Effects on time by gravity or motion

[pervect]

I suspect the usual confusion here, though I can't follow the original poster's argument well enough to be positive I'm right. (There are other issues that cause confusion than the one I'll present, but the one I'll present is very common).

The "usual issue" is that people assume, both explicitly and/or implicitly, that the notion of synchronizing clocks is independent of the observer.

The notion of "synchronizing clocks" is needed, for instance, to determine the idea of "now". When we have a model of space-time, "now" is the set of points that occur "at the same time".

The notion of synchronizing clocks, the notion of "now" in special relativity, is observer dependent. Trying to understand special relativity without realizing this issue inevitably leads to confusion. The bad assumption that causes much confusion is to assume that the notion of "now" independent of the observer, that everyone agrees on what "now" is.

An implicit assumption of the existence of "now" is needed to talk about the rate at which clocks tick when they are at different locations. We compare the time on one clock "now" to the time on a clock at a different location "now".

This usually first shows up in the twin paradox, in flat space-time - a much easier topic to talk about than gravity.

Basically, the only way that in A's frame of reference that B's clock can run slow, and that in B's frame of reference, A's clock runs slow, is when A and B have different notions of "now".

There's a name for this issue, it's called the "relativity of simultaneity". However, just giving the name of the issue doesn't explain it enough so that people who are not already aware of the issue understand it. In general, it seems very hard to talk about this issue in a way that will be understood, but I keep trying.

Things get very complicated if someone tries to understand general relativity without understanding this feature of special relativity. Special relativity is much easier to talk about.

 

[PAllen]

Some of this will be beyond the OP, but I thought it would be useful to quantify 3 effects in question in this thread:

1) Time dilation between two clocks, each with constant proper acceleration, maintaining fixed distance from each other.

2) time dilation (in comparison to a system of stationary clocks as in (1)) due to change in tangential velocity with altitude of clocks comoving with the surface of the earth.

3) Tidal gravity corrections to (1), that is how much the time dilation from top to bottom of a building is diffferent from in a uniformly accelerating rocket of the same altidude.

The relevant quantity I compute is (d/dx)(dτ/dt) for physically appropriate coordinates. This gives a rate of time rate change per meter (in the units I use).

For (1), the exact value of this quantity is simply g/c². This is about 10^-16, thus time rate changes about 1 part in 10¹⁶ per meter. Note that this is the only one of these effects that has been observed on the surface of earth. Even the latest research clocks are not yet precise enough to measure either of the other effects. Let us call (1) the Rindler time dilation as distinct from (3).

For (2) the exact value of this quantity is -vϒω/c², where omega is radians per second for Earth's rotation. If you compare this with (1), specifically (1)/(2), with the approximation that gamma near Earth is close to 1, you get g/rω², which is about 300. Thus this effect is 300 times smaller than the Rindler time dilation. This is currently undetectable, but another order of magnitude improvement research clocks should make this detectable.

For (3), assuming near earth, the time rates are near 1 compared to Schwarzschild t, and that changes in r are very close to physical distance, then the derivative quantity is given by:

(R/2r²) (1 - R/r)^-.5

where R is the Schwarzschild radius and r is the SChwarzschild r coordinate. Noting that with Newtonian approximation, g=GM/r²; and R = 2GM/c², we can write this as:

(g/c²) (1 + gr/c²), using one term taylor expansion of the square root.

This shows the tidal correction to Rindler dilation is about 6*10^-10 times the Rindler dilation. This also means it is about 5 million times smaller than the tangential velocity dilation change with altitude. There is no likelihood of directly detecting this correction near the Earth's surface, in the foreseeable future.