 Thank you very much for invitation to come here and lecture on a topic I've been working on for many years now. This center is really, really great and I think it's a great thing what you guys are doing by bringing different communities together. So let me begin by telling you, it's a bit strange, right? Because if you look at the other lecturers are luminaries in the field and then there's me. So I thought I should tell you a little bit about who I am. So I'm from Argentina. I studied there and then emigrated to the United States in 2000 due to the economic crisis with a football scholarship. So I was a goalkeeper and I have broken every finger in my hands. So I know that, you know, how to persevere, I guess. When you break a finger and you're in the middle of a game, you just close your fist and you just keep on playing, right? So that's a spirit I've also used in physics when unavoidably I get stuck in some physics problem. So I came to the United States and I did, I started with Clifford Will at Washington University. I studied post-Newtonian theory for the most part and experimental relativity there. Then I did my PhD at Penn State at a place, well, now it's called the Institute for Geometry and the Cosmos or something like that before it was called the Center for Geometry. And I worked with Ashtakar on classical relativity, Owen and Pinn on LIGO and Laguna, numerical relativity and Alexander on cosmology. And then from there I moved on to Princeton and to Harvard with postdocs then eventually became a professor at Montana State University where we created a center and an institute for extreme gravity and I am now moving very soon to the University of Illinois to start a new institute for fundamental theory. So I hope it's clear that I'm a physicist, I'm not a mathematician, but I know, so I talked to Paolo extensively about what it is that he wanted me to lecture on so my talks are gonna all be about black holes and he asked me to concentrate on black holes in general relativity. I do a lot of work on black holes in modified theories from the point of view of trying to test or constrain these modified theories with observations. But what I'm gonna concentrate here are black holes in GR and I'm gonna try to, since the audience is very mixed, I'm gonna try to strike a balance between mathematical content and physics although because my background is as a physicist you'll see that most of my explanations are physics first in a sense. I will always give a blackboard talks for the remaining four talks and my handwriting is, I'm gonna try real hard to make it clear but it's not the best so I suggest that people are in the back, you're probably not gonna see anything and there's an entire row of sits over here, feel free to come down like now if you want, this should be informal, you should feel free to ask me questions as I go, okay. What else did I wanna say before starting? Let's see, I think that's pretty good, so let's begin with the syllabus. So I teach a class in the US on black hole theory, it's like an entire semester and I've tried to condense it into like four lectures, so we'll see how this goes. The topics I'm gonna cover, the syllabus if you want. So we're gonna start with the Kerr black hole, that's gonna be today's lecture, we're gonna talk about, obviously I'm gonna define the metric in a few different coordinate systems, we're gonna talk about properties of this metric and eventually we're gonna talk about geodesics, so the motion of a test particle in this metric. Then tomorrow I'll tell you a little bit about something called an embryo, an extreme mass ratio in spiral. So this is when a small black hole gets captured by a supermassive black hole and as the small black hole falls in, it zooms and whirls around the supermassive black hole generating gravitational waves and these are waves that we hope to detect with the LISA mission in 2030. Problem is that it's mathematically very, very, very difficult to solve for the motion to the right accuracy and for the gravitational waves. So this is, I'm gonna tell you where we are at more or less, but it's still an open problem. Three, I'm gonna talk about comparable mass binaries. So here, we're gonna have in mine are two black holes going around each other, typically in a quasi-circular orbit, slowly inspiring due to the emission of gravitational waves that LIGO has detected for example and Virgo and I'm gonna tell you about the mathematical techniques that we use to model the motion of these two black holes and the gravitational waves they emit and then on the fourth lecture, I'm gonna tell you a little bit about black hole perturbation theory. So black hole perturbation theory is here what we're trying to model is after a black hole captures or after a small object falls into a black hole, then the black hole is perturbed and it needs to radiate its excess information, its perturbations and it does so through the emission of gravitational waves and these gravitational waves follow a quasi-normal spectrum. So I'm gonna tell you a little bit about how one calculates that in probably in Schwarzschild because we won't have time to do Kerr, okay? So that's the plan for the next four days. And then I go back to the US and get on a truck and move to Illinois. So let's begin with, obviously if this is an axis labeling difficulty, easy is here, hard is here. So we're gonna start like sort of basic. I know there's a lot of experts in the audience that we're probably gonna know half of what I'm gonna say, just bear with me. If you already know this stuff, this is not ready for you. So Kerr black holes. I always like to start when I lectern this with a quote by Chandrasekhar who said, if you wanna be a good physicist, you gotta know your field or the history of your field, okay? So let's start with a history, a very lightning description of the history. 1916, Schwarzschild finds the first, well, the spherical asymmetric solution to the Einstein equations. The equations had been published in 1915, at the end of 1915, early in 1916. And so he finds the Schwarzschild solution, obviously. Between 1922 and 1926, there's a work by Friedman and a bunch of other people, Lemaitre and so on, working on cosmological solutions. So solutions to Einstein equations that have, well, that represent homogenous and isotropic universe and their expansion. 1930 or in the 30s, so work by Oppenheimer. Later followed, or with Snyder. And the way they were looking here is the collapse of matter distributions and what the end state of that collapse would be. Because, just because you have these solutions doesn't necessarily mean that those solutions are representative of something that happens in nature. So what you have to understand is, are those solutions the end state of gravitational collapse and how generic can you make that statement? Then, pretty much little happens between 39 and 45, for obvious reasons. So I'm gonna put a happy face here. 1960s, after the war, a bunch of things start happening. And it starts happening more or less all at the same time. So you have, and I'll write over here, 1960s. 60s were good. Were good because, not just because that's the year the house I bought in Illinois was built, but also because there were new telescopes that were coming online. And those new telescopes led to new discoveries, new observations that were not really explainable with the status quo. And then, on top of that, there were new differential geometrical results. And there were a lot more scientists who are sort of interested in investigating both general relativity and these new observations that were happening. And so sometimes, and this is, by the way, so this is sort of because of the end of the nuclear era. So we're talking about a lot of scientists moving away from work that they were doing in the 40s related to nuclear physics and trying to look at other problems, new problems that were hot at that time. And so they started investigating these ideas. And this has led to what some people call the golden age of GR. And it was pretty cool because it was golden age, I guess, because a lot of results were coming up pretty much at the same time by people that are the founding fathers, if you want, mothers of the field. In Princeton, you had Wheeler in Cambridge, at least an incomplete list. Cambridge, you had Penrose, and you had Szyama. You have Carter rolling around. In Moscow, the Russian group, you had Soldovich, which I think received a medal from ICTP, right? In Hamburg, you had Jordan, and you had Ellers. And finally in Texas, you had Shield, and Boyer, and Kerr. In 63, using mathematical techniques, Kerr comes up with a solution to the Einstein equations that, later through the work of Newman, was interpreted as the solution representing a spinning black hole. So the solution was axisymmetric, but he had some very interesting mathematical properties. And together with Newman, so you had the work of Carter that essentially led to a physical interpretation, if you want, of these results. So the interpretation here comes in around 64. Okay, so after the 60s, there's a bit of a hiatus in the history. I'm gonna sort of stop here in the history, I'm just gonna tell you in words. In the 70s and the 80s, there's an explosion of particle physics. There's a lot of new particle physics results, new particles, left and right, standard models and all of that. And there's a dearth of data to really verify some of the predictions of general relativity. You were making experiments left and right in the solar system, but solar system gravitational field is very weak. The velocities are very small relative to the speed of light, so the gravitational fields are not strong. So it's hard to actually test some of the more dramatic predictions of general relativity. It was not until really the discovery of the binary pulsar in the 80s and the detection that these two neutron stars, well, actually the neutrons are in a white dwarf, but they have one neutron star that's rotating very fast. And it's producing, it's spin axis, it's misaligned with its magnetic axis. So light goes out through its magnetic axis. And so this magnetic axis since the star is, so I'm the star and I'm spinning about this axis and my magnetic axis over here. So as I spin, I just go like this. And so every time I go by you, you detect a pulse of radiation in the radio. And that thing was called a pulsar. And the pulsars had been discovered prior to the 80s, but the discovery of a very tight pulsar in orbit around a white dwarf allowed people to measure the orbit very precisely and discover that the period of this binary system was shrinking. And this shrinkage of the period of the binary system coincided precisely with the predictions of general relativity of how much a period should be changing due to the emission of gravitational waves by such a compact binary. That eventually led Halson-Taylor to the Nobel Prize in 83, four, something like that. And then we had to sort of fast track another 30 years before we have the next direct detection of gravitational waves with LIGO. And a tremendous amount of theoretical work had to be done in order for that discovery, of course, also experimental work. Of course, you had to build these gigantic machines, bounce lasers, create this interferometer, get the data. But the data that was obtained was actually quite buried in the noise. And the interpretation of the data required theoretical models. And those theoretical models record a solution to the Einstein equations. So a lot of physicists spend a lot of time trying to solve these problems that I'm going to describe here. And the one that LIGO detected first was the solution for a comparable mass binary. Okay, so that's sort of the context. Let's now start with the metric. So a lot of what I'm going to be describing in terms of the metric can be found in a variety of books from MTW, so references. So Miesner, Thorne and Wheeler, who recently updated, I think, keep recently updated his book. There's a book by Wald, of course. I think Bob will be lecturing here later. So if you have any questions about what I said, you can always ask Bob. Chapter seven, right, of his book. There's also a book by Sean Carroll, that's pretty nice. But the book I'm going to be following is a book by, by following. A lot of material I'm going to be describing is in a book by Eric Poisson. Called A Relativist Toolkit, and it's a very handy, thin book. And there's others, of course. Okay, so let's begin with a metric. So for a curve I call the line element, it's described, or can be described, following expression. I'm using a signature that's minus, plus, plus, plus, because that's the right signature. And you can't convince me that that's not the right signature. It's minus 1 minus 2m r over rho squared, it is square. Minus 4m ar divided by rho squared sine square theta times dt d phi. This term is off diagonal in this coordinate system. It's called the gravito-magnetic term. Plays a role of a magnetic field in gravity, sigma over rho square. I'll define all the symbols in a second, okay? Sine square theta d phi square plus rho squared over delta dr squared. This term is really cool, plus rho squared d theta squared, that term's boring. So that's the line element in all its full gory detail in what's called Boyer-Linquist. Make it bigger, I can definitely make it bigger. This is why I said people need to sit in the front. This is Boyer-Linquist. I'll write the next equations bigger, okay? So rho squared is defined to be r squared plus a squared, cos squared theta, delta is defined to be r squared minus 2m r plus a squared. And sigma is defined to be r squared plus a squared squared minus a squared delta sine square theta, okay? This is not the only coordinate system in which you can write the metric, even though it is the most common coordinate system used. And for most of what I'm going to be talking next, I'm going to be using this coordinate system. I'm going to tell you what that m means and what that a means in a second. And before proceeding, however, I think it's useful to present two other versions of this metric in other coordinate systems that are very useful and are used a lot. For example, I was just at the GR Relativity Meeting, the International Meeting in Valencia. And a professor from India, his name I can't quite remember right now, was talking about solutions in the large D limit, because that turns out to have interpretations in a fluid gravity type duality. And he was using the core metric, but in a different coordinate system that he didn't, at the time of working on this, he didn't really know, according to his talk. I'm going to show it to you here, because it's very useful. We use it all the time in Relativity. But before we go there, we need to talk about the ingoing core coordinates. There's two versions of these coordinates, ingoing and outgoing. And so ingoing core coordinates are defined through V equal burning with T plus this R star. The R star is called the tortoise coordinate, which is given by this expression. And then psi is phi plus R numeral. And R numeral is equal integral of A over delta dr. And so these coordinates are useful, just like they are useful in Schwarzschild. Your regular Schwarzschild coordinates just sort of become bad as you approach the horizon. So you need to go into penetrating, horizon penetrating coordinates. This is a set of horizon penetrating coordinates that are actually adapted to the null congruences of your spacetime and the line element. Now, written bigger in these coordinates becomes this, becomes this. So minus 1 minus 2m R over rho squared dv squared. Now there's this off diagonal term dv dr. And just with a number 2, no function there. Then you have the angular part, what I would call earlier, the magnetic piece. That's 2a sine square theta dr d psi. And then you have two other terms, well, three other terms that arise due to this transformation. One, it's a mixing term dv d psi. 4ma over rho square r sine square dv d psi. And then you have two more terms, plus sigma over rho square sine square theta. d psi square and rho square d theta square. That's what the line element looks like. These coordinates are much better behaved across the horizon. So a lot of people investigate them or use them to investigate solutions. And finally, there's the coordinates that were being used in this large d-expansion, which are called Kershiel coordinates. Now Kershiel coordinates are related to ingoing Kershiel coordinates through this transformation. So x plus i y equals r plus i a sine theta e to the i psi. Z is equal to r cosine theta and t prime is equal to v minus r. These coordinates are real, so the Kershiel coordinates are x, y, z, t. The ingoing Kershiel coordinates are v, psi, r, and theta. So here are the maps. And this equation, when I first saw it as a grad student, I was sort of confused, wait, wouldn't this make the metric complex? And now what it's meant here is that obviously you're supposed to match the real part with the real parts. Make a mistake here, I did. Apologize. So for example, if you expand this out, this is cosine psi plus i sine psi. And you carry out the product and it's gonna be a real part to that product. And that's supposed to be equal to x. And then the purely imaginary part supposed to be equal to y, okay? In this coordinate system, magic happens. And the line element is Minkowski plus a function h times a covector, l alpha, a one form, where h is defined to be 2m r over rho squared. And l alpha d alpha is too big to be written over there. So r squared plus a squared over delta dt minus dr plus a over delta. This is written in Boyer-Linckis coordinates. So you have to transform if you want it in Kershiel. All right, so it's very nice because it looks like just Minkowski. Oh, so I didn't say this, but alpha Greek letters range from 0 to 3, because that's what we do. I guess when Bob talks, then Latin letters are gonna range from 0 to 3 anyway. In this case, alpha and beta Greek letters from 0, post you understand it that way. Okay, so that's the line element. Very nice because it looks like Minkowski. And then you have this correction. That's like a conformal factor, sort of, times this extra piece. Okay, and these are three fairly common coordinate systems that people use. And they become very useful when you're trying to study solutions to the Einstein equations, or properties of the car metric. Yeah, are there any questions so far? Far from the fact that I write too small and enter right over here. Yes, I didn't understand your last part. Yeah, why is the mixed term there? No, so there's a dr squared term here. There's a d theta squared, d phi squared, dt squared, and then a cross term. This is dt d phi. I can also tell you that if you try to put a generic ansatz like some function of t and theta in the t component, tt component, some function of t and theta in the t phi component, some function of t and theta in the phi phi component, some function of t and theta in the rr component, and some function of t and theta in your theta theta component, and you throw that into Maple and you ask Maple, find me the solution, and Mipple says, no, I will instead crash your computer. At least as of like 10 years ago when I tried as a grad student. The questions are, the answer to the questions, if you put in like four arbitrary functions or five arbitrary functions, are very, very complicated. They're coupled, they're non-linear, and even modern symbolic manipulation software has trouble solving it. Of course, if you hold the hand of the computer a little bit and you use some of the properties that you expect this metric to have, then it can do it. But just brute force approach, throw it on a computer and let the computer find the solution, does not work. At least didn't as of 10 years ago. Okay, so let's talk about properties a little bit. That was the metric, this is properties. So there's many ways in which it's cast properties, but we're gonna just list a few here that should be obvious. The metric is asymptotically flat, meaning as you do an expansion about I not, so about R goes to infinity, you recover Minkowski. The metric is stationary, but not static. It's a good question, so yes, I can. So the question is, are these solutions that I wrote down here defined in the same domain of your space time are the restricted to different parts of the domain? This goes toward a topic of the maximal extension of the car metric where you have charts that can actually cover all of the domain. So no, the answer is no, they're not well defined in the same domains. In fact, as you would expect, the Boyer-Linquist metric becomes ill defined at the horizon, so connecting region one with region two in your Penrose diagram becomes problematic. You obviously can look at the solution in region two, but those two are not connected. So the idea of going to, in going car coordinates is that you can then connect region one to region two. And the same with car shell coordinates, which is like a simple rewriting. If you want the maximal extension, you have to go to something like Pine-Leveg-Wuster coordinates, but adapt it to a car metric so that you can then connect to the different regions. And the maximal extension is very interesting. Not entirely clear, it's physical, but it's very interesting. And you can find it in detail in Poisson's book or in M2Dalio, but I'm not gonna go into much more detail. I'm gonna concentrate for the rest of my lectures on what we would call region one, which is where observers live and I can make measurements. All expressions are valid in region one. Yeah, so this is a region that actually has access to I naught and it has access all the way up to the horizon. And sometimes, when I'm doing calculations, I'm gonna need to have the coordinates are well defined on the horizon, so then I'm gonna switch to one of these two. Excellent question, yeah. If that didn't make sense, just go la, la, la. It doesn't matter. All right, so what I was saying is that the metric is stationary but not static in the sense that it's time-independent, but it's not invariant under T reversal. It's not invariant under T reversal because there's this term here, okay, dTd phi. And you would expect that the metric of a rotating black hole not be symmetric under time reversal because if I flip the sign on time and I just start going backwards and the black hole would be spinning in the opposite direction it was spinning before. So physically, it shouldn't be time symmetric. Then there's some peculiar parameters that appear here, m and a, okay. Don't have time to prove this, that's why I have a one semester class on this topic, but m turns out to be the ADM mass. It also turns out to be the Comer mass. It also turns out to be what we sometimes call the Newtonian mass, i.e. it's the mass of the object. It's the ADM mass in the sense that it's the total mass energy of your space time. You can define this through an integral that you can then convert into a boundary integral, a surface integral. Turns out you can prove, I think that for stationary space time, the Comer mass is the same as the ADM mass. There's another expression for the Comer mass in terms of killing vectors of your metric. And the Newtonian mass, this is more experimental term, it's the mass you would measure as an experimentalist if you were watching the motion of say a planet around this compact object. It's the mass that enters like the acceleration term, the Newtonian acceleration if you do a far field expansion. A turns out to be the ADM angular momentum divided by the ADM mass, and it's sometimes called the Kerr spin parameter, not to be confused with Kerr spin angular momentum, but we are sometimes sloppy in physics and we call the Kerr spin, both A and J depending on context. The A parameter for all reasons are going to become obvious, it's bounded between one and minus one. So there's a regime of, you can use this solution, you can investigate what happens if A over mass ADM, just say mass here is larger than one, like there's nothing wrong or obviously wrong that happens to the metric here if I just pick A to be 1.1, but then if you look at it more carefully as I'm gonna try to show you next, then problems arise, okay, that I will describe. What else? Oh, there should say that there's killing vectors and killing tensors in this space time, as you would expect, there is a time-like killing vector, the say x alpha dt, and then there's an azimuthal killing vector, say the x alpha d phi, where phi here is supposed to be the angle about the rotation axis and I'm working here in buoyant-inquist coordinates, but you can sort of see that this has to be the case because the metric is stationary, so there should be a killing vector associated with that symmetry, and the metric is also axis-symmetric about the z-axis, about the angle phi, so there has to be an azimuthal killing vector, and then there turns out to also be another object that I'm not gonna write down right now, but I'm gonna tell you that it exists, okay, alpha beta, that's a killing tensor, so it satisfies the killing tensor equations for this space time, so that's a little bit like a hidden symmetry of this metric, it's not directly associated, as far as I know, to any obvious symmetry like this, but it does lead to a conserved quantity, and that fact is very, very important for doing physics and understanding orbits. So let's continue with a few more properties, and we could do this in a purely mathematical way, but since my role here is to stretch your physics brains, we're gonna discuss properties from a more physical standpoint, and we're gonna talk about observers, so there's a class of, there's a few classes of special observers that allow us to understand the physics of metrics, capital what, sorry, except for this is just the coordinates in your chart, no, it could be anything, so it could be like t, r, theta, and phi, it could be v, r, theta, and psi, it could be x, y, z, and t, or sorry, t, x, y, and z, yes, this inequality, yeah, yeah, so m, m in principle can be anything, this inequality, yeah, this inequality says that a over m is bounded between one and minus one, a has to have units of, so j over m is a, but as I'm gonna show you in a second, a has units of mass or energy, I erased it unfortunately, but you can see that from the metric, see, yeah, you're good, okay, oh, what is this symbol here, this is k alpha beta, so the question is what was this symbol, k alpha beta is a killing tensor, so it satisfies the killing equations, so, okay, so a killing tensor is an object that satisfies a differential equation that looks of the form like this, okay, alpha beta is semicolon gamma, where, no, there, equal zero, symmetrized, it's not my notes, I'm like, now I'm just trying to remember, it's similar to a killing vector but it's a generalization to higher rank, so if you want a killing vector, it's a rank one killing tensor and the one I wrote here is a rank two killing tensor and there can be higher rank killing tensors and it turns out that you can show that quantities constructed from the contraction of any killing tensor with the four velocity of a particle are conserved under time evolution, so for example, killing vector contracted onto the four velocity is a scalar and that scalar is conserved, like if you take D by the tau of that quantity, tau being proper time associated with a geodesic whose four velocity you just used to construct this thing, you can show that that scalar is conserved, in the case of T, like if you contracted with a four velocity, you would get the energy, here if you contract five with a four velocity, you would get the angular momentum, here if I take K alpha beta and I contracted with U alpha U beta, I get another conserved quantity that it's also a scalar and that we're gonna use later to the coupled equations. Good question. Also, it's impossible to be completely comprehensive and thorough in the amount of time I have. I'm gonna throw some words here and I'm gonna present some results and I really encourage you to come and ask me later or look up some of these words in the references that I presented, okay. So I was talking about special observers. One of them called the ZAMO. ZAMO is a zero angular momentum observer. It's supposed to be rotating with the black hole. So what do I mean by an observer? When we say observer in physics, what we mean is imagine that you have a test particle that does not really affect the background and then just sort of like sitting there, observing what happens to this, to itself, okay. And so it is in that sense, I'm using the word observer and a zero angular momentum observer is supposed to be something that rotates with a black hole. So typically it's defined through a four velocity. So this U alpha is the four velocity of this observer such that the quantity L tilde defined as the contraction of the four velocity with phi alpha, where phi alpha is a killing vector over here is equal to zero. So it's an observer that essentially measures zero angular momentum. Now, if you go and work out this contraction, I'm using the Einstein summation convention as usual, then you find, and remember that this U alpha here, if you have some trajectory Z alpha, then U alpha is the tangent to the trajectory and tau is the, can be any affine parameter on that geodesic but I'm using proper time because I'm always thinking of massive particles, okay. So if you expand this out because of the form of this quantity and the form of the metric, you get that L tilde is equal to G T phi times T dot where the dot stands for proper time derivative plus G phi phi times phi dot, okay. And you want this to be zero, which then means that phi dot divided by T dot, which is also known as, I'm sorry mathematicians, you're gonna not like this, but we call this the phi dt in physics, okay. And we define it to be an angular velocity omega, which is nothing but just the ratio of minus the component of the metric G phi phi, sorry, G T phi with the component of the metric G phi phi. And you can work this out, you can find that omega is equal to four M A R divided by sigma, okay. So it goes in the far field limit as R goes to infinity, you can do a Taylor expansion about I naught and you find that this thing here, this omega, goes to roughly the spin angular momentum of the black hole, so J divided by R cubed in this case. A few properties of this thing, this omega thing is the angular velocity of this observer, it gets sort of larger and larger and larger as you approach in the horizon until it gets to some sort of critical value at the horizon. This omega is sort of in the same direction as the spin angular momentum of the black hole, as I defined it here. And if you take the limit as R goes to the horizon of the black hole, what you end up getting is the angular velocity of the horizon of the black hole, which is an interesting quantity for a variety of reasons. Okay, so that's what people refer to as a XAML, and that's important because of two additional observers that people like to talk about that sort of reveal the properties of the curved space time. One is the observer called the static observer and the other one is the observer called the stationary observer. So let's talk about the static observer Is there a question? Excuse me, I can't hear you. Seeing if you guys were paying attention. Maybe later? Okay. All right, so let's talk about the second class of special observers called static observers. So static observers are those with four velocity u alpha equal to some proportionality factor gamma or normalization factor gamma times t alpha where t alpha is supposed to be the killing vector associated with stationarity. So in some sense these are observers that are sort of not moving in space time, they're sort of held there. So you should imagine the observer being in some sort of spacecraft being sort of attached to that point in space, but just moving in time only. So gamma here has to be normalized. So if you're using a normalization in my signature to be minus one, then this implies that gamma has to be equal to the time-time component of the metric to the minus one half. I say here these are observers that are sort of held in place as the physical picture you should have in mind. But something sort of curious happens because gamma doesn't really exist everywhere you're in your domain, right? There's a point or a set of points in your domain where gamma goes to zero, sorry, GTT goes to zero. So gamma diverges. So static observers do not exist. Always forget how to write these things, assist. If GTT is equal to zero, which is equivalent to saying at a position r of theta that is equal to m plus m one minus a squared over m squared cos n squared theta to the one half. So whenever you get to this particular point or set of points, then static observers do not exist, which means observers have to actually move in space. They're being forced to move in space. You can't keep them there no matter how strong your spaceship engines are, okay? This thing, this place where you cannot remain still is called the Ergosphere. When GTT is equal to zero, then the static observer zone exists because gamma diverges and GTT is equal to zero you can solve for what that is because I wrote down what GTT was in Boyer-Linquist coordinates and it gives you this solution. So yes, yes, of course. Yeah, I wasn't reading actually, just like saying it. Yeah, yeah, yeah, absolutely. So I was gonna go more into that. So let's make a drawing. I like to make, can you see if I draw down here or is it like too low because of the, no? You can see, okay. So imagine that I draw the black hole and I'm gonna later show you that there's an event horizon just like in Schwarzschild, but it's not quite just as in Schwarzschild. It's different than Schwarzschild, but there's still a horizon. And so let me draw the horizon here. Gonna challenge my skills of drawing a circle. Ah, failed. I will tell my students before they say a defender, this is just practice making a circle before the oral defense. That's supposed to be a circle and that's supposed to be the event horizon, which I'm gonna tell you about later. The turns out the ergosphere is sort of a blade like this. It's like an ellipsoid of revolution. And what this is saying is, if I am here in green, if I am in this like little rocket here, that's my rocket, I have engines are powerful enough to just like keep me there. And I can take all the pictures of the black hole I want and life is good. I can like even observe it with like telescope. And if I get closer, I can still do that. But eventually, if I get here, no matter how much I try to turn my rocket up, I am not able to just stay in place. And what's actually gonna happen is that I'm going to begin to rotate with a black hole. I'm gonna rotate because the space time fabric itself is rotating so strongly that inertial frames begin to rotate. So this is a manifestation of what we would call in the weak field the dragging of inertial frames, but like cranked up to like a million. Talking about massive particles here, yeah. For photons, there's a slightly different limit. Let's talk about that later. I don't think I was gonna talk about the photon sphere, but we can if you want some. So I have one more observer that I wanted to discuss with you, which is the stationary observer. So the stationary observer, stationary observer, an observer with a four velocity at some constant times T alpha plus omega times phi alpha. Where omega here is the same omega I wrote down over there, it's d phi dt, but it's sort of the angular velocity of the, it's not the same omega, ah, good turning. Called this omega bar here. So omega bar is some d phi dt of the observer. And just as before, there's a normalization condition gamma. You require u alpha, u alpha to be equal to minus one. And you can solve for what gamma is. Gamma is equal to g phi phi to the minus one half times two. I'm gonna define what little omega is in a second. Big omega bar minus omega bar square minus g dt over g phi phi. Little omega here, I've defined it to be minus g t phi over g phi phi. So I suppose it's the same as a capital omega that appears here. I'm using a different symbol here, just so that we don't get confused with the omega and the omega bar. So just like before, there is some point or set of points in space time where stationary observers cannot exist because this quantity here diverges. So if gamma, let's say gamma to the minus two is equal to zero, then there does not exist stationary observers. And so you can show that this happens for omega bars that are between some omega minus. So if omega bar is between omega minus and omega plus, then everything is fine, then okay, essentially, okay? So if you crank up your angular velocity enough, then this stationary observers can exist. And so let me plot for you omega bar as a function of say r over m is a tricky result. Omega plus minus here, by the way, is given by the expression little omega plus minus delta to the one half rho squared over sigma sin theta. I mean, you can see that this is just a solution to this quadratic equation for gamma bar because when this thing goes to zero, then you're gonna have problems. So if I plot omega plus, it looks something like this. If I plot omega minus, it looks something like this. Then at some r, they touch, oops, they touch like right here. So that point omega plus is equal to omega minus. And let me denote this point here where they touch as r sub h. Just for kicks, r-ergosphere is here. So what this is saying is that as long as your angular velocity omega bar is somewhere in between these two curves, this one, remember, is omega plus and this one is omega minus, okay? Then you're fine, you can have stationary observers. But eventually they meet and beyond that point you can have stationary observers any longer, okay? So then the question is, what is this r-h? I guess I could do it over there. So recall that r-h is when, or is defined by the condition that omega plus is equal to omega minus. And if you look at that, this is the expression for omega plus minus. These two angular velocities or critical velocities are gonna be the same when these discriminant vanishes. Rho can not vanish because it's r square plus a square cosine square. So it's a sum of two terms that are square. So they're both positive individually. But delta can vanish. So this implies that delta has to be zero and if you solve for delta being equal to zero, you get that r-h is equal to m plus omega minus. m one minus a squared over m squared to the one half. That's right. Well, I slightly lied. There's two solutions to this. There's an r plus and there's an r minus because there's a plus minus here because delta equals zero is a quadratic equation. For r, r-h is equal to r plus. And r minus, sorry, and r minus, and this thing is going to be called the event horizon. I'm gonna explain to you in a second why this is called the event horizon. R minus is another horizon that I'm not gonna go into detail here, which is closer to the singularity than r plus. So r plus is the outermost horizon. This goes back to the causal structure of Kerr and the maximal extension and things like that, which I'm not going to get into because I like to stay outside of the horizon. But just know that in Kerr, there are multiple horizons that arise and they all have very interesting mathematical properties. So how do we know that this is our horizon? Well, turns out, if you say that f is equal to r minus, say, r-h, and you consider the surface, f is equal to zero, then this surface is null, and you can show that the normal to that surface is also null. And more also, you can show that the null expansion, which I'm gonna call theta, so the null expansion of a congruence of geodesics, if you know what that means, vanishes. So that's a statement about there not being any outgoing null geodesics that are timed like, that are being emitted by the horizon, okay? So if you look at bundles of null rays, then this null rays become null on this surface. It's very much like what we do when we prove that there's an event horizon in Schwarzschild, the light cones, do you remember this picture of the light cones, maybe? The light comes sort of like tilting until they become sort of null when you get to the horizon. It's the same idea, but just done mathematically slightly more formally. So f is a function that I'm defining to be r minus rh, and when f is equal to zero, that defines a surface, r equal constant on your space time. That surface you can prove is a null surface, and you can prove that the normals to that surface are also null, okay? And so that's just a mathematical statement. You can also show that the expansion, so how, so if you look at two nearby geodesics in a bundle, and you can ask, you know, as I move forward in time, how much do they separate? Okay, that's like the expansion of geodesics, and now imagine that these geodesics are null rays, okay? You can show that this expansion actually vanishes on the horizon, which is a definition of something called an apparent horizon. Can you repeat your question, sorry? Yes? Is something right? Oh, that's the character of the geodesics switch, yeah, so it switches, and then it switches again after you cross r minus, that's correct. And we're not gonna go there, because I wanna talk about other topics, but yeah, that's a good point. So how do we know that this is a black hole? So far I've told you that there are special observers. I told you that there is this magical, wonderful place called the Ergosphere, where if you try to stay fixed in space, you can't and just rotate, which is weird, because we don't have that in Schwarzschild. I've also told you that there is a place called the horizon, and at the horizon, that's the last point where there can be stationary observers, meaning if I'm inside of the event horizon, I am not going to be stationary anymore, and in fact, you can show that geodesics will focus inside of the horizon, and they will focus toward a point, so I'm gonna grab in a second. What I have not showed you yet is that this is a singular space time that has a singularity essentially in it, just like Schwarzschild has a singularity, and remember, this is general relativity, so I cannot just look at the metric and be like, oh, this metric component diverges here, so therefore this must mean that there is a singularity, like that's not enough, because I can always maybe go to a different coordinate system that's more regular, so what I have to do is you have to calculate curvature invariance, so is this a black hole? So you calculate, any invariant will do, I'm gonna show you my favorite one, which is the crutchman, there's many curvature invariance you can calculate, this one is constructed from the Riemann tensor like so, oopsies, by the way, I can't contract a curvature invariant with a Ritchie scalar, because obviously it's a vacuum solution of the Einstein equations, so the Ritchie scalar and the Ritchie tensor all vanish. So the only thing that I can do is construct invariance with a Riemann tensor, and the Riemann tensor does not vanish, which is a good sign that this is not Minkowski in disguise, better not be Minkowski, so the crutchman is 48 M square over rho to the 12 times R square minus A square cosine square theta times rho to the four minus 16 A square R square cosine square theta. We look, looks wonderful, because it's obviously not zero, like things are not zero, moreover, it is something that when you take the limit A goes to zero, recovers the crutchman scalar for Schwarzschild in Schwarzschild coordinates, so you know that like, well, at least this metric has something to do with Schwarzschild, not just because the line elements more or less look alike, and on top of that, it looks like something that has the chance of diverging when rho vanishes, and if you remember the definition of rho, rho was R square plus A square cosine square theta, and so this rho is equal to zero, occurs, this occurs when R is equal to zero, and, and, when theta is equal to pi over two. That's different from Schwarzschild. You see in Schwarzschild, the crutchman scalar diverges when the Schwarzschild coordinate R goes to zero. Here, the crutchman scalar diverges provided I am on the equatorial plane. Remember, theta is the angle that points from the pole, which is the axis of rotation down, right, and theta is equal to zero, for physicists, means the pole, theta is equal to pi over two, means the equatorial plane. So that's a little bit weird, so here's where we go and we use Kerschild coordinates, we transform, transform, transform, you do the same calculation, and you're gonna, in fact, show very easily that X square plus Y square equals R square plus A square times square theta, and that Z, remember, was defined as R cosine theta. So, R is equal to zero, and theta is equal to pi, is the condition X square plus Y square is equal to A square, and Z is equal to zero, I think. Yes, right? And that's the equation of A, are we going together? A ring on the equatorial plane. So there's a ring on the equatorial plane of this object of radius A, where you have a curvature singularity. And that's interesting because that means that I can go through the middle of the ring, and the curvature does not technically diverge, okay? So what you used to have as a point curvature singularity has sort of somehow spread into this ring structure, okay? How much more time do I, am I done? Five minutes, okay. Good, five minutes, let's do all of the geodesics in five minutes, this sounds, we can do it. Unless there's a question, in which case we'll just answer the question. It's a ring in Kerschild coordinates, yes, very good. Yeah, so you can go to a ellipsoidal type coordinates where the shape would look perhaps a little bit different, but it is, yes, so it is a coordinate dependent statement. In Boyer-Linquist's coordinate, it looks like a point, right? So it looks like the point r is equal to zero and on the equatorial plane. But if you actually take r to infinity, if you actually take m to zero in the Boyer-Linquist coordinate, you don't quite get the Minkowski metric in Cartesian coordinates. You get the Minkowski metric in ellipsoidal coordinates, whereas when you work in Kerschild coordinates, when you take m to zero, you do get the Minkowski metric in Cartesian coordinates. So the x and y and z coordinates that I'm showing here is what you would think of as your regular coordinate axis, but certainly you can always transform to another coordinate axis where the shape will look different. So yes, you're absolutely right. Cosine squared theta, sorry. It's the same row square I had before. That's why they're all squares, so they're all positive, they can't solve each other. Yes, good, good question. So you can go on, you can calculate other curvature invariance, like for example, you can calculate the Pointery-Aggon invariant, which is gonna be like the remand contracted into its duo and you'll find that it diverges on the same place. Also, these only diverges when row vanishes. So in Boyer-Linquist's coordinates, all invariants vanish when row is equal to zero. I think that's true. I'm thinking whether there's a proof of that. They can definitely calculate it one by one, right? Related to, that's right, that's correct. And in fact, once you go to higher than second order curvature invariance, like cubic curvature invariance or quadratic curvature invariance, you can construct it from second order curvature invariance. And you can show that all second order curvature invariance, which are finite, there's not that many. But in fact, there's only two in this case, diverge here. So therefore, all the future ones also diverge. All the higher order. All right, so I'm not gonna turn around so nobody can ask any more questions. And I'm just gonna talk about geodesics for two minutes because this will set me up for tomorrow's lecture on extremist ratio inspirals. So I'm gonna leave this a little bit as an exercise for you guys, for you all. So 1.3 geodesics. For the purposes of here, we're gonna be looking at a time like geodesics. So because we're interested in the motion of a black hole around a supermassive black hole. So you have, you know, the geodesic equation for the trajectory like this. So this is the second derivative with respect to proper time of the trajectory is equal to the Christoffel symbol of your spacetime contracted onto two four velocities. So this is a second order equation and it's a little bit annoying to compute in this particular form. But you can use, instead, and use the fact that the energy which is defined as minus u alpha t alpha the angular momentum which is defined as u alpha phi alpha and this quantity I'm gonna call q which is defined as u alpha, u beta times k alpha beta are conserved. So all of these are conserved. Plus you can use the fact that you know, u mu, u mu is normalized to write down equations for the motion of these objects that look like first order equations. So for r dot, you get, well, you get equations that look more or less like this. Minus a a e tilde sine square theta minus l tilde plus r square plus a square quantity p over delta where p here is defined to be e tilde times r square plus a squared minus a l tilde. And you get similar equations for r dot theta dot and phi dot. These are interesting because these are first order equations for the trajectories of the spacetime for the trajectories for your four velocities for different components of the four velocity which you can then solve on a computer much more easily. And it's also the basis for the evolution of extreme maceration in spirals, right? So the homework for you guys is to compute what the right hand side of each of these equations is by separating by using these conserved quantities. And for extra credit, the homework is to prove that e dot, sorry, d by d tau of e tilde and d by d tau of q is zero. Exactly. And I'm not gonna be handing in solutions but if you wanna show me the solutions, I'll be happy to look at them all. Or if you get stuck, good for just asking questions. Okay, thank you.