07. Satellite clocks: fast or slow?

Compared to a clock on the ground, does an astronaut’s clock on the ISS (400 km up) run fast or slow? Same question for a GPS satellite clock (20,200 km up). Is the sign even the same?

Solution

Opposite signs: the ISS clock runs slow by about $25\ \mu\mathrm{s}$ per day; a GPS clock runs fast by about $38\ \mu\mathrm{s}$ per day.

Two effects compete. Sitting higher in the potential makes a clock run faster (gravitational blueshift, about $+46\ \mu\mathrm{s}/\mathrm{day}$ for GPS, only $+4$ for the ISS); orbiting fast makes it run slower (about $-7\ \mu\mathrm{s}/\mathrm{day}$ for GPS, $-28$ for the low, fast ISS). For a circular orbit both effects pack into one clean formula, $d\tau/dt = \sqrt{1 - 3GM/rc^2}$, to be compared against a ground clock’s $\sqrt{1 - 2GM/Rc^2}$ (ignoring Earth’s spin, a fraction-of-a-microsecond refinement).

Setting them equal gives the crossover at $r = \tfrac{3}{2}R$ — about 3,200 km of altitude. Below that, speed wins and orbiting clocks lag; above it, potential wins and they lead. GPS engineering does not get to debate whether relativity is real: uncorrected, the system’s position error would grow by roughly ten kilometers per day.

Deeper in the notebook: 02. Time Dilation and Clocks (GPS) · 01. Gravitational Redshift