Will a shrinking universe have a reverse arrow of time?

Question

I'm not a physicist so forgive me if this question is silly. I'm reading (actually listening) to Mysteries of Modern Physics: Time by Prof. Sean Carroll.

I'm not sure if the concepts in this book are universally accepted by the physics community or are merely speculative or some measure in between. But i found one concept incredibly interesting and intuitive to understand:
Time itself has no specific direction, time is symmetric but the arrow of time goes in one specific direction and the reason why it goes in that direction (let's call it 'forward') is because entropy increases.

Why entropy increases? Because it was lower before. And why is that? Because it was just a bit higher than yet before that was higher than before... all of that leads to a point in which entropy was as low as possible and couldn't become lower but only increase. That point is supposedly the one our universe started from.

I assume, since the entropy is the amount of degrees of freedom of the information, a universe that inflates is increasing the possible configurations that information can assume and therefore the entropy in increasing. In a deflating universe the opposite is true so the entropy decreases.
Our universe goes from singularity(low) to expanding(hight).
A shrinking one will go from expanding(hight) to singularity(low);

Now my question is: if that's all true then i suppose if the universe was shrinking (as it is allowed and even forecast by some theories like quantum loop gravity) is it legitimate to imagine that the arrow of time would be reversed?
As the entropy would go from higher to lower it would be allowed for a human living in that universe to see a window and think "Oh this was probably shattered glass on the floor BEFORE being a window". That is because the state of shattered-glass is a higher entropy state than being a well refined window glass.

Am i right? And if so why did we ever had the doubt, knowing the second law of thermodynamics, that our universe could've been in a deflating phase?

Edouard · Accepted Answer

Except that it relates an electromagnetic arrow of time to the usual thermodynamic one, this answer's consistent with Nogueira's, and, as no answer's been accepted by the OP yet, I'm wanting to provide a verbal equivalent to it, per the PSE policy of permitting any participant's approval of a number of answers to the same question, and in view of my belief that acceptance of an answer preserves the Q&A.
The standard view, incorporating General Relativity's "spatialization of time", is that entropy tends to increase during the passage through time of anything whose motion can be described as passage through one or more of the three spatial dimensions, regardless of the direction of that passage.
I'm saying that "entropy tends to increase", instead of simply saying "entropy increases", because, for reasons bearing on the inherent uncertainty of the relationship between energy and time, there are intervals of time (usually extremely brief) when it decreases, for quantum-mechanical reasons. Because our nervous systems depend on electrical energy, such variations may be involved in our subjective experience of time's passage as occurring differently in different situations.
Entropy is a form of disorder, representing energy which cannot be transformed into work:  A rough example is the spouting of exhaust steam upward from the stack of a steam locomotive, which occurs regardless of whether the locomotive is moving backward or forward.  Entropy also increases regardless of whether the passage of matter or energy through time is backward or forward.
Cycles of expansion and contraction, in a universe containing mass, were first hypothesized by Tolman in the 1930's, and the problem with his model was the fact that the DENSITY of entropy would increase with each cycle, leaving a universe increasingly disordered.  This does not seem to have happened, given the fact that our observable region is approximately as uniform in every direction as the much older cosmic microwave background radiation.
Cosmological models that analyze contraction (i.e., "shrinkage") in much detail are rare:  One of the few that does is Aguirre & Gratton's 2002 "Steady state eternal inflation", described at https://arxiv.org/pdf/astro-ph/0111191.pdf, which provides for two multiverses each separated from the other by a Cauchy surface, with the arrow of time in one of them pointing in the direction of passage through time opposite the direction represented by the AOT in the other, effectively balancing it.  AG analyze the entropic AOT in some detail, and discuss an electromagnetic AOT as well, claiming that electromagnetic AOTs are explicitly linked to the direction of the expansion. In his profoundly Christian blog, the physicist Aron Wall has pointed to consideration of the electromagnetic AOT as perhaps the simplest means of rendering such explicitly reversed time plausible to us, because of its role in biological neurology.
The simplifying value of the AG model is its elimination of the need for any beginning, which is controversial in some (but not all) branches of western religion.  Most inflationary cosmological models are based on approximations of de Sitter space whose exponential contraction, thermalizing any contents with mass, would necessarily precede any exponential expansion: The well-known Borde-Guth-Vilenkin Theorem, requiring that inflationary models be "on average" expanding, is the result of that consideration, and the AG model was accepted as marginally meeting that criterion, in the last footnote to the BGV Theorem's last revision, which was formulated in 2003.
However, unadulterated de Sitter space does not require the presence of mass, although its expansive nature can only be seen by the motion of markers in it:  In our experience, those markers are the stars, and the changes in the wavelengths of their light as the expansion of space carries them outward from matter that's bound together gravitationally, like our own galaxy and its Local Group.
The link between the thermodynamic and electromagnetic arrows of time remains unclear, and a definitive answer to the OP's question consequently remains unavailable, making it rather a good one.  On an astronomical scale, there's currently no way to tell, with certainty, whether we're in a cosmos like AG's, and, if so, whether we're in the side of it where the relation between the thermodynamic AOT and the electromagnetic one is direct or an inverted duplication:  Science does have a lot to do with replication, as elaborated in many papers (mostly available free on Arxiv) by Lee Smolin, Nikodem J. Poplawski, and others.
This may be a factor in a 2020 paper by John Barrow, at
https://arxiv.org/pdf/1912.12926.pdf, which excludes nearly all inflationary cosmologies from a derivation with "finite action".  The only possible exception I've noticed is Poplawski's "Cosmology with torsion", based on 1929's Einstein-Cartan Theory rather than 1915's General Relativity.

Nogueira · Answer

The probability of the entropy decrease is much more small than the probability of increasing. There are lots of ways to seé that, one way is the following. Assuming that a given macrostate $A$ has microstates $a_i $, and a macrostate B has $b_j $, the probability of A envolve to B is given by

$$
P (Arightarrow B)=frac{1}{N_A}sum_{i, j} P(a_irightarrow b_j)
$$

where $N_A$ is the number of microstates in $ A $. Note that this formula is assymmetrical under time reversal, i. e. exchanging A's for B's. The probability of the reversal thing to happen is given by:

$$
P (Aleftarrow B)=frac{1}{N_B}sum_{i, j} P(a_ileftarrow b_j)
$$

This means that if we assume a time reversal symmetry of the microstates evolution, i.e. $P(a_irightarrow b_j) =  P(a_ileftarrow b_j)$, the macrostate evolution will be assymmetrical if $N_Aneq N_B$. Actually, for the microcanonical ensemble we have:

$$
N_A=e^{S(A)/k_B}\\N_B=e^{S(B)/k_B}
$$

where $k_B $ is the Boltzmann constant and $S $ is the entropy of the macrostate. This means that the ratio between this two probabilities is given by the exponential of the difference between the entropy. This is why the probability of a transition larger if the entropy increase.

Now, in your scenario if the universe getting smaller does not change the basics of probability, the entropy will still tend to increasing.

When you say that "Time itself has no specific direction, time is symmetric but the arrow of time goes in one specific direction and the reason why it goes in that direction (let's call it 'forward') is because entropy increases." you are inverting the two things. Is the opposite actually, time always goes foward and entropy is just one quantity that is sensible by that. When we have a physical law that is symmetric under time reversal, the description of the system is not always symmetric under time reversal, as I showed to you above.

Will a shrinking universe have a reverse arrow of time?

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