 Okay, so I'll start with just a quick reminder of what we were doing last time. So we had this, we had this basic connection within the ADS-CFT correspondence between some measures of entanglement and geometry proposed by Ryu and Takiyanagi where they suggested if I have a state in my CFT and some corresponding asymptotically ADS spacetime, then if I compute the entanglement entropy of some region A, then this is supposed to tell me about the, of this extremal surface in the bulk spacetime that divides the bulk into two parts where the one part has boundary A and the other part has boundary A bar. And so today actually, so the idea was now I wanna consider conformal field theories and I'm only going to assume this formula. Actually for everything today, we don't need any ADS-CFT correspondence or string theory or anything. The idea will be to just assume that for this conformal field theory that for some family of states, including the vacuum state that the entanglement of these states can be represented geometrically, that there exists some spacetime M such that the entanglement entropies for various regions match for the areas in this spacetime. Okay, and I said last time that's a very special property of the states and presumably of the field theories. The space of possible entanglement structures is much larger than the space of metrics. So this is a very special assumption about states but we believe that for states in holographic theories, this is true. And what I wanna do is then, so for these geometries that represent entanglements of CFT states, I wanna understand what kind of constraints to these geometries satisfy? What can I learn directly from properties of entanglement? And so just to remind you of how that would work, the idea is, okay, suppose, so here's the space of asymptotically ADS metrics. Okay, suppose I pick one of these things then I can use this prescription to calculate all of the extremal surface areas. So that puts me, that will give me some function of the subsystems, but then we said last time that not all of these functions can actually be entanglement entropies. So entanglement entropies satisfy various constraints such as sub-editivity, strong sub-editivity, et cetera. So there's only say this green region which could correspond to the entanglement entropies in some consistent field theory and the other regions that can't possibly. So if I find, if I start with a spacetime, calculate the extremal surface areas using the Ryu-Taki-Nagy formula and I find that I'm in this region over here, I find that these areas violate the constraints that entanglements have to satisfy, then I can conclude that this can't possibly be coming, this can't possibly be describing the entanglement of some field theory. Okay, yes. So that's okay, so that's a very important question. That's the same as asking when does ADS-CFT work? So we don't have a complete answer to that. This is sort of another interesting direction for research. How do we characterize states of field theories where the entanglement can be captured geometrically? I, maybe if you ask the question in the discussion this afternoon I could tell you a few ideas, but for now I probably, I'll just say it's an open question. So I'm just going to assume that there are such states and proceed. Okay, so the strategy is going to be to start with, start with a case of just the vacuum state of the CFT and for a holographic CFT then this vacuum state should be dual to pure ADS space time and that should satisfy if we calculate the extremal surface areas in pure ADS space time, that should satisfy all the constraints. And I'm going to start now by considering perturbations. So space times close to pure ADS and ask what are the constraints on those and then we'll talk about more general space times towards the end of the lecture. Okay, so last time I derived the constraint that you have for first order perturbations. Okay, so if I start with a vacuum and then I add some perturbation, there's a constraint that the entanglement entropies have to satisfy and the constraint that we derived is that for some subsystem A as I make this perturbation, the change in the entanglement entropy for region A has to be equal to the change in this expectation value of what I call the modular Hamiltonian and that is just the log of the unperturbed density matrix. Okay, so this would be, in our case, this would be the vacuum density matrix for this region, that's some operator and then as I make the perturbation, the change in the expectation value of that operator must equal the change in the entanglement entropy and this followed just by the definition of entanglement entropy. No, yeah, so the idea is in this formula, you should interpret the H as being fixed and just the state changing. So the H is the modular Hamiltonian corresponding to the vacuum, so this is, I'll say, unperturbed. So I'll give you an example. So this is going to be useful in the cases where we can actually calculate this modular Hamiltonian. I mean, it's always true, but many times the density matrix or the modular Hamiltonian is some very complicated operator that we don't know. But in the previous lectures, we talked about a case where we can compute it exactly. So we're gonna be considering a conformal field theory, our unperturbed state, as I said, is the vacuum state and we're going to take the region A to be equal to a ball shaped region. I'll call it B for ball. And then what I said previously was that we can calculate the density matrix for this special case because this causal diamond or domain of dependence of a ball, you can obtain this region from a Rindler wedge from a half space from the domain of dependence of a half space by a conformal transformation. And so you can obtain an expression for the density matrix, the vacuum density matrix for a ball from the vacuum density matrix for a half space. And that's the thermal density matrix with respect to the Rindler Hamiltonian. So we had this calculation in the previous lecture where we explicitly found what the density matrix for this region is. And so then we explicitly can find what the modular Hamiltonian is. And I can write it down. So the modular Hamiltonian is then just the Hamiltonian that generates this flow in this causal diamond that is the image of the Rindler time flow. So, or the image of the boost operation in a Rindler wedge under the conformal transformations. So this thing is a conformal killing vector. So it's a symmetry generator of the conformal field theory. And explicitly we can write down, okay so if we use the standard know-their procedure to write down an operator, it's then an integral over the region of the zero component of the current associated with this flow, with this symmetry. And this is what it looks like. So remember the usual Hamiltonian, if we wanted to write down the operator associated with just the ordinary time flow in the CFT, it would just be an integral over all of space of T00, which is the zero component of the current. This one is a different symmetry generator and here we have a weighting function. So it's an integral now just over the ball of a weighting function that vanishes at the edge of the ball. So this function here R is the radius of the ball, big R is the radius of the ball, and then little R is the radial coordinate inside the ball. Okay, so this is a weighting function and this defines an operator on this region B. So that's useful now because we have a way to explicitly write down this first law. So this equation now becomes delta S equals delta of the energy density integrated with this weighting function over the ball. So incidentally, this is true for any conformal field theory and it tells you how the entanglement entropy for states near the vacuum is related to the stress tensor expectation value. So what we wanna do is apply this in the case of our holographic theory. Okay, so now we're going to assume that we're talking about a CFT state whose entanglements are related to some geometry. Since we're close to the vacuum state in the CFT, we're thinking about geometries which are going to be close to ADS. So our space time that computes the entanglement will be ADS2 plus some perturbation. Just to write down an explicit formula, we can use this Pheferman-Gramm form of the metric. So this would be the Pheferman-Gramm metric for pure ADS and then if I wanna consider a perturbed space time, I can add some perturbation function. So this is a gauge choice for describing metric perturbation asymptotically ADS metrics. And I've put in this power of Z here for convenience. If I make this choice, then this function H will be well behaved as Z goes to zero. Okay, so our question is if I take this metric and compute entanglement entropies according to Ryu Takenagi, in which cases will I satisfy this? So what does, let me, this right hand side I'm going to refer to as just delta EB. So what does this first law delta SB equals delta EB tell us about H mu nu? So any questions before I go on? Okay, so yes, sorry, I mean entanglement entropy has a definition in field theory and stress tensor has a definition in field theory. So these are, I mean, these are two different things. In principle, I would do different calculations to calculate those. But maybe the content was that I then did a calculation to find out where the modular Hamiltonian is. So I was able to reproduce that. Now I can write exactly what this side should be. So now I think it now it has content once I once I tell you what that modular Hamiltonian is. What, well, I mean, even, okay. So even in CFT, I guess I would say it's too different. I mean, if I told you go and calculate entanglement entropy, here's the state, go and calculate the entanglement entropy for this region. And now go and calculate the expectation value of the stress tensor. I mean, you go and do two separate calculations and then you can check. I mean, it was, I mean, the derivation was somewhat trivial, but for the vacuum, for the vacuum, but now we're making a statement about states near the vacuum. So it's really, I mean, it's really, even when I, even in the general case, remember this is basically the first law of thermodynamics. So if I start with more generally some thermal state, so once I say here's the thermal state for this Hamiltonian, then I do my derivation and I get the first law of thermodynamics. So that has some content. Okay. Now from the, so what we wanna do is translate this to a statement about the gravity side. And we have assumed that the left-hand side here corresponds to the area of some surface. But actually we don't know anything about this, the expectation value of the stress tensor. I'm not assuming, I'm not going to use anything from ADSCFT. So so far we actually don't know how to translate this to the gravity side. But the interesting thing is that I can come up with a rule for that by considering the limit of this equation when the ball is very small. Yes. Oh, just about this form, this Pfeffermann-Gramm gauge. I mean, the physical motivation, one of the physical motivations is that it will turn out, it will turn out that in this gauge, certain relations to the CFT are simpler. But it's just a gauge choice. So if I chose some other gauge, I'd be able to write different formulas. Okay, so I want to consider the implications of that formula, but I'm gonna start with a infinitesimal ball. Okay, so I'm thinking about the change in entanglement entropy for this very small ball in the CFT. And so according to our assumption, the entanglement entropy for that very small ball should be equal to the area of this very small bulk surface, which will tend to be located out near the boundary of the asymptotically ADS space. And so if I take my first law, so now I take this limit where r goes to zero. And on the right hand side, then this is the area of B tilde over 4G and the limit r goes to zero. And what happens to the term here is that in the limit where this ball is very small, the right hand side only depends on the expectation value of the stress tensor at one point, because we're taking a limit around this one point. And so the right hand side, actually you can calculate it. And the leading behavior in the limit where r goes to zero is some power of r times the expectation value of the stress tensor at a point. So this, you get some power of r times the expectation value of the stress tensor at a point. And over here, what you get is something related to the boundary metric. Okay, so you get the area of this infinitesimal, the area of this little surface, you can see that's only going to depend on the metric close to the boundary. And so actually this power that I wrote, had I chosen a smaller power here, there would be some inconsistency because the left side would be finite and the right hand side would have turned out to diverge. If I had I chosen a larger power, there would also be an inconsistency because the right hand side would vanish. So the equality actually implies that the leading behavior of this perturbation comes in with this power. So for metrics that describe entanglement of states, this is the power, this is the first power where you have a difference from pure ADS. Okay, and then when you actually, assuming that once you actually compute the area of this small surface, then it just depends on this function H in the limit where z goes to zero. Okay, so the function H has a finite limit where z goes to zero. And so actually you get what you find is that you derive this relation between the expectation value of the stress tensor in the field theory and the asymptotic metric. So so far it's actually just for one component, E00. And the reason is that we worked in a frame, we worked in a particular frame like the T equals, we were assuming that we were living on the T equals zero slice. When I wrote this formula here, this is a formula for this modular Hamiltonian written as an integral over a slice at T equals zero. So we can say that we worked in this frame associated with observers whose time like vector is just one, zero, zero, zero, or one, zero. But this first law is also true if I look at balls in other frames of reference. So if I had an observer moving at some velocity and considered their frame of reference by Lorentz invariance, that observer should also be able to apply this first law. And so this result really, we should be able to extend it to a covariant expression where a more general vector u appears. And if I translate that to the covariant version, what it looks like is this. And so this is the expression I would get in a general frame of reference. And so this is true for any u. So this is true for absolutely any time like vector u. You can show that that's possible if and only if this is true. There's one step I skipped here. So if I remove the u's, I still have this part with h mu mu. And then I used the fact that this CFT stress tensor expectation value is traceless. That told me that h mu mu must also be zero. And so I just ignored the term. If you are careful and keep all of the coefficients, you get that the constant is, and so you can actually derive, this is the standard formula in ADS CFT for how the stress tensor and the field theories related to the asymptotic metric. So we didn't need to use that. We could actually follow it from this Ryu Takinagi assumption. This is the place where the choice of metric is particularly convenient. If I had chosen another gauge, then this is a more complicated formula. Yeah, so the step I skipped was to say that delta T mu nu, it would be h mu nu minus eta mu nu h alpha alpha. And then if I use the fact that I take the trace on this side, I know it's zero. And that implies that h alpha alpha equals zero. Yeah, so we actually get, from properties of the CFT, I think we get the h alpha alpha equals zero. If I wanted to, I could also impose now the CFT conservation equation, and that will tell me that d mu of h mu nu is equal to zero. So I learned these things, so that's already constrained about the asymptotic metric. Okay, yes. Yeah, right, so far this is just telling us, because we just looked at these little walls, it's telling us about something about how the boundary behavior of the metric has to be if I have a state that describes the entanglement of some theory. Okay, so now this is good, because what I'm gonna do is plug this in to this formula over here. Okay, so now we know how to interpret the stress tensor expectation value on the gravity side. And so now we can actually turn this into a completely gravitational equation. Okay, so this constraint on our first law now implies that the delta of the area of B tilde for G Newton equals two pi, some constant and then h, zero zero. Yes, yes, right. So we're completely focused on ball-shaped regions. And so when you see B, it's a ball. Yeah, so the question was about quantum corrections. And I should say that for now I'm working completely in the large N limit with a classical formula. And I will mention what happens if you include quantum corrections later. So so far it's just using, when I translated from here to here, then it's just using the leading expression. Later, if I wanna include the quantum corrections, then when I go from here to here, there'll be an additional term here that involves the bulk entanglement of quantum fields in the bulk. Okay, so what is this kind of constraint? We've translated this first law into a gravitational constraint. And it's all expressed in terms of this metric perturbation h. So this side is like an integral over B tilde of some local function of h. And the right-hand side is an integral over B of some local function of h. And so if I draw the picture, basically it's saying that if I have a metric which captures the entanglement entropy of some CFT state, and I calculate some integral over here, okay I could write it explicitly schematically. So it looks something like this, some integral over the surface B tilde. I can calculate that in terms of h, and I could calculate this one in terms of h. And those have to be equal. And that will be true for some space times m, and it won't be true for other ones. And so far it's a little bit clear, it's a little bit unclear what that really means. Okay, so it's roughly telling us that something about the metric deep in the space time has to be determined by something on the boundary. Oh, so this index m is, yeah, so I'm in this formula, I'm splitting up the coordinates into t, z, and xi. Okay, so an important thing to emphasize is that this is actually not just one constraint. We get one constraint for every possible ball. Okay, so if I have this metric describing the space time m, this equation has to be true for this ball, but also for this ball, and this ball, and this ball, and balls in other frames of reference as well. So roughly speaking you actually get one constraint. You get one constraint for pretty much every point in this dual space. Okay, if you think of maybe the tip of the ball as labeling which point your constraint is associated with. And so that suggests that maybe we actually have enough, maybe there's enough constraints that you could translate all of these things into some kind of local equation that must be satisfied in the bulk. Okay, and so it turns out that this is the case. Morally speaking, what we're going to do now is basically the same as going from the integral version of Maxwell's equations to the differential version. We're just going to make some application of Stokes theorem and then all these non-local kind of integral constraints are going to turn into some kind of local differential equation. I mean, if I have a geometry m, yeah, I mean, then some point, I mean, there are no points related to it by diffeomorphisms or diffeomorphisms refer to different coordinates I could put, so at the technical level when I wrote down this metric and I wrote down a function h, then I already sort of fixed the diffeomorphisms. Okay, so that would be, yeah. Okay, so here's how it works. And this is basically the same as how you would proceed in converting integral equations to differential equations in electromagnetism. So the idea is we want to apply Stokes theorem. And so it turns out we could find a differential form, which I'll call chi, and chi is built out of this metric perturbation. So chi lives in this region, sigma between the boundary and our extremal surface. And this form has the following properties. So if you integrate it over the surface b, it gives this expression, it gives the right-hand side of that equality. If you integrate it over the surface b tilde, it gives the left-hand side. So integrating it over b tilde gives the area perturbation. And then finally, if you take the exterior derivative, it gives you something which is proportional to a component of Einstein's equation. So d chi equals some positive function, which I won't write down specifically, times this tensor that appears, the tt component of the Einstein tensor, and then times the volume form. So the exterior derivative is a d plus one form, something that could be integrated over sigma. Okay, and then it's sort of obvious what to do now. We just rewrite this first law using this form, so we get that integral of chi over b equals integral of chi over b tilde. And that's the same as saying that the integral over the boundary of that region sigma of chi vanishes. And that's the same as saying that, so now we use the Stokes theorem. So that's the same as saying that the integral over sigma of this differential form d chi vanishes, okay. And so that says that the integral over sigma of this positive function times the component of the Einstein tensor times the volume form has to vanish. And the important thing is that this has to be true for every single region sigma that I could draw. So this integral must vanish for every region. And now it's not quite trivial to then conclude that the function, the integrand has to vanish because the integrand, this function here actually depends on the region. So it's not the same function that we're integrating over all these regions. But there's some trick you can apply some differential operator to this equation to make the integrand independent of the region. So it's a few lines of math. And you can show that the only way for this to be true for any b is, would be for the tensor here to vanish. So this is the linear, this is delta e means it's the Einstein tensor linearized around ADS. So what we've learned so far is that this particular component of Einstein's equations must be satisfied if the geometry is capturing the entanglement of a state close to the vacuum state. And then we use, yeah. No, so when I used the ball of a small size, the conclusion of that was to say that the expectation value of the stress tensor was equal to the asymptotic behavior of this metric perturbation H. And then I used that formula in the general first law to obtain this. Okay, so what that allowed me to do was to replace the expectation value of T mu nu with this asymptotic metric. I should have said this is Z equals zero. And then I imposed this constraint for arbitrary balls. Okay, and that's what we applied Stokes theorem to. And so the integral of this quantity for over any region sigma, any of these hemispherical region sigma has to vanish. And so then we concluded that this tensor component has to vanish. And we're going to use the same trick that we used before. So the reason why we're getting the TT component is that we were working in the frame of reference at this T equals zero time slice. But we could have worked in any time slice. So really the covariant version of this thing that we would get if we thought about arbitrary time slices would be this. And that's only possible if all of these components would vanish. Yes, this? Okay, so the idea was to construct a differential form. So I claim we can find a differential form which is built from H and it's derivatives. Or actually it's first derivative. So it's something that has the property that if I integrate it over B it gives the expression here. If I integrate this form over B tilde it gives the expression here. And if I take the derivative of this form it gives this expression which includes which is just some positive function of the coordinates times this is the TT component of the linearized Einstein equation. So this is something which includes two derivatives on H. So when I write down the linearized Einstein equations you have various terms with two derivatives on H and the tensor structure is some particular tensor structure. And so that's what this is. No, okay, so I just write down a differential form. I didn't write it because it's a little bit complicated but I could tell you off the top of my head I wouldn't know. So yeah, so not quite. So here's the, okay, let me schematically say what I did. So we have this equation, okay. And now I write down chi is equal to H. So I'm just schematically writing down something. Okay, if you want I'll show you the explicit expression that it has a number of terms. Okay, so okay, I write down this differential form. So here is a differential form which I'm defining for you. Okay, now I want you to calculate the integral of this form over the surface and you find this expression. Okay, now take the same form and calculate this and I guarantee that you'll find this expression and now take the same form and differentiate it and then you'll find this expression. Okay, so we just check these three properties about an explicit form that I can write down. And the only reason I didn't write it down is that it wasn't particularly illuminating. And so once I know all of those things about this form then I can go through these steps to conclude that delta equals zero. So you can even, as an exercise just starting from these three properties, you can even try to find out what this is. Probably up to some constants or something, yeah. So yeah, at some point I did this exercise but incident, yeah, so for people that are familiar with this walled formalism, there's a nicer way to say what this form is. Okay, so I'm not going to have time to explain what this is but if you're familiar with walled formalism then it's precisely this kind of combination of these quantities that walled uses to say prove various black hole first laws. Okay, so I should mention a few things. These are the components, these are most of the components of Einstein's equations but there are some other components that we didn't talk about. These ones turn out to be what would be called constraint equations. So you could show that these ones are satisfied, once you have the rest of the components these equations are satisfied as long as they're satisfied at z equals zero. And you could show that they're satisfied at z equals zero as long as h mu mu equals zero and delta mu h mu new equals zero and these are the things that already followed from the tracelessness and conservation of the CFT stress tensor. I'll just mention also that if you had included the quantum correction and I won't have time to go through this fully so had I included this term which says that the CFT entanglement entropy is equal to the area plus the bulk entanglement entropy across the surface. Then it's possible to use a bulk version of this entanglement first law to rewrite this term in terms of the expectation value of the stress tensor in the bulk. And then what happens if you follow through the whole derivation you end up getting precisely the expectation value of the bulk stress tensor as a source term for these linearized equations. No, yeah, so, well let me talk about, the rest of the lecture is about going beyond linearized equations. So yeah, so now I wanna talk about what can you say beyond linear order? And you might, so you might wonder can you get the, can you get the nonlinearized equations? So we've shown that any metric close to pure ADS that captures the entanglement of some state, it must satisfy these linearized unsigned equations. So at the classical level there's just delta emu nu equals zero. So could we show that for just any space time that captures the entanglement which is not close to ADS, can we show that that satisfies Einstein's equations nonlinearly? But the problem is that as a geometrical constraint on space times, so Einstein's equations relate the curvature to some matter stress tensor. And we're kind of starting from some universal property of entanglement and trying to derive constraints on space times. But there's not just one, if I just give you a space time and I ask, does this satisfy Einstein's equations, you can't really check it in general because you don't know what sort of matter fields there would be. I mean there are many possible examples of ADS CFT, there might be lots of different matter fields. Basically any metric satisfies Einstein's equations if you choose the stress energy tensor correctly. So when we're talking about, as we're going to do now, if we're talking about constraints that you get at nonlinear order, they're not going to be specific equations that tell you is this space time a solution of some differential equation or not. Where they're going to be, it will be some kind of inequalities. So the idea would be, okay, you have all the, any equation, any metric is a solution of Einstein's equations with some stress energy tensor. But it may be that some stress energy tensors are impossible to obtain using any kind of matter you could cook up in a consistent theory. So this is the nature of the equations that we're going to get once we go to nonlinear order. They're going to be something that will tell us that here's a metric which is impossible to get in any theory. And that will be something to do with the fact that you can't possibly obtain the required stress energy tensor using a consistent theory of matter coupled to gravity. So the first question is, I mean is there a constraint on entanglements that generalizes this first law that we wrote down? And that would be our starting point for trying to constrain metrics at higher order. And it turns out that there is a very natural constraint we can write down. And it's simply that if I have a large perturbation, so a general state not close to the vacuum state, then the entanglement entropy of this ball relative to the vacuum state, it turns out that it has to be less than or equal to the change in the expectation value of the modular Hamiltonian, of the vacuum modular Hamiltonian. So it's exactly the equation we had before, but now it's an inequality and it applies to any states. And this comes from, this comes from a quantity, thinking about a quantity which is called relative entropy. So it's defined like this. It's a way to compare two density matrices. So we can think of our state and then we can think of the vacuum state. And if I look at the ball, then this one has a density matrix, row B and this one has a density matrix, sigma B. And I might wanna say how different are these two density matrices? So I can compute this quantity. And the nice property of this quantity is that it vanishes if and only if the two density matrices are the same and otherwise it's always positive. And actually furthermore, if I consider a larger region, so I think of one region B and now think of a larger region B2, then this quantity always increases as I go to the larger region. So it's positive and it's monotonic and I can show that it's actually exactly the difference between these two quantities. So by using the definitions that we had so far, you can just check that this definition of relative entropy is equal to the change in the expectation value of the modular Hamiltonian, okay, that I'm calculating from sigma minus the change of the entanglement entropy. So in quantum information theory, this is a measure of distinguishability between these two states. Yeah, so this is a row. So it's not symmetrical. So you should think of it as something which compares the state row to a reference state sigma, okay? If it were sigma, then it would just be the, that would be delta S, that would be the entanglement entropy of this minus the entanglement. But this is a little bit different. Okay, so five minutes or should I? Yeah, okay. So yeah, so I'll just summarize the results and then if you, maybe in the discussion session this evening we can. Okay, so I can just tell you then the results. So it's a quantity that's always positive no matter what perturbation I do, okay? And so that tells me that the first order perturbations, so at first order, perturbing away from ADS, it has to vanish. So if I have some function which is positive and the first order perturbation is zero in any direction and that's another way to see this entanglement first law. At second order, this is where we get the first non-trivial constraint, okay? So at second order, this defines a quantity which actually is symmetric. It's something like a metric on the space of perturbation. So I start with a density matrix and I think about perturbations in various directions and at second order this is a quadratic form on those perturbations and the positivity tells me that this must be positive, okay? And so that would be the first non-trivial thing if I translate that to the gravity side that has to be some positive quantity. What is it? So we think about pure ADS and now we're thinking about perturbations at second order. So we talked about how this region B can be associated with a certain wedge of the bulk, okay? And in ADS that was very well-defined. I was drawing it in the global picture where you had this Rindler wedge here, okay? So in the unperturbed spacetime, there's some killing vector that lives, there's a symmetry and I can define a killing vector which I'll call XC that lives inside this wedge, okay? So there's a, it's kind of a natural, time-like killing vector and so from the unperturbed ADS point of view, this is a natural definition of time that lives inside this Rindler wedge. It's the Rindler time for this wedge, okay? And so at least at the perturbative level, like if I did quantum field theory on, or field theory on the background of pure ADS, then this would be a way to define an energy, okay? The energy associated with that time is some quantity. And at the perturbative level, I can even define that, including gravity, including metric perturbation. So I can write down some standard integral over this region, sigma of T mu nu and this vector. And this would just be the energy associated with that particular definition of time. And at the perturbative level, I can include matter contributions and gravitational contributions. This would be quadratic in the metric perturbation. And this turns out to be what this second order relative entry maps to, okay? So it tells me that this gravitational, this perturbative definition of energy for the region has to be positive, okay? And this is a quantity called, that had previously been considered canonical energy. And it was known that it had to be positive just for pure gravity. If you don't have any matter fields, it was, you could prove that it has to be positive. So the positive of the relevant entropy suggests that for any space time, even if it has matter, there has to be this, for every one of these regions, there has to be this positive energy. But then finally, you can say, well, even at non-perturbative order, even if you're far from pure ADS, on the field theory side, there's this relative entropy quantity that has to be positive. So this suggests that there's actually an energy that you could define for subsystems of your gravitational theory. So this is some arbitrary asymptotically ADS metric. It says that there's some energy, even though there's no killing vectors in this space time, and this is now just some extremal surface, there's an energy I can associate it with this subsystem. And the definition is I can define some vector X, which again behaves like this vector near the boundary. So it behaves like this vector zeta, this conformal killing vector near the boundary. And I just enforce that X behaves like a killing vector close to the extremal surface. So close to the extremal surface, it behaves like this killing vector behaves close to this extremal surface. So it's kind of like zooming in and it looks like a bit of a flat space time and then you define the Rindler time vector. So it's only defined, X is only defined close to the boundary and close to the surface. And in the middle it's just arbitrary. But what you can show is that the energy associated with that definition of time, it doesn't depend on the details of what X does in the middle. So you can argue that there's actually an energy for these arbitrary space time, for these regions of arbitrary space times, that's well defined. And then according to this positivity relative entropy, it must actually be positive. So what it suggests is that there's actually a new positive energy theorem that applies not for entire space times, all of the standard positive energy theorems for ADS or flat space. It's always a statement about the entire space time. But what this suggests is that for asymptotically ADS space times that are physically consistent, any of these subsystems associated with balls on the boundary can have an energy associated with them and it must be positive. So you get sort of an infinite number of positive energy theorems for any asymptotically ADS space time. So I'll stop there and then later today we can ask questions about more details.