 OK, thank you, and welcome back. So I do realize that I'm the last lecture standing in between you and your freedom. So I'll try very hard to stay within the constraints here. So a couple of people asked me last time for some references on the material. So I just wanted to mention that Sean Carroll's textbook on general relativity has some of the material presented. But if you're really interested in differential forms and differential geometry, I recommend a review paper by Eguchi, Gilky, and Hansen. And this is titled something like gravity, gauge theories, and differential geometry. All right, so when we stopped last time, we had just introduced this DRGT theory of massive gravity. So let me just refresh you on what we had said. So I should say that the DRGT, this is the Durham, Gabbadazzi, and Toli model of ghost-free massive gravity. This came out in a couple of papers around 2010. And it was presented in a form very different than the form that I presented here. So it was presented in terms of metrics and usual index notation. And in fact, it was derived in a very different way than the process that we went through. So in fact, we haven't even shown that this is a ghost-free theory yet, which is something that I want to do today. But this DRGT theory takes the following form. So we introduced the dynamical metric, g mu nu, with corresponding tetrides, e mu a, e nu b, eta a b, and the tetrad 1 form, e a, which was equal to e mu a, dx mu. And then because we wanted a Lorentz invariant theory, we also introduced a non-dynamical Minkowski reference metric, eta mu nu. And correspondingly, there were delta function tetrads associated with this reference metric. And we called the 1 form 1, which was just equal to delta mu a dx mu, like so. All right, so in terms of these variables, the DRGT theory took the following form. So s was equal to mp squared over 2. We had the usual Einstein-Hilbert term, which is just a wedge product of the curvature two forms, and the dynamical tetrads, and then also the mass term, which we wrote as a sum from n equals 0 to 4. So you have these five coefficients here, the beta n, which are all independent coefficients. So this doesn't affect the ghost-free nature of the theory. And they multiplied these polynomials, mn, which depended on the e, and the matrix 1, like so. So just to refresh you, we had that s0 was equal to all the wedge products of just the e contracted with the epsilon tensor. And this, we said, was just a cosmological constant. So this is just determinant of e. s1 was equal to the wedge product with one non-dynamical guy. And if you go back to index notation, this is just determinant of e times trace of e inverse times 1. s2 was the wedge product with two non-dynamical reference metrics, c, like so. And this depended on the squares of traces. So there was a 1 half dot e, e inverse 1 quantity squared, minus e inverse 1 squared trace. Let me do the other ones over here. So s3 was the wedge product with three non-dynamical guys, so 1b, 1c, 1d. And this one also you could express. Now is 1 6 determinant of e, e inverse 1 trace cubed. Let's see there's a minus 3 trace e inverse 1 times e inverse 1 quantity squared plus 2 e inverse 1 cubed, like so. And the final guy is this s4, which in four dimensions is non-dynamical. It's just a wedge product, 1, 2, 3, 4 epsilon abcd. And in fact, you can write it out like I did before in terms of the dynamical ease and the 1. This is just because of this trick of being able to extract the determinant in front. But in four dimensions, this is still going to be a non-dynamical term, like so. All right, so the next thing that we want to do is we actually want to show why this is a ghost theory of massive gravity. So we want to identify the constraint that eliminates the extra degree of freedom in this theory. Now there are multiple ways to do it. And in fact, you can do it in the Veerbind language. But what I want to do first is I want to show how you can use this theory to get from the Veerbind language to the metric language and then do the ADM analysis in the metric language. So this is just one particular way of showing that there's a constraint. You can also just do it in the Veerbind language as well. But first let's see how we can get back to metrics again. So before I go on, are there any questions about this or anything we did yesterday? Say it again? Oh yeah, I mean M. Yeah, here. Let me call this S. Sorry. Thanks. OK, so let's count degrees of freedom. So step one is going from E to G. So remember we said last time that the tetrad comes with more components than the metric does. So the tetrad is non-symmetric, so it carries six components, whereas the metric carries only 10 components. But we argue that you could use this overall Lorentz invariance in order to eliminate six components from the tetrad. But now what's happened in these DRGT theories is that while the Einstein-Hilbert term still has this overall Lorentz invariance, so I can transform E and I can transform R. And this is just going to give me a determinant of lambda in front. I've now explicitly broken this Lorentz invariance by adding this non-dynamical reference metric here. So under a Lorentz transformation, the E's are going to transform, but the ones, the non-dynamical guys, don't transform simply because they're non-dynamical. And so I've explicitly broken this Lorentz invariance. So now I can't use this trick to get rid of the extra six components of the tetrad. So the question is then, what does remove these extra six components? So just to say, so under a Lorentz transformation, remember, we have EA going to some lambda AB, EV. And similarly, RAB goes to some lambda AC, lambda BD, RCD. But these terms, the ones don't transform. All right, so let's look at this transformation infinitesimally. So we see that infinitesimally, we can write the variation of E as equal to some omega AB, EV, where now omega is an anti-symmetric tensor as usual. So we have omega AB is equal to minus omega BA, like so. So that's what delta E would look like under a usual Lorentz transformation. And now the statement, so sorry, so I should write it out. I'm just saying that lambda AB, to leading order is delta AB plus this omega B and so forth. All right, but we can use the fact now that the Einstein-Hilbert term is still invariant under this general Lorentz transformation to see the implications of this for the remaining terms. So in other words, Lorentz invariance of S Einstein-Hilbert implies the following. So let me look at the variation of this term under this transformation here. And this tells me that variation delta SEH is going to go schematically like delta L, EH, delta E. Let me restore indices, so delta E mu A here. And then times this variation, so delta E mu A. So in other words, this is equal to delta L EH, delta E mu A times this omega AB E mu B, like so. OK, this is just the Einstein-Hilbert piece. This is separately Lorentz invariant, so I know that this should be equal to 0 here. And I should say, here I'm working in second order form, so I'm assuming that the curvature to form depends on E. And so that's why I'm only varying with respect to E here. OK. But this should generically be satisfied for any anti-symmetric tensor, omega here. So what this expression is telling me is that the trace of this expression here of delta L, delta E, times E, times this generic anti-symmetric guy should be equal to 0. And what that tells me is that this guy times this guy has to be a symmetric tensor. So in other words, since omega is generic and anti-symmetric, this is telling me that delta L, delta E mu A, E mu B. And let me be explicit and lower this index so I can talk about anti-symmetry. Sorry, so I can talk about symmetry, so A to B, C. This should be equal to delta L EH, delta E mu C, E mu B, A to B, A. So this imposes a symmetric condition. And I should say that this is an off-shell condition. So this doesn't depend on E satisfying the equations of motion. It's just a statement that even off-shell, the Einstein-Hilbert term, is Lorentz invariant, basically. All right, so this is an identity. It's not removing any degrees of freedom. It's just a statement about the Lorentz invariance of the kinetic term. But now I can use this statement. So let's use this statement and consider the full theory. So now I'm going to write out the full theory schematically as the action S is going to be equal to the integral d4x of the Einstein-Hilbert piece. Plus, and I'm going to include everything else, all the non-derivative actions, as some potential u. So I'm just going to schematically call it u here. And let me look at the equations of motion of this theory. So the equations of motion are going to tell me that the variation of L with respect to E, sorry, L-E-H plus the variation of u with respect to E, this should be equal to 0, right? All right, but now let me take this equation of motion here and let me multiply both pieces by this factor of E times eta. And then let me symmetrize with respect to the indices. So when I do that, the terms involving the Einstein-Hilbert piece are going to vanish, right? Because this holds identically. And what I'm going to be left with is just a condition on the potential. So this is going to tell me that delta u, delta e, ua, e mu b, eta bc on shell is going to be equal to delta u, delta e mu c, e mu b, eta ba, like so. So this is a symmetric constraint on the tetrad. So notice this doesn't involve any derivatives whatsoever. So I can solve this equation. And it's going to put constraints, in particular, in four dimensions on six components of the tetrad. So this is six on shell constraints. So it's this equation here that's responsible for eliminating the extra six components of the tetrad and getting you back to the right number of components that you would expect in the metric formulation. All right, are there any questions about this? All right, so we can see how this plays out in massive gravity. So in fact, what we can do in particular is we can just take these expressions here and we can plug them into this constraint with arbitrary coefficients beta that appears in front. And the remarkable thing that happens, and this is very specific to DRGT massive gravity, is that you're able to solve this equation independent of the coefficients beta that you pick. So in other words, so for DRGT massive gravity, this expression is solved by the following constraint on the tetrad. So E mu A delta nu B eta AB anti-symmetrized with respect to the mu and nu indices should be equal to 0. And like I said, this is interesting because it's a solution independent of what coefficients you put in front here. Now it's possible that there may be other solutions to this equation, other branches of solutions. However, my claim is that this is the one that lets you get back and forth in between the tetrad and metric formulation. And this is also the branch of solutions that's going to give you a ghost-free theory at the end of the day, like so. OK, questions? All right, so how does this help you getting back and forth between the tetrad and the metric here? So we're going to take this condition, and I'm going to switch now to matrix notation. And in fact, with a little bit of manipulation, I can write this guy in the following way. So in terms of matrices eta, E, and 1, this condition here is equivalent to the condition that eta 1 E inverse is equal to eta 1 E inverse transpose, like so. OK, so I just get this expression here, basically by multiplying by two factors of E inverse on both sides of this expression. And now I'm writing this in matrix notation here. So explicitly, if I write this out, that's going to give me E inverse transpose 1 transpose. I'm going to keep the t here just in case I have a more general reference metric in mind, times eta, like so. OK, so let me use this condition, and let me consider the following expression. Let me see what happens when I look at the quantity E inverse times 1 squared, like so. And I'm picking this quantity here, because that's the quantity that shows up without the square and all these different traces that appear here. So that's a quantity of interest. So I write this out. So E inverse times 1, I just mean E inverse 1, E inverse 1, like so. And let me take this expression, and let me insert the Minkowski identity via the Minkowski metric here. So in between these two pieces here, I'm going to insert an E inverse 1, eta inverse eta, and then just one E inverse 1, 1. So all I've done here is just insert a delta function, like so. OK, so same expression. But now I see that I have exactly this form that appears here in my symmetrization condition. All right, so if I take this expression and I plug in this solution here, what I find is that this is equal to E inverse eta inverse. And now here I have E inverse transpose 1 transpose eta 1, like so. All right, but these terms here, this E inverse eta inverse E inverse transpose, this is just the metric inverse, right? Where is this? Sorry, can you guys see this? Let me write it up here. So I have, what did I start with? I started out with E inverse 1 squared, and I'm saying I can manipulate this to get E inverse eta inverse E inverse transpose times 1 transpose eta 1, like so. But this first piece here, this is just equal to the inverse metric, G inverse. Whereas this piece here, this is just equal to the Minkowski metric, eta, like so. So what I find at the end of the day is that this E inverse times delta squared is just equal to G inverse eta. And what that means is that everywhere in my expressions, where I see E inverse 1, I can replace it by basically the square root of this matrix here. So by a square root of G inverse eta, like so. So here I'm defining the square root in the following way. So square root G inverse eta with indices, say some mu nu, is defined such that when I take two of them, so say G inverse eta mu lambda, G inverse eta lambda nu, this is going to be equal to simply G mu lambda eta lambda nu, like so. So that's what I mean by the square root of this matrix here. All right, so the key point is that it was precisely this solution of the symmetrization condition that allows you to make this replacement in the DRGT Lagrangian. So if you have a different solution for this equation, then it's no longer so straightforward to go in between the tetrad and the metric formulations. But because we have this on-shell condition, it means that in the DRGT action, everywhere we see this E inverse 1, we can replace that by the square root of this G inverse eta and start talking about things in the metric language. Does that make sense? OK, very good. So just to be very explicit here, let me rewrite these guys. Yeah, that's right. Yeah, no, that's exactly right. So in fact, it was formulated in terms of the metric fluctuation around Minkowski. And so it was written as these highly nonlinear expressions in the perturbation H, which turned out that they formally resummed to these square roots, basically. But in fact, they wrote it, not even in terms of these G inverse etas, but I think in the original paper, the object that they were using was some k, which was basically like this G inverse eta minus 1, because that really reflects the first order of perturbation of the metric. Yeah, I think the statement is that in fact, it's not a unique solution. And so I think it's analogous to the fact that when you have the square root of a matrix, the square root is not unique. And so there are different possible solutions. But this is a valid branch of solutions. And as you were saying, so if you pick, let's see, which one is it? If you pick just say beta 1 to be non-zero, then I think the answer is that the solution is unique. And so you can kind of consider the other. If you want, you can consider the other interaction terms as sort of small perturbations away from that, in which case you would always choose this solution here. But I think generically, with generic beta coefficients, you could find other solutions as well. All right, so the statement then is that drgT, massive gravity, you can write totally forgetting about tetrads and forgetting forms. And all this, you can write it as s is equal to mp squared over 2 d4x. This just goes into the usual Einstein-Hilbert term. And now, all these mass terms here, so this is going to be d4x sum over n beta n sn, where now sn is going to be a function of g inverse eta, where we're just making these replacements here. So just to be very explicit, so we're saying s0 is now going to look like root dot g. s1 is going to look like root g times trace square root g inverse eta. s2 is going to look like, sorry, this is about the 1 half. s2 is going to look like 1 half root g trace root g inverse eta squared minus g inverse eta, so and so forth for the rest of them. So this is the drgT theory of massive gravity, just in terms of metrics. All right, so now what we want to do is we want to return to the Boulvard Desert ghost problem. And we want to count degrees of freedom in this theory using the same ADM decomposition that we did before. All right, so last time to count degrees of freedom, we introduced the following variables. So we introduced the lapse variable, n, which was related to the g00 component of the metric. We introduced the shift variable, ni, which was related to the g0i component of the metric. And we also introduced this notation gamma ij for the spatial components of the metric as well. And in addition, we identified a momentum pi ij, which was canonically conjugate to the gamma ij. All right, so now we're going to take the same decomposition, and we're going to plug it in to this drgT massive gravity formula, Lagrangian. So what we find is the following. So as, in fact, Boulvard and Desert predicted, the Lagrangian is going to be nonlinear in both the lapse and shift variables. So the action, and I should say the action in canonical variables, is nonlinear in both laps and shift. So as we said before, what this means is that even though the lapse and shift are still non-dynamical, so even though they still appear without time derivatives, their equations of motion are no longer going to be constraints on the dynamical variables. In principle, you can use them to solve for the lapse and shift themselves. So in particular, what you would find, if you now take the Lagrangian say and vary with the lapse, you would find some function c that depends on both the lapse, the shift, the gammas, and the pies. And this would be equal to 0, whereas variation with respect to the shift would give you some ci that also depends on lapse, shift, gamma ij, and pi ij. So in principle, you could use this expression here to solve for the lapse in terms of the other variables. You could use this expression here to solve for the shift in terms of the other variables, and you wouldn't get any constraints on the dynamical variables. So you have four equations, sulfur, n, and i. So there are no constraints in this theory, naively, which means that you have the gamma ij and your pi ij as the potentially propagating phase space degrees of freedom. So remember that there are six gamma ij and six pi ij. So this gives you six pairs of propagating phase space degrees of freedom, whereas we know we only want five for the massive graviton. So 12 phase space. So again, this is naively. 12 phase space degrees of freedom. So it looks like there might be a ghost in this theory. OK, and in fact, this is precisely what was predicted by Boul云 Desert. They said that any nonlinear completion is going to be nonlinear in lapse and shift, and therefore it's going to look like you don't have a constraint in the theory. So yeah, no. So they're not constraints when you have the same number of variables as equations, because then each equation, you can just solve for its own variable. So the difference with gr is that the lapse and shift appear linearly in the Lagrangian, and therefore when you vary with respect to lapse and shift, you find an equation on the dynamical variables. And so you solve the systems of equations without ever having to specify what the lapse and shift are. Other questions? OK, so what's the way around this argument? So the caveat is the following. So suppose you have the following situation. Suppose it's the case that these four equations are in fact degenerate with each other. And what I mean by degenerate is the following. So suppose that when I vary the action with respect to the lapse, I find some c that depends on only three particular combinations of the lapse and shift, which I'm going to call little ni. So this can depend on little ni, gamma ij, and pi ij. And suppose similarly, when I vary with respect to the shift, I find my constraint ci depends only on the same three combinations of the lapse and shift. So little ni, gamma ij, and pi ij, where here I'm taking little ni to be some function that could be highly non-linear in the lapse and shift variables, like so. Sorry, this is equal to ni of lapse shift. And it can also depend on the fields as well, the dynamical fields. OK, so now this is still true in the sense that the constraint does still depend on the lapse and the shift here. But now because it only depends on three particular combinations of lapse and shift, what that means is that first of all, I can solve this expression here for ni only in terms of the dynamical variables, gamma ij, and pi ij. And now if I take that solution and I plug it into this constraint here, this gives me a constraint that depends only on my dynamical variables as well. So even though naively, it looks like you have four equations that depend on four variables, if they only appear in three certain combinations, then one of these equations is still going to be a constraint. And you're going to be able to eliminate one phase space degree of freedom. And so one can show, in fact, that this is precisely what happens in drgt massive gravity. So in particular, you can do a field redefinition of the following form. So you can take your shift and you can instead use a redefined shift, little ni, that's going to depend on the lapse, the original shift, the gammas, and the pies, like so. So now if you re-express capital ni in terms of little ni and plug into drgt, what you find is the following. So after doing this field redefinition, the action becomes linear in the lapse. So that's the first good thing. But the second and equally important thing is that the ni equation of motion is independent of the lapse. So after doing this transformation, I can get rid of capital N from this equation here, which means that indeed I can solve for little ni in terms of these variables. And now the equation of motion for the lapse, since the action is now linear in the lapse variable, becomes a constraint on the gammas and the pies. So variation with respect to n leads to a constraint. So c gamma ij pi ij is equal to 0. And this constraint removes one phase-based degree of freedom from the theory. All right, so this is a great start. So it means that this theory that naively looked like it propagates 12 phase-based degrees of freedom or six pairs of propagating modes. In fact, there's an additional constraint that removes one phase-based degree of freedom. But of course, you don't want a theory that has an odd number of phase-based degrees of freedom, so you need an additional constraint that removes the momentum canonically conjugate to the degree of freedom that this Hamiltonian constraint removes. So after showing that you have such a constraint in the theory, what you want to do is you want to check for the existence of a secondary constraint that's going to remove the other pair of the phase-based degree of freedom. Does that make sense? OK, so what do we mean by the secondary constraint? All right, so a requirement for consistency on this constraint should be that it's preserved in time. So in other words, not only do you want to constraint, but the constraint should be such that if it's equal to 0 one moment in time, it should also be equal to 0 for all subsequent moments in time. This is just a simple consistency condition. So what you want is that for consistency, you should have the d dt of this constraint is equal to 0. OK, but we can write this expression in terms of Poisson brackets with the full Hamiltonian of the system. So this is equivalent to saying that the Poisson bracket of this constraint with the Hamiltonian should vanish. And I'm going to use squiggly brackets to mean that it should vanish on the constraint surface. So in other words, the terms on the right-hand side can be proportional to the constraint itself. And that's still OK, because as long as the constraint holds, this is going to be equal to 0. So this means on the constraint. All right, so h here is the Hamiltonian. Defined in the following way. So h is going to be equal to d cubed y, some h0 of y, which is independent of the lapse variables, and then minus n times this constraint. So minus the lapse times c, like so. So basically, I'm just taking the drgt Lagrangian in terms of these terms. I'm integrating out the shift variable. And so I'm only left with something that depends on the potentially dynamical variables, the pi's and the gammas, and the lapse appears out in front enforcing the constraint here. OK, so what I want to do is I want to take this expression, and I want to plug in here. And I want to see if this consistency condition generates an additional constraint on my dynamical variables. So there are two possibilities that can happen. So the first possibility is that you find that the constraint doesn't commute with itself. So it's possible that you plug in and you find that the Poisson bracket of cx, cy is not equal to 0, like so. So suppose this were the case, and you plug in here. Sorry, you plug in here. And what this would tell you is that you would get an expression, but that expression would depend explicitly on the lapse. And so because that expression depends on the lapse, you can use this equation here to solve for the lapse variable. And it doesn't represent a genuine constraint on the remaining dynamical variables. So if this is true, consistency condition can be solved for the lapse n. And therefore it's not a constraint on the remaining degrees of freedom. All right, but the second possibility is that the Poisson bracket of c with itself is in fact 0. So you could have c of x, cy is equal to 0. And then what's going to happen is that the term when you take the Poisson bracket with respect to cy is going to drop out. And so you're going to get an expression that only depends on c and the variables contained in h. So in other words, it's completely independent of the lapse. And because of it, this consistency condition is going to represent an additional constraint on the dynamical variables. All right, and in fact, in drgt, you can show that this is again precisely what happens. So if you calculate this in drgt and you look at the Poisson bracket of this constraint c that you found, what you find is the following. So c of x, cy gives you something proportional to c itself. So you have c of x derivative with respect to x of delta cubed x minus y minus c of y derivative with respect to y, spatial y, d cubed x minus y. And because this is proportional to the constraint itself, this is going to be 0 on the constraint surface. All right, so you get an additional secondary constraint. So if we now count total number of phase space degrees of freedom, we started out with 6 coming from the gamma. We had 6 coming from the pi's. We have minus 1 coming from the Hamiltonian constraint c. And we have another minus 1 coming from the secondary constraint, dc dt equals 0. So that gives us a total of 10 phase space degrees of freedom. So we propagate five modes in this theory, precisely what we would expect for a theory of massive gravity. All right, so I should say that this field redefinition that you have to do over here is somewhat non-intuitive and non-trivial. So in fact, in the metric language, it's not particularly transparent that this theory is going to have an additional constraint that removes this ghost-like degree of freedom. It's a bit more transparent in the tetrad language for the following reason. So whenever we have these epsilon tensors, so if I look at some generic mass term, say something of this form, d epsilon abcd, what the epsilon tensor is telling me, along with the wedge product, is that I can only pull out 1, 0, 0 component of any one of these matrices at any given time. So in particular, if the 0, 0 component depends on the lapse and none of the other variables do in the tetrad, what this structure is telling me is it's giving me something that automatically is going to be linear in the lapses of the fields. So that's why this primary constraint works so nicely in the tetrad language. But I should say that there's a caveat that comes with this. So everything looks much nicer in terms of the primary constraint in the tetrad language. But it turns out that things are much trickier with the secondary constraints. And so just because you have the existence of primary constraints in this language, we've learned that it's still important to go through and check and make sure that you have the secondary constraints as well. But it's also to say that, so I argued last time that this wedge product structure is somehow related to being able to write down these ghost-free theories. But it's not actually obvious why that would be the case. So in other words, it's not transparent to just say, OK, anytime I write down something with a wedge product, that's going to be ghost-free. And this is why. So I think both in the case of the lug-flug terms and in this case of massive gravity, this is the structure that emerges. But it's not always evident why that should necessarily be the case. Are there any questions about this? So that was the theoretical success of these DRGT massive gravity. So they eliminated this extra degree of freedom at the fully nonlinear level. So the next thing that you want to check is whether or not these theories exhibit a Weinschein mechanism, and therefore whether or not they're going to be phenomenologically viable in terms of evading, say, solar system constraints. So let's look at the Weinschein mechanism. So in order to do this, I'm going to introduce the same trick that we used before. The Stuckelberg mechanism, in order to study the degrees of freedom in this theory, with the only difference being now that because we have a nonlinear theory of massive gravity, we need to introduce Stuckelbergs that restore the nonlinear diffeomorphism invariance rather than the linear diff invariance that we restored before. But it's fairly straightforward to generalize. So before, we introduced the Stuckelberg fields to restore linear diffeomorphism invariance. So delta H mu nu is equal to d mu xi nu, symmetrized. Here, they should restore the full diffeomorphism invariance. So delta now g mu nu should be equal to covariant derivative d mu xi nu, symmetrized. So the reason why diffeomorphism invariance is broken in these theories, remember, is that because we now have this explicit reference metric that doesn't transform. So you can think of all the diff breaking is coming from this fact that we have this eta mu nu stuck in this theory. So the way that we can restore the dif invariance is if we can add some fields to eta mu nu such that it's going to transform like a tensor under diffeomorphism invariance. And you can do this in the following way. So we take eta mu nu and let's replace it with a nu metric, which I'm going to call, say, g bar mu nu, which is going to be equal to d mu, some phi alpha, d nu phi beta, eta alpha beta, like so. So these phi alpha and phi beta, these are my four Stuckelberg fields. And in order to get this g bar mu nu to transform like a tensor, the Stuckelberg fields are in fact going to transform like a scalars under diffeomorphism invariance rather than a vector. So in other words, we're going to have delta phi alpha is going to be equal to psi nu d nu phi alpha when x mu goes to x mu minus chi mu, like so. OK, so because of this, if we do this trick here, then g, we see that g bar mu nu is going to transform like a tensor under diffeomorphism invariance, precisely because you can think of this term as almost being like a coordinate transformation that's multiplying the metric here. So now if I transform to nu coordinates, I'm going to have a d mu nu that looks like a tensor under this transformation. Exactly, it's not a vector because so you can think of this alpha index as really living in a different space than the mu nu index. And so the alpha is just sort of labeling four scalar fields rather than being a true spacetime vector index. Well, for the full theory, if we introduce these fields. But yeah, so this field, because this is now going to transform like a tensor, the whole theory is going to be diffeomorphism invariant. That's right. And again, we're introducing exactly four fields to restore four diffeomorphism variances. So we're not changing the physical content of the theory. In other words, I can always pick a gauge in this theory where these phi's are equal to x's and get back the same physical theory again. So I'm just introducing gauge degrees of freedom. But this is going to help me isolate the modes of my theory. OK, so g bar mu nu now transforms like a tensor. OK, so because of that, we can construct the covariant object, which I'm going to call capital H mu nu, which is going to be equal to g mu nu, the dynamical guy, minus this new covariant guy, g bar mu nu, like so. And this is just to compare. So before we had the metric fluctuation h mu nu, which was not a tensor, which was equal to g mu nu minus little eta mu nu, like so. All right, so now let me take this expression here for the Stuckelberg fields phi. And let me expand around the identity transformation. So in other words, let me write phi alpha as some x alpha plus some perturbation, which I'm going to call pi alpha. Like so, actually, sorry, let me switch notation. I'm going to call this a alpha. So now I can write the h mu nu in terms of these fields, the a fields, and I get something that looks like the following. So h mu nu is going to be equal to g mu nu and then minus. So if I plug in this expression here, I'm going to get a delta plus a d mu a. So there's going to be a minus delta mu alpha plus d mu a alpha, a delta nu beta plus d nu a beta, all times eta alpha beta, which I can expand out to write. So this is going to be g mu nu minus. From here, I'm just going to get an eta mu nu. But now I'm going to get additional contributions, d mu a nu plus d nu a nu. Sorry, let me do this with a minus sign here. So this becomes minus, minus, and these guys are plus. But now I'm also going to have a minus d mu a alpha, d nu a alpha, like so. Whereas this piece in front here, this is just my usual metric perturbation, h mu nu. OK. And finally, so I can start thinking about these guys as being related to the Holicity 1 modes of the massive graviton. To extract the Holicity 0 mode, I'm going to do one more transformation. So I'm going to write a mu as a mu plus d mu of some scalar field pi. And then I'm going to plug back into this expression here. So let me just write out every term. So capital H mu nu is going to be little h mu nu plus all this extra stuff. OK, let me just write it out. So d mu a nu plus d nu a mu plus 2 d mu d nu pi minus d mu a alpha d nu a alpha minus d mu a alpha d nu d alpha pi minus d mu d alpha pi d nu a alpha minus d mu d alpha pi d nu d alpha pi, like so. And notice, so when I introduced the four a alphas, I introduced four gauge invariances associated with diff transformations. When I do this transformation here, I'm introducing an additional gauge symmetry. So delta a mu equals d mu of some lambda, whereas delta pi is going to be equal to minus lambda. So again, I'm introducing a field, but also a gauge symmetry as well. So again, this pi is just an additional gauge degree of freedom in my theory. All right, so now what I'm going to do is everywhere in the DRGT action, where I see a little h mu nu, I'm going to replace it with this capital H mu nu, which is now a covariant object. And so this is going to restore different variants in the DRGT theories. And I'm going to canonically normalize. And then I'm going to look at the mess of terms that I get by doing precisely this. So in DRGT, replace h mu nu with capital H mu nu and canonically normalize. So you're going to end up with, sorry, just the canonical normalization. So I'm going to introduce an h hat that's 1 half MPh. A hat is equal to 1 half little m MPa. And pi hat is equal to 1 half little m squared MPh, my original pi, like so. OK, so you're going to get a ton of terms when you do this. And what you want to do is you want to isolate the terms that are suppressed by the lowest energy scale, because these are going to be the terms that are most important, most relevant in your theory. And these terms take the following form. So you're going to have terms of the following form. So terms that look like h hat times d squared pi hat to the n, all divided by little m squared MP to the n plus 1. And you're also going to get terms that depend on the vector mode on the A of the form dA hat squared d squared pi hat to the n, divided by m squared MP to the n plus 2. So the important thing is the scale that appears suppressing these terms. So this is the scale that's known as lambda 3. So this is equal to little m squared MP, all to the 1 third power. And all other interactions after you do this transformation are going to be suppressed by a higher energy scale than this lambda 3. So this is already non-trivial. In fact, the original way that the DRGT Massive Gravity was derived was looking for interactions that raise the cutoff of the theory. So raise the scale lambda 3. So any non-linear massive theory you write down that isn't a DRGT Massive Gravity The lowest scale that appears in the theory is not going to be this lambda 3. It's going to be a lambda 5 instead, which is equal to little m to the 4 MP, all to the 1 fifth power. So in fact, this was the way that DRGT Massive Gravity was originally discovered, was simply by looking for interactions that would raise the cutoff of the effective theory. So you can phrase this question independent of any ghost problem in the theory, in fact. So what we want to do in order to study the phenomenology of this theory is we're going to focus on these terms that are suppressed by the lowest scale. And we're going to do one. We're going to perform one more step in order to demix the action. So we want to get the terms as close as possible in the kinetic terms of these fields as close as possible into their canonical form. So let's do one more trick, which is we're going to demix. So first step is focus on terms suppressed by lowest scale. And then we're going to demix by performing the following transformation. So we're going to take our h hat mu nu. And we're going to write it. This is similar to how we demixed in the linear theory as h hat mu nu plus now there's going to be an eta mu nu pi hat. But now we're going to have an additional term that we didn't have in the linear theory, which is going to be a 2 alpha 3, where alpha 3 is just some coefficient, divided by lambda 3 cubed d mu pi hat d nu pi hat, like so. All right, so if we focus on these terms, the Lagrangian takes the following form. So we have L is going to be equal to 1 half h mu nu. So the usual linearized Einstein-Hilbert term, mu nu alpha beta. You're still going to get a remaining term that mixes the h with the scalar modes. And this takes the following form, so minus 8. Another coefficient, which I'm going to call alpha 4, lambda 3 6, h hat mu nu x 3. Let's put these up, mu nu mu nu, which depends on the pi, hats, and derivatives. And then the following term, so there's going to be 6. Let me call this L2 minus 12 alpha 3 over lambda 3 cubed L3 plus 8 alpha 3 squared minus 4 alpha 4, lambda 3 to the 6, L4 plus a factor of 80, alpha 3, alpha 4, divided by lambda 3 to the 9, L5. And then the terms that couple to the stress energy tensor. So we have the 1 over mp, h mu nu hat, t mu nu. But now because we've done this demixing, we also have these terms that couple to the stress tensor as well. So there's going to be a 1 over mp, pi hat times trace of t plus a 2 alpha 3 divided by lambda 3 cubed mp d mu pi hat, d nu pi hat, t mu nu. And the point is that these L2, L3, L4, and L5 are simply the Galilean terms that we're often familiar with in modified gravity. So L2, L3, L4, and L5, these in fact are Galileans for the pi field. So let me just write the first couple of these out for people who maybe aren't so familiar with Galilean theories. Yeah, so I'm ignoring the vector interactions because they're not going to play into the phenomenology here. Yeah, that's OK. All right, so L2 is given by minus 1 half d pi hat squared. L3 is equal to minus 1 half d pi hat squared trace of capital pi, where pi mu nu is equal to d nu pi hat. L4 is equal to minus 1 half d pi hat squared times trace pi squared minus pi squared trace. And L5 is equal to minus 1 half d pi hat squared trace pi cubed minus 3 pi pi squared plus 2 pi cubed trace, like so. All right, so these are the Galilean Lagrangians. And I should say with the exception of this X3 that appears here. So you still have some mixing. In addition to the Galilean terms, you still have this mixing between the spin two component and the holicity zero component. OK, and just to refresh you, so the key valuable features of these Galilean fields is that even though the Lagrangians are higher order in derivatives, the equations of motion are purely second order. And so they don't propagate any extra degrees of freedom. And on top of this, they enjoy this Galilean symmetry, which sends the field pi hat to pi hat with a shift, see some constant shift, plus this Galilean type transformation b mu x mu, like so. All right, and these terms have appeared in other modifications of gravity, in particular in DGP models and other higher dimensional models of modified gravity. But there are interesting theories of modified gravity in their own right as well. And so they were generalized based on these two criteria here. And I should say that this property here, the fact that they're appearing in this limit of massive gravity, is very closely related to the fact that this theory of massive gravity is ghost free. So finding higher order equations of motion for this holicity zero mode in this limit would be naively an indication that your theory had a ghost in it. So this is very closely tied to the ghost free nature of this theory here. OK, but we did all this just so we could study the Weinstein mechanism in this theory. So let's go ahead and look at that. All right, are there any questions before I go on? So in particular, to discuss the Weinstein mechanism, in fact, I'm just going to discuss how the Weinstein mechanism works in one of these Galilean type theories. And I'm going to ignore for the time being the presence of this mixed coupling between the two fields. So if I focus on, say, the cubic Galilean, and I couple this theory again to some point source, so I take T mu nu to be some mass m times some delta functions, appropriate delta functions, what I find is the following. So the equations of motion for the pi field can be integrated, and they give me the following form. So you find that pi hat prime divided by r plus, and now I have additional non-linear contributions suppressed by the scale lambda 3. So plus 1 over lambda 3 cubed, pi prime over r, quantity squared. This is going to be equal to m over mp, 1 over 4 pi r cubed. So this is the mass of the source that I've now introduced. All right, so we see that when I'm far away from the source, so when r is large compared to this inverse scale lambda 3, it's going to be this term that dominates here. And I'm going to find the usual Newtonian form for the potential. So at large distances, we still have that pi hat is going to go like m over mpr. But what this means is that the non-linear terms that appear now in this Galilean theory, so you can think of comparing, say, this term to this term, or I should do it over here, comparing this term to this term here, these are going to be suppressed with regards to the linear terms. So the non-linear terms are suppressed. And they're suppressed by this factor that goes like d squared pi hat divided by lambda 3 cubed. So if I plug in these values here, I see that this goes like m divided by mp lambda 3 cubed times r, like so. All right, so the non-linearities are going to start to become important when this scale here is of order 1. And so we can use this expression here to calculate the Weinstein radius of this theory. So non-linearities are important when this is of order 1, which leads me to define a Weinstein radius of the following form. So rv is going to go like m divided by mp all times 1 over lambda cubed to the 1 third power, or sorry, 1 over lambda 3 cubed, like so. All right, and I can plug in for lambda 3. And this is going to be equal to, I can write this just as g over m divided by m squared, all to the 1 third power. Oh, just because, so at large distances, this term is going to be suppressed by this factor lambda 3. So I can just solve this equation and find the usual Newtonian potential. Yeah, so you also want to look at the solution for h00 as well. So what you'd find when you plug in here is just focusing on the h00 piece. You would also find that this recovers the Newtonian potential. But now what we're worried about is the presence of this fifth force, which is going to be coming from this scalar degree of freedom. And so in fact, we're just focusing on what the additional force is on top of the usual h00 force. So yeah, in fact, I should say the fact that we get a Newtonian type potential for this pi field at large distances means that we do have a discrepancy from the usual predictions of dr at large distances, because this is an additional force now on top of what we're saying, thanks. Are there other questions? All right, so you can plug in values for this radius here. And for a solar mass, this gives you something like 10 to the 15 kilometers. So this is still quite large. All right, so this is the story at large distances. So now we want to see what's happening at distances inside the Weinstein radius. So at short distances, we can solve this expression here, the equation of motion. So basically, we assume that the field pi takes on a power law type form. And if you plug in and you assume that it's this term that's dominating the potential, rather than this term here, what you find is the following law. So you have that pi of r is going to go like lambda 3 cubed rv squared times r over rv, all to the 1 half for r much less than rv. And this is to be contrasted. So I can rewrite this expression there with lambda 3 cubed rv squared rv over r for r much greater than rv. All right, so all this is saying is that because it's short distances, the potential now is going to go like r to the 1 half. This is going to be greatly suppressed compared to a usual Newtonian potential inside the Weinstein radius. So in other words, our fifth force is suppressed for r less than rv, which means that the predictions of gr should be restored. All right, so this is just a very sketchy outline of how this Weinstein mechanism works. But you could calculate in more detail. And these results will hold more generally with the same Weinstein scale, now suppressed by the cutoff lambda 3 rather than the lambda 5. All right, does this make sense? OK, so this is the second success of this theory. So first, we got rid of this extra degree of freedom of this Boulvard Desert ghost mode. And now we're saying that because of the strong coupling of this Helicity 0 mode, the non-linearities act to suppress this fifth force. And therefore, at short distances, we should see no contradiction with the predictions of gr as well. OK, so I just want to make a few more comments about other aspects of these massive gravity theories, just to give you some sense of the current status of the field and the outlook as well. So so far, we've been talking about massive gravity purely within the context of a classical theory. But massive gravity, just like gr, is in fact a non-renormalizable theory. So you have these higher dimensional operators that tell you that you need some UV completion of this theory. Nevertheless, it's completely consistent, just like in gr, to treat this theory as a valid quantum effective field theory below some cutoff scale lambda 3. So in this theory, because our higher dimensional operators, the higher derivative operators are suppressed by this scale lambda 3, that's the naive cutoff of the quantum effective theory. And what's important is that this cutoff, the lambda 3, is at a parametrically larger scale than this Weinstein radius that we found. And so what that means is that there's a regime of validity where we can trust our theory where classical nonlinearities are important, and yet quantum corrections have yet to be important. So this looks something like this. So if we have a following regime where we trust the classical theory as opposed to the quantum theory, this is given by this lambda 3 scale. Whereas we just showed that there's a regime of validity where we trust the linear theory versus the nonlinear theory. And that this scale is given by this Weinstein radius that we just arrived here. So this is this Rv. Sorry, this should be R. Q goes as 1 over lambda 3. And here we should have some Rv, which we said goes as m over mp to the 1 third all over 1 over lambda 3, like so. So as we said, this scale here, this is of order 10 to the 15 kilometers. Whereas this scale here, it turns out to be around the order of 10 to the 3 kilometers, like so. So there's a large region in which we can use the nonlinear theory before we have to worry about quantum corrections here. And just to compare, so this is analogous with what's going on with general relativity. So in GR, we have that quantum effects become important at the Planck scale. So this is given by Rv goes as 1 over mp. Whereas we know that nonlinearities become important at the Schwarzschild radius. So here we have nonlinear versus linear. And this is given by RS goes as 2g over m, like so. So admittedly, these cutoffs here, the scales are much lower than that of GR. But the important thing is that this hierarchy between these two scales, in fact, it can be shown that this remains stable under quantum corrections. So you'll always have this separation of scales in this theory where you can calculate. That's right. It's a very low scale. That's right. That's right. So OK, very good. So yeah, the scale is quite large here. The reason why I wrote naive is because in fact, this is the cutoff that you would expect for in the free space and empty space. You can show that because of this Weinstein mechanism, in fact, effectively around massive sources, you can raise the scale of the effective theory to something of order. I think in the presence of the Earth, the scale becomes of order a centimeter. But it's sort of like a dressed cutoff for the theory in the presence of massive sources. So in fact, you can trust this theory classically at scales that are shorter than 10 to the 3 kilometers. Sorry? That's right. Well, so the, I mean, yeah, quantum corrections are going to appear at a scale that's parametrically lower than the Planck scale. The question is, is it at an observable scale? And the answer is no. Well, so again, around a massive source, you're going to effectively change the cutoff of your theory. So this is why I said naive cutoff here. OK, so that's a brief summary just in terms of the EFT picture. Another open issue that I should mention in this theory is that of superluminality. All right, so we showed that in this special limit, when we introduced these Stuckelberg fields and we look at the most relevant terms, what we find, in fact, is that the holicity zero mode of the massive graviton behaves much like a Galilean. And it's well known that these Galilean theories exhibit superluminal propagation around fairly generic sources, around spherically symmetric sources. So the question then becomes, is this superluminality an issue for massive gravity? So there are several sort of nuances to this question. So first of all, you can ask, well, just because the Galilean exhibits superluminality is the same thing even going to be true of the full massive gravity theory. So do solutions in massive gravity exhibit superluminality? A secondary question that you would want to ask is what kind of superluminality? So we can talk about superluminal phase velocity, group velocity, front velocity. Not all of these are necessarily pathological. And so the question is, what kind of superluminality? And is it pathological? An important question, too, when you're addressing these issues, is the superluminality, is it well defined in the regime of validity of your effective field theory? So often what people do in order to determine the existence of superluminality is they might set up some sort of shock front and try and determine the velocity at which that shock front propagates. But if your shock front requires, say, an infinitely large density, that would be an object that exists outside this range of validity of the effective theory. And so it might not be reliable in telling you whether or not there's actually superluminality in your theory or not. All right, another question you would want to ask is, OK, suppose you can show that there's superluminality and that it's in the regime of validity of the effective field theory, you could ask, does it lead to a causality? So in other words, there's a presence of superluminality necessarily mean that you can form closed time-like curves and thus have some sort of pathology in the theory. And the final question that this raises as well is, does the presence of superluminality in these massive gravity theories, does it mean that there's no Lorentz invariant UV completion for these theories? So there's a nice paper. I'll just give the reference. HEPTH 0602178, which indicates that certain kinds of superluminality in a low-energy effective field theory means that these theories can't have standard Lorentz invariant UV completions. So the fact that you get these Galileans in the decoupling limit and the Galileans look superluminal means that this is an open question now for massive gravity. So these are all sort of open areas of research in the field. No, so that's still a Lorentz invariant theory. So in fact, Lorentz invariant is not enough to guarantee that you're always going to have subluminal or luminal propagation. But in fact, I think in the particular case of Galileans, the easiest way to see superluminality is around a non-Lorentz invariant background. But I think even around Lorentz invariant backgrounds, it's still possible to have superluminal propagation. All right, and let me just briefly, since this is a workshop on cosmology, I've mentioned the cosmology of these theories. So people have been looking at cosmological solutions in massive gravity and perturbations around these solutions. So I'm not an expert on these fields. And in fact, there's probably people in the audience who know more about it than I do. But let me just say that the main issue is that you can show that there are no flat FLRW solutions with this Minkowski reference metric. So this is one of the main results. There are several ways around this issue. So cosmology does not rule out these theories. So one possibility is that you can just have large scale inhomogeneities. So in other words, because deviations from GR are coming in at scales of order, the inverse mass of the graviton, which we take to be cosmological scales, what this means is that even though you don't have homogenous solutions, the inhomogeneities are going to be at order of a Hubble scale, which is not so much a problem for observation. So if you're OK with that, then these could be phenomenologically viable. Other possibilities is that instead of flat solutions, you can look at open solutions, although I believe that these have problems with instabilities. You can instead of picking a flat reference metric, you can also pick a decider or FRW reference metric, although they're apparently run into problems with the Helicity 0 mode of your theory violating the Higuchi bound. So this is something that Sebastian mentioned in his talk yesterday, which is a possibility when you have a massive graviton propagating on a decider background. And then the final possibility is that you can look at extensions of massive gravity. So people have been looking at cosmology, also in bi-gravity theories or quasi-diliton theories, and seeing if these can evade some of the cosmological problems. So I just want to give you a reference for this. So this is all work in progress, and there's a nice review. So in Claudia's review of massive gravity, she gives a nice summary of these issues. So once again, this is the following reference for these things. OK, and so I was hoping to talk a bit more about extensions of these gravity theories, so bi-gravity and multi-gravity. But I see that I'm out of time. But feel free to please come in and talk to me afterwards if you're interested. So thank you.