Relativity and the Global Positioning System

Satellite navigation is the one place where general relativity is an engineering requirement rather than a laboratory curiosity. A GPS receiver fixes its position by timing signals from several satellites and multiplying light-travel times by $c$; a timing error of $\delta t$ becomes a position error of $c\delta t$, so $10\text{nanoseconds}$ of clock error is about $3\text{metres}$ on the ground. The satellite clocks experience two relativistic rate shifts relative to a clock on the Earth's surface, and the two act in opposite directions: special-relativistic time dilation from the satellite's speed slows it down, and the general-relativistic redshift from its higher position in the Earth's potential speeds it up. The gravitational effect wins, leaving a net gain of about $38\mu \text{s}$ per day. That is enormous by navigation standards, and the system is built to remove it.

The two shifts come straight from the results of the previous lessons: the moving clock from special-relativistic time dilation, and the higher clock from the gravitational redshift. The signature is $(-,+,+,+)$.

The two clock shifts

A GPS satellite orbits at radius $r=2.656\times 10^7\text{m}$ (about $20,200\text{km}$ altitude), with orbital speed

Compare its clock to one at rest on the Earth's surface at $R_{\oplus }=6.371\times 10^6\text{m}$. To first order in $v^2/c^2$ and $GM/(c^{2}r)$, the fractional rate of a clock combines the two effects:

the $-v^2/2c^2$ from motion and the $+\Phi /c^2$ from gravitational potential $\Phi =-G{M}_{\oplus }/r$. Taking the difference between satellite and ground clock gives the fractional rate offset

The two terms carry opposite signs and are evaluated separately below. The Earth's rotation adds the ground clock's own small speed; keeping only the leading terms, the two dominant contributions are the following.

• Special relativity (speed). The satellite moves, so its clock runs slow:$$-\frac{v^2}{2c^2}=-\frac{(3.87\times 10^3{)}^2}{2(3.00\times 10^8{)}^2}=-8.35\times 10^{-11}.$$
  Over one day ($86,400\text{s}$) this is $-7.2\mu \text{s}$: the moving clock loses about seven microseconds a day.
• General relativity (potential). The satellite sits higher in the well, so its clock runs fast. With $G{M}_{\oplus }/c^2=4.435\times 10^{-3}\text{m}$,$$\frac{G{M}_{\oplus }}{c^2}\left(\frac{1}{R_{\oplus }}-\frac{1}{r}\right)=(4.435\times 10^{-3})(1.570\times 10^{-7}-3.765\times 10^{-8})=+5.29\times 10^{-10}.$$
  Over one day this is $+45.7\mu \text{s}$: the higher clock gains about forty-six microseconds a day.

The gravitational speedup is roughly six times the special-relativistic slowdown, so it dominates. The net rate offset is

a net gain of $38.5\mu \text{s}$ per day. The satellite clock, left alone, runs fast by about $38\mu \text{s}$ each day relative to the ground.

Why the correction is not optional

A drift of $38\mu \text{s}$ per day is $3.85\times 10^{-5}\text{s}$ of clock error accumulated in $86,400\text{s}$, a fractional rate of $4.46\times 10^{-10}$. Multiplied by $c$, the ranging error grows at

The navigation tolerance is a few metres, so the error crosses it in well under a minute; over a full day it reaches

An uncorrected system would be useless within minutes and wrong by kilometres within a day.

The built-in offset

The correction is applied before launch. A satellite clock nominally at $10.23\text{MHz}$ is set instead to a slightly lower frequency, offset by the fractional amount $-4.465\times 10^{-10}$, i.e. to about $10.2299999954\text{MHz}$. On orbit, where the clock runs fast by that same fraction, it then ticks at the ground rate, and the timing solution stays consistent to the nanosecond. The offset is fixed, not adjusted in flight, because the dominant rate shift is constant for a circular orbit.

Two smaller relativistic effects remain and are handled in the receiver software.

• Orbital eccentricity. A real orbit is slightly elliptical, so the satellite's speed and potential vary around it and the net rate is not perfectly constant. The residual is a periodic term the receiver corrects using the satellite's position; for typical GPS eccentricity it reaches tens of nanoseconds.
• The Sagnac effect. The Earth rotates during the signal's travel time, so the receiver has moved by the time the signal arrives. This rotation-of-frame correction is applied when the light-travel times are converted to a position in the Earth-fixed frame.

The relativistic terms in GPS are not corrections to a Newtonian design; they are the design. The system keeps time to the nanosecond across a fleet of clocks moving at kilometres per second and spread through a potential well, and it works because the metric's time component was accounted for at the level of parts in $10^{10}$. Every position fix a receiver reports is a continuous confirmation of the gravitational redshift.¹²