Loop bounce and geometric entropy (re: Bill A's question)

[marcus]

Bill Alsept started a thread raising the general question---do cosmic models with regularly repeating big bangs conflict with thermodynamics' 2nd Law? (The law to the effect that, where it can be defined, entropy does not decrease, or does so only by rare accident, at irregular intervals if at all.)

The original thread kept getting off track. It's hard for people to stay focused on the central issue which in cosmology is geometric entropy. So I'll start this to consider just that question for a specific cosmic model: the LQC bounce version of the big bang.

According to bounce cosmology, the universe we see may have resulted from the collapse of one not greatly different from ours in a rough overall sense (except that distances were contracting instead of expanding). When the geometric law of gravity (GR) is replaced by a quantum version one finds that quantum effects make gravity repel at extreme densities. Computer simulations then show this causing a collapsing universe to rebound, and produce something which, like the conventional cosmo model, gives good agreement with observation.

Basically the bounce just reproduces the standard big bang model, but without a singularity.

The issue that immediately comes up is geometric entropy. The gravitational field is the geometry of the universe (curvature, dynamically changing distances...). One wants to be able to define the entropy of the gravitational field. Geometric entropy is a major player in the overall entropy picture.

Towards the end of collapse, things are highly clumped with lots of black holes, intuitively the geo-entropy is very high.
By contrast, in the early stages of expansion, geometry is smooth and even, stuff has not begun to condense into clumps---space is filled with nearly uniform hot gas. Intuitively, the geo-entropy is very low.

This intuition, that nice-smooth-even geometry has low entropy and crumpled-warty-pimply geometry has high entropy is based on our experience of gravity as attractive. Because it is attractive, matter always tends to clump, and form stars, galaxies, clouds, clusters, black holes...etc. Clumping makes geo-warts and geo-dimples.

This gives an OK intuition about geometric entropy as long as gravity is attractive. So it works up to a point.

 

[marcus]

But it fails to work when gravity suddenly becomes repellent. I guess you could say that geometric entropy ceases to be well-defined at the bounce.

There is a very brief interval during which gravity is violently repellent, some ordinary rules are reversed, something called "superinflation" occurs: faster than exponential growth of distances. The process of rebound is very interesting and is getting studied a lot.

So the geometry goes in warty and comes out smooth, without geo-entropy ever decreasing because it is not well-defined during a brief interval when it would otherwise do so.

The law does not apply where the function is not defined. An undefined function cannot have negative slope

 

[Chalnoth]

As we've discussed, this doesn't make any sense to me at all. The problem is the relationship between entropy and the number of microstates that can replicate a given macrostate. This tells us that the entropy of a system is a property of that system at a particular time, and we don't have to worry about the dynamics of what happened between two points in time to compare their entropies.

From this, we can say quite definitively that the entropy density going into the bounce is much, much higher than the entropy density coming out of it. The only way to resolve this, as near as I can tell, is to show that the post-bounce volume is so many orders of magnitude larger than the pre-bounce volume that overall the post-bounce entropy ends up being larger. Resorting to special dynamics between these two states cannot solve the entropy issue.

 

[marcus]

... Resorting to special dynamics between these two states cannot solve the entropy issue.

I realize that you don't understand how the issue is resolved, Chalnoth. But people much smarter than either you or I have been studying the bounce in detail for some years and are not worried by the entropy issue you seem hung up on.

I won't try to explain it to you, since I have already. But for anyone else who might be interested I will give it a try.
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The point is basically that the 2nd Law is not an AXIOM about nature, it is something that you have to PROVE mathematically.
If you can't prove it in some setting then it may very likely not apply. And if the entropy is not defined, the law is meaningless.

Geometric entropy is the overwhelmingly dominant form here, that's why we can say the entropy of the early universe is low. So intuition about gas in boxes doesn't work. If we are going to prove the law we need to do it in an abstract general setting.

When you do that you need a MAP of the microstates of the system showing how it is divided up into macrostate regions. A macrostate is a region of states that look the same even though they differ in (invisible) details. Intuitively, a macrostate has high ENTROPY if it corresponds to a large number of microstates. Texas rather than Rhode Island, but even more extreme.

And you need some handle on how the system evolves, how it makes its way around in the map, even if this is just a random walk. The second law says you tend to find yourself in larger regions. Intuitively someone wandering in the US is not likely to find himself in the state of Rhode Island. More likely in Pennsylvania, or Texas.

During the Loop cosmology bounce the map ceases to be well-defined and the dynamics are very different because gravity is repellent.

The math resources you need to prove the 2nd Law are not available. It is not merely violated, it is MEANINGLESS. Without a map, entropy is not defined, so it can neither increase or decrease.

During the bounce, geometry tends to smooth out rather than clump.

If you cannot prove the 2nd Law under the given circumstances, you certainly cannot invoke it as an axiom . Indeed you cannot define the geometric entropy in a situation where gravity changes abruptly from attraction to repulsion. It is hard enough to define geometric entropy even with constant classical gravity. If anyone is interested they might try looking it up and seeing the various attempts that have been made.

 

[Chalnoth]

That explanation is fundamentally wrong because of the definition of entropy. As long as you can identify the state before the bounce with the state after (which would be relatively easy, for example, in a closed universe undergoing a bounce), then you can perform this analysis. The special dynamics that go on in between do not matter.

Now, maybe there is a valid explanation for this within LQC, but your explanation isn't.

 

[bill alsept]

Marcus, thanks for focusing on this question again. I have some overall ideas I am working on and entropy/2nd law was clouding the issue. It seems to me that you and Chalnoth agree other than the details or snap shots of the cycle you may not be talking apples for apples.

Bill Alsept started a thread raising the general question---do cosmic models with regularly repeating big bangs conflict with thermodynamics' 2nd Law? (The law to the effect that, where it can be defined, entropy does not decrease, or does so only by rare accident, at irregular intervals if at all.)

I believe our universe is a repeating cycle and there is no way a cycle like that can work if entropy cannot decrease. As it has also been argued that for a cycle to return to where it started it would have to decrease its entropy the same amount it raised it, right?

QUOTE=Marcus; "When the geometric law of gravity (GR) is replaced by a quantum version one finds that quantum effects make gravity repel at extreme densities. Computer simulations then show this causing a collapsing universe to rebound"

Can you explain more of this in layman terms? thanks


QUOTE=Marcus;The issue that immediately comes up is geometric entropy. The gravitational field is the geometry of the universe (curvature, dynamically changing distances...). One wants to be able to define the entropy of the gravitational field. Geometric entropy is a major player in the overall entropy picture.

It seems to me that the standard concepts of gravity or gravity fields is where the confusion begins.
But sticking to the subject I would say entropy does not need to be a concern if everyone agrees it does cycle back to square one.

QUOTE=Marcus;Towards the end of collapse, things are highly clumped with lots of black holes, intuitively the geo-entropy is very high.

I would argue that because matter would be condensing or clumping more and more at the end that it would only appear to look disorganized. In reality the entropy would be decreasing as more and more particles (the original ones) are nicly pushed, packed and stowed away into black holes. As all the black holes are pushed together into "The Singularity" the entropy would continue to decrease until at some point just before "The Singularity" the effect of gravity would not be strong enough anymore to hold the "The Singularity" together.

QUOTE=Marcus;By contrast, in the early stages of expansion, geometry is smooth and even, stuff has not begun to condense into clumps---space is filled with nearly uniform hot gas. Intuitively, the geo-entropy is very low.

I would say there are four phases to the cycle.
(1) "The Singularity" (with almost zero entropy) would never completely form. If it did motion would stop and so would the cycle.
(2) Inflation (entropy would be much higher but in an overall symectrical pattern as everything in the universe would be moving in the same outward direction.
(3) Contraction (with the highest entropy) which could have been the BB and sourse of CMB
(4) Condensing (entropy slowly decreases)
Of course the last two need more explaining.

QUOTE=MarcusThis intuition, that nice-smooth-even geometry has low entropy and crumpled-warty-pimply geometry has high entropy is based on our experience of gravity as attractive. Because it is attractive, matter always tends to clump, and form stars, galaxies, clouds, clusters, black holes...etc. Clumping makes geo-warts and geo-dimples.

That is why I say the standard concepts of gravity or gravity fields is where the confusion begins.
Sorry guys I thought I was using the quote tab correctly. Hopefuly the way I did it is clear.

 

[juanrga]

Bill Alsept started a thread raising the general question---do cosmic models with regularly repeating big bangs conflict with thermodynamics' 2nd Law? (The law to the effect that, where it can be defined, entropy does not decrease, or does so only by rare accident, at irregular intervals if at all.)

It conflicts if the model is not dissipative {*}. Think on an idealized rubber ball bouncing against the floor, it bounces forever and violates the second law. Now think on a real rubber ball.

In that case the cosmic model is only an approximation (as harmonic oscillators and frictionless fluids)