 Okay, thank you is the microphone adjusted reasonably so people can hear me, but So either it is or no one can hear even here am I asking the question I'll assume it's the former Okay, so this is the second of my talks the the first one had mostly pictures this one is going to have mostly Formulas, but they'll all be quite simple general there aren't going to be any complicated formulas they are going to be Quite general and this what I'm going to tell you today. I mean is all based mainly on work that I did with my former student the vicar air 15 25 Years ago, and that would probably be the best reference for the Details of a few I mean there are a few calculations that do go in and that would be a good place for details So let me just start. I mean I'm Lagrangians and Hamiltonians are going to play the dominant role in What I'm going to be telling you about I'm the ultimate goal is going to be to derive the first law of black hole mechanics and thereby Derive a formula for black hole entropy It's just as easy to do this in a completely general theory of gravity And I will be doing it in that context. Well, the completely general theory of gravity will be one that arises from a Lagrangian as General relativity does with the Einstein Hilbert action I Just want to begin though with You know a statement that my views I mean changed. Well, really quite radically beginning 30 years ago I wrote my general relativity book 35 years ago or so or finished it and at that time, I mean You know while I recognize that Lagrangians and Hamiltonians Certainly play roles in quantization and so on. I didn't I was not really aware that they Classical field theory that they were anything much more than a mnemonic device to Memorize to remember field equations and I had already memorized Einstein's equation So I didn't really think the Lagrangians were particularly important or Hamiltonians important for anything so I you might notice in my book that I stuck Discussion I do have a full discussion of the Lagrangian and Hamiltonian Formulations of general relativity, but I stuck that in an appendix You know I now recognize that the existence of Lagrangians for a theory provides Really important auxiliary structure to the classical field theory so it is a really important thing and you know a lot of the a Lot of nice properties and ability to define usefully define Notions such as mass and so on as I will be very much talking about in this Lecture I really stem back to Lagrangian and Hamiltonian Formulations and in particular the whole symmetries and conservation laws ideas are critically Depend on that so let me the the hard thing to understand in this is well In what I'm going to be telling you there's there's not really any mathematics beyond advanced calculus in what I'll Be doing but keeping a clear picture of what I'm doing and what the Symbols mean and so on is important so I'm actually I'm going to wait So I'm going to first kind of review Particle mechanics Lagrangian and Hamiltonian in the kind of notation that I'm going to use I'm going to explain a little more precisely what I mean by these Variations of things that I'll be writing down all over the place. That's very simple and straightforward, but let me Let me go through the particle mechanics example to kind of familiarize at least the physics people in here with what I'm doing then I'll Say a few words about the field theory case and what I'm Doing there and then I'll do all the stuff that I'm intending to do in this in this lecture so let's Imagine that we have I mean the setup in particle mechanics Lagrangian particle mechanics is that we're given a Lagrangian that is some function I'm just going to do one degree of freedom of Position of the particle and I'm just going to assume here of Velocity it could depend on higher time derivatives in the theories. I'm going to do it could depend on any arbitrary number of derivatives of as long as it's finite number of derivatives of the the field variables, but this is the textbook kind of case and What one does then in mechanics textbooks is consider the variation of the Lagrangian now the Lagrangian is a function of q and q dot so if you vary well, it's nicer to kind of think it think of thing of the of the Description of the particle being a path in time a world line over all time and if at a Given moment of time we vary the Lagrangian it's since it's a function of q and q dot Well, it's going to be DLD q delta q and DLD q dot Delta q dot, but I can rewrite it in a way where I have a term that Only depends on the varied q and not its time derivative or derivative no derivative of q and a term that's a total time derivative now usually this is done by sticking the Lagrangian under an integral in this case with respect to time calling it an action and Varying that and then the manipulations that put this in this form as is an integration by parts where you Integrate by parts to take the derivative the time derivative off of q dot I think it's enormously superior not to muck things up with integrals that probably don't converge anyway and Talk about actions or whatever everything Certainly in the classical field theory I think it's just done much more Simply and nicely by never introducing an action and just talking about the Lagrangian or Lagrangian density that we'll talk about in the field theory case so Setting the coefficient of delta q equals to zero is by definition the Euler Lagrange equations of motion and One would say that your theory can be defined by Or can be obtained from a Lagrangian If in fact the equations of motion of your theory are these Euler Lagrange equations that you get by setting this to zero now in the usual Derivation of the Euler Lagrange equations you also have this boundary term you usually fix Boundary conditions to make this term go away in any case this term is sort of totally discarded in most Discussions and derivations because you're interested in the equations of motion Okay, I'm not the least bit interested in the equations of motion In I mean I'm going to be interested that there are equations of motion of this form and that they're satisfied But I I already know the form of Einstein's equation. I don't care about what this is I'm never going to look at this again. So effectively I'm going to in this entire talk Discard this term as being of no interest. What is of interest is this boundary term That I'm calling theta. Unfortunately. There's a clash of notation with Piotre's lecture who is using this lowercase theta Fortunately for another current this term is a symplectic potential as you'll see and It's just what appears in the in this total derivative is what I'm calling Theta and it's given by this formula. It's traditional in A theory where you only have a first-time derivative to call this Partial derivative the momentum so the theta in this case takes the form of P Excuse me times the variation of Q Now we can obtain something quite important and interesting by taking a Second anti-symmetrized Variation of Q. I'll explain What I mean by the second variation and so on. Let me just get through this to Because this should look at least familiar to physics students. So what I mean is I've done one variation which I'm calling, you know the variation of Q to some In infinitesimal variation to some delta one Q now I'm going to take consider Another variation. I think maybe I should just say what I really mean. I'm going to be I Mean I have these paths and I'm perturbing off the path. I'm going to consider one parameter Families well for one variation one parameter families. So lambda is just some Parameter and by Delta Q. I just mean the partial derivative of this with respect to Land I mean everything is going to be assumed to have smooth dependence on all Variables when I take a When I consider a second variation, I'm going to be Considering a two parameter Variation of the Q So the Delta one Q will be the derivative with respect to lambda one you can see this is not highly Sophisticated and the Delta two Q will be the derivative with respect to lambda lambda two so now what I'm this Theta that I have here depends on the background Q and this first Variation which is now a Delta one Q. So I'm just going to take a Partial derivative of that with respect to lambda two now and then I'm going to Antisymmetrize over one and two. So that's what's meant by this notation. I really good idea to explain it now rather than Wait till later. So that Antisymmetrize variation is what I've Written here if you do the computation in this simple case and this is an interesting Quantity because this is a conserved quantity. It's independent of what time you do the evaluation on That might not actually look totally obvious, but it That is provided the equations of motion are satisfied or in particular these perturbations satisfy the linear linearized equations of motion about the original background path, but You know, this is the equations of motion. This is the theta term if I take a second partial derivative of the Lagrangian here and set the equations of motion Terms to zero. I'm only going to have this this term if I do the Antisymmetrization by a quality of mixed partial derivatives I get zero here and then I'm just going to get the so This time derivative of the antisymmetrization of theta is zero. That's my conservation law now Conservation laws are great and this is a very general conservation law This by itself is not tremendously useful because it depends on two perturbations and usually You're dealing with one perturbation and look one a conservation law for that as well as we'll see tomorrow You can get from this a very nice and very important conservation law from this But that's a further story that will be canonical energy that I'll talk about if the background Has a symmetry Okay, so If of course if I'm looking at this as a kind of functional on the space of pads This thing is incredibly degenerate because it only depends on what p and what delta p and delta q are doing at time T naught I mean now I'm looking at it for it's conserved if these guys satisfy the equations of motion But I can define this for any perturbations and and any you know any paths whatsoever But if I What this is degenerate on is anything any path variations that have no Delta one delta p or delta q at time t naught There are plenty of paths not ones that satisfy the equations of motion that that do that But if I factor out by all of the things that this Omega is degenerate on That defines a notion of phase space so the phase space can be identified with the P's and q's at a moment at one moment of time so that's a finite dimensional space the The space of paths is some horribly infinite dimensional space and now once I use the Lagrangian and this Simplexed structure that you get to define phase space. We can define the notion of a Hamiltonian a Hamiltonian is a function on phase space Well, it we might as well pull it back though to the space of paths Whose variation in this sense is given by the Simplexed product well Yeah, I realize this needs as much explanation as the field theory Kind of explanation. So again, I'm considering this one parameter family of paths and of course my my Atlanta equals zero that's some unperturbed path that I'm considering a Function h on phase space will be said to be a Hamiltonian if under this one parameter variation the Variation of h with the lambda is this Omega that I've previously defined where I stick in this Arbitrary path variation in one of the slots in the other slot. I take stick in the time derivative The time derivative of Q So Another way of writing this would be to say that if I kind of get rid of the arbitrary tangent vector in here and look at this as a well inverse As a symplectic form On the manifold. I mean this this is really saying that the gradient of h I mean, I'm kind of dotting it into this tangent vector here the gradient of h is the Let me put in indices is the symplectic Well, I've gotten rid of an entry in this slot. I mean in this formula. I've stuck it in you know times the well you might call this the Well the Hamiltonian vector field the object that's giving you the flow on phase space This is faith. I mean I've kind of written this in Path space, so this is including when you bring it back to phase space the the momentum motion As well as that so this is the vector field on phase space. That's Telling you how things flow under Dynamical evolution so this is possibly a more familiar To people who've had You know Hamiltonian mechanics from Arnold or something a More familiar form of Hamilton's equations of motion to get the much more familiar form of a Hamilton's equation to motion we just Take the apply the inverse symplectic Form to both sides of this equation and solve for the Hamiltonian vector field Which is telling us how p and q evolve and those are given in terms of Partial derivatives of age, so that's the usual form of Hamilton's equations of motion now if you're given a Hamiltonian and Trying to figure out how the particle moves This is a much more useful form of Hamilton's equations of motion, but in the game that I'll be playing for the rest of the This lecture I mean we I don't care about the equations of motion I already know them in better forms elsewhere anyway But I can figure out what this Symplectic product is from the Lagrangian and what I really want to solve for or try to figure out what it is is the Hamiltonian So this is a nicer way of writing the equation But these are if you solve for the q-dot in this equation That's you'll just get the ordinary form of Hamilton's equations of motion So actually let me write This some of this stuff now down in the field theory case and then let me pause to see if people are happy or unhappy with what I'm saying Because If Everything is really simple if it but it if you're with me on what I'm doing and writing down but It can get very confusing if I if I've You know not clarified what I'm doing so I want to write down analog Expressions to what I just wrote down now in the case where we have Some fields on space time, so I think it would actually be a really good idea for me to draw on one board here some Fixed manifold M. It's apology is completely Irrelevant because all the calculations are going to be local calculations so Pick whatever space time manifold you'd like To work with and then I'm going to denote by just this single letter phi all of the dynamical fields that I'm Interested in considering now. I'm definitely going to be interested in theories of gravity So I'm going to be I'm going to always have in what I do later. It doesn't need to be now I'm going to have a space-time metric And then I might have various matter fields and so on but so Here I'm drawing the space-time manifold and these are all tensor fields Here on the space-time manifold, okay It is useful to keep Distinct from that something that I will be alluding to but only for heuristic purposes some really big space of all field configuration, so a given If we have a particular field, this is space-time. I'm not doing any three plus one Anything like that here If I give you a metric and all the matter fields everywhere on space-time, you know, this will correspond You know to a single point In what I'm thinking of the field configuration space Again, I'm only going to use this for heuristic type purposes But it's nice to think of it this way so I'm not introducing any topology on this or any, you know Bonn-Ox based structure or anything anything like that I Don't need to it's not going to come up particularly or be relevant to anything I will The only thing that I will do where conditions on these fields will be relevant is I will Occasionally as you see take integrals and then I'll need these fields to be Such that these integrals converge and well There'll be some boundary terms arising in those integrals and I'll need asymptotic conditions to make sure those those Are satisfied, but that's all That I'll be that I'll be using okay, so Let's see what you Should have already mastered this slide. So I should while I was talking over here. So I may not Okay, so I I'm interested in Lagrangians One I'm not going to put them under integral signs. I've already said that's one difference that you'll see But another difference that well Peter introduced Lagrangians in the talk yesterday and he introduced them as densities and had you know square root of G's and so on in the Einstein Hilbert or any other gravitational Lagrangian it is tremendously convenient At least from my perspective Not to work with densities, but to take the Lagrangian That's going to be defined here on the space time to be Just writing it down with indices, though I'm not actually going to use indices there I will use boldface letters to denote forms but to take the Lagrangian to be an n form So this is something ready. This is The m I'm taking to be in dimensional because again, there's no Dimension doesn't play any role in and topology doesn't play any any role dimension doesn't play any role and the Form of the field equations do not play any role in anything that I'm going to be saying so there's no reason to make such restrictions, but you're also welcome to Set n equals 4 if you like etc. And you won't lose much Making that that distinction so I'm going to assume that We are given this n form that is a local function So at any point here in the manifold, I'm not in field space here I'm here on the space time manifold. This is a function of Phi This grad is some arbitrary derivative operator. I mean will But We're only allowed That's a different n. That's a k. Let's say finitely many derivatives, but you can have as many as you like okay, and the same Standard classical mechanics Trick that one can do is if you take a variation of L and you know now all I mean by this Delta symbol Is that this Phi well the Phi of course is a function on space time or Tensor fields on space time, but I'm going to consider one parameter families or when I take second variations, I'll be considering two parameter families of it and the the Delta Phi is again just the partial derivative with respect to Lambda now the Delta Phi and Something like the Lagrangian that's a function of Phi. Of course I can take Its variation is just also the partial derivative with respect to Lambda when I go along this one parameter sequence of Phi's the Delta Phi is kind of nicely so this is Phi thought of as a field on all of space time the Delta Phi is kind of nicely thought of as a tangent vector in this field space But again, that's this is all just you know for help in thinking Okay, if I take The variation of L in precisely the sense that I'm taking that is stick in this Into L and take the partial derivative with respect to Lambda again Well if I write it this way it doesn't really look that obvious that you can do it, but if you put it under an integral sign You would immediately be led to do integration by parts manipulations That would put the variation of L into a form where there is something that depends only on the background Times varied Phi's with no derivatives of Phi whatsoever Okay, and then well if we were doing this with a Lagrangian density The way Piot was We'd end up with a total divergence Well again my theta is not his theta, but this is what Piot was writing down the Corresponding formula if we have if we're taking L to be a form instead of a total divergence It's just an exact form So this is some n-1 Form on space time and this is locally constructed from Phi and Delta Phi and you can write out you can work You can kind of do all the integration by parts manipulations and work out what it is So e equals zero are the Euler Lagrange equations of motion and I'm not interested in them I'm going to throw them away, but the boundary term I am interested in and in the same way as I've explained we can take a second anti-symmetrized variation of This theta so now on space time the theta over here this theta is an n minus one form So the omega is also which is just obtained by taking partial derivatives with respect to lambda You know lambda two or whatever That's also an n minus one form on on space time Okay, and now Well, this is all that I had to do in the particle case where this was a Space was You know Effectively zero-dimensional. We were only worrying about things varying in time and this was a Zero form there by or whatever Now with this being an n minus one form If we're going to get a number out of it, we have to integrate over a surface and what I'm going to do is consider this object as this Well, symplectic current or symplectic n minus one form Integrated over a Cauchy surface So Here of course if the Cauchy surface is non compact I would have to assume some Symptatic conditions on the fields that would guarantee that this integral would converge if I want to if I want to work with this So I'm going to implicitly assume that But this is well The corresponding conservation law is that this form is automatically closed and If we have two Cauchy surfaces Then if there is no flux through the boundary Stokes theorem will tell you that the omega is Conserved between here and here of course there might depending on the boundary conditions or asymptotic conditions there might be Flux through the boundary, but anyway, I'm I'm I'm gonna More or less implicitly assume that there's not and when we get to asymptotically flat space times and things like that We'll be imposing boundary conditions so that there isn't any such flux and this will be conserved Okay, and now given this we can again define the phase space of the theory by factoring out all The degeneracies of this omega that is defined by just considering this one Cauchy surface so we get basically get rid of all the field behavior Off Finitely off the Cauchy surface is irrelevant and only is completely irrelevant and only some finite number of derivatives Time derivatives normal derivatives of the fields are going to play a role in the omega Because the original Lagrangian only depended on finitely many derivatives so Whatever the omega depends on or this omega really depends on is going to effectively define the phase space I actually don't know how to define canonical Coordinates in any natural way if we have a theory that only depends on one time derivative then There's a similar obvious way of doing that. There isn't as Far as I'm aware any obvious way of doing that quite generally, but I don't care That's not relevant to anything that still allows me to define the phase space and then I can define a notion of a Hamiltonian but Exactly the same way as I did but now I mean in the particle case You know the only Dynamical kind of direction was time. I mean, you know We were kind of considering a zero-dimensional System with well in our case one degree of freedom or could be finitely many degrees of freedom But now over here in the space-time manifold. I can consider the evolution Along any vector field of my choice I'll call that vector field C and then I will say that well that's a Vector field in the space-time manifold. I'm really Well that induces that motion in When you move the fields along this vector field that of course induces a motion of the fields in phase space and I'm interested in whether there is a Hamiltonian describing that motion, but I have a different notion of Time evolution for every possible choice of vector field. So the there's an important Subscript C here on the Hamiltonian because there isn't such a thing as the Hamiltonian It's the Hamiltonian conjugate to a notion of time translation and time translation can be Chosen to be any vector field whatsoever There's no reason the vector field has to be time-like for example. I mean in order to Ask whether there is a Hamiltonian Conjugate to that so notice I'm not saying that there is a Hamiltonian. That's going to be a major question I'm saying if you can find an an object that satisfies this equation whenever the Phi satisfies the Euler Lagrange equations of motion that you get from the Lagrangian Then I will call this object a Hamiltonian. Okay, so let me Stop there and Yeah, yeah it's The Cauchy surfaces I were drawing here are surfaces in space time So they belong in the same diagram where I had all these other things Good these are that's exactly the kind of question that yeah, yes the in in the Lagrangian here, this is just some arbitrary derivative operator Well, there would be if I have to you I'm about to get to Diffie morphism covariant theories so You'll be happier On this point is when I get to the next slide But for what I'm writing down here and for all the formulas on these pages on the slides I've shown so far the the Nabla is some arbitrary fixed Non-dynamical derivative operator. I mean I can't Take it to be Well, it would be confusing to take that Dynamical because I I want to represent the dynamical fields by tensor fields I could represent it as a dynamical field by taking the difference between the derivative operator and some fixed derivative operator, but that wouldn't that wouldn't help your question I so as much I want to just introduce a fixed derivative operator and I mean people normally would do that because they're working in coordinates and they'd write down partial derivatives and The coordinates, so that's a choice of fixed derivative operator The Lagrangian, I'm sorry No, no you can uniquely The the e is unique The theta is not unique. You could add an exact form to theta, but the e is unique There will be a change in that I will get to what you you know What you can fiddle around with and what you can change by making changes of that sort that's relevant because We'd like to know whether things are unique or have arbitrariness and that that's very relevant to everything. Yeah Yeah Yeah, so that I'm really being formal on I I mean there I'm I mean I that the the factorization I'm doing over here, so I am kind of cheating But the Omega now is an object over here because I've done the integral It only depends on I mean of course I have an arbitrary choice of Koshy surface That came into the definition, but the Omega quantity I can think of as a quantity over here now Yeah, right, so it would more properly be thought of as a two form so it's an the the little Omega is an and this is Can be very confusing the little Omega is an n minus one form on Actual space time the capital Omega would be most naturally thought of as a two form Because it's anti symmetric and it depends on a pair of tangent vectors on this field space But then it's highly degenerate on this field space and so you You know can do the factorization to get to a phase space. No the Omega is Over here is a tensor field so it there's no Yeah, I mean the the the degeneracy subspaces have to be integrable if that's possibly what you're worried about and Formally they are integrable now You know if I really wanted I mean again you could just explicitly write down what the phase space is without having to try to really work with the infinite dimensional Manifolds that I haven't even given a topology to or anything like that and Kind of really do the factorization in the but the Yeah, but I think what's bothering you is completely legitimate that the the the subspaces that you get of Degeneracies have to in fact be integrable and then you can really factor and talk about but again formally they are integrable and If you don't want to do that you could really just explicitly I mean for general relativity I can tell you explicitly what the phase space is and it will correspond to this Yeah, yeah Yeah, yeah, so in fact I will be sort of doing everything Working in a little neighborhood of one point so I'm going to choose I mean everything is going to be very local in every sense. I mean The calculations are all local in space time, but they're all local in field space To so I'm going to choose something when I'm perturbing about a space time I will choose something. That's a cushy surface of the background space time These are very good questions. I mean Okay Yes No, I'm I'm I have no idea if the equations are well posed or not and it doesn't enter anything I'm not saying that that there's a you know I'm Defining a phase space and I'm defining the structure. I'm not Saying anything about existence and uniqueness of solutions to the equations of motion. Okay? I'm just telling you here's the Initial data, but I'm not telling you that if you give me the initial data I can find a solution or it's unique or anything like that. I Mean it would be reasonable to call the phase space to the initial data Space, but I I'm not saying anything about well-posedness Okay, well, I think the rest of this will now be easy so now Onto The theories that I'm interested in which are diffeomorphism covariant theories which Where the Lagrangian is? Entirely constructed from Dynamical fields, there's no background structure now I may have needed to do some extraneous derivative operator as we were saying to write down the Lagrangian, but I want to to if I do a diffeo of all the dynamical fields I want the Lagrangian to change by a diffeo and That actually implies that the Lagrangian can be Rewritten no matter how I first wrote it with a background derivative operator It can always be rewritten in the form. I've written here This is essentially the Thomas replacement theorem of the goes back to the 1930s. I think so I can get rid of the background derivative operator and Rewrite the Lagrangian as a function of the metric the Riemann curvature Symmetrized derivatives of the Riemann curvature. Oh, these are the now the derivative operators associated with the metric so now there's no explicitly no background structure and then the matter fields and their symmetrized derivatives So that's the form the Lagrangian of any diffeomorphism covariant theory has and The diffeomorphism covariant theories are very nice because they automatically acquire another current which is exactly what pewter was Talking about yesterday. So I'm using j for rather than theta for the another current It's associated with some choice of vector field and the definition of it is it's just the Symplectic potential the thing that you got The boundary term that most people throw away foolishly because that's the important thing from the Lagrangian in the varied Lagrangian, but the the theta depends on a perturbation a variation of phi and you stick in depends linearly on that and you stick in the lead derivative of phi with respect to this vector field in that slot and then you subtract off c time Dotted into the Lagrangian this corresponds exactly to pewter's formula If you take the d of this n minus one form Well Yeah, let me I'd better not take the time to go through each derivation even though There is no derivation. That's more than a line or two, but d theta Is given by this formula that has an equation of motion in it and Anyway, if you use appropriate this term When you take the d of that Anyway, you can you can manipulate the formula that I Gave you the the delta there is In that formula is the lead derivative So anyway, you get this Formula which tells you that the j is closed and it's closed for an arbitrary vector field If the equations of motion are satisfied There is a much stronger result though. This is a very standard result. There's a much stronger result That says that Even if you don't satisfy the equations well, okay, let me The fact that that this j is closed for all c Let's consider the case where the equations of motion are satisfied So we're off of phi that satisfies the equations of motion The fact that this object is closed for all c in fact implies that it's exact There's no topology involved in there. It's just a straightforward calculation that you cannot have something That's constructed out of c and finitely many of its derivatives That's closed for all possible Values of c. Well, this is if it's linear in c as it is It can't be closed unless it's exact So that's a separate argument and it's you know, not a totally trivial computation, but it's just a computation So in fact the stronger result is that even when the equations of motion are not satisfied The j can always be written in the form of c with no derivatives Times something that vanishes when the equations of motion hold plus an exact form These c's that come in here into this formula are naturally called for reasons I won't Take the time to get into the constraints of the theory again This has nothing to do with well posed initial value formulation, but the but One could still legitimately call these objects the constraints for general relativity, which I will do in a minute These are precisely the usual constraints. I mean if you Choose a hypersurface the normal If you choose c normal to that that's called the Hamiltonian constraint if you choose Let's see tangential these those are the momentum constraints And this object is going to play a big role in what I'm going to do that You obtain from this another current and that's what we called the another charge So so the another charge is an n minus two form and it's locally constructed out of the dynamical fields And the vector field see and of and of course only finitely many of its derivatives It actually can have more derivatives than you had in the original Lagrangian Because you integrate by parts and get pick up more and things like that, but But it's still only finitely many And in fact you can just chase through The whole procedures that I've given in general here And find that This another charge in fact always takes This form it can have a term That depends on c with no derivatives It can have a term that depends on anti-symmetrized derivatives of c It can have a term that depends on the lead derivative Of with respect to c of the dynamical fields And it can it can of course have an exact form because that you can trivially See however as was raised in a question you Have freedom to redefine things in particular as was already mentioned You could have added an exact form to the Lagrangian You could also have added an exact form to theta And you can certainly add an exact form to to q And when you take that into account You can use that freedom I mean in particular the freedom you have in defining the theta Can be used to just get rid of this term Etc. You you can Show that in fact without loss of generality you can assume that by making by these redefinitions You can assume that q takes this form And indeed you can compute Once you've reduced it to this form What this coefficient x is And strangely enough there's a simple The formula isn't all that simple because the Euler Lagrange equations for a general Lagrangian are not all that simple But rather remarkably and I don't have any understanding of why this is true If you took the Lagrangian Written in this form which you can do And you pretended that the Riemann curvature tensor was an independent Dynamical field having nothing to do with the metric You could then compute the equations of motion for the Riemann tensor Okay Those equations of motion is what comprise I mean, of course, they're not equations of motion and they're not satisfied But that formula For the equations of motion for the Riemann tensor is what this x Is made up of so that's an explicit formula for what x is Okay So now What about Hamiltonians? Is there a Hamiltonian? for Some arbitrary vector field Well Here I have written out the full derivation Of everything remember that a Hamiltonian is We're supposed to We define this sort of symplectic product this way and we're looking for something whose variation Is related to the symplectic product in this kind of way So can we find such an H? Well, I'm going to write down a formula for the variation of the nether current So one of the this part comes Just directly from the definition of nether current Here, I'll just vary that oops But now I'll use the formula for the variation of the Lagrangian and Remarkably that gives me theta terms that come in as a second anti-symmetrized variation One of the variations is the one that i'm considering here. The other variation is Do an infinitesimal gauge transformation associated with c, but this is the anti-symmetrized variation so we get a formula then That says That the symplectic product of an arbitrary variation with a gauge variation Is the varied nether current plus this boundary term Okay, but we also have this formula for the nether current So we can substitute that in here and see that the symplectic product of a variation and lead c phi So i'm I think I didn't say this explicitly but Phi is a solution to the equations of motion here the delta phi is any variation So that's exactly what we need to be delta h Whatever that We get this formula For the symplectic product. So now if we can write this thing Sorry for jumping around if we can write this this is exactly what we Have computed if we can write this as the variation of something Then that something is a Hamiltonian So we've figured out Well, essentially the necessary and sufficient conditions So can we write this as the variation of something? Well, yeah this term. Yes We just pull out the delta. It's the variation of the constraints. So If there's a Hamiltonian There'll be a volume integral. That's a pure constraint form Here we can also pull out the constraints. So there'll be a boundary term That because we're integrating this exact form and i'm using stokes theorem and i'm Not assuming that things go to everything goes to zero asymptotically So I can pull out the delta here. So the whole question is Can I pull out the delta? Here if I can there's a Hamiltonian and we have the formula and if not then Not well this thing you can't in general pull out the delta from Because then this would have been an exact form over here on field space And the symplectic product would have been zero But asymptotically Which is all that we care about for this The omega can go to zero and with asymptotic Flatness boundary conditions at spatial infinity it will go to zero and you can pull this out so for asymptotically flat space times or for many Much more general classes of space times. I mean asymptotically anti-decider will work Lots of other cases, you know may work You can find An object so that when you Just look at this asymptotically the variation Of this boundary term does give you the theta And in that case you have a Hamiltonian for the theory It's pure constraint plus a surface term So if you this is now an arbitrary diffeomorphism covariant theory by the way, right not just general relativity and so if the Equations of motion that I'm calling the constraints are satisfied So on shell as the particle physicists Say then in any diffeomorphism covariant theory if you have a Hamiltonian it will be purely a surface term Of course, if you're doing a compact Cauchy surface where you don't have a boundary Then the Hamiltonian is going to be this pure constraint and will vanish Although I will allow Piotr to make it five because this equation This equation allows you to add a Field independent constant to the Hamiltonian. You're only getting the variation of it So it it can be five, but it has to be five for every solution I'm sticking with zero in the compact case, but you have this surface term So if we use time translations For our vector field. So what would that mean in the asymptotically flat space time? If I take some vector field that is as we have an asymptotic notion of a time translation if I take it If I use that notion The value of the Hamiltonian is naturally what one would call the energy or in Gravitational theories the mass If you choose the asymptote if you choose the vector field for which you're Trying to get a Hamiltonian to asymptotically be a rotation Uh Then the corresponding Hamiltonian well, there's a minus sign Conventionally in that I mean having to do with the signature of the metric and time like versus space like directions That would naturally be called the angular momentum and this would be the formula for the angular momentum now one nice thing happens with angular momentum in fact because The phi you could choose your asymptotic boundary so that phi is tangent to the boundary in which case this boundary contribution automatically vanishes so Completely generally then in that case you have a formula for angular momentum You always have a Hamiltonian and you have a formula for angular momentum. That's just the another charge But that's not true for the Hamiltonian. So let's look at this in general relativity which I think will definitely help make this more concrete and also this Sheds a lot of light on the adm formula for mass An angular momentum versus the so-called comar for well not so called the comar formula for So the Lagrangian in In general relativity is Just the scalar curvature, but I'm taking the Lagrangian to be an in a four form in four dimensions So it's the spacetime volume element Times the scalar curvature is the differential forms version of the Lagrangian It's not hard to compute the symplectic potential, you know do the Going back a bit far, but you can compute what the boundary term theta is when you do the variation of this Einstein-Hilbert Lagrangian And that's what you get for theta if you take a second anti-symmetrized variation of this to get the symplectic current it You know the expression Depending on how compactly the notation you use is you know would take At least half of this slide to write out, but it's completely straightforward to do that Go through the calculations of the another current in the another charge the another current Is what I've written here and Uh, it's not hard to see that it's conserved and not all that interesting in its own right The another charge Is given by this formula which people familiar with the comar formula Would recognize I mean this is of course for an arbitrary vector field here But this would be recognized as the integrand in the comar formula for angular momentum And indeed if you look at an asymptotic rotation the Hamiltonian formula for angular momentum is Just this if phi happens to be a killing field. This is of course the comar formula for angular momentum But it's also the adm formula for angular momentum. I mean adm would write that out in you know slightly different ways with The canonical momentum of the metric and so on in the formulas, but this is a equivalent to the adm expression Okay, if we Go now to the time translation case Then indeed we can find this appropriate boundary term And the boundary contribution, you know this b term that Comes in You know to the Theta, you know that reproduces the theta when we take a variation near infinity So the t dot b comes out to be this Uh The Hamiltonian then again if we're on shell is purely this the surface term and in fact These boundary terms here with the gtt and so on Exactly cancel similar terms that arise in the comar like Formula and what you end up is the with the formula for adm mass uh now If t was a killing field And you take the comar formula but fudge The factor to be a 1 over 8 pi instead of the Correct 1 over 16 pi that would appear from another charge Then in fact in the stationary case You'll get agreement with the adm mass, but if you try to do this in the non stationary case, you'll get complete nonsense I mean you'll your your Answer will depend on your choice of vector field that represents asymptotic time translation symmetry Okay, so now what does this have to do with black hole thermodynamics and so on? Well, we've got all the machinery now set up to Derive the first law of black hole mechanics Not just in general relativity, but in an arbitrary Diffie morphism covariant theory of gravity So let's go back. I was doing general relativity as an example, but let's go back to a general Diffie morphism covariant theory of gravity And if I integrate We have Somewhere here It's got to be after this because this is yeah, we've got well actually I don't Okay here we have this Formula for omega So this is the formula for the symplectic product of an arbitrary product off of some background solution for The symplectic product of an arbitrary perturbation with a gauge perturbation I mean the c of Dynamical fields is just how the dynamical fields change under an infinitesimal diffio. So I'm Totally justified in calling that a gauge transformation so Now let me consider the situation that We left off with at the end of Last lecture Where we have a stationary black hole But I can idealize the final state of the black hole as having a bifurcate killing horizon So off here is infinity Here's the actual event horizon of the black hole But I'm going to take the You know the idealized final equilibrium state of the black hole And I'm going to take a Koshy surface For the exterior so that will Start at the bifurcation surface and then go off To infinity and I assume it's going to be asymptotically flat near infinity So let me take the identity the general identity again I probably shouldn't be paging back and forth This much, but you know, there's a general Identity that comes from this with the j substituted in there I'm going to take that identity and integrate that over this Koshy surface For a stationary black hole and I'm going to get a Correct formula We'll see if that formula is interesting or not I Wouldn't be spending all this time on all these other things if the formula wasn't interesting so Yeah, so let me Of course assume that Let me take the C that appears in these formulas to be the horizon killing field The one guaranteed by the Hawking rigidity theorem That has the property that it That killing field automatically vanishes On the bifurcation surface That killing field as I discussed at the end of The last lecture Is a linear combination of a time translation and a rotation With a constant here in front that's usually called the angular velocity of the horizon Now my My black hole is stationary So the And invariant in particular under the horizon killing field. So in this formula The lead c of phi is zero because My background solution is invariant. My perturbation is an arbitrary perturbation. I'm not Assuming anything like about that, but I am going to assume that my background my perturbation Satisfies the linearized einstein equation and in particular the linearized Constraint equation. So in this formula, this vanishes because that's zero This vanishes because that's zero And all I get Boundary terms are equal to zero But now I'm going to get Boundary terms from infinity and boundary terms From the horizon Well, the boundary terms from infinity we were just we just got through evaluating that's Those are just the well You know, that's just that well, these are not varied Hamiltonians, but Where do I have a varied Hamiltonian? Okay, anyway the boundary terms from infinity are just going to be You know for t is just going to be the varied adm mass And for phi is just going to be the varied adm angular momentum so The boundary term the sum of the boundary term from infinity and the boundary term from the bifurcation Surface, which on these slides. I'm labeling as sigma Vanishes the the These are just give you the varied adm energy and angular momentum From the the horizon Well, you get This is a relatively nasty term but c the theta is but the c vanishes On the bifurcation surface, so you just get the varied another charge So that's what this general formula reduces to in this particular case now if you go back I'm not sure I should go back, but if you go back to the formula for the another charge And vary it as it's supposed to appear there It follows Since the derivative of the killing field comes in that's related to the surface gravity You end up finding that the varied another charge is just the surface gravity of the horizon Times the variation of a quantity that I've written here as s Where s is just this x quantity, which was just The riemann kind of equations of motion now for general relativity The action is just matrix times r. So when you vary that with respect to r You just get metrics and you just end up with an area element here and the s is just One quarter the area when you work through the numerical factors for an arbitrary theory of gravity There'll be more complicated terms with the riemann tensor in it and so on but That is Whoops That is what you end up with In an arbitrary diffeomorphism covariant theory of gravity So let me quickly end now with Since i'm already a minute over I just realized looking at the clock But going back to the subject of black holes and thermodynamics. Whoops You know a body in thermal equilibrium Is somewhat analogous to a stationary black hole Bodies in thermal equilibrium are characterized by a small number of state parameters stationary black holes are Similarly characterized by the very non trivial black hole uniqueness theorems by a small number of parameters We already saw that the surface gravity of a black hole is Constant over the event horizon of a stationary black hole analogous to the constancy of the temperature I've now shown while I've generalized this to allow Maxwell fields and so on but that's fairly straightforward That the variations in mass angle momentum and charge are related to the variation in an area by an complete analog of the first law of thermodynamics Uh, and we've already seen yesterday that the area theorem is an analog of the second law of thermodynamics and It goes beyond that because the analogous quantities in these formulas Well in particular mass and energy Are in fact the same physical quantities We're going to have to wait till my last lecture though to Worry about whether surface gravity and temperature are physically the same quantity and Then we can talk about whether area really represents entropy, but I've already gone three minutes over so I Had better stop here