High School Uniform-gravitational time dilation -- exact or approximate

[pervect]

Let's try the super-simple version. People usually mean the Rindler metric when they talk about a uniform gravitational field. If you mean something else, it's up to you to describe what you mean exactly.

The mathematical techniques usually used to describe a gravitational field are the associated metric. Give the metric, one requires that the metric satisfy Einstein's field equations. So if one wants to use some other notion of what a "uniform gravitational field" might be, other than the Rindler metric, one should write down the metric to communicate what this idea is.

This presuposes that one knows what a metric is. The situation if one doesn't know what a metric is problematic. The problem becomes - how does one verify that the proposed idea of a "uniform gravitaitonal field" is actually consistent with Einstein's field equations? WIthout the proper tools, one can't really answer this question.

The problem of time dilation on the Earth is interesting as well, but of course the gravitational field of the Earth isn't uniform field - loosely speaking, one might say it gets weaker as you get further away from the center of the Earth. It's perfectly valid though to find the appropriate time dilations for this case, using the Schwarzschild metric, and to explore the limit of what happens when the distances are short enough that the deviations from the field are almost uniform.

The basic answer for what happens when one takes the limit has already been hinted at (1+gh)(1-gh) = 1 - (gh)^2, and to linear order (gh)^2 = 0. So to linear order the "time dilations" are reciprocal. So things work out to linear order when one uses approximations valid to the linear order.

 

[PAllen]

Let's try the super-simple version. People usually mean the Rindler metric when they talk about a uniform gravitational field. If you mean something else, it's up to you to describe what you mean exactly.

The mathematical techniques usually used to describe a gravitational field are the associated metric. Give the metric, one requires that the metric satisfy Einstein's field equations. So if one wants to use some other notion of what a "uniform gravitational field" might be, other than the Rindler metric, one should write down the metric to communicate what this idea is.

This presuposes that one knows what a metric is. The situation if one doesn't know what a metric is problematic. The problem becomes - how does one verify that the proposed idea of a "uniform gravitaitonal field" is actually consistent with Einstein's field equations? WIthout the proper tools, one can't really answer this question.

The problem of time dilation on the Earth is interesting as well, but of course the gravitational field of the Earth isn't uniform field - loosely speaking, one might say it gets weaker as you get further away from the center of the Earth. It's perfectly valid though to find the appropriate time dilations for this case, using the Schwarzschild metric, and to explore the limit of what happens when the distances are short enough that the deviations from the field are almost uniform.

The basic answer for what happens when one takes the limit has already been hinted at (1+gh)(1-gh) = 1 - (gh)^2, and to linear order (gh)^2 = 0. So to linear order the "time dilations" are reciprocal. So things work out to linear order when one uses approximations valid to the linear order.

In an early post in the thread, I already exactly worked out the Rindler case, showing the linear formula is exactly correct in reference to any given Rindler observer, producing exact reciprocals due to the precise way each observer's proper acceleration differs.

The wikipedia link posted a few posts ago also establishes the exactness of the linear formula for Rindler using a different method.

 

[PAllen]

I thought it might be nice to post order of magnitude numbers on the change of proper acceleration of stationary observers per altitude for Earth surface conditions. Assuming I have not made any arithmetic errors,

1) If the Earth surface situation were described exactly by the Rindler metric, the change in g per meter of altitude (using meters and seconds) would be about 10^-16 per meter (subtracted from surface g).

2) Assuming Newtonian gravity is exact, it would be about 3*10^-7 per meter (subtracted from surface g). Obviously, the Rindler effect is wholly insignificant against this.

3) The Schwarzschild correction to the Newtonian figure is of order 10^-6 (that is, given a precise Newtonian figure for per meter change, it would be modified by order of 1 part in 10⁶ for Schwarzschild metric). Thus the Schwarzschild correction to the Newtonian figure is still much larger than the Rindler figure.

Thus you can see, Rindler is about as close as you can possibly get to uniform gravity.

 

[pervect]

In an early post in the thread, I already exactly worked out the Rindler case, showing the linear formula is exactly correct in reference to any given Rindler observer, producing exact reciprocals due to the precise way each observer's proper acceleration differs.

No argument here. But sometimes people have funny ideas of what a "uniform gravitational field" really is. Working out the Rindler case exactly won't help if their mental idea of what a "uniform gravitational field" is some other metric than the Rindler metric. Unless both parties can agree on the interpretation of what a "uniform gravitational field" is, they talk past each other.

 

[sweet springs]

Hi. Uniform is not necessary but stationary is a more appropriate condition to deal with, I think. Re:>>30. Zero tx,ty and tz components of metric also.