So I was thinking about this some more on my evening walk tonight, and remembered that the reason I brought this up in the first place is because I've always been fascinated by the notion that everything that happens does so because it maximizes entropy, and that the arrow of time can thus emerge from a timeless view of the configuration-space of the universe: the future is just downhill on the entropy gradients in that configuration-space. So, it seemed suspicious that there should seem to be a second arrow of time built into the structure of spacetime itself, that just coincidentally aligned with that entropic arrow. Why should it be that the geometry of the universe should just so happen to be such that it gets bigger in the same parts of the phase-space where it is also higher-entropy? It seemed like one should be driving the other, and the entropic arrow is clearly the logically prior one since the statistical mechanics behind it apply to abstract systems that aren't even spatial at all, so the entropic arrow had to somehow be responsible for the geometric arrow.
In response to those thoughts I was mulling over what
@Kenosha Kid was saying, about the enlargement of space making room for entropy to increase, and how it seems like the geometric arrow does drive the entropic arrow after all. I imagined to myself a little thought experiment or visualization: a toy system consisting of two points on a line segment, so it would have a very simple 2D configuration space, with obvious (err... note on that later*) peaks and valleys of entropy in it giving an obvious entropic arrow of time. And then, to that model, I added a third variable, and so a third dimension to its configuration space: the size of the line segment. Because higher-entropy states would be available on larger line segments, the entropic arrow of time would naturally point down the dimension of the configuration space that represents the size of the line segment...
...and I was about to say "regardless of what's going on with the entropy gradients along the other two dimensions", but that's not true, and not true in a very important way! It
is true that even if there weren't those other two dimensions of the configuration space, like if there was just an empty line segment with nothing on it, there would still be an entropy gradient down the one-dimensional configuration space, corresponding to the size of the line.
In other words, in a completely empty space, the entropic arrow of time will be toward a larger completely empty space. But
if there's an even steeper entropy gradient in the other dimensions of the configuration space, thereabouts the entropic arrow of time will be angled further away from straight down the dimension of the configuration space representing a larger line segment:
in other words, if there's any process that results in higher entropy faster than making more space, that will happen first.
So merely given that it's
possible for space to expand, that there's not some law somehow preventing it, we should just statistically expect empty-enough space to expand merely because that's the fastest route to the most entropy, but if there are other faster routes to more entropy available, we should statistically expect to see those happen first before we see expanding space.
*Now for that note about the "very simple 2D configuration space, with
obvious peaks and valleys of entropy". I was trying to visualize what that configuration space would look like, where the peaks and valleys of entropy would be, which states of that simple toy system are really more likely or less likely, and something occurred to me. There is exactly one state of the system where the two points are maximally far apart. There are increasingly more states of the system where the two points are increasingly closer together. The most common distance-apart for the two points to be, out of all the possible configurations of the system, is zero. That seemed counterintuitive to me: I initially expected that more-spread-out configurations would be more common, and I started trying to figure out where I had gone wrong. But upon reflection is occurred to me that on that counterintuitive account, we would statistically expect the two points to get closer and closer together... like gravity. And, if I'm not mistaken here, we would still expect
more than two points to get
more evenly distributed, but still for them to get closer together given a particular evenness of distribution.
In other words, on a purely statistical account, we would expect not just clouds of particles to even out, but also to fall together, and for the space surrounding them to get larger, in decreasing order... which is what happens IRL, no?
I know that there are already entropic accounts of gravity (though I don't know a lot about them), but I'm not sure if there are entropic accounts of the expansion of space yet?