While solving Cauchy problem for quasi-linear equations, one of the steps was to pass characteristic curves through points of the datum curve. For that, what we do is we solve a system of characteristic ODE with initial condition so that the point initial condition point lies on the datum curve. Now when it comes to the general non-linear equations, if you want to do the same thing, we did not have a complete system for the characteristic ODE because it involved p and q which were unknown. Therefore, we extended the system to a system of characteristic strip, system of differential equations for a characteristic strip. Now we would like to solve this but now the problem is that characteristic strip is not known on the datum curve. On the datum curve only the values or the initial conditions for x, y, z will be known. Therefore, we need to derive initial strip that is what we are going to do in this lecture. We start with a recap of what we have done so far in the characteristic method for general non-linear equations and then we find the initial strip and then we take the third step which is to define a candidate solution and first steps in the step 3. In the next lecture we are going to conclude the entire all the four steps. So this is to just make you recall the notation ql stands for quasi-linear equations AUX plus BUI equal to C, GE we call it general non-linear equations sometimes people call fully non-linear equations it is f of x, y, u, x, y equal to 0. Now in the solution of the Cauchy problem for general non-linear equations where are we let us look at that. Recall the key steps involving in the solution of Cauchy problem that we proposed for general non-linear equations. Step 1 is obtaining a system of ordinary differential equations for characteristic strip which we have done. Step 2 is finding an initial strip we are going to do today and step 3 defining a candidate solution and establishing that the candidate solution is indeed a solution to Cauchy problem will be taken up in the next lecture. So step 1 was successfully implemented so far. Let us recall the difficulties which are new to GE when compared to ql and whatever the ideas that helped us to overcome them. Difficulties and the resolution in step 1 our analysis was motivated by the quasi-linear equations. Quasi-linear equation ql gave us a characteristic direction. Characteristic direction gave rise to characteristic curves, characteristic curves made up an integral surface. The difficulty 1 for general non-linear equation is the equation does not give away a characteristic direction. A useful idea was we observed that for quasi-linear equations the characteristic direction is the envelope of possible tangent planes. Actually envelope of possible tangent planes is a straight line whose direction is a characteristic direction. So we found that the same idea works for GE as well. A characteristic direction at a point p xyz is given by fp fq pfp plus qfq fpfq denote partial derivatives of f with respect to the variable p and q respectively. And this small pq satisfy f of xyz pq equal to 0 capital P actually is standing for xyz. Now what is the difficulty number 2? Characteristic OD system is incomplete for GE because we were looking at finding a curve having characteristic direction which was given by this dx by dt dv by dt dz by dt equal to fp fq pfp plus qfq. We were looking at solutions to this whose image will be a curve. And with this condition x0 equal to x0 y0 equal to y0 z0 equal to z0. Then this will be a characteristic curve passing through the point x0 y0 z0. Unfortunately in this system pq are involved and pq also depend on the location where you are on the characteristic curve that you are trying to find. So p and q are dependent on xt yt zt. That is why Kara OD that is this system is called Kara OD. It is not soluble. So we need to complete this system that is what we did. We derived a system of OD is for the characteristic strip which is a system of 5 equations. Now we need to find a solution to this. What is the next step? As I told you at the beginning of this lecture is to find solutions to this so that x0 y0 z0 is a point on the datum curve let us say fs gshs. But what will be the initial conditions for p and q? Those are not there. So we need to derive that. That is what is called finding an initial strip. Initial curve that is a datum curve is known fs gshs as s varies in the interval i we are going throughout the curve gamma. Now we need to find ps and qs and ps and qs are not arbitrary functions. They should be such that ps qs minus 1 should be the normal to sorry to a possible tangent plane that we want to find. So it is something tied with the equation the ps qs. For this step we are going to assume that the initial data is c2 fgh are all c2 functions. This is only a temporary requirement. We will see a comment later on saying that okay this is not needed at some point. As I told you to derive the equations we can assume anything that we want. But having got the equations we have to try to show that things work with the minimal assumptions. Assuming fgh is c1 is somewhat reasonable c2 is too much. It is not that much reasonable. But this we are going to do only to derive this an initial strip okay. We will see that. We will also see why or where are we going to need this c2 s. So finding an initial strip. So step 2 in solving Cauchy problem is to find an initial value problem for Kara strip ODE. The datum curve gives rise to the initial conditions for xyz. We need to find initial conditions for p and q. In other words we need to extend the datum curve to an initial strip. The support of the strip will be the datum curve. So finding an initial strip how do we do that. So let zeta t be xt by tz t ptqt be a solution to Kara strip ODE. So it follows from Kara strip ODE that along a characteristic strip d by dt of this is 0. How does it follow? You do the chain rule. So it will be fx into x dash t plus fy into y dash t, fz into z dash t, fp into p dash t plus fq into q dash t. And if you use these equations it turns out that you end up with 0. Therefore d by dt of f of xt by tz t ptqt is 0. So that means this function t going to f of zeta t is a constant function because its derivative is 0 it has to be constant. If it is constant it will be equal to f of zeta of t0 for any t0 but I will choose t equal to 0 because t equal to 0 is where I am going to stay on the datum curve at least x0, y0, z0 will be on the datum curve. Therefore we are going to require that f of zeta 0 is 0 that is what we are going to ask. So if we choose an initial strip zeta 0 which is x0, y0, z0, p0, q0 such that f of that is 0 it means f of zeta t will be 0 for all t in j. So recall that the Cauchy data is given by x equal to fs, y equal to gs, z equal to hs. We are interested in passing a characteristic curve through every point of gamma. Since karastrip ODE is a coupled system involving p and q also apart from x, y, z which only matters to find a characteristic curve to find a characteristic curve you need xt by tz t but the equations involve p and q. So if we first have to determine an initial strip having the datum curve as its support. In other words at every point of the datum curve if this is gamma any point we need to associate a vector I am saying vector because two numbers are there and this should be having this finally this property that this is ps the point is s this is fs gs hs. The strip is fs gs hs ps qs the support is fs gs hs. Since the integral surface must contain a part of gamma on it we should have hs equal to u of fs gs it should hold for s belonging to a sub interval of i. Now once you have that differentiate this equation with respect to s you get h dash s equal to ux into f dash plus uy into g dash by chain rule. Since ps equal to p is supposed to be ux right q is supposed to be uy that should also hold so we demand this ps equal to this should happen qx equal to this should happen therefore what does this imply. The equation that we obtained in the previous slide h prime equal to ux f prime uy g prime now ux fs gs must be p and uy must be q therefore we need to solve for ps and qs in terms of s coming from this equation capital F of fs gs hs comma p q equal to 0 and pf dash s plus q g dash s equal to h dash s. So phi b is this equation and this is the equation of the partial differential equation. The role p and q are actually for ux and uy so that is why we demand these two conditions. So now if you notice this is a function of fs gs hs are given therefore f dash is known g dash is known h dash is known. So what it involves is p and q and a function of s something s. So you can think of this as a function of s pq equal to 0 another function of s pq equal to 0 and you want to solve pq in terms of s. So implicit function theorem says that if we know a special solution for some s equal to s not p not q not which satisfies this as well as this that means capital F of f of s not comma g of s not comma h of s not comma p not q not equal to 0 and p not f dash s not plus q not g dash s not equal to h dash of s not if such p not q not are given and a certain Jacobian condition is satisfied non-zeroness of certain Jacobian we will do the details on the next slides. Then there will exist functions you can express pq in terms of s essentially that is what we want. So there exist functions s going to ps s going to qx of course the conclusions are local therefore these functions will be defined on an interval containing s0 where this Jacobian condition is satisfied and where a particular solution p not q not has been found such that these functions are differentiable and they are solutions to the above system. Moreover they are unique with respect to the above properties implicit function theorem when applicable it gives a unique solution. So when s equal to s not pq equal to p not q not if that is a solution to the system of equations that is exactly this which I have read out earlier. So in order to put our problem in the setup of implicit function theorem we define two functions p1 and p2 as I told you p1 of s comma pq because fs gshs are known it is simply a function of s that is why s pq are unknown quantities I want to solve pq in terms of s. So I have 3 variables here so s varies in interval i pq vary in the interval r cross r I have put but it should be that wherever capital F is defined for this pq right. So it will be the projection to the last two coordinates pq of the domain of omega 5 that is where it it makes sense.  So set up all pq for which f is defined I will consider this function in that case I will not write r cross r I will write something else okay. So with this correction p2 of spq is the second equation that we had on the previous slide we are interested in its solution what is given is s naught p naught q naught is a solution you have to find the p naught q naught such that s naught p naught q naught is a solution. So that is what we assumed now to apply implicit function theorem we need to check that functions are c1 functions phi 1 phi 2 are c1 functions are they yeah with respect to s if fgh or c1 fine this is c1 and capital F itself is c1 therefore composition will give c1 pq are appearing in this and we already assumed f is c1 with respect to all the components therefore pq also will be c1. So phi 1 will be c1 no problem phi 2 c1 s of phi 2 will involve c2 s of f so this is where we need fgh to be c2 okay. In terms of p and q it is linear so it is c infinity so phi 2 is c1 provided we assume the initial data that is datum curve fgh or c2 functions this is where we need fgh to be c2 okay fine. Now the Jacobian of phi 1 phi 2 with respect to pq at the solution that we already found s naught p naught q naught of this system that should be non-zero which is this dou phi 1 phi 2 by dou pq is simply the Jacobian which is here dou phi 1 by dou p dou phi 1 by dou q dou phi 2 by dou p dou q will come now at the point s naught p naught q naught that you can compute from the expressions of phi 1 phi 2 it will turn out to be fp at zeta naught fq at zeta naught f prime s naught g prime zeta naught is this phi tuple fs naught gs naught hs naught p naught q naught this is non-zero we have to assume that assume the above condition on Jacobian okay done now we can apply implicit function theorem it gives us unique functions p equal to ps and q equal to qs which are c1 on an interval containing s equal to s naught let us still denote it by i for convenience if you want you can make it i dash does not make any difference and what is the property ps and qs solve a system phi 1 and phi 2 phi 1 s phi s equal s ps qs equal to 0 phi 2 of s ps qs equal to 0 and at s equal to s naught it coincides with p naught q coincides with q naught thus an initial strip has been determined so this completes successful implementation of step 2 of our program so initial strip consists of the datum curve gamma which is given to us and along with planes planes essentially means the normal to the planes ps qs minus 1 attached at each of the points of gamma okay what we have shown is that if the Jacobian condition is met at an s equal to s naught in a piece nearby that is some part of gamma there we can do this that is what the implicit function theorem said in any case our theorem is going to be local we are always going to fix some point and going to assert that near this point p0 on the datum curve gamma there is an integral surface which contains a piece of gamma therefore this is absolutely fine and initial strips have the datum curve as the support okay if you can find a throughout all the points okay now let us go to the step 3 which is defining a candidate solution to define a candidate solution first thing is we have to solve characteristic strip equation with the initial strip that we have got so here we assume that f is c2 and so let this represent solutions to characteristic ODE satisfying the initial conditions for x, y, z it is fs gs hs for p and q it is small p and small q of functions of s which we have determined in step 2 that is a initial strip that means if you look at the trajectory of the solutions the first three coordinates at t equal to 0 pass through the point fs gs hs on the datum curve and remaining 2 are supposed to be giving you the components of the normal at that point that is why the strip now existence of solutions like this follows from Cauchy-Lipschitz Picard theorem that is why I assumed f is c2 because the right hand side will involve fp fq etc right so to guarantee that right hand side is Lipschitz one easy way to do is assume that right hand side is c1 then it will be locally Lipschitz and then we have existence of unique solutions so that would require f is c2 that is why I have added f is c2 so solutions are unique because of that otherwise you have existence continuity is enough we assumed f is c1 and if you look at the equations of characteristic ODE the right hand side will involve only derivatives of f at maximum right fp fq pfp plus pfq minus fx fz and so on f y they are all assumed to be continuous therefore right hand side is continuous the moment you assume f is c1 which means by Pino's theorem there exists a solution if you want uniqueness you push it c2 if you make a f to be c2 uniqueness is guaranteed so on solutions of the initial value problem that is characteristic strip ODE plus an initial strip we want to solve this initial value problem so without loss of generality we may assume that solutions to the initial value problem which is given by the ODE is a characteristic strip ODE and the initial conditions are given by the initial strip they are defined the solutions are defined for t, s in j cross i where j does not depend on s for quasi linear equations it was possible thanks to the lemma on reparameterization of characteristic curves because the tangential direction does not change under reparameterization on the other hand reparameterization of a characteristic strip need not be a characteristic strip therefore this argument does not hold luckily we have another argument in the case of quasi linear equations we had a second argument which we have given that also allows us to conclude the existence of such an interval j which does not depend on s but now there is a compromise that the interval i needs to be replaced with a sub interval of y but that does not matter because later on what we are going to do is we are going to apply inverse function theorem whose conclusions are anyway local so it does not matter so what is important is that j does not depend on s whether it is s belongs to i or s belongs to some sub interval of i so the solutions are continuously differentiable functions on j cross i by differentiable dependence of solutions and parameters in the theory of ODE's in fact inside the proof of this theorem one needs to get such catch hold of such an interval which does not depend on s so the progress so far has been this first gamma which is given then an initial strip we have obtained that is throughout these points of gamma we have erected these kind of small planes and then characteristic curves going through points of gamma if this is the point p this is the gamma p small gamma p and not only characteristic curves throughout every at any point in the characteristic curve we also got this particular normal to some plane of course the plane is expected to be time in plane later on so for T s in j cross i we got f of x T s y T s z T s p T s q T s equal to 0 and x 0 s is f s y 0 s is g s z 0 s is h s p is p s q is q s so these are coming from initial conditions this is the first thing that we did d by dt f of zeta of t is 0 therefore f of zeta t is constant and we chose zeta 0 such that f of zeta 0 is 0 therefore f of zeta t is 0 so this is also we have proved in step 2 it followed from the way we chose initial strip so summarizing the steps 1 2 and the third one the little bit that we did so far it gives us the following theorem consider the Cauchy problem for general nonlinear equation with Cauchy data assumptions f is c 2 let s be an integral surface for this general nonlinear equation where u is a c 2 function assume that a part of the datum curve lies on the surface yes assume that f g h are c 2 functions which describe the datum curve note these are additional assumptions on f and gamma standard assumptions on them still apply namely f p f q cannot simultaneously vanish and so on and f prime g prime also cannot vanish simultaneously now take a point which is on gamma as well as on s that means that integral surface which is also there in the datum curve that point let p naught q naught be such that the equations are satisfied assume that the Jacobian condition is satisfied at s naught p naught q naught p naught q naught is this s naught is which we have already fixed and here the zeta naught stands for this as before conclusion the surface is the union of the supports of the characteristic strips through the points of gamma of course in a neighborhood of p 0 so this theorem actually generalizes our earlier theorem that we did integral surface is a union of characteristic surface curves integral surface is union of characteristic curves that theorem we proved for Cauchy equations this is generalization of that proof is similar to that same as that so key ingredients here are characteristic ODE and p t equal to this and q t equal to this so now question is the road ahead is smooth because last theorem gives us a hope to find an integral surface exactly like the other theorem which gave us hope in quasi linear equations using the method of characteristics for general and non-linear equations however theorem holds only for u in C2 and Fgh in C2 of i we are looking to solve GE for which a C2 solution may not exist it is unreasonable to expect a C2 solution for a first order PDE to start with it may happen that you you have a C2 solution or C infinity solution but to start with you do not expect that and parameterization of the datum curve is C1 is reasonable but not C2 so would that be a problem answer is no that would not be a problem details on the next slide yes it is true that we used u is C2 of D Fgh is in C2 of i to derive characteristic ODE and initial strip now forget all this and start working with characteristic ODE and initial strip they ever used only to derive these two things right pretend that you are given these two and you are asked to work with them everything will be fine we do not need Fgh in C2 so no need to worry road ahead is indeed smooth from now onwards assume that the system characteristic ODE is given and an initial strip FSGS HSPSQS consisting of C1 functions on an interval containing S0 is given let the functions HTS, YTS, ZTS, PTS, QTS solve the initial value problem and initial strip that will be good enough of course you may not have solutions which are unique but may be correspond to each solution you may get one integral surface who knows that that can happen so we derived an initial strip using which the system of characteristics strip equations may be solved projection of solutions to XYZ space will give you characteristic curves through points of gamma in a bit to extend method of characteristics to GE we come across new geometrical entities so natural question you should be still call it method of characteristics or should we call method of characteristic strips it is okay that does not matter just for you to think ponder about this question so we found an initial strip step 2 of our program is successfully implemented in the next lecture we are going to complete the proof of existence and uniqueness of solutions to Cauchy problem for general nonlinear equations thank you


Hey gamers, have you been playing thousands of hours of League of Legends or Dota or games like PUBG? And you're wondering how you can put all that money and time you invested into it into something that's actually productive Well, here's 10 millionaire life hacks that we learned from playing video games. That's right video games If you want to get more content like this make sure you subscribe to this channel the David Wallum Entrepreneur show Click the button below click the bell so you get all the latest notifications of all the coolest videos from this channel That's it. Let's get on with the show So today we have Kevin Lee, he's the CEO of Brigade Media. He's generated over $65 million in revenue through his marketing company and he also is a avid gamer He's at the top 20 on the server for PUBG What are some of the top 10 life hacks that you've learned from video games? Let's start with the very basic the first one is level and gaining exp Everyone knows in games you have to level you have to fight gaining exp get more levels Learn more skills and your character become stronger. It doesn't really matter which game you play Some sometimes you level inside the game go from level one to level 18 in League of Legends Or you just play the same game more often and you get more experience as a player And then you can start shitting on the new players, which is very fun part of the game That concept is exactly the same way in life In business and in even in career you find a job and as you work stay in the career longer You get more experience and then you can move up In job position. Let's say you started as a entry-level accountant a bookkeeper as you work more you become a CPA You become a tax accountant You give the idea to stay in the business you operate over a period of time You become better at it and you start making more money So the concept of leveling up and experience same thing, right? So if you don't grind for example, if you don't do any of that you get no experience You're gonna stay at a low level and then low levels means what low skills and low ability to make more money Exactly, which leads us to the next one which is skills skill trees and items That's actually a very good analogy for picking the right industry or picking the right sector Right a lot of kids these days graduated from university They don't really know what they want to do and this is the very first thing that they have to do Just like in any game you want to choose where you want to go You want to choose what kind of skill you want to learn so that you can become better at it and go down that path Know your role. How's that working in real life and in the game or about knowing what you're good at going back to the marketing example I gave you right? Let's say if you are a strategist You want to provide consulting services for your company or for your client's company You have to know what you have to do rather than concentrating on the Executional level things like oh, how do I work this at works campaign? What kind of keywords I'm looking for you're more looking at the bigger picture You're looking at brand positioning competitor research. You're looking at the next year Projections you're looking at the product mix So knowing your role will allow you to know what you have to focus on concentrate on and spend your effort on Just like knowing whether you're gonna be a jungler or ADC or tanking or support exactly even going more in depth The type of jungler you are right?  So next one is money and gold. So that's very obvious in business. We need to make money get more gold So how does that apply in the in the game? You get more items through money right the more money you have the better items you can buy It's the same thing in business the more money you're able to accumulate you can choose to use that money to scale your business Having more money just opens up more opportunities both in real life and in game next one is knowing the enemy How's that work? This applies to your competition, right? If you own a business you have to know what their next move is you have to know what kind of trend are going on I work in digital marketing So I have to know at all times what new digital marketing firms pop up here pop up there What is the newest package that they're offering to their clients?  The next one is know the map. This is actually quite similar to Knowing your enemy knowing your enemy is a little bit more specific where you're trying to look at the enemies cool down You know what they're doing what kind of skill they leveled up What kind of items they have and that translates to what your competition is doing knowing the map you're looking at the Market right you're looking at what people are doing in the market in the game You're trying to watch out for gangs if the other team is pushing for objectives Or if you're in PUBG you have to know where all the players are in real life You're just looking at what the competition is doing overall economy economic factors Political factors and knowing that will just always puts you ahead of the competition now We're gonna talk about some strategies in games. The first one is don't quit while you're ahead Don't quit while your head. So that's about pushing your advantage. Let's say you leak most of your team members are half HP Or close with OHP. Well, you're not gonna go all recall back to base and buy item and come back out Because that's when the people are gonna respond you want to push for objectives You want to get more turrets you want to get that free dragon You want to get that free Baron right same thing in business if you're ahead of the competition or if you head in the market You don't just sit back keep all the money in your bank and do nothing You want to reinvest that into the same market opportunity and try to scale try to capitalize on your advantage Next strategy is knowing when to quit knowing when to quit the exact opposite of the other one, right? So it's about not picking a team fight. You can't win You don't want to fight a fed assassin. Let's say Zed when you're half HP ADC, that's just not gonna work You have to run. It's the same thing when you're doing business. You have to choose your battles You have to choose what kind of products to to sell what kind of products to push and what kind of products or services to drop Let's say you offer a certain type of service You're fairly good at it, but then a new competition where someone else does it so much better for a cheaper price Right, you don't want to compete with that person by dropping how much you know You're going to charge for your product because that will eat your margins and you end up losing money Right, you want to do something else, right? Don't fight these price words These are the battles that you don't want to engage in next one is don't just charge in have a plan You know a lot of people who are in league where you know who are stuck in silver stuck in gold is when they get Ahead or they think they get ahead they just run in and feed or they run it out mid very bad People do that in business all the time There's a lot of opportunities in market that come and go and there's a lot of people that blind the follower trend Cryptocurrency blockchain that sort of stuff big data back then across border e-commerce, right? There's all these big words that people think it's just full of money and they can just go in and do whatever and they can make money Well, that's not the case. You have to know what you're doing You have to have a very actionable plan. Don't rush in whenever you smell something good And the last one is practice makes perfect. It applies for everything. You play a game long enough You get good at it. You do business long enough. You get good at it There's a lot of business locally from where we are, right? Restaurants phenomenally, you know, they open up a business They stay afloat for maybe six months to a year and then they disappear They go bankrupt for you to be a successful restaurant or any business They have to keep going where we're from three years or so and then they become very good at it They become very stable. Let's say if you played in season one You will lead a legends and we'll get back then and if you haven't played until what is the season now season nine? Well, you're a new you don't know anything other people are going to destroy you whatever lane you go to Also same concept applies to getting good at one character instead of playing all the characters You have to master your craft so focus and get mastery over a specific market a specific skill or a specific industry All right, thanks Kevin for your wise gaming advice that we can use to make more money Thank you so much for having me in it Thanks for watching the David Wong entrepreneur show if you like this channel make sure to subscribe and we'll see you soon in the next show


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