(gravitational) energy and entropy in an expanding universe

[tom.stoer]

Reading popular books (written by Hawking, Penrose, Greene, Linde, Guth and certainly many more) one finds numerous statements like

entropy was low after the big bang ... Weyl-curvature hypothesis ... entropy increases with time ... black holes violate unitarity and therefore entropy or phase space decreases with time ...

or

the universe was born from a quantum fluctuation carrying zero energy and minimal entropy ... negative gravitational exactly balances positive energy carried by matter and radiation ...

Now reading serious articles and books -- and thinking about the problems in detail -- there are rather different conclusions and serious obstacles:

we do not have a fully developed theory of quantum gravity and are therefore not able to define gravitational entropy; gravitational entropy is still not fully understood - even for very specific scenarios like static black holes - in well established theories like LQG and string theory (fuzzball proposal); there is no theory of thermodynamics in an expanding universe - and even if there were, they would be useless b/c its only thermodynamics; there is no (or at least no universal) definition of energy in an expanding cosmology; there is no (or at least no universal) defintion of gravitational energy at all

So my question is simply this: Are there serious papers or books where these problems are addressed, where precise defintions for energy and entropy are given, and where the above mentioned popular statements are derived or at least motivated mathematically? Or is this pure religion?

 

[tom.stoer]

push

 

[friend]

I think one problem is how to decide if the very early universe was in low entropy or high entropy. How do you decide whether the inflaton field that was very nearly uniformly distributed was in a low entropy state that decays to high entropy, or in high entropy because it everything was in thermal equilibrium? And after re-heating, when the inflaton field decayed to particles, was everything in one state that can evolve to many states, according to the 2nd law of thermal dynamics. Or, was everything already in complete thermal equlibrium with no chance of evolving to a higher entropy state?

 

[tom.stoer]

the problem is not to decide whether there was low or high entropy, the problem is: what is entropy? how can we define gravitational entropy? and how do we define entropy in an expanding universe?

I have never seen the definitions.

 

[marcus]

1209.0065 has a lot of references to earlier work on thermodynamics in a GR context.
For example [1] is a classic from 1934 "Relativity, Thermodynamics, and Cosmology" by R.C. Tolman.
Also references to more recent work by Wald, Padmanabhan, Jacobson and many others. You can get some idea of how far the field has come and where things stand at present.

 

[marcus]

I don't think I can satisfactorily answer your question, but I can give at least some little bits and pieces to help fill in the picture.

This is something very concrete: a recent estimate of the entropy of the universe by a top cosmologist, Charles Lineweaver. You can see how he does it, what's involved, what definitions. This will not completely satisfy you, but it's something definite to get your hands on:

http://arxiv.org/abs/0909.3983
A Larger Estimate of the Entropy of the Universe
Chas A. Egan, Charles H. Lineweaver
(Submitted on 22 Sep 2009 (v1), last revised 25 Jan 2010 (this version, v3))
Using recent measurements of the supermassive black hole (SMBH) mass function, we find that SMBHs are the largest contributor to the entropy of the observable universe, contributing at least an order of magnitude more entropy than previously estimated. The total entropy of the observable universe is correspondingly higher, and is S_obs = 3.1+3.0-1.7x10^104 k. We calculate the entropy of the current cosmic event horizon to be S_CEH = 2.6+-0.3x10^122 k, dwarfing the entropy of its interior, S_CEHint = 1.2+1.1-0.7x10^103 k. We make the first tentative estimate of the entropy of weakly interacting massive particle dark matter within the observable universe, S_dm = 10^87-10^89 k. We highlight several caveats pertaining to these estimates and make recommendations for future work.
Astrophysical Journal. Accepted 11 Jan 2010. 10 pages and 10 figures.

The cosmic event horizon (CEH) is very interesting. it is currently 16 billion LY. If a galaxy is, today, more than that then if, today, we send a light signal to them it will never get there. If, today, there is a supernova explosion in that galaxy then we will never see it even if we wait 100 trillion years. But most of the galaxies which we can see today and study are now more than 16 billion LY from us. So the sky is full of galaxies which we can see and study and admire but which are, today, out of causal contact with us. So there is this CEH, and it is a bit like a BH event horizon, and it has an ENTROPY. So that's something to know about: Lineweaver and Egan's estimate of the CEH entropy.
==============
My impression is that entropy is not yet satisfactorily defined in full GR. But I can't speak with confidence.
The research papers in this area give me the impression that in full GR entropy must be observer dependent, as of course time is as well.

However in standard cosmology the universe is approximated with a very simple model (Friedmann equation) and there is a preferred "universe time" in that simple model. Cosmologists have it easy. They make assumptions that simplify things enormously. So it is not surprising that they can say what they mean by the entropy at a given moment in time, and many other things.
===============

Penrose likes to point out that the picture of increasing entropy in geometry (i.e. GR's gravitational field) is increasing clumpiness, but in a box full of gas the picture of high entropy is uniformity, stuff all evenly spread out.

A uniform gravitational field is LOW entropy, but uniform gas in a box is HIGH entropy. I recall him making a big point about this in a lecture I attended when he was visiting out here. I just mention it FWIW. Apparently geometric entropy is different from other entropy or unintuitive in some respects.
[STRIKE]Have to go. Back later[/STRIKE].
I'm back, but can't think of anything more to say at the moment. Maybe I should first let other people take a try at your question. Of course there is a lot of relevant stuff in
http://arxiv.org/abs/1209.0065
since that's basically what it's about.

 

[tom.stoer]

thanks marcus; you are right, I have to study Rovelli's paper; I'll do that as soon as I'm back ...

 

[haushofer]

I don't have any good contribution to this thread, but I've asked myself the same questions after being exposed by people to the arguments as given in Lawrence Krauss' book, which seems to give the impression that these notions are easily defined. I'm also very curious about how well-defined these popular statements are.

 

[tom.stoer]

thanks for supporting my scepticism (for Hawking popular writing all this no surpsise, but for Penrose it is)

 

[Paulibus]

A simple-minded difficulty I have in figuring out how to define entropy in a gravitational context is this:

In a box of gas it is assumed that individual atoms of the gas have an equal probability of finding themselves anywhere in the box. This means that as they individually explore their possible locations, an equilibrium state of maximum entropy develops when they are uniformly distributed through the box, however inhomogeneously they may be distributed initially.

Isn't this concept of equal probability for microstates fundamental for defining equilibrium and entropy physically, rather than from a classical thermodynamics perspective?

But this restriction to equilibrium on a level playing field, as it were, is absent for gravity. Here inhomogeneity can progressively enhance itself; in gravitational collapse of a particulate disc into planets and a star the 'playing field' is prone to developing potholes of potential as play goes on.

I guess I'm just being obtuse here. But thinking in such simple terms is preventing me from seeing how entropy is enhanced by structure evolution, as Penrose claims it is.

 

[mitchell porter]

The orthodox position is that there is no gravitational entropy outside of event horizons. A horizonless geometry has zero gravitational entropy. But de Sitter entropy complicates this picture.

 

[marcus]

But this restriction to equilibrium on a level playing field, as it were, is absent for gravity. Here inhomogeneity can progressively enhance itself; in gravitational collapse of a particulate disc into planets and a star the 'playing field' is prone to developing potholes of potential as play goes on.

I guess I'm just being obtuse here. But thinking in such simple terms is preventing me from seeing how entropy is enhanced by structure evolution, as Penrose claims it is.

I don't claim to know the answer and I'm not sure there is "orthodox" consensus about all these issues---there may still be open questions about the meeting of GR & thermodynamics.
BUT I can suggest a way to resolve the problem you raise, Paulibus:

when stuff falls together in clumps it releases energy, so you can include the entropy of the cloud of photons radiated by the stuff. If you look at it in this light maybe Penrose idea of entropy of gravitational field is not needed to make the second law hold.

This was the message of a 2005 paper by some people in Norway published in Physical Review D:
http://arxiv.org/abs/gr-qc/0408065
Entropy of gravitationally collapsing matter in FRW universe models
Morad Amarzguioui, Oyvind Gron
We look at a gas of dust and investigate how its entropy evolves with time under a spherically symmetric gravitational collapse. We treat the problem perturbatively and find that the classical thermodynamic entropy does actually increase to first order when one allows for gravitational potential energy to be transferred to thermal energy during the collapse. Thus, in this situation there is no need to resort to the introduction of an intrinsic gravitational entropy in order to satisfy the second law of thermodynamics.
Comments: 9 pages, 4 figures. Phys. Rev. D71, 083011 (2005)

Personally I am reserving judgment about this--I may change my mind about some of these issues.

Mitchell, De Sitter universe HAS an event horizon, so how does dS geometry complicate what you said? You are saying that gravitational field has no entropy unless it has one or more event horizons? The entropy is always associated with an horizon? Off hand I don't see anything wrong. Classically at least.

I think there can be ENTANGLEMENT entropy associated with a bounded REGION. I suppose that entanglement might involve gravitons (both inside and outside the region). At first sight what you said makes sense, at least classically, but I could use more explanation from you. Intriguing issues here!

 

[mitchell porter]

There are arguments dating from the 1970s that stationary gravitational fields without event horizons have no entropy. See Gibbons and Hawking, and Davies, Ford, and Page. (The latter is freely available at http://cosmos.asu.edu/publications/papers.htm.)

I have access to an honors thesis that reviews some of the subsequent history, and I can see that there have been many attempts to develop a concept of gravitational entropy outside of horizons, so perhaps it's a brittle "orthodoxy". But it makes sense to me: that when black holes form, something special happens, new degrees of freedom come into play, and that's where the black hole entropy comes from. But I don't see how to understand the entropy of a de Sitter horizon in this way.

 

[marcus]

There are arguments dating from the 1970s that stationary gravitational fields without event horizons have no entropy. ... But it makes sense to me: that when black holes form, something special happens, new degrees of freedom come into play, and that's where the black hole entropy comes from. But I don't see how to understand the entropy of a de Sitter horizon in this way.

What about the ripples in the field? The gravitons inside a region and those outside.
Our observer might be ignorant about those on one side of a boundary (which might be entangled with the ripples on the other side.)

Could "stationary" be an artificial assumption? In the case of a BH I imagine the horizon to be constantly changing shape in little ways as it interacts with the geometric environment. Perhaps these correspond to the "new degrees of freedom" you spoke of. But even without a BH, just some ordinary bounded bulk region*, I'm open to the idea of the gravitational field in that region (or conversely outside that region) having entropy. Or else because of the universal coupling of geometry and matter perhaps it is ultimately impossible to distinguish--to cleanly split the entropy into two parts as we are imagining doing here.

*OK maybe it has to be a causal horizon, not just an ordinary boundary---still making up my mind. BTW thanks for your discussion--it's helpful.

 

[tom.stoer]

The orthodox position is that there is no gravitational entropy outside of event horizons.

isn't this trivial? a classical el.-mag. field has no entropy, either

 

[marcus]

This just appeared:
http://arxiv.org/abs/1211.0522
Horizon entanglement entropy and universality of the graviton coupling
Eugenio Bianchi
(Submitted on 2 Nov 2012)
We compute the low-energy variation of the horizon entanglement entropy for matter fields and gravitons in Minkowski space. While the entropy is divergent, the variation under a perturbation of the vacuum state is finite and proportional to the energy flux through the Rindler horizon. Due to the universal coupling of gravitons to the energy-momentum tensor, the variation of the entanglement entropy is universal and equal to the change in area of the event horizon divided by 4 times Newton's constant - independently from the number and type of matter fields. The physical mechanism presented provides an explanation of the microscopic origin of the Bekenstein-Hawking entropy in terms of entanglement entropy.
7 pages

 

[Haelfix]

There are many different ways in which people have tried to define it, but there doesn't exist a unique existence proof, or even a quantity that encodes a measure of gravitational entropy in closed form.

As Michael points out, there is a well defined notion of entropy for black holes and systems with horizons.

So what is relatively OK is the following heuristic but highly plausible physical argument:

Start with a homogenous spherical dust cloud, with a very large radius that has all the associated matter degrees of freedom and the associated and countable matter entropy. You can assume thermal equilibrium at the start. This is relatively well defined, and you can assume that the gravitational entropy ought to be small by some measure.

Now, let the dust evolve with time, and after some amount of time, under certain initial conditions conditions the system virializes, and a stellar collapse occurs. Much later, a black hole will form whereupon the matter degrees of freedom are gone and all the entropy will be defined by the horizon. This end state is essentially in thermal equilibrium, and again you have a well defined count.

Therefore in order to preserve some semblance of the 2nd law, there should exist a quantity (the gravitational entropy) that goes up, as the matter entropy goes down during the actual collapse process (which is very far from thermal equilibrium). Further the gravitational entropy must be quite a bit larger than the matter degrees of freedom.

Whats difficult here, is then actually finding a specific quantity that encodes this physics microscopically. Penrose proposed the Weyl curvature as being able to do this, but that doesn't quite work.

Many people have since worked on it, but I don't believe there is an agreed upon measure.

 

[tom.stoer]

I know this argument - and as the discussion shows it gives some reasonable results for the dust case (some dust entropy, zero gravitational entropy) and for the horizon case (zero dust energy b/c matter has vanished, some gravitational = horizon entropy). There are proposals to calculate this horizon entropy. But there seems to be no way to define gravitational entropy w/o horizons.

 

[marcus]

I think the article I just cited in post #16 is relevant

... Due to the universal coupling of gravitons to the energy-momentum tensor, the variation of the entanglement entropy is universal and equal to the change in area of the event horizon divided by 4 times Newton's constant - independently from the number and type of matter fields. The physical mechanism presented provides an explanation of the microscopic origin of the Bekenstein-Hawking entropy in terms of entanglement entropy.
7 pages

 

[tom.stoer]

I think the article I just cited in post #16 is relevant

of course ;-)

 

[yenchin]

See also http://arxiv.org/abs/0711.1656

 

[mitchell porter]

See also http://arxiv.org/abs/0711.1656

That's fascinating. If I ever want to think seriously about a fundamental model for eternal inflation, I'll be returning to this paper, especially to the "SYGLB theorem".

 

[Finbar]

Shouldn't we take more care in distinguishing entanglement entropy obtained by a partial trace over some pure state and a the entropy of a genuine mixed state?

My guess would be that within quantum gravity these two entropies are somehow unified. However from our current point of view, with no complete theory of quantum gravity, we should probably keep them distinct.

One can then always think of space-times without horizons as having entropies due to them being mixed states. To calculate this entropy one would need to know what the underlying microscopic degrees of freedom are of coarse.

 

[marcus]

My guess would be that within quantum gravity these two entropies are somehow unified. However from our current point of view, with no complete theory of quantum gravity, we should probably keep them distinct.
...

Good. However Bianchi makes it clear that he is working within a certain QG context which is certainly legitimate. Part of checking LQG is discovering what the consequences for QFT are if you make some basic LQG assumptions.
The May 2012 paper (the original subject of the thread) was explicitly LQG, and this November 2012 paper can be seen as the corresponding perturbative QG version of the same analysis. Note the role of Immirzi parameter has been somewhat diminished:
==quote from Conclusions==
The derivation presented here for quantum fields on a Minkowski background can be understood as the perturbative version of the calculation of the entropy of the quantum horizon in the framework of loop quantum gravity [94, 95]. In this non-perturbative approach, the entanglement of quantum geometries across the horizon are computed and the entropy S is finite because the theory has no degrees of freedom above the energy scale γ^−1/2Λ_Planck, where γ is the Immirzi parameter. While the entropy diverges when γ is sent to zero, the variation δS corresponding to a change in the horizon energy is in- dependent of γ and reproduces the Bekenstein-Hawking formula by the same physical mechanism discussed in this paper.
...
...
Similarly for other matter fields, so that we may argue that at high energies there is an extra chemical-potential contribution μ to the Bekenstein-Hawking entropy δS = 2π( δA/8πG + μδN) where N is the number of unfrozen matter fields. A similar phenomenon is being investigated in the context of loop quantum gravity [96] and, together with the derivation [94, 95] of the horizon entanglement entropy as in the present framework, it may lead to new physical effects associated to the high energy behavior of causal horizons.
==endquote==

 

[bcrowell]

These may be helpful:

http://www.phy.olemiss.edu/~luca/Topics/grav/entropy.html
http://www.mth.uct.ac.za/~henk/ref_dir/chge.html

a classical el.-mag. field has no entropy, either

Huh? This sounds wrong to me. Don't you just have to introduce some arbitrary coarse-graining, rather than being able to count inherently discrete states as you would in a quantized theory?

 

[Haelfix]

One can then always think of space-times without horizons as having entropies due to them being mixed states. To calculate this entropy one would need to know what the underlying microscopic degrees of freedom are of coarse.

You would think this would be possible, but afaik it tends to hit a wall.

(semi)Classically, it seems that the gravitational entropy saturates and never exceeds the horizon entropy bound. This is played out by various heuristic arguments due to Hawking, as well as by direct calculation where people look for 'extra terms'. For instance, you can ask the question as Davies does in one of the papers linked above, what a thin shell of matter surrounding a black hole contributes. And he finds that it only contributes in so far as a net change in the horizon, or in other words the contribution of a gravitational field without a horizon is zero. Then there are various supersymmetric arguments by Bousso in the 90s. This strongly disfavors the Weyl curvature hypothesis.

I think the current belief is that 'local' entropy counting is very subtle business, due to the presumed holographic nature of the microscopic theory. There should be a lot of very nonlocal correlations that might be invisible is you ask the wrong questions.

 

[tom.stoer]

I agree that the horizon approach is better understood, but it becomes rather obscure when talking about the entropy of the universe as a whole (including its gravitational d.o.f.)

 

[Paulibus]

For a spectator such as myself, this is a very illuminating thread; not so much as regards physics, but in revealing the way knowledgeable folk are thinking about entropy and gravity. To me it seems that when gravitation enters the picture, recourse is immediately made to the concept of horizons, and the entropy associated with them, mainly because microscopic states can't be easily identified and counted, wheras the entropy of horizons is known. Being unable to count is a compelling handicap for the mathematically inclined (especially Penrose)!

An ignored disadvantage of focussing on horizons is that, strictly speaking, they don't exist; they form only in the remote future -- at the end of time, as it were. Nascent black holes are indeed fascinating machines at the centres of galaxies, but are presently sans horizons (and their associated entropies?). And if our universe is in the process of developing a horizon, we'll never observe it!

So, thanks for starting an interesting thread, Tom Stoer, and to Marcus for pointing me at:
http://arxiv.org/abs/gr-qc/0408065
Entropy of gravitationally collapsing matter in FRW universe models
Morad Amarzguioui, Oyvind Gron

which seemed to me sensible, believable and very relevant. Is it then not correct?

 

[tom.stoer]

we have two slightly overlapping threads, so I post my objections here as well. There are several problems

- w/o QG you can't define and therefore you can't count microstates
- w/o thermodynamics you can't define Q, T and dS = δQ / T, therefore you can't identify a macrostate
- w/o a Hamiltonian H (or with H ~ 0) you cannot define E etc.
- you can't define the density operator ρ b/c you neither know the states nor the probabilities for the states

So there us a problem with all definitions of entropy I am familiar with.