Time in classical & in quantum gravity (contd.)

0:00 If I want to see something evolve, I see things evolve, and I want to explain why they are evolving. Okay, I think that's a nice point to break. I guess the next speaker is me, and I think Carol is in charge of the second. In that case, I should announce your iPhone, which I have somewhere. In any case, there is... There is another, Sandhya Motto, which is very nice, where you switch your head from being a speaker to being a chairman, which is, I know my hour, does thou know thine? In any case, the title of the talk is Quantivation of a Relational Quantum Model. Thank you, thank you, Carol. What I'd like to talk about today is primarily some work done In collaboration with Julian Barber about the quantization of a model system that exhibits many of these same puzzles that Carol has been talking about. Nobody more than Carol has convinced us by his work over and over again of the seriousness of the issue of time in quantum gravity. Now I'd just like to make a few remarks to start. We should remember, I feel, that essentially the problem of quantum gravity has two stages. One of them is, is quantum mechanics as it was given to us by Bohr and Heisenberg and everybody else and general relativity as it was given to us by Einstein compatible or not compatible? The answer is yes or no. The second stage is, if no, then what? The first stage has two parts. One of them is, can we solve the equations that we therefore get when we put them together, which are the constraint equations of time and gravity.

2:30 But if there's a second part, supposing that one could solve those equations and one had a space of size expressed in some variables that solved what we could agree on were a realization of the components of quantum gravity, then we need to give it a measurement theory. We have a measurement theory for ordinary quantum mechanics, and as came out clearly on the first day, that measurement theory doesn't extend, at least in an obvious way, to the solutions to these equations in the case of the compact. Now, we'd like a measurement theory that's as good as the measurement theory for ordinary quantum mechanics. And I'm just suggesting, not too strenuously, but I suggest three criteria. It should apply to every solution to the equations. If the measurement theory discusses quantities called probabilities, which being quantum mechanics we would expect it to, those things that are called probabilities should be well defined, which means when the theory tells us what the mutually exclusive possibilities are for some observation, they add up to one, and I say here without non-locality, and I'm setting here up Ted Jacobson for a comment about that. This is referring to these discussions about normalizing probabilities. The third one, the third discussion, the third criteria is that the theory to give us some predictions which are not already predictions from classical general relativity because it's easy to invent ways to because we understand how to make predictions from classical general relativity. Now without going Until Vaughn's discussions, I think that the work of Carroll and many other people brings us to the conclusion that with the exception of this possibility of an eternal time, which is unresolved, We know of no such measurement theory for the solutions to these equations. If not, then what do we do? Then we have to modify the theory in some way, and I'd just like to emphasize that while there are many possibilities, we should remind ourselves that we're now making some fundamental change in our physical principles.

5:00 And we should perhaps think about this. And I'd like to suggest that the problem of time in quantum cosmology is related to the old issue of how do we make a physical theory without ideal elements. And by ideal elements I mean some background structure, some things that act without acting on to use the language that Einstein uses when he discusses general relativity. This discussion goes back to Leibniz's critique of Newton's theory of absolute space, and there's a long history of, on the one hand, people construct physical theories that are good, and then other people point out that there are ideal elements in the theory. In mechanics, the inertial frame is Leibniz, Mach, and Einstein's stress. The absolute time is an ideal element. There's a long history of a discussion of does general relativity contain such things. Certainly it does in the asymptotic and compact cases, which is one reason we might think we could understand the quantum theory in that case. In the compact case, I'd like to suggest it doesn't, although that's a long discussion. Oh, eliminate. Does it eliminate ideal elements? In quantum mechanics, of course, we have, at least under all the interpretations of quantum mechanics that I understand, a necessity to introduce some ideal elements, and we can think of that as Bohr's classical measuring instrument, the split of the world into two parts, the environment we've heard about the first day, and there's many different ways to say it. But every way to get predictions from quantum mechanics involves some notion that the system that we're getting quantum predictions from is only a part of the universe and the rest of it is somehow being used as a frame to make sense of the predictions. And also, and this is the really important point for time and quantum cosmology, the T in the Schrodinger equation is an ideal element that refers to the time as measured in the laboratory outside the system that quantum mechanics was studying. And Jim Hartle has really studied what can go wrong and why you can't make this T be a part of the system and maintain quantum mechanics.

7:30 So I'd like to suggest that when we're trying to understand how to do quantum cosmology, what we're trying to do, understand really, is can we make a quantum theory without ideal elements? We have in general relativity for a closed universe, dynamical classical theories without ideal elements. Can we make a quantum theory without ideal elements? And that is really the issue. I don't have a solution to this question, but I'd just like to frame it in this language. I'd like to suggest that because of this framework, well, because quantum mechanics, as we understand it, is the theory with ideal elements, it's likely that if we can invent a form of quantum cosmology, that that will really have to be something that involves some new ideas. What I'd like to do now is turn to a simple model of a dynamical system which is much simpler than general relativity, but as a dynamical system is a system that is free of these ideal elements and discuss what happens when we try to quantize it. So, this model was invented, as far as I know, independently by Julian Barber and Bruno Bertotti, and then they began to work on it together. They were both very influenced by ideas of Leibniz and Mach and Einstein and so forth, and they wanted to know, could you have a classical dynamical system that has no ideal elements in it? They invented such a system, and I'll describe it briefly, and then I'll describe what happens when one attempts to quantize it, which is the work that Julian Barber and I have been doing. In collaboration, to tell you what the conclusions are, the conclusions are that you can follow the formal steps and find solutions to the analogs of the constraints equations in final gravity, and then you don't know what they mean. We feel this is interesting because it exhibits... It's not surprising but it's a simple dynamical system whose classical dynamics is completely transparent and whose quantum dynamics you can solve and it exhibits how serious is the lack of any measurement theory for understanding... Quantum systems based solely on constraints. Now, their ideas are the following. They called it the Leibniz group.

10:00 They said if you have a system which has no ideal element, it should have these symmetries. First, time should be purely relational. You should be free to rescale time absolutely as you choose. Anything, this is going to be just a Newtonian type system, any dependence on space coordinates should depend only on relative coordinates of particles in the system. There should be no reference ever to an external frame. And therefore there should be a symmetry where I'm free to use, I'm going to use some equilibrium coordinate system to measure the particles at a given time, but I'm free to use a different frame at different times. So I'm free to add to the coordinates an arbitrary time-dependent translation and time-dependent rotation. It's a model. I think I would agree with you and they went through an evolution after this in which they attempted to follow up ideas like that and eventually it led them to the conclusion that general relativity was the theory that you got when you followed that point of discussion. But we'll see that this has enough structure to mimic general relativity in certain ways. The group of transformations, clearly relative distances between particles are invariant and nothing else. Now, in formulating this theory, they had to invent the notion of a derivative, and to illustrate the problem that comes in, so did Leibniz. Let me note this. The data for the theory is a succession of snapshots where you don't know what the frame of the snapshot is. So this is the positions of eight particles at some time, but you're not supposed to have any reference to a frame. This is the positions some time, coordinate time, dt later. The second colored dots are the positions of the particles. But one doesn't know how to orient the two snapshots with respect to each other. How does one set up a dynamical system? Just from the sequences. And their solution is that in each interval of time you should go over the whole gauge group of these spatial transformations until you extremize the square of the differences. So you extremize essentially the total kinetic energy. And doing that is called the intrinsic derivative. Since time is short, I won't describe...

12:30 The definition of the extrinsic derivative is the velocity that you measure In the frame that minimizes the potential energy as you move the frames around, and you do this point by point by point, this is called stacking. Now, once you've done this, you'd like a dynamics that is also independent under this monotonic rescaling of time, and they propose the following action, which I wrote there because I'll need it. They said that kinetic energy is what it usually is using this intrinsically defined derivative, assume there's some potential energy which depends only on relative coordinates, assume it's negative, and take as an action principle the product of the square root of the kinetic energy and the square root of the potential energy. I'll tell you. Clearly, this is just the geodesic principle. And you can see that it's invariant under re-parameterization of the time because this has a 1 over dt squared and its square root, and that's dt. Multiplying it like this may seem a little odd, but it's the only way one can think of to add the potential energy without breaking the invariance of primer scaling. And as I'll show you, this gives the dynamics which in a particular choice of gauge reproduces mutants. It is really beaten in here. That's right. Well, except that Chris's point is still hollow. There is an ideal element here, and that's what Newton was trying to say when he said that there was a universal plan. It's what simplifies the Lagrangian. Yes, but let me show how that happens. So we choose this as the Lagrangian. Let me just note how we return to Newtonian physics. If we look at the equations of motion, they have this form. I assume that I've stacked the coordinate systems in a way that always minimizes the potential energy. Then I get equations of motion like this, and clearly if I can get rid of these factors of t over minus v, I will have Newton's laws. I get rid of it by imposing the condition that I choose the time coordinate such that the total energy is zero.

15:00 In some sense, the ratio of T over V square-rooted is playing the role of the lapse, and I'm choosing something like lapse equals 1. Now, therefore, I get back to Newton's equations, but in a time coordinate which depends on the whole system. It's not an a priori given time coordinate. For different configurations and different classical solutions, the time coordinate runs... By the way, an alternative way of doing it is to start from the ordinary Lagrangian, then eliminate time for the conservative system and come to the Jacobi Principle, and then the Jacobi Principle to put into the hero, which brings me to the Robert S. Lagrangian. Yes, absolutely. And I think that it was Carol who pointed this out to them originally. I was about to say this, but... When we cover Newtonian dynamics, but in a coordinate system where the total momentum and angular momentum is zero and where the total energy is zero. Now, let's look at the Hamiltonian dynamics of this model before quantizing. To choose a particular time. I just wanted this to show that this wasn't some crazy dynamics that you never heard of. It's another way of saying Newton's dynamics for particular systems where the energy, where all the conserved quantities are zero. To quantize it, we go to the Hamiltonian form. I'll do this very quickly because I want to save a minute at the end to make some conceptual remarks. We form the canonical momenta and then we find that there are some constraints. We have a momentum constraint which says the total momentum is zero, an angular momentum constraint that says the total angular momentum is zero. And from the definition of the canonical momentum, we get a Hamiltonian constraint that says that p squared plus v is zero summed over the particles. So this system is analogous in its Hamiltonian form to general relativity. It has a constraint which is purely quadratic in the canonical momenta, and it has some additional sort of spatial type constraints, and this is the theory that we'd like to quantize, and we're going to try to quantize it in the usual way by finding a representation, psi of the coordinates, and applying the constraints.

17:30 Now, this is different, let me emphasize, than the quantization of the Newtonian system, to which it's equivalent to a negate, because in the quantization of the Newtonian system, is that a... In the quantization of the Newtonian system, one has the possibility of getting non-zero values on the right-hand sides of these equations and one can build up states by superposition over these values over these states with non-zero values. Here we're just restricted to states which are in the kernel of the constraints. I don't know. What do you mean the heat-kernel thing? If I did the heat-kernel for a curtain in the can, and I had the IEDP on the other side, and I said, I'm going to do the heat-kernel when it's all set to be equal to what I'm done, what would it be? Let me come back to that later, because I'm not sure. I understand. Okay, let me, what I want to emphasize is that here we have a simple finite-dimensional system. One can study the solutions to these equations in some... Simple models, and we've done a series of simple models with simple forces between the particles and so forth. The problem is the following, okay, how does one give a physical interpretation to the solutions to these equations? And if I have time, I can show you some solutions in some simple cases, but I really want to emphasize the problem. Clearly, one cannot, and this Carol really already emphasized, one cannot use the inner product from the quantization of the Newtonian system Because by solving these constraints, every solution to these is not normalizable in the inner product from the ordinary quantum mechanics. So one has to invent some new inner product in which just the solutions of these are normalizable, and that new inner product has to tell you in some way what the time you're really thinking about in the system is, or perhaps has to. Describe a measurement theory, imagine now that there are many particles, we are some of these particles, we live in this universe, so really make it like quantum cosmology, these are the quantum states of our universe, and we'd like to give a measurement theory that can tell us how to connect our observations with predictions, and I would like to claim that while one can write down lots of solutions to these and lots of simple models, one really doesn't have a way to get

20:00 There are no probabilistic predictions out of the theory, except, say, of the simplest sort, that when you measure all the energies, they add up to zero, or all the momentum. And I think I'll, without giving you examples, I think I'll leave it at that. Let me just conclude by saying that either there is a way to do this... Which there might be one in this model not in general relativity or I'm not saying that if we can solve the problem in this model we can solve it in general relativity. It may be the particular structures of the Hamiltonian constraint in general relativity lead to an internal time which let us solve it, but I think we really need to decide whether we can solve these kinds of problems or we can't. If we can't solve it, then we really have to face, I think, the possibility that quantum mechanics, as we understand it, is something that applies only to portions of the universe, i.e., quantum mechanics necessarily involves ideal elements. And we really are faced with the problem, which I imagine, if this is really the situation, will take... We have a long, long time to realize a solution to of inventing a new way to do quantum physics, which is not ideal. I just want to emphasize that if there is no solution along the lines of the internal time, we're faced with a fundamental difficulty. We really don't know how to do quantum mechanics except in a system that is a portion of another system, and we have to try as hard as we can. So I guess that's all I have to say. I think that Mary should have made that remark, but because he didn't, I should make it. Because Leibniz didn't believe in the ontological existence of time, he dropped the letter T from his name.

22:30 Is that true? He spelled it himself. Just one comment in this sum over histories approach which I'm going to discuss this afternoon. I looked at a model due to banks which is equivalent to, at least classically, to the model you described. And I think I would claim to know how to interpret it. In the sum over histories approach to this issue, which I will describe this afternoon, I looked at a model, which is essentially through the banks, which is classically equivalent to this one, but I had lots of discussions with your collaborator this time. I wonder if you have any comments on the connection between... Sorry, I can't hear you. First of all, I have two comments. One of them is that, as you've said in everything you've written, what you're trying to do is solve this second stage problem. We've accepted that the canonical theory has no good measurement theory, and you're trying to go on and solve the next problem. And we're still trying to really convince ourselves whether or not the canonical theory has a good measurement theory. Doing things with path integrals that have no canonical expression gives a good measurement theory or not for these models, I think is something that we need to discuss in detail. I mean, I think that we are, well, I think it requires some detailed discussion. Julian and I spent a lot of time talking about what was in your paper about this and didn't agree among ourselves. So I think I can't comment on that. If you put up your last slide, you should be able to look at that and see what you're supposed to do, because if you can fix the first mistake, fix the second mistake, and replace the third mistake by a third mistake, you're put on the right side. I think that, as I said before, I think something like that can be done.

25:00 In the gravitational case, you have an explicit proposal for the meaning of that T. In this particular theory, I know of no such proposal. And let me just say, you also are saying you're abandoning the canonical theory. It's really people are fleeing from it right and left, right, and are trying to introduce a new principle of how to do quantum mechanics for the case of the universe. And I am still trying to decide whether or not one needs to flee this sacred ground of the canonical theory. It looks like you're getting there. Yes. Yeah. Can you ensure that I would not twice interpret it if it's not there? Because I don't see why you couldn't. Well, the problem is that in the classical model as given, there's no instruction for what T to put there. In fact, the whole point of the model is that there can't be. Now I would say that that T is your ideal element. It should be determined somehow by the dynamics of the theory. Maybe it should go from your time in the afternoon and let us give chance to Don. Can I just remind you of something that if you do have, in your case, I mean if you do have a stationary state like this thing I think you can define a time if you do a clock. I mean consider the simplest example of his case where you've got three particles in one dimension and you fix the potential to be a negative constant and then and now so the constraints of the total energy is fixed so that means the sum, I suppose the mass is equal, the sums of the three squares of the momentum are summed to a constant which is the negative of the negative potential. And then the domenum is constant. So all you have is these two relative distances. Suppose you take the first relative distance as a clock and then the distance of the third particle from the center of mass of this as my observable. And now I want to ask, given that the clock has some particular reading, what's the observable? What's the distance of this thing from those? And now if I choose states for the clocks, for example, suppose I chose some sort of wave packet state, I project the full state onto this wave packet state, which will now be normalizable in the ordinary Schrodinger inner product.

27:30 Then if I define some other clock states, in this case with these variables the Hamiltonian will be a sum of two parts, one for the relative separation of these and one for the separation of this from that, and then if I want to calculate expectation values for the relative state of this other wave function, and I ask how does that depend on the reading of the clock, then in this particular case where it has this simple thing, then we just get a Schrodinger equation for the motion of the other thing. So, as long as I can divide up the system into one thing that I call a clock and something else I call my physical observable, I can ask what's the correlation between the clock and the other thing, and I can even get an equation relating it. In this simple case, it's a Schrodinger equation. In a more general case, of course, it won't be such a simple equation, and one can ask whether... The fact that it's not exactly the Schrodinger equation gives results that are contrary to observation, but I think in this case it's a very simple description. One can describe this perfectly well in quantum and canonical terms in terms of these conditional expectation values. Yeah, well, do you want to make the comment because you also want to, or is it another question? Yes, I think it would be best if you do it now. Jim, I heard that you agreed to finish five minutes earlier. Well, this is the schedule already given, in fact, so it includes here. Yeah, yeah. Tell them to time it one sentence to five, and then maybe this should be left for a formal discussion. We also look very carefully at this point of view in this paper, so I can, people in the audience can't read all those equations, I know them very well, and I agree in the case that we could agree on an inner product, which is an inner product on the physical Hilbert space, i.e. the states of solutions, but the problem of picking what states within the solutions to the constraints are going to be normalizable and what states are not going to be is the same thing as in some sense deciding what time variable you're going to use. And the choice of the inner product then just depends on what time variable you want to use if you want evolution in terms of time, and the notion of what's a unitary evolution and what's a Hermitian-Hamiltonian again depends. So, I guess, I think it's not, it's not so simple because of the fact exactly that the solutions to the constraints are not normalizable on the obvious inner product.

30:00 And any choice of further inner product depends on having some idea about what you mean by time. There are many solutions, just like in quantum cosmology, you'd like things not to be normal, and that I think is really appropriate to your... The question is this. You were thinking three parts of both. And you were saying, well, let's take the relative distance of two of them. You know that the position of one would not commute with the constraint which tells you that the total momentum is zero. But the relative position of two does commute with this constraint, so you say, well, that's good. However, we have another constraint, which is the Newtonian constraint. The difference of these two does not commute, obviously, with the Newtonian constraint. So the question is to you and also to Tara and to me and to everybody who speaks about time, the same time translated generatively, why do you ask that we want an object that commutes, say, with the three different, the promoter and the constrain, and then we want to introduce a variable with its time, which is by definition something that does not commute with any time? I know that you all know the answer to this question. So this is the philosophy behind that. And what I wanted to do is just a very, very short thing. I want to show that there is a way, I think, a formal way to quantize this model, at least the synthesis version of this model, using Chris Isham's ideas of a group theoretical quantization, or better, using a slightly modification of Isham's ideas. And, well, I take the simplest version of the model, just three particles and constant potential, and that's, well, these are the two constraints, and my main point is what is observable, and I wanted to stress the fact that I want to say that what is observable is what commutes with all the constraints. And now, how do I want to quantize? Well, I take three of these observables that I call S, C, and J, which happen to commute with, be real observable, commute with me. Both these constraints, and moreover, that have a closed Poisson algebra between themselves.

32:30 Okay, this Poisson algebra is the Lie algebra of a group, E2, so I put myself in the framework of Chris. I just quantized the model by looking for a representation of this group, and actually this is one of the groups discussed by Chris, so just look in his, and well, one can discuss how to choose a representation. There are a lot of arguments. That's a representation for the Poisson algebra. I have an Hilbert space, which is an L2 function over the circle, and let me tell you, there's a Hilbert space, there's a physical scalar product here, and I have the operator corresponding to my physical circle, things that I can really measure in this. And while I have predictions, because this has a spectrum which is quantized and so on, so what I want to say, the general description is, well, a general, this is not really Chris' group quantization, because he stressed strongly this morning that he wants the group acting on the entire phase space. This group is not acting on the entire phase space, but is acting on the constraint, on the constraint. What I want to say is that if one uses group theoretical quantization with gauge invariant observable, or geometrically acting on the constrained surface, one gets three results. First, this is a way to obtain the physical scalar product. This is analogous to Dirac quantize, solve the constraint equation, and then search a physical scalar product on the solution. That's analogous to just one of the choices. And then there's no other big with this. I mean, these are just well-defined observable and there's no time problem behind that. One can construct all a theory of well, but I don't want to go any issues, conceptual issues. You say the problem is for us to find the group in the case of canonical, like a formidable path. Absolutely, yes. I want to say that. I think that we should cut it relatively at this moment. I would like to reply as well, but let us do it tomorrow.

35:00 In any case, we need someone to lead us from time to timelessness, which is always achieved after the lunch. And this is Jim York, whose title is Time and Timelessness in Gravity. Could you just holler when I have five minutes left? As a working man, I still will quote Philo of Alexandria, who pointed out, among other things, that it would be correct to say not that the world was created in time, but that time owed its existence to the world, for it's the motion of the heavens that determines the nature of time. That's the sort of thing that I was thinking about in my youth when I figured out that the The way you talked about slicing spacetimes was to call them foliations of co-dimension one, and that's now a commonly accepted way of talking, which I'm very glad to see. And where is this reed or branch that was serving as a pointer? You're going to beat me with it, perhaps. And Bill, I'm not playing peek-a-boo, and this first part we're going to fly through because everybody knows it. Okay. Time and timelessness. This is the time bit. John Stachel also used to look at space-time this way back in his youth. Before I arrived at the notion that what time meant in general relativity was the foliation of co-dimension one, I wondered if it was not just the time-like congruences of curves following clocks. And so, by familiar methods, which I don't have the time... To explain, I will point out that the reason these are difficult to use is because a general one will have a twist, where have I written it there, in red, and so by Frobenius' theorem in generalized nonsense, you can't have everywhere a space-like slice generated by such a thing. Although such structures, which have various names, do have induced positive definite metrics on them, some people call them the induced metric of the space of orbits of the congruence, the associated covariant derivative has torsion and there's no new dynamics in it. I did an ADM analysis of it about a million years ago and there's nothing.

37:30 But kinematical stuff there. So we pass to the notion of using foliations and everybody now knows what that is because of several learned lectures but and we can pass quickly therefore just to remind ourselves that the that the second fundamental tensor has a trace-free part and a trace. The trace-free part apart from the convention The first thing you want to know about the sign is the shear of the normals that are orthogonal to the slices. We imagine it's like a fluid in hydrodynamics. That's the shear and the trace of k is minus what is something with a minus sign in the metric. And you want to know something about space. These are the variables you use because these are the only variables that are natural. They all have many incredibly interesting properties, but they're still referring to the embedding of things, at least as far as the classical theory goes, and when you do the theory of embedding, you use these variables. I mean, you're not committing yourself to doing canonical anything. Now, Carroll's mentioned all the different kinds of time. John Wheeler was the big advocate of the three geometry as the carrier of information about time. The square root of g gamma, the three metric, being an obvious possibility of choice, and for various reasons that doesn't work, and the main objection one has to that is that that's not a scalar, and I'm not willing to entertain the idea that if you fix something that isn't a three-dimensional scalar, that that's an appropriate notion of time, no matter what your formalism may tell you, so I warn you not to do that. Now, the notion of extrinsic time, the history has been reviewed. I'll just mention the particular choice that's proved useful is to control the slicing of space-time. This is very useful in numerical relativity, too. What is funny about this? Yeah, well, it's just that I wasn't going to, it's not a judgment on Don Witt. It was that I had to reinvent this talk just during while listening to everybody else and all the references didn't come immediately to mind. Yes, okay.

40:00 Using the trace of K to control the slicing of spacetime goes back a long way. When it was zero, using it as zero simplifies solving the constraints and it gives a definite way of evolving astrophysically interesting spaces and perhaps quantum geometrodynamically interesting spaces. However, we now know that you can't always have in an asymptotically flat space with a non-trivial topology that the gradient of the trace of k is zero. The main point is that the gradient is zero. And we can get that the trace of k falls off like 1 over r cubed in asymptotically flat spaces, and in fact we really need to do that rather than what you'd expect, 1 over r squared, but we don't have time to go into that. The point is we can get it to go fairly fast to zero. This recouples the constraints and makes life difficult, but I just wanted to point out to people that the equation, the initial value equations and that version of understanding the constraints is still completely valid, it's just it's harder. I mean, every solution of the Einstein equation can be written in the form that they've been written in, familiar to most of you, whether or not trace k can be taken as a constant. Now the reason trace k is interesting for compact cosmologies, which have the topology of some cozy surface across time, is because it's more or less one over the Hubble time. And so it is a kind of variable determined by the motion of the heavens, and it works very well in anisotropic cosmologies. It takes sort of an average Hubble time based on the volume. Rather than just the radius, which one has in a simple cosmology. All these are incredibly familiar ideas to some of the experts here. The reason it works and the reason it doesn't work sometimes, which has been probably interesting, is the equation of motion here. There's a nice elliptic equation for the lapse function. It's the shift dot the gradient, but that is never a very worrisome term whether it's zero or not. Let's say k is a constant and make it zero. Well, the first question is, can you make k equal constant? The second question is, can you keep it constant, like put 3k to 3 equal to 1, say. And if you study this kind of an elliptic equation, you find out that you can, and there's no problem with it, if the rho plus 3 times the pressure rho plus the trace of the stress is positive, that's called a strong energy condition.

42:30 This is positive, and this turns out to be a great equation, and you can solve it until the cows come home. However, in inflation, you have an effective cosmological term which enters the equation as the wrong sign, and this causes some interesting difficulties, not necessarily insuperable in present inflationary models, but in, and I realize this is still sort of time, not really quantum time, but it's leading up to it. The point is, if you had a really highly inhomogeneous inflating cosmology, this could really be a disastrous problem, and in that case I can't and don't have a prescription that will generate a nice foliation. However, I'm working on it and I'm sure other people are interested in it as well. The difference between intrinsic and extrinsic time is summarized by the fact that they're canonically conjugate. There's another term related, which really is the Hubble time, and the minus sign is for giving the same sense and orientation as this case. It's canonically conjugate to a quadratic, one of the quadratic terms in the famous Hamiltonian constraint. And it's conjugate to the one with the minus sign. And it just looks like the answer. I mean, it just looks like the answer, but it isn't. Because when you solve the canonical transformation and redo the theory, the second order operator comes back. This was one of life's disappointments, but there it is. That's the message. The message is it doesn't want to be solved this way. But basically I'm telling you why all my and related other people's great old ideas don't work all the time. Now, talking about... Timelessness, this is the first part of timelessness, and it has to do with the failure of trace k, or that the notion that the epoch that we're in in cosmology can be identified with trace k, and this is some work in progress of Don Witt and Doug Eardley, and this is a counterexample, nearly explicit, with a proof that I regard as correct, of the fact that there exists perfectly decent spacetimes. They don't have any trace k equal constant slice in them whatsoever for any value of the constant, but they're perfectly interesting.

45:00 And you take some Casner universe that expands with k positive, and this is the contractive one here with k negative, and you join them with a wormhole. They have given the argument of how you can make this construction without any matter. You don't even need any dirt or anything to get the joint to go and I am working on a more explicit version of the proof but the proof is good and the idea is a very good one and there's the end of that. You can't always use trace cable constant foliations. Now it may turn out though that there's no causal connection between these regions and so this may not actually be a problem. For quantum, for thinking of building an internal clock out of the variables in theory, one, I just don't know. It's rather likely that there is one. Yeah, it looks very likely there's no causal connection, and here we are, we're either here or we're here, and of course, you know what happens, trace k goes to zero, and here somewhere, time stands still somewhere. Okay, but that may not be a problem. I just pointed out that you let time move forward in there, it'll pinch off. Yeah, in fact, we can make, there's no problem with making time move forward in both of these things. Just put k dot equal to one. But k dot goes to zero, has to go to zero in here somewhere, so time has to stand still in here somewhere. And so it may not be a problem, but it's, the point is, we may be in this bit of the universe, this is similar to many ideas that have been expressed. And there's an end on it. I mean, your internal plot may not be so easily constructed. Did Morrison and Tipler have a topological assumption? Yes, thank you, Jim. Morrison and Tipler's paper could be not so kindly summarized as saying that whenever you can get trace k equal constant slices, you get them. Now, that's unkind. Let me be less unkind and say that in certain topologies there are no topological obstructions, and in fact there are some amazingly inhomogeneous models in which you can get them. But I can't give you a classification yet. I only have this one counterexample, but I think if we can understand John Witt's work, which I put in a recent proposal as D. Witt,

47:30 When it got through the machine, it was DeWitt, so we're going to have to call him Don Witt, I think, to protect his originality in these matters. Could you re-identify that to make them both one space, to run their whirlpool from the castle back into itself somehow? I haven't thought about this. I learned this last week. I mean, you know, I'm going to invoke the familiar thing, okay? Numerical relativity, the familiar thing is we just got this off the computer yesterday, okay, and quantum gravity is we just had this idea last week, and my excuse is I just got this counterexample and started checking the proof in the last few days, so I don't know if you can do it or not. Yeah, yeah, and in fact there's a discrete isometry from one side to the other. It's a beautiful thing, and everybody will be hearing about it. I'll just give you a preview. Now the other part of timelessness comes, for those of you who listened to my talk the other day about describing thermodynamic equilibrium of black holes in the canonical ensemble, this is a slide that reminds us that what I did in that analysis was to eliminate the Hamiltonian constraint and I gave a physical argument for the measure and that physical argument actually I ended up saying that the entropy is the measure and the entropy depends on this R plus variable here, but it should be realized that R plus is a definite functional, functional of the three geometries that comes from solving this problem. So it is, so this measure we construct does not involve, it's physically motivated, does not involve the lapse function in the slightest way. And this is what makes the problem interesting from the point of view of timelessness, is that this remark at the top. Since u, our metric function g is the last function squared, it simply doesn't appear anywhere and it turns out it can be an arbitrary function of two variables. Well this means that the three geometries that you get from solving the Hamiltonian constraint can be stacked together in an infinite number of ways.

50:00 And the infinite number of waves is a sort of like a midi superspace because it depends on an arbitrary function actually of two variables and they have boundary conditions and regularity conditions so but that's still an arbitrary function of two variables. There are a lot of them. So one really has an incredible number of spacetimes just by stacking them all together in the same way and the timelessness is You have no way of knowing what leap of the foliation you're on, and you don't even know whether the proper time between two of these points in phase space, you have no conception of what the relationship is, and this is correct for thermodynamic equilibrium. You should not be able to tell in thermodynamic equilibrium that anything is happening whatsoever, because there's nothing there that can tell you. You can't see anything, and you can't read your watch. You can't tell. And that's what comes out. You're not in thermodynamic equilibrium. Now, this arbitrariness has... Let me point out that it's just not... The other consequence I mentioned in the talk was the... When you construct the curvature, the Einstein curvature in this case, for the problem, you find out that regularity forces the lapse function squared to have an asymptotic behavior in the center, where the black hole is. I've drawn the black hole as if it were an extended object here. And it has a certain definite asymptotic form, and A is a periodic function of time, another positive periodic function. If you compute and find out the Hamiltonian constraint is zero because we solved it, the regularity forces G11 to go to zero at the origin, and I mentioned that that was a necessary condition pointed out by Don Page some years ago for black holes. But the angular stresses, which I've drawn in red here, build up all around the black hole and so there really is an infinite number of black holes. You have timelessness and you have off-shell stresses all built up around the space. And the last aspect of timelessness is the bottom diagram in which I sketch the constraint hyper-surface. And I have the momentum constraint satisfied trivially in this problem, the Hamiltonian constraint is satisfied, and the usual classical picture, which we owe largely to Moncrief to make it rigorous, gives us a direction out of the constraint hypersurface.

52:30 So if I change it to space-based, g and pi are sitting here in this space. And if I go this way, well, I'm just simply not going to get data that generate any solution to the Einstein equation. But if I go, say, this way, it'll just be another slice in the same classical spacetime. Or if I go this way, it'll actually be new data for another vacuum spacetime. Well, now, what I want to point out is that in this thermodynamic equilibrium problem, there's only one point in the phase space. And it just sits there and you don't go anywhere and yet there's an infinite number of space times that get summed over and so that's an aspect of timelessness that I wanted to emphasize and I guess Carol what are you going to do to me now let me let me that's my talk but I want to make a quick remark about We all know that there are 14 such algebraic invariants, and we don't need to let the Ricci tensor be zero, but if we do, there are four, and the two quadratic ones are these that look like the familiar electric and magnetic parts of the Weyl tensor, and these are two invariant expressions. But, of course, they don't give flat spacetime, though they are completely independent of the lapse in the shift, because you can have some plane wave shift. In fact, you can have all 14 invariants equal to zero, and the Riemann Spencer will not necessarily be zero in Lorentzian spacetime. Now, a couple of years ago, I wrote an article and pointed out that this non-invariant combination, e squared plus c squared, which many people have noticed before, This combination is not independent of the Latzman shift, yet if it's zero on a Cauchy slice in a Ritchie flat space-time, then the Riemann tensor on that slice is zero, and by using the uncontracted Bianchi identity, one can prove that the Riemann tensor stays zero in the domain of dependence of the data. My final remark is to remind you what the vacuum is.

55:00 We've all been talking about foliation independence, and the last time I heard flat spacetime, a Lorentzian signature, minus plus plus plus, was a pretty decent vacuum state, and if you cut it up in a wild way, you have to satisfy those equations, and when somebody can produce me a connection whose natural curvature If you give me those, then I'll be totally convinced that you have the perfect variables. Now, our base variable gives hat-flat, and that's a fantastic result. Oh, they do. Now, if I ask you if I misunderstood you, then the variables are about as nice as they can be that way. So one can actually explicitly deduce these equations. In real relativity, they have to use that. Now in Euclidean space, it's even easier than that. Okay, so anyway, that's an argument strongly in favor then of those variables. It's a combination.