### On Price on the Arrow of Time

Apologies for the lack of posting recently. I had made my mind to write my long-delayed review of Huw Price's book Time's Arrow and Achimedes' Point, but rereading it made me think so much about the issues it raises that I decided insteadt to write two or more separate posts discussing them. My evaluation of the book is simply that it is a must read if you are intereseted in the philosophy of time and/or conceptual problems at the foundations of physics.

Price assumes more or less the "atemporal" or "block universe" perspective in the philosophy of time, and discusses the asymmetries to be found within time, their physical and conceptual relation to each other and the possible ultimate explanations for them. It is a commonplace that all the asymmetries we see in ordinary life (broken glasses that do not reassemble spontaneously, etc.) are ultimately traceable to the second law of thermodynamics and the growth of entropy. (The fundamental dynamical laws are time-symmetrical, with the exception of some very specific particles subject to weak interactions and that so far as we know play no role in ordinary physics.) Price explains very clearly that the growth of entropy means only the tendency to macrostates of greater probability, and as such requires no particular explanation; what does require an explanation is the surprising fact that entropy is so low to begin with, when the "natural" state of the universe is a thermal equilibrium with maximum entropy. Boltzmann suggested once that the original state of the universe was really a high entropy one and that our low-entropy universe was just product of a fluctuation. This proposal has unacceptable quasi-solipsistic consequences, explained by Sean Carroll some weeks ago: by its reasoning it is far more likely that I have sprung directly into existence by a small fluctuation that created my present brain, not by a large one that created a whole universe in which my brain could eventually appear (the "Boltzmann's Brain" problem).

Price also does an excellent job of noting the "double standard" many authors commit in discussing time asymmetries: applying different criteria of plausability in different temporal directions. For example, a common assumption is that "incoming influences are independent": sistems interacting and with no common cause have their properties uncorrelated before the interaction but correlated after it. This seems entirely plausible and commonsensical, but if dynamical laws are time-symmetrical there is no good justification for it. A priori we should find the correlation of systems prior to interacting and uncorrelation afterwards to be just as possible. For macroscopic systems the principle can be justified from the second law and shown to arise from the special low-entropy state at the beginning of the universe (instead of positing it to explain the second law, as some physicists used to hope), whereas for microscopical systems the situation is more obscure (more on this later). Price is extraordinarily good at thinking in "atemporal" terms and uncovering hidden assumptions and double standards in the way we usually think about these questions.

The third chapter of the book contains a discussion of the so-called Arrow of Radiation (we see "retarded" waves going out from emmiting systems and not "advanced" wave focusing spontaneously in absorbing systems, a situation which is the time-mirror image of the first and just as possible on dynamical grounds). Price explains the well-known Wheeler-Feynman theory for explaining this asymmetry and gives a thought-provoking reinterpretation and reassessment of it. Alas, he does not discuss the how his version of the theory fares when quantum electrodynamics replaces classical electrodynamics. Can anybody point to a good discussion of a quantum version of the Wheeler-Feynman theory, if such thing exists?

The fourth chapter discusses cosmology, adressing the fundamental question: "What could be a possible explanation for the unnatural, extraordinarily low-entropy state of the early universe?" After considering and subjecting to criticism the opinions of many well-known physicists (the names of Davies, Penrose and Hawking come up repeatedly) Price concludes that there are four possible alternatives:

1) The anthropic approach: there is a multiverse with vast, vast, number of other universes, with all possible initial conditions, and ours is just a lucky one that got a sufficently-low entropy initial condition to allow for life to develope. Two objections to this view are that it postulates a huge number of unobservable entities, and that it is not clear whether a universe like ours is really the "least costly" way of creating life; if it isn't, we may face a Boltzmann's Brain problem here.

2) The asymmetric law proposal. Roger Penrose is the main contemporary defender of this idea, which essentially postulates a fundamental asymmetric physical law that constrains initial singularities (like the Big Bang) to have low entropy but puts no such constraint on final singularities (like stars collapsing into black holes, or a possible eventual Big Crunch). The main problem with this idea is how arbitrary it seems, especially if the dynamical laws are time-symmetric. (It may become more palatable if a future Unified Theory shows the tiny time asymmetry in weak interactions to derive from some deep time asymmetry at a fundamental level, but this is, I think, pure speculation at the moment. Do string theory folks out there have any idea about the possible status of T-symmetry in M theory?

3) The Gold universe (named for cosmologist Thomas Gold, the first one to propose it). There is a fundamental constraint that makes singularities have a low entropy, but it applies symmetrically. If the universe recollapses to a Big Crunch entropy would revert and decrease in the collapsing phase (at least from the point of view of our present standpoint; from the point of view of observers in the collapsing phase it may seem that the universe in expanding and increasing entropy, if as it seems likely our psychological arrow of time depends on the thermodynamical one). Also black holes should are constrained to have low entropy. Price favours this kind of view for its attractive stmmetry, but most physicists reject it. I will discuss below who has the upper ground in this discussion.

4) The "corkscrew" view. Like a factory that produces an equal number of left-handed and right-handed corkscrews, it might be that the fundamental theory of the universe is time-symmetric but only allows strongly time-asymmetric realizations. Price understands Stephen Hawking's present views (after retracting from a Gold-like view in the 1980s) as an example of this kind of view. Hawking claims that by using his "no boundary proposal" for Euclidean path-integral quantum gravity it can be proven that one extreme of the universe must have low entropy. He further agues that the connection between the thermodynamical and psychological arrows of time warrant calling the low-entropy extreme the "initial" one instead of the "final" one; and that once we have the universe securely expanding to a higher-entropy future we can use ordinary statistical arguments that say that a reversal of entropy is overwhelmly improbable, so the final state will be high-entropy instead of low-entropy.

Setting aside the anthopic view for the reasons outlined, Price favours the symmetrical Gold view (3) over Penrose's and Hawking's asymmetrical views, or at least argues that the Gold universe deserves a more serious consideration that most physicists are willing to give it. The standard argument against the Gold view is that the reversal of entropy requires events of huge statistical imporbability (broken glasses spontaneosly mending themselves, etc.) in the contracting phase. Price argues that this rebuttal invoves a Double Standard: we know already that the argument does not apply in one time direction, so why apply it confidently in the other? From an atemporal perspective, a glass breaking is a trajectory in configuration space precisely as improbable as a glass "unbreaking". We assume that an initial boundary condition of low entropy, whether imposed by fiat as in Penrose or dynamically as in Hawking, can override statistical considerations in one direction; so why not in the other? If we have a thoery saying that a "natural" initial boundary condition is a low entropy one, then the theory will have more symmetry, therefore be more plausible, if it also applies to final boundary conditions. Price devotes a lot of space to showing that there are (and are likely to be) no empirical contradictions to the Gold universe, at least until and if the universe actually starts recollapsing.

I think love of symmetry takes Price to far away in this. His reasoning neglects a very simple fact: we have lots of empirical evidence supporting a low-entropy past, none for a low-entropy future. Thus an explanation for the low entropy past is the only thing we need -and according to Ockham's Razor, we better take the simplest explanation that we can find. If we take Penrose's "Weyl Curvature Hypothesis" or Hawking's dynamically generated low-entropy singularity to be satisfactory explanations, then the question is: does the additional "simplicity" of temporal symmetry gained by extending them to the future as per Gold's universe compensate the additional "non-simplicity" of this making the universe's history far statistically unlikelier than it is without this extension? Statistical considerations are usually very powerful; they enable us, after all, to make reliable predictions from the Second Law in all ordinary situations. It is my opinion that they should not be overriden by a mere desire for symmetry. Absent any empirical evidence that the future is low-entropy (as we have regarding the past) the statistical unlikeliness of such a future takes precedence for me over symmetry, making me reject Gold's universe.

There is an additional consideration: it seems pretty certain that the universe will not in fact recollapse to a Big Crunch, but continue its expansion indefinitely at an ever greater rate, driven by dark energy. Morevorer, it seems to me as plausible as not that this is not an "accident", but a necessary feature of universe. (To have a Big Crunch, the dark energy content of the universe would have to be much lower and the matter energy content much larger; but the closeness of the actual values of both quantities, which are of the same order of magnitude, suggests that there may be a dynamical reason for it, which would make recollapse impossible). If this speculation is solid, then either Penrose's or Hawking's theories would support the "corkscrew" result of a naturally asymmetric universe. Price mentions the possibility of the universe expanding forever, but does not give it the importance he should in my opinion. (The book is written before the discovery of dark energy.) Price argues that even if the unverse does not recollapse there are local collapses to black holes; but I am not sure if these local singularities must be subjected to the same conditions than the global one. (For example, does anybody know if Hawking's "no boundary condition" can be applied to a black/white hole as well as to the universe? If yes, there is indeed a question of why does it not imply that collapsed black holes have low entropy; but I suspect the answer must be no, or Hawking would have realized this!)

There is a fifth possiblity which Price does not discuss because it is more recent than his book. It is suggested by Sean Carroll and Jennifer Chen in this article, which Sean is fond of citing in his Cosmic Variance posts. It claims to give a model for the emergence of the universe as a statistical fluctuation in a high-energy background, but avoiding not only the usual problems of the Anthopic Principle but also, and crucially, the Boltzmann Brain problem as well. I do not really understand how it manages to do so; it has appearently something to do with a distinction between entropy and entropy density, but I am not sure how this does the required job. I have asked Sean about it yesterday in this comment, but the question seems to have been lost among the discussion of sexism in science that highjacked the thread.

My next post on Price's book will discuss how he frames the question of "independence of incoming influences" with respect to microsystems, and the radical interpretation of quantum mechanics and Bell's theorem this leads him to. Stay tuned.

UPDATE: In case you are interested, this paper contains most of the the ideas of Price on cosmology I have discussed here. Check his list of publications for more material on this and other philosophical problems. I find Price to be one of the most interesting contemporary philosophers.

Price assumes more or less the "atemporal" or "block universe" perspective in the philosophy of time, and discusses the asymmetries to be found within time, their physical and conceptual relation to each other and the possible ultimate explanations for them. It is a commonplace that all the asymmetries we see in ordinary life (broken glasses that do not reassemble spontaneously, etc.) are ultimately traceable to the second law of thermodynamics and the growth of entropy. (The fundamental dynamical laws are time-symmetrical, with the exception of some very specific particles subject to weak interactions and that so far as we know play no role in ordinary physics.) Price explains very clearly that the growth of entropy means only the tendency to macrostates of greater probability, and as such requires no particular explanation; what does require an explanation is the surprising fact that entropy is so low to begin with, when the "natural" state of the universe is a thermal equilibrium with maximum entropy. Boltzmann suggested once that the original state of the universe was really a high entropy one and that our low-entropy universe was just product of a fluctuation. This proposal has unacceptable quasi-solipsistic consequences, explained by Sean Carroll some weeks ago: by its reasoning it is far more likely that I have sprung directly into existence by a small fluctuation that created my present brain, not by a large one that created a whole universe in which my brain could eventually appear (the "Boltzmann's Brain" problem).

Price also does an excellent job of noting the "double standard" many authors commit in discussing time asymmetries: applying different criteria of plausability in different temporal directions. For example, a common assumption is that "incoming influences are independent": sistems interacting and with no common cause have their properties uncorrelated before the interaction but correlated after it. This seems entirely plausible and commonsensical, but if dynamical laws are time-symmetrical there is no good justification for it. A priori we should find the correlation of systems prior to interacting and uncorrelation afterwards to be just as possible. For macroscopic systems the principle can be justified from the second law and shown to arise from the special low-entropy state at the beginning of the universe (instead of positing it to explain the second law, as some physicists used to hope), whereas for microscopical systems the situation is more obscure (more on this later). Price is extraordinarily good at thinking in "atemporal" terms and uncovering hidden assumptions and double standards in the way we usually think about these questions.

The third chapter of the book contains a discussion of the so-called Arrow of Radiation (we see "retarded" waves going out from emmiting systems and not "advanced" wave focusing spontaneously in absorbing systems, a situation which is the time-mirror image of the first and just as possible on dynamical grounds). Price explains the well-known Wheeler-Feynman theory for explaining this asymmetry and gives a thought-provoking reinterpretation and reassessment of it. Alas, he does not discuss the how his version of the theory fares when quantum electrodynamics replaces classical electrodynamics. Can anybody point to a good discussion of a quantum version of the Wheeler-Feynman theory, if such thing exists?

The fourth chapter discusses cosmology, adressing the fundamental question: "What could be a possible explanation for the unnatural, extraordinarily low-entropy state of the early universe?" After considering and subjecting to criticism the opinions of many well-known physicists (the names of Davies, Penrose and Hawking come up repeatedly) Price concludes that there are four possible alternatives:

1) The anthropic approach: there is a multiverse with vast, vast, number of other universes, with all possible initial conditions, and ours is just a lucky one that got a sufficently-low entropy initial condition to allow for life to develope. Two objections to this view are that it postulates a huge number of unobservable entities, and that it is not clear whether a universe like ours is really the "least costly" way of creating life; if it isn't, we may face a Boltzmann's Brain problem here.

2) The asymmetric law proposal. Roger Penrose is the main contemporary defender of this idea, which essentially postulates a fundamental asymmetric physical law that constrains initial singularities (like the Big Bang) to have low entropy but puts no such constraint on final singularities (like stars collapsing into black holes, or a possible eventual Big Crunch). The main problem with this idea is how arbitrary it seems, especially if the dynamical laws are time-symmetric. (It may become more palatable if a future Unified Theory shows the tiny time asymmetry in weak interactions to derive from some deep time asymmetry at a fundamental level, but this is, I think, pure speculation at the moment. Do string theory folks out there have any idea about the possible status of T-symmetry in M theory?

3) The Gold universe (named for cosmologist Thomas Gold, the first one to propose it). There is a fundamental constraint that makes singularities have a low entropy, but it applies symmetrically. If the universe recollapses to a Big Crunch entropy would revert and decrease in the collapsing phase (at least from the point of view of our present standpoint; from the point of view of observers in the collapsing phase it may seem that the universe in expanding and increasing entropy, if as it seems likely our psychological arrow of time depends on the thermodynamical one). Also black holes should are constrained to have low entropy. Price favours this kind of view for its attractive stmmetry, but most physicists reject it. I will discuss below who has the upper ground in this discussion.

4) The "corkscrew" view. Like a factory that produces an equal number of left-handed and right-handed corkscrews, it might be that the fundamental theory of the universe is time-symmetric but only allows strongly time-asymmetric realizations. Price understands Stephen Hawking's present views (after retracting from a Gold-like view in the 1980s) as an example of this kind of view. Hawking claims that by using his "no boundary proposal" for Euclidean path-integral quantum gravity it can be proven that one extreme of the universe must have low entropy. He further agues that the connection between the thermodynamical and psychological arrows of time warrant calling the low-entropy extreme the "initial" one instead of the "final" one; and that once we have the universe securely expanding to a higher-entropy future we can use ordinary statistical arguments that say that a reversal of entropy is overwhelmly improbable, so the final state will be high-entropy instead of low-entropy.

Setting aside the anthopic view for the reasons outlined, Price favours the symmetrical Gold view (3) over Penrose's and Hawking's asymmetrical views, or at least argues that the Gold universe deserves a more serious consideration that most physicists are willing to give it. The standard argument against the Gold view is that the reversal of entropy requires events of huge statistical imporbability (broken glasses spontaneosly mending themselves, etc.) in the contracting phase. Price argues that this rebuttal invoves a Double Standard: we know already that the argument does not apply in one time direction, so why apply it confidently in the other? From an atemporal perspective, a glass breaking is a trajectory in configuration space precisely as improbable as a glass "unbreaking". We assume that an initial boundary condition of low entropy, whether imposed by fiat as in Penrose or dynamically as in Hawking, can override statistical considerations in one direction; so why not in the other? If we have a thoery saying that a "natural" initial boundary condition is a low entropy one, then the theory will have more symmetry, therefore be more plausible, if it also applies to final boundary conditions. Price devotes a lot of space to showing that there are (and are likely to be) no empirical contradictions to the Gold universe, at least until and if the universe actually starts recollapsing.

I think love of symmetry takes Price to far away in this. His reasoning neglects a very simple fact: we have lots of empirical evidence supporting a low-entropy past, none for a low-entropy future. Thus an explanation for the low entropy past is the only thing we need -and according to Ockham's Razor, we better take the simplest explanation that we can find. If we take Penrose's "Weyl Curvature Hypothesis" or Hawking's dynamically generated low-entropy singularity to be satisfactory explanations, then the question is: does the additional "simplicity" of temporal symmetry gained by extending them to the future as per Gold's universe compensate the additional "non-simplicity" of this making the universe's history far statistically unlikelier than it is without this extension? Statistical considerations are usually very powerful; they enable us, after all, to make reliable predictions from the Second Law in all ordinary situations. It is my opinion that they should not be overriden by a mere desire for symmetry. Absent any empirical evidence that the future is low-entropy (as we have regarding the past) the statistical unlikeliness of such a future takes precedence for me over symmetry, making me reject Gold's universe.

There is an additional consideration: it seems pretty certain that the universe will not in fact recollapse to a Big Crunch, but continue its expansion indefinitely at an ever greater rate, driven by dark energy. Morevorer, it seems to me as plausible as not that this is not an "accident", but a necessary feature of universe. (To have a Big Crunch, the dark energy content of the universe would have to be much lower and the matter energy content much larger; but the closeness of the actual values of both quantities, which are of the same order of magnitude, suggests that there may be a dynamical reason for it, which would make recollapse impossible). If this speculation is solid, then either Penrose's or Hawking's theories would support the "corkscrew" result of a naturally asymmetric universe. Price mentions the possibility of the universe expanding forever, but does not give it the importance he should in my opinion. (The book is written before the discovery of dark energy.) Price argues that even if the unverse does not recollapse there are local collapses to black holes; but I am not sure if these local singularities must be subjected to the same conditions than the global one. (For example, does anybody know if Hawking's "no boundary condition" can be applied to a black/white hole as well as to the universe? If yes, there is indeed a question of why does it not imply that collapsed black holes have low entropy; but I suspect the answer must be no, or Hawking would have realized this!)

There is a fifth possiblity which Price does not discuss because it is more recent than his book. It is suggested by Sean Carroll and Jennifer Chen in this article, which Sean is fond of citing in his Cosmic Variance posts. It claims to give a model for the emergence of the universe as a statistical fluctuation in a high-energy background, but avoiding not only the usual problems of the Anthopic Principle but also, and crucially, the Boltzmann Brain problem as well. I do not really understand how it manages to do so; it has appearently something to do with a distinction between entropy and entropy density, but I am not sure how this does the required job. I have asked Sean about it yesterday in this comment, but the question seems to have been lost among the discussion of sexism in science that highjacked the thread.

My next post on Price's book will discuss how he frames the question of "independence of incoming influences" with respect to microsystems, and the radical interpretation of quantum mechanics and Bell's theorem this leads him to. Stay tuned.

UPDATE: In case you are interested, this paper contains most of the the ideas of Price on cosmology I have discussed here. Check his list of publications for more material on this and other philosophical problems. I find Price to be one of the most interesting contemporary philosophers.

## 6 Comments:

I have not yet seen a satisfactory explanation of the second law. If the laws of dynamics are time-symmetrical,

entropy should be a constant of motion.

By Anonymous, at 12:26 PM, September 04, 2006

Why? The usual account is that entropy is a property of a macrostate, measuring the (log of the) number of microstates compatible with it. If the initial condition is a very "special" and "orderly" microstate (corresponding to a macrostate of low entropy) then time-symmetric dynamical laws at the micro level are statistically garanteed to increase the entropy of the macrostate. The only problem is why should entropy be low to begin with.

By Alejandro, at 6:12 PM, September 04, 2006

Hello, I just discovered your blog. I like your review of Price's book, and I'm looking forward to further episodes....as I understand it, the Carroll-Chen work does depend crucially on the distinction between entropy and entropy density. So it can indeed explain why the initial entropy was low. But I agree that it doesn't really solve the Boltzmann's Brain problem, which is a problem for *all* theories that rely on random fluctuations. Anyway, this whole random fluctuation business looks very implausible to me. You might enjoy the papers by Andreas Albrecht and RM Wald on the arxiv.

By Brett, at 4:15 AM, September 15, 2006

I'm not sure why "The only problem is why should entropy be low to begin with" is such a big problem. Ordered states are much simpler than entropic states. It seems to me to be much more natural for the universe to have started in a simple state than in a massively disordered state - where would all that disorder have come from? Now, disorder has all sorts of sources, ranging from statistical chaotic effects to quantum perturbations, but BEFORE that?

I also contest the idea that a high entropy state would prevent entropy from increasing. I doubt any equilibrium state involving complex or interacting matter is stable as T->infinite. Sooner or later everything is going to shoot off into the void and no longer interact with anything else.

Indeed, it seems to me that the only reason that didn't happen immediately - i.e., the initial "low entropy" state being essentially equivalent to the final "high entropy" state - is the quantum perturbations. So, I'm suspicious of any explanation of the arrow of time that ignores them.

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