# Hubble Constant and Entropy

1. Feb 27, 2015

### Jimster41

Do I understand correctly (in general terms) or wildly incorrectly if I imagine that the constant of expansion and the second law of thermodynamics are very closely connected, or even that the constant of expansion is potentially the source of the second law?

2. Feb 27, 2015

### wabbit

I don't know, but I doubt a broken glass would spontaneously reassemble itself if the universe wasn't expanding. Of course I can't test this:)

3. Feb 27, 2015

### phinds

Why is your line of reasoning for this?

4. Feb 27, 2015

### Demystifier

You are wrong. But you are in a good company, because Hawking made a similar mistake (and later called it his biggest mistake) when he concluded that the thermodynamic arrow of time will reverse its direction when (and if) the Universe will start to collapse.

5. Feb 27, 2015

### Jimster41

Well now I need to figure out what I'm getting wrong. I have been imagining that the expansion has an effect on the size of phase space?

6. Feb 27, 2015

### Jimster41

Does the expansion of the universe affect the size of phase space?

I can picture collapse (shrinking phase space?) allowing for entropy increase as long as the phase space (the universe) still contains an additional state that would be higher entropy than current state.

But isn't an expansion of state space a pre-requisite for entropy increase. At some step i doesn't there have to be a phase space incrementally bigger than the one occupied by a system at step i-1? Otherwise probability of state i-1 remains 1 at step i.

Seems a system could move spontaneously from a high entropy to low entropy state in a given state space, then have room to move back toward higher entropy. But that's only a possible interim. At the limit doesn't the probability of maximal entropy win? Doesn't that mean that any system that is increasing in entropy can only be doing so because at some point it was able to move into an expanded state space - one with higher entropy states available.

Lord I am confused.

Last edited: Feb 27, 2015
7. Feb 28, 2015

### Jimster41

So am I "not wrong" if I believe that expansion, or some introduction of additional degrees of freedom, which allow for states of higher entropy, is required for entropy increase. This just seems definitive to me. I hadn't thought about this, but I can see that this does't mean entropy follows expansion lock-step - a large increase in DOF would provide "headroom" for the arrow of time to carry on in the same direction. And if at some point phase space starts getting smaller it would't matter until the available higher entropy phases were exhausted - a situation that in no way sugests a backward running clock...

I thought this was Penrose's conundrum of the great collapse looking like a bunch of colliding "white holes" (penrose' term) rather than a reverse Big Bang.

8. Feb 28, 2015

### wabbit

9. Feb 28, 2015

### marcus

Thanks for the link to Baez easy-going tutorial on entropy-with-gravity-included
http://math.ucr.edu/home/baez/entropy.html
That in turn leads to another fine informal rap on the virial theorem
http://math.ucr.edu/home/baez/virial.html
It's worth repeating: Baez is a really good explainer.

BTW there is something else that by contrast is comparatively subtle and hard to grasp--the observer-dependence of entropy. Especially (since you mentioned gravitational entropy) in a GR context, where the entropy of the gravitational field (geometry) enters the picture.
I don't feel competent to summarize the situation so will just refer to work by prominent people like Don Marolf, Robert Wald, Thanu Padmanabhan.
Wald's and Padmanabhan's earlier papers are cited in this 2003 paper by Marolf et al.
http://arxiv.org/pdf/hep-th/0310022.pdf
http://arxiv.org/abs/hep-th/0310022
Notes on Spacetime Thermodynamics and the Observer-dependence of Entropy

==quote wald's: http://arxiv.org/pdf/gr-qc/9901033v1.pdf ==
The comments in the previous paragraph refer to serious difficulties in defining the notions of gravitational entropy and total entropy in general relativity. However, as I now shall explain, even in the context of quantum field theory on a background spacetime possessing a time translation symmetry— so that the “rigid” structure needed to define the usual notion of entropy of matter is present—there are strong hints from black hole thermodynamics that even our present understanding of the meaning of the “ordinary entropy” of matter is inadequate.
...
...
I believe that the above puzzle suggests that we presently lack the proper conceptual framework with which to think about entropy in the context of general relativity. In any case, it is my belief that the resolution of the above issues will occupy researchers well into the next century, if not well into the next millennium.
==endquote==
This is just my interpretation but I think that the concept of entropy is not absolute, but is relative to the observer and in particular to the observer's MAP OF PHASE SPACE which shows which collections of microstates (are indistinguishable in that observer's experience and) constitute the partition into distinguishable macrostates.
Marolf et al speak of the observer's resolution. The observer's map RESOLVES phase space into macro regions, where the macroscopic variables that he cares about are the same and microscopic differences can be ignored. We cannot expect two observers to agree on the entropy unless they share the same resolution.
But Marolf et al explore a deeper observer-dependence where even two observers that have the same resolution may differ about the entropy.

Basically I suspect the gist of what these researchers are saying is that you cannot expect the "2nd law" to apply if you change observers or switch perspectives too radically. (and we need progress in defining the phase space of geometry so that the entropy of the geometric state can be included in the total entropy.)

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10. Feb 28, 2015

### wabbit

Hah I knew I shouldn't have used that expression or you'd open a whole other can of worms !
But thanks for the pointers : ) and yes Baez rulez

11. Feb 28, 2015

### marcus

Indeed he rulez!
I'm feeling quite hopeful that the problems represented by that can of worms can be resolved (maybe even while-U-wait)
It's not enough to consider just the phase space of a bunch of particles moving around in a rigid geometry. You have to also include the phase space of the geometry they are moving in. I think Jimster started a good thread largely because he starts to grapple with this by bringing in the idea of EXPANSION. How does that affect thermodynamics? We don't have a general relativistic thermodynamics. It's a major outstanding problem, discussed in general terms in this 2012 Rovelli paper http://arxiv.org/abs/1209.0065) and in a bunch of subsequent ones by Rovelli and others.

The most recent one (google "rovelli compact phase space" ) argues that at least in the Euclidean 3d case including a positive cosmological constant makes the phase space of geometry discrete in a certain sense---the corresponding Hilbert space when you quantize becomes finite dimensional.
That's a remarkable idea and it would greatly simplify the thermodynamics. Still must be extended to 4d and to Lorentzian case.

Another good sign is a fat new paper by Hartle and Hertog. What they say is rather similar to some things in recent Rovelli et al papers but they say it resoundingly with considerable grandeur.
Quantum geometry is the real thing. Classical GR is just an approx that works in certain situations and not in others. Quantum gravity may know how to "tunnel" from a black hole to a white. From collapse to expansion. (http://arxiv.org/abs/1502.06770) It appeared Friday, it reminds me a lot of a 2014 Haggard Rovelli paper (http://arxiv.org/abs/1407.0989).

Last edited: Feb 28, 2015
12. Feb 28, 2015

### wabbit

TBH I need to look at this more closely, I know at some point one is supposed to utter the magic words "gravitational deegrees of freedom" but it is still a little frightening to me..,

Edit. I'm trying to think of this in 2+1d, spherical space, empty but for a gas of pointlike massive galaxies, strictly gravitation only.

Last edited: Feb 28, 2015
13. Feb 28, 2015

### ChrisVer

Well that is so badly written (as a form), that my initial positive intentions of reading that article disappeared after the first couple of equations.

14. Feb 28, 2015

### wabbit

I wonder if the "Shape dynamics" approach of Koslowki & al. that you mentioned elsewhere might provide a basis for an interesting alternative way of looking at this ?
http://arxiv.org/abs/1302.6264
http://arxiv.org/abs/1501.03007

15. Feb 28, 2015

### Jimster41

Thanks Marcus.

I'm always trying to shrink this stuff. For completely selfish reasons...
I know it's anathema to progress being made

For better or worse my current cartoon (that I'm trying to check) is:

quanta of spacetime geometry are the open set that is our universe
expansion of the quanta's phase space, creates geometry phase probability gradient, entropy is gravity.
matter and energy (things and all the moving things do) are emergent spacetime geometrical systems, on that gradient

I keep wondering about gravitational clumping backwards where the expansion of phase space makes what was even, a step before, now relatively speaking, an uneven - clump. I'm just struck by picturing it that screwed up way. Is that inconsistent with emergent structure, or not at all? And what does non-local randomness (Bell) have to do with it all... Surely something.

Not going to say I follow the proofs or derivations, in detail, but in terms of grappling with concepts...
Unruh effect: I saw it in Verlinde. It was a key take away from that for me. His entropic gravity depends on it I thought.
Virial Theorem: new to my eyes but, I believe it, and it seems consistent with an answer I got recently from Penrose which is that for matter and energy in a g-field, clumped is higher entropy than evenly distributed. But then I'm confused when he says at the end of the expansion of the universe, matter and energy are all totally distributed, and that's maximum entropy.

16. Feb 28, 2015

### marcus

Hi Jimster, just to be clear about the background to this problem let me quote the first sentence of the abstract of CR's 2012 paper
http://arxiv.org/abs/1209.0065
General relativistic statistical mechanics
Carlo Rovelli
(Submitted on 1 Sep 2012)
Understanding thermodynamics and statistical mechanics in the full general relativistic context is an open problem.
==endquote==
It's sort of important, I think, that it is an open (unsolved) problem. people with a lot of experience and knowhow and intuition haven't gotten it figured out yet. I think progress is being made though. On a fixed rigid non-interactive curved geometry it IS solve, and that can provide useful approximations but it is not how nature works. Anyway that was the first sentence of the abstract. The idea is amplified in the first two sentences of the introduction section
==quote http://arxiv.org/abs/1209.0065==
Thermodynamics and statistical mechanics are powerful and vastly general tools. But their usual formulation works only in the non-general-relativistic limit. Can they be extended to fully general relativistic systems?
==endquote==

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17. Feb 28, 2015

### marcus

Coming at this from another angle there's the recent Hartle Hertog paper. It's apt to be influential as the two of them separately and jointly have co-authored many papers with Stephen Hawking and James Hartle is a big name on his own. I'll quote their conclusions:
http://arxiv.org/abs/1502.06770
Quantum Transitions Between Classical Histories: Bouncing Cosmologies
James Hartle, Thomas Hertog
(Submitted on 24 February, 2015)
36 pages, 6 figures
==quote from conclusions==
It is an inescapable inference from the physics of the last nine decades that quantum mechanics is fundamental and that classical physics is an approximation to it that emerges only in certain limited circumstances. Classical physics is an approximation that holds when the probabilities predicted by a system’s quantum state are high for histories exhibiting correlations in time governed by classical deterministic laws.

In particular, classical spacetime is an approximation in a quantum theory of gravity holding in limited circumstances specified by the quantum state. We should therefore not generally assume classical spacetime, or classical backgrounds. Rather we should assume a quantum state and derive when and where the classical approximation holds in a background independent manner.

It is common to assume that the classical approximation to quantum mechanics holds until the classical equations become singular or Planck scale physics is predicted by them. A lesson of this paper is that this assumption is not generally reliable. In the examples of barrier penetration, the growth of fluctuations, and bouncing universes classical prediction breaks down in regions of configuration space where the classical equations remain well defined...
==endquote==
It looks to me as if Hartle and Hertog, in their treatment of BH, are considering black tunneling to white hole---like a kind of barrier penetration---with some nonzero probability. This possible BH explosion before the conventional Hawking evaporation time is up. Likewise in a cosmological bounce they consider different expanding universe outcomes might have different quantum probabilities. Some of their diagrams or figures are intended to illustrate what they have in mind.

Last edited: Feb 28, 2015
18. Feb 28, 2015

### marcus

I'm hopeful that Wald's estimate that it might take many decades for physicists to understand thermodynamics (and entropy in particular) in a quantum and general covariant (i.e. general relativistic) context is pessimistic. The reason I'm hopeful is the Rovelli Vidotto paper I mentioned back in post#11.
It's the one you get by googling "rovelli compact phase".
The idea is that not only does a system of particles have a phase space representing the individual positions and momenta but also geometry itself has a phase space and thanks to the intrinsic curvature constant this phase space is compact and upon passing over to quantum states acquires a certain discreteness reflected in the geometry's Hilbert space of states being finite dimensional.
http://arxiv.org/abs/1502.00278
Compact phase space, cosmological constant, discrete time
Carlo Rovelli, Francesca Vidotto
(Submitted on 1 Feb 2015)
We study the quantization of geometry in the presence of a cosmological constant, using a discretization with constant-curvature simplices. Phase space turns out to be compact and the Hilbert space finite dimensional for each link. Not only the intrinsic, but also the extrinsic geometry turns out to be discrete, pointing to discreetness of time, in addition to space. We work in 2+1 dimensions, but these results may be relevant also for the physical 3+1 case.
6 pages

Last edited: Feb 28, 2015
19. Mar 1, 2015

### Jimster41

I understand. It is helpful confirmation that the cartoon I have is a good one (for daydreaming about) if I know it includes real puzzles, and isn't predicated on some obvious misconception.

My interpretation of the above quote and the first couple of pages, is that there is not yet a proven model for ensemble treatment of geometic quanta. It could be quite different from the model for particles. So I think I'm hearing you. I can't wait to hear more about what is discovered.

Some unsanctioned thought w/respect to effect on my cartoon's utility (for me). 1. If particles are emergent from s.t. geometric quanta-ensembles, then surely there is some meaningful similarity. If the classical case is contained by the fundamental case, then classical is a useful proxy in the mean. (I know that's a weak use of the cartoon to prop itself up). 2. But what about the non-classical limits and thereabouts? I get it, you can't rely on the classical statistical Thermo model. I feel immensely lost in the limits anyway... black/white holes, beginning and ending of cosmological cycles (bounces), entangled events... Being able to ruminate (while reading on the subway) about these puzzles pieces is the purpose of the cartoon in the first place. So that's okay. 3. Most importantly, and this is a big question I would love help on, I see Louiville's theorem and the basic notion of "entropy" as as a fundamental mathematical/logical dynamic/property of any phase space. There is always a probability gradient (though it can be at zero) for any phase space, and all my cartoon posits is that we are in a s.t. Quanta phase space, where there there is a probability gradient - which we experience (in the classical projection) as the 2nd law (plus aforementioned staggering puzzles). Is that seriously wrong? Maybe I just can't picture the obvious exception - and I'm being way way too naive for even cartoons.

Compactness: when you mentioned that awhile back, I tried to grok it, but got confused as to why it's a requirement for a model of s.t. Geomtry "spaces". Totally naive, but I interpret Bell's theorem as suggesting non-compactness is actually more of a required feature of a model of observation to-date (doesn't entanglement show that in one step from any spot you can wind up in exactly the a same, different and at least ostensibly random, spot). I may be conflating "compactness" with some other similar "bounded and well behaved" topological definition, but it seems like the difference between "arbitrarily distant random spot in the space" and "infinity" is a question of degree of bizarreness. And I don't believe in infinity anyway.

On this subject of non-locality, I saw a paper in your long "intuitive content..." Thread that I need to go back and find, something about how superposition of the quantum state probability wave from distant points in the past light cone can bring forward both space-like and Time-like information. Huh? ...that sound fascinatingly illegal. I am hoping to get more of what you are alluding to in the black holes tunneling to white holes.

As always the exaustive work you do in tracking and paraphrasing all the material you do, is very, much appreciated. Every time I get on here I lose, like two days.

20. Mar 1, 2015

### wabbit

I would like to mention that, in my view, while Rovelli's treatment does extend to the quantum situation, he starts with a strictly classical setup which is interesting in its own right, and perhaps sufficient here : if we go back to your original question about the second law and expansion, this is a well-posed question in a purely classical context, and I believe we should expect to be able to answer it without any need to venture into the more difficult realm of quantum gravity - especially as the question doesn't require us to go anywhere near a final singularity where GR becomes problematic.

I cannot myself provide an explanation beyond some crude attempt at paraphrasing Baez, but I find his analysis of the entropy of a collapsing cloud of dust, marcus' mention of the phase space of geometry, or Rovelli's considerations about thermodynamics in classical GR, more illuminating for the original question than any mention of quanta of spacetime.

21. Mar 1, 2015

### marcus

Wabbit, I think you are right! If we just focus on Jimster's original question in post#1 we should be able to answer in a purely classical context.
I should apologize for broadening the topic to include the general problem of getting a general relativistic thermodynamics (this is a problem which has proven recalcitrant but which I see hints may be solved by including geometric quanta---Jacobson's 1995 paper showing GR equation as equation of state of unspecified microscopic degrees of freedom.)

There are really exciting questions here, and exciting new results! Jimster's original question, by putting GR and thermodynamics together can lead into very interesting areas of discussion!

But first let's just focus in a restrained way on the original question. You are right about it being answerable in purely classical context.

IS IT RIGHT to "imagine that the constant of expansion and the second law of thermodynamics are very closely connected, or even that the constant of expansion is potentially the source of the second law?"

If "constant of expansion" means Lambda the cosmological curvature constant then the answer is no. Lambda is not responsible for the expansion and it could have been zero without any great qualitative difference. It is just the cause of a slight acceleration in the scale factor which first began, as far as we know around year 7 or 8 billion.

So what else could "constant of expansion" mean? The Hubble growth rate H(t) = a'(t)/a(t) is not constant. It is a time varying function that is used to DESCRIBE the expansion. It is not the CAUSE of anything. So again the answer is no.

It is easy to answer the original question. But that is somehow unsatisfying. There remains the challenge of rephrasing the question.

Could the fact that distances are increasing rather than decreasing be the source of the "arrow of time"? I think Demystifier already responded. The answer is no. The Friedmann equation gives an amazingly good fit to data and it has solutions which involve contraction as well as expansion.

The problem does arise there, though: how do you define entropy in full GR? It is a serious question. How do you make sense of thermodynamics in a fully relativistic context (where there is not even a global time). Whose entropy? Does the phase space include geometric degrees of freedom? Yes it must because it must be background independent. Nature does not have a fixed prior geometry.

So if Jimster does not want to broaden the discussion and transform the question, then it seems the answer is simply "no". The constant Lambda is not the "source" of the "2nd law" or the "arrow". The Hubble growth parameter H(t) is not actually a constant and is a descriptor keeping track of growth. It is not a "source". Also the mere fact of growth cannot be the "source" either because some solutions have distances grow and then contract.

I feel the question ought to be broadened and rephrased, and rephrased if possible in a clear way. There is something interesting at the heart of this and it is unsatisfactory to simply leave it in the original form, with a simple answer of "no".

22. Mar 1, 2015

### marcus

Jimster, would you accept the answer "no" to your original question as phrased in post #1, and accept the following revision?

The question is really about the relation of dynamical geometry to thermodynamics. One possible question that is clear and can be explicitly stated is this: In 1995 Ted Jacobson showed that the Einstein GR equation is a thermodynamic equation of state describing the collective behavior of a host of microscopic degrees of freedom. His paper has so far not been refuted in the 20 years since it appeared. He did not say what the microscopic DoF might be. One can then ask: assuming Jacobson was right and GR, our theory of dynamical geometry, is a thermodynamic EoS, what are the degrees of freedom whose collective behavior it describes?

When I google "einstein equation of state" the first 6 hits are to this paper:
http://arxiv.org/abs/gr-qc/9504004
Thermodynamics of Spacetime: The Einstein Equation of State
Ted Jacobson
(Submitted on 4 Apr 1995)
The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation δQ=TdS connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with δQ and T interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air.
8 pages, 1 figure

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23. Mar 1, 2015

### wabbit

Agreed, its just that I felt there might be more to say at the easier level:

If we start with "does the Second Law hold locally in a contracting universe far away from any singularity", I believe the answer is "Yes". Even this however is somewhat mysterious as the phase space appears to shrink and entropy to have nowhere to hide. This leads to the cloud collapse discussion.

Then if we ask "does it hold globally" I suppose the answer is more ore less "Yes as long as you manage to define it properly in GR". This leads to Rovelli.

But still in both cases I don't yet have a clear understanding of how exactly the "phase space paradox" is resolved - your mention of the phase space of geometry, or equivalently(?) gravitational degrees of freedom, seems to hold the key, but I wouldn't mind knowing a little more about how it opens the lock:)

Edit: just to clarify, written before I saw your post about Jacobson.

Edit2 : and just to add another can of worms : are we really surprised that entropy is a difficult subject in GR cosmology, when energy is already a tough one ?

Last edited: Mar 1, 2015
24. Mar 1, 2015

### marcus

It's good to have a balanced attitude and try to increase one's understanding in small steps. I may have gotten excited and generalized the question prematurely. Let's try pulling back to where we have a fixed solution to the GR equation and a global universe time. Friedmann time. A preferred class of observers who are at rest wrt the ancient light.

The problem with GR is that you don't even have a preferred time until you have an observer, and to have an observer worldline you first need a solution. So before you know it you have committed yourself to a fixed prior geometry. It is not GR anymore, it is conventional thermodynamics on a fixed curved spacetime. I think that was covered by Tolman in a book he wrote circa 1930.
No guarantee that nature obeys conventional thermodynamics on curved spacetime. We don't even have the conservation of energy, and systems at equilibrium with each other do not necessarily have the same temperature (as Tolman famously pointed out). Various "laws" and intuitions we have are compromised.

there must be an online tutorial for thermodynamics on curved spacetime so one doesn't have to go to the library and hunt down Tolman's book. that would address the question of what measure to use on phase space. why and how do we think that the volume measure on phase space should involve metric expansion? It seems obvious that one should use the metric but I feel too ignorant in this area to risk a guess. Demystifier would have a crisp answer.

Last edited: Mar 1, 2015
25. Mar 1, 2015

### Jimster41

Wabbit, I will go look at Baez more closely. By s.t. quanta I kindof thought I was playing loose but close enough to Marcus' reference to "phase space of geometry"

I appreciate very much you guys trying to help me reinterpret my own question, so answers, clarification can be had, or at least confusion is exposed. I know I'm not using the right terms, and so huge amounts of needed clarity just aren't there. So yeah man, I'll take a no any day with some why...

To wit, I'm still digesting. I have to go try to follow where you guys are going (which is a scramble). And by all means I'm interested in the discussion wherever you think it should go. And please don't let me slow you down.

I need to try more with the Friedman Equation. But last time I did, and just now when I wiki'd the Metric Expansion of space and cosmological constant. I get stymied right off by the use of the term "density" of matter and "pressure". One page said that the expansion of the universe can be accounted for by pressure (the density of matter and energy) and gravity. I don't understand this...If gravity is the geometry of space-time which gives matter - mass and energy, and therefore defines "density". How is there density and pressure (aren't these classical thermodynamic notions) from geometric phase space that - we don't really have a statistical thermodynamic model for...? I think this gets at why I have so conflated the notion of the expanding universe with some phase trajectory of s.t. geometry in geometrical phase space, which must have some notion of entropy - or probability gradient. Only that seems fundamental.

Seriously I need to think about all this. I'm missing so many basics. It is what it is. I'm listening. And I'm pushing through Susskind's books "Theoretical Minimum" (really great) as fast as I can.

Last edited: Mar 1, 2015