 Okay, let's continue with the last lecture of Sasha. Thank you, works. Today I will be talking about something quite unrelated to the subject of the first three lectures, namely in the, let me remind you first we discuss this universal infrared aspects which do not depend on the UV details of the theory and that's why they're so universal. And of course, whatever the theory is we know that at long distances it reduces to Einstein theory but of course for string theory as well it's very interesting what happens at intermediate energies or high energies. Namely, what is the UV completion of gravity? What is the theory of high energies? Of course, this is a very hard question and especially we don't have an experiment. So a version of the question which seems to be promising and very popular is to understand what are the consistent low energy effective field theories which admit a UV completion. So sometimes this consistent, this question is called I guess landscape versus swamp land which is a landscape being a low energy effective field theory which do admit UV completion and the swamp land being the theories which looks a priori they look fine but if you try to make them gravitational and make them UV complete you run into some problems. Of course, figuring out what are the borders between swamp land and landscape it's a very hard problem but today I would like to talk about the version of the problem which is a classical which is that we will not don't want to talk about quantum swamp land but sort of consider a classical theory. Say we start with the Einstein theory and we add to it some high derivative correction. Let me write an example, some famous example but you can imagine all kinds of terms. This term is called the Gauss-Bonne term. It has two prominent features first in four dimensions when D equal to four, it is topological and also if you consider equations of motion and some background for perturbations are second order still. So it's a well it's one correction but the question which I would be having in mind is completely general which is that imagine that you have some theory with many high derivative corrections maybe extra particles, et cetera, et cetera and the question is how can we constrain these theories? What is the landscape of consistent classical theories of gravity? So important point here when I want to emphasize that this question is classical. By this I mean that we have a say Planck scale which controls quantum corrections which would be very high and there is an extra scale which is more like a string scale which controls this extra high derivative corrections. So I will be always working to leading order in the Planck scale but to say to order in string scale that's an important point. And of course an example of such theory which has this kind of corrections is a string theory. That's one example. And from the studies of these questions there are two sort of two messages maybe that one can see is first that this type of high derivative corrections for theory to be consistent. You need the higher spin particles. You need an infinite number of higher spin particles. The higher spin I mean here spin larger than two. And let me remind you that graviton is a spin two. That's a graviton. It's the first thing which I would be talking today and it's related to again some as the previous lectures was some modern twist on the old topics. This will be also some modern twist on the old subjects, old general relativity experiments. And the second message which I don't have time so much to explain is that if you think about this classical theories of high spin particles in certain things they have strings, these theories of strings. It would be nice to turn it into a theorem and to prove that if you wish that the string theory or strings is this universal mechanism of having consistent interacting theories of high spin particles. It's not that you, well there are probably many ways to end up in string theory but from this point of view you start with this problem, you correct clavities and you want it to be consistent and you see that the only way you can do it is to have strings classically. Of course I'm not proving here that string theory is a theory of the universe because it's a classical problem. But if you can do something like that for the full quantum theory of course that would be something very great for string theorists. So that's the two messages and I will talk today about this because it involves many sort of simple and classical experiments structures. If you have any questions? This was a motivation, very good. So by this I mean that it has at the moment that it has infinitely, infinitely many, say asymptotically linear retro trajectories and all the other predictions of effective theories of strings but I don't know how to prove that. So far I know only that you can show that the four point amplitude in some universal region looks like when it's a anode. Okay, that's... That's what I mean, yeah. That's what probably, well, ideally one would like to show more. That's what I mean at the moment. The object which I would like to discuss today, it's scattering amplitudes, it's a nice object because if you're dealing with actions there are field definitions. But if you want to talk about invariant data it's packaged in the scattering amplitudes. The object which we would like to discuss is a two to two scattering and the S matrix is usually written like this. It's one plus I.T. It's unitary, well, I will not use it too much. Placed in this sense and the ZT has a sum of a momenta times an amplitude. Amplitude is a function of Mandelstam invariance. Okay, what is the source of the problem? Why is it that suddenly we started with a theory and we would like to constrain it? So what would be where would the juice is coming from? How do we get these constraints? How do we get these claims? To do that it's useful to recall some basic facts about scattering amplitudes and then and then we will find that what is special about gravity in high spin particles is it's high energy behavior. And we will see that the high energy behavior is very constraining. Recall that if you have a, let me start with something simple that you have two particles and you exchange particle with a spin J and show you can compute the amplitude. Then this diagram behaves as T as to the J. This is a known fact. Well, if you simplest example is consider a scalar field then you have take this diagram it will be equal to one over T which is at large has the case like as to C zero. Now you can consider instead of a useful formulas to or useful experiment would be to consider instead of scattering in momentum space. Imagine two wave packets. So we have particle one and two which are separated by some distance B. That's now you imagine two wave packets which in the transfer space separated by distance B. And yep, you can, if we add the diagram it will be one over S, yeah. But at large energy this dominates, yeah. Now in the impact parameter space, sorry this guy's called impact parameter in the impact parameter space the S matrix takes this form one plus say I delta. Delta is called a phase shift and this one is the same one but this one is secretly a bunch of delta functions and this one is literally one and the phase shift is related as follows so the scattering amplitude that's one over two S integral over transverse transfer momentum and you take an amplitude S and S and T minus Q squared. So S you think about it as just the energy of scattering and Q is the exchange momentum. So if you do it for your transform, oh, something is missing. You do it for your transform as respect to exchange momentum you get a phase shift. Now where is the problem, where the problem comes from? So the problem would like to impose is causality. For example, that's we want our theory to be causal and we'll motivate it in two steps. Let me first think, you can think of scattering as a, especially if it's a high energy scattering whereas there is no much deflection going on particles just acquire some phases which then you can think of scattering as a propagation through the medium. So imagine I send the many particles and then you send the probe. You can think from the point of view of these particles this other particles is just the medium and then we would like this medium to be such that the propagation is causal. So it's not faster than speed of light. A simple model to think about when we think about propagation through medium through the medium is to think about some signals. Imagine you send, you have a black box. In this case, this black box can be made of gravitons. So this black box made of gravitons transforms in going signal to out going signal. Then, well, let me consider the following model and you can then show that it's a good model for this. We can imagine that the black, this black box does transform the signal through some S-metrics. It is diagonal in the Fourier space. So F out omega is equal to S of omega. F in. And let me imagine that, and this is related to the unitarity of the original S-metrics, that this medium cannot increase, if you wish, the amplitude of the signal, namely that in this language, the unitarity is a statement that the norm of the L2 norm of the outgoing signal will be less or equal. Now, if you, you can ask, okay, what are the constraints, what are the constraints from causality and this unitarity on this S-metrics? Well, that's a simple problem. Some of, probably you've all seen that, that it's a, for example, if you take an in-going signal that would be zero on negative times, then you want the outgoing signal to be also zero on negative times. So if you send some signal from time zero, you cannot get outcome before. That's a statement of causality here. This implies that this S-metrics, it is analytic in upper half plane in omega. And together, if you, if you, but you can imagine that S of omega is something which is analytic in upper half plane, but it can be some growing function. But you can show also that this unitarity, it's a simple exercise that it's, if you go to the upper half plane, that it's less than one. It can only decay. Otherwise, you will get that the norm will be growing. So this is a kind of medium we would like to have. And now we can think of the scattering amplitudes if we imagine this set of scattering events. I, let me repeat again. Everything what I'm saying here, I consider the signal model where the incoming signal is transformed to the incoming signal through the S-metrics, which is a function of omega. I assume that it's unitary. And by unitary here, I mean that the norm of outgoing function is less or equal than the ongoing function. And now I impose that this S of omega is causal. Namely that for incoming signal is zero, the outgoing is zero, and then that it's unitary. And I'm saying that the condition that S of omega should satisfy, it's analytic in upper half plane. First condition. And moreover, it decreases in the upper half plane, so it's less than one. When you go to the upper half plane. For example, you cannot have something like, it was i omega cube. This is analytic in upper half plane, but it blows up in some directions in upper half plane. On the other hand, you can have something like this. This is analytic in upper half plane, but it decays in upper half plane. This is related to the difference between particles spin two and higher, as we will see now. Now it's a little bit of work, but morally you can see it even from here that you can think of the scattering events as this medium and that providing this S of omega with S of omega being controlled by delta. And then we see that the condition of this type that S of omega is less than one becomes in our model, something like one plus i delta S of B, the less or equal than one in upper half plane. And say S is S naught plus some i gamma. Recall again that we said that if we exchange a particle of J and this model is a good at high energies, so particle just rush through each other without deflection. So what happens if we imagine that this is we exchanged a particle with spin J? Well, this thing becomes one plus i C S J minus one plus equal than one. Now you can see the following things. Well, first you can consider, and here I'm working to first order in C because I assume that this is a small phase, otherwise I have to say, I assume that the coupling is weak, so I'm working to first order in C. Now you see that something special happens for say, let's take, let me take say J equal to and this S, this is Mandelstam S. It's a roughly times some number, a product of energy of a particle of a probe times the energy of a particle of a medium. That's a scattering event here, delta, delta, delta, which we repeat many times say. And this energy of a probe plays a role of this omega here and you get that the causality constraint become the analyticity of this thing in upper half plane in this way. You can think, you can connect this scattering through with a signal model and you get that the phase shift should be analytic in upper half plane in the S variable. And you see that if you take J equal to two, that this is imply, this implies that C is positive. And as we will see, this is related to the fact that gravity is attractive. Then if J larger than two, this is always violated. You will always find the direction in a complex plane where it grows, no matter what C is, is bad. And this is related to the fact that higher spin particles are problematic by itself. So it's very constraining. As soon as you add one higher spin particle, you should add infinite number of them. Because imagine you just added one higher spin particle, then it grows like S to the J, then you will violate this, we are in trouble. That's why when we add one higher spin particle, they always come infinite number of them. Yeah, so the, well, it fixes roughly that you have to resum, and when you resum, it becomes softer, like in string theory. So this is not a correct formula anymore. We have to resum infinite number of particles and the resum in phase is in such a special way, such that the high energy amplitude becomes small. Well, maybe it's not the correct model, but you can imagine something. Oh, it's resum X to the N, and there is a finite radius of convergence, but the large X asymptotic is controlled by information. Now you can ask what happens, yeah. Yeah, here, here, this is in the upper half plane, so it's not physical region. It's, well, it should not because it's a two to two scattering and you can create other particles. So if scattering is purely elastic, it will be one, but generically it will be less than one. You can ask what happens to J equals zero and J equals one. Well, for J equals one, this is a pure phase, and to first order in C, the absolute value will be one. So if we exchange a photon to first order in C, this is not a problem, and if J equals zero, if we exchange a scalar, you can, if you carefully go through this model, so I kind of use the connection of the scattering with the signal model, and you can show that for the scalars, this connection is not, doesn't quite work. But the message from here is that we see that when we have a graviton, we have fine potentially, and when we have hard spin particles, we have to worry. This is the source of these constraints on the classical landscape of classical theories of gravity and high derivative corrections, and the idea will be roughly that as soon as you start adding the hard spin, as soon as you start correcting the gravity, you can find a regime where the C becomes negative, or you will violate causality in this medium, and then when you will try to fix a problem, we would have to add all of these guys, but each of them is by itself problematic, so we'll have to infinite number of them, and then the problem is fixed. That's the idea. And then we can study the theories of these hard spin particles by themselves and try to see what are the conditions that they re-sum up correctly and don't spoil the high energy behavior. Let me explain a little bit more why is it related to causality, and what do we mean when we talk about causality and gravity? Because while in gravity, you can say the spacetime fluctuates, so what is, what do I mean when I say causality and here's a notion of causality and gravity, which I'm having in mind, is what is called the asymptotic causality, and this is due to Gao and Walt paper about time delays and the statement of asymptotic causality is, well, the idea is, as we also discussed in the previous lectures, is that far away, so the spacetime fluctuates, but if you go far away, you have a rigid asymptotic structure. For us, it was Minkowski space, and the idea is that we cannot send signals faster than what is allowed by the asymptotic structure, or is allowed by the asymptotic causal structure. So Walt and Gao, they actually were proving theorems but you assume, you start with GR and you add some matter which satisfies our energy condition and then you can show that when you say, let me draw, the simplest, the cleanest thing is to consider ADS, so ADS is a gravitational box, so asymptotic structure is a boundary of the cylinder, it's just Minkowski, it's a flat space, so you have a light cone, and now you can imagine that you added some matter here in the middle of ADS. You can imagine sending signals either through the bulk or you can imagine the signal through the boundary, next to the boundary, and the idea, the theorems of Gao and Walt and the notion of asymptotic causality is whenever you're going through the bulk, it can only slow you down, it cannot speed you up. You can never go faster through the bulk than what is allowed by the boundary. You cannot arrive somewhere outside of the light cone. In the context of ADS CFT, it is especially clear because if you have two CFT correlators, then you know that the commutator would be zero when there's space like separated, that's a notion of causality. So asymptotic causality is this kind of notion. That asymptotic correlators, asymptotic operators should commute, and when you go through the bulk, it can only slow you down. Simplest example, if you take, so you can see that if you take a negative mass Schwarzschild, it will speed you up, but if you take a positive mass Schwarzschild, it will slow you down. And this slowing down, slowing down of signals when going through the bulk, it has a name, and this slowing down is called Shapiro time delay. What is Shapiro time delay? Well, as you know, there are three, yes, and the idea is that we take this effective theory of gravity, gravity plus high derivative corrections, and we impose this asymptotic causality and see what are the constraints on the corrections. What is the Shapiro time delay? Well, as you might have heard, there are three classical tests of general relativity, which is a shift of mercury perihelion, deflection of light, and gravitational red shift of light. And so in 1964, Shapiro posted a paper called the Force Test of General Relativity, and the idea is that if you send the signal to Venus or Mars, and it reflects back when it passes by the sun, it will slow this signal down. And you can measure this effect. It's 1000, so 10 to the minus fourth of a second, something of this order, I think. And this is an example of this slowing down when going through the bulk. So let me compute quickly for you a Shapiro time delay. Let me consider a metric which corresponds to two Schwarzschilds. So I consider the metric, which is Minkowski space, plus I added the Schwarzschild solutions. Of course, it's not an exact solution of Einstein equations for the usual two sources, but it's a good approximation, and we can imagine doing that. And imagine I put, say, I have some transverse plane, and I put one source at impact parameter B, and another source at impact parameter minus B, if you wish. And now we will send the particle from infinity passing between these two Schwarzschilds. So it will go from minus infinity to plus infinity. And because it's a symmetric, it's symmetric, there is no deflection, because there is symmetry. If you put one of them, there will be a deflection. But if you just, if you choose a symmetric configuration, what happens is the light does not deflect, but the Shapiro time delay accumulates. So what we will find is that we will now compute the time, time it takes light to propagate from minus infinity to plus infinity in the presence of these two masses. Let me do this computation. So there are this R i. So R one is simply, say, x minus B, and R two is x plus B, it's a usual coordinates. Now, we can imagine that the problem, we can localize problem in this transverse plane and study the propagation of light in this metric. So for light, this thing should be zero. And say, in this case, we can call this direction Z. Then both this R, R they are equal to B square plus Z square. If I consider a position of the particle, this difference is simply B square and this is a coordinate Z. So we get the equation of the type. If we say that it's a geodesic, we get that one minus two RS over R D minus three. You can get it. Check that this is a correct thing. Easy square one plus two RS over R D minus three. So it's a little exercise to plug this and check that it's correct. And now from this, we can compute the change. Well, we can find the delta T as an integral over DZ from minus infinity to plus infinity. And we have a, well, from this, we get question for DT over DZ. We compute DT over DZ at R minus three. That's AT over DZ when RS is equal to zeros. It's just Minkowski. So this is a relative delay compared to propagation in flat space. Well, and you get something which looks like some numbers, which we don't care about times over B to the D minus four. Here's some number depends on D. So if you consider propagation of light, you get is that it's get delayed and the delay is a one over B to the power D minus four. An interesting exercise, which I will not do here, is to repeat the same thing for the particle which moves with velocity V. So here we've had a particle which moves with the speed of light. That was a photon. If you repeat the change for the particle with the speed V, you will get that it's one over V, one over V. And here, curiously, you get minus delta T of four. Okay, what is this formula again? This is a Shapiro delay or advance for massive particles which move with the velocity of V, which is expressed in terms of Shapiro delay of photons. Notice that there is a one over V square factor in the brackets. So when the particle is very slow, when V is small, the delta T is actually negative. If you take a massive particle, it gets slow, it trails faster and there is a critical velocity after which we can get delay. The fact that these slow particles get Shapiro advance and they travel faster does not contradict our symptotic causality because the symptotic causality is formulated in terms of the fastest signals, which is a light and the light gets delayed here. But there is this curious fact. No, sorry, let me, if you wish, let me write it. That's the right answer. Now if you set V to one, you recover delta T of four things. This is a classic results from the seven case, which is a computation of a delay from a planet. When the light passes by the planet, it gets delayed. If a slow particle passes by this two planets, it gets advanced. But asymptotic causality structure only cares about the fastest signals, which are controlled by the propagation of light. Here we get that this Shapiro time delay is positive, because the gravity is attractive. Now I would like to tell you about the relativistic version of this argument. So imagine that we would like to propagate next to a relativistic planet, which moves with the speed of light. It will, or just some photon. Then we can consider a source in Einstein equations, which corresponds to, let's say we have, as before we have in Kowski space, whatever x d minus one and v plus x d minus one. And now we add the source to this. So imagine we have a relativistic source, which is this. It's some particle which moves along the light cone with some momentum pu, and it's localized in the transverse plane. There is a famous exact solution for this source. You can show that if you add two things, so there is the exact solution is known, and it's usually referred to as a shock wave solution. Or yes, and this h you can find, it's simply the same kind of h, that same kind of Shapiro delay we had. It's a number, which depends on d. It's localized at u equals zero. And instead of rs, which roughly measures the mass of the planet in the units of g, you have now the momentum of the source in the units of g, and you have the same r transverse in this space to the d minus four. Now, two comments. First is that this is the exact solution to find the equation. It's the first comment, not the linear. And the second comment is that actually you can find if you add any type of corrections to Einstein theory, you can write any polynomial terms, whatever you like, this is still an exact solution. That's a peculiar fact, which you might wonder about, but then I will try to explain another explanation why it's, is not so maybe surprising that it happens. In this language, what happens now if we send another probe along through the shockwave? What happens is that here is our particle. So the shockwave is localized at u equal to zero. And find the geodesic and you will find that where this particle jumps through. It's get shifted by some delta v. And if you try to embed it in a global space of Minkowski, you'll see that in the original Minkowski, that's what will happen, but now it's get delayed. It arrives later. This delta v, which simply if you compute is equal to g, pu, and again bd minus four times the number, is a relativistic Shapiro time delay. Okay, that's fine. It's a nice shockwave solution. It's exact, so far, so good. There is something curious happening if we try to translate this computation. This is just, we computed the light geodesic passing through this exact solution. It is curious and instructive to repeat the same computation using the language of scattering amplitudes. Let me do this quickly for you. And let me consider for this purpose and exchange. What is this, what is the physics of this? What is the physics of this geometry? Well, the physics is very simple. This, it's a propagation of one particle on the gravitational field of the other relativistic particle. In the language of scattering amplitudes, what we have is simply, this is a, say, our probe. This is a source. And they interact through graviton exchange. Let's imagine that all these particles are scalars. Then the result for this, not written, but Feynman-Dyger, is simply eight pg s square over t. That's, you can check. And now let's compute the phase shift. As I try to explain, if we, if we, this are wave packets with separation b. Yes, and here, sorry, I had to explain. You could have asked, what is b here? So this is a two plane and this is a, there is a transverse plane. And I consider the source as the origin of a transverse plane and I send a probe somewhere here. It's a distance b. To avoid deflection, I can again consider two shocks, which are symmetric, but I neglect deflection for this purpose. That's what is b here. And this is the same as the impact parameter just by definition. Now we had to do this, we do the integral over the transverse momenta, e i q b. And here we have s square minus q square. Remember that t is equal to minus q square. Well, this integral you can take, it's trivial to do. You will get exactly the same, you'll get exactly the same d minus four times the same number. And if you remember that s is something, p u, which is the momentum of the source, that's the momentum of the probe. And then you can imagine that you start some plane wave, then you add this phase. Then this phase can be recast as p v times delta v. And this formula exactly coincides with this formula. So you can either do the classical geometry computation, or you can take the scattering amplitude, compute the phase shift, and then take a plane wave and propagate it with add to this phase shift. And you can see that the effect of this phase shift is exactly the effect of shifting the wave packet by delta v. So there is an equivalence between this computation as classical geometry and the scattering amplitude computation. However, something you can ask, what is the dual to this fact? Remember I told you that this is an exact solution in any theory of gravity you can check. This geometry is very special. It has some special symmetries, such that it's protected against correction to the Einstein theory. What is the dual of this fact in the scattering amplitude language? And the answer to this is the following, that when you compute this integral, you can see, let me choose some b, some component, b1, q1. You can compute this integral by closing the contour, and you see that the phase shift is controlled by the residue of the amplitude of the amplitude at t equal to zero. However, there is a simple fact that if you have an exchange of the particle, you have a pole, obviously, at this position, say. There is a fact that the residue of the amplitudes at this pole is equal to the square of a three point amplitude. This is an optical theorem for three level amplitudes. The residue of the pole, it's the same as imaginary part. You have something like that. So this is, if you have heard about optical theorem, that is an optical theorem for three level amplitudes. And now, since this delta is controlled by the residues of the amplitude, you can ask, what is this three level amplitude? And the curious fact about three level amplitudes, which I will talk about next, is that they are completely fixed by symmetries, up to a number. No matter what your theory is, of gravity just lowering symmetry fixes them. And in this way, you see that the Shapiro time delay is controlled by three level amplitudes, and these three level amplitudes are fixed by symmetries, no matter how you correct your theory. And this is a dual of this fact that this is an exact solution for any corrections to the theory. And after you realize this, you can ask the following question. Okay, let's now forget about the actions and geometries. We're computing the objects, which is completely fixed by three point couplings. So we can take the general gravitational theory with three point couplings, compute Shapiro time delay, and see what's going on in this most generic case. This is the idea. Here is a little comment. It's probably confusing for those who haven't thought about this problem, but notice that we start with a purely transverse momentum. So it's a Euclidean momentum. Q square is positive in the physical domain. And then we are computing the residue at a point where Q square is equal to zero. And we are computing it because we say evaluate it in a complex plane somewhere. So let me rewrite it as a Q1 square plus Q transverse. And the residue is at where it's equal to zero. So it's effectively Q1 becomes complex. And making a momentum complex is the same as doing a weak rotation. When we go from Euclidean plane to Minkowski space, we say take P naught to IP naught. So here while doing this computation, effectively we switch from say Minkowski signature to what is called mixed signature, which is two minuses and say pluses, effectively it's in quotes. And this is related to the fact that kinematically if you consider say this are three massless particles. If you consider a decay of massless particles in Minkowski space, they are kinematically prohibited. So just write energy momentum conservation. You will see that the massless particle can only decay linearly in Minkowski space. However, in a mixed signature when we have two times roughly that is allowed kinematically. And the moral of this computation is that not on, so the moral of this computation is that the Shapiro time delay is controlled by effectively a computation as a mixed signature due to the simple fact that moreover it's fixed by three point amplitudes. So it's a very clean observable it because it's completely fixed by symmetries. That's why it is so nice to study. Let me explain why is a little bit, why the three point amplitudes are universal. So let me consider now three point amplitude. And you probably will hear a lot about them next week in the lectures by Marcus Sprudlin. University three point amplitudes. Imagine that we have three massless particle which say one decay into two others. So we have momentum conservation and each of them is massless. Well, first you can immediately see by taking square of this equation move P2 or P3 to other side and take a square. You see that PI times PJ is equal to zero. That's a first thing. And now if we have a scattering which involves gravitons as we discussed, let's say we have a gravitons then it should be gauge invariant which is that we can shift this guy. If there's some graviton with momentum P we can shift that P mu lambda nu plus lambda nu. Now let's consider this process which has two scalar particles interacting with the graviton and let's try to write some amplitude. Let's say it's P1, P2, P3. Let's try to write some three point amplitude. Well, the answer is that there is only one thing you can write, P1 nu, P2 nu. This and some number in front. That's it. That's a result in a full quantum gravity. There is nothing else you can have. This just follows from Lorentz invariance and gauge invariance. This lambda is a parameter of the theory. So it's a coupling constant that you should measure in the experiment with its all loop corrections, et cetera. And that was exactly when we were taking residues. We effectively sum over all the states of the gravitons that we exchange and so we get this S square here by taking a square of two amplitudes, of three point amplitudes. You can repeat the same exercise for the gravitons and let me write you the result. If you repeat the same exercise for the gravitons, so now take three gravitons instead. The result is that the three point amplitude for gravitons, it has three structures. One of our Einstein theory, the one was roughly Gauss-Bonne and the one roughly of R-Cube. Riemann-Cube you can add to the action. Yeah, this is two scalars and a graviton. Yeah. Notice that this is gauge invariant because if you shift epsilon by P because of PIPJ zero, it is gauge invariant. And here I can put quotes because it's really what I'm using here is symmetry. I'm not using any particular action. It's completely robust. Just based on symmetry in any theory of gravity, this is the answer. And now as I try to explain to you the Shapiro, this phase shift when we consider scattering for gravitons is roughly given by this A3 square times one over B, D minus four. And write the precise formula. Now you can start playing the following game. Imagine you have the most general three point amplitude and you can, and you, we have a, so if this, if the Einstein theory is one, we have two corrections, alpha two and alpha four, say. So Shapiro time delay is a function of alpha two and alpha four. But it also, it is also, since we are scattering, if we are scattering gravitons, it also is sensitive to the polarization tensors of our gravitons. You can polarize them differently. This was something new compared to the scalars. Now you can ask, what is the condition that it's positive? And the answer, which I, it's a simple computation. You can know the three point amplitudes, you take the same kind of integral, you compute and you see that if alpha two and alpha four are non-zero, then you get the delta is negative, which is a violation of causality. That's the first conclusion. So as soon as you, if you correct the three point vertex of the gravitons and it's, then you get into this trouble. It's the first thing. And a second you can ask, how do I fix this problem? Well, one way to fix this problem is set this parameter to zero. That's fine. But that's a boring solution. And then you can say, okay, let's try to fix this problem by adding new particles, almost done. And then you can show that the only way you can fix this problem is by adding of particles with spin larger than two. Basically, you can repeat the same exercise. You start with three point amplitudes of particles with some mass and some spin. And then you see that unless, if particles have spin two or less, they will not help you. That's the only guys that can help you as particles with spin larger than two. And then by this argument, which I discussed in the very beginning, is that you cannot add one of them. If you add one of them, you violate this bound. You violate causality. Then you have infinite number of them. And then you can ask the following bootstrap question. Imagine you have an amplitude you have an amplitude with higher spin particles which generate phase shift, which does not grow faster than S, according to this bound. And it's a unitary amplitude. What does a class of theories which, what does a class of theories which have this property of higher spin particles? And this is a bootstrap problem. It's completely rigorous mathematically. You should just be able to classify such theories. And the hope is that these theories will be theories of strings and there is some evidence to that. So let me reiterate, we consider this simple extension of this classical general relativity experiment which is Shapiro time delay. We found that if you modify gravity in a very clean fashion, in a very gauge invariant fashion by changing the three vertices, generally we violate causality and then the string modes naturally appear. The same experiment, it doesn't matter if you can run an ADAS or in this theater space in cosmology, it doesn't, it's kind of insensitive to that. If we have these particles, we will have this problem. And this is an example of this warm plant like question where we started with corrections to the theory and at first glance it looks completely fine, but upon further investigation we saw that there is some problem. Of course, it's almost classical. It would be great to get such bounce in a quantum level and probably heard recently about this weak gravity conjecture which is exactly a safe time of question. Thank you.