Cosmological Aspects of Time — Part 2

0:00 The answers to Boltzmann's problems lie in the heavens. How much time do I have left? All right. Okay, so the last topic I want to talk about is black holes. So, classical general relativity theory leads us to believe that the gravitational collapse of stars of up to ten solar masses or so will produce black holes, and there's good evidence that such objects do exist throughout the universe. Also, there's good evidence of supermassive black holes at the core of most galaxies. And there's some theoretical speculation about primordial black holes that would have a tiny mass. And here's the relevance for our discussion. It may be that time itself is infinite in the future. The universe may just go on expanding forever and forever. forever. But for those unfortunate observers who fall into a black hole, their future is finite. And here's what's called a conformal diagram of a black hole that's formed through spherical gravitational collapse. Now, a conformal diagram distorts metrical distances, but it causal relations and it presents them in a way that you can just read these causal relations off at a glance so here spatial infinity has been brought in to a finite distance scribe plus is called future null infinity that's you can think of that as the terminus of outgoing light rays so we're here's the center of symmetry, here's collapsing matter that eventuates in a singularity, and the dashed green line is the black hole horizon. So an observer who's unfortunate enough to pass through the horizon of the black hole is trapped. He can twist and turn however he will. He has only a finite

2:30 amount of proper time before it crashes into the singularity. Now, a second way in which this is relevant, there's the absence of evidence for white holes, which you can think of as just the time reverse of black holes, and now time is going up. if this absence of evidence for white holes turns out to be good evidence for their absence then we'd have quite an interesting time asymmetry lots of black holes but no white holes in the universe and one can wonder whether this asymmetry is de facto or whether it's based on some as yet unknown laws. So far, all classical general relativity, but now let me bring in quantum effects. First, there was Hawking's discovery that if we put a quantum field on a classical general relativistic space-time background with a black hole, the quantum field will thermalize. So it appears then as if the black hole is not black but that it radiates with a thermal spectrum. Then second, as a result of the radiation, the black hole loses mass and its horizon area shrinks, and so one wonders what the upshot of this process is. Well, if the black hole evaporates entirely, and if we can describe the result of evaporation using a space-time of classical general relativity, the conformal diagram would look something like this. So it's the original diagram I gave you with this extra piece added. Sigma is a time slice through the post-evaporation portion.

5:00 Now, the black hole evaporation doesn't help the poor guy who's fallen into the black hole. He's still snuffed out of existence. And something even more remarkable and disturbing takes place. the quantum field will undergo a pure to mixed state transition which necessarily implies a loss of unitarity and this is usually referred to as Hawking's information loss paradox. So the imprecise way of formulating the idea is something like this. Throw your computer or the Encyclopedia Britannica or however you encode your information into the black hole, well, that information is lost. It's not going to be available to the observers who exist in this post-evaporation regime. But the precise way of formulating the idea of information loss is, I say, through this pure-to-mixed state transition. And let me try to make this intuitive, at least to the people who know a little bit of quantum mechanics. So in ordinary quantum mechanics, you know that if you have a state that gives you correlations between observables associated, say, with this region of space and that region of space, then if you trace out the degrees of freedom associated with these observables and restrict these observables you'll get a mixed state so it's not a pure state but a mixed state and something like that is going on here consider the observables associated with a region interior to the black hole and the observables associated with a post-evaporation time. First of all, those observables are going to be correlated. Why? Well, there's a common cause. The Hawking radiation will affect both sets of observables

7:30 and establish correlations between them. Second, you can read off the diagram that this region is relatively space-like And so the basic axioms of quantum field theory says that, therefore, those observables commute. And that's all we need to prove the result that if we restrict then the state to the post-evaporation time slice, it will be mixed. So if we started off with a pure state, we'll get a mixed state in the post-evaporation phase. And this necessarily implies a loss of unitarity and, by the way, a violation of time reversal invariance. Now, it's this loss of unitarity that drives some quantum field dearest nuts, so nuts that they're willing to believe six contradictory things before lunch in order to try to resolve the paradox. So they're willing to believe things like black hole complementarity. I would argue that there isn't any real paradox here at all. And that unitarity is not essential to quantum field theory, so there's no big paradox to be resolved. Okay, let me summarize by giving you my guesses about the questions I raised. Is time finite or infinite in the future? My guess would be that the accelerating expansion of the universe we see is caused either by a genuine cosmological constant or quintessence and hence we'll get eternity with a big chill. This of course is assuming classical general relativity theory. If we open Pandora's box of other possibilities and contemplate extra dimensions for space

10:00 I mean, I wouldn't even have any guesses as to what was right. As for the beginning for time, my guess would be that time does have a beginning. The universe started in a big bang about 14 billion years ago. This, again, is assuming classical general relativity. If we open up Pandora's box of possibilities, I think the most plausible possibility I've seen for the quantum gravity treatment of this issue comes from loop quantum gravity, and it does give a sense in which one can extend through the initial singularity, although whether this actually allows us to talk in a meaningful way about what happened before the Big Bang singularity, I think is open to question. And in between, my negative conclusion would be that cosmology does not solve Boltzmann's problems and the origin of the thermodynamic arrow is still very much a mystery. As for black holes, I do believe they populate the universe, and my guess would be that there is a black hole, white hole asymmetry, but whether or not it's a de facto asymmetry or the indication of some deeper law-like asymmetry, I don't know. As for black hole evaporation, my guess would be that insofar as we can trust the calculations from semi-classical quantum gravity, we do get a pure to mixed state transition in black hole evaporation, but this is no cause for hysteria. There's no genuine paradox about the information loss paradox. box. And if we open up Pandora's box, well, who knows? I don't really have any guesses. Okay, thank you.

12:30 I want to thank John for this very interesting talk and now time for questions Thanks for this very interesting talk, two comments and one question, first comment regarding the accelerating universe if you grant that it's true that the expansion is accelerating there could be three possibilities one is a kind of dark energy as you mentioned it and this is probably the most probable issue but there could be something wrong with gravity that is for example the extra dimension so the Friedman equations would not hold or there is a be something wrong with our assumption that is we do not live in a homogeneous universe so if we would live in an underdance bubble the environment could also cause these accelerating effects. Right. Before you can I just address this third possibility that you mentioned. Recently it's been claimed by what's his name, Kolb and his co-workers Yeah, there's a way to have expansion without a dark energy. And, yeah, the idea, as you pointed out, is some kind of back reaction effect. In fact, we don't live in an exactly homogeneous and isotropic universe. The Friedman-Walker-Robertson model is an idealization. You have to do some kind of smoothing. Well, first, smooth out the matter-energy distribution You know, then smooth out the metric and then put these two smoothed out objects in Einstein's field equations. Well, the field equations may not be satisfied by these objects you've got by doing these two kinds of smoothing, essentially because they're nonlinear equations. And so subtract the difference, and you can regard that difference as an extra piece of the stress-energy tensor. And you find that that stress-energy tensor may not satisfy the strong energy condition and hence may give rise to accelerated expansion.

15:00 I'm highly skeptical of this because there are no-go results that apply to non-homogeneous but non-rotating universes. You can generalize the deacceleration parameter to these non-homogeneous cosmologies. And using Einstein's field equations and, you know, the Raycharduri effect, you can prove a no-go result that says you cannot have accelerated expansion, you know, without either having something like dark energy or enough rotation, which, you know, and we think that there isn't rotation. So I'm highly skeptical of this way out, but it's still something that's very much under discussion. And the preprints are flying back and forth on this now. So I agree, this is a very exciting development. I'm skeptical too, but this no-goat theorem is already refuted, as I saw recently by Kolb and Kohlver. So it's an open issue. is regarding the beginning. Great that you mentioned Luke-Quentin Gravity. It is not published yet, but Abbe Ashtekar told me some weeks ago that they now can not only evolve through the singularity, but they reach another classical space-time. So there would be a time before time in the ordinary sense. Yeah, okay. Right. It's important that when you extend through eventually get to something that can be described in terms of classical general relativity. But you're still gonna have that period in between where you're gonna have something that can't be described by classical general relativity theory. So my worry is, if we don't have, as it were, an unbroken time series that can be described classically, in what sense can we speak about the before? Right. So that leads me to my question. If it would be true as a matter of fact that there is some bouncing solution, that is that we can walk through the singularity and came back to a classical space time which comes in from infinity as a collapsing one, what would be your philosophical judgment about this?

17:30 because in some sense it's not an explanation why is there anything. And so the other point is about the initial conditions. If it would be true that it's not just a statistical fluctuation, how could a philosophical satisfying explanation even be well perceived to solve this? Is there anything about this? I mean, my tendency would be not to try to do the philosophy first, but to, you know, wait and see what these guys produced and then try to provide an interpretation of it. So, I mean, I don't conceive of philosophy as, you know, a source of all answers on these things. In fact, I see it just the opposite, that the business of philosophers of science is basically that of trying to interpret and understand what the physicists have produced. So I wouldn't try to anticipate. I would just wait for several years. So I'm sorry I can't be a... Two small questions. One, when you said that for in between the problem of time remained open if accessed on the Boltzmann path, you didn't talk much about what they call black hole thermodynamics. So would you rather think that this is not more than an analogy, so not really a genuine type of thermodynamics? first question. And the second concerns is unitarity. Of course, you had no time to go more into detail. But was the problem that, so to say, if I throw in my phone book and all my information, then I somehow make a part of the space of those who survive the

20:00 black hole evaporation non-unitarity, despite being space-like, at space-like distance from them. Was that the problem about unitarity? Why some people worried that you somehow spoiled the unitarity in a region that's space-like separated? Okay, these were the two questions. Okay. Well, yeah, on the first question, no, I do think that black hole thermodynamics is, you know, genuine thermodynamics. I think they're talking about the thermodynamic entropy in the ordinary sense of a black hole. And I think Hawking's discovery of black hole radiation underwrites this because it shows that you can assign a temperature to black holes. And yeah, so I think they are doing genuine thermodynamics. Yeah, but I wasn't, on the second question, Sure, I understood what the question was. I mean, I think many quantum field theorists simply think that unitarity has got to be a feature of any good quantum theory. And once you lose that, then things just go so haywire, they're willing to try virtually anything to avoid it. I mean, I think you can do quantum field theory in the algebraic approach in a way that doesn't suppose unitarity. So you can talk about evolution in the sense of an automorphism of the algebra of observables. And then it's always a question of whether or not that automorphism is unitarily implementable. In some cases, it will be. In some cases, it won't. And the cases where it won't be unitarily implementable cover not just the exotic things about black hole evaporation, but even more normal things. I mean, if you propagate a Klein-Gordon field, say, in Minkowski spacetime, and you think about time slices, which are not the usual, you know, flat slices, but curvy, curvilinear space-like slices, then the dynamics in that case won't be unitarily implementable.

22:30 So, but, you know, no big deal. Okay, Hugh's going to rake me over the coals because he thinks cosmology does resolve Boltzmann's problems. Well, first of all, I wanted to thank you, John, for a very nice talk. But, yeah, I mean, my question does concern this thing that you're skeptical about. the relationship between the condition of the early universe, the homogeneity of the distribution of matter in the early universe and familiar thermodynamic phenomena. I mean, it's always seemed to me that the most plausible kind of way to argue for that link goes in two steps. I mean, first of all, you argue that the familiar stuff depends on the presence of the sun. Secondly, you argue, not in thermodynamic terms, but in basically sort of astrophysical terms, evolution of galaxies and stars terms, that the evolution of galaxies and hence of objects like the sun depends in a very critical way on the smooth distribution of matter. And as I understand it, the argument is that if the distribution had been just a little bit more irregular, then the tendency would have been for the bulk of the matter to fall into black holes and not form galaxies. And I guess I got that from people like Penrose and Davies and so on. Now, I mean, the nice thing about putting it that way is that certainly the second step and arguably the first step, too, you don't have to mention thermodynamics, so you don't get into any of those worries about defining measures and any of that stuff. You don't even have to use the term entropy, as I mentioned in my talk yesterday. I just wondered, as it were, which of those steps you find implausible if it's put that way? Well, look, I certainly don't want to deny what is a truism, namely, if we have some interesting asymmetry, whether it's a temporal one or some other kind, and we can't explain the asymmetry in terms of fundamental laws, then the asymmetry has to come from initial

25:00 conditions. And, you know, maybe the initial conditions can be traced back to the early universe. So, I mean, I don't doubt that because I don't doubt truisms, right? I mean, my skepticism just extends to the way of trying to implement this truism by using the Boltzmann concept of entropy as applied to the entire universe. So I don't doubt that some kind of story of the type you told is largely correct. and it may be that the disagreement is much less than it looks at first sight as it were because I think I'm quite happy with the idea that the right way to tell the story is really not in thermodynamic terms applied to the whole universe. Right. I'd just like to say I think what Hugh said there I would agree with very largely and I forget who it was but somebody was making the point very nicely that one doesn't really have to worry. Was it one doesn't really have to worry too much about calculations of entropy there just are manifestly asymmetric things going on all around us and one doesn't have to sort of get stuck in I can't remember whether it was you somebody said it so I think that seems to me to be a very good thing I'd like to say something about loop quantum gravity according to Lee Smolin I'm more or less the godfather of that thing but I have to say reservations about it in particular I don't think to be quite honest really they've got as much as they say because the really difficult problem in quantum gravity is the so-called Hamiltonian constraint and the only way and they haven't really solved that problem at all and the only way they can do their calculations is by taking an extremely simple toy model which I don't think is really getting to grips with the thing at all so I would be I don't want in any way to discourage people from working in loop quantum gravity because it's a serious thing with some very good people working in the field. But I think one has to be a little bit aware of some of the claims that are made there. Right. Okay. Okay. Can I just comment on that before you go on? I agree completely. But I mean, the reason I concentrated in loop quantum

27:30 gravity is that, you know, once you open up the Pandora's box of possibilities, it's hard to control them. So, I mean, I wanted to open up the box a little bit and loop quantum gravity is the to a quantum theory of gravity that sticks as closely as possible to classical general relativity and then tries to quantize it. So that's why I concentrated on that approach. My own guess would be that loop quantum gravity probably isn't crazy enough to be true. I mean, the true theory of quantum gravity will probably be something much wilder. But I think the other virtue of discussing it is that it's much closer to being a genuine scientific theory than so-called string theory is. I mean, agreed, there are various ambiguities in quantization. They don't quite know how to handle the Hamiltonian constraint and so on. But it's something that, you know, at least approaching a genuine scientific theory. So that was my reason for concentrating on it rather than other possibilities. Perhaps I could just make a quick response to that last point. They're all really bent on having a discrete underlying thing for space-time, which I think is also a huge assumption. I mean, when quantum mechanics came in, you didn't destroy the configuration space of Newtonian mechanics. You just quantized it. They are actually making a very radical change to the configuration space of general relativity by having loops wandering around in a bare manifold, which is also another reason why I have reservations about it. But could I perhaps just make a little plug for my own idea of thinking about cosmology? The thing that, and why I think that actually the illusion of time can arise, if you take the Wheeler-DeWitt equation seriously, and there's lots of theories that would lead to exactly the same result, it is just a fact that the wave crunching of the universe is going to be static, period. So how on earth do you get our impression of the illusion of the passage of time? Now, the thing that really strikes me about a configuration space, and particularly a relative configuration space, is that it is highly symmetric. There is a uniquely distinguished point. If you have the Einsteinian ontology,

30:00 it's where everything is sitting on top of each other. It's the point from which the Big Bang starts. If you have what I would much prefer, which is a scale invariant theory, that singularity does not belong to the shape space of the universe because a point is not a shape. The equivalent of that distinguished point is actually the most uniform state that you can possibly have. Now, I believe that if you are solving a static Schrodinger equation, a time-independent Schrodinger equation, on a highly asymmetric configuration space, which has this uniformly distinguished space, that is going to have a profound influence on how that wave function turns out. And that is my belief in the ultimate origin of the... that the only way to explain the arrow of time is to get rid of time altogether. in the arrow of time with curves going backwards and forwards. There's never going to be a classical resolution of the arrow of time. I believe that timeless quantum cosmology will give the answer. You may well be right. No, it was only a small question, relatively. You talked about, let's say, as a first solution of, If you go, if you, if you begin for, for example. No, I have to, to, to, to, to, to, to reformulate it. It's about the thermodynamic approach, yeah. And you, and the thermodynamic approach was firstly mere, through statistical thermodynamic approach. But then, it's the H-theorem, which in the end of his life, Boltzmann was put. And this H-theorem is nothing else, that the entropy, which formerly was a thermodynamic, but statistical through concept,

32:30 but suddenly uses as a prediction. Yeah. And that, of course, is just what you said, but you didn't mention that. And then, of course, it came what kind of prediction. And this prediction was basically an individual expectation. And that, of course, was a great disappointment of him. I'm from Vienna and I know of course what he has written so this would be a point but then of course we could say his whole so predictive power just this is a way it cannot be used and he goes back to a concept which momentarily is in the social science of C concept the concept of prediction as Yes, I just didn't have time and I also didn't want to poach on other speakers who talked about the age theorem and the various objections, the reversibility objection and the recurrence objection. But there was quite a good discussion in several of the talks that I went to. And I'm not sure that I really have anything new to add on this topic. Are there any further questions? If there aren't, I think we want to thank John again, and this is then the end of the session for this morning. Thank you. Thank you.