From Eternity to Book Club: Chapter Three

By Sean Carroll | February 2, 2010 8:00 am

Welcome to this week’s installment of the From Eternity to Here book club. Next up is Chapter Three: “The Beginning and End of Time.” Remember that next week we’re doing two chapters at once, Four and Five.

For those who missed them, here’s the Science Friday discussion, and here’s the Firedoglake book salon with Chad. I should also point to some substantive review/discussions: Wall Street Journal, New Scientist, USA Today, and Overcoming Bias.


For the most part, people interested in statistical mechanics care about experimental situations in laboratories or kitchens here on Earth. In an experiment, we can control the conditions before us; in particular, we can arrange systems so that the entropy is much lower than it could be, and watch what happens. You don’t need to know anything about cosmology and the wider universe to understand how that works.

But our aims are more grandiose. The arrow of time is much more than a feature of some particular laboratory experiments; it’s a feature of the entire world around us. Conventional statistical mechanics can account for why it’s easy to turn an egg into an omelet, but hard to turn an omelet into an egg. What it can’t account for is why, when we open our refrigerator, we are able to find an egg in the first place. Why are we surrounded by exquisitely ordered objects such as eggs and pianos and science books, rather than by featureless chaos?

This chapter is a fairly straightforward review of the modern understanding of cosmology, with a particular eye on those issues that will become important later in the book. We zip through the expansion, structure formation, and dark energy. There I got to tell a fun personal story of my wager with Brian Schmidt. At least I think it’s fun — including personal stories is not my natural tendency, but at the right moments it can help to humanize all the forbidding science. Hopefully this was one such moment.

A few topics go beyond the standard cosmology summary. I discussed the Steady State theory a bit, because it’s a relevant historical example when we will much later turn to the question of what the universe should look like. I also dwell a bit on vacuum fluctuations and dark energy, because those will pay a crucial role in my personal favorite explanation for the arrow of time. And we close the chapter with a very brief overview of the evolution of entropy. It has to be brief, because we haven’t laid nearly enough groundwork to do the job right. This is a conscious choice, which may or may not work: rather than simply progressing on an absolutely logical path from foundations to conclusions, I felt free to mention points that would be important later, on the theory that they would come as less of a shock if we had established some familiarity. Again, hope that worked.

Tom Levenson, who is an actual writer, advised me to omit “smoking a pipe” from the caption to Figure 7, on the theory that what is shown should not also be told. I left it in anyway. It’s my book!

  • Oded

    This is a very minor point. In an endnote, you mentioned that there is about 1 calorie of dark energy in each cubic centimeter, and then proceeded to calculate how much that is for an average household, etc. But later, in the same chapter, when you showed the 10^120 problem, you said there is only 10^-15 joules in each cubic centimeter! what gives? did you mean cubic kilometer when showing the calorie calculation? it just barely works out then. Or did i misunderstand and these are not the same number?

  • Lawrence M Kuklinski

    Hello Sean,

    Chapter #3 was a fun one.
    I was able to grasp some concepts and follow most of your reasoning.

    Early in the chapter, and in the Teaching Co DVD as well, you downplay the balloon and raisin bread analogies of our universe. After reading your thoughts I agree with you that they are terrible, but can you supply a visual concept that is accurate and easy to understand?

    On page 52 you describe a post big bang era called “Recombination”.
    What combined before the re-combination?

    You indicate that the big bang may have been a transition period from “elsewhere”.
    Also entropy goes lower the further back we travel to the big bang.
    How low can entropy exist pre big bang?

    Finally, does entropy exist in intra-atomic space and can it be zero?

    I look forward to your discussion.

    Thank you,

    Lawrence Kuklinski

  • Tom Allen

    I have a question about the expansion of the universe. Very early on (but later in the book) we have the era of inflation driven by the postulated inflaton field. Presently we have the era of acceleration driven by dark energy. My question is what is the driver for expansion between these two regimes? Does it just fall out of Einstein’s equations?

    Another way to frame my question is to ask why, when inflation was finished, the expansion rate did not become exactly zero.

  • Sean

    Oded– It looks like I just made an arithmetic error in footnote 48, good catch! There are roughly 10^{-8} ergs per cubic centimeter in the dark energy, and 10^7 ergs per joule, so 10^{-15} joules per cubic centimeter. And a calorie is about 4 joules. (This would never happen if everyone just agreed to use electron volts.)

    Lawrence– I don’t have any suggested analogies to replace the balloon and raisin bread; my suggestion is to think about the actual universe, and not appeal to analogies in this particular instance. Recombination is the time when electrons and nuclei combined for the first time; the word is borrowed from non-cosmological contexts.

    Empty space can’t have entropy in the conventional sense, because there aren’t any degrees of freedom there. Gravity changes this in subtle ways, because spacetime itself is dynamical; for example, black holes have very large entropy. As for what happened before the Big Bang, stay tuned for Part Four!

    Tom– Yes, according to Einstein’s equation the rate of expansion is proportional to (the square root of) the total energy of the universe. It gradually decreases as the universe expands and dilutes, unless dark energy is dominating, which doesn’t dilute away.

  • Tim van Beek

    Maybe I may try to answer some questions and let Sean correct me.

    @Lawrence M Kuklinski
    1. Question: “Early in the chapter, and in the Teaching Co DVD as well, you downplay the balloon and raisin bread analogies of our universe. After reading your thoughts I agree with you that they are terrible, but can you supply a visual concept that is accurate and easy to understand?”

    My opinion:
    The main criticism is that we picture bread and baloon as embedded in an ambient space in which they expand into. Do you know the movie “Poltergeist”? There is a scene where the mother tries to run down the corridor to reach the door of the room of her daughter, who is in danger. As she runs down the corridor, the corridor expands and the door moves away from her. The point is that space itself, inside the house expands, without the notion of an ambient space outside the house (and in fact without the house, if pictured embedded in the garden, expanding).

    2. Question: “How low can entropy exist pre big bang?”

    My opinion:
    Any theory about what was before the “big bang” is purely speculative, as Sean mentions in his book (I don’t remember the page). The answert to this question would be also purely speculative. (There are some models that suggest answers, but these should really be explained by an expert like Sean).

    3. Question: ” Finally, does entropy exist in intra-atomic space and can it be zero?”

    My opinion:
    We have to agree upon a specific model to answer that. You can always define entropy if you have a system with degrees of freedom. If we take classical spacetime with quantum fields on it, that gives you degrees of freedom in every region of spacetime, even in regions that have a spacial volume smaller than an hydrogen atom (the volume of a hydrogen atom is a priori an ill defined concept, but let’s put that aside for a moment, clearly we are speaking of a regime where quantum field theory on a classical spacetime gives a pretty good approximation).
    The entropy of our system would be zero if it had exactly one state. In case of our spacetime with quantum fields this would not be possible, because you cannot suppress vaccuum fluctuations: You’ll always have virtual particles and the existence of these gives you more than one microscopic state. (That point seems to me to be more subtle than my answer suggests, would like to be corrected on that one :-)

  • Metre

    On page 50, you correctly state that “there is no special point corresponding to where the BB happened.” which is true. I always thought of the BB as happening at every point, since they were all squeezed together initially (kinda, sorta); i.e. the BB happened everywhere. That helps me understand why each point today appears to be at the center of expansion to an observer located at that point – it is! The universe is expanding away from every point, and every point is a local center of expansion. Is that view correct?

    Figure 7 alternate captions – 1) “Edwin Hubble, staring at the camera” 2)”Edwin Hubble, sitting at a desk” 3)”Edwin Hubble, wearing a suit!” (Just having some fun with that caption – but it is your book so you do get to caption it as you wish!)

    So far, I like the book. I’m on chapter 8 and will try to stay ahead of the discussions. The chapter on entropy (8) is really scraping the rust off that old undergrad statistical mechanics course I took last millenium. It was so long ago, I think Boltzmann himself taught the course (actually, it was Gaurang Yodh, now at UCI, then at U of MD).

  • Sean

    Tim– All in favor of everyone taking a shot at other people’s questions. Your answers sound good to me.

    Metre– Sometimes I say that, sometimes not. Usually I try “the Big Bang is a moment in time, not a point in space.” It would be great to see some actual research on what explanations work the best; right now all I have are feelings and anecdotes.

  • Lawrence Kuklinski

    Hi Sean & Tim,

    Thanks for the comments. My knowledge of the subject is low, but learning the unknown makes it exciting.
    One question: what does “degrees of freedom” mean?

  • Sean

    Perfectly good question. A “degree of freedom” is just “something that can move around.” In classical mechanics, there are no degrees of freedom in empty space — there’s nothing there! Solid objects have various degrees of freedom: they can move through space, and also rotate in various ways. In quantum field theory, empty space does have degrees of freedom (quantum fields exist, even if they’re in their vacuum states), which makes things confusing.

  • Bee

    Sean: I bought your book! Out of bad consciousness for using your lecture notes but never buying your GR book.

    So here’s a question: I’ve been looking recently for a reference clarifying which structures do and which don’t participate in the Hubble flow. You mention this so by the way, but I really couldn’t find something useful. (Oh, I found a paper explaining why atoms don’t expand, but that was about it.) I gather the warm-hot intergalactic medium does partly expand and partly collapse, and my guess is that it’s somewhere on the sup galaxy-cluster level that structures start to expand with the Hubble flow, but any reference would be much appreciated.

    Some other remarks:

    – I was disappointed you didn’t mention Olber’s paradox.
    – Why is the recombination called REcombination? I’ve always wondered. (I just saw somebody asked that above already.)
    – Thanks for sparing me the expanding ballons/raisin muffins etc.
    – About Hoyle’s C-field: if you have the C-field in addition to the usual expansion, do you get accelerated expansion?
    – Page 63, pt 1. Sorry, I missed n>5 steps in this argument. Will you elaborate on this later? Didn’t you just tell us that in the early universe curvature is strong enough so GR breaks down? Now you’re telling us to forget about gravity? Also, so far entropy has pretty much been a particle-state counting, what am I suddenly supposed to make of the entropy of the gravitational field?

  • Sean

    I agree that buying copies of the book helps everyone feel better about themselves.

    The difference between participating and not participating in the Hubble flow is not a cut-and-dried one. Early on, almost all particles are moving apart from each other; some groups pull together and become bound, while on large scales expansion continues. You might look at something like this paper for more details.

    I’m not sure what the rules for the C-field are, as they never really made sense. I think it acts basically like a cosmological constant, but one that creates matter all along. In the steady-state model, expansion is definitely accelerated at all times, because the Hubble parameter is constant. (Steady!)

    On page 63, that meaning of “early” is “before structure formation,” not “near the Planck time.” The mutual self-gravity is not important, so the entropy of the gravitational field is simply not significant. We’ll certainly talk about this much more carefully in Part Four (although it remains a murky subject deep down).

  • Lawrence Kuklinski

    Recombination is the moment when electrons combined with protons(and other nuclei) once and for all. This is from the Glossary of Dark Matter etc Teaching Co.
    If this is the first attempt at Recombination than why not call it “Combination”. The prefix “re”-combination to me implies that there was a prior commbination which did not complete or decayed and there for another combination or recombination had to happen.
    It sounds like I am being nitpicking on semantics but I still remain opaque in my understanding.

    Thank you.

  • Jolyon Bloomfield

    Now, just convincing myself that I remember the terminology correctly, following recombination (and a stint in the dark ages), we have reionization, where the recombined stuff is ionized for the first time thanks to the first stars starting to put out ionizing radiation. Is there a re-recombination following that? (something niggles at me telling me “duh, of course, that’s where the CMB comes from!”, surface of last scattering or something?)


  • Susan

    On page 50 you say—So if we extrapolate the universe backward in time to a state that was denser than it is today, we would expect such an extrapolation to continue to be good; in other words, there’s no reason to expect any sort of “bounce.”— What does bounce mean in this context?

  • Wondering

    @12 Lawrence

    This page:
    agrees exactly with you and calls ReCombination a misnomer for the reason you state.

    However, i think i remember reading that everything is in an active or rapid flux of combining/re-ionizing just before the onset of the recombination era, so it might be fair to say that for any given atom there were ‘combinations’, however brief, until the ‘last one’ which is the REcombination.

    Don’t quote me though, lol.

  • Sean

    The actual reason why it’s called “recombination” is because astrophysicists always used that word to mean “when an electron joined with a nucleus.” It’s less accurate in this context, because there was no prior era when all the atoms were combined. However, an individual electron actually combines with a proton and then gets dissociated again many times before it finally settles down; so if you really want to be a nitpicker, “re”-combination is still perfectly accurate.

    Jolyon– The CMB does come from the surface of last scattering, at a time around 380,000 years after the Big Bang, which is the period of recombination we’re talking about right now. After stars light up there is a period of (somewhat incomplete) reionization, which we’re still in today. That has a subtle effect on the CMB, but not a huge one.

    Susan– By a “bounce” I mean the idea that the universe started out contracting, then stopped, then started expanding again into what we currently see as the Big Bang. We’ll talk about that more in the final chapter of the book.

  • J

    I have 2 questions: In note 48 you explain that the vacuum energy is high entropy so we can’t get much useful work out of it (I assume that’s why the conservation of energy seems so ironclad even though it apparently isn’t), but how is it then that the universe itself gets so much useful work out of it? I mean accelerating galaxies around seems like work to me.

    The second question is do cosmologists ever play around with W=10^88. I mean do they ever go, “what if it was 10^66 or 10^95 at the boundary?” and then evolve a universe forward from there?

  • Sean

    You can’t get work out of the vacuum energy in the sense that you can’t convert it into any other kind of energy. (At least, if it is the truly lowest-energy vacuum; a false vacuum can decay into matter and radiation with higher entropy.) It does cause particles to accelerate away, but the vacuum energy itself doesn’t dissipate in the process — you’re not extracting energy from it.

    The 10^88 number isn’t very fundamental; it’s an artifact of the current age of the universe. That number is the entropy in matter + radiation within a comoving volume equal to the current Hubble volume. At an earlier time the Hubble volume would have been smaller, and “the matter + radiation entropy within the observable universe” would also have been correspondingly smaller.

  • Pope Maledict XVI

    It was very kind of you to describe all of those reviews as “substantive”. But it is possible to be *too* kind. In particular, that New Scientist review was laughable. The conclusion: “We shrug and say, that’s just the way it is. Sometimes it is best not to scratch explanatory itches.” This from someone who claims to be a professional philosopher. I wonder if he uses that line each time an undergraduate asks him a question he can’t answer. Jeez. Once again NS sets the upper limit to which lameness can aspire.

  • Sean

    Describing something as substantive doesn’t mean I agree with it. Craig Callender is a respected philosopher who has written good articles on the philosophy of time, but he has this one idiosyncratic opinion that the low entropy of the early universe is simply a brute fact to be accepted rather than explained. I find it weird, but it takes all kinds.

  • Bee

    Hi Sean:
    Thanks for the reference, I’ll check that out. Reg the C-field, ah. Hadn’t been clear to me from what you wrote it’s accelerated expansion indeed. Best,


  • Spaceman Spiff

    A tiny nit to pick. On page 61 (and similarly noted within Note #45) it says that neutron stars are “stars that used to be white dwarfs but collapsed further under the pull of gravity.”

    The standard wisdom is that neutron stars form via the collapse of the electron degenerate iron-rich cores of massive stars (~10-25 solar masses at birth), initiating their demise, largely instigated by densities reached that energetically favor the iron undergoing electron capture (with the extra resulting neutrons stabilized by the Fermi electron sea), thus destroying hydrostatic equilibrium within the core. I have never seen a reference to this pre-collapse iron core as a white dwarf.

    In the scenario that produces Type Ia supernovae mentioned in this book, the carbon/oxygen white dwarf takes on mass accreted from its normal stellar binary partner. As the mass approaches the Chandrasekhar limit, the carbon density in the deep interior exceeds those necessary to induce pycnonuclear reactions of the carbon, resulting in a run-away thermonuclear bomb (it is my understanding that the critical density for pycnonuclear reactions is beneath that for electron capture in the case of carbon). GR effects also soften the equation of state in white dwarf stars nearing the Chandrasekhar mass limit, and I am sure the above is a skeletal description at best. Maybe an expert out there will chime in. In any case, the detonation models predict and the observations reveal that Type 1a supernovae do not leave behind compact remnants.

    In a nutshell, white dwarf stars do not become neutron stars, at least not in any of the standard models.

    Nevertheless, the reading has been great fun!

  • Spaceman Spiff

    The commonly made statement (in the popular science literature, but even in astrophysics colloquia) of how the “theoretical expectations” of the energy density in the quantum vacuum exceeds that which we measure in dark energy by “120 orders of magnitude”, which then leads to the question as to why the “vacuum energy is so much smaller than it should be”, is puzzling to me. The problem would seem to be misplaced. If the energy density in the quantum vacuum were anywhere near as large as predicted, we wouldn’t be around to ponder the problem — the universe would have long ago rapidly inflated to oblivion. Right? Isn’t it more the case that we ought to reconsider/ponder more deeply what we think we know about vacuum energy? Because all of our estimates would seem to be non-starters for the universe we find ourselves in. This issue must have come up over some beers, or something.


  • Sean

    Spaceman Spiff– A fair complaint about the neutron stars. White dwarfs become neutron stars in parameter space, but not as a function of time.

    About vacuum energy, I think you’re not disagreeing with the conventional wisdom. The expectation from our understanding of effective field theory is that the vacuum energy should be at the Planck scale. But it’s not, so there is something incomplete/incorrect about effective field theory in this particular case. It’s also true that if the expectation were correct, we wouldn’t be here talking about it. That fact may or may not play a role in explaining the disagreement.

  • Philoponus

    If I follow the arithmetic on page 63 correctly, the increase in entropy in our universe is mostly due, not to the expansion of our universe, to the proliferation of super-massive black holes. The early universe had none(?), and now we have 100 billion of these critters (one per galaxy). So is the increasing entropy of our universe fundamentally a question of why we evolved so many large black holes?

  • Pingback: From Eternity to Book Club: Chapter Three « JackTales()

  • Sean

    Philoponus– Black holes are important, but they’re not the whole story; they’re simply the end-state of gravitational collapse whose entropy is easy to calculate (and extremely large). The evolution of black holes is part of a more comprehensive story, in which high-entropy states at high density are very lumpy, but the real early universe is very smooth. (Note the question is certainly not “why we evolved so many black holes,” which is easy to answer — it’s “why weren’t all sorts of black [or white] holes to begin with?”)

  • Lorianna

    I am a complete novice so pardon me if this question seems silly. I’m going to keep the question shorter than I want to since I don’t really know what I’m talking about. You can thank me later for that. Why do physicists think the graviton exists and if it does what does that mean for general relativity? Is gravity a force or not, or both?

  • Sean

    Lorianna, this probably isn’t the best place to get into this topic. But roughly: general relativity predicts gravitational waves, and quantum mechanics is correct, and quantum mechanics says that when your classical theory has waves your quantum theory has particles. So essentially every physicist who has thought carefully about the question believes that gravitons exist. It doesn’t mean anything at all for general relativity, which is only a classical theory. Whether or not you call gravity “a force” or not is largely a matter of taste; the GR perspective suggests that you don’t have to think of it as a force, but it doesn’t really matter.

  • Fill

    I tried to skim the comments to see if my question had already been asked, but I didn’t see it. So apologies in advance if this is a retread.

    Why is it that the early universe, which was essentially a homogeneous(to 1 part in 100,000) mixture of elementary particles and radiation was extraordinarily low in entropy? Shouldn’t that be a high entropy configuration? Is the low entropy due to the unification of the forces or the amount or the free energy in the system?

    How much of the increase in entropy is simply due to the increase in size of the universe?

  • Sean

    Fill, we’ll talk about this more in Part Four. But the increase in entropy is not due to the expansion of the universe; the universe can be high-entropy at any size, although the configurations will look different. At high density, when gravity is important, a high-entropy configuration will look very non-smooth. Think of a collapsing universe; there’s no reason at all to expect structure to smooth out.

  • Fill

    Yeah that actually makes perfect sense.

  • RantingNerd

    Sorry to be late catching up here (I just got the book a few days ago :-)).

    If lumpy = low entropy for the universe, then how does a cloud of intersteallar hydrogen (which is fairly smooth) collapse into a higher-entropy star, which is much lumpier? (I am clearly missing something here, but Stat Mech was always my weakest subject.)

  • Sean

    Lumpy means *high* entropy, if the density is high enough. Once the universe expands (quite far in the future), high entropy will be smooth again.

  • sonasi

    Why is Sean doing his book club on the cosmic variance blog? I vote for way less Sean and more of everyone else.


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Cosmic Variance

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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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