> is there a reason why the Big Bang couldn't have happened uniformly in the dimension-3 hyperplane
It's a really good question. I think it has to do with how space expanded after the big bang (which is to me still a mysterious subject).
To avoid confusion, let me call "time" the arrow of time as perceived from within a system, and "TIME" the coordinate axis of spacetime that has the opposite sign from the other three.
The early days of the universe were full of heat and radiation everywhere throughout space, which seems like a high entropy scenario for sure. But entropy was able to increase as the universe cooled because space itself expanded adiabatically (and so the configuration space became larger).
In other words, we end up with an entropy gradient that points along the TIME axis because the volume of space expands along that axis. Like space-time is shaped like a pyramid rather than a cube, with the pointy axis in the TIME direction.
I don't know enough general relativity to say for sure, but I expect that that pyramid shape (expanding towards later times) is probably a consequence of TIME behaving differently than other spatial dimensions. If spacetime shrank volumetrically toward the galactic north, in a fairly uniform way throughout all times and throughout the universe, then we'd have a different conclusion. (This might seem like the case near a black hole).
I guess another thing to remember is that TIME and space really are different things. For example, conservation of angular momentum means the universe has constant angular momentum at all times, but not necessarily in all directions of space.
> In other words, we end up with an entropy gradient that points along the TIME axis because the volume of space expands along that axis.
That's a good insight, I hadn't thought of it that way.
> I guess another thing to remember is that TIME and space really are different things. For example, conservation of angular momentum means the universe has constant angular momentum at all times, but not necessarily in all directions of space.
Yes, I realize they are different, but I feel like the issue is more that don't understand the relationship between the special properties of TIME and those of time.
It's a really good question. I think it has to do with how space expanded after the big bang (which is to me still a mysterious subject).
To avoid confusion, let me call "time" the arrow of time as perceived from within a system, and "TIME" the coordinate axis of spacetime that has the opposite sign from the other three.
The early days of the universe were full of heat and radiation everywhere throughout space, which seems like a high entropy scenario for sure. But entropy was able to increase as the universe cooled because space itself expanded adiabatically (and so the configuration space became larger).
In other words, we end up with an entropy gradient that points along the TIME axis because the volume of space expands along that axis. Like space-time is shaped like a pyramid rather than a cube, with the pointy axis in the TIME direction.
I don't know enough general relativity to say for sure, but I expect that that pyramid shape (expanding towards later times) is probably a consequence of TIME behaving differently than other spatial dimensions. If spacetime shrank volumetrically toward the galactic north, in a fairly uniform way throughout all times and throughout the universe, then we'd have a different conclusion. (This might seem like the case near a black hole).
I guess another thing to remember is that TIME and space really are different things. For example, conservation of angular momentum means the universe has constant angular momentum at all times, but not necessarily in all directions of space.