If you put an atomic clock on the moon and left it for 4 billion years what would be the time difference between that clock and one on Earth?

Answer 1

The derivation of the equation needed to answer your question is a bit intense, so I'm going to skip it and just write down the result:

∆tE/√[1 - (3GMi/ri)/c2 - v2/c2] = ∆tM.

∆tE is the elapsed time of an atomic clock on Earth.

∆tM is the elapsed time of an atomic clock on the Moon.

G is the gravitational constant = 6.67428 X 10-11 m3/(kg s2)

c is the speed of light in a vacuum = 299,792,458 m/s

v is the average orbital speed of the Moon = 1022 m/s

Mi and ri are the sums of the masses and radial distances, respectively, of all of the objects with a gravitational influence on the moon. In this answer, I'm going to approximate these two values by only considering the masses and radial distances for the Earth, the Moon, and the Sun. Because of this approximation, the value for ∆tM that I'm going to list below will a bit lower than the actual value.

So, after plugging in all of the numbers into the above equation, with ∆tE = 4 billion years, the value for ∆tM is 4000000059.5 years. So the atomic clocks would be about 59.5 years apart.

Answer 2

The time difference between an atomic clock on the moon left for 4 billion years and one on Earth would be extremely small due to the high precision of atomic clocks. Factors such as gravitational time dilation and relative motion would cause a difference of only a few milliseconds at most.