 Okay, welcome back. We we are ready for the third lecture by Robert world. Okay, let me Check the microphone again. Is that Reasonable okay again this time people are even nodding so okay, so This is the third talk, which I'm going to be talking about Dynamic and thermodynamic stability of black holes, but let me back up the To the last few slides of the previous talk, which I rush through in the last minute or two because that kind of summarizes Where we are and will also indicate where we're going although so Again, you can Think of some and now I mean you can think of a of a black hole that settles down to a stationary final state as being Certainly in a way analogous to a body an ordinary, you know non-general relativistic Physics in your laboratory, you know a box that you've just filled with a gas If you wait a while it will settle down into a thermal equilibrium state. I will talk more about that in fact today and Something I didn't Emphasize I mentioned it very quickly right at the end when I was rushing to go through these slides but just as Bodies in thermal equilibrium and again, I'll talk a little bit more about this in a few minutes are characterized by some small number of state parameters for this Box of gas just the energy Well, the only thing that's really important about the box is its volume as far as the Properties of this particular system a gas is concerned the number of particles in the gas would be another state parameter Well analogously to that it is a non-trivial fact that in general relativity in four space-time dimensions There are the stationary black holes comprise only a small family by a parameterized by well in the Vacuum case parameterized only by two parameters mass and angle momentum I mean Nico has already told you a lot about curb black holes. That's what I'm talking about there the unique Stationary vacuum solutions of Einstein's equation at least in four dimensions if we allow electromagnetic fields Which I won't be talking about but I'm just including it here for generality then black holes could also have a net electric charge and The results I showed you already about black holes are really remarkably direct analogs of Standard laws of thermodynamics now why these laws of thermodynamics are called the laws of thermodynamics. I mean is more Historical or what but anyway they are features of thermodynamics. I mean I Whether they should be considered the fundamental principles of thermodynamics or not I think one could argue about But it's anyway, this is a nice way of comparing the two. I've there is a so-called 0th law of thermodynamics which Well really says that if you have a body that's Locally in equilibrium you can define a notion of temperature I think it's usually assumed that all the matter one is considering on very small scales has reached Equilibrium and then to globally be in a true equilibrium state the Temperature of that body has to be constant Well in black hole physics if we have a stationary black hole Then as I explained in the first lecture it's event horizon will be a killing horizon and in that context Where the black hole is stationary? We can define the notion of a surf of surface gravity and then it's a non-trivial theorem that that surface gravity has to be constant over the event horizon of the of the black hole what I spent the the entire lecture last time telling you was that there is a First law of black hole mechanics I Didn't include electromagnetic fields that didn't in thereby include Charge and the potential at the horizon In the derivation so you can ignore this term, but I'm as I say I'm putting it in for To show you the further Generality that one can one can derive this in the same manner as I did if you add magnetic fields as matter so that law said that in fact The variation in mass and angular velocity of the horizon times variation of angular momentum Which was this surface term we got from infinity integrating this identity or this equation involving the symplectic product of Of an arbitrary perturbation with a gauge transformation Taking the diffeomorphism to be the horizon killing field We got a surface term at infinity which was the difference between those terms and we got a Surface term at the horizon which I explained would evaluate to this in general relativity But this then is a formula that says if you have a stationary black hole and perturb it The change in its mass and change in angular momentum And if you include this change in electric charge will be related to the change in the area Of the event horizon of the black hole This is very analogous to first law of thermodynamics the First law of thermodynamics will always have a variation of the energy term here, and it'll have temperature times variation of entropy Appearing here, and then what appears here depends on what systems you are considering I've written it for a box of gas or whatever, but You know these are often referred to as work terms or something They're of exactly the same character as these terms if you Considered a rotating system in thermodynamics. You'd have a term just like this if you Considered a charge system in thermodynamics. You'd also have a term just like that so these are remarkably analogous and then even more remarkable is the analogy Between the area theorem that I explained to you again the very first lecture Which is says that simply says that the area of the event horizon of a black hole never decreases with time There aren't very many laws of that character of physics laws that tell you things are non decreasing with time One of them is the second law of thermodynamics, which says which asserts that entropy is non Decreasing so this is a remarkable analogy, and if you look at the analogous Quantities well, I mean I've written them in a form where you can kind of Read off the analogous quantities, and I mean let's not forget the The advance and the the laser pointer is working, but the The laser pointer part is but that isn't no Well, we can look at this slide for the analogous quantities are mass, and well Just disappear. Yeah No, I have no idea what happened, but Okay, yeah, so let's not forget the zeroth law that has surface gravity and temperature as analogous Now we've got well we have area and entropy as being analogous and They come in the same way with surface gravity and temperature We have mass and energy as analogous, and then we have these various work terms so one thing that's also quite striking is that mass and energy are the same quantity the same physical quantity in General relativity on the other hand The temperature of a black hole in classical General relativity is clearly absolute zero so the physical temperature, so that would seem not to have a lot to do with surface gravity in terms of I mean the mathematical Relationship is perfect as I described it, but the physical relationship isn't and then that would also Call into question perhaps Whether one should take seriously the idea that area should physically represent entropy So I will return to these issues In the last talk if if we finish early I'll even start it maybe in in this talk but I'm going to move on now to Staying within classical general relativity extending this analogy actually Considerably further in terms of well going beyond just sort of first law stuff and actually showing you that Thermodynamic stability, which I'll have to define for you is in fact equivalent to dynamic stability of a of a black hole in general relativity now I'm in this talk I will Restrict to general relativity rather than an arbitrary theory of gravity But I will allow the dimension to be arbitrary which will be useful for getting more examples than just the Kerr black hole But before I start on this I thought since I didn't really Leave time for discussion and questions at the end of Last talk this might be I mean since I've just summarized the main conclusions just if anyone had any Questions before I go on any reasonably quick questions because I Shouldn't end this in a rush. Okay, so what I What I'm going to talk about now here is well stability of black holes and also For reasons that are Important in terms of extending the thermodynamic analogy. I'm going to talk about black brains Which I'm going to define in a second so black holes well in four dimensions Kerr black holes are Physically very relevant and one would like well one believes they're physically relevant But they wouldn't be unless they're stable that is if you perturb them They will go back to being Kerr black holes may be with slightly different parameters But they won't disintegrate or become naked singularities or and become naked singularities or do Other things and I think I don't really have to motivate You know the the fact that understanding the stability of black holes is something of Interest by a black brain in this talk. I'm just going to mean some higher dimensional so Dimension D which would be the dimension of the black hole where we want to consider Plus some extra Spatial dimensions space time so if we have a black hole in D dimensions described by a metric DS square D that I've written down That's only supposed to tell you that I have a metric in mind here I don't know writing it down may not be the quite the proper terminology But anyway, if I add extra flat dimensions So just add You know take the Cartesian product of whatever manifold I had here with RP and with the flat metric on RP That's what I mean by a black brain. So that's a Solution that'll be obviously a solution of Einstein's equation if this was in D plus P dimensions Of course, it's not asymptotically flat, but You'll see why I want to consider these because at least from the point of view of thermodynamics stability They'll be interesting and what I'm going to introduce based on the formalism that I Showed you in the last talk is a quantity called a canonical energy Which I will argue Or show it well show slash argue the positivity of this quantity in The case of axi-symmetric perturbations Rotationally symmetric perturbations. I'll explain where that restriction comes from in due course That is the positivity of this quantity. It's a quantity quadratic in the perturbation of the black hole that if this is positive for all perturbations then The black hole is stable at least in some sense and if this quantity is not positive definite on All perturbations of the black hole Or is not positive then then The black hole will be unstable in a certain sense more specifically if this is non negative Then it's fairly straightforward to show that there don't exist any Exponentially growing linear perturbations. That's a weaker result than the black hole Non-existence of exponentially growing modes is a weaker result than Linear stability, but it's it's at least a result very much in that direction and on the other hand if This can be if this canonical energy can be made negative for for any allowed perturbation then That perturbation if you gave the initial data for that perturbation It could not settle down to a stationary black hole state at late times So in that sense the black hole would have to be unstable And in fact if that perturbation is of the form of a time derivative of another perturbation Then you can prove there has to be in fact an exponentially growing solution so I Mean this I think is of some interest definitely for just the general Study of black hole stability. I mean you just have a single quantity that Whose positivity or lack thereof is in these weak senses Telling you about the stability, but what I'm the reason I'm giving a you know spending one of my four lectures on this is that I'm going to show you that the positivity of this quantity is in fact Precisely necessary and sufficient for thermodynamic stability of a black hole and that really relates further the dynamic and thermodynamic properties of black holes and well Further indicates the I mean basically if the area is a maximum for the given is a local maximum for What I'm saying here is if the area which represents the entropy is locally Maximum you know in terms of second variation for fixed Mass and angle momentum then your black hole is stable or fixed mass angle momentum and charge if you have those parameters So I mean again, I'm going to be using the general formalism that I'd introduced last lecture. So that really would a lot of what I'm saying applies to other theories of gravity, but the specific Results and certainly the you know particular form of the canonical energy Will just be for general relativity that however, there's no difficulty doing that in d dimensions and Then it's quite easy to include matter fields and also other asymptotic conditions Asymptotically ADS etc You know there's that that's a completely straightforward generalization of the things that I'm saying Okay, so the hard part I think the hard part of yesterday's talk was explaining the notation and the Framework or something and the hard part of this talk is explaining is Forgot about general relativity for now. Just what do we mean by thermodynamic stability and where does where do those notions arise from? Well as long as you know how to fix it will be okay. I'll be on this slide for a few minutes anyway Okay, so now we're in the context of out of general relativity ordinary Physics and I want to consider a finite system, but a system of course with a very large number of degrees of freedom That has a time Translationally invariant dynamics, so it has a Hamiltonian. That's time independent Okay, then Automatically the energy which is just the value of the Hamiltonian will be conserved There may be some other conserved quantities and there may be some also just some Parameters in the Hamiltonian you might be putting your System in an external magnetic field and the magnetic field might then be a parameter Appearing in the Hamiltonian or whatever so there will be some number of well energy always will be there, but there may also be some additional state parameters as they're called that will be Unchanged during the dynamical evolution. I mean Maybe you'll want to change your magnetic field, but I'm going to analyze what's going to go on with the magnetic field Constant then you can change it later After we've done the analysis to do if you want to do an experiment with that so underlying the thermodynamics is the Idea and statistical physics for that matter. I mean is the idea that if so now I'm drawing phase space here This is not space time or anything like that. We're Putting that aside for a minute, but this whole board is phase space over here, I'm looking at some surface where these State parameters are constant. Well if these are parameters in the Hamiltonian then they're just constant anyway But you know we might have a conserved angle momentum or some other Conserved quantity here, so the the dynamical trajectories are going to Live this is a dynamical trajectory now that I'm drawing on this energy shell Let's just pretend that energy is the only Parameter, so we just have an energy shell in phase space That I'm drawing here the key underlying assumption is that this Trajectory is well. I use the term that I made up effectively ergodic this It well it It typically won't be literally true that this orbit passes arbitrarily close to Every point on the energy shell and if it did that wouldn't really be too relevant to physics Anyway, because the amount of time it would take to get close to every point is way longer than any Observation time in any laboratory, but the idea is that this does kind of fill up The energy shell in phase space or it inhabits a good sampling of the energy shell in phase space And it it well if you have strong assumptions about The ergodic behavior and average over all time It is a theorem well under strong hypotheses that in fact the amount of time that Any orbit will spend in a given volume Will be proportional to that volume right, so Systems will tend to spend Equal times in equal volumes So if I want to know what fraction of the time would I expect to find the system in the region that I've drawn if I take the phase space Volume on this energy shell of this region and divide by the volume of the energy shell That is that should be the right answer and I'm going to assume that's the right answer in what I mean by effectively ergodic Again, this is definitely not Mathematics here its argument. Well, I mean there's plenty of mathematics in ergodic theory and in various theorems of this sort But in the application here, okay, so now the Key idea in well certainly with regard to the second law of thermodynamics and so on is That I mean again, this is a system with a huge number of degrees of freedom which I Mean nobody could keep track of but even if one could it wouldn't one wouldn't typically want to what one is Interested in is what you might call macroscopic observables now What macroscopic observable you choose is up to you or whatever? I mean, but you know again if you think of a box of gas you're not going to Observe the positions and velocities of all of the gas molecules But you're you certainly may observe some average behavior of them And the idea is that there will be a lot of well There will be a huge number of states in this phase space. I suppose I can Color this in with some different color Which with respect to at least the observable you're considering will be Macroscopic will be I mean I'm using the word macroscopic to refer to whatever Reasonable observable you've chosen will be macroscopically indistinguishable Whereas You know over here in this region this might well be macros got I mean this is Macroscopically distinguishable From this so if you looked at the gas you would notice the difference between it being in one of these states or in one of these States, but you wouldn't notice the difference between the gas being here or here Nor would you notice the difference between the gas being here or here? okay, and so now with respect to this sort of coarse-graining Well the coarse-graining is kind of by its nature discreet because I'm really talking about these states being Indistinguishable, I mean that will give some You know discreet volumes that are sort of declared to be indistinguishable but I'm going to define well in the classical case the entropy of any state like this one to be well Typically be a Boltzmann constant thrown in but the log of the volume of the energy shell occupied Excuse me by all the states that are Macroscopically indistinguishable from that in the quantum case, which is I think what I've written up there then it would be the just the logarithm of The discreet the total number of discreet quantum states within some slightly thickened energy shell You know that would again be Macroscopically that would be indistinguishable with respect to the Macroscopic like observable that you've chosen Okay, and now there Will thereby get a Function which I will then I will now treat as though it was continuous or smooth although it kind of has to be You know step function like the way I've Defined it here breaking things up into volumes, but you know all of the awkwardness is Mitigated substantially when you consider that you're You know you've got at least you know 10 to the 25th degrees of freedom in the system and so on I mean so you know it's You know there's a very high dimension thing and you know Anyway so I'm gonna think of the entropy well as a function then on this energy shell. Let me put back in my Xi's I'm not sure why I took them out You know the other state parameters and of course you know at other state parameters. I can do the same thing and define the entropy there Okay, so now I've gotten up to What I mean by a thermal equilibrium state so if I oh well okay before I do that right so this is the Explanation of the second law of thermodynamics namely if you are starting out in a low entropy state like this one That you can see is low entropy because there's only a little volume on the energy shell of Indistinguishable states and you start orbiting around while you're bound to enter again You know the disparity in volumes is typically incredibly enormous again in this high dimensions So but you're kind of bound to Since you're spending equal times and equal volumes to evolve to a state That's much more that that corresponds to something that has much much more phase space volume So if you start in a low entropy state you're bound to evolve to a higher entropy state and What if you start in a high entropy state? Well, you're bound to just evolve around pretty much in that high entropy state. I mean eventually if you wait you know Age of universes way too short if you wait a lot longer than that Yes, you'll evolve out of that and into this low entropy region but You're if you're in a high entropy Well, if you're in the maximum entropy state, you're going to stay there for all intents and purposes and If you're in a low entropy state, you're going to evolve to higher entropy. So that's the That is the Explanation of the second law of thermodynamics just from the point of view of ordinary mechanics or whatever Okay, but now I can define a thermal equilibrium state well normally one would be concerned if in doing thermodynamics with stable thermal equilibrium states and really just consider the maximum Consider the Maximum volume region which would be essentially the volume whose volume would be essentially the entire volume of phase space and Call that a thermal equilibrium state, but I'm interested in unstable thermal or possibly unstable Thermal equilibrium states as well. So what I'm going to consider is any anything on this energy shell any Point region or whatever that is that extremizes the entropy again There is this disparity of thinking of entropy as a kind of you know step function like thing and Looking at extrema of it And so on in and I'll be taking derivatives and so gradients of the entropy and so on but again in You know, I mean If you look at anything in physics, you know on the Planck scale, it's probably not a smooth function Right, I mean you've got some fudging to do in all cases. It's more blatant here But I'm I'm just going to go ahead and do the same sort of fudging so So I'm going to call a state of course a state is a whole Family of states because all of them are you know that have the same Macroscopic observable are considered equivalent, but I'll consider a state to be at an extremum a Thermal I'll call it a thermal equilibrium state if the entropy does not vary to first order as I change You know as I move on the energy shell So if you're in a thermal equilibrium state well by definition things don't change When you move within the energy shell, that's my or within the constant state parameter shell So the only way it's going to change to first order is if you change the state parameters so the if we define So we get the first law of thermodynamics as a completely empty kind of trivial statement out of this which is just saying that the well, I can The entropy is going to then add a thermal equilibrium state just Depend on the state parameters. I mean it's written in a slightly Funny way, it would be better a little clearer if I put the delta s on the left put it as 1 over t times delta e and then had these other terms, but all I'm saying is that the variation the first variation of s only depends on the state on the change of the state parameters and these Coefficients are just then the partial derivatives. So I've happened to in order to write this in the more conventional Form I've put delta e. I'm viewing e as a function of s and x i Around this thermal equilibrium state, but these are just the partial derivatives That occur and this relation of course holds for all perturbations because if you don't change the state parameters To first order you don't change the entropy by definition of what I mean by a thermal equilibrium state right I Warned you this was going to be the hard part of the talk, right? Okay, so now we can define the notion of Thermodynamic stability, which is that if on This energy shell the s is a local maximum not just an extremum But it's a local maximum For of course Variations on the energy shell now. I'm only going to second order variations So I only have to keep the state parameters fixed to first and second order if the entropy is a is is a Maximum then I will call this a stable for thermal equilibrium state and indeed that state should be stable because the entropy is Locally maximum you'd have to deep you you'd have to find a much smaller volume to evolve into in order To get out of That thermal equilibrium Okay, well, maybe I'll go on and finish the thermodynamic stuff and then I should stop to see Whether there are any questions about this because as I say the GR part will be quite easy I think this is the part that I think is Incredibly unfamiliar to almost everybody And we're for which I don't know of terribly good Treatments and certainly not Not the standard thermodynamics or statistical physics textbooks or whatever Okay, but now I can so this is my definition of thermodynamic stability which it You know if my box of gas is in a thermodynamically stable state Then it should not be changing when I look at it And in fact shouldn't if I perturb it a little bit itch without changing the state parameters It ought to go back to what the way it was just by these Ergodic like arguments that I've just been making Okay, but I can rewrite this in a somewhat nicer way because here I have to keep the state parameters fixed to first and second order but in view of this first law identity If instead I consider this quantity again or Purposes I've written it would make a lot more sense to write the Delta squared s and 1 over t Delta squared e and so on But if I look at this quantity This quantity doesn't if I If I allow myself to change the the state parameters at second order I'm not going to allow you to change the state parameters at first order But if I allow you to change them at second order then The second order change the the change induced from that second order change is just going to be like this first order change That this first law applies to and the change that I get from second order variations of the energy and the other state parameters is going to make this quantity zero so if instead of what I'm saying is that if instead of Just looking for the entropy to be a maximum if I look at this quantity and make that That second order variation well positive because I put a minus sign here Then I can just check this for variations that just keep the state parameters fixed only at first order So that's a nice technical advantage Okay, so I've been talking about finite systems, but in people like in thermodynamics to talk about spatially homogeneous systems Which are necessarily infinite they of course have infinite energy and so on but you can Define per unit volume Quantity, you know corresponding quantities and of course write down the same sort of laws and write Make the same sort of definitions, but now if you want stability In you know in the exactly the same sense of the entropy being locally maximum At fixed state parameters Well, we already have this criteria where we only had to fix the state parameters at first order But now we don't really even have to fix the state parameters at first order Because in this infinite homogeneous system we can always borrow Enter if if we want to make a perturbation that decreases the energy I can make a very long wavelength perturbation that decreases the energy in this region but increases it Further away and the second order variations are not going to care about the sign of the Change, you know, whether I increased or slightly increased or slightly decreased The energy and again if you make the wavelength being enough the Inhomogeneities that you're introducing are not going to matter so You need you'll need this to hold First thermodynamic stability even if you consider Perturbations that change the energy and state parameters at first order So you get a sort of stronger criterion necessary criteria well necessary and sufficient really but you You get a stronger criteria for thermodynamic stability in this infinite system case and indeed a necessary condition then for thermodynamic stability is that That in particular when you do change the state parameters when you just consider some Change of state parameters the entropy change better be Negative the entropy better be locally a maximum. So if you just look at this matrix of second derivatives Let's forget about the Xi's and just worry about the second derivative of the entropy with respect to energy that better be Negative in other words if that Is positive or if this more generally if this matrix has a positive eigenvalue, then you have Thermodynamic instability and again the argument is that you need this you don't need to impose this restriction of Fixing the state parameters at first order because of the ability to do this Interchange so if we just consider the the case of The energy is the only state parameter this condition is equivalent to having With this condition for instability is equivalent to having a negative heat capacity So if you have any infinite system that has a negative heat capacity So when you if you add energy that lowers the temperature then that system is going to be thermodynamically unstable because basically what's going to happen is part of the system you can make Part of the system a little hotter and part a little colder well You can add energy To a part of it having taken it away from the other part But this part is now going to get colder and more energy is going to flow From the colder part to the hotter part. I mean that's the argument as to why with the infinite homogeneous system You need this condition and this negative heat capacity Is part of it so And a homogeneous system with a negative heat capacity has to be Thermodynamically unstable. Okay, that is not true for a finite system and Stars in Newtonian gravity have a negative heat capacity if you add energy To a star in just Newtonian gravity well or in general relativity. I mean a nice stable star You know like the Sun except I don't want to have the source of thermonuclear energy at the center of the Sun So just imagine the the Sun as a equilibrium ideal gas Like object if I add energy to the Sun it will expand and cool And its temperature will go down if I add energy It has a negative heat capacity that has nothing to do with the thermodynamic or other stability of the Sun because the Sun is not an infinite homogeneous system but if you have an infinite homogeneous system then this is a criteria and That's why I want to consider the black brains because they are homogeneous. I mean they're homogeneous in Some infinite direction, which is all you need to make this kind of argument Okay, so that's the hard part of the talk and now we're on to the dynamic and thermodynamic stability of black holes but let me pause for a minute to see if if anyone is Sufficiently outraged by anything that I've said in the last Well nearly a half hour probably talking about thermodynamic stability and thermodynamic properties to Complain vociferously. Yes on the I couldn't chemical Cannot I mean I'm gonna I'm I'm gonna get to that that has nothing to do with the ordinary thermo. That's what I'm gonna talk about the rest the Remaining 40 40 minutes Yeah Okay, well, why don't I move on and you're probably more interested in black holes anyway So we've just seen that black holes and the same for black brains are in fact thermodynamic systems with these analogous quantities again, the other conserved quantities or state parameters would be angular momentum and Charge if charges if you have them I've put a Subscript I on angular momentum here Because if we're in higher dimensions, then there are many independent planes and so there can be You know many independent angular momenta, but again, you can ignore that if you want so they for a finite system I E a black hole The condition if we take this analogy seriously and apply the thermodynamic results The condition for thermodynamic stability. I mean, this is Really just a definition so I suppose you can't really argue would be that if you have perturbations that keep Well, I'm gonna set the charges to zero now if you keep the state parameters fixed For all perturbations then this quantity better be positive I mean that is the if I take the condition I just described to you for thermodynamic stability and convert that to a condition for black holes That's the condition and what I'll show you is that this is in fact Equal to the canonical energy so the positivity of the canonical energy is in fact The condition for thermodynamic stability, but then I'll also show you that it is the condition in this weaker senses for dynamic stability, but black brains are homogeneous, so we have this additional stronger criteria that this That it'll be necessary for thermodynamic stability that This quantity Well, this matrix be negative negative definite I guess Or at least it can't be positive or can't admit a positive eigenvalue and in fact For the black brain system, I will show you this condition also is is Necessary for the canonical energy to be positive by exactly mirroring the thermodynamic argument I gave of borrowing energy or other state parameters from one part of the system to the other and so that means that Well, let me give you an application of this which if that this doesn't motivate any Interest I'm probably not going to succeed in in in motivating interest in anything else. I say so if you Compute the heat capacity of the Schwarzschild black hole Well, you can compute the sign of the heat capacity by just taking the second derivative of the entropy the area with respect to the energy and Given that the area is 16 pi m squared this computation yields 16 pi which is bigger than zero. That's a positive eigenvalue But that doesn't tell you that Schwarzschild is unstable it just tells you Schwarzschild the Schwarzschild black hole has a negative heat capacity But what I'm telling you is that that tells you that a black brain any black brain Made out of Schwarzschild in particular the black string, which is just the five-dimensional Hmm, I don't know what happened there So you just take a Schwarzschild black hole add an extra dimension to get a five-dimensional solution so you could kind of picture that again not in a space-time diagram, but if this is The spherical horizon of a Schwarzschild black hole. We've now added an extra dimension and Doesn't look too good, but anyway we have That's what the black string looks like. We know already that the black string is unstable That that was actually proven by Gregory and LaFlam in the 1990s, but the computation got more this computation that taking the second derivative of the area of 16 pi m squared with respect to m and noting that 16 pi is greater than zero is A proof that the black string is unstable. So now Let me So now let me show you this and by the way define canonical energy and all that now So this is all things that we had yesterday I've actually taken these slides from another talk so the notation there's slight notational differences But this is the Lagrangian the Einstein-Hilbert Lagrangian as an inform This is the fundamental equation of varying Lagrangian to get Euler-Lagrange Equations and this boundary term which I denoted with a capital theta Last time. This is what that theta is the same formula as I showed you yesterday Then you get the symplectic current by taking an anti-symmetrized second variation of that And you define the symplectic form I call that capital omega yesterday by integrating the current over a Cauchy surface. I didn't specialize this formula to general relativity, but This is the explicit formula for the symplectic form the symplectic the integrated symplectic current in general relativity. It's just the anti-symmetrized product of the metric perturbation on the three slice and The momentum perturbation on the three slice where the momentum perturbation is defined the momentum Excuse me is defined in terms of the extrinsic curvature by just You know Removing this I You're not removing the trace. This isn't trace-free, but you're subtracting off this Trace term and I've written this now as a density You know to be integrated rather than as a as a form in this Formula. Okay, this is the another current that I introduced and Piotr also introduced in his talk Associated with a diffeomorphism X and this is the formula I gave you last time that the another current always can be written in this form in terms of constraints and the nether charge and Then this is the Well what I'm calling now the fundamental very variational identity that you get by varying the nether current under a perturbation and equating this to this and using the definition of the of the symplectic form in this Equation I haven't made any Restrictions at all in the equations. I wrote down. I assumed that the background on which you were Proterbing what was a solution of the equations of motion? I'm not assuming that now And the main reason is I want to be sure to have an equation that I can take another derivative of and Be And be still valid I don't want to drop a term that might Might have a non-zero derivative and the reason is well so again where The metric is our dynamical variable here, and I'm in these variations considering a one-parameter family of metrics and The Delta G is just the derivative With respect to the parameter But in today's talk, I'm going to go a step further and take second derivatives, so this is Evaluated at lambda equals zero, but I'm also going to be concerned with taking Second derivatives. I took second variations Yesterday and they're in these formulas, but then that was those were two independent variations Now I'm going to be wanting to consider some one-parameter family of solutions I guess I should put indices on both sides if I'm going to do it and I'm going to want to take Actually take second derivatives with respect to my one parameter in there, but that's getting a little bit ahead of the game But that's why I've kept all terms in that formula So I explained to you yesterday that a Hamiltonian would be a function on the phase space or the field configuration space whose variation satisfies this formula and that gives rise With the boundary term at infinity to the formula for ADM conserved quantities now I spent some time yesterday getting rid of this delta to try to and Doing something about the fact that this wasn't automatically delta of something to get an actual formula For h sub x the ADM mass of the angle momentum I don't actually have to do that bother doing that now because I'm only concerned with the fact I'll only need these varied quantities in any of the formulas I'm about to define of course if this wasn't the variation of some quantity that might be a problem So it's good to know that it is but I don't need to Introduce that be that I had and so on because I'm only going to be concerned with this and then as I explained in the last lecture if I consider the horizon killing field and integrate so Now I am going to first for the first time in this talk draw a spacetime diagram with a bifurcate killing horizon of a black hole and a surface that goes from the bifurcation surface of This killing horizon out to infinity and if you integrate the fundamental identity there With x chosen to be the killing field You get the first law of black hole mechanics and again This term is zero because x is a killing field This is zero because the background satisfies the equations of motion This is zero because the perturbation satisfies the linearized constraints So all you get is this term the boundary term from infinity gives you the ADM quantities and the boundary term at the horizon gives you the Gives you the Kappa delta a and if you have many if the horizon killing field is a Involved the sum of rotational killing fields, then you just get the sum in higher dimensions appearing here Okay, so now to move on to canonical energy finally and give you the the results We Want the quantities that we're dealing with to be gauge invariant now the It will turn out that this Canonical energy will not automatically be gauge invariant. I mean as I haven't defined it for you as I'm about to Define it but to make Something gauge invariant you can make things gauge invariant by imposing gauge conditions In which case they're gauge invariant with respect to any remaining gauge freedom and there is an important and Very physically natural gauge condition to impose on the perturbation which can For where we're only considering the positive non-zero surface gravity case with the bifurcate Killing horizon this one can show you always can impose so there's no physical restriction on the perturbation which is that The that the Well in the background horizon, of course, there is zero expansion of the generators In the perturbation there might be this this what you're calling this surface. Well, it might not even be A null surface anymore, but if you it what we're we're only concerned with perturbations of the initial data What we're going to demand is that the convergence Or expansion is what I've written there of these Generators at the location that we that was previously the horizon That that expansion vanish now what that actually guarantees is that to first order What you were what was the horizon of the black hole is still the horizon of the perturbed black hole So anyway, we impose this gauge condition and then our definitions will be Gauge invariant and finally here is the definition of canonical energy. So this is the Symplectic product which requires two perturbations You've seen this before Today and yesterday with the two perturbations you integrate the Symplectic current over a Cauchy surface This gives you a conserved quantity But it's not Tremendously useful since it's a conserved quantity for a pair of perturbations But now we can choose as the pair of perturbations when we have a background symmetry whatever perturbation we're interested in and The time derivative the lead derivative of that perturbation So this gives a conserved quantity for a single perturbation and This is a formula that you get for that quantity. So how did I get that formula? Well We have this Variational identity. I'm going to take one more derivative with respect to lambda now because this term is zero I'm only going to get a contribution when the lambda derivative hits the G and That when integrated is going to give me the canonical energy This is not going to give anything because the equations of motion are going to be satisfied to all orders And so I'm just going to get these boundary terms But the boundary terms are just the mass angle momentum and area Quantities so when I take another derivative, I'm just going to get the second derivative of those Quantities coming in so the canonical energy to find this way is in fact Equal to exactly the quantity that has to be positive for thermodynamic stability So what does that have to do with? Dynamics stability well With a fair amount of work. Well, some of the some of these are fairly easy But well, it's first of all Even though I'm really interested in this for a single perturbation. You might as well look at this as a quadratic form on pairs of perturbations and viewed in that way, it's conserved For the same reason as the symplectic product is conserved. It's not hard to show that it's symmetric It's with this gauge condition imposed its gauge invariant well with respect to perturbations that keep the area and the ADM linear momentum of all things fixed But we're going to be interested in perturbations that keep the state parameters fixed which will automatically keep the area fixed by the first law and Without loss of generality. We can do a Lorentz boost to get rid of any Perturbation in the linear momentum so there's no restriction physical restriction on the perturbations here okay, and now the least trivial of these things is that the this canonical energy is in fact degenerate only on perturbations So this is the statement of degeneracy. I mean that this is zero for all So we're only interested in Perturbations that keep the state parameters fixed to first order and this we can do for free and on that subspace This is non degenerate if and only if the gamma is a perturbation toward another stationary and axi-symmetric black hole so On this subspace of perturbations Because it's non degenerate it's either positive definite or it can be made negative Now I didn't Yeah, well, okay, that that's all true Okay, so if The e is positive definite We can't have exponentially growing modes because the energy this e would have to grow exponentially with those exponentially growing modes and but it But e is constant so you might ask what is positive definite in this have to do with it Well, if it wasn't positive definite you could have something growing Exponentially and something else negatively growing exponentially so that they cancel each other and keep the e conserved. I mean that You need the positive definiteness to have mode stability on the other hand if if you could make e negative and If the e were to settle down to another stationary axi-symmetric black holes at late times then you'd get a contradiction because non-trivial e with a lot of work you can prove these Flux formulas of canonical energy the canonical energy flux through null infinity is just the bondy news Squared and the canonical energy flux through the horizon is just the shear squared and the canonical energy then can only decrease but if you were to asymptotically approach a Stationary black hole of the same Parameters it has to be of the same parameters because there can't be a first order at first order because there can't be a first-order change in those parameters given that we Didn't put in any first-order change in the first place The e can only decrease but it would have to go to zero if you were going to approach a stationary black hole At late times so this I didn't I've said axi-symmetric in the right places here but axi-symmetry is critical in this formula because this flux is positive for the time translations at infinity this flux is positive for the horizon killing field And only in the presence of axi-symmetry are the fluxes associated with the time translation and the horizon killing fields equivalent because You know we have This relation between the killing fields, but if everything is axi-symmetric We won't have any fluxes associated with the axial killing fields okay, so Let me just quickly Mentioned to you since I already built this up. I don't if we were to consider black brains Well, you have to do some work, but you can If you have a perturbation that changes the angular momentum the mass or the angular momentum and Decreases the canonical energy if you have that for a black hole. That's not of any interest That doesn't do anything for you, but if we take one of those black hole perturbations That decreases the canonical energy Multiply that by either the ikz Then we have to we've modified the data, so we have to Readjust it a bit to satisfy the constraints and we have to control the fact that we don't Change the perturbation all that much to resolve the constraints if we take k to be extremely small Then the new data will satisfy The required conditions, but it will still have negative canonical energy, so it will be unstable So what I'm saying here is that if you have a black brain based on a black hole That has negative canonical energy that changes the state parameters Then for the corresponding black brain you can find a perturbation that makes the canonical energy negative but doesn't change the state parameters to first order and That by these arguments Gives you an instability so have We done everything with linear stability of black holes does it just reduce to the computation of canonical energy well There's a It's restricted to axi-symmetric case The stability and instability results are weaker than what you'd want and we're only doing linear theory to so one Has the restriction there, but it's also not that easy to tell whether something has positive Whether a black hole background has positive canonical energy or not and I've saved this sort of for the last To show you the explicit formula for canonical energy So in this the notation here is that the h is the metric of the initial data surface over here of the background spacetime the Q is the perturbed metric the Pi is the Canonical momentum of the background spacetime and the P is the perturbed canonical momentum and This is the first page of the expression for canonical energy This is the rest of the expression so what Prebu and I though were able to show is that you can break this up into two pieces depending on their t-fi reflection symmetry and in fact the T-fi Antisymmetric part which would naturally be called the kinetic energy That one can that's a very simple subset of the terms there compared to the full expression That we've been able to show is always positive definite for any black hole or black brain background that Enables us to show that if the potential energy can be made negative you actually will get Well with for suitable perturbations exponential growth in time if that is negative Okay, but what I've once again just as last time run one minute over which isn't too bad So Amazingly at least with respect to the axi-symmetric perturbations dynamic and thermodynamic stability for a black hole are equivalent well equivalent in quotes because I'm using weak notions of stability and instability here, but But nevertheless, I think it's really remarkable that the laws of black hole thermodynamics Extend to stability arguments and the thermodynamics stability of a black hole is Directly relevant to its physical dynamical stability so That's it for today and tomorrow. We're on to the quantum version of black hole thermodynamics