 All right, so yeah, this is the title of my talk. I wanted to make a point of just mentioning my collaborators here, one of whom is somewhere in the audience, and this is the archive number, so look it up, I think it's great. And yeah, since I only really have of order 20 minutes, this is gonna be a pretty sort of rapid tour through I think a very interesting subject, but one which also has a lot of confusing pitfalls and technical details, so it'll be of the kind of cartoon variety, but if there's any very basic questions, please interrupt me, and also of course afterwards, I'd be happy to talk about some technical details. Okay, so let me begin. So I wanna just flash the take home message before I try and rush through everything because I wanna make the point clear and just you have something to take home. So my main focus here is that there really is a tension when you're talking about spinning objects, and when I say spinning objects, I mean macroscopic spinning objects like a space station or something. So when I use the word spin, I'm not talking about spin one half, like an electron, that kind of spin, so macroscopic spin. There's a real tension between Lorentz invariance and describing spin, so we like Lorentz invariance, number one because we think it's a symmetry of nature, but also because it's crucial in building things like theories of gravity, in particular general relativity and coupling general relativity to matter. So we have this nice prescription that we learned from classrooms where once we get something which is nice in Lorentz invariant, we essentially promote the Minkowski metric to g mu nu, and we promote derivatives to covariant derivatives, and this is called minimal coupling, and so we know how to take a nice relativistic field theory, let's say, and couple it to gravity. And now the problem is that spin is a little confusing because macroscopic spin, we like to think of as a three vector, and so it's a three vector, not a four vector, so what do we do? And so there's this tension here, and I want to describe a resolution of this tension, how to think about this problem, how to construct it in an organized and intelligent way so that the physics is nice, and the method of constructing this is what's known as the nonlinear realization of symmetries or the coset construction, and I'll sketch a little bit of that here, and once you do this, you get a nice low energy effective field theory for slowly, and I'll describe this in a little bit, rotating objects coupled to gravity. And so this is a natural extension of what Walter Goldberger was talking about in his discussion where he had point particles exchanging gravitons, compact objects in his case. Now we can also describe point particles with some rotational velocity exchanging gravitons. So this is my bad pun of the day. Why should we care? First of all, things in astrophysical objects do spin, and some of them spin very, very quickly, so I was doing some Wikipedia searching last night, and I found this amazing pulsar with this really sexy name, but the thing that you should pay attention to is the fact that it spins around in 1.4 milliseconds. So this is something of approximately two solar masses, and the surface of the neutron star roughly moves at about a third the speed of light. So it's pretty rapidly rotating. We need to understand how we incorporate strong gravitational effects in a situation like this. Here's a little map of where it lives. I have no idea where this group of stars is. Nope, have I wandered too far? There we go, okay. So when we talk about spin in general relativity, we usually have one thing in mind. We have the care solution, which is this beautiful object here, and it's so tricky and complicated that even though people were, of course, interested in rotating stars and planets, it took until 1963 to find the solution. It wasn't even clear if there would be an exact solution, but here is one, a nice vacuum solution, but it's pretty much useless when we wanna actually do astrophysics. So as long as we're kind of far away from care, you usually have another object or maybe some disc of dust or something like that, and so even as beautiful as this thing is, it's not particularly useful for actually doing astrophysical computations. So in particular, if you're interested in the binary problem that Walter Goldberger was discussing, what are we gonna do? So just as a quick reminder, having really accurate gravitational templates is a very, very useful thing. Rather than just doing numerical simulation, you can do analytical analysis as far as possible, and this gives you a better way of basically scanning the possible gravitational templates, and this is crucial for actually extracting the signal from gravitational wave telescopes, and so this is something that we want, and in particular when you have a separation of scales, there's some effective field theory slash perturbative methods that we really can't employ, and that's what we wanna see if we can sort of pull out of this today. Okay, so let's actually talk about spinning objects. Again, macroscopic spinning objects. So, well, when you're talking about a point particle, we're lucky enough to have, oh, there should be a derivative right here, we're lucky enough to have the four velocity, right? And this four velocity is a nice four vector, and we know how to put it into the action for the point particle, which is just basically the proper time, so integral detail minus m, and we vary this action under this constraint. We don't let the four velocity change magnitude, okay? But when we have spin, like I said, in the very introduction, we have this three vector, okay, it's a little unclear what to do. Maybe we can promote it into some larger, manifestly, you know, Lorentz and Varian object like some anti-symmetric tensor, S, but the problem is that there's no simple, unique constraint, so these kinds of constraints are what's known as spin supplementary conditions, and they're basically designed to get rid of these extra degrees of freedom that you've introduced by making this thing nice and Lorentz and Varian, okay? So we don't have quite as simple a story here. We have something which is sort of ambiguous and kind of confusing and it's unclear what the best choice is, so on and so forth. So we wanna try a different perspective. So our viewpoint, our viewpoint, meaning my collaborators and myself, is that this object like this ballerina here is really describing the motion and the dynamics of gapless degrees of freedom. So for instance, if I take the translational motion, right, it doesn't take any energy to excite a little bit of translation, and it's the same thing with rotations. So in this sense, these things are gapless, and notice that they're also associated with symmetry, or generators of symmetry transformations. So for instance, translations are associated with the usual translation generators, and rotations, of course, are associated with spacetime rotations. And so there's this interesting connection which should remind us of the situation of spontaneous symmetry breaking, right? So we have a situation where acting with spacetime generators creates fluctuations on our vacuum state. So in this case, our ballerina, if we act with a rotation generator, we rotate her slightly, okay? And here's the symmetry breaking pattern. We have an unbroken time translation. This is the ballerina sitting at rest. We have some residual rotations encoding the residual symmetry of the object. So for instance, if she was completely spherical, there'd be some residual rotations in here, but we also have broken generators that are spatial translations that gives rise to our three positions, and we also have rotations and boosts. And so we break all these things, and we want to find a way to come up with the low energy dynamics of these goldstone modes, okay? So the symmetry breaking pattern is gonna give us this, and we wanna find, it turns out that there's a mathematical construction which really is just a big crank that we can turn and outfalls the low energy effective action essentially for these degrees of freedom. And it goes under the name of Don Linear Realization of Cemetery's or the CCWZ Construction. Now this is something which is very, very well developed and super precise for internal symmetry breaking patterns, but when you have symmetry breaking patterns that also include spacetime symmetries, it becomes a little trickier, and there's been a lot of work in the last five years in particular to understand exactly all of this. And I think it's fair to say that the situation is pretty well understood, and really it is a crank that you can turn and outfalls interesting stuff. And what it essentially amounts to is that you have these gold stones, which I'm just gonna generically label by pi, and they're derivatively coupled because they're gold stones. And I need to find a clever way of dressing these things in a way that transforms nicely. And exactly what this means is that it has to transform under the unbroken residual symmetries. And so you find some way of dressing your fields and this construction gives you exactly that dressing. And once you have these objects, you throw them in the Lagrangian and you're off to the races. And so if you really wanna see exactly how this is done, please take a look at the paper. I think it's relatively pedagogical. You can check it out. There's some nice examples in there. Like even the point particle, for instance, is something you can analyze from this perspective. And of course, the usual examples from particle physics like pions and stuff. Okay, so you turn this crank, this wonderful thing, and outfalls a action. And the degrees of freedom of this action are my positions, my three oiler angles. And then since our whole point of this thing was to try and couple to gravity, there's a procedure for understanding how that works as well. And we have the graviton coupled to our degrees of freedom. So here, I've written it in a nice manifestly Lorentzian invariant way. This is the usual point particle term. And now there's this other term which should look vaguely familiar, especially because I am using sort of usual notation. This thing here is gonna be the moment of inertia tensor. And then we have these objects here which are derivatives of my oiler angles along with the velocity and other objects. So it's this bundle right here. And what's important is that it's made up of these, sorry, I'm being a little schematic here, that these matrices, these Lorentz matrices, lambda, and these lambdas essentially define a frame. So when the dust settles, you realize that your action is basically given by some frame. Part of this frame is picked out by this condition here. So the four velocity dotted into the frame is equal to zero. And this is basically telling you that this frame you sort of have to boost into the frame. And then this frame can have any rotation that it wants. And so if you really sit down and start thinking about this thing, you can be convinced that indeed it's the right object. And the coupling through gravity comes in because there's some parts of the fear bind in here. And then this is also the spin connection omega. And so when you expand the sky, you get coupling of my low energy degrees of freedom like my orientation variables and my translation variables with the graviton, which is exactly what I want. Now, since this is an effective field theory and unlike the point particle, I can actually write a whole bunch of extra terms. And so this dot dot is something I'm gonna address in just a moment. Okay, so as a reality check, just to make sure that I'm not completely just smoking mirroring you. So let's turn off gravity. Let's go to the rest frame of the object, meaning it's not moving at near relativistic speed. So it's just spinning right here in front of me. A lot of things simplify. And what I end up seeing is I see that my action is just given by this. So this is the usual action that you can find in something like land down lift shits mechanics for a rotating body where this capital omega, which is just sort of a rewriting of this nabla alpha is my angular velocity in the frame of the particle. And so this is the right leading order Lagrangian for a rigid body. If I vary this action with respect to the Euler angles alpha, I find that I get the Euler equations. So this really is the dynamics of a rigid body. And then the question is, so what exactly are these dots? Because these dots are not something that Landau actually bothers to put in his mechanics book. So are we crazy? What's exactly going on? Well, you know that maybe you're not completely crazy because the subject title of that chapter is rigid bodies, but as we know, nothing is actually truly rigid, especially when you have relativistic physics going on. Thank you. And in particular, the next term is gonna go of this nabla alpha to the fourth with some constant in front. And we can just do some dimensional analysis. And we realize that this constant is gonna scale like something like our angular velocity squared over some other velocity squared. Other, sorry, frequency squared, excuse me. And that's this omega not here. And so the question is, what could this omega not be? And from the effective field theory point of view when you're going down from high energy and you're trying to integrate out things and find this low energy physics with the minimal amount of degrees of freedom, you realize that when I went from describing a planet or a comet or a chair or something to just a couple of degrees of freedom, meaning just the rotational degrees of freedom and the translational ones, I've actually missed out on all the internal possible dynamics of these objects. These things can twist, right? They have vibrational modes, all of that kind of stuff. But importantly, these modes, especially for generic rigid bodies like a steel bar or something, are gapped. And so they have some characteristic frequency. And so when I go below this characteristic frequency, I can integrate them out. And what do they do? They introduce higher derivative couplings. That's precisely what these things are. So what should we expect this omega to be? Well, it's gonna be related to the typical frequency of those kinds of objects, right? And so this extra term, which you can write, really gives me the correction to rigid body dynamics due to distortion of the body itself. So slight flattening or something like that. A correction, if you will, to the moment of inertia tensor. And so this is a beautiful thing and you can check how the Euler equations are changed and you get interesting forms of procession and everything. We can throw back in gravity. We can have things moving at relativistic speeds. Everything fits together. And the only thing we have to remember is the thing controlling the derivative expansion is that we don't spin too much faster than the normal modes. Because if we start spinning too much faster than the normal modes, then we thought we had this sort of rigid body and it just, I don't know, breaks apart or completely becomes a disc. Something very drastic. So as long as we don't do that, we're safe. We can calculate stuff. It's wonderful. And in particular, we can just start doing perturbation theory and now we have these new vertices, a la Walter and company. And we can compute new gravitational potentials due to this new coupling. And so we can compute things like the spin orbit and the spin spin potentials for things like rotating black holes or neutron stars or whatever. So I think this is a nice way to view this, not missing piece. There are other ways that you could have handled this. But this is a really sort of elegant way to keep track of these orientational degrees of freedom when you're doing these kinds of computations. And they'll end up in your final effective action just like your point particle. OK. So let me flash, once again, the take home message. Because that's, like I said, what I'd like you to leave with. So again, there's this Lorentz invariance is crucial for GR, but spin is a little funny in that perspective. And so what do we do? Well, we sidestep the whole issue. And we use this coset construction to build this effective field theory, this effective really in a true sense of the word, effective description of the dynamics. And it works beautifully for astrophysical objects. And we also get this wonderful sort of extension of classical physics that is like a really nice interpretation. So I think it's easy to compute with, very useful for astrophysical computations, and has a nice physical picture. So thank you very much.