You can get the ratio of clock rates relatively easily by knowing GR.

Start with the Schwarzschild solution to EFE:

Use the fact that

Then:

produces, by division with

:

Write the above for two locations , at radial distances

and

and you get:

Divide the two:

where

i=1,2. If the orbit is circular, then

but this is not the usual case.

The above equation relates the proper times of two clocks , located at different radial distances and moving at different angular speeds. It is the foundation of the calculations for setting up the frequencies of the GPS clocks (such that they are synched up).

Since the above (exact) formula is rather complicated, the GPS calculations use a power series approximation, making use of the fact that

. The formula encapsulates the effects in:

1. gravitational potential difference , via the term in

2. radial speed difference (if it exists), via the term in

3. angular speed difference, via the term in

In general, the formula is expressed in terms of ratios of frequencies, not clock rates: