 Today, I didn't do any formulas, pretty much, but just animations. Today I'm going to make up for it, and it's going to be mostly formulas. But the idea is to try to see how, you know, gravitational waves don't come out magically out of metric theories and Einstein equations, but rather they express some basic physical truths, okay, some basic physical facts in the theory. So in preparing for this lecture, I looked around at what kind of, you know, explanations and what kind of developments you can find. And there are probably a couple dozen, a couple dozen, right? If you look in various books or the web, everybody is doing a kind of a variant on doing linearized field equations for the gravitational field, the gravitational waves and so on. So I quickly, so I'm giving you a bit of a mix of different treatments and different ideas, but if you pick up one of these three books pretty much, you cannot go wrong. The first one, Einstein, Gravity is a nutshell, it doesn't have that much about gravitational waves, but it's a very nice book about gravity that's all about physics and intuition and how things fit together. It's also written by field theories, so it kind of has that kind of approach and Z is interested in symmetry, in groups, in putting things together, you know, writing Lagrangians out of symmetry. So it's a very pleasant thread, very nice. The second one is a recent textbook in gravitational waves by two colleagues of mine at the University of Wisconsin, Milwaukee. It's somewhat formal in its treatment development of equations, but at the same time it's rather complete and it's not two verbose, okay? So it gets there quickly. So I do recommend that book. Although there's a couple formulas in there that come out of my papers and they don't cite me, so I'm a little, they should have said that's the valesinary formula, the valesinary catar formula or something like that. Anyway, the third one is a long-awaited textbook. So a modern classical physics application of classical physics is a course that has been taught at Caltech for the last 20 years maybe. And the book was always going to come out next year. So it seems now it's slated for February, but all this time you've been able to find basically the entire book in chapters on the web. And so that's what I was looking at also. So if you Google something like a PH, physics 136 Caltech, you'll get it. And it's rather interesting. It's a very brief and quick roll through pretty much all of classical physics. It does general activity in one chapter, okay? And gravitational waves in the second chapter. It does statistical physics in one chapter. Perhaps it's not the first book you should read about any of these things. But if you want to go back to it in kind of like the notation from the viewpoint that's useful for astrophysics and for gravitational waves, that's pretty good. Okay, so lots of formulas, but the point is that these are the things I want you to remember if you know them already. If you've had a course in general activity of gravitational waves. So maybe if you don't, I want you to take away from today. And these are all physical statements, okay? So in the history of general relativity, Einstein came up with the gravitational waves quadruple formula pretty early in 1916. Although he had a mistake, he first didn't believe in them, he believed in them again. So this was quite a bit of confusion about the subject and about this topic for many years until probably the 1950s. And so it took that long to get a solid physical understanding of gravitational waves and to take them from a rather formal aspect of general relativity to something physical, okay? Now, because of that, statement one, okay? So most treatments of gravitational waves focus on the metric. You take the metric which is the basic general relativistic quantity disguised as the geometry of space-time. You look at small perturbations of the metric around either flat space-time or maybe some background. And you have this object H for gravitational wave. And you start playing with it and you show that it obeys a wave equation. And that's a gravitational wave. That is what Einstein did and that's what confused Einstein because general relativity has this coordinate invariance, okay? So the equations don't change whatever coordinate frame you use to write them. But it means also that this freedom that you have to, you can mistake this freedom for actual physics. So when Einstein was writing equations for gravitational waves in 1916 and 18, he was finding some solutions that look like waves but they were just really coordinates. And he found those either didn't propagate or didn't seem to carry energy. So to cut it short, it's the easiest way to do the equations is to look at the perturbations of the metric. But that can be confusing because you have a problem in choosing coordinates. So another viewpoint that Pirani and later Thorne took is that you can take the curvature, okay, as your basic physical element. The curvature involves the derivatives of the metric. It's embodied by the Riemann tensor, which unfortunately has four indices. Okay, so you have to work with something with four indices, whereas in electromagnetism, two are okay. But then you can work at the level of the Riemann tensor. You can show that curvature also obeys a wave equation. And now curvature is something more physical, right? It's something that we can relate to the Covarian differentiation and so to the transport of vector. So a physical, geometrical operation on your curved space time. So that means that this thing then that's traveling like a wave is really a physical quantity. Okay, so that's one. Then we're going to see that the wave that curvature manifests itself is as tidal forces. So tidal forces I think that don't push on the center of mass of some physical system, but just pull it apart or together, just like the moon is doing on the earth or so on. Then we're going to see that gravitational waves are transverse. So they act in a direction of Thogelmann to the direction of propagation. And they do carry energy momentum. This was another big point of confusion for Einstein and for many. General relativists, many relativists in the first years of the theory, because to attribute some energy momentum to gravitational waves, you have to give something away, which is locality. So you cannot exactly localize energy in gravitational waves. You have to do some averaging. If you accept to do that, however, everything is fine and the energy momentum of gravitational waves is very real since, after all, you can detect them and any detection of a physical phenomenon must involve some transfer of energy. And finally, we're going to go to mass, to gravitational wave generation. So gravitational waves are emitted by... The dominant emission is from a time-dependent quadruple of mass. And because of that, the prototypical binary system has a quadruple, it's going to lose energy to gravitational waves and it's going to have an accelerating spiral. And you think that 0.6 is kind of obvious, right? After all, that's how we think that gravitational waves... That's how we established they existed from the House Taylor-Polser. But that was also a big point of contention because it was a little embarrassing to think that the Keplerian orbits or at least the orbits of planets and binary stars were not just closed stationary solutions, you know? They would go on forever. So in fact, many thought that, yes, you can have gravitational waves, but somehow the system, a binary system where you have two masses going around each other, because of some internal consistency, maybe that system doesn't emit because things balance out and so you keep this, you know, this comfortable, stationary, eternal configuration that has helped astronomy for so long. Okay, so let's go into formulas then, but you'll recognize some of this. And the point in having them there is that, you know, kind of look at the, yes, number four. So the energy momentum in gravitational waves? In locality, is that right? Yes, it can only be localized if you average over wavelengths. Okay, that's kind of like the physical viewpoint. No, I think if you look at the point in vector in EM, if you look at the energy momentum tensor for electromagnetism, you can pretty much have a well-defined energy momentum at any given point in the field, actually, or in a wave. It's true there's a gauge freedom, okay? So the potential, you can change the potential and you can use, you can apply transformations to it, but energy momentum classically, at least, is well-defined. If you go to quantum, then, you know, maybe you have to take an expectation value, but you say that's right. So, okay, let's go back here. A metric theory of gravity, Rafael was describing that. Yesterday, basic element is the metric, okay? So your description, your generalization of Pythagoras theorem that tells you if you move away in coordinates, how much proper distance you make, of course, here, we're talking without pseudo-Romanian manifolds, so there's a time direction that has the output assigned, but it doesn't really change much. From the metric, or even just from the idea of a curve spacetime, you come to the idea of covariant differentiation. So what is that? You can look at that rather formally, but the main point is that if you're just looking at the scalar field, just, you know, numbers as a functional space, it's not a problem to put those on a curve manifold. However, if you look at vector, you run into a problem because a vector is defined in a local tangent plane, and the local tangent plane changes as you move across the manifold. So you have to do something particular to take derivatives of vectors or to move vectors around and compare them. The easiest way to think about it is to, you know, to lose Riemann for a moment and go back to Gauss and think of a manifold as embedded in a larger flat space, in a Euclidean space. In that case, all that covariant differentiation does is, you know, you have a vector here, you want to move it here, you just move it rigidly in the embedding Euclidean space, and you just project it back, flatten it back onto the manifold. If you write that into formulas, you know, you write in a derivative, you take in a derivative, both of the components of the vector and of the kind of like the basis field, and you have three terms, and you just lose the terms that's normal to your manifold, okay? So, and then the piece that you have is the standard coordinate partial derivative, and a piece with these gammas, you know, the Christoffel symbols that kind of are related to the change in your basis vectors. This is not a tensor, right? They're symbols, but together, these things transforms like a tensor, and this was a big achievement when mathematicians came to it. It was actually Narici and Levi-Civita, working at the beginning of the 20th century, because once you have this derivative, you can take all of your standard physics, classical physics that you use to write in either in Euclidean space or in Lorentz space, and you can just bring it into any curved space time. That's what general relativity does. This is how, if you have a metric, of course, Riemann actually showed that you don't need an embedding space. All the information that you need is already in your metric, and therefore you can write the Christoffel symbols from just the metric. And these equations down here, actually I should start getting into the Socratic thing, but these are, okay, what's that? This is the velocity, okay, of an observer, the velocity of a particle. And then what does this equation represent? Who remembers GR? Come on, yeah? Is there, yeah, is there a geodesic equation for a time like particle? So you're transporting the velocity as straight as you can. That also gives you a trajectory with a maximal proper length, okay, in space. So that's the equation of motion pretty much in the absence of other forces. If you add the M or so on, you could put them on the other side here. Okay, that's parallel transport. Now, I didn't start my timer, so I'm going to give myself... Okay, Riemann, what's curvature? We know how to transport vectors. The way that curvature works is that if you transport a vector around two different paths, you're going to end up with different vectors at the end, okay, so the transport of vectors doesn't commute with doing different paths. We're going in different directions. In general, it certainly doesn't on the earth, okay? So if you take this vector U and take it along two other directions in different orders, you're going to end up with different vectors, and the difference between those is encoded by the Riemann tensor. You can also write it formally as just, you know, the commutator of covariant derivatives, and this is a linear operation in the vector W. Again, you can write this as derivatives of some products of the Christoffel symbols if you're working with a metric, but this, right, although there are symbols, there are indexes, and the formulas are always confusing, it's a physical thing, okay? It's something that expresses how space-time is curved. Now, the Riemann tensor has lots of components, but actually, they're very redundant, okay? So there are lots of symmetries and lots of internal relations in the Riemann tensor that end up, that express kind of like its internal consistency and how curved space-time is consistent with itself when you study its curvature. And that consistency is actually what gets you, eventually, the wave equation for curvature and the four gravitational waves. So, for instance, what's called, what's known as the first Bianchi identities is something where you're just taking permutations of indices, the cyclic permutation of indices, the raise index is where you apply the metric, but it doesn't really mean, that's not too important here. The point is, again, this can be seen as a geometrical fact, which is that if you're taking, the most general case of taking alphabeta gamma is going along three directions, okay, in a curved space. So if you do this Riemann difference, the difference between taking, moving a vector along two different paths, along all three axes, you end up with three different vectors and they do have to form a triangle, okay? The differences between these three vectors, just because the dimensions close upon themselves. So that's what these Bianchi identity shows. There are also internal symmetries, so for instance, Riemann is anti-symmetric in the first two indices, is anti-symmetric in, which is kind of obvious, right? Because it is just the order in which you do the commutator. It's anti-symmetric in the second two indices, which is not that obvious because it's saying that the vector that you're transporting and the vector you end up with also have this kind of anti-symmetric relation, but this comes out of, comes out of the fact that this is the curvature for a given metric and in a metric, you can put tensors together to make scalars and so by conserving, conserving that, you get this and also, also a little more surprising, Riemann is symmetric under exchange in the first pair of indices with the second pair of indices. So you start putting these things together and the number of independent components go down to 20, I think. Now, second Bianchi identities, this is the most complicated formula I have here and they are, it's a somewhat second order thing. There are many interpretations of this, many ways to look at it. If you just, just a statement is that the, if you take a combination of three derivatives of Riemann, covariant derivatives, again permuting indices somehow, you get zero. Okay, so it's an identity equal to zero. For instance, a beautiful way to look at it is to kind of do this work using Carton's differential form theory which leads you to express this identity saying that the boundary of a boundary is zero. If you go to gravitation, Mr. Tom Wheeler, that's how they do it. I kind of like this view of it, which is you're doing again this transport, this Riemann transport of vectors and you're doing it around all sides of a cube. So doing it on the bottom side, for instance, on the high side, the difference between these two is what the derivative is of Riemann. You do it on all of them and then you end up going around, transporting around each edge of the cube twice in different directions, so that must be zero. So again, this looks formally like a magic identity. In practice, it's something related to the consistency of curvature and how it's expressed in geometry. So in practice, it's hard to not be formal at some points. When you derive equations at some point, you're just juggling indices and substituting things but it's good to remember that all the things have physical meanings, okay? And then we go to, I think you've seen this in your courses but also the last couple of days, Einstein's equations. So Einstein struggled to find the right object describing curvature that gives you gravitational physics and but finally he found it in the Einstein tensor. So Richie, you get by saturating one of the indices here, collapsing two of the indices of Riemann. You can do one more and get the Richie curvature which is a scalar and then this combination is a combination that is divergence free so it's appropriate to match up with an energy momentum tensor which needs to be conserved locally in curved space time. So these are Einstein's equations and the other principle that Einstein used to put these together of course is to try to go to the Newtonian limit. Then you've got a very simple energy momentum tensor. The dominant piece is the T-00 where you have the density of mass and that you need to go back into to match up with something that's kind of like the Poisson potential. Okay, Richie, Richie and Einstein. Now, let's go to a weak field. Let's go to linearized gravity because we're going to postulate that gravitational waves are going to be a very weak phenomenon and we'll start actually with waves on a flat space time just on a Minkowski standard metric. You don't need to remember any of these but just to show that if we do this hypothesis then we can replace all covariant derivatives just with ordinary derivatives and that the expressions for curvature are somewhat simpler to first order in age. These are the infinitesimal coordinate transformations so which are appropriate, the kind of gauge freedom that's appropriate to keep at this level. Now, one thing here that's very, very interesting is that this object, Riemann for linearized field is actually gauge invariant with respect to these transformations. So that's kind of what's nice in working at the level of Riemann instead of the level of the linearized metric because in that case the choice of a gauge very much changes your linearized metric but here it doesn't. So we're working with something physical. It's not in Tali invariant. If you do a Lorentz transformation here if you do rotations of course it has to change in the right way but it is with respect to coordinate transformations. So that's comforting it means we're doing something physical at this level. And this is my punchline but I was going to ask you how do you write wave in mathematics? And you've seen it, right? So that's how you write wave in mathematics. So something like this in three plus one dimensions. The job then is to show that the Riemann tensor actually obeys a wave equation like that. Okay, so here we need a little bit of formalism but we go back to the second Bianchi identity. Alpha, beta, gamma, delta, mu plus alpha, beta, delta, mu, gamma plus alpha, beta, mu, gamma, delta. Okay, and then in a vacuum. Okay, we're working outside of a source. We're working just where the waves propagate. We have that the Regi tensor is zero. Okay, that's just Einstein. Einstein's equation in the vacuum if t is zero. Now, that means that we can, what we can do is to take Riemann with run raised index. We're actually going to take this equation, alpha. We're going to collapse by saying that alpha is mu. Okay, so we're going to multiply by eta alpha mu. So that gives us R mu, beta, gamma, delta, comma mu plus R mu, beta, delta, mu, gamma plus R mu, beta, mu, gamma, delta equals zero. So I've just rewritten the Bianchi identity and multiplied it by the metric so that I'm collapsing the alpha and mu indices. So now what do you see that just by the fact that Regi is zero, I can throw away this term and I can throw away this term. This one is different though because it doesn't make Regi because the index that's the same is the index that I'm differentiating with respect to. So what this says that is that Riemann, the divergence of Riemann is zero and actually because of Riemann's internal symmetries, Riemann is divergence free on every index, every single index that you do this collapsing gives you zero. Okay, Riemann's divergence free in every index, that's kind of nice. So let's take, then let's start from here again and let's take another derivative, alpha, beta, gamma, delta, mu, mu and we're going to build the Dalland version, okay? Take a sum over derivatives over the same index and then we have alpha, beta, delta, gamma, delta, mu, gamma, mu plus alpha, beta, okay? And since these are just regular derivatives in this weak field limit, we can turn around, switch over these two derivatives. So these ends ends up being a divergence of Riemann goes away, this is another divergence of Riemann goes away. So we left with this, okay? Just as we promised, alpha, beta, gamma, delta, zero. So that means that the Riemann tensor admits wave-like solutions and propagates like a wave and all that I needed was the internal structure and the internal consistency of curvature plus the Einstein equation. So this is true in Einstein's theory in vacuum for a weak field. Okay, so questions? Yes? In what? What did I do where with the bottom line here? Okay, I took just the derivative of this, of the Bianchi second Bianchi identity with respect to mu. So see this mu, there's one more mu there and then two terms are just derivatives of divergencies, there's zero and the other term is just the wave operator, Dalland version or Riemann. Okay, oh, and now I can give you some literature actually. So I was talking yesterday about this book by Italo Calvino who is a very beloved Italian author and it's called Cosmic Comics. It's not quite sci-fi, it's more like this whimsical stories about science, really, deeply science. He clearly was learning about cosmology and about generality and was trying to express what he learned in a literary way and there's this story, which is really about geodesic deviation and it's about this guy who's falling, actually not even just the equation, it's also about the equivalence principle because it says we were falling all the time, you can read this while I talk, we're falling all the time, couldn't even tell if we were falling, we might have been just standing still in vacuum, in nothing, so that's the equivalence principle for you but there were three of them and there was the protagonist and this beguiling character Ursula HX who he'd really like to get close to and this enemy of his, Lieutenant Fenimore. So all the while he's kind of hoping that his freefall is going to get him closer to Ursula but worried when it seems that instead she's going towards Lieutenant Fenimore. So that's geodesic deviation, okay? So you're falling freely but since space time may be curved, he doesn't really know at this point, lines that are parallel could converge or diverge. So for the formulas, this is the, not surprisingly, this geodesic deviation equation involves Riemann, involves the known commutation of covalent derivatives because if you move a little away from your time like geodesic, that's a differentiation. If you move towards the future it's another differentiation so that's how it looks. But I think that's kind of important physically is that you can look at it geometrically as curvature but physically this is really analog to tidal fields in Newtonian theory. Okay, if you just have acceleration due to a potential, so to the grad of the potential and if you now look at the relative acceleration of two particles that are both in the same field, well, that relative acceleration is going to be proportional to the second derivative of your potential, okay? So to a mixed second derivative times the separation. So this Eij, the second derivatives, is a tidal field. Now this equation is analog to the equation terms here although there are more indices. In particular, if your particle is at rest then this velocity vector is going to be 1 0 0 0, okay? And then the components of Riemann that are going to give you relative acceleration are just going to be the 0 I 0 J, the time spatial, time spatial components. So this vector here for a particular rest or in cases where velocities are very small so things that are moving very non-relativistically, very slowly is entirely analog to the Newtonian tidal field. Any questions here? This is central to thinking about gravitational waves in a physical way, okay? So there's actually, yeah, there are analogs, that's... So Kip Thorne and his students, the last thing that he did before retiring and becoming a Hollywood icon was to try to look again at this phenomenology of tidal fields in a physical way and he came up with this idea of tendex seas and vortices, vortex seas, and what the tendex seas is, you have a field so you cannot quite do, like with the electric field, you cannot quite do lines of force, but you can look at the eigen directions and eigen vectors of the field and try to draw them and in this tendex lines that express the content of the tidal that will show you where you'd be either stretched, okay, for the positive ones, the red ones, or where you'd be compressed if you're oriented along those lines. So it's a graphical way of expressing the content of the tidal field and there's also a... The counterpart to it is that if you look at the Riemann components with the three indices, okay, those coupled to velocity in the geodesic deviation equation, so they're kind of like a magnetic potential in a way, they're a torque, they're a frame drag, and that's what he calls them a frame drag field, which give you the rotation of a gyroscope. So again, if you're in this vortex field, the local frame drag field will tell you how you're rotating. And there's actually, if you put this thing together just effectively in the Bianchi identity, what you get is equations that are formally the same as electromagnetism. And you can push this analogy forward and pretty much you have the same equations except for some coefficients. This is called gravito electromagnetism and is where you're looking at the space, at the, sorry, electric and magnetic parts of the Riemann tensor and its effects on motion. Of course, this breaks symmetry, you shouldn't really separate this thing, it's better to look at the whole thing, but if you're moving slowly, it's not a bad way to do it. Okay, what now? Now then, we have established that Riemann propagates as a wave, so we're going to say that it propagates along a specific direction, along z. And we're going to try to understand what the effect on particles is of having such a perturbation of curvature. And the idea is that, so that's the question that, so if we just define this H, this H is not a metric at this point, it's just my definition of the gravitational wave field. Define it as the second integral or vice versa, we define this as a time, space, time, space component of Riemann as the second derivative of this. You see that the effect on local particles is just going to be proportional to this gravitational field times their displacement. Okay, is this clear? Yes, say again, sorry. Can't hide our field. Of extrinsic curvature. So again, you're looking at an embedding and, I'm sure, yeah, I'm sure you can do something like that. And it's, by this point, we're back into three plus one into our space time and so on. And it's, we don't want to go, what do you need to embed from three plus one? 11, 12, or in general? 10, 10, right, so. Okay, so then the, what does this look like? Okay, so that's the time, space, time, space components of Riemann, we're going to look at waves that are propagating along Z. And so you can look at these components individually or the components of the gravitational field HGW and for instance, if you look at a ZZ component of it, what effect is it's going to give you? The fact is going to be that if you have particles that are displaced along the direction of propagation a little bit, they're going to get more displaced when the field is positive or less displaced. So it's going to be a longitudinal oscillation in the motion of nearby particles immersed in this wave. If you look at the longitudinal at the kind of like XZ or YZ, well that's going to be again a motion that moves particles in a mixed direction. Okay, and if you're going to look at the XX or YY or XY field, well those are effects that are going to take a ring of particles in a plane orthogonal to the propagation of the wave and they're going either to expand it and squish it like a circle or to do an ellipsoidal effect or squishing along one direction and lengthening along the other one. So it's kind of, that's kind of the way that you want to look at these components by taking little matrices in XY, you look at a component that has the same XX and XY components that has the opposite of them and then is along the diagonal. So that's a symmetric tensor. So between these three, these are the only ones that I can write, exhaust the entire freedom that I have to write components. Now that's not what I had promised. Okay, there's more modes, there's more polarization modes that I told you there are in general relativity and that's because I've actually left something back. I'm taking just any tidal gravitational wave tensor propagating along the z direction. I have not imposed the Einstein equations. Okay, if I go back and impose the Einstein equations, which you can do by taking Richie again and looking at sums and differences, you end up showing that the longitudinal transverse and transverse spin zero components of the gravitational field have to be zero in GR. If you have a different theory, if your field equations are not quite in vacuum this, but maybe there's another term coupling, I don't know to Richie or another field or whatnot, then you can have all five modes in general, but in GR you only have these two transverse modes. You see this, it says spin zero, spin one, spin zero, spin one, spin two. This is kind of like the field theoretical classification of this perturbation and one way to look at it is by looking at what happens when you do rotations in the plane orthogonal to the direction of propagation. Spin zero means that, okay, let's take this thing that gives you just an expansion of a circle and shrinking of a circle alternatively. If you do a rotation, what's the rotation? Any rotation brings it into itself, basically. It's just invariant with respect to rotation, that's spin zero. If you've got this longitudinal transverse, well, you need a full rotation of 360 degrees to go back to the same perturbation, so that's spin one. If you have this real GR mode, gravitational wave mode that are ellipses like that, if you do a rotation 180 degrees, that brings you back into yourself again. So that's why spin two, 360 degrees divided by 280, that's the return angle is known technically. So spin is a group, right? A group of concept in field theory. So we're down from six independence degrees of freedom to just two in general relativity. And just to give you an idea, to reiterate how they behave. So you can write this title field as a sum of two pieces, a plus and a cross polarization, in this case H plus and H cross. These are the polarization tensors which I was trying to write here. So one is a trace-free, symmetric trace-free around the diagonal. The other one is again symmetric and trace-free. And if you have a say a sinusoidal function for the plus polarization, it's going to take a circle of particles and it's going to squish it alternately along the two dimensions. And the cross polarization will do the same, but it will do so at a 45 degree angle. If you look at the forces at the title field, that's kind of how they look like. And these forces, once you integrate them, they give you this kind of motion. This is how this thing transform with respect to rotation in the plane orthogonal to propagation. So this is the physical effect. I would think it was my statement three of transverse effect of the waves, of gravitational waves as they propagate. Everything good here? So see, this is, we're not talking about metrics here. We're actually talking about real physical motion of nearby particle in a local, in the proper frame. If you put a particle at the middle, it doesn't move at all because we're looking at tidal relative motions here. And that's the effect that gravitational wave detectors have to try to see. It's a physical thing, okay? So local nearby particles are moving with respect to each other. You want to look at the vortex, at the 10-dex field for it is not very inspiring, but that's how it looks like. And the colors will alternate, as opposed as you change the phase of the wave. Now, if you go back to one of the more standard treatments of the gravitational waves, there's a big deal about this TT, transverse traceless gauge that gives you a special form of the metric. And the idea is, if you look at what the metric is that gives you this tidal effect on this geodesic deviation on nearby particles, you'd look like something like this, okay? Your time, space, time, space, Riemann components would end there in the T0, zero component of the metric because you need them there to take the right derivative and end up with a geodesic equation. So this is a metric that is a proper reference frame because at the origin is just Lorentz. It's just, it's locally flat. And then as you move away from it, there's this tidal, this curvature that manifests itself into a tidal field. So that's very physical, especially if we look only at the geodesic deviation equation and we don't try to do distances here. However, another, that's appropriate for instance, if you're talking about the LIGO, Virgo, and you want to think of the mirrors, at the end mirrors as moving due to forces. So you can say, okay, I'm in this proper frame. That's what I'm going to describe it. Say however you want to describe Lisa. Okay, Lisa is big enough that these expansion, local expansion of the metric is not correct anymore because Lisa is as a size comparable actually to the gravitational waves themselves. So to describe something like that, you probably want to work with a description of, with a metric description that's constant, that's homogeneous across the entire extent of the wave, across space. So what you can show is that you can do a transformation of variables that takes you into a metric that has this form, okay? All that you need to do is to match the curvature locally and that's all your physics and therefore the, the resulting metric expresses the same geometry, the same curvature and the same physics. And this is very nice and very simple. And in particular the, the gravitational wave polarizations components appear only in the space section of the metric. And then what this, this is interesting because if you go in and do the geodesic equation for a particle that's initially a rest in the metric, it doesn't move at all. So if you have two nearby particles that are just going along the, along their way and a wave comes, the coordinates don't change. In this matrix particles don't move. What happens is that the coordinates are moving, okay? And therefore physically the distances are changing in the same way as we saw before being stretched and squeezed, but we're also moving the coordinates with the distances so that the, you can describe your particles as not changing at having constant coordinates effectively. So the implicitly it's the coordinate field here that's being stretched and squeezed together with the particles. And then the way that, what the particle will see is that the distance to the nearby one changes because to make the distance you have to integrate along the metric. You know you have to do Pythagoras theorem with the metric. And so in that description of the particle are not moving but it's the distance between them that changes because the gravitational wave is coming by. So see this description is dual to the one, the other one but in terms of what you can observe physically it's entirely equivalent. You can either have, you can either measure your distances with the Lorentz metric and have things move or you can have them not move and change the distance between them because the metric changes like in cosmology. Any question here? I think all of GR does kind of this schizophrenia almost where Einstein was so proud that the theory was generally covariant you can use any coordinates and so on. In practice when you work, you want to work with it you have to pick the best coordinates to describe your physics and then you really try to stay attached to them and get the physical meaning from them. In some cases you can pick more than one and they give you complimentary descriptions of the same physics but if you do the numbers and if you write what your experiment must come out it must come out the same. So that's the TT frame for you. Let's go on to talk about energy momentum content of gravitational waves. This was as I was saying the origin of much confusion because Einstein was trying to wrote almost immediately an object that he called an energy momentum pseudotensor and it was a pseudotensor because if you applied coordinate transformations to it it would move energy effectively. For something like a wave you could either put the energy where the wave was at the top or where it crossed zero it was quite confusing. And therefore people suspected for a long time that this was only the waves carrying no energy at all because the energy seemed only to come out differently depending on how you did coordinates. The physical somewhat pragmatic resolution to it is to think of gravitational waves as a small short wavelength perturbation on a smooth curvature solution on a smooth background. And then again you can split the field into two pieces a background and a wave like perturbation you can do the same for Riemann. And the important thing is that there's going to be an averaging operation you can think of it like one of these smoothing filter, right? So you're just going to do a local integral a bit like doing Photoshop when you want to blur images. That smoothing operation takes you from the entire tensor back just to the background component of it. So now we split also the Einstein tensor G remember Ricci plus a combination of the metric and of the Ricci curvature into terms that are background terms that turns that's linear in your gravitational wave perturbation field and a turn that's quadratic. And then you can apply Einstein's equation and so the background field doesn't see background field could be cosmology for instance or it could be just I don't know the geometry around the black hole, okay? So it's changing very smoothly with a large radius of curvature so that it's on solution. If you look at this equation at linear order in H you get Ricci equals zero which is what gives you the propagation, the wave propagation equation for Riemann. If you look at the second order, if you include the second order you have that the time space average of terms that are quadratic in the gravitational waves come in together with the background. Since these terms are quadratic they don't cancel out when you do an average, okay? Just like when you go from a cosine to a cosine square you take the integral, it doesn't cancel out. So that means that if you take things up to second order in the perturbation then you have to solve for the background curvature together with that. That means that this perturbation is going to affect the background curvature. If you put it on the other side of the equation, I just put this on the other side with a minus sign you can think of it as an energy momentum tensor as if it were matter in a sense. So you can rewrite your Einstein equation for the background curvature as being sourced by an energy momentum tensor that comes from the piece of the field that's quadratic in the gravitational wave perturbation. So it's, certainly Einstein's equations are non-linear so everything has to work together but if you break up things into orders you can push them on either side and you can say that, again, you have a content of energy in gravitational waves in this perturbation H that's going to curve space-time. That's what energy does, okay? So this is a physical effect, this is a very real thing but you can only do it if you take this average over many wavelengths of the gravitational waves. That's actually, yes, yes, yes. And the way you see it in the equations is that since we're expanding over a curved background we can't quite use normal derivatives. We have to use covariant derivatives with respect to the background so that changes a bit, that gives you a lensing, for instance, you do something called the geometric optics approximation to derive, to show that the wave field moves along geodesics of the background curvature and Redshift is another manifestation of that if you look at the time-space components, the time-time and time-space components. So yes, okay, so now if we want to do this for a wave we just have to plug back Riemann into the second order pieces of Einstein. I don't show them here because they're kind of messy but you end up with a very simple expression for the energy momentum tensor of the wave which is exactly analog to what you have in electromagnetism, in fact. You have derivatives of the potential field squared and sum up. So you get a symmetric expression with respect to the plus and cross polarizations and if you put in, so a wave that's propagating along the z direction would only have components in time z and time and zz and they're all equal to the same. If you put in back in constants of c, g and so on and go dimensional, you see that the content of energy and gravitation wave is actually pretty high. So the flux of a wave of 10 to the minus one strain, something like what we detect with LIGO at the kilohertz, okay? So something that's oscillating kind of fast is on the order of 0.3 watts per meter squared which is something like the luminosity, the flux that you get from the full moon. But as I was saying yesterday, although there's a lot of energy, it doesn't, space time is very stiff. So a lot of energy like that only gives you a strain of 10 to the minus 21 and the fact in the motion of relative particles is only that big and you have to spend a billion dollars to actually see it and detect it. But then you're very happy. Now, let's go to the final hassle which is to look at the generational gravitational waves. So I thought I had, yeah, that's a very good historical books about physics, about just this kind of things that I was telling you. The initial debates about whether gravitational waves are real, the history of the quadruple formula for the generational waves and it's really a pleasant, very pleasant reading. So to, unfortunately, to describe wave generation, I kind of have to go back to talking about the metric. Okay, and it's a little unwieldy to do it in terms of the remount tensor. So just allow it for me. And this is the thing that all the textbooks would usually have in the first page which is if again you're looking at a linearized perturbation of the metric, you can put it into a form that looks like a wave equation if you do some things. In particular, the thing that you want to do is to build what's called the trace-reversed metric perturbation. So you're just subtracting the trace of the, this trace H of the field from itself that kills some terms, makes it simpler. And you want to go in a gauge, use your freedom in doing coordinate transformations to again simplify things and you want to go into a gauge where a divergence of the field is zero. If you do that, you have something that looks like a wave equation. The nice thing is that you know how to solve that, okay? You know how to solve that, for instance, for the electromagnetic potential using a retarded field solution which is just an integral of your energy momentum tensor on the side over the extent of the source divided by this one over the retarded radius, effectively. The distance on the light had to propagate to get to you. So my laser is dying, but if, so that's exactly equivalent and that's exactly analog to the expression for the electromagnetic field. However, that involves for the Hij's, which are the part that I want because they do the tidal field. It involves the stress part of the energy momentum tensor and stress is always something that's hard to think about, at least for me. It's the zero, zero component of energy momentum. Okay, that, I know what it is, the zero J, okay, that's momentum, but IJ, I start to think, okay, pushes, what does it do? So however, you can use just the conservation of the energy momentum tensor, which is here, to kind of relate the stress to just the energy energy, the zero, zero component. You use this magic tool of integration by parts which always works in theoretical physics and it pops out these two coordinates, okay? So you can rewrite this integral as an integral over only the mass density effectively. The energy density multiplied by local coordinates. So this is a mass quadrupole, right? This kind of integral. And then, this is just a definition. I'm going to define this as the mass quadrupole. I get a twice a time derivative to enforce conservation and so my solution for the trace reverse gravitational field is going to be the second derivative of the quadrupole. I've also simplified this time-retarded combination of coordinates into just the radius because I'm assuming that the source is moving very slowly, okay? So the final result, I add this TT combination to it because, do I have that? Yes, there's something very expedient which also has an interesting physical meaning which is we saw that for a wave that propagates along z, let's see, I only care about xx, xy and xy and xyy. I don't care about the other components. And in fact, I only need those components to build the Riemann tidal field. So that means that whatever metric, whatever coordinate gauge I have that's close to Lorentz, I can just get this TT thing with the nice form for the metric just by taking only those components and subtracting out the trace. So it's just a projection operator. So that's what this TT means. So now, what's missing here? So we took, we were generating a gravitational wave using a local distribution of matter, for instance a binary that's moving slowly. However, how is that binary moving? By gravity, right? It needs its own local gravitational field. Why is it not going to go around? It's just going to stay there. So this thing is not internally self-consistent because there's no place for it to describe the actual gravitational content of the field itself. So this description, as I've given here, would be okay if I was moving the masses in some other way. So for instance, there's a in the Caltech astrophysics, what they call the Interaction Room. There's a little contraption that somebody built which is a gravitational wave generator. It's a dumbbell. There's two little masses connected to the rod and there's a motor that makes you rotate. And there's a nice gauge that actually says, you can increase the speed, you can set the speed. And I should have taken a picture of it. And the part that I like best is that it actually says caution, gravitational radiation. So of course you don't get too much out of that. So if you do that, the description is perfect. Because you have other forces that are moving the masses and those forces, there are some stresses, internal stresses, but probably they're very small and this description is fine. But if it's the gravitational field, the gravitational potential that's actually moving things around, maybe you have some trouble. And that's what was the origin of confusion because Einstein's equations had this formula. Actually he had a mistake by a factor of two that Eddington fixed later. But he had that, but it wasn't clear whether that could be applied to something like a binary which is bound by gravitation itself. So that's why people said maybe the gravitation is going to match even things out so that actually you don't have gravitational wave emission. Maybe the retarded field locally and so on just fixes everything and we've got a beautiful, self-consistent solution. The solution to this took a long time. It was only settled really, I think, in the 70s and 80s. Although Landau and Liffschitz already may be in 1950 or 47, in their textbook had a derivation based on a full nonlinear expansion of the Einstein tensor. And they said, OK, it's all fine. It actually works in any case. If you just have to formally, it works. But it wasn't clear to the Americans, at least, and the Brits and the Germans that that was actually true. So one of the ways to resolve this problem is to look at what's called a two-lane scale expansion or sometimes a matched asymptotic expansion. And the idea is the following. Is that if you're far enough, if you have a self-gravitating system, that's going to be your source of gravitational waves. If you're far enough, you're just seeing the waves and the description or propagation of the waves just applies. There's no problem there. If you go close enough, you can go to a region where the metric, where you're far enough from the metric, that the metric is going to look Newtonian. It's going to be just a weak perturbation on Lorentz with the dominant G00 term. That's effectively one, what is it? Minus twice the Newtonian potential. That's also a very accurate description of the metric of spacetime in there. It's going to be time-dependent because these bodies are moving. So what you can do then is you're going to match these two descriptions, these two metrics in this weak field region in the near zone, where both things are accurate, both the description of the wave going out and both the end the description of the Newtonian field. If you do that, that's how it is in symbols. So don't worry too much about it. But the point is that the Newtonian potential can be expanded in multiples. The first multiples is, of course, just the total mass of the system, which gives you your basic Coulomb potential. Then there's a dipole term that you can set to 0 by setting the center of motion of the system just at the origin. And then there's a quadrupole term. The same is true for the wave solution. If you write it as a spherical symmetry, it's also going to have an object that looks like a quadrupole with the right derivatives and so on. So if you match the two, you end up with exactly the same equation that you had originally. And this tells you two things. It tells you that if you have a slowly moving, we need slow motion in this. And weakly gravitating system, then Einstein's formula just applies as it is. You can actually prove to great accuracy that the mass quadrupole tensor that generates waves is just the integral over the mass density that you expect. If the source is strongly gravitating, so say a neutron star or so on, you cannot do that anymore. You cannot reliably take that integral. However, you can still go to this weak gravity zone and you can read off the Newtonian potential there and match that again to the quadrupole that generates the wave. So that becomes like an effective quadrupole potential that sources the waves. This is also what happens when you do post-Newtonian theory. So when you're taking corrections to just Newtonian physics, you want to account for the energy of the field and so on. In effect, you're building a multipolar expansion of the potentials of the metric and you're matching them to the source of the gravitational wave. And that matches, creates your effective mass quadrupole tensor. Questions here since I think I may have brought myself up a little. Five, okay. Okay, so then we have to look at the binary, prototypical source, very simple description. Two masses, different masses separated by some separation vector. We just compute the quadrupole tensor. It's going to have the separation, the reduced mass, right mu, and the phase of the rotation. Take a double derivative, pulls out two factors of the orbital angular velocity. Let's just say this is just rotating at some orbital angular velocity. Plug this into the quadrupole formula. You get the two polarizations. If you put the rotation in the xy field and put yourself at the top looking down, it's actually very simple because you're already in the TT frame. It's already only x, y, and z and no confusion. If you go at an angle, you have to do some transformation of the basic vectors to project to the TT frame and that gives you this cosines of the inclination. And you see using Kepler's formula, you can relate of course the angular velocity to the velocity itself. And so you get a very simple expression for the gravitational wave polarizations which have a velocity squared. And just two phases of the rotation. Okay, so that's just a sinusoid. The entire wave from is proportional to mu to the reuse mass. And there's a one over R tempered by of course constants that gives you the decline of the waves in a spherical fashion going out from the source. Now, gravitational waves carry energy. So we can use the other piece that we did. Okay, so the energy momentum content of the gravitational waves, we can compute the luminosity by integrating that across the sphere, taking the waves across the sphere. And that tells you how much energy is being carried away by the waves. So the expression is actually rather simple for that wave and it involves the power of velocity to the 10th and it involves a constant which is the larmor luminosity which is a huge number. Okay, the instantaneous gravitational luminosity for a binary is actually a huge number. And there's something interesting about that expression. So this eta is this dimensionless mass ratio. So I'm giving it away a bit, but what's interesting here? It has no mass left in there. So this is just a mass ratio and it is a velocity. So this luminosity is only a function of how fast things are moving, but it's not a fast of how heavy they are. So that means among other things that if you have the final inspiratory two black holes, the instantaneous luminosity from that system is going to be the same no matter whether the small black holes or large ones. However, the inspiratory large ones takes longer. So the overall amount of energy that I'm going to radiate is much, much bigger. But instantaneously all binaries have the same luminosity. Then you can do something, you can compute your time to coalescence by just by matching the derivative effectively. But the workhorse for this kind of computation is this, an energy balance equation where we're going to take the expression for energy in the system. There's just the Newtonian energy, okay? So velocity minus potential. And you're going to match the derivative of that to these luminosity of gravitational waves. So you're basically physically, you're describing how fast the energy of the system is changing because you're losing energy at infinity. This is the simplest way to look at this. You could also look at what the instantaneous effect is of gravitational wave emission on the points. And that's much, much more complicated. That would be called the radiation reaction computation. It's also somewhat gauge dependent, but this is the simplest way and that's how people have been doing this thing in the trade. And so from this energy balance equation, you can compute then what the evolution of frequency is and what the evolution of phase is. So that's my final expression for this Newtonian binary that's emitting gravitational waves due to quadruple, the quadruple term only. And again, H plus and H cross, the two polarizations, there's a V squared, which is the instantaneous strength of the radiation. There is a mass here, a reduced mass, and there's a cosine of a phase which is evolving. And this evolution with V to the minus five with V is the instantaneous velocity is the dominant Newtonian term. So in practice, we're going to use this in fact to try to estimate parameters in the LIGO detection on Thursday, but in practice, you want to be a little more precise than that. And so you do what's called the post-Newtonian expansion. So, okay, formula's getting a little more ambitious there, but the point is that the energy and your flux, your luminosity are going to have corrections with respect just to the Newtonian expansion because after all, there's general activity, there are no linear interactions between the masses and so on. So those corrections are written in terms of the velocity squared. And this piece here is what I had back here for energy, just minus one mu V squared. And then the corrections written as an expansion as a series in velocity squared. The first term here would be the first post-Newtonian correction to the energy. Likewise for luminosity, this is the dominant term which is of order V10. Okay, and then you have corrections. And then you can take these things, do the balance equation and plug them back into your phasing and you get this is an expansion for phasing as a function of again velocity. And this is very similar to what's used in actual gravitational wave searches. Yes, I'm sorry, I can't hear you. Is it independent of masses? Whether this is independent of masses. Or all just for the basis here? So these energy and fluxes are normalized. So the flux as we said is independent of mass. This energy, when you write it like that, is divided by mass. So it's energy density or energy divided by mass. So these expressions only depend on mass ratios, these eta's. However, when you put them together to make a waveform, the mass does provide the scale of it. So if you go to 10 times the mass, you're just stretching time effectively by 10 times and you're changing energy also. But it's symmetric in this way, yeah. So the other thing that I didn't show you is that there's a very specific combination of masses that enters in the frequency change as a function of time. That's what's known as the chirp mass. We talk more about it on Thursday. But then this kind of expression is what gives you then these post-Newtonian waveforms, which are very good descriptions of the spiral, as far as the spiral is slow enough that things are evolving slowly. Remember, to define energy momentum, we needed to take averages over several wavelengths. So that means that this energy balance, I can only do it if it's proceeding slowly with respect to the evolution of frequency. When things start to change very fast at the end, you need to do something else either analytically or using numerical activity computers and you'll get a merger, something that's not like a, it's not like a circular binary at all, it's quite complex. There are of course many complications here if you bring in spin, spin is very interesting because it brings its own dynamics and so on. That's the basic, your circular waveform. Okay, so going back to my scheme, we went through all of them, was a bit of a slog and there are indices, there are formulas and so on. But for each of them, if you spend a couple of hours looking at the equation and matching them and so on, I think you can convince yourself that it corresponds to a very basic physical statement. So it's all physics, it's not mathematics and they're very physical things, you can build experiments that measure gravitational waves and you can look at binaries and see that they evolve because of them. So general relativity over a hundred years went from being mathematics to physics. So that's my story for today. Thank you. Thank you.