Newsgroups: sci.physics
Date: 11 Nov 1995 10:54:02 -0800
Organization: University of California, Riverside
In article <480ttr$q1u$1@mhafn.production.compuserve.com> Dale Jenkins <100575.143@CompuServe.COM> writes:
Edited by Jack Sarfatti
The conclusion of this paper is that the no boundary proposal can explain the existence of a well defined thermodynamic arrow of time. This arrow always points in the same direction. The reason we observe it to point in the same direction as the cosmological arrow is that conditions are suitable for intelligent life only at the low entropy end of the universe's history.
"Remember Hawking's theory about time flowing the other way when the Universe passes the mid-point and starts contracting? (At least, I _think_ it was Hawking's, though he might have been citing someone else.) Broken cups would mend themselves, then leap back up onto the breakfast table in defiance of the very force responsible for the contraction. Doesn't sound very logical to me."
It wouldn't work quite that way. This idea was apparently thought of by a guy named Gold, by the way; Hawking considered it for a while, then abandonded it; but his recent argument against it has been questioned by Huw Price and also H. D. Zeh. Zeh has a *very* interesting paper on it, which I haven't gotten around to reviewing in This Week's Finds.
Here's some stuff from old issues of This Week's Finds on the Gold universe. To see old issues, by the way, you can now check out
http://math.ucr.edu/home/baez/README.html#TWF
From Week 2:
5)
"The laws of physics do not distinguish the future from the past direction of time. More precisely, the famous CPT theorem says that the laws are invariant under the combination of charge conjugation, space inversion and time reversal. In fact effects that are not invariant under the combination CP are very weak, so to a good approximation, the laws are invariant under the time reverseal operation T alone. Despite this, there is a very obvious difference between the future and past directions of time in the universe we live in. One only has to see a film run backward to be aware of this.
There are are several expressions of this difference. One is the so-called psychological arrow, our subjective sense of time, the fact that we remember events in one direction of time but not the other. Another is the electromagnetic arrow, the fact that the universe is described by retarded solutions of Maxwell's equations and not advanced ones. Both of these arrows can be shown to be consequences of the thermodynamic arrow, which says that entropy is increasing in one direction of time. It is a non trivial feature of our universe that it should have a well defined thermodynamic arrow which seems to point in the same direction everywhere we can observe. Whether the direction of the thermodynamic arrow is also constant in time is something we shall discuss shortly.
There have been a number of attempts to explain why the universe should have a thermodynamic arrow of time at all. Why shouldn't the universe be in a state of maximum entropy at all times? And why should the direction of the thermodynamic arrow agree with that of the cosmological arrow, the direction in which the universe is expanding? Would the thermodynamic arrow reverse, if the universe reached a maximum radius and began to contract?
Some authors have tried to account for the arrow of time on the basis of dynamic laws. The discovery that CP invariance is violated in the decay of the K meson, inspired a number of such attempts but it is now generally recognized that CP violation can explain why the universe contains baryons rather than anti baryons, but it can not explain the arrow of time. Other authors have questioned whether quantum gravity might not violate CPT, but no mechanism has been suggested. One would not be satisfied with an ad hoc CPT violation that was put in by hand.
The lack of a dynamical explanation for the arrow of time suggests that it arises from boundary conditions. The view has been expressed that the boundary conditions for the universe are not a question for Science, but for Metaphysics or Religion. However that objection does not apply if there is a sense in which the universe has no boundary. We shall therefore investigate the origin of the arrow of time in the context of the no boundary proposal of Hartle & Hawking. This was formulated in terms of Einsteinian gravity which may be only a low energy effective theory arising from some more fundamental theory such as superstrings. Presumably it should be possible to express a no boundary condition in purely string theory terms but we do not yet know how to do this. However the recent COBE observations indicate that the perturbations that lead to the arrow of time arise at a time during inflation when the energy density is about 10^{-12} of the Planck density. In this regime, Einstein gravity should be a good approximation.
[I'll skip some more technical stuff on the validity of perturbative calculations in quantum gravity...]
One can estimate the wave functions for the perturbation modes by considering complex metrics and scalar fields that are solutions of the Einstein equations whose only boundary is the surface $S$. When $S$ is a small three sphere, the complex metric can be close to that of part of a Euclidean four sphere. In this case the wave functions for the tensor and scalar modes correspond to them being in their ground state. As the three sphere $S$ becomes larger, these complex metrics change continuously to become almost Lorentzian. They represent universes with an initial period of inflation driven by the potential energy of the scalar field. During the inflationary phase the perturbation modes remain in their ground states until their wave lengths become longer than the horizon size. The wave function of the perturbations then remains frozen until the horizon size increases to be more than the wave length again during the matter dominated era of expansion that follows the inflation. After the wave lengths of the perturbations come back within the horizon, they can be treated classically.
This behaviour of the perturbations can explain the existence and direction of the thermodynamic arrow of time. The density perturbations when they come within the horizon are not in a general state but in a very special state with a small amplitude that is determined by the parameters of the inflationary model, in this case, the mass of the scalar field. The recent observations by COBE indicate this amplitude is about $10^{-5}$. After the density perturbations come within the horizon, they will grow until they cause some regions to collapse as proto-galaxies and clusters. The dynamics will become highly non linear and chaotic and the coarse grained entropy will increase. There will be a well defined thermodynamic arrow of time that points in the same direction everywhere in the universe and agrees with the direction of time in which the universe is expanding, at least during this phase.
The question then arises: If and when the universe reaches and maximum size, will the thermodynamic arrow reverse? Will entropy decrease and the universe become smoother and more homogeneous during the contracting phase?
[I'll skip some stuff on why Hawking originally thought entropy had to decrease during the Big Crunch if the no-boundary proposal were correct... and why he no longer thinks so.
] The thermodynamic arrow will agree with the cosmological arrow for half the history of the universe, but not for the other half. So why is it that we observe them to agree? Why is it that entropy increases in the direction that the universe is expanding? This is really a situation in which one can legitimately invoke the weak anthropic principle because it is a question of where in the history of the universe conditions are suitable for intelligent life. The inflation in the early universe implies that the universe will expand for a very long time before it contracts again. In fact, it is so long that the stars will have all burnt out and the baryons will have all decayed. All that will be left in the contracting phase will be a mixture of electrons, positrons, neutrinos and gravitons. This is not a suitable basis for intelligent life."
From Week 26:
1) Cosmology, time's arrow, and that old double standard, by Huw Price, 26 pages, in LaTeX with 3 figures appended as postscript file, available as gr-qc/9310022. (Written for Time's Arrows Today Conference, UBC, Vancouver, June 1992; forthcoming in Savitt, S., ed., "Time's Arrows Today," Cambridge University Press, 1994.)
Why is the future different from the past? Because it hasn't happened yet? Well, sure, but that's not especially enlightening, in fact, it's downright circular. Unfortunately, a lot of work on the "arrow of time" is just as circular, only so erudite that it is hard to spot it! That's what this article takes some pains to clarify. I think I will be lazy and quote the beginning of the paper:
"A century or so ago, Ludwig Boltzmann and others attempted to explain the temporal asymmetry of the second law of thermodynamics. The hard-won lesson of that endeavour---a lesson still commonly misunderstood---was that the real puzzle of thermodynamics lies not in the question why entropy increases with time, but in that as to why it was ever so low in the first place. To the extent that Boltzmann himself appreciated that this was the real issue, the best suggestion he had to offer was that the world as we know it is simply a product of a chance fluctuation into a state of very low entropy. (His statistical treatment of thermodynamics implied that although such states are extremely improbable, they are bound to occur occasionally, if the universe lasts a sufficiently long time.) This is a rather desperate solution to the problem of temporal asymmetry, however, and one of the great achievements of modern cosmology has been to offer us an alternative. It now appears that temporal asymmetry is cosmological in origin, a consequence of the fact that entropy is much lower than its theoretical maximum in the region of the Big Bang ---i.e., in what we regard as the *early* stages of the universe.
The task of explaining temporal asymmetry thus becomes the task of explaining this condition of the early universe. In this paper I want to discuss some philosophical constraints on the search for such an explanation. In particular, I want to show that cosmologists who discuss these issues often make mistakes which are strikingly reminiscent of those which plagued the nineteenth century discussions of the statistical foundations of thermodynamics. The most common mistake is to fail to recognise that certain crucial arguments are blind to temporal direction, so that any conclusion they yield with respect to one temporal direction must apply with equal force with respect to the other. Thus writers on thermodynamics often failed to notice that the statistical arguments concerned are inherently insensitive to temporal direction, and hence unable to account for temporal asymmetry. And writers who did notice this mistake commonly fell for another: recognising the need to justify the double standard---the application of the arguments in question `towards the future' but not `towards the past'---they appealed to additional premisses, without noticing that in order to do the job, these additions must effectively embody the very temporal asymmetry which was problematic in the first place. To assume the uncorrelated nature of initial particle motions (or incoming `external influences'), for example, is simply to move the problem from one place to another. (It may *look* less mysterious as a result, but this is no real indication of progress. The fundamental lesson of these endeavours is that much of what needs to be explained about temporal asymmetry is so commonplace as to go almost unnoticed. In this area more than most, folk intuition is a very poor guide to explanatory priority.)
One of the main tasks of this paper is to show that mistakes of these kinds are widespread in modern cosmology, even in the work of some of the contemporary physicists who have been most concerned with the problem of the cosmological basis of temporal asymmetry---in the course of the paper we shall encounter illicit applications of a temporal double standard by Paul Davies, Stephen Hawking and Roger Penrose, among others. Interdisciplinary point- scoring is not the primary aim, of course: by drawing attention to these mistakes I hope to clarify the issue as to what would count as adequate cosmological explanation of temporal asymmetry.
I want to pay particular attention to the question as to whether it is possible to explain why entropy is low near the Big Bang without thereby demonstrating that it must be low near a Big Crunch, in the event that the universe recollapses. The suggestion that entropy might be low at both ends of the universe was made by Thomas Gold in the early 1960s. With a few notable exceptions, cosmologists do not appear to have taken Gold's hypothesis very seriously. Most appear to believe that it leads to absurdities or inconsistencies of some kind. However, I want to show that cosmologists interested in time asymmetry continue to fail to appreciate how little scope there is for an explanation of the low entropy Big Bang which does not commit us to the Gold universe. I also want criticise some of the objections that are raised to the Gold view, for these too often depend on a temporal double standard. And I want to discuss, briefly and rather speculatively, some issues that arise if we take the view seriously. (Could we observe a time-reversing future, for example?)"
[And now let me jump forward to a very interesting issue, Hawking's attempt to derive the arrow of time from his "no-boundary boundary conditions" choice of the wavefunction of the universe. (See "week3.") I found this rather unsatisfying when I read it, and had a sneaking suspicion that he was falling into the fallacy Gold describes above. Let's hear what Price has to say about it. - jb]
Our second example is better known, having been described in Stephen Hawking's best seller, A Brief History of Time. It is Hawking's proposal to account for temporal asymmetry in terms of what he calls the No Boundary Condition (NBC)---a proposal concerning the quantum wave function of the universe. To see what is puzzling about Hawking's claim, let us keep in mind the basic dilemma. It seemed that provided we avoid double standard fallacies, any argument for the smoothness of the universe would apply at both ends or at neither. So our choices seemed to be to accept the globally symmetric Gold universe, or to resign ourselves to the fact that temporal asymmetry is not explicable (without additional assumptions or boundary conditions) by a time-symmetric physics. The dilemma is particularly acute for Hawking, because he has a more reason than most to avoid resorting to additional boundary conditions. They conflict with the spirit of his NBC, namely that one restrict possible histories for the universe to those that `are finite in extent but have no boundaries, edges, or singularities.'
Hawking tells us how initially he thought that this proposal favoured the former horn of the above dilemma: `I thought at first that the no boundary condition did indeed imply that disorder would decrease in the contracting phase.' He changed his mind, however, in response to objections from two colleagues: `I realized that I had made a mistake: the no boundary condition implied that disorder would in fact continue to increase during the contraction. The thermodynamic and psychological arrows of time would not reverse when the universe begins to contract or inside black holes.'
This change of mind enables Hawking to avoid the apparent difficulties associated with reversing the thermodynamic arrow of time. What is not clear is how he avoids the alternative difficulties associated with it not reversing the thermodynamic arrow of time. That is, Hawking does not explain how his proposal can imply that entropy is low near the Big Bang, without equally implying that it is low near the Big Crunch. The problem is to get a temporally asymmetric consequence from a symmetric physical theory. Hawking suggests that he has done it, but doesn't explain how. Readers are entitled to feel a little dissatisfied. As it stands, Hawking's account reads a bit like a suicide verdict on a man who has been stabbed in the back: not an impossible feat, perhaps, but we'd like to know how it was done!
It seems to me that there are three possible resolutions of this mystery. The first, obviously, is that Hawking has found a way round the difficulty. The easiest way to get an idea of what he would have to have established is to think of three classes of possible universes: those which are smooth and ordered at both temporal extremities, those which are ordered at one extremity but disordered at the other, and those which are disordered at both extremities. If Hawking is right, then he has found a way to exclude the last class, without thereby excluding the second class. In other words, he has found a way to exclude disorder at one temporal extremity of the universe, without excluding disorder at both extremities. Why is this combination the important one? Because if we can't exclude universes with disorder at both extremities, then we haven't explained why our universe doesn't have disorder at both extremities---we know that it has order at least one temporal extremity, namely the extremity we think of as at the beginning of time. And if we do exclude disorder at both extremities, we are back to the answer that Hawking gave up, namely that order will increase when the universe contracts.
Has Hawking shown that the second class of universal histories, the order--disorder universes, are overwhelmingly probable? It is important to appreciate that this would not be incompatible with the underlying temporal symmetry of the physical theories concerned. A symmetric physical theory might be such that all or most of its possible realisations were asymmetric. Thus Hawking might have succeeded in showing that the NBC implies that any (or almost any) possible history for the universe is of this globally asymmetric kind. If so, however, then he hasn't yet explained to his lay readers how he managed it. In a moment I'll describe my attempts to find a solution in Hawking's technical papers. What seems clear is that it can't be done by reflecting on the consequences of the NBC for the state of one temporal extremity of the universe, considered in isolation. For if that worked for the `initial' state it would also work for the `final' state; unless of course the argument had illicitly assumed an objective distinction between initial state and final state, and hence applied some constraint to the former that it didn't apply to the latter. What Hawking needs is a more general argument, to the effect that disorder--disorder universes are impossible (or at least overwhelmingly improbable). It needs to be shown that almost all possible universes have at at least one ordered temporal extremity---or equivalently, at most one disordered extremity. (As Hawking points out, it will then be quite legitimate to invoke a weak anthropic argument to explain why we regard the ordered extremity thus guaranteed as an *initial* extremity. In virtue of its consequences for temporal asymmetry elsewhere in the universe, conscious observers are bound to regard this state of order as lying in their past.)
That's the first possibility: Hawking has such an argument, but hasn't told us what it is (probably because he doesn't see why it is so important). As I see it, the other possibilities are that Hawking has made one of two mistakes (neither of them the mistake he claims to have made). Either his NBC does exclude disorder at both temporal extremities of the universe, in which case his mistake was to change his mind about contraction leading to decreasing entropy; or the proposal doesn't exclude disorder at either temporal extremity of the universe, in which case his mistake is to think that the NBC accounts for the low entropy Big Bang.
[And, eventually, Price concludes that there is indeed something lacking in Hawking's attempts to derive an arrow of time. - jb]
------------------------------------------------------------------------- I have said this many times here and there, but I'll say it again. For a good introduction to these issues, read
2) The Physical Basis of the Direction of Time, by H. D. Zeh, Second Edition, Springer-Verlag, 1992. ISBN 3-540-54884-X or 0-387-54884-X
Zeh is one of the most clear-headed writers I know on this vexing problem. Interestingly, Price acknowledges Zeh in his paper.