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

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.

• 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. Jun 26, 2014 at 21:15
• I have once again (!) no idea why in the world this legitimate beginner GR question is considered not mainstream by 5 closevoters. Jun 28, 2014 at 21:23
• 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. Jun 28, 2014 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. Jun 29, 2014 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 ... Jun 29, 2014 at 11:21

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.

• 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.
– user10851
Jun 26, 2014 at 9:59
• @ChrisWhite: yes, true. I'll clarify my answer. Jun 26, 2014 at 10:02
• 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". Jun 29, 2014 at 9:54
• @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. Jun 29, 2014 at 9:58
• I also voted to reopen. Jun 29, 2014 at 9:58
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