Hafele-Keating Experiment

The Hafele-Keating experiment, carried out in 1971, put four cesium atomic clocks aboard commercial airliners to test Einstein’s two relativities: the time dilation linked to speed makes the clock run behind, the gravitational shift linked to altitude makes it run ahead.

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How to measure it in class

Relativistic effects are far too small for a smartphone’s clock: they are on the order of a nanosecond, whereas a smartphone measures at best a millisecond, a million times coarser. However, FizziQ can be used to reproduce Hafele and Keating’s experimental approach - synchronize clocks, separate them, compare them on return - and to calculate the expected effects numerically.

Steps:

Synchronize two smartphones on the same time reference, then start a FizziQ stopwatch on each at the same instant, triggering both simultaneously.
Separate the two devices for several hours, one motionless in the classroom, the other taken along by car, train or bicycle, and record the speed with FizziQ’s GPS.
Compare the two stopwatches on return and measure the difference: it is on the order of a few tens of milliseconds, entirely due to the drift of the quartz oscillators and not to relativity.
Calculate, from the average speed recorded by the GPS and the duration of the trip, the theoretical time dilation in special relativity: for 100 km/h for 3 h, it is about 5×10⁻¹² s.
Compare this theoretical difference with the actual precision of the smartphone and conclude on the type of clock required: only an atomic clock, stable to within 10⁻¹³, makes the experiment possible.

Scientific activities on this topic

Parabolic flight and microgravity - Explore the effects of a flight on the physical quantities measured on board, as an extension of Hafele and Keating’s airborne experiment.
Clock synchronization: speed of sound - Put into practice the synchronization of two devices, a key element of the Hafele-Keating protocol.

Learn more

The 1971 protocol

Joseph C. Hafele, a physicist at Washington University in Saint Louis, and Richard E. Keating, an astronomer at the United States Naval Observatory, put four cesium atomic clocks aboard ordinary commercial flights - they simply bought tickets, including for the clocks. The eastward trip lasted about 65 hours, including 41 h of flight; the westward trip, about 80 hours, including 49 h of flight. On return, the clocks were compared with those that had remained on the ground. The total budget of the experiment remained modest, which makes it a textbook case: a test of relativity conducted with commercial means.

The two effects, and why they oppose each other

In special relativity, a clock moving relative to the reference frame ticks more slowly. Since the airplane travels at about 900 km/h, its clocks run behind those on the ground. The effect is proportional to v².

In general relativity, a clock placed higher in a gravitational field, and therefore at a higher gravitational potential, ticks faster. At 10,000 m altitude, the airplane’s clocks run ahead of those on the ground. The effect is proportional to the difference in altitude.

These two contributions are of the same order of magnitude for an airliner, which makes the experiment particularly instructive: the sign of the result is not given in advance.

The east-west asymmetry, signature of the experiment

The most often omitted point is that the relevant reference frame is not the ground, but the non-rotating geocentric frame. Yet the ground itself rotates with the Earth, at about 465 m/s at the equator. An airplane flying eastward adds its speed to that of the Earth’s rotation: its total speed is greater, the special-relativity effect wins, and the clock loses time. An airplane flying westward flies against the rotation: its speed in the geocentric frame is lower than that of the clock remaining on the ground, the gravitational effect dominates, and the clock gains time.

The published results, in nanoseconds, are as follows. Eastward: −59 ± 10 ns measured, versus −40 ± 23 ns predicted. Westward: +273 ± 7 ns measured, versus +275 ± 21 ns predicted. The change of sign between the two flights is the true signature of the experiment. A presentation announcing only a single effect, “the airplane’s clocks run behind”, is incomplete and wrong for the westward flight.

What the experiment did and did not demonstrate

Given the error bars, particularly for the eastward flight, the precision of 1971 was modest. The experiment did not “prove” relativity on its own; it confirmed its predictions with transported macroscopic clocks, which no previous experiment had done. It has been redone several times since, notably in 1996 for its twenty-fifth anniversary with much better precision, and current measurements with optical clocks detect the gravitational shift for an altitude difference of only a few centimeters.

GPS, an everyday application of the same calculation

GPS satellites orbit at an altitude of 20,200 km, at about 3.9 km/s. Their clocks run behind by about 7 µs per day because of their speed, and run ahead by about 45 µs per day because of the altitude. The net balance is about +38 µs per day. Without correction, the positioning error would accumulate at a rate of about ten kilometers per day. The frequency of the onboard clocks is therefore deliberately offset before launch to compensate for this effect. Every use of a smartphone GPS is thus a permanent verification of both relativities.

Orders of magnitude

Effect measured aboard a commercial flight: a few tens to a few hundreds of nanoseconds over two days. Stability of a cesium clock in 1971: about 10⁻¹³. Rotation speed of the Earth’s surface at the equator: 465 m/s. Cruising speed of an airliner: 250 m/s. Total relativistic correction for GPS: 38 µs per day, or 4.4×10⁻¹⁰ in relative value. Cumulative effect over a career: an airline pilot who has logged 20,000 h of flight has aged about 50 microseconds more than a sedentary person who stayed at sea level, the altitude effect winning over the speed effect.

Formula

Time dilation in special relativity, for a clock of speed v:

Δt’ = Δt · √(1 − v²/c²)

where:

Δt: duration measured in the reference frame (s)
Δt’: duration measured by the moving clock (s)
v: speed of the clock in this frame (m/s)
c: speed of light in vacuum, 2.998×10⁸ m/s

For v ≪ c, which is the case for an airplane, the relative difference simplifies:

Δt’/Δt − 1 ≈ − v²/(2c²)

The negative sign reflects the fact that the moving clock runs behind.

Gravitational shift in general relativity, between two altitudes separated by Δh in a uniform gravitational field:

Δt’/Δt − 1 ≈ + g·Δh/c²

where:

Δh: altitude difference, counted positively upward (m)
g: gravitational field strength, 9.81 m/s²

The positive sign reflects the fact that the higher clock runs ahead.

Total effect on an onboard clock, algebraic sum of the two contributions:

Δτ/τ ≈ g·Δh/c² − v²/(2c²)

The result changes sign depending on whether the altitude term or the speed term wins, which explains the asymmetry between the eastward and the westward flight.

Application examples

For a flight at 10,000 m altitude, the gravitational term is g·Δh/c² = 9.81 × 10,000 / (3.00×10⁸)² ≈ 1.1×10⁻¹², an advance of about 3.9 ns per hour of flight.
For that same airplane at 250 m/s, the speed term is v²/(2c²) = 250²/(2 × 9.0×10¹⁶) ≈ 3.5×10⁻¹³, a lag of about 1.3 ns per hour: here altitude wins, but the Earth’s rotation speed modifies the balance depending on the direction of flight.
Eastward, the onboard clocks lost 59 ns; westward, they gained 273. Same airplane, same altitude, opposite effects.
The clocks of GPS satellites are set before launch to 10.229 999 995 45 MHz instead of 10.23 MHz, so that they tick at the right rate once in orbit.
An uncorrected GPS would give a position error growing by about 11 km per day, which would make it completely unusable in less than an hour.
Modern optical clocks measure the gravitational shift between two points separated by 33 cm of elevation, turning general relativity into a geodesy tool.

FAQ

Q: Why do clocks run ahead going west and behind going east? A: Because one must reason in the geocentric reference frame, in which the ground is already rotating at several hundred meters per second. Eastward, the airplane adds its speed to that of the rotation: the speed term dominates, the clock runs behind. Westward, the airplane goes against the rotation and therefore moves more slowly than the laboratory on the ground: the altitude effect wins, the clock runs ahead.

Q: Can this experiment be redone with a smartphone? A: No. The effect is on the order of 100 nanoseconds, whereas a smartphone’s quartz oscillator drifts by several milliseconds per day, ten thousand times more. Only atomic clocks, stable to 10⁻¹³ or better, make the measurement possible. However, the protocol can be reproduced and the expected effects calculated.

Q: Does this experiment test special relativity or general relativity? A: Both at once, and that is its whole interest. The speed term belongs to special relativity, the altitude term to general relativity. Since they are of the same order of magnitude and of opposite signs for an airliner, the experiment can only be explained by taking both into account.

Q: Does the twin paradox apply here? A: Yes, in a very attenuated version. The clocks that traveled eastward came back 59 ns behind those that stayed on the ground: they really aged less. The situation is not symmetric because the onboard clocks underwent accelerations and changed gravitational potential, which is not the case for those that remained on the ground.

Q: Should you reset your watch after a long-haul flight? A: Only for the time zone. The relativistic effect on a Paris-New York flight amounts to tens of nanoseconds, tens of millions of times less than a second. No watch, even a very high-end one, is capable of displaying it.

Reference Frame - GPS Geolocation