Can a difference in the "speed of time" introduce acceleration? [closed]

Question

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.

1. 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

2. (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

3. 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

4. (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?

5. 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?

(Edit below)

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.

Answer 1

As it stands your question is rather hypothetical. You introduce a difference in the time without describing the physics behind it, and without any mathematical model to describe the phenomenon it's hard to make any useful comments.

However something like what you describe happens in the real world, and yes it does cause acceleration. In General Relativity the trajectory of a freely falling body is described by the geodesic equation:

$$ {d^2 x^\mu \over d\tau^2} + \Gamma^\mu_{\alpha\beta} {dx^\alpha \over d\tau} {dx^\beta \over d\tau} = 0 $$

The $\Gamma^\mu_{\alpha\beta}$ terms are the Christoffel symbols. If we stick to Cartesian coordinates$^1$ the Christoffel symbols are only non-zero when spacetime is curved so for flat spacetime the geodesic equation simplifies to:

$$ {d^2 x^\mu \over d\tau^2} = 0 $$

which just gives us a straight line in spacetime so there is no acceleration. Offhand I can't think of a (realistic) metric where only the time coordinate is curved and the spatial coordinates are flat. However in such a metric some of the Christoffel symbols involving time would be non-zero, and the result would be that the geodesic would no longer be a straight line i.e. the freely falling object would accelerate.

$^1$ as Chris points out, in polar coordinates some of the Christoffel symbols will be non-zero even in flat space.