Einstein in Your Pocket
⏱️ Relativity
Question
Does relativity actually matter for your phone?
GPS satellites carry atomic clocks, and your position is computed from how long their signals take to reach you. But their clocks tick at a different rate than ground clocks, for two reasons: special relativity (they move at 3.9 km/s, which slows their clocks by ~7 microseconds per day) and general relativity (they are 20,200 km up where gravity is weaker, which speeds their clocks by ~45 microseconds per day). Net: +38 microseconds per day. The satellite firmware is pre-corrected for this before launch.
The math
SR: −v²/(2c²) = −7.2 µs/day. GR: +GM/c² × (1/R − 1/r) = +45.7 µs/day. Net +38.5 µs/day. Light travels ~300 m per microsecond, so uncorrected the position error grows by ~11.4 km EVERY DAY. The satellite clocks are pre-set to 10.22999999543 MHz (not 10.23 MHz) so they tick at exactly the right rate in orbit.
Why it matters
Your phone's GPS fix is a relativity experiment you run every day. If Einstein were wrong, GPS would be useless within a day - and yet it works, to the metre, for billions of people.