It's due to Gravitational Time Dilation -
https://en.wikipedia.org/wiki/Gravitational_time_dilation

According to the calculator at
https://www.omnicalculator.com/physics/gravitational-time-dilation clocks
on the moon tick about 6.6e-10 faster than those on Earth, or about 0.057ms
per (Earth) day.

On Tue, 31 Jan 2023 at 12:58, Kevin Rowett via time-nuts <
time-nuts@lists.febo.com> wrote:

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@lists.febo.com wrote:

Anybody studied the influence of the Sun's gravity on clocks in GNSS satellites? The field might change slightly by 40,000 km distance when the sat is closer to the Sun than later on the opossite side of Earth. Is this measurable on the clocks?

.md

Kevin,

That moon link is making the rounds through the 'net. It's an
interesting topic what to do with SI units and UTC timekeeping for Mars
and Moon.

and how much.

1. The obvious example is pendulum clocks since they are directly
   related to g. Period T = 2pi sqrt(L/g) so a 1 ppm change in g is a 5e-7
   change in frequency. This is why you have to recalibrate a good pendulum
   clock if you move it up or down a few floors (g depends on elevation).
   The same is true if you move it north or south (g depends on latitude).
   The very best pendulum clocks are also affected by tides (local g
   changes due to sun and moon).

2. OCXO are also sensitive to acceleration, including g, the
   acceleration of gravity. Check the 2g-turnover spec on the datasheet.
   It's not uncommon to see 1e-10 or 1e-9, which means its frequency
   changes by 1 ppb if you rotate (literally, turn over) the oscillator.
   This is why you want to place your best quartz frequency standards on a
   very solid table or rack or floor, without vibration, and especially do
   not tilt them, not even by a millimeter.

3. The best atomic clocks and all optical clocks are good enough that
   relativistic effects appear (both velocity and gravity). The frequency
   changes by 1.1e-16 per meter change in elevation. This is why you can
   demonstrate gravitational time dilation using portable cesium clocks and
   a mountain. Go up one km, stay for a day, and the clock comes back 10 ns
   ahead (older) compared to the clock you left at home.

The UTC timescale is based on the SI second defined at sea level on
planet earth. Cesium clocks tick at a different rate depending on the
mass and radius of the planet. So compared to a clock freely floating in
space far far away from any mass, a clock at the surface of the earth
runs 6.95e-10 slower, which is about 60 us per day, and a clock on the
surface of the moon runs 3.13e-11 slower, which is about 2.7 us/day. The
difference is about 57 us/day, as the article mentions.

/tvb

On 1/30/2023 12:34 PM, Kevin Rowett via time-nuts wrote:

On Tue 2023-01-31T15:34:16+0100 Marek Doršic via time-nuts hath writ:

IAU 2000 resolutions B1.3 and B1.5 codify the understanding of the
spacetime transformations from the potentials and metric.

https://www.iau.org/static/resolutions/IAU2000_French.pdf

This particular effect is small, and I am not sure that the GPS clocks
are stable enough to reveal it.

--
Steve Allen                    sla@ucolick.org              WGS-84 (GPS)
UCO/Lick Observatory--ISB 260  Natural Sciences II, Room 165  Lat  +36.99855
1156 High Street              Voice: +1 831 459 3046        Lng -122.06015
Santa Cruz, CA 95064          https://www.ucolick.org/~sla/  Hgt +250 m

Hi

Early on the design of the folks in charge didn’t think that the impact on the GPS clocks would matter.
Somebody did the math and found that the tuning range on the Cs standards was not adequate to
compensate for the relativity issues. They modified the standards before they got to far down the
production process.

How accurate is that? I heard the same story from multiple folks back in the late 70’s early 80’s. Each
version was pretty specific about names and numbers. I’ve always assumed it was true.

Bob

Hi

If you want to look at TV shows, there’s one staring a vehicle loaded to the gills
with 5071 Cs standards and batteries. Our very own TVB also stars in it along
with the Cs standards and his car…..

As far as I know, he’s pretty much the pioneer in amateur implementations of
atomic clock relativity experiments.

Bob