Hypothetically, lets say we have a space divided equally into two adjacent areas where (somehow) in one of the areas time goes by at half the speed as the other area. Or specifically, when a clock in the fast area shows 1 minute having gone by, a clock in the slow area shows only 30 seconds having elapsed.
In order for the speed of light to be constant, would lengths in the fast half be compressed to half the size of the slow half? My guess: Yes
(assuming (1) is correct) Let's create two identical objects, one in each area, positioned as close together as possible without passing into the other area, each given an equal velocity in a direction parallel to the boundary between the two spaces. Q: Over time would the distance between the objects increase? My guess: No
OK, (2) was all kinds of lame. So to make it more interesting lets put a strip of space between (but not touching) the two objects where time flows at the average rate between the two areas (so 75% as fast as the "fast" area). Then lets connect the two objects with a rigid bar of negligible size and mass that goes right through the "average speed" area. Then at the exact center of the bar (equidistant between the two objects) we give the compound object a specific velocity - again in a direction parallel to the boundaries between the areas. Q: Would the compound object rotate? My guess: yes - but it's just a hunch
(if (3) is correct) Depending on the direction of the rotation, since you would constantly get the two objects changing which area they are in, would the compound object just keep spinning faster, accelerate toward the slow area, or accelerate toward the fast area?
I've totally been ignoring mass on purpose. I realize that the same things that cause time dilation also cause a change in mass. Would taking mass into account change the results of any of the above?
I realize that this is all hypothetical and is a situation that cannot exist naturally (which is why I tagged it as a thought-experiment). What I want to know is whether the time-dilation portion of a space-time curvature introduces an acceleration above and beyond that caused by gravity, or whether it gets cancelled out by the length distortion, or if the curvature of space-time is what causes the acceleration of gravity and it would make no sense to talk about the effects of time-dilation separately. The thought experiment is just my effort to figure out exactly what the interactions between time and gravity are.