# Second law of thermodynamics implies a linear cosmology?

If one applies the second law of thermodynamics to the Universe[1] as a whole then one might expect that the entropy of the Universe always increases as time goes forward (or more accurately that the entropy never decreases).

But the second law of thermodynamics is based on time-reversal invariant physics so that one would also expect that the entropy of the Universe should always increase as one goes backwards in time as well.

Both these conditions are met simultaneously by a linearly expanding cosmology such that the scale factor $a(t)$ is given by:

$$a(t) = \frac{t}{t_0},\tag{1}$$

where $t_0$ is the present time.

According to Raphael Bousso's Covariant Entropy Conjecture, the future entropy of the Universe is bounded by the area of the event horizon at a distance $d_e(t)$ given by:

$$d_e(t) = \int_{t_0}^\infty \frac{c\ dt}{a(t)}.$$

Substituting the equation (1) for a linear cosmology into the above expression we find that the distance to the event horizon, $d_e(t)$, diverges logarithmically as $t \rightarrow \infty$. Thus in a linear cosmology the future entropy of the Universe is not bounded.

Similarly the past entropy of the Universe is bounded by the area of the particle horizon at a distance $d_p(t)$ given by:

$$d_p(t) = \int_0^{t_0} \frac{c\ dt}{a(t)}.$$

Substituting the equation (1) for a linear cosmology into the above expression we find that the distance to the particle horizon, $d_p(t)$, diverges logarithmically as $t \rightarrow 0$. Thus in a linear cosmology the past entropy of the Universe is not bounded either.

Is this reasoning correct?

Does this show that such a linear cosmology is interesting even though the current evidence seems to point to an accelerating Universe with a finite future entropy?

A universe with a finite future entropy leads to strange hypotheses like Boltzmann brains so it seems reasonable to consider other alternatives (see Leonard Susskind's latest talk for example).

1.Instead of "Universe" I should have stated my argument in terms of the causal Universe defined by the volume inside the event/particle horizons of a particular observer.

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– Ben Crowell Aug 30 '13 at 19:34

Is this reasoning correct?

No.

But the second law is based on time-reversal invariant physics so that one would also expect that the entropy of the Universe should always increase as one goes backwards in time as well.

It's not based on other laws of physics, it's based on boundary conditions. In the early universe, at least as early as the times we can probe with observations, a region at least as big as our past light cone had an entropy much lower than maximum.

Both these conditions are met simultaneously by a linearly expanding cosmology[...]

You haven't given any evidence to support this assertion, which is false.

According to Raphael Bousso's Covariant Entropy Conjecture, the future entropy of the Universe is [...]

No, this paper says nothing about the entropy of the universe. The entropy of the universe is infinite if the universe is spatially infinite.

Does this show that such a linear cosmology is interesting [...]

No. Linear expansion is not a possible realistic cosmology for the reasons given in my answer to your previous question.

[...] even though the current evidence seems to point to an accelerating Universe with a finite future entropy?

Current evidence does not say that the universe has finite or infinite entropy. That's because current evidence does not determine whether the universe is spatially finite or infinite. If it's finite now, it has to be finite at all future times (because GR doesn't allow topology change). Likewise for the infinite case.

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A linear cosmology can contain matter provided the overall equation of state is $p=-\rho c^2/3$. – John Eastmond Aug 30 '13 at 19:37
@JohnEastmond: Thanks for your correction. That's true. I've edited my answer appropriately. – Ben Crowell Aug 30 '13 at 19:41
I guess I shouldn't have stated that Bousso's bound limits the entropy of the Universe. But it does limit the entropy inside the event horizon of an observer in an accelerating Universe. – John Eastmond Aug 30 '13 at 19:43
I don't know if I agree with your assertion that GR doesn't allow topology change -- singularity formation is certainly a topology change in most interpretations. (though i agree with the bulk of your answer and upvoted it) – Jerry Schirmer Oct 30 '13 at 1:03

Some scientists believe that the second law of thermodynamics is not true for very large structures like complete galaxies.

If this is true then it would not imply a linear cosmology.

By the way - a bit off-topic: Many scientists believe that it is not true for very small structures so a microchip gaining electrical energy by cooling down the environment could be possible.

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Some scientists believe that the second law of thermodynamics is not true for very large structures like complete galaxies. Not true. – Ben Crowell Aug 30 '13 at 19:37
Martin, please give references when you state such things. – fffred Aug 30 '13 at 23:02
@Ben Crowell: Yes, it is true. I didn't write that ACCEPTED scientists believe this. The scientists believing this are definitely NOT ACCEPTED by the science mainstream. However they exist! I saw an interview with an university professor in television some years ago. He said that he lost his reputation as good scientist because of stating this. – Martin Rosenau Nov 2 '13 at 6:53