GPS and relativity
This is an extension of the school topics of time dilation and the gravitational field — beyond the syllabus. Change the satellite's speed and altitude: the special-relativity effect (motion) slows the clock, the general-relativity effect (gravity) speeds it up; the net result is about +38 μs/day. Without correction the position error grows by kilometres each day.
I will give you a target daily clock correction — tune the speed v and altitude h to reach it. Δt = Δt(GR) + Δt(SR).
What just happened?
A clock on the satellite ticks differently than on Earth for two independent reasons. Motion (special relativity) slows it down, while the weaker gravitational field high above Earth (general relativity) speeds it up. For the GPS orbit gravity wins and the clock runs fast by about +38 μs per day.
Special-relativity correction (motion):
— the faster the clock moves, the slower it ticks (time dilation). This is a negative term: the satellite “loses” time relative to the surface.
General-relativity correction (gravity):
— the weaker the gravitational field (higher above Earth), the faster the clock ticks. This is a positive term. Both effects have opposite signs, so they partly cancel.
Net effect and position error:
GPS determines position from the time the signal takes to reach the receiver. An uncorrected 38 μs is the light path — a dozen-odd kilometres of error every day. That is why the satellites’ atomic clocks are deliberately “detuned” at the factory, so that once relativity is accounted for they keep correct time.
Why this matters for your exam:
GPS is the cleanest example that both theories of relativity operate in everyday technology — they are not an abstraction. It extends the topic of time dilation (the term) and the effect of the gravitational field on the passage of time (the term): you can state the sign of each effect and compute its scale.
Remember: motion slows the clock (−), a weaker gravitational field speeds it up (+); for GPS the sum ≈ +38 μs/day, and the position error is c·Δt ≈ 11 km per day.
Theory & formulas
Two relativistic effects
A GPS satellite clock runs differently than on Earth for two reasons:
- motion (SR) slows it down,
- the weaker gravitational field in orbit (GR) speeds it up.
Relative components
Slowdown from speed and speed-up from gravity:
Net result
In the GPS orbit the gravitational effect dominates — net the satellite clock runs fast by about per day. Without correction the position error would grow by a dozen-odd kilometres per day, so the clocks are “detuned” at the factory.
Symbols: — relative change of the clock rate, — satellite speed, — orbit altitude, — gravitational acceleration, .
Frequently asked questions
›Why does a clock in orbit run differently?
Motion (SR) slows it by −v²/2c², and the weaker gravitational field (GR) speeds it up by +gh/c².
›Which effect is larger?
In the GPS orbit the gravitational effect dominates — net the clock runs faster (about +38 μs/day).
›How does GPS correct for it?
The satellite clocks are “detuned” at the factory so that, once relativity is accounted for, they show the correct time.
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