[time-nuts] Re: gravity fields affect time keeping?

[alan bain]

It's a consequence of general relativity.

The simplest way I can think to answer this question is to think of a
point mass in a spherically symmetric situation  (as one would expect
around a point mass in a vacuum) and solve Einstein's equation which
in this case (point mass is handy) is just

G_{\mu\nu}=0

After some boring algebra with tensors in spherical co-ordinates which
some examiners seem to think it is interesting to see if you can
reproduce, you arrive at a metric

ds^2 = - (1-r_s/r) c^2 dt + dr^2/ (1-r_s/r) + r^2 (dtheta^2 + sin^2
theta dphi^2)

r,theta,phi are usual spherical co-ordinates, c is the speed of light
in vacuum and the Schwartzchild radious r_s = 2 G M /c^2 (which is
found by comparing with Newtonian Gravitation, G is Newtonian
coefficient, M is mass of point mass). So if we stay in the same place
and compare time

ds^2 = - (1-r_s/r) c^2 dt

And ds^2 = -c^2 d (tau)  where tau is time measured at a stationary
point arbitrarily far from the mass.

So

\delta t_{on planet} = \delta t_{clock infinitely far away} sqrt ( 1-r_s/r).

So the closer one gets to the point mass the slower time goes.

I don't really know any maths-free explanation of this (unlike for say
special relativity when there are good no-maths explanations). Would
like to know one if someone does...

Alan

On Tue, 31 Jan 2023 at 13:00, Kevin Rowett via time-nuts
<time-nuts at lists.febo.com> wrote:
>
> From an article about moon time keeping:https://www.nature.com/articles/d41586-023-00185-z <https://www.nature.com/articles/d41586-023-00185-z>
>
> The author says
>
>
> “...Clocks on Earth and the Moon naturally tick at different speeds, because of the differing gravitational fields of the two bodies. …”
>
> I’m curious about what type of clocks are affected by local gravity, and how much.
>
> Anyone familiar enough to go into detail?
>
> KR
>
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