The First Law of Thermodynamics for Kerr-Anti-de Sitter Black Holes

0:00 I'm going to generate Pritt's 60th birthday, and my title is Olog, and it's actually just the time you used to something you did, which I last made in short. Before doing so, let me say that the work I'm about to prescribe is joint work with This is Chris Cope and Martin Perry, who are certainly the old scientists of the data with Chris, and he's also said to me, it's just a really good question. The working question is based on the last three, it was before, but in fact, the most important research is this paper, again, with Chris Cope, but also Tom Blue and Dr. Payne and Dr. So, well, it's an obscure topic on the Facebook and a very long time call. It's partly to put off questions from the audience about excellent theories that you're talking to. Professors from obscure islands of the world, okay. But I think the topic is especially appropriate So most well-known papers is this famous paper with Steve Edwards and Steve David Scott, quantum field theory and antidepressants space time, and written in the 70s, a very prescient paper because we all know that nowadays people are very interested in antidepressants space, and that's what we'll be talking about. Now, that paper actually predates the interest in the atmosphere space by people working in supergravity. But what was very important for the development of idea of supergravity is because the atmosphere is supergravity, so it was influential at the time, but of course its influence goes So it's not now because nowadays we're all supposed to be chan, what we call it, banter of the NDSC. I think our funds are now which I'll see later, and so it's not important. In fact, the whole subject is quite interesting, because what we're trying to do here is to bring together what I call thermodynamics and the whole subject together, which Chris is very active in his family not so much trying to think about it, but elucidating the conceptual

2:30 because there's time to get as many useful resources on this subject. So that's another subject. You can't find the gravity about which you will, the truth inside it. And you can't figure it out. Nothing. But that's the way we've done this together. So, most of the talk will be a little bit technical and I'm going to skate over the technical details by not giving any detailed derivations. Okay, so, what's the basic idea? The basic idea is, on the face with, we'll start off, with simple relativity. So we know, and have known since the work of Frankenstein and others, cryptidoo or a thing that started the same off, all those people including Stephen and Brandon Carter and so forth, that there's such a thing as the first law of thermodynamics called black holes. And actually it's kind of a triviality for ordinary black holes. And of course, if the cosmological cost is zero, it's easy to establish something which I'm going to call the Smaar-Bibs of Buchan relation, which is that the energy, using a co-marked or using some integral, sorry about it, the energy is a combination of the surface gravity of the black hole, around this area, that's the rotational piece which you can think of on the sunset as kinetic energy. Omega is the angle of velocity in the back of a hole, and higher dimensions are maybe more than one angle of velocity. I'll come to that shortly, and here is the angle of momentum. Triviality from this, you use dimensional analysis to deduce this formula, which is of course, a form of the first law of thermodynamics, if you interpret A as a multiple of the entity and Kappa as a multiple of the potential. So that's absolutely a too bad argument, you don't have to work too hard to prove that. So no problems there. As I said, you can do this in higher dimensions, the relevant black pill solutions were found remarkably in spite of analytically and were found by Meyers and Perry, a beautiful paper in the 80s, astonishing that you can write down an actual formula, if you can, we'll come to that later, and you can derive all these laws and it just rolls on as you would expect, and that's going to be a . On the other hand, if you tell us that our total constant is not zero, then you can't

5:00 prove this as an extra term, in fact, this is actually infinite, and moreover, you can't Because you've introduced a new length scale, this is one of the length squared dimensions. You can't use simple dimensional analysis and Euler's theorem you get from here to here even if that was correct. So you can see there's an issue. Is this formula true for appropriate E, et cetera, et cetera, for black holes with a cosmological constant? That's the interesting question, and we'll start to discuss whether that's true or not in that one. Well, in four dimensions, we've had available to us solutions for a long time. First written down, I believe, by Brandon Carpenter in 68. Here is a metric. It solves the Einstein equations. We've got a logical term, which will be interesting in case it's negative. And we introduce a next scale, L, which is basically the radius of curvature of angular which is the solution which you would get if you put the various parameters equal to zero which are in this method. For example, there are, there is a n parameter, a lower case n, a mass-like on D made, but not actually a mass parameter. And there is also a parameter delay similar to the solution. And important in this story is this thing called, well we call it cascade, but I think it's actually called design. in the last time you check out the space, you can look at the real physics. Now, from this trivial view to where the horizon is, it's the root, which we call R plus, of this polynomial term. And then you can work out the area, and you can work out the surface of gravity, very mechanical, no problems. We can also work out angular velocities. The black hole should be thought of as rotating, relative to a non-rotating frame at infinity, and the relative angular momentum between the black hole and infinity. There's another even introduced, which is to forget about In fact, these oil increased coordinates are not rotating, that appears to have been done a lot in the literature and appears to have given rise to confusion.

7:30 So let's just emphasize what I mean here. So roughly speaking, you should think of these spaces as having a boundary infinitively time-like. This is the conformal boundary of the space-time, and in the interior, this was actually called the bulk. In the case of pure angular space, if you put the pyramid in at a zero, we would still get a solution, but it's not an obvious solution. It's actually rotating with respect to the obvious frame. So when we do our calculations, at least we in our papers have moved to a we wish to discuss energy and energy changes relative to a non-rotating frame. You can clearly do so relative to a rotating frame, but there will be extra terms. It's simpler, we thought, to move to the non-rotating frame. So, what are the questions that we'd like to answer here? These are really straightforward and were answered long ago. They are, what are the angular momentum and mass for this system? Now, you can find that in a variety of ways. The first people who asked this question were from a Hamiltonian formulation. These are the things that generate rotations and time translations. In other words, they are values of moment acts for the Hamiltonian theory. There are other definitions of these objects. a famous paper where they discuss the stability radius. And another definition given by Ashton-Magnon, which involves integrating the biotensor over the boundary of infinity. I'll discuss the relationship with those later. Now, I think it's good that this and this are the same. Okay, having got those formulae, it's now something we can check. we've got E, we've got J, we've got T, actual factor at least, we can just differentiate it. From that, we can deduce what the entropy is. Now, note that we don't just assume the entropy is important in an area. By this calculation, we prove it because we take D of it and it fits the second and third and first law.

10:00 And if you do that, you find that this is a quarter of the area. Very straightforward elementary calculation. You'll also find some other things which are interesting, which is that if you take this combination, E minus TS minus omega J, which those of you who have studied the dynamics and recognized as the term dynamic potential for our system, which is what would be appropriate if you're considering a system at fixed temperature at angular velocities, and that could be due to all the other quantities, and that actually is up to a factor of the temperature, which is the gravitational action. So I'll talk more about the gravitational action shortly, but this relation is basically central to the idea that there's some connection between, on the one hand, thermodynamics, and on the other hand, some functional integral involving gravity. It's the central assumption of the whole theory, and what we're saying here is that we have verified it in this special case. I'll talk more about how this thing is evaluated, whilst it's evaluated, I think, by talking Hunter and Ted Robinson. However, we don't agree with everything, they say. Not all alters agree in this subject. In particular, those alters are getting different values we put a prime on the energy. I think they agree with the energy momentum. They actually got this formula with s the quarter of the area, which is nice. But, the other prime is the area of velocity which has not been corrected. It's an uncorrected area of velocity. But a simple calculation reveals that TDS finds that we can try and deject. I'm sorry, there should be a plus there because that's the way it goes. It's not exactly different. Sorry about that. First of all challenge. I'm obviously going to do a course somewhere else. It's not a boy, every time you realize you can make the second stage when you have to talk so I don't know, you just can't do it again. Okay, so it's not an exact different angle, which is a peculiar.

12:30 Other people looked at this from them, using what's called the Brown-York method, which I'll come to later, and they basically agreed with RE, not the E prime. Curiously enough, Silva, excuse me, Julia, had a long procedure for getting these objects, these asymptomatic generators. He agreed with this formula, then he noticed that it's e, not e prime, is required for the . So a bit of a puzzle, a bit of a mystery there, but it seems all to be sorted out. Now, I think that Steve probably will agree. But the whole theory works rather nicely, and that's very satisfying. Okay, so that's now in a nice state. What about five dimensions? Now five dimensions is much more interesting from the point of view of the present glory about the ESCF T-Torris process. I think that's why it's reluctantly, we live in five dimensions. Probably it's better to live in 11 dimensions. So the first requirement is a solution of the equations. And here it is, it's an astonishing solution, written down in this paper, I'm asking the other day how he got it and he just smiled and said he didn't know or something. Anyhow, it's a solution, you can check. This is the exact solution. Five, it satisfies this. There's a four here. And again, you can crack the handles. It depends on two parameters A and B. that's because there are two ways of rotating in five dimensions. There are two orthogonal planes with respect to a little bit around which you just rotate. So there are two angular velocities, and we write as amigas of a, amigas of b, like this. And again, we will insist that we do so relative to a non-rotating frame at infinity. And all those are unconversial and straightforward.

15:00 for them. So now the question is, oh, and there's two zis, and kind of skates, but they're subtracted like this. So now the question is, what are the energies of the rest of them? So, I'm going to make a bold assertion, which is, I think, true. Let's define E to be this thing. Now, it's quite complicated, because it's involved in psi A's in another peculiar fashion. And you can see the J's, the whole thing is symmetric between interchange A and B, of course, but the J's involve both the psi A's and psi B's in a very, very good way. Okay, well, you accept that and you arrange it. Now, I should say, these formulas are very messy. They're not things you can do in your head. Indeed, they're not even things you can do on a . We use mathematics. So the other collaboration that you have mentioned here is, of course, So what is true is that if you take that, take it's D, if you were in first here, and then if you get this formula with S equals a quarter of A, so you prove once again that that's the right formula again to me. And you also get this relation which we've chosen to call the quantum statistical relation quantum statistics, and similar things. And there's actually an action which these people worked out, using an action. And that's the formula. And this action is worked out using what's called the boundary subtraction method. And which is the algorithm, I'll just say what that is. So these angular momentum agree with Comar expressions. And presumably agree, this is what I wrote last night, with Kermar, Abertasar, Hampton mass, et cetera, et cetera. But we didn't explicitly check that. There will be some software receiving on that at the end of the point. But anyhow, that's the obvious thing that ought to be the energy, because it satisfies the right formula. So now what is the background subtraction method?

17:30 You take a finite version of your manifold, x sub epsilon, and as the whole name says, epsilon will always make it small, this will go to infinities if epsilon goes to zero. It has a boundary. Now that action for that is by this expression. That's the one we need to get the Einstein equations out. And so we can certainly work that out for a finite system. But we want it for an infinite system. And for the infinite system it diverges. And so we've got to regulate or make finite. Here is one procedure. Now if you're critical you'll see this procedure is almost a procedure that you can find it under the details. The first thing is you fill in your boundary with a fiducial metric. In our case it would be AES itself and then you work out I-acidon. but for finite. Then you figure the metric you're actually interested in, and we call that I side an actual. Then take the difference because you want the difference in actions, and you take the limb, that side I'm going to spin it to zero, and you hope you get a finite action. So that's what we did, that's what Hawking, Hunter and Mr. Taylor Robinson did, and that's the answer you get. So that seems quite nice, except other authors appear to disagree. Now they have reasons for this disagreement which have to do with the interpretation of the dual theory, about which I won't say an awful lot so they gave a bunch of expressions with crime but crime is wrong they don't agree with the expressions that we do so one of the great achievements of this paper is I've taught very few things I could teach Chris Pope but one of the things I did teach him was to lay tech boxes

20:00 And they actually found this formula as true, but it is the case that this is exactly parental. And you might, J in crime might be replaced by the J that we use, It doesn't really any way of getting out of that using the prime. These are the prime, in some sense, absolute omegas. They're not relative to a non-rotating frame, perhaps, infinitesimally. So I'll leave Stephen to explain in detail what went wrong there, but maybe it's elaborated. But it seems that our expressions at least satisfy the truth. Now, one, Johnson had a go at this game. They got this expression, which is even more complicated than our expression, and had some interesting problems. The things that we got all vaguely should be going to pure ADS. It seems to be a reasonable thing, the energy of pure ADS will be zero. But in the ADS CFD correspondence, ADS can have a certain zero-pointed energy, at least the boundary field theory should be. So then it's a plus, but when you put the rotation parameters to zero, and you also put the mass to zero, you don't get zero. You get three quarters of the, you get some zero point energy, which corresponds to the mass of the current period. And this recovers the result in the non-repentant case, and so that's meant to be able to fix. Now, actually pointed out quite rightly, that it's a bit of a puzzle making a generator of SO be not zero for the solution for which it vanishes, for which it's invariant. And there doesn't really seem to be any obvious way of making this compatible with SO in there, sorry, for two. And that's not accidental because, as we should see, conformal advantage is broken in this set-up, and it's safe.

22:30 They used the boundary counter-telling procedure, which I would have, I'm sure, to get these results. And they also got an action, which can be written as I-5 double-prime. It's not even the I-5 prime or anybody else's action. So that's what they got. Now we can check out their thermodynamics. Does it work? And the answer appears to be no. There didn't seem to be any way we could get their formula to make the right kind of sense. They did find that this was true in regard to the non-trivial check of the ADS-BFT correspondence. So that one has to be turning out right to be the area. But this is not the exact differential. And moreover, if you were to take the action and compute all of the quantities you should, because it's the free energy or the thermodynamic potential. So from the thermodynamic potential, you can get the energy, get the angular momentum, get the entropy. If you were to do that, you do not get an orderly area, you don't get your angular momentum, and you don't get your energy. So there seems to be something a bit fishy with formulae. Now, how did they get those formulated is the question one is asking, and what's the resolution? Well, I'll try to give some of the resolution in the next few evidence, but I don't think we claim to have understood all this, but we do understand potential pitfalls, and I'm going to show you some of these potential pitfalls. OK, so the main part is, let's recap. We have this situation here where we have a manifold x. We complete it through a compact manifold, which is the joint union of this piece. That's if we complete it in the boundary. And on this x bar manifold, we have a metric which is conformal to the original metric. And we need to deal the conformal factors advantage.

25:00 That is what Roger calls conformal classification. Now it's important. From that we can produce a metric on the boundary with h bar, which is a restriction of g bar in the boundary. But it's important to understand that the metric on the boundary is defined only after conformal rescaling. We can at any stage in our calculation replace omega by F omega, and that would give us a new H, S squared. There's not an F that's a band, but it's a kind of one of the cases. Now, it is in odd bulk dimension, it's known that, but I will show it to you in a moment, the so-called boundary counter-regulated action and the regulated stress tensor from which these authors retain their energy and angular momentum depend on the conformal factor. Because of the existence of what are called anometers, this is the analogue in the filtering on the boundary of, I'll just inform the knowledge for that. Okay, so now what you do in this game is you do the following. So this is a small cartoon that I expanded later. You've basically got the conformal boundary, and you've got a metric, G bar. And what you do is you do a series expansion across the distance away from the boundary. And you notice already that that requires a choice of conformal factor because the proper distance will depend on conformal factor and then you study the limit as epsilon goes to zero of your equation. That's the basic idea and you can see that that inherently depends on the choice of conformal structure upon the boundary. In more detail, what is done in these papers is to set the thing up like this to compute, as I indicated earlier, the action. Now at xylon, the regulator has proper distance along here with respect to a particular g-bar. Compute this guy. Now for finite xylon, on zero, we can associate a boundary stress test which was done by Brown and York. Differentiate this object with respect to the metric on the boundary.

27:30 And that would be your stress test. Now, as xylon goes down to zero, both of these guys will diverge. So the question is, how do we regulate it? The restriction that is used in the literature is to do an expansion of the powers that are sign on and then subtract off certain universal counter-turns. The counter-turns being obtained by expanding the Einstein equation to the power series and the boundary. these capsules are expressed entirely as a majority of the boundary and it's not embedded. And it's this procedure which gives a non-vanishing energy for AES, the zero-point energy. How does it do that? Well, having got the stress tensor on the boundary, suppose the boundary is itself a product of an R time factor with something spatial called S, and suppose you've got a killing vector on the boundary, well then this is a conserved stress tensor, or should be. So we saturate it with the killing vector and integrate over a slice of the boundary and that gives an energy with respect to every killing vector on the boundary. That's the one that these guys compute. And of course it depends now on how they set the whole theory up. So to see the pitfalls, let's say there are some potential problems, let's see what those are. Those potential problems are, to reiterate, it depends on choice of conformal representative on the boundary, and the odd dimensions that causes problems. Of course, at technical level, the implementation depends on the headband correctly, and you need normal coordinate system with respect to the conformal boundary. And, you know, that may be a bit technical, maybe you can make a slip. Finally, it's not obvious that the stress tensor on the palm rig should be the variation of the action, the regulated action, because it's a question of interchange of limits. You could take the limit of the stress tensor or you could take the variation of the limit of the action. A final point which could come up is it's not obvious that this each will is actually the thermodynamic being energy.

30:00 And the reason for that is the turbulent red shifting phenomenon which I will describe shortly. So it may or may not be that these are the reasons why these all are different. We don't really know that yet. Okay, well, just to remind you of some technicalities now. Y-elinquished coordinates are not only rotating at infinity, but they're also ellipsoidal. So it's actually a bit of a mess. We can reproduce asymptotically spherical coordinates, which we call wise and happy coordinates. And notice the angular coordinates have to be shifted. So, you can use these formulas to express the metric in these coordinates asymptomically. And you see that what you get is a metric which depends in part on this metric here, which is in fact a round metric on this metric. And it also depends on the delta theta, which sums up a trivial angle of theta that we need to know. Now, the conformal completion suggested by these all of those techniques to be a good one, is just to take one over y as a conformal factor, and y goes to infinity. And then you can work out the boundary metric, and it's a very nice metric, it's minus the t squared plus this round square metric. And in particular, the genealogy is fixed with ultra-stagging technology as well. So that's one boundary matrix that you can use, and that's the one they use, and they have a reason to do it. The one used by a was to use one over R, and one over R is a whole lot easier to use because you don't have to do this for them. But by simple calculation, if I say simple, all of these are very vividly, shows that what you get is something conformal to the boundary. And notice that is now not constant. So that might cause you to have some worries, because from an old result of relativity,

32:30 if G naught is not constant, then the temperature varies in respect to the position. So it's not quite clear what the temperature is actually. So when you differentiate with respect to temperature, it's not quite clear what you're differentiating. So it could be, we don't claim it is, it could be that that's the source of ambiguity in this procedure. Okay, well that's more or less the story in five dimensions, but my title promised more, so I'll say a few words about the higher dimensional case. Actually we got into this by a very circuitous route, so firstly in the paper that Stephen wrote, they rather astubiously not only don't view the details of how they made this solution, but they say it would be an interesting problem or a confusing problem to find higher dimensions. And of late, that's been quite interesting because these solutions can be used in the five-dimensional one to produce interesting other types of metrics by taking limited cases. In particular, it's possible using these metrics to have infinitely many, some kind of double infinite sequence of Einstein metrics on S2 times S3, which is the sort of interesting fact. There are so many metrics on S2 times S3. So that was done by some Japanese, and we thought, well, we could do it in higher dimensions if only we knew the metric in higher dimensions for the Kerr atoms of the metric. So in higher dimensions, the case with one rotation parameter non-zero was written down by Hawking under And, as I said, they left it as a problem to find the general case. So this problem will be solved by using a certain line to check value. A part of it was on page, which is a kind of useful thing to have. And it doesn't matter, but we focus on two separate things. We might actually rather care, you've got to make a man's stats here, so how do you make a man's stats? So what we actually found was all the metrics in all dimensions can always be written as

35:00 the square of a null vector and the null vector is tagged to a geodesic congruence. And the merit of doing that is the well-known fact that if the metric takes that form, then the fluid prices many rise. So in order to check actual computer, in other words, we write a metric like this. We raise an index with the metric d, background metric. Actually, it doesn't matter whether we use g or g0 here, as you can check because of the condition. So I didn't make this . And I said 0 . That's great for that reason. So in other words, this H, which makes the indices, actually sanitized the linearized Einstein equations about the background. So it's somewhat easier to compute those, and that's what we did. we dreamt on the Landsat by putting together a lot of information and calculated. Now we did it up to dimension 11. We could have gone further, but we could get it out. But anyhow, the end result is we got the solution. And they depend on, as you would expect, because your power of the airplane is going over two rotation parameters, a sub i, the space-time dimensions, and they depend. And as constants, you get a bunch of design factors coming in into the metric. Now, I'm not going to write down the metric. It's a disgusting mess. You can read that paper. But you can calculate it. And that's what we've done. So the first thing you can calculate easily is the angular momentum. And it turns out that's never contentious. Everybody seems to agree. and the term arc will be easy to do it. Look at that formula. Script N here is the area of volume of 10 minus 2 sphere. You can compute the area, compute angle velocity rather than finishing, and then the state values. But if you pick these E, depending on the E of the lot less, then they'll satisfy the score. So we certainly identified what are the correct and we checked first of all. And actually we did a bit more, actually it can be calculated and we can verify this nice formula.

37:30 So that's all looking pretty good. And now I come for equations, the postscript. So the conclusions. First of all, we're now in good shape to check black holes thermodynamics and indeed lots of other possibilities of black holes in a higher which is a non-vanishing cosmological term. It doesn't have to be negative if that was our application, but if you're interested in positive cosmological term, the solutions are available. And as I mentioned, by taking limits and other forms, you get a whole bunch of other solutions which may be For example, the first in-home gene, Einstein metric you mentioned before, was discovered by the montague, precisely by filling around the cylinder. So you get interested in it, do you make it? Now as far as the thermodynamics are concerned, it seems that all these results agree nicely with the original proposal of Stephen and I made that we just write down the action and then call it the potential which as long as we use the background subtracted action function. That seems to work. However, there are many questions which hang over this boundary stress tensor method. And they may well affect constructs or attitude to the ADS. I'm not saying that that's not a terrible correspondence. But what I did find is that it requires the calculations that people did do require more detailed information. Let me briefly say that the whole story was set off by these co-workers, which was that the boundary theory we normally think of is a quantum field theory in a static space, and the idea is what happens if the space is rotating, you've got a rotating quantum field theory, And their idea was that we should be able to make a structure between the rotating

40:00 boundary and the interior. And at least the high temperature there seems to be some of those that hangs together We have a lot of low factors, but we're saying it's a lot more needs to be done if that's really to take seriously as a precise correspondence rather than a qualitative correspondence. Now, I was delighted that Abbe was coming to this meeting and not speaking. Not speaking because he was relaxed enough to talk to me about these problems. And he told me about a paper that we had, unfortunately, overlooked. and also a lot of interesting information and i'm talking about this paper today and so this is stock press news which should be taken with a grain of salt and the point here is that there is a nice paper of gas and man i don't agree with anything in paper but anyhow in that paper they did a calculation of a comparison of two different types of mass that you can assign to a space. One is using the stress tensor method that I described to you earlier, and I think they used R coordinates. They did it in all dimensions in the case that they had just one rotational parameter because when this paper was written some time ago, there were not solutions available except for the five-dimensional and he complicated in higher dimensions. But in all higher dimensions in one parameter, they computed two separate things, or three. They computed the angular momentum by the boundary stress tensor method. Nobody seems to have any problems with that, right? Modular, so, well, more than anybody seems to be able to. Then they used a method suggested by Abbe, originally in four dimensions, but then generalized, one of those, you take the biotensor restricted to the boundary, take the electric part, in other words, saturate two of the indices with a normal vector. That gives you a symmetric tensor here, which is divergence free on one index. And so from that, you can construct an object by integrating over this surface here and using a killing vector. So that's

42:30 And the issue is, how is that related to what we wrote down and how is it related to all the other masses? Now I believe the correct statement in dimension four is that his mass, the upper data mass, the Antonian mass, various other kinds of masses, all agree, there's no problem. And that's partly, perhaps, because of the fact that there are conforming anomalies in the three-dimensional field. And how that's corresponding. Now what is it in these cases, in fine dimensions for example, well in fine dimensions they do this formula and it looks completely wrong. It doesn't appear to be equal to an alpha. So, well, that's very puzzling. But actually, I then noticed that they used a non, they used a rotating killing vector. You should obviously use a non-rotating killing vector. I mean, I feel you shouldn't, because we are here, energetic with respect to the inertial program. And how to do that, and this is a quick calculation, so I've made a lot of wrong, and it seems to me, it probably agrees, first time counting agrees, with the specialization of our result. So that would indicate that the E that I wrote down in the square box, which I claimed is great, is also the same as absolute . That seems to be a good indicator than if we could work out what the average data mass is in real time with that. But we don't really know yet. There's much to be done. So as Karol Foukesh talks, I look back and I look forward. This talk has been down to it. Looking forward, as you can see, there's much more to be done. Thank you. So to make a comment, I think half of it I already mentioned to you for this. This is sort of a more general framework which involves so-called isolated horizons. And it has been generalized to . And in that framework, one calculates quantities associated with horizon, etc. Now, the question that one asks, also in that framework, one can calculate angular momentum . energy. And the energy actually is supposed to be, I will tell you, it's supposed to be generator of some time concentration, basically. So you consider a time-like vector field,

45:00 and on the boundary, the question is what is the boundary condition for this time-like vector field. And it's going to be some multiple of L plus some multiple of the rotation, several And so you just say that let there be these rotation parameters. And now I ask that the flow should be Hamiltonian. The first interesting result is that the flow is Hamiltonian. If and only these coefficients of the L and of the rotations are functions only of area and area moment. And secondly, if Hamiltonian, if and only the one form you wrote down, the right hand side of the first floor is exact. The statement is that the requirement that the flow be Hamiltonian restricts your vector field, and then the energy associated with that vector field is the energy that you can calculate by construction compared to the first law. In this case, the statement would be that the omegas that you have used are all as functions of the blackboard parameter. They're all consistent. The flow generated by a vector field, which agrees with your time-consultational horizon, would be anatomical, whereas that constructed by Stephen and Hunter and so on would not be . And the second remark is that this is something I just realized that if there is a consistent anatomical then, in fact, in the stationary context, this horizon energy constructed by the data, as I said, is equal to the energy . It's a good playground for all of these different techniques and ideas that people have been using. I mean, I think that's a general point. You might not think physically, but this is an interesting setup in high dimensions. But if you've got some general technique for studying problems in relativity, here's a new example to try and out. Would there be any, if you had a correspondent term, would there be multiplications or whatever? Yeah, I mean I just don't know what would happen. I mean that's, you know, there are people who have general procedures for getting into the world, etc. Which you could apply to this case. I don't know what the answer would do. I don't even know what wall would get to this situation.

47:30 That's another problem that one can try to make out. I suspect that I, Mr. Tartt, answered the question I was going to ask, non-villainous relations that arise into infinity. But two tests for a conserved one, for a definition of energy is that it should actually be conserved on one hand, and the other that it should have something to do with generating a time translation in one or another sense. So when you say, you know, through your list, which of those satisfy these conditions, or which... Well, dimensionally all, I think it's not at all. No, I meant in your... I mean, you said they all agreed at the convention before, but you say in the . And do you think the various definitions of energy? Yeah. Well, I think that the broker of boundary conditions now, we were having a discussion, Abby and I, about this. There were papers that were crucial on this, which sped out the kind of boundary conditions under which various definitions of mass are equivalent in this context. I have not fear, I think he's actually worked out since John was in high election, but he hasn't, I mean, what could he principle to turn the handle? My impression was that the boundary conditions satisfied by these solutions are such that in principle, all of the different definitions should occur in science. But I think that's not Abby's recollection, so at this stage. But I thought you just told those people that . No, no, no, no, no, that's separate. What is certainly clear is that the boundary stress tensor method is excluded from that. The boundary stress tensor method may have its virtues, but it does not seem to be what? A relativistic natural use. So it doesn't produce something which would be conserved for a general solution. I think not. I mean, I don't want to make a dogmatic say, I think the main problem with that is that the energy is not so, it's not so much population. It's just, you give it a symbolically flat space-time and that energy will depend on which particular conclusion you do. So you don't have a definition from our point of view. But even from the field theory point of view,

50:00 there's an ambiguity. The point is that these are field theories which are with a conformant anomaly. That means there's no way of making a physical statement without specifying a boundary metric. So even if you were just an ordinary quantum field theorist, and you were asked, given a metric, what is the energy of the quantum field theory in that? You would not know, even though it was a conformity bearing theory, because of the anomaly. It would depend on which choice you made. So it's that which is regarded as support for the ADF safety correspondence. Because you have an ambiguity here, they have apparently the same ambiguity there. So that's certainly true, and the same kind of calculations reveal the ambiguity. The issue is whether you can use the extracts and physics with the ambiguity. There are no more questions next. Thank you.