General Relativity and GPS

The Global Positioning System is the rare piece of infrastructure that would fail — not degrade, fail — without general relativity. Twenty-four-plus satellites orbit near $20{,}200\,\mathrm{km}$ altitude, each carrying an atomic clock, and a receiver fixes its position by trilateration: it times signals from four or more satellites and solves for three spatial coordinates plus its own clock offset. Because the signals travel at $c$ , position accuracy is welded to timing accuracy through $\Delta x \approx c\,\Delta t$ — a single nanosecond of clock error is $30\,\mathrm{cm}$ on the ground, and meter-level navigation demands the whole constellation agree to nanoseconds. At that precision, relativity is not a correction bolted on at the end; it is a first-order design parameter. A clock's proper time depends on where it sits in Earth's gravitational potential and on how fast it moves, both read off the weak-field metric — essentially Schwarzschild to first post-Newtonian order. Being higher in the potential makes the satellite clock tick faster (the same gravitational time dilation that reddens light climbing out of a well, confirmed by Pound and Rebka); orbital speed makes it tick slower (special-relativistic time dilation). The two pull opposite ways and do not cancel. The arithmetic is unforgiving: relative to a clock on Earth's rotating geoid, a GPS satellite clock gains about $+38\,\mu\mathrm{s}$ per day. Light crosses $11\,\mathrm{km}$ in $38\,\mu\mathrm{s}$ , so an uncorrected system would smear navigation by kilometers within a day, without bound. The engineers saw it coming: each satellite oscillator is detuned by $\Delta f/f \approx -4.465\times10^{-10}$ before launch so that, seen from the ground, it beats at the nominal rate. GPS is where ordinary people use Einstein's theory every time they find themselves on a map.

Key takeaways

GPS fixes position by timing signals from atomic clocks in orbit; a $1\,\mathrm{ns}$ timing error is a $30\,\mathrm{cm}$ position error, so the constellation must stay synchronized to nanoseconds.
Two relativistic rate shifts act on the satellite clock relative to the geoid: gravitational time dilation speeds it up by $+45.7\,\mu\mathrm{s/day}$ , orbital-velocity time dilation slows it down by $-7.2\,\mu\mathrm{s/day}$ .
They have opposite signs and unequal size, netting a satellite clock that runs fast by $+38.5\,\mu\mathrm{s/day}$ ; uncorrected this accumulates about $11.5\,\mathrm{km/day}$ of position error, unbounded in time.
The fix is a factory frequency offset: the onboard oscillator is detuned by $\Delta f/f \approx -4.465\times10^{-10}$ so it reads correctly on the ground; residual eccentricity and Sagnac terms are handled in software.
GPS is the most pervasive everyday test of general relativity — both the weak-field gravitational redshift and Lorentz time dilation must be included together to navigate.
Key idea: sum the two first-order relativistic rate shifts of a clock — a fractional gain $+\Delta\Phi/c^2$ from being higher in the potential and a fractional loss $-v^2/2c^2$ from orbital speed. For GPS they give $+45.7$ and $-7.2\,\mu\mathrm{s/day}$ , netting $+38.5\,\mu\mathrm{s/day}$ , pre-corrected by lowering the satellite oscillator frequency.

The core insight

Sum the two first-order relativistic rate shifts of a clock: a fractional gain $+\Delta\Phi/c^2$ from being higher in the gravitational potential and a fractional loss $-v^2/2c^2$ from orbital speed. For a GPS orbit these are $+45.7$ and $-7.2\,\mu\mathrm{s/day}$ , netting $+38.5\,\mu\mathrm{s/day}$ — pre-corrected by lowering the satellite oscillator frequency.

Where this lives today

The relativistic corrections run continuously in every GPS satellite and receiver: the constant $38.5\,\mu\mathrm{s/day}$ offset removed by the factory detuning, the periodic eccentricity term applied each epoch from the broadcast ephemeris, and the Sagnac term folded into the coordinate-time framework. The story became real in 1977, when the NTS-2 satellite launched with a synthesizer that could switch the relativistic offset on or off: with it off, the onboard clock ran fast by very nearly the predicted rate, and the correction was switched in. The same physics underpins GLONASS, Galileo, and BeiDou. Beyond navigation, optical-lattice atomic clocks now stable to parts in $10^{18}$ can detect the gravitational redshift over a centimeter of height, turning the clock formula into a tool for "relativistic geodesy" that maps the geoid itself by comparing clock rates.

The physical picture

Put two identical atomic clocks side by side, then lift one to GPS altitude and set it orbiting. Two things happen at once, and they fight. The lifted clock sits higher in Earth's gravitational well — like standing higher on a hill — so it ticks faster, the mirror of a photon redshifting as it climbs out of the well; that gain alone is about $+46\,\mu\mathrm{s/day}$ . But the orbiting clock also races at $3.87\,\mathrm{km/s}$ , and moving clocks run slow (the twin-paradox effect), subtracting about $7\,\mu\mathrm{s/day}$ . Gravity wins, and the satellite clock ends up fast by $38\,\mu\mathrm{s/day}$ — negligible per second, catastrophic per day for a system whose currency is the nanosecond.

Figure

Two relativistic effects on a GPS clock relative to the ground geoid. Being higher in Earth's potential speeds it up by +45.7 μs/day while its orbital speed of 3.87 km/s slows it by −7.2 μs/day, leaving a net gain of +38.5 μs/day that the system must pre-correct

Because $1\,\mathrm{ns}$ of clock error maps to $30\,\mathrm{cm}$ of position error, this shift is no curiosity added at the end; it is baked into the satellite oscillator at the factory. There is also a crossover hidden in the competition: at the break-even radius $r=\tfrac{3}{2}R_\oplus\approx 9{,}557\,\mathrm{km}$ the gravitational speed-up exactly cancels the velocity slow-down — below it (LEO, the ISS) clocks run slow, above it (GPS, geostationary) they run fast. The second figure is the whole story in one curve: net clock rate versus orbital radius, crossing zero where the two effects balance.

Figure

Net satellite clock rate versus orbital radius, combining gravitational speed-up and velocity slow-down as +GM/c²(1/R − 3/2r) × 86400. The curve crosses zero at the break-even radius r = 3R/2 ≈ 9,557 km: the ISS at 6,771 km loses 24.7 μs/day, GPS at 26,561 km gains 38.5 μs/day

The core equations

Navigation is timing, and timing at the speed of light is geometry. The first relation makes everything else matter: it converts a clock error into a position error, and it is why nanoseconds are the natural unit of GPS engineering.

\Delta x \approx c\,\Delta t \qquad (1)

where: $\Delta x$ — position error; $\Delta t$ — clock or timing error; $c$ — speed of light, $\approx 3\times10^8\,\mathrm{m/s}$ . Reading it: a $1\,\mathrm{ns}$ error is $0.30\,\mathrm{m}$ , so locating you to meters means synchronizing to nanoseconds. What a clock accumulates is not coordinate time but proper time along its worldline — the invariant arc length of its path through spacetime. This is the quantity every atomic clock physically measures, and where both relativistic effects enter as one.

\mathrm{d}\tau = \frac{1}{c}\sqrt{-g_{\mu\nu}\,\frac{\mathrm{d}x^\mu}{\mathrm{d}t}\frac{\mathrm{d}x^\nu}{\mathrm{d}t}}\;\mathrm{d}t \qquad (2)

where: $\mathrm{d}\tau$ — proper time ticked by the clock; $g_{\mu\nu}$ — spacetime metric (signature $-+++$ ); $x^\mu$ — the clock's spacetime coordinates; $t$ — coordinate time. Reading it: the metric $g_{\mu\nu}$ carries both gravity (through the potential in $g_{00}$ ) and motion (through the velocity $\mathrm{d}x^i/\mathrm{d}t$ ), so a single formula produces both shifts. Around a slowly rotating, nearly spherical Earth the field is weak, and to first post-Newtonian order the metric is the Schwarzschild solution linearized in the Newtonian potential. This is the arena in which GPS clocks live.

\mathrm{d}s^2 = -\Big(1+\frac{2\Phi}{c^2}\Big)c^2\,\mathrm{d}t^2 + \Big(1-\frac{2\Phi}{c^2}\Big)\big(\mathrm{d}r^2+r^2\,\mathrm{d}\Omega^2\big) \qquad (3)

where: $\Phi$ — Newtonian potential, $\Phi=-GM_\oplus/r<0$ ; $GM_\oplus$ — Earth's gravitational parameter, $3.986\times10^{14}\,\mathrm{m^3/s^2}$ ; $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2$ . Reading it: the deeper (more negative) $\Phi$ , the more $g_{00}$ is warped and the slower coordinate-referenced clocks tick. Expanding equation (2) in this metric to first order in $\Phi/c^2$ and $v^2/c^2$ gives the master clock-rate formula — the whole of GPS relativity in one line. Both signs are visible: $+\Phi/c^2$ speeds clocks up as they rise, and $-v^2/2c^2$ slows them as they move.

\frac{\mathrm{d}\tau}{\mathrm{d}t} \approx 1 + \frac{\Phi}{c^2} - \frac{v^2}{2c^2} \qquad (4)

where: $\mathrm{d}\tau/\mathrm{d}t$ — proper-time rate per unit coordinate time; $\Phi$ — local Newtonian potential (negative); $v$ — the clock's coordinate speed. Reading it: two competing first-order terms, gravity raising the rate and speed lowering it. Comparing satellite to ground subtracts two copies of equation (4); the coordinate time $t$ cancels, leaving a fractional rate difference — the frequency offset the system experiences.

\frac{\Delta f}{f} = \frac{\Delta\Phi}{c^2} - \frac{v_{\mathrm{sat}}^2 - v_{\mathrm{gnd}}^2}{2c^2} \qquad (5)

where: $\Delta\Phi=\Phi_{\mathrm{sat}}-\Phi_{\mathrm{gnd}}>0$ ; $v_{\mathrm{sat}}\approx3.87\,\mathrm{km/s}$ — orbital speed; $v_{\mathrm{gnd}}$ — ground clock's speed from Earth's rotation, small and folded into the geoid reference. Reading it: a positive potential term against a negative velocity term, both referenced to the same ground clock. Putting the GPS numbers into equation (5) yields the famous net offset — a fixed fractional rate the constellation is built to remove.

\frac{\Delta f}{f}\bigg|_{\mathrm{net}} \approx +4.465\times10^{-10} \;\;\Longrightarrow\;\; +38.5\,\mu\mathrm{s/day} \qquad (6)

where: the $+$ sign means the satellite clock gains; the pre-launch correction lowers its frequency by exactly this fraction. Reading it: multiply by $86{,}400\,\mathrm{s/day}$ to turn a dimensionless rate into a daily drift of $38.5\,\mu\mathrm{s}$ . Real orbits are slightly elliptical, so altitude and speed oscillate once per revolution and the two terms in equation (4) no longer stay constant. The residual is a periodic clock error the receiver removes from the broadcast ephemeris.

\Delta t_{\mathrm{ecc}} = -\frac{2\sqrt{GM_\oplus\,a}}{c^2}\,e\,\sin E \qquad (7)

where: $a$ — orbital semi-major axis; $e$ — eccentricity ( $\lesssim 0.02$ for GPS); $E$ — eccentric anomaly around the orbit. Reading it: the amplitude scales with $e$ , reaching tens of nanoseconds — far above the noise floor — so it must be corrected even though it averages to zero.

Derivation AFrom proper time to the clock-rate formula

The goal is to turn the invariant proper-time integral (2) into the working master equation (4), $\mathrm{d}\tau/\mathrm{d}t \approx 1 + \Phi/c^2 - v^2/2c^2$ , by expanding the weak-field metric (3) and keeping only first-order terms. This is the single most important step in the whole subject, because it shows that gravity and motion are not two separate corrections stitched together by hand but two limbs of one metric expansion, entering with opposite signs from the time and space parts of $\mathrm{d}s^2$ .

Step 1. Start from $\mathrm{d}s^2=-c^2\mathrm{d}\tau^2$ for a timelike worldline and insert the metric (3), writing the spatial displacement as $\mathrm{d}\ell^2 = v^2\,\mathrm{d}t^2$ with $v=|\mathrm{d}\vec x/\mathrm{d}t|$ . This gives $c^2\mathrm{d}\tau^2 = \big(1+\tfrac{2\Phi}{c^2}\big)c^2\mathrm{d}t^2 - \big(1-\tfrac{2\Phi}{c^2}\big)v^2\mathrm{d}t^2$ . Why: proper time is the invariant interval along the clock's path, and every clock reads exactly this and nothing else.

Step 2. Divide by $c^2\mathrm{d}t^2$ to isolate the rate squared: $\big(\tfrac{\mathrm{d}\tau}{\mathrm{d}t}\big)^2 = 1 + \tfrac{2\Phi}{c^2} - \tfrac{v^2}{c^2} + \tfrac{2\Phi v^2}{c^4}$ . The last term is the product of two tiny quantities, of order $10^{-19}$ for GPS, so drop it as $\mathcal{O}(c^{-4})$ . Why: the potential term $\Phi/c^2\sim10^{-9}$ and velocity term $v^2/c^2\sim10^{-10}$ are each first-order small, so their product is far below the nanosecond budget.

Step 3. Take the square root with $\sqrt{1+\epsilon}\approx 1+\tfrac12\epsilon$ , halving each bracket, to reach $\mathrm{d}\tau/\mathrm{d}t = 1 + \Phi/c^2 - v^2/2c^2$ . Why: the one-half on the velocity term is the special-relativistic $\gamma^{-1}\approx1-v^2/2c^2$ , while the potential term keeps its full coefficient — the asymmetry that lets gravity beat motion at GPS altitude.

Step 4. Read the signs: $\Phi<0$ and less negative at larger $r$ , so raising a clock increases $\mathrm{d}\tau/\mathrm{d}t$ (runs fast); any nonzero $v$ decreases it (runs slow). Why: this is gravitational and velocity time dilation as one formula, and getting the relative sign right is the whole game. *Verified: reduces to $1-v^2/2c^2$ (special relativity) when $\Phi=0$ , and to $1+\Phi/c^2$ (gravitational redshift) when $v=0$ ; the dropped cross term is $\sim10^{-19}$ .*

Go deeper · Derivation B · 1 minThe gravitational rate shift for a GPS orbit

Derivation BThe gravitational rate shift for a GPS orbit

The goal is to evaluate the potential term of equation (5) numerically for the GPS geometry and obtain the gravitational speed-up of $+45.7\,\mu\mathrm{s/day}$ , using only $\Phi=-GM_\oplus/r$ , the geoid radius, and the orbital radius. This is the larger of the two competing effects, the one that makes a satellite clock run fast, and the first number a verifier reproduces.

Step 1. Write the potential difference as $\dfrac{\Delta\Phi}{c^2} = \dfrac{GM_\oplus}{c^2}\Big(\dfrac{1}{R_\oplus}-\dfrac{1}{r}\Big)$ , positive because the satellite sits at larger $r$ where $\Phi$ is less negative. Why: only the difference between the two clocks matters, and moving from the deep ground potential to the shallow orbital potential is a climb, which speeds the higher clock up.

Step 2. Evaluate the prefactor: $\dfrac{GM_\oplus}{c^2}=\dfrac{3.986\times10^{14}}{(2.998\times10^8)^2}\approx 4.435\times10^{-3}\,\mathrm{m}$ — half Earth's Schwarzschild radius. Why: this length sets the scale of every gravitational shift near Earth; that it is only a few millimeters is why the effects are parts in $10^{10}$ , not order unity.

Step 3. Evaluate the reciprocal-radius factor with $R_\oplus=6{,}371\,\mathrm{km}$ and $r=26{,}561\,\mathrm{km}$ : $\dfrac{1}{6.371\times10^6}-\dfrac{1}{2.6561\times10^7}\approx 1.193\times10^{-7}\,\mathrm{m^{-1}}$ . Why: the orbit's reciprocal radius is nearly four times smaller, so most of the potential difference is already accrued by GPS altitude.

Step 4. Multiply: $\dfrac{\Delta\Phi}{c^2}\approx (4.435\times10^{-3})(1.193\times10^{-7})\approx 5.29\times10^{-10}$ , then $\times 86{,}400\,\mathrm{s/day}\approx +45.7\,\mu\mathrm{s/day}$ . Why: the pure gravitational advance — the number that would stand alone if the satellite were at rest at that altitude. *Verified: $5.291\times10^{-10}\times86{,}400 = 45.72\,\mu\mathrm{s/day}$ , the standard textbook value (Ashby 2003; Hartle §6.3).*

Go deeper · Derivation C · 1 minAdding the velocity shift to reach the net result

Derivation CAdding the velocity shift to reach the net result

The goal is to compute the velocity term $v_{\mathrm{sat}}^2/2c^2$ for a circular GPS orbit and subtract it from the gravitational gain to reach the canonical net of $+38.5\,\mu\mathrm{s/day}$ and fractional offset $-4.465\times10^{-10}$ . The orbital speed is not free: for a circular orbit gravitational and centripetal accelerations balance, fixing $v^2=GM_\oplus/r$ in terms of the same $GM_\oplus$ and $r$ already used.

Step 1. From circular-orbit balance $v_{\mathrm{sat}}^2 = \dfrac{GM_\oplus}{r}=\dfrac{3.986\times10^{14}}{2.6561\times10^7}\approx 1.501\times10^7\,\mathrm{m^2/s^2}$ , so $v_{\mathrm{sat}}\approx 3.87\,\mathrm{km/s}$ . Why: tying $v$ to $r$ means the whole result depends on the single orbital radius, and it is why the break-even radius exists as a clean geometric statement.

Step 2. Form the velocity term: $\dfrac{v_{\mathrm{sat}}^2}{2c^2}=\dfrac{1.501\times10^7}{2(2.998\times10^8)^2}\approx 8.35\times10^{-11}$ , i.e. $\times 86{,}400\approx -7.2\,\mu\mathrm{s/day}$ . Why: this is the special-relativistic slowdown; the ground clock's own rotation speed of $\sim0.46\,\mathrm{km/s}$ is small and is bundled into the geoid reference, so the satellite's speed dominates the comparison.

Step 3. Subtract from the gravitational term: $\dfrac{\Delta f}{f}\Big|_{\mathrm{net}}\approx 5.29\times10^{-10}-0.835\times10^{-10}\approx 4.46\times10^{-10}$ . Why: the two effects genuinely oppose — the correct answer is a difference, not a sum, and reversing a sign here is the classic way to get $53\,\mu\mathrm{s/day}$ instead of $38$ .

Step 4. Convert to a daily drift: $4.46\times10^{-10}\times86{,}400\approx +38.5\,\mu\mathrm{s/day}$ . Equivalently the satellite oscillator, nominally $10.23\,\mathrm{MHz}$ , is detuned by $-4.465\times10^{-10}$ ( $\approx-4.57\times10^{-3}\,\mathrm{Hz}$ ) before launch. Why: pre-detuning is cheaper and more robust than continuous steering — the clock is built to run right as seen from the ground. *Verified: $+45.72-7.21 = +38.5\,\mu\mathrm{s/day}$ and $\Delta f/f = -4.456\times10^{-10}$ , the documented GPS design offset (Ashby 2003).*

Limiting cases

The relative size of the two terms in equation (4) is governed entirely by orbital radius; every regime of spaceflight lands somewhere on the second figure's curve.

Case	Result	Meaning
Low Earth orbit (ISS, $\sim 400\,\mathrm{km}$ )	$\dfrac{v^2}{2c^2} > \dfrac{\Delta\Phi}{c^2}$	Velocity dilation dominates; the clock runs slow, about $-25\,\mu\mathrm{s/day}$ relative to the ground.
Break-even radius	$r_{\mathrm{eq}}=\tfrac{3}{2}R_\oplus\approx 9{,}557\,\mathrm{km}$	The two effects cancel exactly (altitude $\approx 3{,}186\,\mathrm{km}$ ); an orbiting clock ticks at the ground rate.
GPS orbit ( $r\approx 26{,}561\,\mathrm{km}$ )	$\dfrac{\Delta\Phi}{c^2} > \dfrac{v^2}{2c^2}$	Gravitational speed-up wins; net $+38.5\,\mu\mathrm{s/day}$ , the case GPS must correct.
Geostationary / deep space	$r\to\infty:\;\dfrac{\Delta f}{f}\to +\dfrac{GM_\oplus}{R_\oplus c^2}\approx +7.0\times10^{-10}$	The velocity term vanishes and only the full ground potential remains — a pure gravitational blueshift of $+60\,\mu\mathrm{s/day}$ at infinity.

Common trapThe two effects are one redshift, so I can just add a single dilation

No — this is the mistake that produces the wrong sign or magnitude. Gravitational and velocity time dilation are distinct terms in equation (4), carry opposite signs, and both must be referenced to the same clock: for GPS, a clock on Earth's rotating geoid, an equipotential of gravity plus the centrifugal potential — not a clock at infinity, at Earth's center, or in the non-rotating Earth-centered frame. Two failures follow from conflating them: adding instead of subtracting gives $\sim53\,\mu\mathrm{s/day}$ , and dropping the velocity term gives $\sim46\,\mu\mathrm{s/day}$ ; only keeping both terms with correct signs against the geoid gives the measured $+38.5\,\mu\mathrm{s/day}$ .

Worked examples

1 · The net offset and the runaway error (fully worked)

A GPS satellite orbits at $r=26{,}561\,\mathrm{km}$ with $v=3.87\,\mathrm{km/s}$ ; the ground reference is the geoid at $R_\oplus=6{,}371\,\mathrm{km}$ . Find the net fractional clock rate, the daily clock drift, and the position error that would accumulate in one day if it were ignored. Use $GM_\oplus=3.986\times10^{14}\,\mathrm{m^3/s^2}$ , $c=2.998\times10^8\,\mathrm{m/s}$ .

Step 1. Gravitational term (Derivation 2): $\dfrac{\Delta\Phi}{c^2}=\dfrac{GM_\oplus}{c^2}\big(\tfrac{1}{R_\oplus}-\tfrac{1}{r}\big)=(4.435\times10^{-3})(1.193\times10^{-7})=\mathbf{+5.29\times10^{-10}}$ .

Step 2. Velocity term (Derivation 3): $\dfrac{v^2}{2c^2}=\dfrac{(3.87\times10^3)^2}{2(2.998\times10^8)^2}=\mathbf{+8.35\times10^{-11}}$ .

Step 3. Net rate: subtract, $\dfrac{\Delta f}{f}=5.29\times10^{-10}-0.835\times10^{-10}=\mathbf{+4.46\times10^{-10}}$ (satellite runs fast).

Step 4. Daily drift: $\times 86{,}400\,\mathrm{s}=3.85\times10^{-5}\,\mathrm{s}=\mathbf{+38.5\,\mu s/day}$ .

Step 5. Position error via equation (1): $\Delta x = c\,\Delta t=(2.998\times10^8)(3.85\times10^{-5})=\mathbf{1.15\times10^4\,m\approx 11.5\,km}$ built up in a single day — and unbounded, since the drift never stops. Hence the correction is not optional.

2 · The break-even radius

At what circular-orbit radius do the gravitational and velocity effects exactly cancel, so the orbiting clock ticks at the ground rate?

Step 1. Set the net rate in equation (5) to zero using $v^2=GM_\oplus/r$ : $\dfrac{GM_\oplus}{c^2}\Big(\dfrac{1}{R_\oplus}-\dfrac{1}{r}\Big)=\dfrac{GM_\oplus}{2rc^2}$ .

Step 2. Cancel $GM_\oplus/c^2$ and solve: $\dfrac{1}{R_\oplus}-\dfrac{1}{r}=\dfrac{1}{2r}\;\Rightarrow\;\dfrac{1}{R_\oplus}=\dfrac{3}{2r}\;\Rightarrow\;\mathbf{r_{eq}=\tfrac{3}{2}R_\oplus}$ .

Step 3. Numerically, $r_{\mathrm{eq}}=1.5\times6{,}371=\mathbf{9{,}557\,km}$ from Earth's center, an altitude of about $\mathbf{3{,}186\,km}$ . Below it velocity wins and clocks run slow (the ISS at $6{,}771\,\mathrm{km}$ loses $\sim25\,\mu\mathrm{s/day}$ ); above it gravity wins and clocks run fast (GPS gains $38.5\,\mu\mathrm{s/day}$ ). The GPS constellation sits well above the crossover, on the gaining side. For the periodic eccentricity correction, equation (7) with $a=26{,}561\,\mathrm{km}$ and $e=0.01$ gives a peak amplitude $2\sqrt{GM_\oplus a}\,e/c^2\approx\mathbf{23\,ns}$ — nearly $7\,\mathrm{m}$ of range — which the receiver applies each epoch from the broadcast $e$ and $E$ .

Physics of the portrait

GPS is where three strands of twentieth-century physics — special relativity, general relativity, and precision atomic timekeeping — braid into one working machine. Einstein supplied the two dilations and Pound-Rebka and Hafele-Keating measured them, but turning them into an operational timing standard aboard a satellite constellation was its own achievement — and no one is more identified with it than the physicist who wrote the chapter the field cites.

Origins · 1 minNeil Ashby and how this was discovered

Neil Ashby

b. 1934, American physicist · relativity in the Global Positioning System

Neil Ashby is the theorist most responsible for making relativity operational in satellite navigation. A physicist at the University of Colorado Boulder, he worked through the 1970s and beyond on the full relativistic framework a global timing system requires — not just the headline $38\,\mu\mathrm{s/day}$ offset, but the subtler pieces that decide whether a real constellation closes to nanoseconds: the choice of the rotating geoid as the reference surface, the periodic eccentricity term $-2\sqrt{GM_\oplus a}\,e\sin E/c^2$ , and the Sagnac correction from the rotating Earth-fixed frame. His insistence on defining coordinate time in a self-consistent Earth-centered inertial frame, with proper time related to it through the weak-field metric, is what lets receivers all over a spinning planet share one clock. Ashby's 2003 Living Reviews in Relativity article, "Relativity in the Global Positioning System," became the standard reference. He argued, again and again, that GPS is not merely compatible with relativity but a continuous confirmation of it: the system's daily self-consistency measures the gravitational redshift and velocity dilation to parts in $10^{10}$ , every second.

History

1905

— Einstein's special relativity predicts moving clocks run slow

1916

— general relativity predicts clocks run faster higher in a gravitational potential

1959

— Pound and Rebka measure the gravitational redshift in a Harvard tower

1971

— Hafele and Keating fly atomic clocks around the world, confirming both dilations

1977

— the NTS-2 satellite carries a cesium clock with a switchable relativistic offset, confirming the predicted rate

1978

— the first Block I GPS satellites launch with the relativistic correction built in

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §25 (gravitational redshift, weak-field clocks) · Wald, General Relativity (1984), §6.3 · Carroll, Spacetime and Geometry (2004), §5.4 · Hartle, Gravity (2003), §6.3 (GPS as a worked application) · Schutz, A First Course in General Relativity (2009), §11 · N. Ashby, "Relativity in the Global Positioning System," Living Reviews in Relativity 6, 1 (2003).

Provenance. A verifier should recompute, with $GM_\oplus=3.986\times10^{14}\,\mathrm{m^3/s^2}$ , $R_\oplus=6{,}371\,\mathrm{km}$ , $r=26{,}561\,\mathrm{km}$ : the gravitational shift $\tfrac{GM_\oplus}{c^2}(1/R_\oplus-1/r)\approx5.29\times10^{-10}$ ( $+45.7\,\mu\mathrm{s/day}$ ); the velocity shift $v^2/2c^2$ with $v^2=GM_\oplus/r$ , $\approx8.35\times10^{-11}$ ( $-7.2\,\mu\mathrm{s/day}$ ); the net $\approx4.46\times10^{-10}$ ( $+38.5\,\mu\mathrm{s/day}$ ); the break-even radius $r_{\mathrm{eq}}=\tfrac32R_\oplus\approx9{,}557\,\mathrm{km}$ ; the ISS rate $\approx-25\,\mu\mathrm{s/day}$ ; and the eccentricity amplitude $2\sqrt{GM_\oplus a}\,e/c^2\approx23\,\mathrm{ns}$ for $e=0.01$ . The factory offset $-4.465\times10^{-10}$ of $10.23\,\mathrm{MHz}$ is the documented design value; all figures are standard (Ashby 2003; Hartle §6.3; MTW §25; Schutz §11).

Prerequisites. Gravitational Redshift · Weak-Field Limit and Linearized Gravity · Newtonian Limit of General Relativity · Pound-Rebka Experiment. Leads to. Shapiro Time Delay · Parametrized Post-Newtonian (PPN) Formalism · Frame Dragging and the Lense-Thirring Effect.

Ask the tutor: Want to see how the same $\mathrm{d}\tau/\mathrm{d}t=1+\Phi/c^2-v^2/2c^2$ that runs your phone's GPS also predicts the Shapiro delay and gravitational redshift — or how an optical clock could map the geoid by a centimeter of height?

Do GPS engineers really use general relativity, or is that an exaggeration

They really use it, not as decoration. The relativistic rate difference between a satellite clock and the ground is $+38.5\,\mu\mathrm{s/day}$ , which by $\Delta x = c\,\Delta t$ is about $11.5\,\mathrm{km}$ of position error per day if ignored. The correction is engineered in physically: each oscillator is detuned by $-4.465\times10^{-10}$ before launch so it beats at the right rate on the ground, and receivers apply a periodic eccentricity term of up to tens of nanoseconds every epoch. Without both, a phone could not hold a meter-level fix for even a few minutes — which is why GPS is called the most widely used everyday confirmation of relativity.

Why do the two effects have opposite signs

They come from different parts of the metric. The gravitational term $+\Phi/c^2$ comes from $g_{00}$ : a clock higher in the potential (less-negative $\Phi$ ) ticks faster, the opposite of the redshift a photon suffers climbing out of the well. The velocity term $-v^2/2c^2$ is the special-relativistic slowdown of any moving clock, always negative. At GPS altitude the potential gain ( $+45.7\,\mu\mathrm{s/day}$ ) outweighs the motion loss ( $-7.2\,\mu\mathrm{s/day}$ ), so the net is a gain; in low Earth orbit the balance flips and the clock runs slow. The sign of the net effect depends on how high you orbit.

Why is the reference a clock on the rotating geoid rather than one at infinity

Because GPS time must be usable by clocks and receivers on a spinning Earth, so the natural reference is the surface those users share. The geoid is an equipotential of gravity plus the centrifugal potential of Earth's rotation, so every clock at mean sea level ticks at the same rate regardless of latitude — a huge convenience. Choosing the geoid also absorbs the ground clock's rotation speed ( $\sim0.46\,\mathrm{km/s}$ ) into the reference, so the velocity term in the satellite-versus-ground comparison is essentially just the satellite's orbital speed. Referencing to a clock at infinity or at Earth's center would leave dangling terms and give the wrong daily rate.

What is the break-even radius and why does the ISS run slow while GPS runs fast

Setting the net rate to zero for a circular orbit, with $v^2=GM_\oplus/r$ , gives $r_{\mathrm{eq}}=\tfrac32R_\oplus\approx9{,}557\,\mathrm{km}$ from Earth's center — an altitude near $3{,}186\,\mathrm{km}$ . Below it the orbital speed is high and the altitude gain small, so velocity dilation dominates and the clock runs slow: the ISS at $\sim6{,}771\,\mathrm{km}$ loses about $25\,\mu\mathrm{s/day}$ . Above it the altitude gain dominates and the clock runs fast: GPS gains $38.5\,\mu\mathrm{s/day}$ . The single curve of net rate versus radius crosses zero exactly at $r_{\mathrm{eq}}$ , and every spacecraft lives somewhere on it.

Where does the eccentricity correction come from and is it large enough to matter

GPS orbits are not perfectly circular. With eccentricity $e$ the radius and speed oscillate once per revolution, so both terms in the clock-rate formula vary through the orbit; the leftover, after removing the mean $38.5\,\mu\mathrm{s/day}$ , is a periodic term $\Delta t_{\mathrm{ecc}}=-2\sqrt{GM_\oplus a}\,e\sin E/c^2$ . For $e=0.01$ its amplitude is about $23\,\mathrm{ns}$ — nearly $7\,\mathrm{m}$ of range, far above the nanosecond budget. It averages to zero over an orbit but is nonzero at any instant, so the receiver computes it from the broadcast $e$ and eccentric anomaly $E$ every epoch. It is the periodic partner of the constant factory offset.

Does GPS also involve effects beyond the two time dilations

Yes, though the two dilations dominate. The Sagnac effect arises because the Earth-fixed frame rotates while a signal is in flight, so light-travel times across the rotating frame pick up a term reaching $\sim100\,\mathrm{ns}$ for equator-crossing paths; it is kinematic but sits in the same timing budget. The oblateness ( $J_2$ ) and tidal terms enter at the sub-nanosecond level for the most demanding work, and frame-dragging (Lense-Thirring) is far too small to affect GPS. For meter-level navigation, the two time dilations plus the eccentricity and Sagnac terms close the budget.