As the old saying goes, "It ain't the heat, it's the humidity". In this case you can say "It ain't the force, it's the potential". Two clocks, originally synchronized at some point, begin to show a difference in tick speed when one is moved to a different gravitational potential. This has been experimentally verified, originally, by researchers at Harvard University and, more recently, by the GPS clocks orbiting the earth. Now the gravitational force at the exact point you suggested is indeed zero, but the potential is not. You have to calculate the amount of energy one needs to move the test clock out to where the standard clock is residing, usually taken to be infinity. While the force at the exact midpoint is zero, as you move away from it the force begins to grow at first then, when the distance is bigger than several radii away from the orbiting pair, begins to fall off as 2Gm/r2
. The potential, the total integral of Fdr, is finite and hence will affect the clock rates.
(published on 03/30/2010)