The entropy of the universe? (attempts to define gravitational entropy)

[marcus]

An important component of the total entropy is the entropy of the gravitational field. There have been various attempts to define this, some relating it to the extent of structure formation or the degree of inhomogeneity.

Here is an example:
A measure of gravitational entropy and structure formation
http://arxiv.org/abs/astro-ph/0111502
Here's about the author, Manfred Leubner:
http://homepage.uibk.ac.at/~c706102/me.html
http://homepage.uibk.ac.at/~c706102/my_research.html
http://homepage.uibk.ac.at/~c706102/allpub1.html

Can you supply links to what you think is a better definition? Or can you make up a better way to quantify the geometric entropy of the universe? Any comments on Leubner's, or others' attempts?

 

[marcus]

Some definitions of entropy use a horizon, like a black hole horizon. The entropy depends on the observer and is a measure of the observer's ignorance of what is behind the horizon.

Is it logically necessary to have a horizon in order to make the concept of gravitational entropy meaningful?
=========================

The reason that grav. entropy is associated with inhomogeneity/structure is that if you start with evenly distributed matter (imagine a vast dust cloud causing an approximately uniform field) the dust will begin to nucleate by its own gravity, and gather into regions of overdensity and underdensity. Eventually forming stars planets galaxies etc.

As long as gravity is universally attractive, things will tend to coagulate clump cluster and collapse. This structure formation is necessarily accompanied by an increase in entropy.

In this imaginary example, I suppose the maximum entropy configurationn would be for the whole dustcloud to have condensed into a single massive black hole.

Anyway, when we are talking about gravity/geometry what represents the low entropy configuration is uniformity. As long as gravity is attractive, uniformity is unstable and over time will evolve structure. A PF poster named Oldman referred to this process as curdling. The world tends to curdle. The curdled states are the high entropy ones.

 

[marcus]

When gravity is loop-quantized quantum corrections to the classical equations are found which have the effect of making gravity repellent at very high densities. The change is very abrupt.

You can find the Friedmann equation with quantum corrections in for example Ashtekar's June 2010 review http://arxiv.org/abs/1005.5491 .

You can calculate if you wish the density at which the sign of gravity reverses, according to this theory. As I recall it is somewhere around 1% of Planck density. The bounce itself occurs at about 40% of Planck.

There is already a substantial phenomenology literature devoted to how to test Loop cosmology---the main kind of bounce cosmology currently being researched. It is likely to be extensively tested in the next decades and may be falsified.

In any case the abrupt gravity reversal preceding the bounce itself flips the entropy, relative to the degree of inhomogeneity/structure. Under attractive gravity an inhomogeneous clumped field is high entropy, but with the switch to repellent gravity it is redefined as low entropy. Concentrations of curvature try to smooth themselves out and the gravitational field tends towards uniformity.

If you like to think of it this way, in this particular quantum cosmology model it is as if a giant slide projector is suddenly switched on and projects a different map on phase space. There are now different regions of different sizes---the system continues to evolve towards higher entropy as before. It is simply that higher entropy now means more spread-out-ness, more even-ness.

Then after bounce when the density gets back down to around 1% Planck the quantum corrections stop dominating and we are back to a classical attractive gravity. The slide projector switches off and we have the old map of phase space back again. The system continues to evolve towards higher and higher entropy which means structure formation (but it is starting out low because it has been uniformized.)

This is a natural consequence of the switch to repellent gravty at high density, which is what causes bounce in the first place. Nothing appears to be put in by hand. The conclusion is robust, or as some people say "generic."

 

[twofish-quant]

As long as gravity is universally attractive, things will tend to coagulate clump cluster and collapse. This structure formation is necessarily accompanied by an increase in entropy.

This is a deep and extremely important fact, that is a sign of some deep truth about the universe. Unfortunately/fortunately no one quite knows what that means.

 

[Dmitry67]

In any case the abrupt gravity reversal preceding the bounce itself flips the entropy, relative to the degree of inhomogeneity/structure. Under attractive gravity an inhomogeneous clumped field is high entropy, but with the switch to repellent gravity it is redefined as low entropy. Concentrations of curvature try to smooth themselves out and the gravitational field tends towards uniformity.

Very interesting.
And the 'singularity' of the non-rotating BH appears to be a time-like object with the density about 1% of Planks?
Whats about the informational paradox then? For an external observer, who does not care about the singularity, it does not change a lot.

 

[marcus]

Very interesting.
And the 'singularity' of the non-rotating BH appears to be a time-like object with the density about 1% of Planks?
Whats about the informational paradox then? For an external observer, who does not care about the singularity, it does not change a lot.

I can't extend this to BH. Basically this is a quantum version of the Friedmann equation, it does not seem to apply directly to BH. I don't understand why. with cosmology there is no outside, with BH there is. BH appears to be harder to describe the evolution of. Less progress.
I would suggest you have a look at Ashtekar's cosmology paper
http://arxiv.org/abs/1005.5491 .
I'm falling asleep, have to go.

 

[Dmitry67]

But what is a motivation for the big bounce if we know, that even if Universe is closed, it still be expanding forever? So gravity will never ever be repulsive?

Of course, it kinda gives you an infinite time lasting (-inf,+inf), not just (0,+inf) as in standard Big Bang model. But what do you think about the t<0 area?

Even we assume the block time, we somehow believe that the Universe had 'evolved' from the big bang, so the initial conditions are somewhere at t=0. In the Big Bounce model either the boundary conditions are at t-> -inf, which is weird, or at t=0, which means that in t<0 world everything is pre-determied, evolving to some fixed state at t=0.

 

[marcus]

But what is a motivation for the big bounce if we know, that even if Universe is closed, it still be expanding forever? So gravity will never ever be repulsive?

Of course, it kinda gives you an infinite time lasting (-inf,+inf), not just (0,+inf) as in standard Big Bang model. But what do you think about the t<0 area?...

The significance is not philosophical (as I see it) but instead simply has to do with empiricism. It gives a robust feature of the theory that phenomenologists can devise ways to test, and observationalists can actually test (probably in the next decade or two.)

I did a search recently that turned up around 30 bounce phenomenology papers that appeared 2009 or later specifically focused on testing the Loop early universe model.

Another paper to look at is Ashtekar Sloan's March 2011 "The Probability of Inflation"
http://arxiv.org/abs/1103.2475
The bounce gives a way to put probability measures on the initial conditions at the precise beginning of expansion. With the classical model you cannot say anything because the beginning does not exist, it disappears in a singularity.

So Loop bounce cosmology gives a mathematically tractable replacement for the classic singularity that MAY BE WRONG and which has observable consequences to look for in the ancient light and which you can do calculations with.
You can say, for example, what was the all-time maximum value of the Hubble rate H(t).

(If you read the June 2010 review 1005.5491 then you saw that.

So to respond to your comment, I think the bounce has no philosophical motivation, but rather it gives some empirical traction. Using it one can calculate and observe and test and possibly falsify the Loop theory. The bounce was not something anybody wanted or tried to get. It appeared around 2001 and then when the model was revised in 2006 it was still there, and it persists in a robust way whatever assumptions and analytical/numerical methods are used. It has attracted the professional attention of the "theory testers" (the phenomenologists.)

(the same is not true for the Loop BH, there is still no clear picture of BH collapse, in Loop, that I know of---it is something to work on. And the opportunities for testing are, I think, much more meager. With early universe modeling one has the CMB always staring one in the face. A huge amount of information. WIth BH collapse what do you have? Only perhaps an occasional flash of gammarays that one cannot understand and the unfulfilled promise of some gravity waves that might or might not come from a BH collapse/merger...Like a blind man listening for a footstep.)

 

[Chalnoth]

Well, as I was saying earlier, I don't see how a bounce cosmology works with entropy, regardless of the definition (provided it goes up with time).

My reasoning is this:
LQC predicts that if you have a collapsing universe, that collapse will, at sufficiently high densities, produce a repulsive force that smooths out and produces inflation. Apparently, this collapse is generic, producing an inflating universe under a broad class of collapses.

My problem with this is that if this is actually generic, then one should be able to simply reverse the arrow of time and show that it predicts a similarly-inflating universe into the past, which, by definition, would have increasing entropy into the past. This seems like a blatant contradiction in the theory.

Furthermore, there is the problem that the most dense point is also expected to be an extremely low entropy configuration, but it had to have come from the far future of an earlier universe, which would have been a high entropy configuration by definition.

So I strongly strongly suspect that when this is investigated further, it will simply come apart at the seams.

 

[Dmitry67]

My problem with this is that if this is actually generic, then one should be able to simply reverse the arrow of time and show that it predicts a similarly-inflating universe into the past, which, by definition, would have increasing entropy into the past. This seems like a blatant contradiction in the theory.

BTW I find this picture very beautiful mathematically
If Universe is infinite, or if MWI is true, then ALL potential configurations exist in both worlds (t>0 and t<0), which simply means that the configuration of the Universe U is an 'even' function of the cosmological time t:

U(-t) = U(t)

But then these both worlds are the same world!

 

[Max™]

I was thinking about the first time I heard bounce models tossed around.


If you treat the vacuum energy/stress-energy tensor as a functional tensile strength for spacetime, something comes to mind.

If the curvature of the universe kept tending towards flatness as it expanded, what would you get if you assumed there was a maximum flatness/size, beyond which the energy driving the expansion would dwindle until it was not enough to overcome to tensile strength of the vacuum itself... what then?

I know in a simple to imagine model of a tightly rolled up rubber sheet let loose to unfurl, the acceleration as it strove towards flatness eventually wouldn't be able to overcome the resistance of the sheet towards being stretched further. Wouldn't it then "antibounce", to use a vaguely Loopy term, as the tension induced by the initial stretching/the vacuum energy itself began to dominate the shape of the sheet again?

In such a state where matter had decayed, leaving an extremely entropic mush of neutrinos and little else, what would there be to prevent it from folding back up?

By the time the neutrino density became significant enough to actually matter, the contraction could have accelerated to a point where there was nothing preventing it from folding back into an arbitrarily compact state, besides the effects of gravity triggering a rebound at sufficient densities, as Ashtekar worked out?

 

[yenchin]

Is it logically necessary to have a horizon in order to make the concept of gravitational entropy meaningful?

As long as gravity is universally attractive, things will tend to coagulate clump cluster and collapse. This structure formation is necessarily accompanied by an increase in entropy.

I don't think horizon is necessary to define gravitational entropy. Presumably when we say that the very early universe has very low entropy (with respect to gravity), there should be a way to describe this geometrically, and there has been such an attempt:

http://arxiv.org/abs/0711.1656
[more details at http://arxiv.org/abs/0711.1656]

It is stringy, but you might find it interesting.

 

[Chalnoth]

BTW I find this picture very beautiful mathematically
If Universe is infinite, or if MWI is true, then ALL potential configurations exist in both worlds (t>0 and t<0), which simply means that the configuration of the Universe U is an 'even' function of the cosmological time t:

U(-t) = U(t)

But then these both worlds are the same world!

It may sound neat, but if correct it's a complete killer for the idea, because it requires massive fine-tuning.

 

[Dmitry67]

Fine tuning? Of what?

 

[bcrowell]

It may sound neat, but if correct it's a complete killer for the idea, because it requires massive fine-tuning.

I would be interested to hear you explain more about this.

On a possibly related point, plain old standard, classical cosmology has a massive fine-tuning problem because the initial entropy of the universe was so low. This fine-tuning problem is not solved by inflation.

 

[Chalnoth]

I would be interested to hear you explain more about this.

On a possibly related point, plain old standard, classical cosmology has a massive fine-tuning problem because the initial entropy of the universe was so low. This fine-tuning problem is not solved by inflation.

That's sort of the point. If my reasoning is correct, then LQC also would be incapable of solving the problem, for the same reasons.

 

[Chalnoth]

Fine tuning? Of what?

Entropy is the logarithm of the number of microstates that can replicate a given macrostate. As a result, low-entropy configurations are rare: there are many, many more ways to organize a universe in a high-entropy configuration than there are to organize one in a low-entropy configuration. Therefore, specifying a universe in a low-entropy configuration requires fine tuning.

 

[Fra]

Some definitions of entropy use a horizon, like a black hole horizon. The entropy depends on the observer and is a measure of the observer's ignorance of what is behind the horizon.

Is it logically necessary to have a horizon in order to make the concept of gravitational entropy meaningful?

My considered opinion is that any meaningful (intrinsically constructible measure) must like you say be observer dependent, meaning there always is some kind of "communicaction interface", "boundary", "screen" or "horizon" (all different names of some general abstraction).

I think of the disorder measure of a system A, as construced, represented and manifest on the observer side of the interaction interface.

This is why I personally thing that notions of objective global non-observer related god given measures of disorsder of anything (gravity or elsewise) are realist constructions that are out of place. I find it irrational and I don't really understand how it merges with an instrinsic observer.

What is the point and use, of an abstraction that is not constructible from the point of view of a real inside observer?

One simple collorary of the above personal view, is that there was no such thing as a low entropy big bang. The entropy of this created is only low, as measured relative to present observers. But if we want the proper intrinsic measure - ie how it was measure back them - we need to SCALE the measures and the observer back to big bang, and then by the conjecture that there was no complex observers during those events, neither could anyone measure this extremely low entropy - and thus there was no fine tuning problem at all.

Instead there was an evolution. I think it's a fallacy to use the measure defined TODAY, applied to BB, and then translate that into a fine tuning problem. The fine tuning problem is apparent, and not real. This is like trying to measure the radius of my coffe cop using a meter stick from the future. In neither way does it make sense.

/Fredrik

 

[Fra]

As long as gravity is universally attractive, things will tend to coagulate clump cluster and collapse. This structure formation is necessarily accompanied by an increase in entropy.

This is a deep and extremely important fact, that is a sign of some deep truth about the universe. Unfortunately/fortunately no one quite knows what that means.

If I may throw in an personal view.

As I see it, the universal attractive of gravity might very well originate due from the universal properties of any interacting and reaction systems in a game, to come to agreements. To negotiate and compromise, means to approach each other. And I don't just mean to play with words here, I mean it in the deepest sense.

In the information geometric pictures, a distance metric between two observers can be constructed (in several ways but still) to conceptually be a measure of the amount of disagreement between them. Thus spacetime might be just the information space, in between observers (separated from the internal degres of freedom).

This means that regardless of internal structure, ANY two interacting systems are bound to have an attractive force. The only way to screen, is to cut the communication, and thus cut the causal connection. But of course we cant' do that.

This may also suggest that spacetime isn't a thing "in between observers", it's better thought of as holographically existing behind the horizons implict as structures in the view of the enviroment. This also is the home of the entropy.

It's still an open issue to work this out, but I personally think something in this direction is highly likely to be right. So since some years back it's how I think of this, and it's my working hypothesis whenever I think of gravity and spacetime.

Edit: So from the point of view of a third observer, seeing a gravitational system (or a disgreeing system) it seems clear why the system is agreement (ie the one forming sturctures and clumps) are more likely. At least conceptually this is clear I think.

/Fredrik

 

[marcus]

...
On a possibly related point, plain old standard, classical cosmology has a massive fine-tuning problem because the initial entropy of the universe was so low. This fine-tuning problem is not solved by inflation.

Good point! Loop cosmology addresses the problem in a novel way. There is a good summary on pages 15 and 16 of Rovelli's December 2010 review paper
http://arxiv.org/abs/1012.4707

I recommend the paper, it is well-written and aimed at an audience of physicists who are not specialists in the subject. It covers a broad range of LQG topics and gives some history of the theory's antecedents. Easy way to get some general understanding.

The application to cosmology is covered neatly in a couple of pages and a few equations.

========================

The essential feature is that when gravity is quantized Loop-style some quantum corrections appear which become important at high density and which change the Friedmann equation in a simple way that depends on a critical density (estim. about 40% of Planck). Gravity becomes effectively repellent at high density---as I recall this begins when density reaches a few percent of critical.

While gravity is repellent the normal evolution is from "lumpiness" to uniformity. It's a kind of seredipitous result: the contracting phase ends up by preparing appropriate initial conditions for the expanding phase.

At this point Abhay Ashtekar is the primary authority on Loop cosmology so if anyone wants to go a little deeper than Rovelli's December review of LQG they should probably look at Ashtekar's most recent review http://arxiv.org/abs/1005.5491 and Ashtekar's latest research paper (March 2011)
http://arxiv.org/abs/1103.2475 on the probability of inflation in Loop cosmology.

 

[bcrowell]

Good point! Loop cosmology addresses the problem in a novel way. There is a good summary on pages 15 and 16 of Rovelli's December 2010 review paper
http://arxiv.org/abs/1012.4707

Interesting -- thanks for the link, Marcus!

-Ben

 

[marcus]

I hope it's useful! Bear in mind that Rovelli (especially in this review paper but also generally) is cautious and reserved. He only takes it to the point where quantum corrections make gravity repel at high densities, and does not follow through beyond that.

And that may be the wisest thing for now, to go just that far and not speculate further.

Ashtekar goes into a little more detail about what the quantum regime right around the bounce could look like.

(Assuming Loop bounce works out as a model and does not get falsified by observations in the next few years---everything we're talking about assumes it survives testing, which it clearly might not!)

 

[StateOfTheEqn]

In a closed expanding universe consisting only of photons it can be shown that dS is proportional to dR/R. The latter expression can be considered a time-independent Hubble parameter. The implication of this to our universe needs to be explored.