The science
Clocks run slower the deeper they sit in Earth's gravity. A clock a height h higher runs faster by g·h/c², about 1.9 × 10⁻¹⁶ for the 1.7 m between your feet and your head. Over 80 years (2.5 × 10⁹ s) your head ages about 0.47 microseconds more than your feet.
GPS satellites orbit about 20,200 km up. Weaker gravity there makes their clocks gain about 45.7 µs a day; their orbital speed makes them lose about 7.2 µs; net, about +38.5 µs a day. Light travels about 11.5 km in that time, so without correcting for relativity, positions would drift by around 10 km a day. The satellites' clocks are deliberately set to tick slow before launch to cancel it.
Sources
- Chou et al., Science 329, 1630 (2010): optical clocks and relativity, a 33 cm height difference.
- Ashby, Living Reviews in Relativity 6, 1 (2003): relativity in the Global Positioning System.
