just put two cesium clocks, one on the moon and one on earth the will diverge by 21 milliseconds per year
from ChatGPT:
Approximate Calculation:
The difference in the gravitational potential between the Earth and the Moon can be estimated. The time dilation factor is given by:
tMoontEarth1+c2
tEarthtMoon1+c2
where:
is the difference in gravitational potential,
cc is the speed of light.
For Earth:
Earth=GMEarthREarth
Earth=REarthGMEarth
For the Moon:
Moon=GMMoonRMoon
Moon=RMoonGMMoon
Where:
GG is the gravitational constant,
MEarthMEarth and MMoonMMoon are the masses of the Earth and the Moon,
REarthREarth and RMoonRMoon are the radii of the Earth and the Moon.
Given these, you can compute the difference in the potential, and hence the time dilation. The gravitational potential difference is dominated by Earth's mass, which is much larger than the Moon's.
Result:
After computing the relevant factors, the clock on the Moon would run slightly faster than the clock on Earth. The difference is approximately 0.66 parts per billion.
Over the course of a year (which is about 3.16×1073.16×107 seconds):
t0.66×109×3.16×107 seconds
t0.66×109×3.16×107 seconds
This results in a divergence of approximately 21 milliseconds over the course of a year, with the clock on the Moon running ahead of the one on Earth.
This is a rough estimate; more precise calculations would need to account for various other factors, but this gives a good ballpark figure.
