Unsolved problems in spacetime structure / breaktime conversations

0:00 One, two, three, four, one, two, three, four, one, two, three, four. Listen to this on the other tape and use it to integrate reflections on Colin's article on love here. Please take your seat. It's my pleasure to introduce the next speaker, Professor John Ehrman from the University of Pittsburgh, and he's going to talk about the answer to problems in space-time philosophy. So, this talk will have exactly zero logic content. This is perhaps not a bad thing. We've been getting a very rich, very meaty diet of logic talks. Most people need a little break to digest, or you can take a nap. Okay, I thought the most useful thing I could do for the conference was to outline some unsolved problems in space-time philosophy and in choosing which problems to talk about. I had two things in mind. First of all, I learned problems with some real philosophical content. And second, I learned problems that had some connection to the foundation's issues in physics. But from there on, it was a matter of personal preferences. So here's one of the ones I chose. First was a time machine. Second was time finite or infinite in the past or future. Third was a branching space-time. Fourth was modern cosmology, which in asymmetry problems, by its Hawking's information loss paradox, a genuine paradox, and does its solution require believing six contradictory things before lunch, and finally what is general coherence. I'm not sure how far I'll get it, but let's proceed. Now, I'll be very brief on the topic of time machines,

2:30 because Chris Berthridge discussed this in detail yesterday, but I'll spend just a couple of minutes outlining the issue for the sake of those who want his talk, and it will also serve as advertisement because on Monday there's going to be a workshop that will take up the issue of his time. So there's a rather large physics literature on time machines and I'm not talking about a fringe literature because the physical review, the physical review letters routinely publish articles on time machines. But you will search this literature in vain for a definition of time machines. And so I want to point out the difficulty in getting the definition. So, suppose that we're in a space-time that possesses a partial portion surface. So, this is a space-like slice without edges. So, in some sense, it slices through the entire space-time. And it also has a property that no future-directed time-like curve intersects the surface more than once. Okay, and now on this surface we specify the instructions for our time machine. Whatever else a time machine is, it's some kind of device that operates for a finite amount of time in a finite region of space-time. And then as a result of the operation of the time machine in the group, the crazy thing it does, closed time-like curves appear in the future. Now, here's the basic problem. In what sense are the closed timeline curves due to the operation of the time machine? In what sense does the time machine produce or manufacture the closed timeline curves? And I think you can see the problem quite readily because due to cannot be taken to mean due to in the sense of causal determinism because the chronology that contains the closed time-life curves is necessarily going to be disjoined from the future domain of dependence of this partial coast. So the future domain of dependence consists of all those points where the state can be uniquely determined

5:00 from the state on the initial slice. Now, so to speak, the next best thing would be it would be proved that every extension the future domain of dependence of sigma contains closed timeline curves, because then there would be a clear sense in which the existence of closed timeline curves would be due to the time machine, all of the particular details, perhaps not due to the existence of closed timeline curves, would be due to the operation of the time machine. So is there any hope of proving that every extension is going to have closed timeline curves? Well, here's a case where there appears to be some out. This is a two-dimensional version of what is called the Taub, not space-time, it stands for Newman, Putti, and Tandorino. So the space-time is the surface of the cylinder. Down here in the Taub region, everything is normal. We have a partial Cauchy surface, which is what the light comes look like. But as you go up the cylinder, the light comes tip over until you reach the future Kochi horizon of sigma. That's the future boundary of the domain of the pendants. And then if you go into the non-region, you eventually get to a place where the light comes to tiptoe over and not that you have close time on curves. Okay, now, is it plausible that we can, so, so, uh, uh, not space-time is one extension of tau space-time. Is it plausible that we can show that every extension of tau space-time will obtain close-time by Kirkus? Well, no, not without some kind of side conditions, because we can always just artificially, surgically remove points from the null extension that would block the condition close to the null curves. we can try to get rid of this kind of thing by requiring that we'll only consider extensions which are maximum further extended but if you can do

7:30 various tricks to to get around that so the point is you would like to come up with some non-question-begging notion of a suitable extension and then prove that there exist some cases of general localistic space-times where every suitable extension of a future domain of independence contains close timeline curves. Then you would have succeeded in showing that time machines are at least conceptually possible, they violate various laws or energy conditions to include a separate issue, but at least you would have shown that at that time machines are conceptually possible. Or, if you could take a more negative slant on the issue, and instead of trying to give a definition of that machine, You can just search for a necessary condition for the operation of a time machine and then use that necessary condition as part of a no-go result that shows it's impossible to build and operate a time machine. Of course, this approach is not going to be very helpful if you're striving for a go result rather than a no-go result. And in addition, the alleged necessary conditions that appear in the literature, a condition that the future coasting and horizon become actively generated are not obviously necessary conditions. I will try to explain this. You can ask Chris what it means. So my conclusion is that this subject of time machines is in a very unsatisfactory state, and I think this is an issue in which philosophers of science can do some real contribution by offering some of conceptual clarification. OK, onto the second problem. Is time finite or infinite in the past or the future? And of course, the first task is to give some sort

10:00 of analysis of what it means in terms of space-time structure for time to be finite or infinite future. And here's an obvious first try. Let's say that space-time is, say, finite in the past, if and only if for any choice of now, as given by a space-like hypersurface without edges, every past-directed timeline curve with future endpoint on this now slice, and no past endpoint has a proper length less than some. bound, say, 14 billion years. All right, this seems fairly plausible, but there are a number of problems. First of all, the definition is going to be vacuously satisfied for space-times that do not admit any global time slices. An example of such a space-time would be the neural space-time, and for the logicians, yes, this is the same So, no logic content, but maybe some logic connection. Okay, second problem. Just take ordinary Minkowski space-time and truncate it in the past. say, delete all of those points on a surface that represents, say, yesterday and below. Intuitively, the space-time should count as finite in the past, but it doesn't according to our definition. Let's consider this space-like slice, which is a hyperboloid that fits inside the future light. on that definition doesn't count this space time in the past and a third problem again start with ordinary Minkowski space-time and then just surgically delete these regions and people I'm doing this all the way back to the past and it has to be intuitively. Time itself in the space-time is not finite in the past. It goes on and on and on. But the definition of the planet has been finite in the past.

12:30 Because if you start from any point and try to go backwards in time along a time-like curve, within a finite amount of proper time from one end to one place, it will lead to change. So I think it's not so easy to give a general definition of what it means for time to be finite or infinite in the past or future. And since I'm talking about unsolved problems, I'm not going to try to solve it. I'm just going to push on because I want to talk about the implications of our best available cosmological theories for the question of whether it's not time is finite or infinite. And so fortunately, I can bypass the problems I just mentioned because standard cosmology describes the large-scale structures based on using the Friedman-Robertson-Wonger models. And these models, there's an obvious and natural choice of time metrics. There's no problem within this restricted class of models time to be planning for the future. Now, not so long ago, meaning until the end of the 1990s, the situation was quite straightforward. So here's a diagram of the behavior of the scaling factor, sometimes called the radius of the universe, in these FRW models. They come in three flavors in the k-plus-one models. These are the models where the space slices have constant positive curvature, so space will be topologically closed. The k equals zero models, the space slices are flat, and the k equals minus one models where the space slices have constant negative curvature. And all of these models, time is finite in the past. when the universe begins with a big bang, and then you might say, well, what happened before the big bang? I'll say something about that in a minute, but let's live in the future direction. So you can just read off this diagram that time is finite in the future if we live in a cable plus one universe, because such a universe expands to the maximum volume

15:00 and then retracts to a big crunch. Whereas if we live in a cable zero minus one universe, the universe expands forever, and there's a turn to be stretching out on the floor. Okay, so this was the situation up until the end of the 1990s, and now we just had to wait for the observational cosmologists to tell us which one of these cases held. And then all hell broke loose with the discovery of the accelerating expansion of the universe. To get accelerating expansion in a Friedman-Robertson-Orleford model, you need so-called dark energy, for which the equation of state, which is just the ratio of the pressure of the matter of energy to the density, is less than minus a third. And this violates what's called a strong energy condition that is thought to apply to the normal forms of matter. So to get this accelerated expansion, we need some kind of weird matter of energy. We don't know what it is, but the possibilities can be put into three categories. If W equals minus one, then we have a true cosmological constant. That's the gizmo that Einstein introduced in 1917 and later denounced as his greatest blunder. So it may well turn out that this is not a blunder after all. The second possibility is W is strictly between minus the third and minus one. This is called quintessence. And the other possibility is that W is less than minus one. and what are the implications for the fate of the universe? Well, suppose a true cosmological constant is responsible for the accelerating expansion. Then, regardless of whether we're in a k equals 0, plus 1, or minus 1 universe, the universe will expand forever. However, it's very doubtful that critters anything like us can take advantage of this eternity

17:30 because an observer co-moving with the expansion sees an increasingly empty, dark, and cold universe. So critics think like us are eventually going to die out. So that's why I say we have eternity with a big chill. If the cosmological constant is zero, and quintessence is responsible for the accelerating expansion, times it was a big chill, assuming that the equation of state doesn't change with time. The third possibility is that there's no cosmological constant, but phantom matter is responsible for the accelerated expansion. Then we get something really interesting. We get a big rip followed by a big smash. is the great thing that eventually all gravitation and bound systems are just ripped apart. And it's smashed because static matter has the weird property that as the universe expands, the density of the static matter gets greater and greater, and in a finite amount of time, it gets infinite. So we just splat against this infinite density. And then there are all kinds of mix-and-match possibilities. So for example, suppose that there were a tiny negative cosmological constant, so small that we couldn't attack it by any current experiments. And suppose the quintessence is responsible for the accelerated expansion, then eventually, as the universe expands and the quintessence becomes diluted, the negative cosmological constant universe will start to re-collapse. On the other hand, if that matter is responsible for the accelerating expansion, then lambda will never dominate and get the same verdict as here. So as you can see, the situation at the end of the 90s was relatively simple. Now it's relatively complicated, but we're still looking out that in the fullness of At this time, observation cosmology will narrow down these possibilities. Now, I promised to say something about the beginning, what happened before the Big Bang.

20:00 Well, in classical general relativity theory, that's not a meaningful question because there's no physically meaningful way to extend through the initial Big Bang singularity. However, it's one of the dreams of quantum gravity that somehow getting a quantum theory of gravity may smooth out the singularities of classical general relativity theory. And the proponents of what's called the blue quantum gravity claim that they have succeeded in resolving the initial Big Bang singularity. And this claim is based on two results. First, they're able to show that the quantum operator corresponding to the inverse scale factor is bounded from the bottom, in which in the case they are controlling the infinities. and second, they're able to show that the quantum evolution continues through the classical singularity. But note that the continuation passes through a Planck regime where the space-time of classical general relativity is not even approximately valid. So one could wonder in what sense we can speak of events prior to the Big Bang you know, a nice time series in the sense of a class of industrial or . So I think this calls, I think, for some philosophical cogitation. I've lost count already. This is number three. Number two. Number three. Okay, branching. space-time. There's a metaphorical sense in which space-time can branch. So consider the ensemble of all physically possible space-time models, so all the

22:30 models that satisfy the laws of physics group they are. It could be the case that two or more of these models will agree up to some given time, but disagree in the future, okay? So, there isn't any literal branching, there's no individual space-time literally branches, it's rather that there's a branching, so to speak, within the ensemble of possible space-times. So here, branching space-time is just a kind of metaphor for indeterminism. in a state up at some given time doesn't determine the state at a later time. Now, some philosophers are not satisfied with this metaphor, so they want to talk about space-times that literally are a branch. And I just want to point out that if you try to go from the metaphoric to the literal sense, then you're going to run into various problems. So I just want to remind you of the definition of what a relativistic space-time is. So it's a manifold on which lives Lorentz's signature metric, and it's defined on all of the ball points of the atom, and initially I'll start off by assuming that the manifold is household. So given any two distinct points, you can find this point. Okay, so here's the problem. branching space-time in the literal sense is going to imply a topology change. So here's that sort of upside-down traverse universe. As we go forward in time, the universe splits into these two legs. So we change from a connected spatial topology to a disconnected and the various no-go theorems in classical general relativity on topology change, and here's one of them from the 1960s so the theorem says consider a compact spacetime with boundary

25:00 which consists of a disjoint union of two and sigma1 and sigma2, then if the space-time is time-orientable and contains no closed time-like curves, then sigma1 and sigma2 have the same topology. So time-orientable just means that you can define a nowhere-manishing continuous time-like vector. So you have a continuous division of the lobes, so the light comes into the forward lobe and pass the lobe. Okay, so if you want branching, you better have topology change, and if you want topology change, then you're denying the consequence. We have to deny some of the conditions in the antecedent, and let's see what the possibilities are. Well, you could give up on either time-orientability, or you could allow closed-time light curves in time machines, but the people who talk about branching space-times, I don't think want to give up either. They assume that there is a consistent time-directionality and a consistent time order. you can go to spatially open universes so Gerosch's theorem only applies to the case where you have compact spatial topology, but what if it turned out that we live in a K equal plus one, Friedman, Roberts and Walker, that's consistent with the most recent cosmological data. And in addition Gerosch's theorem can be generalized to apply to changes in the open universe. So I'm not sure that this is a good way out. You can give up the assumption that M is a manifold. For instance, that it's locally Euclidean, but it's hard to see how you can do relativity theory. You did that. You could allow the metric to be undefined at some points of the manifold, but then what you're doing is introducing singularities, either by hand. to erase the metric at various points, but that seems very artificial.

27:30 Or the singularities might be genuine singularities in the metric in the sense that, for example, some curvature scalar blows up as you approach the point, but then it seems rather miraculous that these genuine singularities would appear at just the right places for the branching. Now, I think the most plausible thing to give up is the Halstorff assumption, and you can certainly have manifolds that are non-Halstorff, but you're going to cause a host of problems. For example, you're going to have bifurcating geodesics. In particular, you're going to have bifurcating timeline geodesics. So we'll have geodesics, timeline geodesics for the same initial point, and the geodesics will agree up to some branching time, some branching time, and then have different endpoints. Now, it's one of the assumptions of classical general relativity that a test particle that's pre-falling in the gravitational field will follow a time-lapse gdc. Well, if we have branching theories, what's the thing supposed to do? Is it supposed to go up on this branch or that branch, or does it go both ways? Also, if you give up the Halstorff assumption, it's hard to have global conservation laws. I mean, normally, if you assume a Halstorff manifold, you can, under appropriate conditions, integrate up global conservation laws to get global conservation laws. But if you allow non-Halstorff manifolds, then this is going to be a problem. So, my question would be, is there really a compelling reason to move from the metaphorical sense of branching space to the literal sense? And if there is, what price are you willing to pay for the literal sense of branching?

30:00 Okay, I think I'm up to number four, asymmetries of time and asymmetries in time. So, time itself seems to be asymmetric, it seems to have a third-directionality, and And there's certainly lots of interesting asymmetries of physical processes that take place in time. And a widely shared dogma here is that these temporal asymmetries are somehow grounded in the entropic behavior of physical systems. If you subscribe to this dogma, and you can take entropy in Boltzmann's sense, then you have to face up to a pair of problems. So just granted the outset that we've succeeded in showing that if at some initial time the Boltzmann entropy is low, then it's very, very probable that at some appropriate later time, the Boltzmann entry will have increased. This is the second discussion. First problem, what justifies the posit of the initially low entry? And even more pressing is the asymmetry problem. If this is true, and if the micro-dynamics of the governing system are time-reversal invariant, then it's also very probable that the corresponding earlier time on the entropy was higher. And this kind of symmetry here simply contradicts our normal expectations. So, take a country, for example, a glass of water, I wish there was an ice cube, I wish I had it right now. So, what we would normally expect is that if this system remains thermally isolated for, say, the next five minutes,

32:30 and we look at it five minutes later, the ice cube will be partially melted, and the water in the glass will be correspondingly colder. And we also expect that if the system had remained thoroughly isolated for the preceding five minutes, and if we looked at it five minutes earlier, we would have found that the ice cube would have been even less melted and the water in the glass even warmer. But that's not what this says. our expectations are asymmetry we seem to get a symmetry of behavior in the entropy so what's the solution? well now comes a second dogma namely that cosmology comes to the rescue and resolves both of Boltzmann's problems so the story is supposed to be that the universe began with the Big Bang, okay, that's standard cosmology, and shortly after the Big Bang, the entropy of the universe was very, very, very, very low. Supposedly, somehow, this initial low entropy of the universe accounts for the sort of asymmetries of the observant entropic processes now. So here's one expression of this fogma from an article like Christ. this discovery about the cosmological origins of low entropy is the most important achievement in the half-century science. It's a bit over the top, but you get the idea. Now, I have two objections here. First, I'm not sure this discovery is a genuine discovery, because it seems to me that claims about the low entropy of the early universe are very not even false. I have in mind here Pauli's remark, he was talking about somebody's theory, I forget whose it was, and he dismissed it by saying, it's not even false. I don't intend this kind of haughty dismissiveness here, but I think it's quite likely that these claims are literally not even false. And second, I think even if they're true,

35:00 they don't really explain the thermodynamic behavior that are of interest to us now. OK, so what would you have to do to make good on the dogma? Well, here's the way the Boltzmann apparatus works. So you have a dynamical system consisting of a state space, and you have some sort of deterministic flow on this state space, and you have a measure that's invariant of the flow. When we talk about Boltzmann entropy, you've got to do some kind of course-working of the state space. So a macrostate corresponds to some region of the state space. And then the Boltzmann entropy of a macrostate is proportional to the log of mu. Okay, and so the statement that the Boltzmann entropy increases in time means that the microdynamics is such that it takes the system through a sequence of microstates that actualized macro states that correspond to larger and larger volumes in the state states. Okay, so what we want to have to do to make good on the sort of dogma that we found in the true price quotation is to show how to apply this apparatus to cosmology. Now, what do we get? Do we get some kind of calculation? Well, what we get is hand-waving and diagrams. I took this from Roger Penrose's book. I think it's The Emperor's New Mind, is it? So, Penrose says that in order to reduce a universe resembling the one in which we live, the creator would have to aim for an absurdly tiny volume of a phase-based impossible universe to the hundred and twenty-third of the entire volume of the situation under consideration. Now, where does Penderos get this number from?

37:30 Well, not from applying it out as I sketched, but from a kind of heuristic calculation. Now, granted, the heuristic calculations of first-rate theoretical physicists like Roger Penrose count a lot more than even the most precise calculations that I can do still one would like to see just a model calculation that would give us some compliments that we're on the right track but when we try to do these model calculations we don't get what we want so here's a here's a concrete example so This is a model that was studied by Hawking and Page, so it's a Friedman-Robertson model coupled to a scalar field that could drive inflation, which we apparently see in the early universe. Now here the state space is particularly simple. It's a four-dimensional space. It's coordinated by the field, its canonical momentum, our old friend the scale factor, and the momentum canonical to the scale factor. And the dynamics is Hamiltonian dynamics on this phase space, although it's not normal Hamiltonian dynamics because we've got a constraint which says that the Hamiltonian vanishes. So we can go down to the constraint surface, where the, this is the region of phase space, where the constraint is satisfied. And so now we're down to three dimensions. And because of the presence of this constraint, he has some gauge freedom. And we can get rid of that by just factoring out the gauge orbit. So now we're down to a two-dimensional reduced phase space. And what's the relevant measure here? Well, there's a natural measure that's invariant under the dynamical flow. I mean, it is technically the pullback to the reduced base base of the natural symplectic form. And now comes the problem. This natural measure doesn't normalize. And you can't blame the lack of normalization on the universe being infinite, because this is true even if we live in a spatially finite capable of the plus one universe.

40:00 Now, you could try to salvage something here. You could still try to define some kind of partial probability measure on a macrospace in the following way. You see the probability of a macrospace is zero if the corresponding phase volume has a finite measure. And the probability is one if the corresponding phase volume has infinite measure and has complement on a finite measure, and otherwise it's undermined. So at least you have some kind of partial probability measure on macro states. Now, I mean, my own guess is that when we consider the macro state for our early universe, it's kind of fine to fall into this third category. Even in the best case for the dog, that is, case one, we have a problem that if the probability of the initial macrosate is zero, then the conditional probabilities, at least in the usual sense, are going to be undefined. what if it turned out Miraboli did to the Boltzmann entropy of the macro state of the early universe is well defined and has a very, very low value I would still question the relevance first of all if the entropy of the very early universe has a well defined low value the gravitational contribution. OK, so think of a standard hot Big Bang model. We have a thermalized state that's homogeneous, more or less homogeneous, more or less isotropic. Intuitively, we would think that that sort of state has a high infinity. Well, the answer is no, because gravitation likes to clump matter.

42:30 So a homogeneous, smooth state would correspond gravitational entropy. Okay, but gravitational entropy, I think it's just largely irrelevant to the tournament behavior systems of interest to us now, ice cubes and glasses of water, because gravitational entropy for both reasons of time scale and length scale, this seems irrelevant. And in addition, the kinds of systems we're concerned with are relatively small systems that although may be thoroughly isolated or something not dynamically isolated. Things are being shaken and jiggled by streetcars and buses passing by. And so the entropy of the early universe, I think, places very little constraint on the initial state for the kinds of thermodynamic systems we were concerned about. So my conclusion is we have a, it's almost back to squared one, we're still struggling with the very same problems that Boltzmann wanted. Okay, black holes and the Hawking information paradox. Okay, so classical general relativity theory predicts the formation of black holes for example due to the gravitational collapse of stars, and there's good evidence that there are lots of such black holes in the universe, and also good evidence that there are supermassive blackheads at the cores of most galaxies. So here's what's called an informal diagram of a black hole that's formed by a spherically symmetric gravitational collapse. A conformal diagram distorts metrical distances, but it allows you to read off at just a glance the causal structure. So the convention is null directions are drawn at 45 degrees.

45:00 Spatial infinity is brought in to a finite distance, so you can see the whole picture at a glance. Scribe-plus is a future null infinity you can think of as a terminus of outgoing white rays. And so here's the center of symmetry, here's the matter that collapses to form a singularity, and here's the horizon of the black hole. And here's some unfortunate observer who falls through the black hole horizon. such an observer is turned. He can wiggle, however he will turn on his rocket motors, but in a finite amount of time his life is on hand with the singularity. Okay, so this is all just classical general relativity. What happens if we put a quantum field on such a space-time background? And one of Hawking's great discoveries is that on such a space-time background, a quantum field thermalizes. So, in fact, a black hole is not black, but it seems to radiate with a thermal spectrum. And because of this radiation, a black hole will lose mass and will eventually evaporate. So how do we picture the upshot of black hole evaporation? Well, here's an attempt to do it on the assumption that we can still use classical space-time and general relativity to represent such a thing, which is a big assumption. Okay, so we still have the singularity. Our unfortunate observer is not helped by the Machola evaporation. He still is not. But we get this extra piece of the diagram, and I put in a post-evacuation time slides. Okay, so what's the information loss paradox? Well, first, the sort of intuitive, imprecise, really, of stating the paradox. Think of the byron years before we had the internet, And if you wanted to look up some piece of information,

47:30 you have to go to the Encyclopedia Britannica. I mean, I know it's hard to imagine that. Now, throw the Encyclopedia Britannica into the black hole. Well, all that information just disappears, and it's unavailable to the observers of the state outside of the black hole. That's, you know, the very hand-waving formulation. Here's the more precise formulation. If this is the correct way of depicting black hole evaporation, then we're going to have a pure to mixed state transition in the state of the long field. And here's the argument for that. Consider some observables associated with the region interior to the black hole and observables associated with a region in a neighborhood of this post-evaporation time slice. Those observables are going to be correlated by the quantum state. Because we have a common cause here. that the radiation is going to hit both of those regions and the common cause will establish correlations. Second, since this region is locally space-like compared to this region, the associated observables are going to commute because of the basic assumptions of the quantum field theory that the observables would support on regions that are on the space-like commute. Okay, and so now we have all of the ingredients we need for the formal statement of the paradox. So it's plausible that the quantum state on the global algebra of observables gives correlations between the algebra of observables the black hole, and the algebra associated with a post-evaporation time slice. Second, these algebras commuted because the regions are relatively space-like. And then it's just a simple lemma to show that if you restrict the state to the algebra associated with the post-evaporation time slice, that state will be a mixed state.

50:00 So if the initial state, that is a state associated with some pre-evaporation time slice, is pure, then we get a pure-to-mixed transition, and such a transition necessarily implies a loss of unitarity, and by the way, it also implies a violation of time-reversible variance. But this loss of unitarity is what drives some kind of field theorists to absolute distraction. You know, such distraction that they're literally willing to believe six contradictory things before lunch in order to resolve the paradox. And I refer to so-called black hole complementarity, which the government tried to formulate. But my take on this issue is that we don't have a genuine paradox because unitarity is not essential to quantum field theory. In the algebraic formulation of quantum field theory, we can discuss dynamics in terms of automorphisms and how to do this, and those automorphisms may or may not be unitarily implementable. If they're not, well, too bad. We can still do respect a little amount of quantum field theory. On the other hand, if you think that there is a genuine paradox here, I think you have to go to the full theory of quantum gravity to try to resolve it. And again, the people that work in loop-like gravity have recently claimed that they do have a changement resolution. This is all a summit speciality. It's all in pre-traumatic form. But here's the claim. The claim is that the diagram I gave with black hole evaporation earlier is not correct. Here's the correct diagram. They claim that the black hole singularity is resolved. and they claim that the quantum evolution continues through the classical singularities, so a pure state remains a pure state and all the problems are resolved.

52:30 Our unfortunate, uh, uh, observer, uh, well, his world might still deserve, uh, dissolves into some kind of quantum, uh, bone in this, uh, deep, uh, pond regime, but he can die happy, uh, knowing that the, uh, that there's no, uh, no paradox. Okay, I think what I'm going to do is just mention the last problem, but not try to go into the table because I don't want to run over time. This is the problem of what is general covariance. about 10 years ago my colleague John Norton wrote a very nice review article entitled something like general covariance colon 80 years of controversy it's now going on 90 years and there's still controversy covariance means, I want to distinguish between what I'll call formal general covariance and substantive general covariance. And of course, I've chosen the labels with MELS and forth of that. So formal general covariance is a restriction on the form in which we write a theory, but it doesn't place any substantive restriction on the content of the theory. So the formal requirement is just to rewrite the laws of the theory in a form that's co-variant under arbitrary coordinate transformations. Now, we can satisfy this requirement even for Newtonian or special relativistic theories if we choose to write these theories in intensive notation. Substantive general co-variance is a different beast, then it does place substantive restrictions on the content of the theory. And here's the way I formulate it as a conjunction of two requirements. First of all, the theory should exhibit a diffeomorphism invariant. So let me assume that the models of the theory in this form have a differentiable manifold and then various geometric object fields

55:00 living on that manifold. So, I mean, for concreteness, just think of the object fields, I think scalar fields are back in terms of tension. And then diphenomorphism invariance means that if this model satisfies the loss of the theory, then any model of this form will also satisfy the loss of the theory, where D is an arbitrary diphenomorphism, and D star is the drag along of the object that was by that diphenomorphism. And second, and this is a really meaty assumption, this video-morphism invariance is a gauge symmetry of the theory. Now, if I add, I simply mean that all of these models are regarded as being descriptions of the very same sort of situation. So in other words, we have a lot of descriptive redundancies in the theory, so we have a many-to-one correspondence between the descriptions of the theory and the actual physical situation. So the gage freedom comes in the choice of the description. Okay, well, this is a rather empty requirement unless I've got a criterion for recognizing gage freedom. And I claim that we do have a satisfactory account of this for a very broad range of theories. Dirac worked out such an account that applies to any theory whose equations of motion are derivable from an action principle. And this class of theories includes most of the theories discussed in modern physics. It doesn't include everything, but it includes 95% of the cases. And my claim would be that if you apply this Dirac account of gauge symmetry in conjunction with my definition of substantive general covariance, then you get the conclusion that the typical Newtonian or special relativistic theory does not satisfy substantive general covariance. It may satisfy the formal requirement, but it doesn't satisfy substantive general

57:30 covariance, whereas Einstein's general theory of relativity does satisfy the substantive requirement of general covariance, because Diffie-Morphism invariance indicates symmetry in general relativity theory. So there is something special and unique about the Einstein's general theory of relativity, it's the first theory in physics to satisfy the substance and requirement of general covariance. Paradoxically, this turns out to be the source of a tremendous number of headaches in the foundations of relativity, and especially when you try to I've tried a long time to read the theory, but Chris is writing a dissertation on that and can tell you all about it. Now, if I saved enough time, I would have gone on to try to illustrate how this actually works and then point out that there is a kind of lingering problem, despite my proselytizing for this account. It's still not quite clear what kind of a variance is, but I think I'd encourage you on enough some business to put in the sky there. Thank you. Thank you very much. Are there any questions, remarks? Yes, you mentioned the evacuation diagram, and a year ago, Hawking, in Dublin, redrew his statement. What is the status of that? Yeah, I asked a few people about this. There was this famous lecture in which Hawking, as it were, took back his original claims and on the basis of this, he even settled a bet with someone who had it at Caltech, his name, like Pergotten. But to my knowledge, Hawking's calculations have never been published. They aren't even available in preprint form.

1:00:00 And the two physicists I trust very much, Bob Wold and Bill Unruh, were at the lecture and they couldn't understand exactly what the calculations were, how they succeeded in resolving the paradox. So I think we're just going to have to wait until Hawking decides that it's finally time to publish this material and then we can have a chance to digest it and try to understand it. So it's, yeah, they made a big splash in the newspapers. But what is the meaning of his claim? Oh, I mean, as I read the article, the claim was that the spectrum of the radiation wouldn't be thermal, so that there would be still some information available to the observers who were out of space. More questions? The title of your talk was And one of the poems of this involved in the question of whether it's space and spine energy and a spine and spine energy. My thoughts turned then to the trying to get a lot of which tries to show that these government actions into the world. So it seems that in the last 150 years or something like that. Speculative cosmology has gone more. I actually want to answer this question. Yes, I think that's right. In the last few decades, it keeps me a remarkable transformation in cosmology. It was a domain that used to be filled with fairly unbridled speculations. But now the observational constraints are becoming good enough that it's beginning to look like something approaching an empirical science. There's still plenty of room for speculation, and I try to point out that we still have a large range of possibilities. And the possibilities I discussed were just the possibilities that exist within classical general relativity theory.

1:02:30 So, you know, you can go beyond that and then you look for other explanations of the accelerated expansion, for example, extra dimensions, whatever, when you just open up a Pandora's box. So there's still plenty of room for speculation, but we're getting tighter and tighter empirical controls. I think it's not beyond the realm of hope that we might actually be able to, maybe not in my lifetime, but in your lifetime, come up with, you know, an empirical resolution, another question, is space finally or infinite, is time for life or infinite? Of course, in 200 years, there's probably another scientific revolution that will overthrow all of this. I mean, you know, settle the module of at least the present period of science. So yes, it was a very anti-constant talk. Concerning the problem of branching spacetime, maybe somebody who would want to bend a little branching in spacetime would say that, well, I'd take it that spacetime is not the host of one boat, that's fine for me. And there's a splitting or a spifurcating of geodesis is exactly how I imagine indeterminism's work. So there's a particle traveling on this geodesis, and some probability goes one way, and some probability goes the other way. What do you think are such a reply by virtual computer launching space-times? Yeah, I can see how much space-times might a lot of them to model some kinds of indeterminism, but I suppose the kind of indeterminism they're trying to model is, you know, that I flip a coin, and the coin comes up either heads or tails, like one branching. So fine, but what about other things for which there is

1:05:00 I mean, I hope to meet up with my wife later this evening, but, you know, if there's a branching, you know, to account for the indeterministic outcome of the claim, and she happens to be around, you know, too close to the indeterministic experiment, she may go on, and other branching won't lead up. So, I mean, I'm sure even, you know, jury read things so that everything will work out fine, but I think you have to put in a lot of stuff by hand that isn't controlled by the, you know, the physics as we know it. Gee, I think we are out of time. It doesn't mean that you have to stop discussing. So let's thank the speaker again. OK, let's put it down. I would like to repeat. So, one of our participants, this was the guy yesterday, because at the airport he was a drone. And now we decided to write a letter to him, in which we completed a quick recovery. Some of them already signed it, and if you agree, please sign your name for the Civil Center. Thank you. I'd like to make a short announcement that there was a question by Jesse Alamo about the end of this talk about the future of the universe. The tomorrow's invited lecture at C30 by Jula Garvey will continue where John has stop now. So please come if you are interested. I don't know anything. I will never the last question of all of you. I'm sure you are. I'm sure you are.

1:07:30 Let me keep your hands. Ten. Nine. Yeah. ...of a black hole that forms a sphere in the sea of the glass, and it has a totally flat, uh, uh, uh, space, and, uh, and it's very beautiful. Yeah, yeah, yeah. and then we put our stock on that yet we're going to go for it. We also want to thank you for your food. Dr. Schmeck, we should share your favorite questions. I just want to concentrate on as simple as possible for the case to remove as many distractions as possible when discussing the information loss for business. I have no idea what a person to be in a rather realistic case of a... Yeah, I think it's funny, I think it's funny. Yeah. Oh. Yes, it was. Right. I mean, I don't know. Yeah, I just don't know what I mean, I should have thought about one of those things when you hear it, which is trying to lick at all, you know, to remove as many distractions as

1:10:00 as possible. I mean, to discuss the knowledge, the information about the paradox, I think it's really interesting to me. There's a whole lot of clarification between the people who come from the background of general relativity theory, personal status theory, that comes from the particle physics side, and the people who come from the general relativity side, you know, Paradox, we can't tolerate this, we have to have black hole complementarity and all these other... This is the result of the process. Yeah. It was Kip Thorne, wasn't it? No, no, no. It's not Kip Thorne. He's the man who's famous for making bets, of course. One of the originators of free theory. Oh, yes. I remember reading that article. Oh, okay. Could have... Not Mike Green, not Chance, none of these people. No, sorry. At the end of this conference, there will be a movie, and that guy showed up in the movie. just two things about the this dublin lecture of hawking's hawking gave a talk at chris eysham's 60th birthday they had a conference for six chris eysham's 60th birthday in october last year at imperial uh where he again well gave a similar shortened version of the dublin lecture and there was quite a lengthy discussion afterwards in which he was pressed, you know, very hard on this. And the gist of his answer was that, well, you know, I got my very best PhD student writing it all up. Well, said very best PhD student gave an account of his results at the Newton Institute meeting in December on quantum gravity and quantum information theory, quite a bit of which was about the black hole paradox. I didn't understand a word of it, I've got the whole thing on tape, so you might want to listen to it sometime. So I just, you know, useful people, archivists. And do you remember what his name was? Yes, I do, but not, I do, I've got it written down. I can't remember my own name at the moment

1:12:30 because I've been out of sleep for two days. I can do better than that. I can look it up on, I mean, I can let you know tomorrow as soon as I have a chance to go and look it up on the computer and just go on to the archives. um he gave it was it was a very it was it was technically the most impressive talk of the meeting but i mean it was it was pretty well no yes oh yes no this was this was considerably quite a long time after this talk in dublin this was in december last year at the newton institute no surely the dublin talk was earlier earlier last year wasn't it early in 2004 this was yes this was just before Christmas last year the Newton meeting on quantum gravity and quantum information was 14th to 19th of December it was just before Christmas and the talk at Eichem 60th was I think it was about the end of September middle or end of September the Dublin lecture came under discussion of both of those and it was certainly sometime after who was at the original... And who was also at the Newton meeting. ...was quoted in the newspaper saying, I couldn't understand Hawking's claims, but there may be something to it, since Hawking isn't a dumb guy. And so I wrote to Bill and said, I know I can never get your highest accolade that he's not a dumb guy, say he's not a completely dumb guy. This is rather like the famous stories about Pauli, isn't it? I'll be honest, you're not a complete fool. First time you met Heisenberg. Anyway, sorry. More anecdotes. Right now, I'm kind of about my lecture yeah sure I'm later absolutely absolutely what is your lecture it's about logical

1:15:00 Do you tell me exactly where physically in Budapest is Budapest? Right, but not that. Particularly in the field of science. Yeah, and we are here at a meeting in Budapest. Yeah, it is. Oh, the Buddhists, of course. Right. Do I show, if I have a map of Budapest here, anyway, because what I have to do sometimes while I'm here, and probably only happens that someone has to do it, is to arrange accommodation for myself as a tandem here. So can you show them very roughly where it is? We are here, nowhere, this is not the vast, there's the largest island, this is, we're down here, we're down here. This is just a disaster. This is the galaxy. Oh, it's not so much far. Oh, it's not so much far. Oh, it's not so much far. So it's not here? And the History and Philosophy and Science department? Yes, it's here. Well, good, so if I could stay in the same conversation where I am now, in fact, that should be perfect. Thank you very much. Good, thanks. We'll talk, don't worry. Thank you.

1:17:30 But then, when you look at that, the reason how the problem is, is that it's a huge way to do a little bit. And that's why you want to make a big change. And then the construction business model, and it's probably one of the people that we have, of course, the larger part of the, that we have, uh, micro-stakes, initials versus It's just that you're going to do it. Thank you. Well, yeah, I mean, something about, I mean, I'm kind of a fan of the, the, the, the gravage. I'm probably not crazy enough to be curious, you know, in the kind of the theory of the I don't know. We didn't take it.

1:20:00 Thank you. Thank you. Thank you. Thank you. What else have you done?

1:22:30 I'll give you a call. Hey, hey, that piece. You want the right comment down? Yes, we're all done now. It's now recording, it's on double time, it's on, and it's an automatic reverse, so you don't have to change the side over, when it gets to end of that side.