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

[john.haine at haine-online.net]

There's an excellent book called Gravity From The Ground Up by Bernard Schutz:

 

https://www.amazon.co.uk/Gravity-Ground-Up-Introductory-Relativity/dp/0521455065/ref=sr_1_1?crid=253QEUS0PJDXU <https://www.amazon.co.uk/Gravity-Ground-Up-Introductory-Relativity/dp/0521455065/ref=sr_1_1?crid=253QEUS0PJDXU&keywords=gravity+from+the+ground+up&qid=1675248211&s=books&sprefix=gravity+from%2Cstripbooks%2C82&sr=1-1> &keywords=gravity+from+the+ground+up&qid=1675248211&s=books&sprefix=gravity+from%2Cstripbooks%2C82&sr=1-1

 

…which gives a fairly straightforward derivation of the small-field approximation to GR including gravitational time dilation – actually the latter comes just from a very simple thought experiment with observers and falling clocks, and once you have that the small field approximation follows quite easily.  That is a/k/a Newtonian Mechanics…  The book is quite expensive but there’s a lot of good reading to keep one busy.

 

John.

 

-----Original Message-----
From: alan bain via time-nuts <time-nuts at lists.febo.com> 
Sent: 31 January 2023 13:40
To: Discussion of precise time and frequency measurement <time-nuts at lists.febo.com>
Cc: alan bain <alan.bain at gmail.com>
Subject: [time-nuts] Re: gravity fields affect time keeping?

 

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 < <mailto:time-nuts at lists.febo.com> 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> 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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