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 $+45.7\ \mu\mathrm{s}/\mathrm{day}$ for GPS, only $+3.5$ for the ISS); orbiting fast makes it run slower (about $-7.2\ \mu\mathrm{s}/\mathrm{day}$ for GPS, $-28.3$ for the low, fast ISS). The two are not independent: on a circular orbit $v^2 = GM/r$, so the speed toll is always exactly half the satellite’s own potential term, and the contest reduces to comparing $2/R$ with $3/r$. For a circular orbit everything packs into one clean formula, $d\tau/dt = \sqrt{1 - 3GM/rc^2}$ — and the 3 is no fudge: a circular Schwarzschild geodesic obeys $\Omega^2 = GM/r^3$ exactly, so $d\tau^2 = \left(1 - \tfrac{2GM}{rc^2}\right)dt^2 - \tfrac{r^2\Omega^2}{c^2}\,dt^2$ collapses with the 2 from the potential and one more $GM/rc^2$ from the orbital speed. Compare 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, satellite time would walk away from ground time by those $38\ \mu\mathrm{s}$ — eleven kilometers of light-travel — every single day. (A constellation-wide offset partly hides in the receiver’s own clock solve; what fails outright is the tie of GPS time to UTC and the satellite-to-satellite terms, which is why the correction is designed in.)

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