Question of three clocks
Imagine three ideal clocks.
Clock A is on the surface of the Earth. The other two, B and C, are side-by-side in space, remotely distant from the Earth's gravitional well. As I understand it, clock A on the Earth's surface will be running slower than the two out in space, which will both be running at the same rate.
Now imagine that the two in space become separated. Clock B is tethered so stays the same distance from Earth. Clock C is not, and falls very slowly, picking up spee
With the rotations and boosts above of the Lorentz tranformations we can now examine the different clocks under proper acceleration which I will detail below for each case. (I will need to set one observer at CoM instead of sea level for simplicity)
four velocity
\[u^\mu=\frac{dx^\mu}{d\tau}=(c\frac{dx}{d\tau},\frac{dx}{d\tau},\frac{dy}{d\tau},\frac{dz}{d\tau})\]
\[A=\frac{du}{d\tau}\]
without going through all the relations a constant acceleration along the x direction with