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Relativityundergraduate

Why GPS Needs Both Relativities

Also known as: GPS clock correction · Satellite time drift

A GPS satellite clock runs fast by 45 microseconds/day because it sits higher in Earth's gravity well (GR), and slow by 7 microseconds/day because it orbits fast (SR). The net +38 microseconds/day, uncorrected, would shoot your position off by ~11 km per day. GPS only works because engineers pre-tune the clocks for relativity.

Δtt=ΔUc2⏟GR, faster−v22c2⏟SR, slower≈+38 μs/day\frac{\Delta t}{t} = \underbrace{\frac{\Delta U}{c^2}}_{\text{GR, faster}} - \underbrace{\frac{v^2}{2c^2}}_{\text{SR, slower}} \approx +38\ \mu\text{s/day}
Live simulation
warming up the physics…

A satellite orbits Earth while a running counter accumulates the net +38 microseconds/day; bars show the GR (faster) and SR (slower) contributions, with the orbital radius slider shifting their balance.

Equivalent forms

gr term
ΔtGRt=GMc2 ⁣(1RE−1r)\frac{\Delta t_{GR}}{t} = \frac{GM}{c^2}\!\left(\frac{1}{R_E}-\frac{1}{r}\right)
sr term
ΔtSRt=−v22c2\frac{\Delta t_{SR}}{t} = -\frac{v^2}{2c^2}
The most everyday proof of relativity: two competing effects that almost cancel, but not quite — and your phone depends on the leftover.