Cosmological Aspects of Time — Part 1

0:00 No? Not working? Yes, working. Okay, good. Thank you, Jan, and thanks to the organizers for making me part of this wonderful occasion. I was going to talk about cosmological aspects of time, but John Perry's talk inspired me so much that I decided to talk about McTaggart. You know that when McTiggart walked down a hall, he walked with a sort of funny sideways scuttle with his back to the wall. This was because he was physically deformed, and at his public school, the other boys would taunt him and kick him in the backside. So this was a sort of defensive maneuver. But being the good Englishman he was, he left quite a bit of money in his will in fond memory of his old school. I think this is what made England great. Well, I could tell you many more McTaggart stories, but maybe I should go to my topic of cosmological aspects of time. And I think the topic is now ripe for discussion because over the last few decades, cosmology has made remarkable strides in transforming itself from basically a rather speculative enterprise into something approaching an empirical science. Now, as we'll see, there's still plenty of room for speculation, but we can now discuss questions about cosmological aspects of time with growing specificity and with a growing hope that we'll be able to settle these questions in the foreseeable future. So, the first issue I want to discuss is whether time is finite or infinite in the future. And until quite recently, the answer was quite simple. It was geometry is destiny. And here's what that means. Orthodox cosmology describes the large-scale structure of the universe in terms of the Friedman-Walker-Robertson models. and these models come in three flavors,

2:30 k equals plus one, k equals zero, and k equals minus one. The k equal plus one models describe universes in which the space slices have constant positive curvature. k equals zero models describe universes in which the space slices are flat, and k equal minus one describes universes with constant negative curvature. of the behavior of the scale factor in these universes. And you can just read off the diagram that if we live in a Kegel plus one universe, the universe will expand to reach a maximum volume and then re-collapse to a big crunch. But if we live in a Kegel zero or Kegel minus one universe, then the universe will expand forever and eternity stretches out before us. So now it's just a matter of empirical determination to decide which of these universes we live in and thus whether or not time is finite in the future. At least that was the situation until the end of the 1990s when the accelerating expansion of the universe I can't quite get all of this on here. Every December, Science Magazine announces a breakthrough of the year, and in 1998, it was the accelerating universe, obviously a cartoon of Einstein. One wonders what Einstein would have made of this. the accelerating expansion as being caused by a cosmological constant or some surrogate for the cosmological constant. But when Einstein introduced the cosmological constant in 1917, it was not to get an accelerating universe, but to get a static universe. And Einstein later denounced the cosmological constant as his greatest blunder. So one wonders whether he would have been amused or this cover of Science. So what I wanna give you now is a very brief history of the ups and downs of the cosmological constant.

5:00 Einstein had two motivations in 1917 for introducing the cosmological constant. First of all, he found that the original form of his general theory of relativity was not consistent with what he deemed, what he called Mach's principle. Second, in 1917, he believed on the basis of very scanty evidence that on the large scale, the universe is uniform and isotropic and that it's static. But the original form of the field equations in general relativity did not allow for such static models. So he inserted the so-called cosmological term allowed him to construct the static Einstein universe. Now, over the next few years, Einstein fell out of love with Mach's principle. According to Julian, that was a bad mistake. So that knocked out one of the motivations for the cosmological constant. And then in the 1930s, Hubble's redshift observations indicated that the universe is not static, but is in fact expanding. And this knocked out the second motivation for introducing the cosmological constant. And after that, Einstein would have nothing further to do with the cosmological constant. the cosmological constant, or lambda, as it's to simply refuse to die, and here's one of the reasons. Again, a diagram of the behavior of the scale factor in a Friedman-Walker-Robertson model, and you can just see from the diagram that the age of the universe, that is the amount of time that's elapsed from the time of the Big Bang, is less than the Hubble time, which is this time. The Hubble time is just the inverse of the present value, that's the knot here, of the Hubble parameter which is defined this way. So the age of the universe then is strictly less than the Hubble time. But the values Hubble reported in the 1930s

7:30 for the Hubble parameter translated into an age for the universe of less than two billion years, which is less than the age of the stars as estimated from theories of stellar formation. And it's even less than the age of the Earth as estimated by radioactive decay. So this created something of a minor crisis in cosmology in the 1930s. And now enter stage right, one of the great champions of lambda, George Lemaitre, who's sandwiched here between Einstein and Millikan. Lemaitre found that with the use of a positive cosmological constant he could produce a model that has a scale factor with this kind of behavior. The universe expands and then the expansion slows down to a bare crawl. This is because the value of the cosmological constant which you can think of as giving a repulsive force that repulsive almost exactly balanced by the attracting force of gravity, and then lambda wins out, and the universe begins expanding faster. Now you can adjust the model so that this coasting or loitering phase is made as long as you like, and so then you can just see from the diagram that we can now make the age of the universe larger than the Hubble time, and in this way we can resolve the age problem. Lemaitre also thought that this kind of model would solve the structure formation problem, but I won't have time to talk about that. Here's the other great champion of lambda during this period, Arthur Stanley Eddington, who wrote, I would as soon think of reverting to Newtonian theory as dropping the cosmological constant. His reasons for this were rather different from Lemaitre's. First of all, Eddington said, if lambda is zero, then the observed velocities of galactic recession

10:00 have to be postulated ab initio. This might be true, but it can scarcely be called an explanation of the large velocities. And so I think here we have an early expression of the intuition that was later to drive inflationary cosmology, namely that a good scientific explanation shouldn't have to rely on special initial conditions. Eddington also thought that relativity theory implied quantities are relative and he thought length is relative particular particular standard, namely curvature, but Einstein's field equations tell us that in empty space, the Ricci curvature scalar is four times lambda, so if lambda is equal to zero, then R has to be equal to zero, and so to drop the cosmological constant, Eddington wrote, would knock the bottom out of space. So here we have the slightly nutty character of English genius, with my apologies to my English friends. Okay, as far as I can tell, Eddington's ideas were not very influential, and as time wore on, Hubble began reporting smaller and smaller values for the Hubble parameter, so this shifted the estimate of the age and so the age problem became less pressing. And so lambda was just put back on the shelf for a number of years. And then in the early 1970s, it was taken off the shelf again in an attempt to explain the observed red shifts for quasi-stellar objects, but it was later realized that that explanation didn't work very well, so lambda was put back on the shelf again. But still, Lambda just refused to die, and the motivation came from what's called inflationary cosmology, which postulates that in the very early universe there was a period of enormously rapid expansion which was caused by an effective cosmological constant that was produced by something called the inflaton field, which I'm not going to talk about in any detail.

12:30 The initial motivation for inflationary cosmology philosophical. The proponents of inflationary cosmology thought that the standard models of the universe did not offer a very satisfactory explanation of the observed homogeneity and isotropy. The standard model simply has to postulate that as part of the initial conditions, and the intuition was this is not a good explanation because it has to rely on very of initial conditions. And so inflation gives one a mechanism that allows the universe to start off in a more or less arbitrary state and then this inflationary mechanism smooths things out to produce the universe we observe. And it was later found that inflation offers a rather natural explanation for the nearly scale-free spectrum of density perturbations encoded in the microwave background radiation. But there was a big problem. The simplest models of inflation, which you can think of as just a Friedman-Robertson-Walker model with an initial inflationary period inserted in the early universe, The simple inflationary models predict that the density parameter will be driven towards unity. And the density parameter is the sum of the part due to mass and the part due to the cosmological constant. Now, estimates of omega matter that are derived from observing the motions of galaxies and galactic clusters tell us that the present value is about 0.3, not one. So if inflation is right, this seems to require either a positive cosmological constant or some form of matter that's a surrogate for the cosmological constant. Now for a number of years, this was simply a pious hope or a prayer. it seemed that the prayers of the inflationary cosmologists were answered.

15:00 In that year, two independent teams reported on red shift measurements made on what are called type 1A supernovae that indicated that the expansion of the universe is not slowing down, but indeed it's accelerating. And I won't go through the arithmetic, but the conclusion is that in order to have accelerated expansion in these models, we need either a positive cosmological constant or we need a large amount of strange matter, strange because it violates what's called a strong energy condition that's thought to be satisfied for normal forms of matter. popular discussions, you run across the term dark energy, which is just a term that covers both of these possibilities. Now, given the little potted history I gave you of the ups and downs of the cosmological constant, the reaction might be, well, let's take a few aspirin and wait a few years and it will all just go away again. I think that's not going to happen this time and here's the reason. So I indicated we have these measurements on supernovae which seem to imply we need a lot of dark energy. There are also x-ray observations of galactic clusters which give the same conclusion. They're very measurements on the cosmic microwave background radiation, which tell us that the total value for lambda is very close to one, whereas the dynamics of galaxies tell us that the part of omega due to normal matter is only about 0.3, so the conclusion is about 70% of matter energy in the universe has to be in the form of dark energy. And then there's something called the integrated Sachs-Wolf effect that I won't even try to talk about. But the overall point is we now have several independent lines of evidence all pointing to the same conclusion.

17:30 So there would have to be some kind of nearly miraculous conspiracy among these independent lines of evidence if they all pointed to the same wrong conclusion. So I think lambda or some surrogate for lambda is here to stay. And now I wanna bring the discussion back to my initial question of whether or not time is finite or infinite in the future. And in the context of orthodox general relativity theory, there are three main possibilities. So this W parameter is just the ratio of the pressure to the density of the dark energy. And there are three main cases. If W is exactly equal to minus one, then we have a true cosmological constant, the thing Einstein initially introduced in 1917. 17. So if this possibility is realized, then Einstein's blunder wasn't a blunder after all. Second possibility is W is exactly between minus the third and minus one. This is usually referred to as quintessence. And the third possibility is W is strictly less than one. And then we have phantom matter. And here are the implications for the fate of the universe. if a true cosmological constant is responsible for the accelerating expansion then geometry is no longer destiny because regardless of whether we live in a k equals zero, plus one, or minus one universe the universe will expand forever so we get eternity but we get eternity with a big chill because any observer co-moving with the accelerated expansion we'll see a universe that is increasingly dark, empty, and cold. And so it's dubious that critters, anything like us, can survive to appreciate this eternity. Second possibility, what we're seeing is not a cosmological constant, but quintessence. And assuming that the equation of state for the quintessence doesn't change, then again we get the same conclusion, there will be eternity with a big chill.

20:00 Third possibility is that what we're seeing is not a genuine cosmological constant, but a phantom matter. and then the fate is that we're going to have a big rip in which all gravitationally bound systems will eventually be torn apart and then we're going to have a big smash because the funny thing about phantom matter is that its density increases as the universe expands and the density reaches an infinite value in a finite amount of time. So eventually, we'll just smash against this infinite density singularity. And then there are various mix and match possibilities. I mean, here's one. Suppose that there were a tiny negative cosmological constant, so tiny that present experiments couldn't detect it. Well, and then suppose that quintessence is responsible for the accelerating expansion. Well, eventually lambda will come to dominate and the universe will start to collapse. On the other hand, if phantom matter is responsible for the accelerated expansion, then lambda will never dominate and we get the same verdict as in 3, that is, we'll get a big rip followed by a big smash. So the possibilities are now much more complicated than they were just 10 years ago. But still, one can hope that within the foreseeable future, observations will rule out some of these possibilities and we'll be able to know whether or not the universe will go on and on forever, even though we and our progeny may not be around to enjoy eternity. Okay, let me now turn to the opposite question. Does time have a beginning? Is it finite in the past? And orthodox cosmology says yes

22:30 in these Friedman, Walker, Robertson, Big Bang models. No time-like curve that starts from now can be extended into the past for a proper length greater than about 14 billion years. And the simple inflationary models don't change this conclusion. Remember, the simple inflationary models are just the Friedman, Walker, Robertson models with an inflationary era inserted in the early universe. And although there's an enormous amount of expansion during the inflationary era, the era lasts just a short time so that this estimate isn't changed. And then just as an interesting historical footnote, I'll mention that there's a sense in which steady state cosmology also implies that time is finite in the past. And this is rather curious because the term Big Bang was a term of derision that was coined by by the advocates of steady state cosmology. But what they didn't realize was that the model that they were working with, if it's properly understood, implies that time is finite in the past. So, steady state cosmology uses what's called the Sitter space-time, and if you can think in five dimensions, you can picture the Sitter space-time as a four-dimensional hyperboloid embedded in a five-dimensional space. Well, it's hard to draw that, so here I picture two-dimensional Minkowski spacetime, one space dimension and one time dimension embedded in a two-dimensional space. So it's this hyperboloid. Now, steady-state cosmology writes the line element for the Sutter spacetime in a special form in which the space sections are flat. well, here's what the flat space sections look like in the Sitter space time. And you can just see diagrammatically that you can neatly foliate the upper half of the hyperboloid with these flat sections, but the foliation can't be extended into the bottom half. And so that just has to be thrown away. And so the model that's presupposed by SETI say cosmology is time-like geodesically incomplete in the past,

25:00 and in that sense, time is finite in the past. Ah, but there's a footnote here. So let's forget about this, get back to orthodox cosmology. So the universe began with a Big Bang about 14 billion years ago, but what about before the Big Bang? In orthodox general relativity theory, that's not a meaningful question because as you approach the Big Bang, the scale factor goes to zero, indicating that there's a genuine space-time singularity, and there's no physically meaningful way in classical general relativity theory to extend through that singularity. But still, one can ask, well, is there some way of avoiding this initial singularity and continuing time into the past? Or second, if we can't avoid the singularity, is there some way of resolving it, for example, in quantum gravity? So I want to briefly take up these two issues. So I talked a bit about the simple inflationary models, which are just Friedman-Walker-Robertson models with an inflationary era inserted in the early universe. But they're more complicated inflationary models called internal, eternal inflation. And so the picture is something like this. Some regions of the universe thermalize and then they expand and other regions thermalize and maybe some of these thermalized regions merge. but the background of non-thermalized portions of the universe is expanding much faster, so the thermalized regions never fill up the entire universe. The entire universe just goes on inflating forever. Now you can turn this around and ask, can we have eternal inflation to the past? And in that way, can we avoid the past singularities? And it's rather interesting that advocates of inflationary cosmology have attempted to prove no-go results that say, in effect, that no, you can't have eternal inflation to the past that avoids an initial singularity.

27:30 However, recently George Ellis and his co-workers had pointed out that these no-go results are based on overly restrictive assumptions. In particular, the no-go results assume that we're either in a Kegel 0 or Kegel minus 1 universe, but the latest observations are compatible with a Kegel plus 1 universe. Some of the no-go results also assume that the Hubble parameter is strictly greater than zero for all times. What Ellis has shown is that if you relax one or both of these assumptions, you can get eternal inflation to the past in such a way as to avoid the initial singularity. Here are a picture of the behavior of the scale factor in three of the models he found. And interestingly enough, in two of these models, you can, the scale factor has a minimum value. And you can set this minimum value high enough so that you avoid any sort of Planck scale physics. So in these models, not only do you avoid an initial singularity, you can, if you like, avoid quantum gravity effects. Well, I think this is an interesting finding, but these models don't seem to describe the actual universe we see around us. So let me turn to the second issue of whether it's possible to somehow resolve the initial Big Bang singularity and extend through it possibly using some effects from quantum gravity. Now those of you who are not familiar with quantum mechanics may not realize that quantum mechanics has the magic ability to smooth out singularities of classical physics. So just to give one example, imagine a system of Newtonian point mass particles that are moving around under their mutual Newtonian gravitational attractions. Well, such a system can exhibit two kinds of singularities.

30:00 First of all, there can be collision singularities where two or more of these particles collide and then the solution just blows up. Or even more interestingly, there can be non-collision singularities. The particles can accelerate themselves off to spatial infinity in a finite amount of time and the solution just disappears. If you quantize this system, both of these singularities are smoothed away. The quantum mechanical solution is well defined for all time. So some of the people who work in quantum gravity have the hope that the proper theory of quantum gravity will somehow smooth away the singularities of classical general relativity theory. And the proponents of what's called loop quantum gravity that they have succeeded in resolving the initial Big Bang singularity of the Friedman-Walker-Robertson model. So loop quantum gravity tries to produce a quantum theory of gravity by quantizing general relativity theory. And this approach doesn't get all of the publicity that the string theory does, but it's made considerable progress. So this claim to have resolved the initial Big Bang singularity is based on two results. They show that in their quantization, the operator for the inverse scale factor has a spectrum that's bounded from above, indicating that the infinities are being controlled. And second, they're able to show that the evolution continues through the classical singularity. And I put evolution in scare quotes has mentioned in the canonical approach to quantum gravity, which was pursued by loop quantum gravity, there isn't any evolution in the normal sense, but what one can do is to pick out some variable that can serve as a clock variable, and then you can measure the evolution of other variables relative to the clock variable. Okay, this is an interesting result, but I just want to point out that the continuation

32:30 through the classical singularity passes through an intense Planck regime where the space-time of classical general relativity is not even approximately valid. So one can wonder in what sense we can speak of events that are prior to the Big Bang, since we don't have any normal kind of time series as given by any space-time with which we're familiar from special or general relativity. Okay, so much for the end of time Let me talk about in between. And I first want to talk about an issue that's come up many, many, many times during the conference, namely asymmetries of time and asymmetries in time. And as several speakers have pointed out, we need to distinguish between these two things. So it seems that time itself is asymmetric because it's ordered by this asymmetric relation of earlier and later. But there also seem to be asymmetries of physical processes in time. And a widespread dogma in both physics and philosophy is that one or both of these kinds of asymmetries are grounded in the entropic behavior of physical systems. Now, if you subscribe to this dogma and you want to take entropy in Boltzmann's sense, then you have to confront a pair of problems with which Boltzmann struggled, but never satisfactorily resolved. So let's just take it as a given of the discussion that we've succeeded in showing or making it plausible that the microdynamics governing thermodynamic systems is such that if at some initial time the Boltzmann entropy is low, then it's very likely that at some appropriate future time the Boltzmann entropy will be higher. Let's just take that as a given. Okay, but two problems. First, the initial state problem.

35:00 What justifies the posit of initially low entropy for the system? And then even more difficult is the asymmetry problem. Because if this is true, and if the microdynamics governing the system are time reversal invariant, then it will also be very likely that at the corresponding earlier time that the entropy was higher than at the initial time t. And this contradicts our normal expectation. So just to take a concrete example, imagine an ice cube put in a lukewarm glass of water. We would expect that if that system remains thermally isolated for the next five minutes, say, at the end of the five minutes we look at the system, will find that the ice cube is partially melted and the water is correspondingly cooler. But we would also expect that if the system had remained thermally isolated for the previous five minutes and we looked at the system five minutes earlier, we would find that the ice cube was less melted and the water was correspondingly warmer. But those expectations are contradicted by this symmetry. So here we encounter another dogma, namely that cosmology comes to the rescue and resolves Boltzmann's twin problems. So the story is supposed to be the one I've been telling, namely that the universe began with a Big Bang, and then very shortly after the Big Bang, the entropy of the universe was very, very, very low. Shortly after, well, if you believe, if you don't believe in inflation, you can take shortly after the Big Bang to be any time in the early universe if you like. If you believe in inflation, you'll want to take this time to be shortly after the inflationary era ended reheated, but that doesn't really matter. Now, I have two qualms here. First, it seems to me that the claims about low entropy of the early universe are very

37:30 likely not even false. So this is my second Austrian reference. Do you know who I'm referring to? No, not no. What? Pauly, yes. Pauly was told about somebody's theory, I'd forgotten whose it was, and he said he didn't want to spend any time on that. That theory is not even false. Now, Pauly was a great physicist, but not a very nice man, and I don't intend any of Pauly's haughty dismissing this. But I'll tell you what I mean by this. and second, I think that even if the claims were true, it's not obvious that they would explain the asymmetries in the thermodynamic behavior of systems of interest to us, but first let me explain this not even false remark. So here's what one would have to do to make good on the dogma. So here's the kind of mathematical apparatus that we'd be working with. The dynamical system consists of some state space and some deterministic flow on that space, and then we have a measure on the space that's preserved under the dynamical flow. And then to talk about Boltzmann entropy, we have to do some core screening. So a macro state corresponds to some volume of the state space, and then the Boltzmann entropy of a macro state is set proportional to the log of the volume of the state space corresponding to that macro state. So the statement that Boltzmann entropy is increasing in time corresponds to a sequence of micro states that actualize macro states that correspond to larger and larger volumes in the state space. Okay, so to make good on the dogma that cosmology resolves Boltzmann's problems, apply this apparatus to our cosmological models and show that when we calculate the Boltzmann entropy for the relevant macro state for the early universe, we get a very, very, very low value. Well, have any such calculations been done?

40:00 Usually what we get are not concrete calculations, but hand-waving. So here's a drawing from Roger Penrose's book, I think The Emperor's New Mind. And it says, in order to produce a universe resembling the one in which we live, the creator would have to aim for an absurdly tiny volume of the phase space of possible universes. So 1 over 10 to the 10 to the 123rd. third. Now, where does this number come from? I mean, does it come from applying the kind of apparatus that I just sketched? And the answer is no. It comes from a kind of heuristic argument. Now, granted, heuristic arguments from a physicist of Penrose's caliber are usually more reliable than the most exact calculations that 99% of the people in this audience, including myself, could do. But still, one would like to see some concrete calculation in some model that would give us confidence that this dogma is on the right track. Well, here's an example of a concrete calculation that was done in a rather simple model in Page, so they discussed a Friedman-Walker-Robertson model coupled to a scalar field that can drive inflation. And in this case, the state space is very simple. It's just a four-dimensional space. It's coordinatized by the field phi, its canonical momentum, our old friend the scale factor, and its canonical momentum. And dynamics, it's just given by the Hamiltonian dynamics of the system, we have a constraint that says in effect that the total Hamiltonian vanishes. So we can go down to the constraint surface on which this constraint is satisfied. That's a three-dimensional surface. And then we can factor out the gauge freedom that's present by going to a reduced space space, and that's a two-dimensional space. And what's our volume element? Well, there's a gadget called the symplectic form,

42:30 and if you do the pullback to this reduced phase space, you get a natural volume element, which is invariant under the dynamical flow. Okay. The problem is the measure that we get doesn't normalize, And the failure of normalization cannot be blamed on the fact that we're in a model in which space is infinite because this normalization, the failure of normalization holds even for K equal plus one models where space is finite. Now, things aren't absolutely hopeless because even though the measure doesn't normalize, we might be able to define at least a partial probability measure on the macro states. So we could do something like this. We could say the probability of a macro state is zero if the measure of the corresponding phase volume is finite. The probability is one if the measure of the phase volume is infinite and the measure of the complement is finite, and otherwise it's undefined. Now, my own guess would be that when we try to compute the probability of the relevant macro state for the early universe, we'll be in this worst case, number three. But just assume that the best case for the dogma holds, namely case number one, where the probability is zero. then the problem is conditional probabilities in the usual sense are not going to be well defined. So we won't be able to talk about what the probability of some future state is given the initial state. Okay, now this is just one model and perhaps more realistic models will give a result that's more congenial to the dogma that cosmology solves Boltzmann's problems. But even if the entropy of the early universe

45:00 is well-defined and has a low value, I'm skeptical of the relevance of this to the problem at issue. Because if the entropy of the early universe is low, it's because of the gravitational contribution. of an initial thermalized state that's relatively smooth, intuitively one would think that such a state had high entropy, well the answer is no, it has low entropy because you've neglected gravity, gravity likes to make things a clump, and that's the reason that the entropy is low in the early universe. But for the sorts of thermodynamic systems we're concerned with, the gravitational component of entropy is not relevant. And second, it has to be remembered that Boltzmann entropy is a global quantity. It applies to a macro state of a system in this state. In this instance, the system is the entire universe, so we're talking about the Boltzmann entropy of a macro state of the entire universe. And as such, it places only a very weak Thank you.