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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.

Freudian Slip
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    It appears some do not feel my question is about mainstream physics. The purpose of my question is to understand the current mainstream understanding of the physics of time and gravity. I edited the question to explain. – Freudian Slip Jun 26 '14 at 21:15
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    I have once again (!) no idea why in the world this legitimate beginner GR question is considered not mainstream by 5 closevoters. – Dilaton Jun 28 '14 at 21:23
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    Check the revision history, the original version of my question was... muddled by an attempt to deconstruct the question into a series of thought experiments that ultimately invalidated the feature I was trying to get an answer to. I'm hoping my last revision will fix this. – Freudian Slip Jun 28 '14 at 21:42
  • @FreudianSlip: the edit makes your question completely different, and incidentally makes my answer somewhat nonsensical. I'd be inclined to roll back the edit and post a new question. Note however that the split into space and time coordinates is down to the choice of coordinates, so your middle paragraph doesn't make sense. – John Rennie Jun 29 '14 at 10:02
  • @JohnRennie please dont roll back the question. But maybe you could add to the 3 reopen votes such that Prof. V.Moretti who is an expert in such topics (among other things) can expand his interesting comment below your answer to an answer too? BTW is it really true that in GR applied coordinate transformations can change the signature of the metric as you say, for example from (1, -1, -1,-1) to (1,1, -1,-1) and the choice of time and space directions can be choosen arbitrary? That would surprise me ... – Dilaton Jun 29 '14 at 11:21
  • @Dilaton: I have already voted to reopen. When did I claim coordinate transformations could change the signature? All I said (well, implied) was that they could change which dimension was timelike. – John Rennie Jun 29 '14 at 11:35
  • @JohnRennie (2 above), I had thought about doing that, but apparently I'm on notice for low quality posts (because my only 2 questions are on-hold/closed), and want to try getting them re-opened so I don't have to worry about getting banned from asking questions – Freudian Slip Jun 29 '14 at 18:32
  • @FreudianSlip you won't get banned from asking questions, we don't have that feature enabled on this site. And asking a new, good question has a significant effect on how close you are to any sort of automatic question restrictions. I agree that it would be more appropriate to roll this question back and ask a new one with the edited version. – David Z Jun 29 '14 at 20:16
  • @DavidZ Understood – Freudian Slip Jun 29 '14 at 20:35
  • New question: http://physics.stackexchange.com/q/122392/2451 – Qmechanic Jun 29 '14 at 21:13

1 Answers1

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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.

John Rennie
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    Flat spacetime in non-Cartesian coordinates also has nonzero Chrisoffel symbols. For example in spherical coordinates, 9 of the 64 symbols are nonzero. Not that this thwarts the argument. –  Jun 26 '14 at 09:59
  • @ChrisWhite: yes, true. I'll clarify my answer. – John Rennie Jun 26 '14 at 10:02
  • The only condition I can think of that would cause a difference in the speed of time would be gravity, and that already introduces an acceleration. I'm just curious about whether time dilation adds any additional acceleration to a system – Freudian Slip Jun 26 '14 at 14:28
  • @FreudianSlip: gravity generally curves both time and space, but if you could come up with a gravitational field that only curved time then yes it would cause acceleration even in flat space. In other words time curvature alone can cause acceleration. The only problem is I don't know any way to curve time without curving space as well. The other way round is easy as the FLRW metric curves space but not time. – John Rennie Jun 26 '14 at 15:58
  • @JohnRennie: Wait, how can it curve space but not time? Wouldn't that change the local speed of light? I thought the whole reason that both time and space needed to curve were to preserve the constancy of the speed of light. – Freudian Slip Jun 28 '14 at 18:50
  • @FreudianSlip: the local speed of light is always $c$ because spacetime is always locally flat. – John Rennie Jun 28 '14 at 18:53
  • @JohnRennie: I realize you must be correct, my ignorance on the subject is astounding. But if something can curve space and not time, then the only thing you would need to introduce a curvature in time w/o space would be to negatively curve the space in an area. Or is that the part that isn't possible? – Freudian Slip Jun 28 '14 at 19:04
  • The OP is NOT of the mark, as what goes into the Einstein equations is curvature of spacetime and not curvature of space only. – Dilaton Jun 28 '14 at 21:25
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    It is a pity that i cannot answer. The answer is important because it reflect exactly what is going on in our actual universe! Our universe is a FLRW spacetime with (large scale) flat spatial sections! However galaxies accelerate, in this sense acceleration is due to "time curvature". – Valter Moretti Jun 29 '14 at 09:54
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    @V.Moretti: I was a bit surprised the question was closed. It's a confused question but the underlying concept is interesting. Anyhow I've voted to reopen - we just need two more votes. – John Rennie Jun 29 '14 at 09:58
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    I also voted to reopen. – Valter Moretti Jun 29 '14 at 09:58
  • @JohnRennie I've rolled back my edits and asked the revised version as a new question, in case anybody wants to take a crack at it. http://physics.stackexchange.com/questions/122392/can-a-curvature-in-time-and-not-space-cause-acceleration – Freudian Slip Jun 30 '14 at 02:27