# Effect of gravity on time dilation.

1. Sep 22, 2011

Can anyone tell me What is the relationship between the orbital radius of a satellite and the time gained per second due to its weaker gravity at its altitude?
In relation to that how much time does the International Space station lose per-second due to weaker gravity at its altitude?
Also What about a GPS? How much time does it gain if it is on Earth?

2. Sep 23, 2011

### Bill_K

The apparent clock rate of a satellite will be affected firstly by its orbital motion. From Newtonian mechanics and assuming a circular orbit, mv2/r = GMm/r2, implying the orbital velocity is v2/c2 = GM/c2r and γ = 1/√(1-v2/c2) ≈ 1 + GM/2c2r.

According to Wikipedia the Schwarzschild radius of the Earth is 2GM/c2 = 8.9mm. The radius of the Earth is r = 6378 km, so for a surface-skimming satellite, γ ≈ 1 + 3.5 x 10-10.

Secondly, the effect of the gravitational potential. ds2 = (1-2GM/c2r) dt2, so ds/dt = √(1-2GM/c2r) ≈ 1 - GM/c2r. Larger by a factor of two, but what matters here is that you must take the difference between the gravitational potential at the surface vs the potential at the orbit.

3. Sep 23, 2011

### ?????

See attached.

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