Physics · Gravitation and the universe · Grade 10

Special relativity

Why does a clock on a GPS satellite run faster than one on Earth?

The situation

In the streets of Tashkent the navigator on your phone finds your position to within a few metres. To do this it measures the travel time of signals from GPS satellites at an altitude of 20 000 km: a 1 microsecond error in the clock means a 300 m error. But the atomic clock on the satellite runs differently from a clock on Earth: it moves fast and sits where gravity is weaker. If Einstein's theory were ignored, how far off course would the navigator go in one day?

Topic

Special relativity. Why time runs differently on GPS satellites

Key ideas

Time dilation
Length contraction
Lorentz factor γ

Formulas

γ = \frac{1}{\sqrt{1 - v^{2} / c^{2}}}

The Lorentz factor: for $v ≪ c$ , $γ \approx 1$ ; for $v \to c$ it grows without limit.

Δ t = γ Δ τ, L = \frac{L_{0}}{γ}

Time dilation (the time $Δ τ$ shown by a moving clock is $γ$ times less than the time $Δ t$ on Earth) and length contraction — only along the direction of motion.

x^{'} = γ (x - v t), t^{'} = γ (t - \frac{v x}{c^{2}})

Lorentz transformations: the coordinate and time in the frame of a ship moving at speed $v$ . For $v ≪ c$ they become the Galilean transformation: $x^{'} = x - v t$ , $t^{'} = t$ .

\frac{Δ τ}{τ} \approx - \frac{v^{2}}{2 c^{2}} + \frac{G M}{c^{2}} (\frac{1}{R_{\oplus︎}} - \frac{1}{r})

The GPS clock: speed slows it down (special relativity), weaker gravity speeds it up (general relativity).

N = N_{0} e^{- t / (γ τ_{0})}, τ_{0} = 2.197 μ s

Decay of muons: seen from the Earth, their mean lifetime is $γ τ_{0}$ .

Theory

Postulates: from Galileo to Einstein

Galileo's principle of relativity: any mechanical experiment performed in a uniformly moving (inertial) ship gives the same result as one performed at rest; velocities simply add, and time was assumed to be the same for everyone. But Maxwell's electrodynamics predicts the speed of light as $c = 1 / \sqrt{ε_{0} μ_{0}}$ — without naming any reference frame; where one observer measures only an electric field, another also measures a magnetic field. In 1887 the Michelson–Morley experiment detected no motion of the Earth relative to the “aether”. Einstein (1905) adopted two postulates: 1) in all inertial frames all laws of physics, electrodynamics included, are the same; 2) in a vacuum the speed of light is the same for every inertial observer — $c$ , independent of the motion of the source. According to Galileo, an observer on the Earth should measure the speed of the ship's photon as $\sqrt{c^{2} + v^{2}}$ (compare in the light-clock measurements), but experiment always gives $c$ — which is why time and distance turn out to be relative. The limit of classical mechanics: for $v ≪ c$ , $γ \approx 1 + v^{2} / 2 c^{2}$ ; at the speed of a GPS satellite (3.9 km/s) the difference is only $\approx 8 \cdot 10^{- 11}$ , while at $0.6 c$ it is 25 %.

Lorentz transformations, simultaneity, interval

From the two postulates follows the rule for converting the coordinates of an event from one inertial frame to another — the Lorentz transformations: $x^{'} = γ (x - v t)$ , $t^{'} = γ (t - v x / c^{2})$ , and in directions perpendicular to the motion $y^{'} = y$ , $z^{'} = z$ . Conclusions: 1) between two events happening at the ship clock's own position ( $x^{'} = 0$ ), a time $Δ t^{'}$ passes on the ship and $Δ t = γ Δ t^{'}$ on the Earth — time dilation; 2) if we mark the positions of the ship's ends simultaneously on the Earth, $L = L_{0} / γ$ — length contraction; 3) events that are simultaneous on the ship but a distance $Δ x^{'}$ apart happen on the Earth with a time difference $Δ t = γ v Δ x^{'} / c^{2}$ — simultaneity is relative (the Minkowski diagram and the addition of velocities are in the “Relativistic dynamics” experiment). A quantity that is the same for all observers is the interval $s^{2} = c^{2} Δ t^{2} - Δ x^{2}$ ; for an observer moving with the clock $Δ x^{'} = 0$ , so $s / c$ is the clock's proper time $τ$ . After every tick of the light clock, the results table calculates the coordinates of the tick event in both frames and the interval. The twin paradox: the twin who flew into space and came back returns younger than the one on the Earth — the two twins are not equivalent, because at the turn-round he accelerated and changed reference frame.

Light clock

Einstein's postulate: the speed of light $c$ is the same for every inertial observer. Inside the ship the photon travels vertically between the mirrors. For an observer on the Earth, however, the mirrors move sideways during the photon's flight — the photon covers a longer diagonal, but its speed is still $c$ . So, seen from the Earth, each “tick” of the ship lasts longer: $Δ t = γ \cdot 2 L / c$ . The pilot inside the ship notices nothing — for him the clock, his heart and chemical reactions are all as usual.

How do muons reach the Earth?

Cosmic rays create muons in the upper layers of the atmosphere (≈ 15 km). At rest their lifetime is 2.197 µs — even at the speed of light they would cover only ≈ 660 m in that time. According to classical physics almost none of them should reach the Earth. But at $0.995 c$ , $γ \approx 10$ : seen from the Earth, the muon's lifetime is ≈ 22 µs, and ≈ 10 % of them arrive. For the muon itself the layer of atmosphere is $γ$ times thinner — the two descriptions give the same result.

GPS and the two relativities

A GPS satellite flies at ≈ 3.9 km/s: special relativity slows its clock by ≈ 7 µs per day. But it is at an altitude of 20 200 km, where the gravitational potential is higher than on the Earth: according to general relativity the clock runs ≈ 45 µs per day faster. The total is ≈ +38 µs/day. Light travels ≈ 11 km in 38 µs — uncorrected, the navigator would go wrong by ≈ 11 km every day. That is why, before launch, the satellite clocks are set on the Earth to 10.22999999543 MHz instead of 10.23 MHz. The satellites of Russia's GLONASS system orbit at ≈ 19 100 km, going round once in about 11 hours 15 minutes (check with the “GLONASS” button): their clocks are corrected in the same way. Modern phones use signals from several systems (GPS, GLONASS, Galileo, BeiDou) together.

Model simplifications

GPS orbits are circular, the formulas are first-order for a weak field; the receiver's motion with the Earth's rotation (≈ 0.1 µs/day at the equator) is not included. All muons are assumed to arrive almost vertically at the same speed and to lose no energy on the way. For visibility, time is slowed down or sped up: in the light clock 1 s = 10 ns, for muons 1 s = 10 µs, for GPS 1 s = 30 min; the satellites, detector and particles are not to scale.

Did you know?

Einstein published the special theory of relativity in 1905, in the paper “On the Electrodynamics of Moving Bodies”.
In 1971 J. Hafele and R. Keating flew caesium atomic clocks around the Earth on ordinary passenger aircraft: the clocks flown eastwards fell behind the ones on the ground by about 59 ns, and those flown westwards gained about 273 ns — in agreement with the prediction of relativity.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

The Solar System — Grade 7
Scales of the universe — Grade 9
Newton's cannon: an artificial satellite — Grade 10
Law of universal gravitation — Grade 10
Relativistic dynamics — Grade 10
Kepler's laws — Grade 10
Physics of stars — Year 1
The expanding universe — Year 1
Black holes — Year 4