The Hafele-Keating Experiment: Atomic Clocks, Airplanes, and Time Dilation

Relativistic time dilation — the slowing of time for objects in motion or in strong gravity — was a mathematical prediction of Einstein's theories for more than half a century before anyone confirmed it with a clock you could hold in your hands. In October 1971, physicist Joseph Hafele and astronomer Richard Keating put four cesium-beam atomic clocks into the passenger cabins of commercial airline flights and sent them around the world, east first, then west. When the traveling clocks came home and were compared to reference clocks that had stayed at the US Naval Observatory in Washington, they disagreed with each other by nanoseconds — exactly the nanoseconds that Einstein's two theories of relativity predicted. The result was not a surprise to theoretical physicists. But demonstrating it with clocks on a passenger jet was something else entirely.

What Einstein Predicted — Two Effects, Not One

Understanding why the Hafele-Keating results were in the exact direction and magnitude they were requires separating two distinct relativistic effects that both apply to a flying clock simultaneously.

Special Relativity: Kinematic Time Dilation

Einstein's 1905 special theory of relativity predicts that a clock moving relative to an observer runs slow compared to a stationary clock. The relationship is given by the Lorentz factor:

Time dilation: Δt_moving = Δt_rest × √(1 − v²/c²)

For small velocities (v much less than c), this approximates to:

Δt_moving ≈ Δt_rest × (1 − v²/2c²)

The faster a clock moves through space, the slower it ticks relative to a stationary clock. This is not a mechanical effect of motion shaking the clockwork — it is a fundamental property of spacetime itself.

A commercial jet travels at roughly 900 km/h (560 mph), or about 250 m/s. The speed of light is about 3 × 10⁸ m/s. The ratio v/c ≈ 8 × 10⁻⁷. The kinematic time dilation from a jet's velocity alone is extremely small — on the order of a hundred nanoseconds per day — but perfectly calculable.

The direction of this effect is always the same: a moving clock always loses time relative to a stationary one. An eastward-flying clock and a westward-flying clock should both lose time due to velocity.

But they do not lose the same amount, because of Earth's rotation.

Source: Einstein, A. (1905), "Zur Elektrodynamik bewegter Körper" (On the Electrodynamics of Moving Bodies), Annalen der Physik, 17, 891–921 — the special relativity paper, which first derived kinematic time dilation.

The key subtlety: Special relativity's kinematic time dilation is measured relative to an inertial frame — a non-accelerating, non-rotating reference frame. Earth's surface is not such a frame: it rotates at about 465 m/s at the equator. To calculate clock rates correctly, both the ground clocks and the flying clocks must be compared from an inertial frame — roughly, a non-rotating frame centered on Earth.

Relative to this inertial frame:

• Ground clocks at the equator are moving at ~465 m/s due to Earth's rotation.
• An eastward-flying jet moves in the same direction as rotation, at ground speed + aircraft speed ≈ 465 + 250 = 715 m/s relative to the inertial frame.
• A westward-flying jet moves opposite to rotation, at approximately 465 − 250 = 215 m/s relative to the inertial frame.

The eastward clock is moving faster (relative to the inertial frame) than the ground clock. The westward clock is moving slower (relative to the inertial frame) than the ground clock. Both clocks experience velocity time dilation, but the eastward clock experiences more of it and the westward clock experiences less — so the westward clock actually loses less time than the ground clock, meaning relative to the ground clock, the westward clock appears to gain time.

General Relativity: Gravitational Time Dilation

Einstein's 1915 general theory of relativity predicts that clocks in weaker gravitational fields — at higher altitude — run faster than clocks in stronger gravitational fields. This is called gravitational time dilation, and it is a consequence of the equivalence principle: the effects of gravity are equivalent to the effects of acceleration.

Gravitational time dilation: Δt_high = Δt_low × (1 + gh/c²)

Where g is the gravitational acceleration, h is the height difference, and c is the speed of light.

A clock at altitude gains time relative to a sea-level clock. The effect is tiny for ordinary altitudes: for a height of 10 km (33,000 ft, typical cruising altitude), the fractional time gain is approximately:

gh/c² ≈ (9.8 m/s²)(10,000 m) / (9 × 10¹⁶ m²/s²) ≈ 1.1 × 10⁻¹²

Over a day, this amounts to roughly 96 nanoseconds gained per day at cruising altitude. Both the eastward and westward flying clocks spend similar amounts of time at altitude, so both gain roughly the same amount of time from the gravitational effect.

Source: Einstein, A. (1915), "Die Feldgleichungen der Gravitation" (The Field Equations of Gravitation), Sitzungsberichte der Preussischen Akademie der Wissenschaften, 844–847 — the general relativity paper; gravitational time dilation follows from the metric solution.

The direction of this effect: both clocks gain time relative to the ground clock.

The Combined Effect

Put the two effects together:

• Eastward clock: Kinematic dilation causes loss (moving faster than ground), gravitational dilation causes gain (higher altitude). Net effect: a small loss, because the kinematic loss exceeds the gravitational gain.
• Westward clock: Kinematic dilation causes gain (moving slower than ground, relative to inertial frame), gravitational dilation causes gain (higher altitude). Net effect: a relatively large gain.

The two effects work in opposite directions for the eastward clock and in the same direction for the westward clock — which is why the two flights produce dramatically different outcomes despite nominally covering similar distances at similar altitudes.

The Prediction — Numbers Before the Experiment

Hafele published his theoretical predictions in Science in July 1972, alongside the observational paper. The predictions for the full around-the-world flights were:

Direction	Predicted change	Predicted component
Eastward	−40 ± 23 ns (lose time)	Kinematic: −184 ns; Gravitational: +144 ns
Westward	+275 ± 21 ns (gain time)	Kinematic: +96 ns; Gravitational: +179 ns

Source: Hafele, J.C. & Keating, R.E. (1972), "Around-the-World Atomic Clocks: Predicted Relativistic Time Gains," Science, 177(4044), 166–168 — the prediction paper.

These numbers carry substantial uncertainty because the commercial flights did not follow simple circular paths at fixed altitude. The planes varied altitude and speed, made stops, and flew over terrain with varying local gravity. Hafele and Keating had to integrate the relativistic effects along the actual flight paths using logged flight data — altitude, speed, and heading recorded at regular intervals.

The Apparatus — Cesium Beam Clocks on Economy Tickets

Cesium-beam atomic clocks were the most precise portable timekeepers available in 1971. They work by locking an oscillator to the hyperfine transition frequency of cesium-133 atoms — a transition at exactly 9,192,631,770 cycles per second, which is by definition one second in the International System of Units.

Cesium-beam atomic clock — a timekeeping device that uses the resonant frequency of cesium-133 atom hyperfine transitions (9,192,631,770 Hz) as its frequency reference. Accuracy: on the order of 1 nanosecond per day in 1971-era portable models.

Hafele and Keating used four HP 5061A cesium-beam frequency standards. Each clock weighed approximately 30 kg (66 lb) and required mains electrical power, which meant the aircraft needed to supply power to the instruments during flight. The clocks were transported in standard airline seats — they each had their own ticket.

The experiment was not cheap, but it was remarkably affordable by the standards of fundamental physics. Hafele later reported that much of the initial funding came from a research grant, and some of the flight costs were covered by the US Naval Observatory, which maintained the reference clocks.

Source: Hafele, J.C. & Keating, R.E. (1972), "Around-the-World Atomic Clocks: Observed Relativistic Time Gains," Science, 177(4044), 168–170 — the observational paper, giving instrument details and results.

The reference clocks — three cesium-beam standards — remained at the US Naval Observatory in Washington, D.C. throughout the experiment.

The Flights — October 1971

The eastward circumnavigation was completed between October 4 and 7, 1971, on scheduled commercial flights. The westward circumnavigation followed immediately, October 13 to 17, 1971.

Each direction used multiple flight legs on commercial carriers — Pan American, Japan Air Lines, and others — with layovers at airports along the route. The clocks were kept running continuously; the power supply situation during layovers required careful management.

During the flights, Hafele or Keating (the team split duties across the two legs) logged the flight data: the plane's altitude and ground speed were read at regular intervals and recorded, along with the latitude and longitude. These data were essential for the post-flight calculation of the predicted time differences along the actual flight paths, not a simplified circular orbit.

Immediately upon returning from each direction, the flying clocks were transported to the US Naval Observatory and compared against the reference standards. The comparison was done by detecting the beat frequency between the flying clocks and the reference clocks — two nearly identical frequencies produce a very slow beat that can be timed with high precision.

Source: U.S. Naval Observatory (1972) — internal records of clock comparison procedures; cited in Hafele & Keating (1972), Science, 177, 168–170.

The Results

Direction	Predicted	Observed
Eastward	−40 ± 23 ns	−59 ± 10 ns
Westward	+275 ± 21 ns	+273 ± 7 ns

The westward result was in spectacular agreement with prediction — the observed value was within 1 nanosecond of the central predicted value. The eastward result agreed in sign (the clock lost time, as predicted) and was within 1.5 standard deviations of the prediction, well within the stated uncertainties.

Both the sign and the magnitude of the effects matched general and special relativistic predictions. The experiment simultaneously confirmed:

1. Kinematic time dilation (special relativity)
2. Gravitational time dilation (general relativity)
3. The correct sign for each effect in each direction

A purely kinematic prediction (special relativity alone, without general relativity's gravitational term) would give the wrong answer. A purely gravitational prediction (general relativity's gravitational term alone) would give the wrong answer. Only the full combined calculation produces results consistent with what was observed. The experiment validated both theories simultaneously.

Source: Hafele, J.C. & Keating, R.E. (1972), Science, 177(4044), 168–170 — Table 2 gives the observed and predicted values; the agreement is discussed in the conclusion.

The Critics — Controversy and Reanalysis

The Hafele-Keating experiment is sometimes cited in discussions of experimental precision in fundamental physics, but it has also attracted criticism that is worth taking seriously.

Louis Essen, the British metrologist who co-developed the first accurate cesium atomic clock in 1955, argued that the experimental precision was inadequate to draw the strong conclusions Hafele and Keating claimed. Essen pointed out that the individual clock stability was not sufficient to resolve the predicted nanosecond differences with the confidence the authors implied.

More substantively, A.G. Kelly performed a reanalysis of the original raw clock data and argued in 2000 that the four traveling clocks disagreed with each other substantially — by amounts comparable to or larger than the predicted signal — and that the final averaged result depended sensitively on which clocks were included in the average.

Source: Kelly, A.G. (2000), "Hafele & Keating Tests: Did They Prove Anything?", Physics Essays, 13, 616 — the most detailed critical reanalysis of the raw data.

The physics community's response has been that even accepting the criticism, the Hafele-Keating result was directionally correct and the subsequent experiments that replicated and extended it are far more precise and definitive. The experiment is best understood as a convincing demonstration-of-principle, not a precision measurement.

The Successors — Cleaner Confirmation

The Hafele-Keating results were confirmed, refined, and extended by several subsequent experiments that addressed the limitations of using commercial airline routes.

The Maryland Experiment (1975–1976): Carroll Alley and colleagues at the University of Maryland flew an atomic clock in a small aircraft over the Chesapeake Bay at known, controlled altitudes for pre-specified durations, comparing it against a ground-based clock in real time via laser link. The results confirmed gravitational time dilation at the level of 1% experimental precision.

Source: Alley, C.O. (1979), "Relativity and Clocks," in 33rd Annual Symposium on Frequency Control, IEEE — summary of the Maryland time-transfer experiment.

GPS Satellite Clocks: The Global Positioning System provides continuous, ongoing confirmation of both relativistic effects. GPS satellites orbit at an altitude of about 20,200 km (12,550 miles) and travel at about 3.87 km/s (13,900 km/h). Without relativistic corrections:

• Gravitational time dilation makes satellite clocks run fast by about 45,900 ns/day.
• Kinematic time dilation makes them run slow by about 7,200 ns/day.
• Net uncorrected drift: about +38,700 ns/day.

At the speed of light, 38,700 ns/day corresponds to a position error accumulation of about 11.6 km per day. An uncorrected GPS would be useless for navigation within hours.

GPS satellite clocks are pre-compensated at manufacture for the expected relativistic drift and the satellite software applies ongoing relativistic corrections. The GPS system works because general and special relativity are correct, with an accuracy requirement of nanoseconds per day.

Source: Ashby, N. (2003), "Relativity in the Global Positioning System," Living Reviews in Relativity, 6, 1 — the definitive technical review of relativistic effects in GPS.

Optical clocks (2010–present): The NIST and PTB groups have used strontium and aluminum optical lattice clocks — accurate to better than 1 part in 10¹⁷ — to measure gravitational time dilation between two clocks at different heights on the same bench, separated by as little as 33 cm (13 in). General relativity's prediction of time dilation with height was confirmed at this tiny scale.

Source: Chou, C.W. et al. (2010), "Optical Clocks and Relativity," Science, 329(5999), 1630–1633 — NIST demonstration of relativistic time dilation between aluminum ion clocks at different heights and velocities.

The Timeline

Date	Event
1905	Einstein publishes special relativity; kinematic time dilation predicted
1915	Einstein publishes general relativity; gravitational time dilation predicted
1938–1941	Ives-Stilwell experiment confirms transverse Doppler shift (kinematic time dilation in spectra)
1955	Louis Essen and Jack Parry build the first accurate cesium atomic clock at NPL
1959	Pound-Rebka experiment measures gravitational redshift directly at Harvard
October 4–7, 1971	Hafele-Keating eastward flight
October 13–17, 1971	Hafele-Keating westward flight
July 14, 1972	Both Hafele-Keating papers published in Science
1975–1976	Maryland experiment extends and refines the result
1978	First GPS satellites launched, incorporating relativistic clock corrections
1995	GPS declared fully operational; relativistic corrections confirmed daily
2010	Chou et al. demonstrate relativistic time dilation at 33 cm height difference

What People Get Wrong About Time Dilation

"The eastward and westward clocks should behave identically because they both fly at the same speed." This is the most common misconception. The relevant speed for kinematic time dilation is not ground speed (speed relative to the ground) but speed relative to an inertial frame. Because Earth rotates, the eastward clock moves faster relative to an inertial frame than the ground clock does, while the westward clock moves slower. The two flights produce different kinematic effects.

"Time dilation means the astronaut actually ages slower — the effect is subjective." The effect is real and objective. When two clocks come back together and are compared side by side, they physically show different times. The atoms in a twin who travels at high speed complete fewer quantum transitions than the atoms in a twin who stays home. It is not an illusion.

"Only special relativity matters for everyday GPS." Both special and general relativity matter, and they act in opposite directions. Neither alone gives the right answer. The gravitational effect (general relativity) is about 6.4 times larger than the kinematic effect (special relativity) for GPS satellites, and they partially cancel each other.

"Hafele-Keating proved special relativity." It confirmed both special and general relativity simultaneously, and it was not the first test of either. The novelty was using macroscopic portable clocks — a result that is viscerally convincing in a way that spectroscopic measurements of Doppler shifts are not.

FAQs

Q: How large are the time dilation effects in the Hafele-Keating experiment?
A: The eastward flying clocks lost 59 ± 10 nanoseconds over the full eastward circumnavigation. The westward clocks gained 273 ± 7 nanoseconds. These are tiny fractions of the total elapsed flight time (about 41 hours for the eastward trip), but well within the detection capability of cesium atomic clocks.

Q: Why did the eastward clock lose time while the westward clock gained time?
A: Two effects are at work. Kinematic time dilation (special relativity) makes faster-moving clocks lose time relative to the inertial frame. Gravitational time dilation (general relativity) makes higher-altitude clocks gain time. The eastward clock moves faster than the ground clock relative to an inertial frame (Earth's rotation adds to its velocity), so its kinematic loss exceeds its gravitational gain — net loss. The westward clock moves slower than the ground clock relative to an inertial frame (rotation subtracts from its velocity), so both effects add up to a net gain.

Q: Were the Hafele-Keating results ever disputed?
A: Yes. Critics, including metrologist Louis Essen and analyst A.G. Kelly, argued that the individual clock uncertainties were too large relative to the predicted signal, and that the raw data showed substantial scatter. The physics community generally accepts the result as directionally correct and a proof-of-concept, while acknowledging the limitations. Subsequent experiments with purpose-built aircraft and GPS data provide far more precise confirmation.

Q: How does this relate to GPS navigation?
A: GPS satellites must correct for both relativistic effects to maintain meter-level positional accuracy. Without relativistic corrections, GPS clocks would drift by about 38,700 nanoseconds per day, accumulating a position error of about 11.6 km per day. GPS works precisely because relativistic time dilation is real and has been correctly quantified.

Q: Could you do this experiment with a wristwatch?
A: Ordinary quartz wristwatches are not nearly precise enough — they drift by thousands of nanoseconds per second, far more than the ~100–300 nanosecond total signal being measured. You need an atomic clock, which maintains accuracy at the nanosecond level per day, to detect the relativistic effect from a single circumnavigation.

Q: What is the real-world significance of gravitational time dilation beyond GPS?
A: Gravitational time dilation is relevant wherever precise timing is needed at different altitudes. High-precision experiments in fundamental physics — tests of the equivalence principle, searches for variations in fundamental constants, gravitational wave detection — all must account for gravitational time dilation. The LISA space observatory and planned optical clock networks explicitly incorporate general relativistic timing corrections.

The Takeaway

The Hafele-Keating experiment moved time dilation from a mathematical abstraction to something you could hold in your hands — cesium clocks that literally came back showing different times depending on which direction they flew. The numbers were small enough to require atomic clocks to measure, but they matched Einstein's predictions within experimental precision. Every GPS receiver quietly recapitulating the same calculation, 24 hours a day, is a more precise vindication of the same physics. Time is not absolute. It is a clock's record of its own path through spacetime, and that path depends on how fast you move and how deep in a gravitational well you travel.

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Read next:

• The Pound-Rebka Experiment
• The Michelson-Morley Experiment
• The Cavendish Experiment