Something had been bugging me about Sean Carroll’s Arrow of Time FAQ, and it was probably because his answers were too pat. For example, after acknowledging that the entropy of the universe is not well-defined, he writes:
If you don’t understand entropy that well, how can you even talk about the arrow of time?
We don’t need a rigorous formula to understand that there is a problem, and possibly even to solve it. One thing is for sure about entropy: low-entropy states tend to evolve into higher-entropy ones, not the other way around. So if state A naturally evolves into state B nearly all of the time, but almost never the other way around, it’s safe to say that the entropy of B is higher than the entropy of A.
It sounds like here he’s saying that entropy can be defined as that which always increases in time. Akin, perhaps, to Boltzmann’s assertion in his Lectures on Gas Theory that the direction in which entropy increases is always the “future” in the same way that the direction pointing away from the centre of the earth is always “up”. But if that is the case, we’d hardly need a cosmological explanation of the second law. After all, even if entropy had undergone a monotonic decrease from the big bang till now, we shouldn’t (by Sean’s argument) interpret it as decreasing with time. We’d either redefine entropy to be something else that we conveniently found to at least mostly increase with time, or invert our direction of time — interpret time to be increasing towards the Big Bang. Either way, we can’t be sure that there is a problem that needs to be solved by a cosmological solution. If we can redefine entropy once we find that it has not been increasing with time the way we want it to, then why can’t we now just redefine entropy to be something else other than the inconvenient thing that seems to increase in time persistently for all systems despite our microdynamically time-reversible laws? If we can invert our interpretation of the direction of time to fit the direction of increasing entropy, then there is just no need for a cosmological solution — just reinterpret the local entropy minimum to be when time ’started’!
This is why we shouldn’t blithely commit ourselves to saying that entropy’s principal characteristic is that it increases with time. Presumably there are other characteristics of entropy we hope to retain as we search for a definition of the universe’s entropy. So the fact that we can’t find a proper definition for the universe’s entropy should bother us — if we can’t find anything that resembles other aspects of entropy other than its increase with time, then we can’t use our makeshift definitions of the universe’s entropy to back up the claim that the beginning of the universe was a low-entropy state. It certainly couldn’t be anything other than a low entropy state if we’ve defined our criteria for a definition of entropy to be one which characterizes the Big Bang as a low entropy state.
As I commented over at Cosmic Variance, I recommend this John Earman paper for an account of how all the cosmological definitions of entropy to date have been unsatisfactory.
December 27, 2007 at 5:47 am |
“It sounds like here he’s saying that entropy can be defined as that which always increases in time.”
It may *sound* that way, but I’m pretty sure that that is not what he means. What he means is that one does not know how to give a precise definition of the entropy of the universe, where that entropy is some measure of the “genericity” of its state, but *whatever* that precise definition turns out to be, the entropy of the early universe will surely [on the basis of observed facts] turn out to be extremely small, and this smallness will demand an explanation.
Also, I do not understand your usage of “overhyped”. If you think that this problem gets too much attention, I can assure you that you are very mistaken. I could, for example, give you the name of a *very* famous physicist who is apparently unaware that this problem exists at all.
December 27, 2007 at 7:39 am |
But I’m questioning exactly why we are entitled to think that ‘the entropy of the early universe will surely turn out to be extremely small’. If we have no working definition of the entropy of the universe, then we have no basis for insisting that the entropy of the early universe will definitely turn out to be extremely small. If we’re taking ‘the entropy of the big bang is extremely small’ as an axiom, well then the explanation for why it is so is just that we have assumed it.
I think it’s overhyped because I think that if one does not have a working definition of the entropy of the entire universe, then it’s pointless trying to solve the problem of why it started out so small — there might well be no such problem, after all. It’d pretty much be as if physicists went around tearing their hair out upon coming across a novel system which is claimed to violate the conservation of energy but for which a non-problematic definition of energy has not been formulated. I think people should show clearly that there is a problem first before running around trying to solve it, and I have seen no reason to think the early universe had an extremely low entropy, given the major problems with every single definition of cosmological entropy that has popped up so far (Earman’s paper makes exactly this point).
I know only too well that many, many physicists have no interest in or awareness of such ‘fundamental’ problems. I do think that the problem of why entropy seems to increase in the same direction of time for the vast majority of subsystems of the universe is not getting enough attention from the physics community. But there we know there really is a problem because the definitions of entropy for a wide range of systems that are part of the universe (just not the universe itself) are fairly robust. Cosmological entropy, however, just isn’t anywhere as well-defined as thermodynamic entropy.
December 27, 2007 at 7:52 am |
OK, let’s drop all talk of entropy altogether. “The state of the early universe,including its spatial geometry, was non-generic to a fantastic degree.” Do you agree with this statement or not? If not, how do you account for the fantastic uniformity of the cosmic microwave background?
The point is that it is hard, as a technical problem, to *quantify* the idea that an exactly round sphere is a non-generic shape among all possible shapes on the topological sphere. But our inability [for the moment] to deal with difficult technicalities should not blind us to the obvious fact that the round sphere *is* in fact a very “special” shape. The early universe was certainly “special” in this sort of sense, and this has to be explained.
December 27, 2007 at 9:20 pm |
The cosmic microwave background is supposedly explained by inflation, and is generic in the light of the inflationary hypothesis. A quite different issue from whether the universe began in a low entropy/probability/whatever state, which is the question Sean was concerned with. I do not think we have any reason to believe that the ‘initial state’ of the universe was non-generic to a fantastic degree.
December 27, 2007 at 9:59 pm |
As Sean explains, inflation itself can only begin with *very* special initial conditions. As Andreas Albrecht says, inflation “passes the buck” back to special initial conditions *before* inflation. So if you believe in inflation, you should believe in initial “non-genericity”. Note that Earman agrees with this.
I finally was able to read the Earman article. While I usually admire his work, this is not one of his better efforts. Far better is the article by Robert Wald in the same issue of the journal. He admits that there are problems in giving a precise definition of “gravitational entropy”, but goes on to explain why the universe must have begun in a very special state. Note also that Earman doesn’t deny that the universe began in such a state, he only takes issue with the use of Boltzmann entropy in such discussions; which, as above, is interesting but not really relevant. Huw Price has some excellent articles [on arxiv] explaining all this.
January 3, 2008 at 11:03 am |
Thanks for the pointer to Wald’s paper. The Hollands/Wald argument that a universe that inflates can do so only from a non-generic state is persuasive. But in his SHPMP paper, Wald writes:
“For example, in chaotic inflation, the initial conditions needed to produce an inflating patch in the early universe are very “special”; most regions would not inflate and would not evolve to a universe that looks anything like ours. Of course, it is true that, nevertheless, some regions are bound to inflate. Indeed, if the universe is infinite, the probability of having an inflating patch (and , indeed, infinitely many such patches) is 1.”
He then goes on to argue that the chaotic inflation hypothesis strictly implies only that inflation is a possible explanation for the uniformity of the early universe (unless the universe is infinite). Thus it seems that the question of whether inflation (for our observed part of the universe) is probable or merely possible depends on our estimates of the size of the universe and the ‘probability’ of inflation as measured by phase space volumes occupied by ensembles of universes. So once again it’s not obvious that inflation requires a non-generic state of the entire universe — if our universe is large enough, it is likely that some patch of it inflates.
I don’t think Earman ‘agrees’ that inflation implies initial non-genericity. He writes that he does not “pretend to be able to adjudicate such conflicting claims [of the inflationary cosmologists and the Hollands/Wald camp]“.
Finally, it’s worth pointing out that even if the universe began in a ‘non-generic’ state, it’s not clear that this is at all relevant to the ‘arrow of time’. In his post, Sean says the arrow of time is encapsulated by the second law of thermodynamics, which describes a monotonic change in thermodynamic entropy with time. So once you concede that there is no coherent notion of cosmological thermodynamic entropy currently available, then it’s not obvious that cosmology can say anything about the oriigin of the arrow of time.
Certainly, the fact that the universe begain in a ‘non-generic’ state could be a problem (some, like Craig Callender, don’t think it is). But this is not the problem of the cosmological arrow of time, which remains ill-defined so long as ‘the arrow’ = Second Law and cosmological entropy is also ill-defined.