Weyl Curvature Hypothesis and entropy

[ryokan]

What's the actual status of the Weyl curvature hypothesis?
Is there any best explanation to the early low entropy?

 

[meteor]

Hi,
try this:
http://arxiv.org/abs/gr-qc/0408065
it exposes the problem of how the universe can evolve starting from its initial state, and the solution given by Penrose, by introducing the concept of "gravitational entropy"

 

[ryokan]

Hi,
try this:
http://arxiv.org/abs/gr-qc/0408065
it exposes the problem of how the universe can evolve starting from its initial state, and the solution given by Penrose, by introducing the concept of "gravitational entropy"

Thank you, meteor.

 

[hellfire]

In the introduction of the paper it is written:

In terms of classical thermodynamics, this means that the early universe must have been in a state of (near) thermodynamic equilibrium. From the usual definition of entropy, we can state equivalently that the early universe must have been one of (near) maximal entropy. Thus, according to the second law of thermodynamics, the universe could not have evolve beyond the initial state, since any such evolution would mean a reduction in entropy.

But, nevertheless, we know that matter eventually breaks up due to gravitational attraction and ends up forming structures such as galaxies, stars, planets, planetary clouds etc. This is an evolution in the direction of a less homogeneous distribution of matter, and hence towards a lower entropy state. It appears therefore as if the evolution of structures in the universe breaks the second law of thermodynamics.

A possible solution to this problem was suggested by Penrose [2] in 1977 by introducing the concept of gravitational entropy…

I do not understand this argumentation. This state of maximal entropy mentioned (during nucleosynthesis), did correspond to a specific volume (Vo) and to a specific phase space (No). In a later time the universe expanded, the volume under consideration grew, Vi > Vo, and the phase space became larger Ni > No. Although the entropy So was indeed maximal for the state during nucleosynthesis with its phase space No, the later states had a greater maximal entropy (Si > So) because of new phase spaces Ni (Ni > No) in bigger volumes (Vi > Vo). For me there is no apparent contradiction between evolution of the universe and the second law. Probably I am missing something.

 

[Mike2]

Wouldn't the concerns of entropy apply only to the light cone of the observer? I mean by definition we cannot obtain any information (thus entropy) from outside the light cone, right? And isn't the light-cone a type of horizon beyond which we cannot see. So wouldn't the entropy/area calculations of black-holes apply to the surface that we will be able to see to in the future?

I wonder if the universe expanse faster than light, then it may be that information is escaping our horizon to the limit of the surface area/entropy would allow, which may force negative entropy effects.

 

[ryokan]

In the introduction of the paper it is written:


I do not understand this argumentation. This state of maximal entropy mentioned (during nucleosynthesis), did correspond to a specific volume (Vo) and to a specific phase space (No). In a later time the universe expanded, the volume under consideration grew, Vi > Vo, and the phase space became larger Ni > No. Although the entropy So was indeed maximal for the state during nucleosynthesis with its phase space No, the later states had a greater maximal entropy (Si > So) because of new phase spaces Ni (Ni > No) in bigger volumes (Vi > Vo). For me there is no apparent contradiction between evolution of the universe and the second law. Probably I am missing something.

Since beyond the Universe we cannot talk on any phase space, I don't think that your argument be correct because of, although the Universe expand, the whole of phase space would be ever included in it from the beginning. If the early Universe is in thermodynamic equilibrium,it would remain near maximal entropy forever, with independence of expansion.
Could an early low entropy be "created" by local asymmetries due to fluctuations and inflation?
Nevertheless, my knowledge on Physics is very limited and probably I am missing more things than you.

 

[hellfire]

ryokan, I am not sure to understand your point about the phase space, but let's forget the phase space and focus on the entropy and the volume. If one assumes a ideal gas with N particles at thermal equilibrium at a temperature T one has an entropy (afaik):

S ~ N (ln V + ln N)

Thus, if N remains constant and V increases S must increase assuming the next state is also in thermal equilibrium. If the relation above is correct and can be applied at least in a rough approximation, then I fail to see the problem with the second law. Every next state in time will have a greater maximal S (which corresponds to thermal equilibrium). May be there is something completely wrong in what I am writing here, or may be there is some subtlety if one takes into account gravitation... some guesses?.

 

[Mike2]

Every next state in time will have a greater maximal S (which corresponds to thermal equilibrium). May be there is something completely wrong in what I am writing here, or may be there is some subtlety if one takes into account gravitation... some guesses?.

I'm still curious... if we take an system inside a volume with a particular entropy, how fast can it change? I assume that no information can leave or enter the light cone. So is the maximum amount that entropy can change governed by surface area change whose radius expands with the speed of light? So what would happen then if one starts low or zero entropy of a system in a small volume?

 

[hellfire]

I'm still curious... if we take an system inside a volume with a particular entropy, how fast can it change? I assume that no information can leave or enter the light cone. So is the maximum amount that entropy can change governed by surface area change whose radius expands with the speed of light? So what would happen then if one starts low or zero entropy of a system in a small volume?

Are you claiming that the definition of entropy makes only sense withtin causally connected volumes? I think one can always consider a ‘bigger’ volume and ask which grade of order/disorder/entropy it contains, regardless whether it is causally connected or not. (I may be wrong because I am guessing). Anyway, I do not see how this could help to understand why there is a conflict between 2nd law and evolution of the universe.

 

[ryokan]

ryokan, I am not sure to understand your point about the phase space, but let's forget the phase space and focus on the entropy and the volume. If one assumes a ideal gas with N particles at thermal equilibrium at a temperature T one has an entropy (afaik):

S ~ N (ln V + ln N)

Thus, if N remains constant and V increases S must increase assuming the next state is also in thermal equilibrium. If the relation above is correct and can be applied at least in a rough approximation, then I fail to see the problem with the second law. Every next state in time will have a greater maximal S (which corresponds to thermal equilibrium).

If so, in an hypothetical Universe evolution to a big crunch (if critical density were enough) it would have a reduction of entropy following a reduction in the Universe volume, with all its consequences, one of which would be a change in the time's arrow.

 

[hellfire]

If so, in an hypothetical Universe evolution to a big crunch (if critical density were enough) it would have a reduction of entropy following a reduction in the Universe volume, with all its consequences, one of which would be a change in the time's arrow.

Good remark. Yes, according to this formula it would. Any closed and gravitationally bound system which obeys this formula would decrease its entropy (its volume would always decrease). It seams to me that if one wants to save the 2nd law one has to take into account gravitation. But I do not know how gravitation has to be considered in this case and in case of expansion.

 

[ryokan]

Good remark. Yes, according to this formula it would. Any closed and gravitationally bound system which obeys this formula would decrease its entropy (its volume would always decrease). It seams to me that if one wants to save the 2nd law one has to take into account gravitation. But I do not know how gravitation has to be considered in this case and in case of expansion.

The Weyl's curvature hypothesis wants to solve this problem. That was the reason for this thread.

 

[Mike2]

Are you claiming that the definition of entropy makes only sense withtin causally connected volumes? I think one can always consider a ‘bigger’ volume and ask which grade of order/disorder/entropy it contains, regardless whether it is causally connected or not.

Let's see, I think entropy compares the order of a state compared to that order of that state at a different time, right? It certainly sounds like causally connected regions to me. What use is it to compare the order of some state compared to some other that is not at all causally connected to it?

Anyway, I do not see how this could help to understand why there is a conflict between 2nd law and evolution of the universe.

"evolution"... How can we consider how things change without considering how it state of order changes. Evolution is how entropy changes things.

 

[ryokan]

Evolution is how entropy changes things.

Only entropy do not seems to induce biological evolution. Rather it is the biological information which conducts evolution although increasing the entropy.

 

[Mike2]

I'm still curious... if we take an system inside a volume with a particular entropy, how fast can it change? I assume that no information can leave or enter the light cone. So is the maximum amount that entropy can change governed by surface area change whose radius expands with the speed of light? So what would happen then if one starts low or zero entropy of a system in a small volume?

And if the entropy inside 3D sphere of a 4D light-cone is governed by the surface area of that 3D sphere, then what would be the entropy of a sphere shrunk down to the size of that Plank scale?

 

[hellfire]

The Weyl's curvature hypothesis wants to solve this problem. That was the reason for this thread.

Which problem? According to the formula given above there is a problem for isolated systems which decrease in volume. But the universe expands, and expansion implies a growth of the maximal entropy (entropy in thermal eq.) Obviously I am missing something but I do not see what. It would be nice if someone could explain or give a link to an explanation of the problem which leads to this hypothesis. I just want to understand.

 

[ryokan]

Which problem? According to the formula given above there is a problem for isolated systems which decrease in volume. But the universe expands, and expansion implies a growth of the maximal entropy (entropy in thermal eq.) Obviously I am missing something but I do not see what. It would be nice if someone could explain or give a link to an explanation of the problem which leads to this hypothesis. I just want to understand.

If in any time after the Big-Bang, the Universe's entropy is maximal because it is near its thermal equilibrium (although this maximal entropy were increased by increase of V in expansion, following the formula in your post), what would be the origin of low entropy? It seems that in our solar system, Sun is the basic source of low entropy. It seems, then, that gravitation have a role in the origin of an early low entropy. But also bacause of gravitation, entropy will increase when stars age. There is so a time's arrow in the role of gravitation on entropy that, if I understand well, would be due to a different status of Weyl's tensor along time.

 

[moving finger]

Good remark. Yes, according to this formula it would. Any closed and gravitationally bound system which obeys this formula would decrease its entropy (its volume would always decrease). It seams to me that if one wants to save the 2nd law one has to take into account gravitation. But I do not know how gravitation has to be considered in this case and in case of expansion.

Hi. I'm new to this forum so forgive me if I'm going over old ground. I have a particular interest in cosmological entropy, so here is my small contribution :
The formula quoted in this thread implies entropy decreases with volume in a gravitating system but it is correct ONLY for a system which is in thermodynamic equilibrium with its surroundings. In fact what happens is that a gravitationally collapsing system loses energy (usually via radiation), and the decrease in entropy of the gravitational system is more than made up for by the increase in entropy of the surrounding universe through the loss of (radiated) energy. Thus the overall entropy increases. See http://math.ucr.edu/home/baez/entropy.html for a detailed explanation.

Thus the 2nd law is not violated. Gravitational attraction does cause an overall increase in entropy.

 

[moving finger]

If in any time after the Big-Bang, the Universe's entropy is maximal because it is near its thermal equilibrium (although this maximal entropy were increased by increase of V in expansion, following the formula in your post), what would be the origin of low entropy? It seems that in our solar system, Sun is the basic source of low entropy. It seems, then, that gravitation have a role in the origin of an early low entropy. But also bacause of gravitation, entropy will increase when stars age. There is so a time's arrow in the role of gravitation on entropy that, if I understand well, would be due to a different status of Weyl's tensor along time.

My take (in a nutshell) : The dominant cosmological force just after the Big Bang (during inflation) was not attractive (gravity), but repulsive (anti-gravity?) which caused a sudden expansion of space/matter/energy for the first instant of time. This inflation resulted in an extremely homogeneous distribution of mass/energy AT A TIME when gravity was not the dominant force in the universe.

When gravity is not dominant, the state of maximum entropy is in fact a uniform, homeogeneous distribution of mass/energy. Thus towards the end of the period of inflation, the entropy of the universe was the maximum that it could possibly be, given the prevalent circumstances.

Then inflation ended, and gravity kicked in. Once we have gravity as the dominant force, then a uniform, homeogeneous distribution of mass/energy is no longer the state of maximum entropy. In other words, the goalposts were shifted at the end of inflation - and entropy has been trying to catch up (ie increasing) ever since.

In other words : At the end of inflation, entropy was both maximal (in the absence of gravity) and at the same time minimal (in the presence of gravity). It was the sudden end of inflation and the consequent rise of dominance of gravity which flipped entropy from being maximal to minimal.

Why is the Weyl curvature hypothesis needed?