 Okay, we move to quite different topics and Rachel Rosen will speak about modified gravity the theory part Okay, so thank you very much and thanks to the organizers for organizing this this terrific school so far So yeah, so I'm gonna be discussing various theoretical aspects of modified gravity I'm gonna focus on just giving some some general background What's the motivation outstanding issues and then talk about recent developments and in particular? I'm gonna focus on recent developments in massive gravity and in its closely related cousins So Galileans by gravity and multi-gravity I wanted to refer you to these two excellent reviews by Kurt Hinterbickler and Claudia de Rom That talk about a lot of the outstanding issues and and recent developments in the field All right, so why modify gravity? Why should we be talking about modified gravity? So most frequently people talk about modified gravity in the context of trying to find a solution To the cosmological constant problem and so here I mean both the old and new problems So why is the universe accelerating and why is the acceleration so small? So our understanding of cosmic acceleration comes basically from gravitational from indirect observations of gravitational effects and so it makes sense that we would ask Maybe the solution to these problems lies in modifying gravity at very large distances So that's that's sort of the frequent motivation is can modified gravity tell us anything about cosmic acceleration But something sort of interesting happens when you when you try to modify gravity Which is that it turns out that even from a theoretical standpoint it turns out to be remarkably Difficult to write down consistent modifications of general relativity So I think in fact what's happened is that before we can even really discuss this question in great detail We have to sort of address a much more basic if no less compelling question Which is are there consistent and well-motivated modifications of general relativity and the usefulness in answering this question Beyond answering Anything in particular about cosmic acceleration is twofold. I mean first of all it turns out that these lead us to two very fundamental questions In field theory, so we're really asking about what are the the fundamental constituents and forces that can go into a Gravitational theory and the second more practical reason why why this is even an interesting question Is just simply that if we're testing general relativity already Be a precision cosmology on these very large distances. We really should know what the alternative theories are So what are the other possibilities besides gr that we could be observing? Okay, so I'm gonna focus I'm gonna be interested in long-distance modifications of general relativity Like I said, I'm gonna be focusing on massive gravity and its close cousins Galileans by gravity Many other theories of modified gravity exist in the literature My main reason for for restricting to these is that they're conceptually. They're very simple theories of modified gravity and yet even though they're simple they exhibit many of the usual theoretical Obstacles that one encounters when trying to modify gravity and also much of the rich phenomenology that you can get from these theories So here I'm gonna take What's often referred to as sort of a particle physics approach to modifying gravity? Which is I'm really gonna use the tools of a Lorentz invariant field theory And and in fact the fact that we can even do this for general relativity is somewhat non-trivial So it's it's sort of fun to see that Right it was in 1915 right that Einstein formulated general relativity and formulate it In terms of by identifying the gravitational field with a metric tensor of Riemannian geometry It wasn't until 1939 That Fierce and Pauli First tried to formulate this theory in terms of a Lorentz invariant field theory So that's a very long separation in time and what's somewhat remarkable So so Fierce and Pauli went about it trying to write down just the theory of a free massless spin-2 particle They then looked at general relativity. They took the linearized limit So they looked at the theory at lowest order in the fields They showed that this in fact is coincided with the theory of a Lorentz invariant massless spin-2 particle And still there was some skepticism that these two theories had anything to do with each other So we take it for granted nowadays That GR of course is you know, and we'll talk about this the unique Interacting theory of a Lorentz invariant spin-2 field at low energies But but this fact was was originally I want to emphasize somewhat somewhat non-trivial all right, so Today so I should mention so I said I'm going to use the tools of Lorentz invariant field theory There's also a whole host of modified gravity theories that are Lorentz violating and in fact, it's it's easy to to Sort of evade I shouldn't say easy It's more easy to evade some of the usual obstacles if you're willing to give up Lorentz invariants But for the purpose of this talk, I'm gonna I'm gonna focus on Lorentz invariant modifications of general relativity All right, so the idea is the following so I'm gonna take this this particle physics approach which means that I'm gonna pretend that we're we're field theorists who've just discovered the force of gravity So we've just discovered that that two masses can attract each other And we want to write down a theory that can describe this new force that we've discovered So how would we go about doing this? So we should start with the simplest possible theory and that would be this theory of a scalar field, right? So can we write down a theory of a spin-zero particle that can accurately describe the force of gravity? Well, we want to describe a long-range force So that compels us to look at a massless scalar field So we could write down a theory of some massless scalar field phi like so All right And because we want to couple because we want to describe a force of gravity the scalar field should somehow be coupled to the stress energy tensor of matter But because phi carries no indices it can only couple to the trace of the stress tensor So I have to add some coupling like phi t to my Lagrangian, and let me also introduce a coupling constant g like so All right, so is this a viable theory of gravity? Well, what I can do is I can calculate the potential between two static objects mediated by the scalar field here and when I do that what I find is the following so I find the potential v of r Which goes like minus g squared over 4 pi m1 m2 Divided by r. So in fact, I do recover the correct Newtonian potential in this theory I said I was interested in a long-distance modification But I could also think of having a sorry a long-distance theory of gravity So a theory of gravity that mediates a long-range force But I can also add a small mass to this theory as well and see how that changes the potential That I want to describe so going from a theory of a mass spin-zero particle To a massive spin-zero particle the effect that this is going to have on the potential is I'm going to find a yukawa suppression At large distances like so. So if this mass is small enough It's still possible that this theory of gravity could could could Be consistent with our with our observations Okay, but the problem with this theory is that I've only coupled phi to the trace of the stress tensor here And we know that relativistic matter has a traceless stress tensor So in particular for electromagnetism trace of t is equal to zero and what this means is that the scalar field Doesn't interact then with relativistic matter So what this is going to tell us is that this theory you're going to see no bending of light in a gravitational field So it's only non relativistic matter For which I see this interaction So just based on this we can rule out the possibility that the theory of gravity that we see today is Mediated by a singular it's a single scalar field like so Yes Yeah, so if you want to stick to a linear theory where you just have say phi interacting with t You would have to couple say like a d mu d nu T mu nu and then because the stress tensor is conserved this would give me zero by integration by parts Yeah, so that that would be higher order So it's it's possible that you can write down these terms. This would be the leading order effect here in a linear theory Yeah, this would this is an inconsistent. This is inconsistent. I think you're still not going to get the correct Bending of light. I mean, I think you would really have to tune it in order to Because it because that's going to be a much Suppressed compared to this coupling which you would see for matter and what you really want is the ratio of the two to be of order unity okay So spin zero gravity fails on observational grounds If we're taking this this particle physics approach to gravity, we're saying, okay We should be we classify particles according to their spin so spin zero half one etc We want to use Bosons to mediate long-range forces the next particle We should try is we should try and write down a Spin one theory of gravity Okay, but spin one so this means writing down some Lagrangian That depends on some field a mu and say its derivatives d mu a nu like so But for spin one gravity we can really use our intuition from electromagnetism We know automatically this theory would tell us that positive masses would repel each other and because of this We can right away rule out a spin one theory of gravity All right skipping spin two for the moment What about higher spins so can I get have a theory of gravity made up of spin three or Greater Well, we know that we have very powerful no-go theorems Largely due to Weinberg that forbid interacting theories of high spin particles So in fact, we see that if we want to describe gravity, we're really forced To look at spin two fields and within spin two we have these two possibilities We can consider either a massless spin to or a massive spin to all right So since we're trying to mediate a long-range force, let's start with the simplest case and let's consider Just the theory of a free massless spin to particle. All right, so this was the action that was written down in 1939 By fearsome Pauly and for a massless particle it takes the following form So we have s is equal to The following and we can couple to matter in the following way introduce a 1 over mp h mu nu T mu nu like so so here h mu nu you can think of the metric fluctuation So if I take g mu nu and I expand around some flat background So g mu nu equals a to mu nu and here I've canonically normalized so that this is a to mu nu plus h mu nu over mp This is in simply the Einstein-Hilbert action at lowest order in h like so All right now h mu nu is a rank two symmetric tensor Which means that it carries ten components with it on the other hand We know that a massive spin to field should only propagate two physical degrees of freedom right to polarization So plus two holicity and minus two holicity as well. All right, so what gets rid of these extra components? Well the linearized theory Of a massless spin to particle enjoys a linearized if you morphism invariant So this action is invariant under delta h mu nu equal to partial derivative d mu psi nu plus d nu Ximu like so and it's precisely this gauge invariance that's responsible for removing the extra components of h mu nu And giving you the two propagating degrees of freedom of the massless graviton so via diff Invariance we get two degrees of freedom All right We can ask the same thing that we asked over here and we can ask what's the phenomenology then of this theory of Basically a free massless spin to field coupled to t mu nu in this way so in order to do this Let's introduce a point source So let's couple our theory to some t mu nu that's equal to m Delta zero mu delta zero nu delta cubed x like so And I'm going to work in the Lorenz gauge Also known as the Dondar or harmonic gauge Which means that I'm going to set d mu h mu nu minus one-half d nu h equal to zero All right, so in this gauge I can solve for the components of h mu nu and what I find is the following So in spherical coordinates I get h zero zero is equal to minus m over 8 pi m playing r and I get h i j is equal to m over 8 pi m p r So h zero zero we can identify with the Newtonian potential After dividing by the appropriate factors of 2 minus 2 so we get Sorry in this language. It's just 2 So we find the Newtonian potential phi of r is just going to be equal to minus g m divided by r so here g is Just the usual Newton constant and I'm using units such that g is 1 over 8 pi m blank squared All right, so again, we see that in this theory we recover the correct Newtonian potential for a static point mass All right, so this is an initial success From h I did I j we can also calculate the potential psi of r Which is going to be equal also to minus g m over r and from these two Potentials we can calculate the angle of deflection of light in a gravitational field and namely the angle alpha is Going to be related to Just do proportional to 2 times 1 plus gamma Divided by the impact parameter B where this gamma here is the ratio of the two potential psi of r Divided by phi r like so so for this theory here. We simply have the gamma is equal to 1 So that alpha carries a factor of 4 over B And in fact you need a gm over here as well. All right, so in this theory of linear Gravity we do in fact recover the correct bending of light that we observe so phenomenologically This does in fact seem to be a viable theory of gravity All right, so it succeeds where the spin zero case fails All right, but this isn't actually the end of the story and the problem is the following So the problem is that you can't consistently couple Linearized gravity to a dynamical stress tensor. So here we took t mu nu to be some point mass Right proportional to some m times some deltas and we completely ignored the dynamics of t mu nu here But if you have a dynamical t mu nu then in fact that theory over there is Inconsistent and the reason for this is the following so we have this linearized diff invariance And the in order for this term here to be diffeomorphism invariant it means that t mu nu Has to be exactly conserved like so this is what this theory tells us But remember that t mu nu Refers to just the stress tensor of the matter sector of our theory It doesn't contain any gravitational energy in it at all And so what this expression is telling you is it's telling you that matter fields can't exchange energy with the gravitational field All right, which we know is inconsistent So it's inconsistent with the equations of motion So if you calculate the equations of motion of your matter sector you're going to find that they don't satisfy this condition And in addition conceptually it's inconsistent in a sense that it would tell you that say a point point particle in a gravitational field Would not accelerate all right, so we know that this can't be correct gr couple to a dynamical source is Inconsistent all right, so the way around this problem is that you want to modify This stress tensor t mu nu so that it includes the gravitational energy as well as The energy of your matter sector because then these two sectors will be able to exchange energy with each other All right, so how do we fix this? We want to take our t mu nu and We want to replace it with t mu nu plus Some theta mu nu here Where now this theta mu nu is going to correspond to the stress tensor Associated with this part all right, so we know how to calculate that that's simply going to be Essentially it's going to contain terms that are like variation of d l 2 d d mu h alpha beta d nu h Alpha beta like so Now these terms are quadratic in the fields they're quadratic in derivatives and they're quadratic in the field h as well Which means that in order to have this term showing up in my Einstein's equation So the equations of motion here are basically we have linearized Einstein tensor g mu nu Goes like t mu nu and now we want to add this theta term on the right-hand side of the equation So this term is quadratic on the field in order to have that show up in my equations motion It means that I have to take my action s S 2 and I have to add to it terms that are cubic now in the field in order to generate this term here so this piece Is quadratic in the fields it came from s 2 But now in order to generate the term in the equations of motion It means that I have to add some s 3 to my action as well All right, but you can start to see what the problem is so if I now have some terms here that are cubic in H So I'm taking this action and I'm adding terms that go Say is some d squared H cubed something like that these are going to generate additional terms in my stress tensor as well So it's going to generate terms That go of order H cubed Like so and so once again I'm going to have to add terms higher order in my action in order to generate these guys and of course this goes on And it goes on infinitely and what you find when you do this procedure To all orders the action that you end up with is uniquely up to boundary terms the action of general relativity All right, so this procedure leads you To general relativity I should say those so so this was successfully carried out by desert And this is in 1970 and in fact the way that desert did it is using first-order formalism And also using the inverse metric Rather than the metric he was able actually to carry out this procedure in a single step So in fact in first-order forum and using inverse metrics, there's only one term That you have to add to the action. So it's a much simpler way of dividing this but conceptually it's the same principle here That's what's going on Yes, so you want to add this term because you don't want the stress tensor of matter By itself to be conserved you want the stress tensor of matter plus the gravitational stress tensor to be conserved because that's telling you then That your matter feels and your your gravitational field can exchange energy with each other by itself Yeah, it's just a theory of a free massless spin to field. Yeah, okay, so this procedure Leads you uniquely to general relativity So this this leads us to the the claim the moral of the story Is that general relativity in fact is the only consistent Poincare invariant low energy theory of a massless spin to field And I should say that this program was initiated and carried out in many ways by other people besides desert So in particular I can refer you to the five papers of Feynman Gupta Weinberg and also a paper by Buller and desert Okay, so to go back to our original question of trying to write down an appropriate field theory that describes gravity We ruled out spin zero and spin one on phenomenological grounds We could rule out higher spin based on theoretical consistency for a massless spin to particle We now have a unique consistent theory. So the only piece that's missing now is massive gravity So what about a massive spin to field can this possibly describe the gravitational field that we see in our universe? so so you want to Know so that the problem with higher spin is that you want to couple to a conserved tensor So for I should say it's for mostly from it's the no-go is for massless high spin particles So for massless high spin particles, you need a gauge invariance, which means that when they couple to your matter sector They have to couple to some conserved current and there are simply no high spin conserved currents that we can write down So our starting point for the massive spin to is going to be again the theory the linearized theory of Fierce and Pauli here for the massless spin to But now we're going to add to it a non-derivative interaction. So mass terms So to this theory we can add terms of the following form to describe a massive spin to We can add terms that look like Say an M1 squared H mu nu H mu nu or we can add terms that look like say some M2 Squared H squared where H is the trace of H mu nu here So these are the two terms that we can write down that are quadratic in the fields that contain no derivatives And these are going to act as mass terms for our field All right But the thing about writing down these mass terms is we said that in order to get the right number of degrees of freedom For the massless graviton. We needed to have this linearized if you morphism invariance. So this d mu x i nu d nu X i mu by writing down these terms here We're explicitly breaking this diffeomorphism invariance and because of that We're going to have extra propagating degrees of freedom in the theory now by itself This isn't actually a problem because we know in fact that a massive graviton should propagate more holicities than the massless graviton So a massive graviton contains not only the plus minus two holicities But also the plus minus one and a zero holicity as well So in fact a massive graviton should have 2s plus one Or five degrees of freedom All right, but how many degrees of freedom does this theory actually carry? So now the equations of motion are going to look like the following so the equations of motion for this theory I have the linearized Einstein tensor d1 mu nu and now I'm also going to have pieces coming from the mass terms as well So I'm going to have an m1 squared H mu nu plus an m2 squared Eight a mu nu times h Like so And let's ignore the coupling to matter for a moment All right, so these are my equations of motion now if I take the divergence of this I know because this because the original massless theory does have this gauge invariance That's telling me that d mu d mu nu By itself is identically zero right and as a consequence I have a constraint equation for the h mu nu so m1 squared d mu h mu nu plus m2 squared d nu h Should be equal to zero so these are on-shell constraints on the components of h mu nu So my index runs over zero one two three So in fact these are four equations so they represent four constraints on the ten components of h mu nu So I have ten components minus four constraints and these are going to give me six Propagating degrees of freedom Okay, but this is one too many right we wanted a theory that only described the five Policities of the massive graviton so we have an extra component now that's floating around in our theory So what can we do? So in the theory of Fierce and Paulie They pick particular values for this m1 squared and m2 squared here, so in particular they had that M2 squared is equal to minus m1 squared which is equal to M squared so this term here takes the following form so you get minus One half h mu nu h mu nu to h squared like so Okay, so why does this help the issue? Well, let's go back to the equation of motion here and let me take the trace of this expression So taking the trace I get that g1 Mu mu is going to be equal to so now here. This is going to look like minus m squared minus 8m u to h like so So when I take the trace of this expression, I'm going to get a minus M squared this is going to give me an h and this is going to give me a d a minus d as well So in fact, I can write this as plus m squared d minus 1 h Is equal to zero But g mu mu. I know what the trace of this is so when we calculate this is just simply equal to d mu d nu h mu nu Sorry minus box h like so On top of which we also have our original constraint equation So this is one set of equations. We also have the original constraint equation, which is now telling us that m squared d mu h mu nu minus d nu h is equal to zero So if I take a derivative of this expression This is m squared d mu d nu h mu nu minus box h is equal to zero But this is precisely the linearized Einstein tensor, right? So that's telling me that this piece here on this constraint. I can set this equal to zero So using this constraint I see that in fact I have another constraint in my theory, which is telling me that m squared h is equal to zero as well All right, and that's precisely because of this particular tuning of the mass terms relative to each other All right, so now I have a theory where I started out with the ten components I still have these four constraints the four on-shell constraints that I had before But now in addition I have this trace-free constraint as well. So I have ten minus four minus one Or five degrees of freedom in this theory, which is what I want So this is the this is the fierce Pauli theory of linearized massive gravity And it propagates the right number of degrees of freedom All right, we're gonna do the same thing we did before and we're gonna ask about the the phenomenology of this theory I'm basically the simple problem of what happens when I couple this to a point source So this is the only relation in the masses that gives you an additional constraint. Okay, so again, let's consider this point source So T mu nu equals n delta 0 mu delta 0 nu d cubed x like so And once again, we can calculate the components of H given this point source So we find the following we find h zero zero is equal to minus four-thirds m over eight pi M-plank R e to the minus mr, which gives us the Newtonian potential phi of r Which is equal to minus four-thirds GM over R e to the minus mr And we also have Hij is equal to two-thirds now m over eight pi Mpr e to the minus mr So that psi of r Is equal to minus two-thirds GM Over R e to the minus mr All right, so the yukawa suppression is something that we expect, right? That's what we expect to happen when we add a mass to a theory exactly what happened in the case of the spin zero particle here So once again, we find that at distances that are large compared to the inverse mass of the graviton We're gonna see this suppression the surprising thing is These factors of four-thirds and these factors of two-thirds Compared to the case of the massless spin two particle So these are extra numerical factors and what they mean is that in the M goes to zero limit of this theory I don't in fact recover the same solutions as I had for the massless spin two So M Goes to zero limit Doesn't recover Now you could say is this really such a problem right because I have Newton's constant out here in front And I could absorb this factor of four-thirds into Newton's constant and not see anything different in the M goes to zero limit But once again the answer lies in what happens if you calculate The the deflection of light around a massive source. So remember we had that That alpha goes like two times one plus gamma Over the impact parameter V Where gamma was the ratio of psi over Phi So now it's going to give me a one half rather than a one right and that's Independent of whether or not I've absorbed this factor of four-thirds into G or not So now this is going to give me a two times one plus one half or A two times three-fourths versus two times two that we had so these two theories differ from each other by a factor of three-fourths All right two times three-fourths. So this goes like three halves versus four For the case of the mass less spin-to-field. So this is telling us That even for a very small mass We would be wrong when we observe the bending of light by a magnitude of order one Basically, all right. So this is a manifestation of what goes by the name of the V dvz discontinuity So this is Van Damme Veltman and Zakharov In 1970 or 72 and the reason for it is actually not that hard to understand. So remember that In order to describe the massive graviton we had to introduce extra degrees of freedom into the theory, right? So the point is that even when I take the this massless limit the massless limit doesn't get rid of the extra degrees of Freedom it just sends their mass equal to zero and so I even in the massless limit I end up with a theory that has more degrees of freedom than the original theory of a massless spin-to And we'll see that in fact it's the holicity zero mode of the massless of the massive graviton that's sticking around And giving rise to this discrepancy here But I should say that that So this this discontinuity is important both on phenomenological reasons Because it would seem to rule out massive gravity is a viable theory of gravity But also theoretically it's a little bit unsettling right because it means that in nature You would be able to tell the difference between a graviton whose mass was truly zero and a mass who's on a graviton Whose mass was arbitrarily close to zero as well All right, so how do we see where this is coming from? So there's a nice way to look at the origin of this discontinuity And that's by using what's known as the Stuckelberg trick or the Stuckelberg method which has the following idea So remember that this this mass term here Broke the different variants the linear different variants of the massless theory So what we can do is we can introduce extra fields into our theory precisely as to restore The different variants the linearized different variants But because we're introducing fields basically in one-to-one correspondence with the symmetries that we're introducing We're not actually introducing any additional degrees of freedom into the theory So this theory has the exact same physical content as the theory of massive gravity with broken different variants It's just a way to sort of keep track of the additional degrees of freedom of the massive graviton So hopefully this will become more clear, but the idea is That we take our original h mu nu and We replace it with an h mu nu and I'm going to canonically normalize so I'm going to have a plus 1 over m d mu a nu Plus d mu a nu Sorry, do you knew a mu plus 2? Over m squared D mu d nu phi like so So this was a original field that represented a massive graviton now We'll see that this field is going to correspond to the holicity plus minus two modes This field will give us the holicity plus minus one and this field will reflect the holicity zero mode of the massive graviton And the reason that what I can do this So I have these new fields a mu and Phi so it would seem like I'm introducing additional degrees of freedom into my theory But in fact this new theory where I replace h mu nu with this combination of fields enjoys these two gauge transformations, so My theory is now invariant under the following transformation. So under now delta h mu nu Is equal to d mu psi nu Plus d nu psi mu While simultaneously now transforming the a field so delta a mu is going to be transformed by minus m X i mu when I perform this transformation on h And in fact besides this sort of restoration of linearized different variants I'm also going to introduce a u1 symmetry. So by writing the fields in this way I also have an invariance delta a mu Is equal to d mu lambda As long as I perform a simultaneous transformation of phi So that delta phi is equal to minus m Okay, so because I've introduced the same number of gauge invariances as I have fields What this means in principle in basically is that I can pick a unitary gauge In which I set these new fields equal to zero and I recover The original theory of massive gravity. So these are really pure gauge degrees of freedom That I've introduced here. So they're not changing the physical content of the theory in any way All right, so what is our new theory look like after doing this transformation? So this is going to help us isolate the origin of this V dvz discontinuity So you can think of this as being like a holicity decomposition of the spin-2 fields So it's going to help us identify In a particular limit the holicity plus minus two plus minus one and holicity zero modes of the massive Graviton, so we're decomposing essentially decomposing The field into its holicities So in the absence of sources the trace of H is zero on shell. That's right. Yeah, this is so this is this is the on-shell Constraint, sorry, which H are you referring to? Yes That's right In the massive gravity theory on shell Okay. Oh, yeah, so I have the equation the motion g1 mu nu is equal to some t mu nu and this contains h mu nu is in it No, that's not necessarily equal to zero. It depends what gauge you're in Okay in vacuum. Yeah, okay. Yes In a particular gauge for example. Yeah. Yes That's right. This is a now an on-shell constraint equation No, I mean the field h mu nu is the same It's just here this corresponds to a massless graviton and here it's a massive graviton That's right That's right. Yeah, but this is so so in in massive gravity. We can't even talk about gauges, right? So it's it's not a statement of whether you can go to a gauge in which h is equal to zero This is just always true now on shell Yeah, in fact you can't choose a gauge in this sense, but this is just always equal to zero now Yeah, other questions Okay, very good. So we're doing this this field redefinition here And we get the following form for our Lagrangian. So I'm just plugging this in To that action down there and we find that L is going to be equal to The M equals zero L and in fact all the new terms that we're going to get are Only going to be coming From the mass term here as well And that's because the kinetic term was already invariant under this linearized diff And you can think of these extra fields here as being completely absorbable by a diff transformation So we shouldn't get any new terms that depend on the fields coming from the kinetic part of the theory It's only from the mass terms that we're going to get fields The a mu and the phi that show up in our new Lagrangian So the moon the new Lagrangian M equals zero Sorry, the new Lagrangian is going to be L of M equals zero plus these extra pieces. So now we're going to have minus one half M squared h mu nu h mu nu where now this is our new h mu nu minus h squared We're going to have a kinetic term for the spin one field for the a mu So there's going to be an f mu nu f mu nu minus 2m h mu nu d mu a nu so this term mixes The holistic two components with the holistic one component or I should say the h with the a's d mu a mu Minus twice now we have a term that mixes h mu nu With the phi so d mu phi D nu phi minus h squared D squared phi And again if we assume that are that we're coupling to a stress tensor that's conserved We're going to get no contribution from these new fields coupled to the stress tensor because we can just undo those with a gauge transformation So we're just going to get the usual one over mp h mu nu T mu nu like so Okay, so this is our theory after after performing this field redefinition. All right So now we what we're our motivation is to sort of understand the origin Of this vdvz discontinuity. So let's see what happens now if we take this theory Where we've introduced these these pure gauge fields and let's take the m goes to zero limit of this theory and what we find is the following so This term is going to drop out completely as well as the mass term as Well and so what we're left with is L Is equal to the original Lagrangian at zero mass We have the kinetic term For a mu but now a mu is decoupled from every other field, right? So this stands by itself and in addition we have minus twice the term that makes it as h mu nu and d mu d nu phi minus h d squared phi like so and Plus the master or sorry plus the the interaction term like here Okay, so now we can really see in this massless limit that this is a theory that propagates five degrees of freedom, right? So we have a Lagrangian for a massless spin-to field Which is going to propagate two degrees of freedom In addition, we're adding to it a Lagrangian of a massless now spin one field So this is going to propagate an additional two degrees of freedom corresponding to holicity plus minus one And this field is decoupled from all the other fields So it doesn't interact with the holicity to The holicity zero or with the stress tensor as well But in addition we have this scalar mode an extra one degree of freedom So these are really the five degrees of freedom of our original massive graviton now in this massless limit Thank you. I mean Yeah, it's it's Yeah, you're seeing Sorry, so so what's happening here is in fact these aren't these other degrees of freedom are not high-energy Degrees of freedom so what what you're thinking of is is say the goldstone goes on decoupling from the radial mode, right? Which is a heavy degree of freedom in the theory here These are really massless degrees of freedom that survive in this limit the the new a mu So those those aren't the photon so there's there are new massless spin one particle Spin one. Yeah, it's not a photon. That's right. Ah, no So this so again, I'm just putting in some conserved stress tensor by hand That's meant to represent some coupling to a conserved source in this theory So it acts as a spin one, but it doesn't couple to this T mu nu So it's not acting like an additional force an additional gravitational force That's right. That's right. Okay, so the the final step what we want to do here Is we want to put this in a in a standard form By demixing the holicity two with the holicity one here So we have this this linear mixing between the two and let's let's demix them So in order to do that we can perform the following transformation. So we can now write h mu nu as H mu nu plus five Times some a to mu nu. So this is just a field redefinition. I'm not doing anything funny here And plug it into this expression and what you find is the following. So under this replacement L goes to L of M equals zero now with this new h mu nu Once again, the spin one part is untouched so f mu nu f mu nu The mixing term becomes a pure kinetic term for the spin zero part. So I haven't canonically normalized So this is minus three d mu phi squared But now because I've done this transformation I'm gonna pick up an interaction between the holicity zero mode and the trace of the stress tensor here So now in addition to having a one over mp H mu nu T mu nu. I'm also going to have a one over mp Phi t right So now I can really see what's going on in this theory I have a free massless spin to a free massless spin one a free massless spin zero But the spin zero couples to the trace of the stress tensor of this theory in this massless limit So it's this coupling here that we can see is the origin of This vdvz discontinuity So even in a massless limit you have an additional field that's interacting with your sources and giving rise to an additional force Yeah in the presence of matter So yeah, so it's going to be constrained equation, but it's going to depend on on trace of T mu nu Yeah, I mean in the vacuum. It's the same theory Okay, okay very good. So that's that's exactly where we're heading So the it turns out that a possible resolution to this vdvz discontinuity Could be that that higher order terms become important And can can screen this this fifth force at short distances, but that's that's exactly what we're heading. Yeah, okay so The origin of this vdvz discontinuity Sorry, the the presence of this vdvz discontinuity would seem to rule out Massive gravity as a as a viable gravitational theory However as was just brought up. This is just a linearized theory of massive gravity So as we saw before for the case of massless gravity Nonlinearities were in fact needed in order to restore the theoretical consistency of the theory now here It's not a matter of theoretical consistency so much as phenomenological viability, but at the same time It's worthwhile asking what happens if we introduce nonlinearities into this theory can this help can that help us out at all? With this vdvz discontinuity So that motivates the search For nonlinear massive gravity all right so our starting point in a theory of nonlinear massive gravity is going to be the following So instead of writing down The linearized theory of a spin to plus this fierce Pauli mass term We're going to start by writing down the Einstein Hilbert term Which is fully nonlinear and then adding to it the linearized fierce Pauli mass term and just exploring and seeing what exactly this theory gives us So our action Now is going to be Integral d4x Usual Einstein Hilbert term and now we're going to add to it Written in terms still of the metric perturbation h mu nu minus 1 4 m squared 8 a mu alpha 8 a new beta h mu nu h alpha beta minus h mu alpha h nu beta So here I'm being very explicit That the indices of the fluctuation h are always going to be raised and lowered with the Minkowski reference metric in this theory whereas This term here of course is expressed fully nonlinearly in terms of g mu If we want of course we can always expand this term in terms of 8 a mu nu and h mu nu Over M. Plink in which term in which case this is just going to give us a series of terms that contain two derivatives and ever-increasing powers of h Right, so this is just going to give us some d squared h squared plus d squared h cubed and So forth so I'm sort of mixing notations here with the g and the h It's also worthwhile pointing out that the reason that we were writing it in this way If you want to write down a nonlinear mass term for the graviton You're always going to have to introduce an additional reference metric into your theory in other words There's no way of introducing in the nonlinear theory a non derivative interaction That only uses g mu nu and no other tensor and the reason for this is the only things that you can construct out of g mu nu by itself are just trace of g Which is equal to a number or determinant of g Which is just a cosmological constant So in order to write down a non derivative interaction for g mu nu We're in fact forced to have this additional metric structure here Okay, and just to say so this is the usual of course Einstein Hilbert term that now instead of the linearized different variance enjoys the full different variance Delta g mu nu is Equal to now d mu Psi nu Symmetrized okay, so let's take this theory and now we're going to do once again the exact same thing we did before We're going to introduce a static point source And see what we get So in fact before doing This exercise for the massive case Let's just remind ourselves what happens if we run the same argument now for the nonlinear theory of massless gravity So if I just look at the Einstein Hilbert term and I introduce a static point source and once again I Go to the Lorenz gauge So the solution that I find can be expanded in the following way So this is for massless gr So we find that h zero zero is equal to minus two GM over R but now Written in terms of an expansion one minus GM over R plus terms that are higher order in GM over R and Similarly h ij is equal to plus GM over R one plus three GM over four R plus additional terms Delta ij like so Okay, and of course we know that that in general relativity we can Complete the sum of these terms and we just get the usual short child solution The reason why I'm writing it in this way is we see that at lowest order We recover of course the Newtonian potentials But we can also see the scale at which non-linearities become important in this theory Right, so this is telling us that in the presence of some source of mass M The non-linearities are going to be important when my distance to this source is of order the short shield radius, right? of Rg Which is of course 2gm and for scales. We know that this is roughly of course three kilometers When M is of order a solar mass like so Okay, this is a usual story for non-linear massless gr Now if we do the massive case instead, so now we're going to include these terms and again solve for little h What we find is the following so we have h zero zero is equal to We find the same four-thirds factor of course that we did before so minus four-thirds 2gm Over r and now we have one minus one sixth GM divided by little m to the fourth r to the fifth plus higher order terms Whereas for h rr We now have minus four-thirds 2gm over r One over m squared r squared One minus I have a 14 here GM Over m to the fourth r to the fifth Plus higher order terms. Okay, so what's happened here is That this scale has changed so the scale at which non-linearities become important now when you have a theory of massive gravity It's much lower than in the case for the massless spin to and in particular We can read off that non-linearities become important when now we have R Is of order what's known as the Weinstein radius Which in this case is going to be equal to? GM divided by little m to the fourth All to the one fifth power like so Okay, so a few things to notice here. So we're always interested in a very not always We're generally interested in a very light massive graviton We usually take that mass to be of order the Hubble scale because we're interested in cosmological effects So if we just take these expressions here and now again We plug in a solar mass for our source and the Hubble scale for the graviton mass what you find for this Weinstein radius is 10 to the 19 Kilometers so huge compared to what we expect for non-linearities in gr So this is well outside the solar system, of course Which means that if we really want to talk about say the vdvz discontinuity And whether or not this is an obstruction for the phenomenological viability of the theory then we need to know the effect of these Non-linearities inside our solar system first All right, so this this mechanism was first described by Weinstein. It goes by the name of the Weinstein mechanism And the idea is the following first notice that because The Weinstein radius scales with inverse powers of the graviton mass that when you take the little m goes to zero limit So when you take the massless limit of this theory the Weinstein radius goes to infinity, right? So as little m goes to zero RV Goes to infinity so what this means is that if you have a non-linear theory of massive gravity And you take the massless limit in fact there becomes no regime in which you can trust the linearized Theory so all the statements that we were making before Become invalid in the m goes to zero limit of a non-linear theory of gravity So then it becomes the case That if the non-linear interactions are such that this extra force gets screened within the Weinstein radius Then the predictions of massive gravity in fact can be reconciled with what we actually observe in the solar system So the idea would be that Suppose we have some source and and a radius outside that source up to RV this Weinstein radius and in this regime Non-linearities are going to be important But it's possible that we could introduce non-linearities in such a way that the predictions of dr are restored at short distances Whereas outside the Weinstein radius I you would be able to use the linearized theory of Massive gravity and there the predictions would deviate from dr But they might not be obstructions to the phenomenological viability of the theory All right, so this goes by the name of the Weinstein mechanism Are there questions about this? Yeah, that's right. Yeah. Yeah, you would need to know the full non-linear theory and solve What part of it? Yeah, no no so that's the idea is that it's it's it's just saying that if you had such a theory then you would be able to To evade these so then the questions that people want to solve is can you write down a theory that does this so this is the goal Okay So this is the aim so the question now becomes can you write down a non-linear theory of massive gravity that does in fact Have this property so that general relativity is restored at short distance, but before we even get there again There's potentially a problem So remember for the linear theory of massive gravity We had to pick the mass term such that we generated this additional constraint on the equations of motion And it was this constraint that got rid of the extra degree of freedom So this was necessary to having a healthy theory that propagated the right number of degrees of freedom Now we're talking about adding Non-linear interactions to this theory and we should ask what happens to the constraint in this case So if I just take these two terms and add them together Do I still have an additional constraint in this theory? All right, so this was a problem that was investigated by Bulwar and desert In 1972 and so there's a nice way to study degrees of freedom in these non-linear theories in particular to count degrees of freedom In the non-linear theories so in order to do that. I'm gonna adopt this EDM framework for for gravity so this is Bulwar and desert and in the EDM framework I'm gonna perform the following decomposition. So I'm gonna introduce a function known as the laps and Which is related to the G zero zero component of the metric I'm gonna introduce three functions known as the shift and I Which are related to the G zero I? components of a metric And finally, I'm just gonna call the spatial components of the metric gij. I'm gonna label them as gamma ij So this is just a decomposition of the usual metric. So the laps Contains one component the shift Contains three components the gamma is a symmetric three by three matrix. So it contains six components So this is just another way of rewriting all ten components of the metric as well All right, but let's take a look at what happens when I just look at general relativity Within the context of this decomposition here So first we're gonna count degrees of freedom in GR So if I take usual massless general relativity, and I perform this decomposition what I find is the following So my action s I can write in terms of canonical variables as mp squared over 2 integral d4 x Pi ij gamma dot ij Minus n times some function c which depends on gamma and pi minus ni ci times some function that depends on gamma and pi so now here the pi ij are the momenta Canonically conjugate to the gammas The laps in the shift appear linearly without time derivatives and these functions c and ci Depend only on the gammas, and they're canonically conjugate momentum pi All right, so in other words If I were to write down the Hamiltonian of this system, I would see that it's just equal to nc Plus n i now this form is useful in the following regards So the key point is first that the laps in shift appear without time derivative So they appear non dynamically they have no canonically conjugate momenta. They don't represent propagating degrees of freedom So n and i are non dynamical Whereas the gamma and the pi represent six times two potentially propagating degrees of freedom Okay, but the laps in shift ended on I on top of being non dynamical They also appear linearly in the Lagrangian right and what that means is that they act as Lagrange multipliers So their equations of motion are gonna enforce constraints on the remaining Dynamical degrees of freedom. So if I vary this action with respect to n and n i I'm gonna find the constraint equations c which depends only on gamma and pi is equal to zero and ci Which depends only on gamma and pi is equal to zero like so So the laps and the shift in general relativity Enforce for constraints on the dynamical fields. So if I count phase-based degrees of freedom I start out with six times two I have 12 degrees of freedom. I Have four constraints Enforced by the laps and shift and in addition these constraints generate the diffeomorphism in variance of the theory So I can use the diffe invariance of the theory To remove an additional four degrees of freedom So now when I add this up, I see that I'm left with four Which is equal to two times two phase-based degrees of freedom for this theory So this is the right number of degrees of freedom for a propagating massless spin-2 particle All right, so this is the usual story in general relativity. So now we can see what happens If we take the same action here, sorry the same decomposition here and we plug it into our theory of massive gr So where we just had the Einstein-Hilbert term plus this Fierce-Powley mass term So now after doing this decomposition The action is going to have the following form so s is equal to mp squared over 2 d4x We get the same thing from the kinetic piece Gamma ij dot But now from the mass term we get the following additional pieces. So there's going to be a minus m squared over 4 Delta ik Delta jl like so Okay, so now we observe the following things so From the mass term, we're getting additional pieces That depend on the lapse and the shift right so we have this n squared piece and we have this ni and jp so The end the lapse and the shift still appear non dynamically But now they also appear non-linearly Which means that their equations of motion are no longer going to be constraints on the additional variables if I vary with respect to lapse and shift I'm going to find equations which depend on lapse and shift and I can use those equations to solve for the values of lapse and shift As a function of the other variables But they're no longer going to be constraints and in addition By adding this mass term we've now explicitly broken the non-linear diffeomorphism invariance of the massive gravity theory so diff Invariance is broken So now when we go to count degrees of freedom in this theory we find that we still have the six times two potentially propagating degrees of freedom from the the pi and The gamma But now we have no constraints and we have no different variants So in fact at the end of the day, we're just left with 12 propagating phase-based degrees of freedom Or six normal propagating modes So five of these are going to be the holistic components of the massive graviton But now we have this extra mode that we had tried to get rid of in the linear theory and we see it's back again For the fully non-linear theory All right in addition to this so in addition to just having this extra degree of freedom What bull are in desert argued is that the Hamiltonian of this theory is in fact unbounded from below So not only is this an extra mode, but it's a it's a pathological mode And it always comes with a wrong sign kinetic term and that's it represents truly an instability in the theory an inconsistency in the theory All right, so it's possible. So we're starting. This is analysis Of the very specific theory right where we started with general relativity We added this Fierce-Powley mass term to it, which was quadratic in the fields. You could ask, okay So this is the analysis for this theory. What if I add higher order terms in H to the theory? Is it possible that I can kill this this ghost mode through these non-linear? Editions, so what about now we start with Einstein Hilbert we add these terms That were crudgetic in the fields so h mu nu Squared h squared and now what if we start adding terms non-derivative terms They go of order h cubed So what bull are in desert argued in these papers is that these terms generically Are going to suffer from the same problem as just the massive gr theory that we were studying before So they argue that these non-linear terms Can't remove this extra degree of freedom But it turns out that there are loopholes in these initial arguments and these loopholes have been successfully exploited In the last few years to write down consistent ghost-free theories of massive gravity So I should have said this extra degree of freedom because of this paper by bull are desert goes by the name of the bull are desert ghost But at this point in our story In 1972 I could say we were left with the questions of Is there a non-linear theory of massive gravity that exhibits the Weinstein mechanism? Is it free of the Bower desert ghost? So these are the questions that I'll answer next time