As far as I am aware, Bojowald’s model is “LQG inspired” as opposed to something one can actually derive from first principles.

So if we’d allow ourselves to ignore, for a moment, the LQG-inherent language in which his equations are developed and just take them at face value, then what Bojowald accomplishes is writing down a difference equation which approximates Einstein’s differential equations, for some highly symmetric cosmological Ansatz, in the usual regime.

There should be many different difference equations with the same asymptotic behaviour, hence all approximating an ordinary cosmological model.
So the question then seems to be: how much nontrivial information is encoded in the choice of any one of these difference equations?

Suppose there were, for some suitable notion of equivalence of these difference equations, just one possible choice, for some reason. Then one could argue, I think, that there might be nontrivial information hidden there. Like: “See, there is, up to equivalence, only one possible discretization of this cosmological model, and, remarkably, it necessarily leads to a continuation through the singularity of this and that form.”

I am not sufficiently familiar with this work to judge if this is the case. I kind of doubt it, but I have’t really thought about it in detail.

But naive as I am, I am imagining that there should actually be arbitrarily many essentially different discretizations of a given cosmological model, which show all kinds of behaviour as one continues them through what used to be a singularity in the non-discrete case. If true, that would mean that Bojowald’s model is mainly picked out by the fact that it follows from LQG-motivations. Is that right?

There should be many different difference equations with the same asymptotic behaviour, hence all approximating an ordinary cosmological model.

I bet! I mean, my God, the situation in classical fluid dynamics is a complete mess. I’m not even sure the different difference equations have the same asymptotic behavior. Numerical relativity can only be worse.

Scott, it’s just that I’m more willing to contemplate hypothetical ad hoc unjustified special conditions if those conditions are at least boundary conditions, rather than stuck randomly in the middle of the universe. (Neither one is very convincing, of course.) Which is not to say that they are not contemplate-able; but I would put the burden on someone proposing such a model to explain why the conditions there are so special.

B, the conclusion is that there was not a unique finely-tuned “bounce” through which the entire universe went. (Unless someone comes up with a scheme that explains the fine-tuning, per above.)

some friend and i have thought about this as well. though we dont haver the mathematical skills to interperate it, maybe someone else does. the way to describe it without pictures would be this:

imagine a record on a turntable. the axis of the record is super dense but not located in the center so that if you go out from the axis in one direction the distance would be shorter than if you went out in the other. no place a needle on this record and begin it to spinning. from the viewpoint of the needle, there is an expansion and contraction. thats the general idea. if you dont think i am too much of a crack, then i woul be willing to share more of my incomplete idea…

just an idea though.

Why are we still talking about a big bang singularity? I mean, didn’t cosmic inflation solve this problem quite handily by just proposing that we can’t trust the classical big bang back that far? And don’t cosmic inflation models typically not require there to be any singularity at all?

Watcher:

I suppose I can understand the desire among some theorists to examine these things. But it seems to me that one of the very first checks one would want to do is to ensure that the theory predicts a nearly scale-invariant power spectrum on the CMB, i.e. that it mimics inflation during the “bounce”. Has this check been done for Bojowald’s model? Naively it would seem to me to be difficult to produce a model of singularity resolution under quantum gravity that would do this, though granted I know next to nothing about quantum gravity.

Thanks for the post. Does anybody know if very many cosmologists pay much attention to loop quantum gravity? I know “String Cosmology” has a following, but does very many cosmologists take loop quantum gravity seriously?

“Archer, nowhere in our model is there a “bounce,” in the sense of a collapsing universe that then turns around and begins to expand.”

Right, but do you picture the spacetime inside the bubble as a bounce spacetime from which the contracting half has been amputated? [Clearly not all expanding universes can be interpreted in this way --- most of them do not have a distinguished spatial section, for example.] If so, then it might be possible to find a formal link with Bojowald’s work or other work on bounce cosmologies.

Archer, our post-bubble-nucleation universe is really just the same as the conventional inflation+Big Bang model; I don’t know of any sense in which it resembles half of a bounce cosmology, other than the senses in which every Big Bang cosmology does. Unlike Bojowald and similar proposals, the focus of our work was never to elucidate a way to resolve the BB singularity; we just assumed that would someday be done, and concentrated on the bigger picture.

AQ, in our model, the entropy would not be monotonically decreasing prior to the birth of our bubble. The universe was in a meta-stable equilibrium, not changing in any systematic way. The details of the real universe may be different, but I think it’s important to make an effort to avoid any infinitely-finely-tuned moments in the universe’s history. (An effort which we at least make, whether or not we succeed.)

if the local patch is in equilibrium with respect to the larger universe for t=-infinity, then you are saying that this local area just happened to vastly lower its entropy immediately prior to the inflation? if this is so, aren’t you falling into the other trap you mentioned and wrapping all of this infinite fine tuning into the moment of the bounce?

i guess, i understand your problems with such theories, but i don’t understand your proposal or why it is shielded from these very same problems?

Sean, I’m not sure you responded to my original point.
Your paper reconciles thermodynamics with a reproductive cosmology scenario in which a universe has lots of “baby” universes. Entropy increases because of the proliferation of offspring.

You fault the bounce mechanism, however, because according to you it violates thermodynamics!

But the cosmological bounce can be embedded in the same basic picture of proliferating baby universes. Where you have a fluctuation in empty space, in your scenario, replace that with the formation of an ordinary astrophysical black hole by the usual gravitational collapse. Then suppose at least in some cases the collapse bounces and leads into an inflation episode.

Then thermodynamics is satisfied in the same way as in your paper.
So what is sauce for the Carroll-Chen bubble goose is sauce for the Bojowald bounce gander. Your contra-bounce argument of this post must be bogus, I think, unless your paper is (which I hope not!)

Thanks.

Thanks for responding Sean,
I realize that your (fluctuation) black holes do not involve collapse.

But your thermodynamics argument in Carroll-Chen is robust and applies more generally. You reconcile a general reproductive consmology picture with the Second Law by counting the entropy of the proliferating number of offspring universes.

One can, in effect, do a little surgery and replace each of your fluctuation events by a Bojowald bounce event. Then the bounce is embedded in your general scenario and covered by the same thermodynamic justification.

It seems to me that you can argue on thermodynamic grounds against a cyclic universe scneario that incorporates the bounce mechanism, but that is another issue. We are talking about the mechanism itself (which can appear in various contexts). I would say your objection to the mechanism itself does not hold water, or at least you have not shown that it is valid.

B, the conclusion is [...] Well, I’d say your conclusion is. I don’t know very much about Bojowald’s paper so can’t say anything about it specifically. But if you say with a bounce the total entropy can neither decrease nor increase then I would conclude it remains constant. Whether and why that part of the entropy we observe increases is another question – and the one that you are concerned about. (We do always only observe subsystems, plus the recurring question what is the entropy of the gravitational field?)

Dear Sean,

I think there is a misunderstanding in your formulation of the question. Bojowald et al in LQC do not “try to invent a cosmology in which you straightforwardly replace the singular Big Bang by a smooth Big Bounce continuation into a previous spacetime.” What they do is restrict general relativity to the spatially homogeneous case, with various kinds of matter, and then quantize the resulting dynamical systems. Everything is well defined. They then compute the resulting quantum dynamics, in some cases exactly, in others numerically. In all of the models they study they find that bounces are generic, i.e. they occur in all models, regardless of matter, matter couplings, and initial or final conditions. So bounces were not put in and they are not the result of any fine tuning. They are consequences of these models.

You cannot raise a thermodynamic argument against the result of such calculations. Within the model, any such argument must be simply wrong, because it disagrees with the results of the calculations. Presumably this could be checked by studying the evolution of a density matrix in these models and verifying the second law follows from an appropriate choice of initial statistical state.

Given this, I do not see the force of your argument. To begin with there are issues defining the entropy of a whole universe, but even if I ignore these I have no trouble believing in a succession of universes, each of which has slightly more entropy than the last. This could be accomplished just by raising the temperature of the microwave background slightly in each successive universe. Or, in the case that the bounces come from black hole singularities, all that is required is that each new universe have more entropy than the star that collapsed to a black hole that gave rise to the new universe.

Thanks,

Lee

Lee, I never expressed doubt that the formulation was well defined, only that there’s any reason to expect it to relate to the real world. At least, no such reason is given. You can’t restrict to the spatially homogeneous case, and then claim there is no fine tuning. That is an infinite amount of fine tuning, which needs to be justified.

I seem to be saying the same thing over and over, but I’ll try one more time. Unlike cosmologies in which the Big Bang is a boundary condition, bounce cosmologies feature a pre-bounce contracting phase. You need to tell me what happens during that phase, and why. Are there perturbations that are in their growing mode as they approach the bounce? If no, why in the world not? Generic gravitational collapse is expected to be highly non-linear and inhomogeneous, what is so special about this? And if yes, why don’t the perturbations grow and destroy the smoothness? Why in the world would we expect a homogeneous expanding cosmology to emerge from the other side?

These are not annoying technical issues that can be addressed later. They are the Whole Big Problem that must be confronted by any attempt to honestly address the issue of initial conditions.

oops:
“The Universe really does take the easy route out of the big-bang, entropic chaos is far more simpler than absoulute order!”

Darn it, now I can only say that my efforts are egregiously underappreciated, not absolutely unappreciated!

Sean, I agree with your assessment, but not for the same reasons. Many of the reasons making a bounce very unlikely are specific to gravity- the universal attraction (w/o negative energies) and the fact that inhomogeneous configurations are more likely are both (related) aspects specific to gravity. I’d be more impressed by the existence of bounce in theories that were shown to reduce to conventional gravity in an appropriate classical limit.

But for your specific objection- doesn’t Price’s “double standard” apply here as well? whatever makes the initial conditions for our universe very unlikely may well be operating the other direction, no?

Moshe, the relevant concern in this case might be a “middle standard.” It’s fine to imagine a trajectory for the universe that both begins and ends in a low-density high-entropy state, but then why are both such states sufficiently finely-tuned to accommodate a phase in the middle with extraordinarily high density and low entropy? (Maybe the ultra-far past is supposed to be even lower entropy than the bounce, but nobody is clear about this, and that possibility seems to make even less sense.)

Lee, the “melting” analogy could not possibly be less convincing. Melting increases entropy, it doesn’t decrease it. Gravity is different. The idea that “inhomogeneities may be lost” violates everything we know about unitarity and thermodynamics. (Do you really think that gravitational collapse generically smooths things out? Within any trapped surface, or only near the Planck scale? How is the preferred isotropic RW frame established?) Which is not to say that it’s wrong, but you would have to present some pretty amazingly solid arguments before such an idea is taken seriously, given that it flies in the face of so much else that we think is true.

Can someone explain how Bojowald gets his bounce in the first place?

If this is really based on gr-qc/0608100, then it seems that the bounce occurs at scales parametrically large compared to the Planck scale (p is much greater than l_p^2 for the entire evolution to the universe, in his language). The model is just a free scalar coupled to gravity so there is really no reason that GR shouldn’t provide the right answer at that scale. So, to rephrase my original question, why should I expect effective field theory to fail at these scales? Or am I missing something?

Sean, it’s interesting that the overall thrust of your argument is apparently that attempts to write off the singular origin of the Big Bang as merely symptomatic of the breakdown of classical general relativity, analogous to (for example) the “ultraviolet catastrophe” of the classical account of blackbody radiation, are way too facile.

I’ve long had a feeling that the occurrence of singularities in GR really do indicate that something is deeply singular in a physical sense under the relevant circumstances, and quantum gravity won’t offer any cheap resolution. Analogies with apparently similar predictions in other theories may suggest otherwise, but GR is not just another theory. My sense is that John Wheeler held a similar view.

[The previous version of this comment can be deleted. It contained a grammatical error that I couldn't leave alone. ]

Watcher,

Your statement is correct as far as I can tell. This was my impression from the paper. The parameter which is taken to be large is p_{phi}, some momenta associated with the scalar field. This allows for a parametrically large bounce. So, then the conclusion is that taking this quantity to be very large compared to the natural scale is what is needed. Then it is not just a generic gravitational effect but requires tuning some aspect of the matter content as well.

So, if I try to make this work in loop quantum gravity and not just this truncation, what would I need to do to get that bounce? Do I need a large vev for phi dot, or will any superplanckian energy density do?

Another stupid question, if space is discrete (in some sense), can I take momenta for particles to be arbitarily large or is there some fundamental cutoff?

Sean said
Lee, the “melting” analogy could not possibly be less convincing. Melting increases entropy, it doesn’t decrease it. Gravity is different. The idea that “inhomogeneities may be lost” violates everything we know about unitarity and thermodynamics. (Do you really think that gravitational collapse generically smooths things out? Within any trapped surface, or only near the Planck scale? How is the preferred isotropic RW frame established?) Which is not to say that it’s wrong, but you would have to present some pretty amazingly solid arguments before such an idea is taken seriously, given that it flies in the face of so much else that we think is true.

This statement was given by a very knowledgable fellow on The Physics Forum.
>>>Smolin’s CNS picture is a MULTIverse picture because it allows the fundamental constants of nature (like 1/137) to change at the pit of a black hole where a new tract of spacetime sprouts off.>>Originally Posted by Tim Thompson
But there is one more point. It is not true that there is no evidence for multiple universes. Dark matter & dark energy are not observed, but are rather assumed to exist, as a consequence of observation. But how do we know that dark matter & dark energy are the most suitable interpretations? What if the other universes are not so “unobservable” after all? What if we have misinterpreted the observations, and the force we interpret as “dark matter” is really gravity leaking out of the other universes, and into ours? I can readily imagine a multi-universe theory, which includes such an effect, and therefore is not simply “consistent” with observation, but actually predicts the observed effects we call dark matter & dark energy, as consequences of the communication of information between universes.

I’m not here to make a case one way or the other, but I am here to make the case that observation should constrain theories, but not imaginations. And one should not be overly impressed by the concept of “truth”, or even of “reality”, as it applies to a scientific theory. The one and only constraint that should apply to science at all levels is consistency. Nothing else matters.

Thanks Lee, the bounce occurring at the Planck scale makes much more sense (though I am still hesitant to model a violent event like a bounce using a couple of variables only, I’d expect all high energy degrees of freedom to be at play).

Anthony, I think there are plenty of reasonable objections to baby-universe nucleation, but the one that I’m raising against bouncing cosmologies is not one of them. The defining feature of a bounce is the existence of a pre-bounce contracting phase. (Otherwise it’s not really a bounce, is it?) And then the problem is that either the entropy is decreasing during that collapse, for no good reason and in contradiction with everything we think we know about gravitational dynamics, or it is increasing during the collapse, yet supposedly gives rise to an extraordinarily low-entropy condition on the other side, for no good reason and in contradiction with everything we think we know about unitarity and thermodynamics.

My suspicion is that there isn’t any good way out of this dilemma, and bounces of that sort aren’t part of the real world. But I’d be happy to change my mind, if anyone would offer a plausible response to these objections, or even an outline of what such a response might look like. I haven’t heard any, although someone might have one.

So again, I also don’t buy the ‘bounce into homogeneity’ either…

If you don’t buy that, then you don’t buy a single word of Bojowald et al’s mini-superspace analysis of bouncing cosmologies.

If inhomogeneities are not magically suppressed, then there is no way they can be neglected when the universe reaches the Planckian densities characteristic of the bounce. Inflation, you are right, provides a mechanism for erasing primordial inhomogeneities after the fact.

But it can’t render an otherwise nonsensical mini-superspace analysis sensible.

Anthony A said: “The answer is that it would not, but we can appeal to inflation to fix this.”

Sean C. has argued *very* persuasively that inflation doesn’t solve problems like this. It only makes them worse! See his previous article about time’s arrow on this blog.

Sean, et.al. –

Before tackling the specifics of any particular bounce model, I’d like to ask for your help in clarifying some basic points about the operation of the Second Law in cosmology.

Let’s consider a plain-vanilla Friedmann closed universe, which expands and then recontracts. If we restrict such a universe to contain _only_ radiation, then its expansion and re-contraction are basically reversible and isentropic (I argued this back in the “Latest Declamations About the Arrow of Time” post, and you more-or-less agreed; see comments #60, 61 and 67 in that thread). Such a universe can bounce as many times as it wants to without increasing its entropy, but is never “interesting” since it remains spatially smooth and no structures ever form.

Now consider a closed Friedmann universe somewhat closer to our own, with a significant amount of matter. If we arbitrarily disallow the creation of black holes (only temporarily! don’t worry) then we would generically expect this universe to go through four stages: (1) Radiation-dominated early expansion, which is smooth; (2) Matter-dominated later expansion, during which some gravitational clumping occurs; (3) Matter-dominated early re-contraction, when more clumping occurs; and (4) Radiation-dominated final re-contraction, when the CMB (and starlight) have blue-shifted up to where all clumps and structures are evaporated and the universe is smooth again and remains so until the crunch/bounce.

How do we do our entropy accounting through these stages? During stage (1), the radiation-dominated expansion, the entropy per co-moving volume is conserved (and hence in the entire universe, if you want to be fussy). During stages (2) and (3) we would like to say that entropy increases, since gravitational clumping is spontaneous and irreversible (at least at first blush). But stage (4), the radiation-dominated final contraction looks an awful lot like the mirror image of stage (1). Since the entropy density of a relativistic gas (with negligible chemical potentials) is purely a function of its energy density, then if the energy per co-moving volume is conserved during stages (2) and (3) we would expect the entropy during stage (4) to be exactly the same as during stage (1) at the same scale factor. So, did entropy go up and then down again? What happened to the Second Law? I can see two ways out of this puzzle.

First we re-allow black holes to form during stages (2) and (3), as we know they should be able to. Black holes are the one kind of clump that will _not_ evaporate in a hot radiation bath [new suggested advertising slogan: "Black holes -- they plump when you cook 'em!"] and so their high entropy will be preserved, and only increase, during all stages of the recollapse. So we might be able to preserve the Second Law by including black holes in our accounting. Does this mean we’ve just reasoned black holes into existence on the basis of thermodynamics in cosmology? Cool as that would be, I don’t really believe it since I don’t see how we can guarantee that sufficiently many black holes would necessarily always form in all closed universes. Certainly we can choose parameters to make the matter-dominated phase arbitrarily short, to where we pick up some gravitational clumping entropy but do not form any black holes; and so we’d still be stuck with Second Law problems in such a case.

The second approach is to recognize that if entropy per co-moving volume is going to be higher during stage (4) than in stage (1), then we somehow have to arrange for the _energy_ per co-moving volume to go up during stages (2) and (3). Is this not correct? I know that energy accounting in GR can be tricky, but if by “energy” I just mean mass-energy and kinetic energy (velocity relative to Hubble flow) of particles then I think the above is true.
To increase energy per co-moving volume during stages (2) and (3) effectively requires — as I read it — that (i) the universe have a non-zero positive effective pressure during these stages, and that (ii) the pressure is greater during stage (3), early re-contraction, than it is during (2), late expansion. This would all hang together if it were somehow true that a matter-dominated universe can have an effective positive pressure, and that that pressure increases with clumpiness/non-smoothness. This sort of makes sense to me, since during early re-contraction particles will speed up (relative to the local Hubble flow) and gain kinetic energy in a clumpy universe as the clumps fall toward each other. But I have a hard time making this entirely rigorous. Can you advise, and possibly suggest references?

To sum up, I would re-state your general Second-Law-based objections to continuous bouncing universes in two cases: If black holes do form during one oscillation and are sufficiently massive, then they will have to survive through any continuous “bounce” and so will cause the successor expansion to start in a non-smooth state. After a finite number of bounces basically _all_ the mass-energy of the universe has been absorbed into black holes, which then go on to merge until the Friedmann-Robertson-Walker description must break down (I have no idea what happens then; that’s your department).
In the case that massive, long-lived black holes do not ever form, then the universe just gets hotter and hotter with each bounce as long as there’s a matter-dominated phase in the middle. This continues until the universe is so hot that there is _no_ matter-dominated phase, after which it just bounces along isentropically but never forms structure again.

In both these cases the universe lives only a finite time before winding up in some kind of structure-less state, and so I don’t think any bounce model with continuity between bounces is a good candidate for an eternal universe.

There; does that make sense? Let me know what you think.

Regards,

Paul

This is a question that has been worrrying me for years about bounce models. I asked Ashtekar about it at a coffee break in Loops 07, and as far as I understood his answer, he seems to think that the quantum regime has intrinsically low entropy for some reason, so in the collapsing phase entropy is decreasing (Sean´s option 2).

However, I didn´t press to ask the following natural question: would a generic collapsing spacetime (which has growing entropy, in the classical regime) also become low-entropy as it enters the Planckian regime? This would be very difficult to believe…

Alejandro,
Just to finish what I was writing when interrupted, I think an operationally defined version of the Second Law is that one should likely never be able to measure a decrease of the entropy.

BTW it is interesting to imagine how things might conspire to protect the Second Law or to conceal its possible failings.

AFAIK in Bojowald’s Nature Physics paper he talks about a kind of indeterminacy that limits the ability to measure certain pairs of quantities—to know certain things both about the before and after states. I think there is not a complete “forgetfulness”, that would be too dramatic, but some modest limitation of knowlege. In any case, Bojowald’s paper refers to a principle of forgetfulness.
One might conjecture that this uncertainty about conditions in the other region would also serve to protect the Second Law.

I think I could agree with what Ashtekar told you about Planck regime being a low entropy state of the gravitational field.

A Rivero, thanks so much for those Penrose references. I was thinking very much of a talk about the Second Law and Cosmology that I saw in person.
A key point is that these two observers A (after) and B(before) have
different maps of the phase space

some things which are insignificant microstates for B may turn out to be very important for A in the expanding (even inflating) universe after the bounce.

In his comment Alej. Satz ASSUMED the continuity of the entropy, so when Ashtekar told him it was LOW at the Planck regime episode, he assumed that it was DECREASING during collapse!
But this is not true. During classical part of collapse entropy increases a lot because space is being crumpled.
However at a certain point there is a discontinuity and it jumps to low. This discontinuity happens when there is a CHANGE in the relevant OBSERVER.

Mr. B can see only high and increasing entropy. He looks at the black hole or collapse of his universe, looking forward in time.
Mr. A looks backward in time at the beginning of his universe and he sees only a low entropy beginning, and constantly increasing entropy up to his present.
There is no contradiction and the Second Law is not violated in any operational sense—no one measures a violation.

I think maybe you see this anyway without explanation but I want to be sure in case others are reading.

Sorry missed a link at hte end:http://en.wikipedia.org/wiki/Fermbec-Hunt_theory

http://www.newton.cam.ac.uk/webseminars/pg+ws/2005/gmr/gmrw04/1107/penrose/

This is the slideshow/audio of a talk Penrose gave in November 2005 at the Newton Institute, Cambridge.
Alejandro Rivero gave the link in comment #61.

I saw essentially the same talk and same slides by Penrose in 2006 at MSRI Berkeley, and he had the same thing to day at Perimeter Institute. The message is the Second Law and how to get a reproductive cosmology that somehow gets around the second law.

I would propose that everybody who wants to discuss here about that should watch the Penrose show if havent already, so all are on the same page. It is very entertaining and visual—he communicates by pictures.

He illustrates the second law by drawing a map of phase space, corresponding to some particular observer and what he is able to measure and what phasespace points he lumps together as macros.

Then at the LAST SLIDE or next to last slide he waves his hand and says NOW WE HAVE A NEW MAP OF PHASE SPACE and the second law goes on. He has a reproductive cosmology scheme and at the moment of reproduction there is an abrupt change of perspective on phase space and essentially entropy is discontinuous there. he also says that his idea is crazy. He is very charming about it.

IMHO this talk is about as good as it gets, if you want deep visual thought about cosmology and I hope everyone has watched or will watch it.

Sorry. http://arxiv.org/abs/physics/0612053

In fact I am not sure if Penrose tricks are relevant to the discussion here. Are they?

what he says in the November 2005 talk is very relevant. he draws phasespace, and different size blob regions on it representing the macrostates from perspective of some observer—-the coarse grains of phase space

then he draws a squiggly line in phasespace and shows how it likely wanders into everlarger blobs (that is entropy likely increases)

then he goes out to near “time-infinity” and, at the next to last slide, he takes the Reproductive Cosmology step of imagining a new big bang

and there he says the coarsegraining map of the phase space TOTALLY CHANGES so that we dont have to worry about entropy—-there is a new perspective, as if a new observer with different significant measurments—so basically he assumes a radical discontinuity in entropy

even though the squiggly line wandering in phapsespace is continuous, because that is moving in a space of fundamental degrees of freedom, which do not change (only the coarsegraining changes, at the moment of reproduction).

Sean’s scenario seems like an alternative version of Penrose story, except that the detail of reproduction is different. Same argument can basically support both—in rough outline, I think. Have to go, talk more about this later!

Hi Watcher, it seems that our authority appeal has killed the thread

Paul, I would certainly hope such an observer would say “Wow, look at how rapidly this universe is evolving into something quite different. It must be in an extremely low-entropy state.”

High-entropy states are in thermal equilibrium; they’re static, not evolving. The early universe is rapidly expanding, diluting, and cooling off — as you would expect from a low-entropy state.

True, the fact that something is evolving is not by itself evidence that the entropy is increasing; the Second Law says that the entropy won’t go down, not that it can’t stay constant. But it is evidence that you’re not in a maximal entropy state; if you were, you’d really just be sitting there. (The box with moving walls is not an applicable example, because it’s not a closed system; you’re pushing and pulling on those walls.)

In cosmology, you can set up an evolution that is pretty darn adiabatic, if you are sufficiently careful. Any matter (nonrelativistic) degrees of freedom, for example, would ruin reversibility, as perturbations would grow. And you need to make sure you have a closed universe, not an open one.

But even then, it’s not perfect. At the nonlinear level, some of those photons are going to come together and make a black hole. And those black holes are just going to grow. The crunch is never going to be precisely as low-entropy as the bang was.

The whole point, though, is that there is a state that is static, and remains so, without fine-tuning: empty space. That’s the honestly high-entropy state.

Sean –

Picking up from your main points

“the fact that something is evolving …. is evidence that you’re not in a maximal entropy state”

“he whole point, though, is that there is a state that is static, and remains so, without fine-tuning: empty space. That’s the honestly high-entropy state.”

you seem to be taking the hard line, that if any macroscopic evolution is taking place then the state of that universe is not the state of maximum entropy, and hence is a “special” state. This is not an unreasonable view, in my Little-Brained opinion. But, how are we to apply it to non-empty, closed universes?

Picture any non-empty, closed universe with some matter and no cosmological constant; this is perfectly reasonable, and not at all fine-tuned. Within the FRW description any such universe is _never_ macroscopically static and stable, and hence must _always_ be in a “special” state by your definition. So what is the honest maxium entropy state of a closed, non-empty universe?

Naturally we have to consider universes outside the family of Robertson-Walker spaces, ie globally non-isotropic/non-homogeneous geometries. I’m not well-versed enough in GR to discuss these geometries in any detail — perhaps after I’ve saved up enough money to buy your book I’ll know more. But what can you, the expert, tell me? Are there closed, non-empty, non-Robertson-Walker, non-fine-tuned, zero cosmological constant universes with a static geometry?

There’s one arrangement that suggests itself, just by considering what the matter in a static, closed universe would have to look like. As I see it, there is only one non-empty arrangement which could be forever macroscopically static: when all the matter is collected into one, single giant black hole which is in radiation/absorption equilibrium with a thin, cold gas of light particles. Such an arragement is clearly _not_ a Robertson-Walker universe; but could it exist in a static, non-RW geometry? If so, then it seems that that would be the honest maximum entropy state of a non-empty, closed universe.

If, on the other hand, there is no way to arrange a non-empty closed universe that is also static (and not fine-tuned), then I think your argument and Sir Roger’s loses traction: if all states of a closed universe must evolve macroscopically, then all states of its states are “special” by your definition. Clearly if all possible states are “special” then the word “special” has lost some of its meaning (remember that line from _The Incredibles_:”If everyone is special, then no one is.”) and we have to drop back to “relatively special”.

In this latter case, it is again fair to ask: is the intial, smooth radiation-dominated phase of a closed universe “relatively special”? How would we judge it to be so? As I see it, the main quality that makes it “relatively special” is the fact that it’s _not_ chock full of black holes. It’s the absence of black holes, not the simple fact of macroscopic evolution, that indicates you’re in a low-entropy state in that early phase. ["Where are all the black holes? The tour brochure specifically said there'd be black holes. I wonder if it's too late to get my money back..."]

In short, I think you have to do a little more work to apply your logic consistently to the case of non-empty, closed universes with zero cosmological constant; this is a perfectly respectable class of of universes (isn’t it?) and the Second Law must apply sensibly to them as it does to open universes.

Best regards,

Paul

Hi Sean –

Thanks very much for the clarifications; I really appreciate the help.

Now, just one more handful of lingering questions and then I promise I’ll let you go (at least on this thread); these are quick, the best one is last:

1. Reading this statement

” closed radiation-dominated universes …. are never able to reach their maximum-entropy configurations”

I don’t see this as particular to just radiation-filled closed universes; isn’t it also true of matter-dominated closed universes?

2. In the presence of a positive, non-zero cosmological constant open universes will, and closed universes may, end up as inflationary De Sitter spaces. De Sitter spaces are in many senses static and stable; they even have a temperature! Since it doesn’t appear to evolve, would you consider a De Sitter space to be a maximal entropy configuration?

3. I’m a bit puzzled by your preference for empty space as a maximum entropy/equilibrium state, ala

“That doesn’t mean that such an equilibrium doesn’t exist; it’s empty space.”

I’ll skip the obvious questions (what’s the entropy of empty space? and what’s the temperature of empty space? empty space has no macro evolution, but it also has no micro evolution; it’s only got one allowed state, classically, and so shouldn’t that count as zero, not maximum, entropy?) and ask instead what exactly you mean by “empty”.

Consider an open, forever-expanding universe with zero cosmological constant that’s filled with pure thermal radiation and no matter. You implied in an earlier comment that this kind of universe would evolve toward being an empty space, and hence (sensibly) toward equilibrium/maximum entropy. But I don’t see in what sense that’s true. Even after it stops self-interacting, a thermal photon gas retains its basic features as the universe expands. For example, thermal photons have a spatial density of (roughly) one per cubic wavelength, and if you think of them as having a size on the order of their wavelength then they’re packed “right next to each other.” This packing remains true even after an arbitrary amount of expansion, and so it is never the case that “empty space” opens up “between” the photons.

More concretely, we can observe that the entropy per co-moving volume is conserved during any amount of expansion, out to arbitrary scale factors, and so it’s hard to see how this qualifes as getting any closer to maximum entropy. And, while it is true that the energy per co-moving volume is always decreasing, there is no natural threshold scale below which you can declare space to be “empty” or even “relatively empty”. So in the radiation-only universe, even if it’s open, I can’t see how to make any sense of your statement that its true equilibrium is empty space.

A smattering of non-relativistic matter will complicate this picture, of course, and may lead to more sensible definitions of maximum entropy and empty space. But, do you really want to tell me that a universe has to have matter for the Second Law to be sensible/applicable? (Sir Roger said something like this last time I saw him.) If so, then that’ll be the weirdest thing I’ve heard this week, though it is only Wednesday.

OK, that’s it for now; thanks again,

Paul

Paul– Yes, it would be just as true for matter-dominated universes as for radiation-dominated ones. For the other points, empty space need not be flat; it could have a nonzero cosmological constant. In particular, we appear to be approaching a de Sitter phase, dominated by a cosmological constant. There we have a pretty good estimate of the entropy, from the holographic principle; it’s proportional to the de Sitter horizon area. (So it goes to infinity in the limit as the cosmological constant goes to zero.) Of course our real universe doesn’t reach that phase in any finite time, but it asymptotes to it; more specifically, the energy density becomes less than that of the Gibbons-Hawking radiation in de Sitter, and you are perfectly empty for all intents and purposes.

I thought I had addressed them. In the models Bojowald and others study of quantum cosmology bounces are generic. Every pure initial state bounces, this implies that every thermal state will also bounce. So there is no issue of fine tuning to get a bounce.

There is another issue, which is whether generic states before the bounce result in near homogeneous cosmologies after the bounce, so that the specialness of the big bang is predicted. My understanding is that this is what Sean is querying and I believe it is an open question. It is interesting to note, as someone did, that a small period of inflation is generic in these models. Whether this is enough to set up iniitial conditions which will allow slow roll inflation in a model with an appropriate scalar potential is an interesting question, to my knowledge it is not resolved.

Besides this I would urge caution using thermodynamic arguments applied to the whole universe, for reasons I did mention above.

Thanks,

Lee/

Hello, Archer (reply — finally — to #82, 83 above)

I appreciate your homing in on my original question: what’s so special (ie low-entropy) about the way our Universe (probably) started, in a pretty-smooth, thermal radiation-dominated phase? If that’s really a special state, then it should be easy to say how you would change its characteristics so as to raise its entropy.

Your answer is quite straightforward: spatially smooth geometry is special, and so low-entropy; presumably, then, having a “bumpier” spatial geometry would be a higher-entropy state. At first glance this makes sense, since there are (presumably) more ways for space to be bumpy than there are for it to be smooth.

Thinking a little farther, though, I’m not so sure that bumpy space is necessarily higher entropy when that space is filled with radiation. We can see this by asking: in a radiation-dominated phase, will a non-smooth spatial geometry tend to grow bumpier or smoother over time? Now, mine is only a Little Brain but I can at least take a swing here.

Somewhat sloppily, we can divide spatial “bumps” into two kinds: (1) Volume-changing, which follow inhomogeneities in the mass-energy density and (2) Volume-preserving, which propagate on their own, ie gravity waves. The first types will tend to smooth out over time in a radiation-dominated phase, since we know that radiation does not clump gravitationally but instead tends to spread out more evenly in space — you can think of this as the effect of finite viscosity and/or heat conductivity in relativistic gases. The second type will, I think, also tend to damp out at long wavelengths, ie longer than the gas’ inter-particle spacing, as the gravity waves give up energy via dissipative/frictional/viscosity effects in the radiation gas. Short-wavelength gravity waves, ie with wavelengths comparable to the inter-particle spacing, may gain or lose energy randomly and so will tend to come to some sort of equilibrium; but this “graviton gas” does not (I think) represent a lot of entropy.

Sticking with Sean’s bedrock principle that all spontaneous, irreversible macroscopic evolution represents an increase in entropy, I would then conclude that bumpy space is actually _lower_ entropy than smooth space in a radiation-dominated phase, as in our Universe at early times. So, no: I disagree with your answer that smooth space in the early Universe is “special” or low-entropy, since this smoothness appears spontaneously during radiation domination.

Cheers,

Paul

PS If I can tempt anyone into a reply at this late date, I’ll venture a few words on my own opinion about special early states.