going to test this What this means is that you are not able to predict any future any future any of your future You can't predict how things are going to happen once you cross because your eyes and so okay, so this is a big It's a big conceptual problem, right? It's not necessarily related to quantum effects. It could be Horizons could be somehow Classically unstable. We don't even know that but in the solutions we have they're all there. They're always there Of course, there's other problems Some of them are related to the horizons. I think these are minor problems For example the statement that we have Hawking evaporation Because we have horizons which may lead or not to information loss Or not, okay. I think it's a minor problem in the sense that large wackels Basically behave as minkowski. This is really just equivalence principle, right? So if you take a supermassive guy The effects that an observer feels as he crosses the horizon should be should be basically negligible. It just use equivalence principle however in 2018 we are still testing equivalence principle, okay, so So even in that sense testing horizons is almost like At the same footing as testing equivalence principle with the added bonus that you still are also testing whatever is inside the horizon Cal second had a very nice sentence for this that I love to use Which is if you're going to tell me that you find in the universe a spot We from which light never comes in and within which you're hiding a singularity where you know Nothing of what's going on if you're making this extraordinary claim, then I'm going to demand of you Extraordinary evidence for that. Okay So I think I and you should start thinking about the evidence quantitative evidence that you have for horizons there's other reasons for at least Trying to test the presence of horizons but I But you don't even need to believe any of the crack potty statements about modified theories of gravity or resolving singularities or whatnot. Okay This the only statement that you're given is the following You have a detector You have a detection you have data Can you quantify the compactness of the object that you saw? How compact is the object? Where is the surface of the object? Okay, this is basically what I'm interested in I'm going to try to convince you now that If you find a more sensitive detector What you're going to be doing is basically constraining The compactness of your object to be closer and closer to that of a buckle you can never get to the horizon To the limit you're just approaching it. So in this sense, it's like doing particle physics, right? Better and larger accelerators are going to probe smaller and smaller distances Very good Actually, this was a question my mother asked me When she heard about the news she knows nothing about science. Okay, but she's she heard about bloggers She's asking what the hell how do people know where is the surface of the object? How do they measure to know it's a buckle? That it's black I can understand we haven't seen it. Well, where is the surface? right Now there's a lot of questions to address And I think it's fair to say that all of these are open issues, which is why I have the least here, okay? So the first question is do we have alternatives? So we're going to need to quantify this and and to quantify the the statement that we have an horizon The easiest way is to have alternatives because then we can see the differences. So the question is do we have alternatives? and if we have them Do we know that they form dynamically? Just in the same way that what goes do when we collapse neutron stars. Does this happen? Are these objects stable because a neutron star can leave for millions of years can merge with another one and form a buckle if I for if I have an alternative That leaves for one second then it's not a very good alternative to a buckle, right? So I need to make sure that it's stable and there's some funny developments here, okay? And finally we have seen stuff So does the do these alternatives look like anything like Waggles if I use a telescope? Use a normal telescope or if I use Lego do they reproduce? Waggles in that sense or not Now the alternatives are too many to mention I'm going to put down a few. I think I sent you a notebook also I was full of notebooks that I took me a lot of trouble. I Think in one of the notebooks I build one of the alternatives and it's a boson star. Okay, so Among the I don't know dozens of alternatives to buckles alternative means It's an object that can be massive and can be very compact. Okay? The most I think the most one of the most attractive models are these guys Objects that you build in a very nice elegant theory a Theory which is well posed we can do evolutions Some of these objects are stable as I'm going to discuss we can collide them. Okay? One of the examples are boson stars. You just take a minimally coupled scalar field Just the one I discussed yesterday. Okay? It has a mass term. So it's massive These scalars can form stars Okay, they form compact configurations which are bound by gravity They don't collapse because there's pressure. There's kinetic terms Of pressure in the field equations and that's why they sustain There's an isotropic star. So if most of the neutral star models that we have assume Anisotropy of the material basically that pressure in the radial direction Equals pressure in the tangential direction. Okay, and this gives you the usual neutral stars and neutral stars don't Usually don't have ergo regions nor light rings Okay So they are not that compact, but if you add an isotropy you can easily Build more compact models Then there's of course all the exotica And the exotica means You take wormholes. Okay. For instance, if you find exotic material or some alternative theory of gravity You punch a hole in the space time you build what people call wormholes Most of them were and are known to be dynamically unstable. Okay, but still it's an alternative. That's very compact There's graver stars. So you take a decider interior and if you're able to somehow Match this to an asymptotically flat exterior, then you have what people called a graver star So it's a gravitationally bound star with the decider core. Okay, they can be as compact as you like But they have big problems like most of the guys here the big problem being you do not have a Theory from which they come you assume you do some matching you build a solution You don't really you can't really know the underlying theory and perform evolutions with this. Okay people in string theory Produce all sorts of things. I'm not criticizing it. I love this one of them argues that black holes are just a kind of average of a large ensemble of horizonless objects these are called fuzz balls So you could you could think about the horizon as the average value of some fluctuation of a large number of fields, right? Each of them has no horizon, but the average looks like a buckle Unfortunately, we do not have any first ball construction in a syndotically flat four-dimensional space time in our universe. There are some constructions very simple constructions in higher dimensions So I think all of these by the way are I Think very very exciting open issues to work on Okay, so don't take me wrong if I if I if I say that string theories do all sorts of things So there is a huge effort going on into trying to build Some of these objects in four dimensions and I think we really really need this Some people in particular or Java argued that there could be there's no fundamental reason to have the curve bound the curve bound if you remember in four dimensions was a smaller or equal to M and These argument was there's nothing in string theory that points to a special To anything special happening at this limit So perhaps the universe produces objects which are spinning above the curve bound and somehow The effects of the singularity are shielded In some way, we don't know. Okay, so we would have a quantum if a quantum object Very compact, but again, we don't really have a theory to describe this And then there's all sorts of things collapse polymers to two holes that appear in higher Curvature corrections to the field equations and so on and so forth. So I told you that one of the best models are Boson stars and the reason is There's a question. Oh Yeah, very good question So the question is have any of these been simulated the answer is yes one so so so so there's Really, there's decades of work here because we know really nothing the only thing we know Are boson stars. These are minimally coupled fields Scalars we it has a well-posed initial value problem. So we can just take equations and evolve them And there's an added value of scalar fields some models of dark matter Actually rely on very light fields. So you see there's motivation to even look at this stuff in other contexts now Before I go to the evolution. I think I have let me check Yes Before I go on to the evolution of the only case we know of There's a nice way to kind of parametrize all our ignorance. Okay. There's names for this and the names Again crazy names, but anyway, I I think I introduced one of them, but a nice way to parametrize this for For our discussion, okay Is the following so we're only interested in compact objects, okay? So I'm going to be measuring how much my object Deviates how much the surface of the object is Distant from the event or from the event horizon corresponding to that mass, okay now you put this in a diagram where you have compactness and The timescale so or the time that it takes a photon to get from the light ring to the surface of the object Okay Right, you see I'm sitting at the light ring I send this laser beam and I measure how much time the photon hits us takes to hit the surface and get back to me I'll tell you why this is important Now if you use this there's all sorts of beasts right neutral neutral star sits somewhere here then you have Exotic compact objects, which is everything else. Okay, all of these are exotic compact objects Basically anything that's as compact or more compact than a neutral star Okay, if the object has a light ring So the surface sits at the photos here or inwards we are going to call these ultra compact objects and there's a reason for For putting the photos here there, okay If now now now comes the interesting part Clifos if the surface of the object is Such is at the point where? This is complicated, but so I'm at the light ring. I shine a laser beam I measure the time it takes the laser beam to reach the surface and get back to me Okay, if this time is larger Then the time the instability timescale of the photon orbit that we computed two classes ago One class ago Then I'm going to call these objects a clean photosphere object. Why is this important? because if the surface is Located so deep inside the potential well that photons take a longer time that the than the instability timescale For all purposes for all dynamical purposes these objects behave as buckles because you see The response of buckles was in the light ring, right? So if you throw something at this object This guy is going to respond at intermediate timescales exactly like a black hole Right, so this will be a cliff like clean photosphere object very good now Do they form and do they exist can we evolve them? Yes, that's a good question. I'm going there You mean in all these other objects Yes, yes, that's a good question. I'm going there But so let me first Tell you that we know that some of these objects form they grow and they grow to be compact this one example Was on stars have been around, you know, since the 40s or 50s cow built the first solution in 68 and I'm showing you here an example of what we did. Okay, you take a scalar field So really you only have kinetic term In the action Was a mass term. This is it. Okay, because of course the Einstein Term and you let this guy go so you do the following you build an initial profile It's Gaussian for the scalar field and we've all these are the initial conditions and you let it go So this is what happens This is field as a function of position and this is energy density as a function of position Okay, you let the field go you see that the field oscillates more or less wildly Okay, that's a frequency actually the frequency in all of these stars is dictated by the mass of the field. Okay But the energy density Actually, it's roughly stationary. Okay. These are real fields. I mean, this is really a simple theory. Okay The name of these objects is not really both on stars people call it oscillotons because they oscillate Slightly if you put this a complex field or two scalar fields You're able to build a truly stationary solution and that's both on stars. Okay Now you can start playing around with this and you can build you can find all the stationary solutions and build a mass radius diagram In the same way that you do for neutron stars. Okay, this is the mass radius diagram. Okay And you see the following Stars both on stars with a large radius of very large guys. Okay are very dilute. They're Newtonian objects almost. Okay But then as the mass grows so this is mass here This is radius in some units. Okay as the mass grows the the radius decreases So the compactness of these objects is increasing This is for a scalar. This is for a massive vector. Okay Actually, I think two months ago a similar diagram was done for massive tensors. So they exist as well. Okay these solutions These was on stars with only this term are Never as compact as to be Yuko, so they never have a light ring. Okay If you add self interaction terms here five to the four five to the six and so on you easily Build a boson star which has a light ring. Okay, so this is easy to do. So the question is Sure, it's fine to build this diagram But if I let the field go is it going to form a star which lies here or here? Is it going to be very dilute or not? Okay? We don't know this but we know the following if I take two stars here So I take one star with this radius and this mass and I collide it with another star These stars are going up the mass radius diagram. Okay, the mass is larger. So they are going this way and We found something interesting we and other people found the following if you if your star is sitting here And it decretes a small amount of scalar field Okay, you would expect something nasty to happen because well This is the maximum mass Okay to the left of this diagram usually in neutral stars Usually the star is unstable and it collapses to a buckle Okay, so you'd expect that if you're sitting here with a star and it eats a bit more material It's going to collapse to a buckle Right the mass cannot grow. It's at the peak. So it has to do something else The funny thing and this is shown actually in this movie these are two such stars The total mass of this system is larger than the maximum possible mass. Okay? what we saw instead is that There is no black information the two stars just released a huge amount of scalar field and They sit down eventually Ride close to the maximum mass point. So the idea is a star grows by accretion Okay, and at the very late stages is where it's basically hanging over this maximum mass point Okay, so this is a not it seems like a natural way to build compact configurations Very good And now to the big question are these objects stable or not? By the way, all or almost all the things that I'm saying now You can study them using the tools of last two classes or the notebooks Okay stability just to linear stability analysis in exactly the way I described. Okay Now there was an inter yes Yes, okay, so that's a good point. How much massive are we talking about? so the mass the maximum mass of a boson star depends on the mass of the field with which it's made of okay, so if you take a field with With a mass 10 to the minus 9 electron volt for example, then you're able to build a star Which roughly is one solar mass? Okay, so if you want to build the galaxy then you're talking about fields of 10 to the minus 21 electron volt or so so of course Actually, the question is very good because this takes us to another item, which I I Didn't even write down which is the following so Okay, I want to mimic I want to build a boson star that looks like a one solar mass Waco, okay, and then for that you tell me. Oh, yeah, there is in the universe a field with a mass mu Equal to 10 to the minus 9 and I don't involve let's say, okay And then that's fine. I will build a once over mass object That looks like a buckle the problem is That we have seen dark and compact objects across nine orders of magnitude in mass, right the mass Radius diagram of Michaels look something like this So I need to be able to produce I need to have a theory That gives me something that looks like this face diagram, right? It can't be this one is not good enough Because maybe it mimics this guy, but it doesn't mimic this so I need a theory that does this for me and We don't know this yet. Okay You see what I mean, maybe you maybe you mimic a one solar mass object But then you're going to lose the 10 to the 6 object that we see in the center of galaxies So this is still a big problem Now the stability of these objects is an interesting issue a few years ago three years ago Actually, so it's probably 2015 joke here show the following if you take any Object any compact object, okay and If the object has a light ring So it's more compact than 3m Okay Right then if you kick the object So you do this to the object then the fluctuation the wave that you produced Will live on on time scales that are never Smaller than one over log of t. Okay, so they live for a very very long time. This is the point and His argument then was well Then we can show using energy arguments that if fluctuation leaves longer than or As much as one over log of t then non linearities will pile up and destroy the object, okay so there's a strong There's strong reasons to believe that Objects which have light rings are non linearly unstable Okay, I think these answers Some of your non linearly. Okay. These are non linear arguments Actually, there's a funny Joe told me he was visiting Lisbon and he told me that this result was known in minkowski spacetime If you try to trap light in a maze, so this is a maze. Okay, if you try to trap light here Then exactly the same decay Exactly the same thing happens. So light will leave the maze at a pace no Faster than one over log of t So in the end it's like the light ring really works as a trapping point in your space. Okay, very good We don't really know what's what's going to drive The the instability we don't really even know the end state some Conjectures are that the stars going to collapse because the the linear modes look like look like cylinders and there's a mechanism called The Dyson Shanderzacker for me mechanism that should basically disrupt the star This is non linear if you want rotation to a very compact object You're going to do exactly the same thing that you do with Kerr. You're going to produce Ergo regions, okay The problem with Ergo regions is that they have negative energy states there. Okay, and let me tell you why this is a problem If you throw away Into a curve buckle The wave is going to enter the Ergo region. It has a negative energy there. Okay So this is the Ergo region of Kerr. It's going to enter here. So I throw a wave It has energy smaller than zero here Okay, but negative energy states are not allowed in flat space time Okay, so when the wave leaves the Ergo if it leaves the Ergo region it has to come up with a positive energy, okay This could be a problem, right? Because if there were no with the horizon, it's fine this negative energy state can go in and that's the end of it Actually, this is why Michael's radiate right Hawking radiation is really just a negative energy state inside the horizon. Okay In Kerr Michael's this is fine the wave enters here negative energy dumps into the hole if you take if you remove the horizon Then you have a big problem Because now you have a wave That's sitting inside the region with a negative energy the wave is going to travel to the origin There's a huge and difficult barrier there. It's going to be reflected and it wants to leave What happens if it leaves the negative energy region? It has to leave with a positive energy right It can't Energy is conserved. So what happens is when the wave hits the Ergo region boundary It leaves behind a state which has even larger energy Larger negative energy, right? So you see what you're doing. You're actually creating an instability Right So this is called the Ergo region instability is the arguments were put forward a long time ago by Friedman But recently Machides gave a really nice mathematical rigorous argument for the instability So any object with an Ergo region and no horizon is going to be linearly unstable Okay So my conclusion from this would be most likely if you try to compactify an object and produce Light rings in the space time then the space time is unstable. Okay And many people will be concerned with this. I am not very concerned Because I am also unstable Okay, my lifetime is around I hope 100 years Most likely much less But still that's not a reason why I shouldn't be here, right? so this to tell you that Instability by itself is not a big issue the timescale is the issue we need to know timescales We don't know timescales, okay? We know nothing about timescales for this problem. So this is a big to the least So finally, I think I have still 10 minutes. Let me tell you a little bit about How these objects can or cannot mimic wuckles, okay? So let me start by telling you that Hawking radiation is of course something that looks very unique to Space times with horizons. It's not Okay, if you want a bit of dynamics to the space time you always produce mixing of modes So you always you're always able to at least mimic some of the features of Hawking radiation So very compact objects, which are even slightly dynamical They will also emit radiation and there's a nice work by Padmanaban showing that it's even thermal in some stages, okay? So it's not exclusive of of wuckles What about imaging there's telescopes? They can look at at the space time and they seem to form an image one of the examples is this one This is how so this is how a black hole Surrounded by an accretion torus, okay, and this this these events happen would look like to a telescope What's going on? Okay, this is how it would look like to a telescope for example the event horizon telescope, okay? The luminosity would look brighter here just because the the waggle is spinning in this direction So there's beaming of light here, okay? Right and Doppler shifting that makes this fraction look more luminous This is how The your telescope would trace the image of the torus. These are 95 percent level drawings, okay Curve buckle surrounded by a torus This is exactly the same thing, but you replace the buckle by a boson star. It's a spinning boson star Roughly with the same mass with a slightly different spin Well the frequencies here that's not relevant, okay, and you could tell me well. Wait a minute. This looks different Right because a boson star has no horizon some photons cross through the center some of those photons are here but for any Observation that we're going to have in the next 40 years There's no way a telescope is going to suffer to distinguish this from this guy. There is no way the resolution is just too bad and the Model dependency that we that we have in a Christian disc or torus are just way too large to be able to discriminate these two pictures Okay, now there was an interesting proposal a few years back To really distinguish wackles from any other object the argument was really nice. Okay, the argument was the following Most of the black holes in the universe are surrounded by a Christian discs. Okay matter is here This is the isco you remember there's nothing Within the isco and then matter falls occasionally. It's false. It's been accreted. Okay, and The argument was the following I look at the center of our galaxy Okay, I see a dark spot and I see a very deep a Christian disc here Okay Automatically, this is a proof that goggles exist. Why? Because well a Christian discs have been here for millions of years Whatever is here has been here for millions of years Matter is falling into the object. It surely must have gained thermodynamic equilibrium, right? So Whatever goes in Must come out. It must reach an equilibrium, right? So you throw a photon, but after millions of years this guy gives is giving back what's getting in Okay, and because we don't see any right spot here. This is basically ruling out any alternative. Okay now The problem of these type of arguments is one It assumes that whatever is coming out is always coming out in the electromagnetic window. Okay so this is a minor issue, but I think a bigger issue is Gravitational lensing its lensing of whatever is coming out if you take an object, which is very compact So so let me let me do this experiment I'm an observer falling to a compact object, and I'm throwing light rays all over isotropically. Okay If I'm here, so at 6m, this would be the East coast of a Schwarzschild buckle then All the photons shot in these directions Escape the central object. Okay, but as you get closer then the opening angle gets much tighter Okay, so if I'm close to the surface Then basically all of the photons that I sent So all of the photons send sent within this direction are going to fall back in fact the opening cone so if I'm sitting here and Only photons shot within this cone of aperture Delta will escape and Delta is basically equal to epsilon Where my surface is at 2m 1 plus epsilon. Okay, so very compact objects very small epsilon Very small opening angle Almost all of the light that the object tries to meet gets back into the object There's no way you can reach thermodynamic equilibrium Okay Very good So finally, I think I still have five minutes Okay, gravitational signal. So we have seen gravitational waves There's at least four different ways we can test Wackles the answer by the way is let me give you already the final punch line We do not know what object that is. It's compatible with anything that has That is a cliff or anything that has a surface deep inside the light ring Explains all of LIGO observations. Okay So far so we need a lot more work and a lot more detections to actually put More meaningful constraints, but let me tell you what you can do. Okay So you have two objects. They're in spiraling as you saw in the neutron star case They are going to tidally deform one another So one of the things you can do is well, let me see how much black holes distort one another You can measure this you can compute this for the earth. It was done a hundred years ago by a guy called love Okay, and love quantified the deformation of bodies using something that we call tidal love numbers a Tidal love number of a black hole is Computed in the zero frequency limit of all the equations we wrote down. So it's a really really simple calculation, okay You basically just impose so you take a star Your star is sitting here or a black hole whatever and you want to measure how much that star Which at zero order is at rest how much it's going to deform this guy. Okay In our language in the language we use yesterday This is basically the same as requiring as looking for static fluctuations So you set omega equals to zero, okay? We saw the conclusion that we saw was that there was no hair. There were no possible Static fluctuations in black hole space times But remember there were no possible static fluctuations, which were regular. Okay Now what you're going to be looking at is a fluctuation that at large distances looks like the field produced by this guy Okay, right. So it's irregular. We know it grows. It grows like R to the L Okay, and you basically then can use it's very simple. It's a one-line equation. Okay, but the end result is That the tidal of numbers of black holes Are zero. It's a remarkable result in gr. Okay You you they can move you can change the shape of black holes But that doesn't change the way the black hole Effects gravitationally nearby objects. They are zero. They're the only objects. We know with zero Title deformability. Okay, so a simple way to test if the object is a black hole is looking at gravitation with data and estimate The title of numbers of the objects there right in the same way So if you have Two guys which are spinning or not spinning if they are not curved black holes They are going to be something else This means that all of the multipolar moments of this object curved black holes are deformed right if you make an object spinning It's not vertically symmetric. It has a multipolar gravitational field, which is not trivial Okay, and that's going to affect the other black hole or the other object in a different way And you can test this with gravitational waves, but since we discussed so much ring down Let me tell you how we can use how we should use ring down to test the nature of the final object. Okay So this is our ring down we merge two guys to black holes They always produce ring down. In fact, we know it's it's coming from the light ring so there's something trivial you know has to happen if Instead of a black hole here So instead of your space I'm being vacuumed all the way to the left if you put a surface here Okay, the ways that you generate when your object crosses the light ring Have to hit the surface and I have to give you some signal. Okay, right? So if you think about the photosphere as this wall right Whenever a star crosses this wall it produces a burst of radiation. This is this first Right This first Okay, but now If there's a surface if there's nothing there the burst that went in It's lost forever right and you see basically this Nothing, okay If there's a surface there you expect at least a fraction of that burst to hit you back to come back Cross the light ring and into the detector. So you expect Echoes of the original signal and this is in fact what you get okay If you plot the effective potential for a black hole it looks like this a star crosses the peak This is the light ring burst of radiation a fraction into the horizon a fraction into LIGO If you have a boson star a wormhole a Compact star whatever the burst that you generated in the light ring a Fraction goes to LIGO a fraction goes in but now it's not lost forever Right, it hits the surface of the object. It comes back a fraction escapes the original the Photosphere hits the detector again. So you're going to get a series of echoes of the original signal This is here. So black line is Black hole waveform red line is a wormhole signal, okay a wormhole that has a light ring That's very very compact. You basically get copies of the original burst as later times Okay, I think I'm out of time, but let me tell you just very quickly That these bursts Wait, wait, wait were claimed to be seen in LIGO data last year actually they were claimed to be Present in LIGO data at a very high significance very recently, okay The model that was used to do the searches was roughly okay This was a study by Nyayesh of shorty The analysis was more or less verified by LIGO groups But the significance was lower than was claimed. Okay, there's still something in the data But way too low to have any significance to claim detection, okay So the modeling of the signal is something we need to worry about We don't know how to model properly these compact objects because we don't have the objects, right? The thing that went less well in their analysis was the estimate of random fluctuations that look Like repeating signals. Okay, and this is what gives a lower significance to the detection and I think I'll stop here So maybe we're continuing the afternoon