eduKate Learning Manual · Relativity × Time Metrology × Earth Rotation · JC → Edge · Synchronise → Fly → Compare → Decompose
Wait, What? Take Atomic Clocks on Commercial Flights and They Return With Different Times
In everyday life, “one second” feels universal. Relativity says otherwise: elapsed proper time depends on motion and gravitational potential.
In 1971 Joseph Hafele and Richard Keating carried cesium atomic clocks around the world on commercial airliners, first eastward and then westward, and compared them with clocks that remained at the U.S. Naval Observatory.
The airborne clocks did not return perfectly synchronised with the ground reference. The direction and magnitude of the shifts were consistent, within experimental uncertainty, with the combined predictions of special and general relativity.
synchronise clocks → place them on different spacetime paths → airborne clocks move faster/slower relative to an inertial frame and sit at higher gravitational potential → return clocks → compare accumulated proper time → separate velocity and gravitational contributions.
The Big Question
How can two clocks that are individually accurate disagree after travelling around Earth — and why does flying east differ from flying west?
Quick Answer
For weak gravity and speeds much smaller than c, a useful approximation for clock-rate difference is:
dτ/dt ≈ 1 + Φ/c² − v²/(2c²)
where Φ is gravitational potential and v is speed in an appropriate inertial frame.
Flying higher increases Φ, so the airborne clock tends to gain time relative to a lower clock. Moving faster increases v², so the airborne clock tends to lose time. Because Earth itself rotates, an eastward airplane usually has greater inertial speed than the ground below it, while a westward airplane can have lower inertial speed than the same ground reference. The two relativistic effects therefore add differently for eastward and westward flights.
What You Will Learn
- what proper time means
- why atomic clocks can resolve nanosecond-scale effects
- how velocity time dilation enters
- how gravitational potential affects clock rates
- why Earth rotation matters
- why eastward and westward flights are not symmetric
- how clock transport differs from the twin-paradox cartoon
- why clock drift and synchronisation uncertainty matter
- how later clock experiments improved the test
- why GPS requires both special- and general-relativistic timing corrections
Part 1 — A Clock Measures Its Own Proper Time
In relativity, each clock accumulates time along its own worldline. That elapsed time is called proper time.
Two clocks can start together, follow different paths through spacetime, and later reunite with different accumulated proper times.
No clock needs to malfunction. The disagreement is the physical prediction.
Part 2 — Special Relativity: Speed Slows the Moving Clock
For constant speed v in flat spacetime:
dτ = dt√(1 − v²/c²)
For v ≪ c:
dτ ≈ dt(1 − v²/2c²)
Greater inertial speed means slightly less proper time accumulates.
Airliner speeds are tiny compared with c, so the effect is only tens to hundreds of nanoseconds over long flights. Cesium clocks made that scale measurable.
Part 3 — General Relativity: Higher Clocks Run Faster
Near Earth’s surface, raising a clock by height h increases gravitational potential by approximately gh.
The corresponding fractional rate change is approximately:
Δf/f ≈ gh/c²
so a clock at cruising altitude tends to run slightly faster than a similar clock at sea level.
This is the same gravitational time/frequency principle tested by the Pound–Rebka experiment, but now the receiver is an atomic clock carried through the gravitational field.
Part 4 — Why Earth Rotation Changes the Directional Prediction
The ground is not stationary in an inertial frame. Earth rotates eastward.
At a given latitude, a ground clock already has substantial eastward velocity around Earth’s axis. An eastbound aircraft adds to that inertial velocity. A westbound aircraft subtracts from it.
Therefore:
- eastward flight: stronger special-relativistic slowing, partly opposed by gravitational speeding at altitude;
- westward flight: reduced inertial speed relative to the rotating ground can make the velocity term favour a gain, while altitude also favours a gain.
This is why direction around a rotating Earth matters.
A Quantitative Scale Window
At h = 10 km:
gh/c² ≈ (9.8)(10,000)/(9 × 10¹⁶) ≈ 1.1 × 10⁻¹²
Over 40 hours, that gravitational contribution corresponds to roughly:
(1.1 × 10⁻¹²)(1.44 × 10⁵ s) ≈ 1.6 × 10⁻⁷ s = 160 ns
The velocity contribution is of a similar nanosecond order, so both must be included.
The Historical Carrier — Around the World in 1971
Hafele and Keating flew portable cesium-beam clocks around the world aboard commercial aircraft. The flights went eastward and westward on separate journeys, after which the transported clocks were compared with reference clocks that remained at the U.S. Naval Observatory.
The results were published in Science in 1972. NIST’s modern historical summary notes that all three clock sets differed and that the measured shifts agreed with combined special- and general-relativistic predictions within the experimental margin of error.
The experiment became memorable because relativity left the accelerator and astronomy laboratory and boarded ordinary passenger aircraft.
Part 5 — Why This Is Not Simply “Moving Clocks Run Slow”
That slogan captures only special relativity.
The aircraft also flies higher, where gravitational potential is greater and clocks run faster relative to lower clocks.
A correct prediction must therefore keep at least two ledgers:
velocity contribution + gravitational-potential contribution.
Ignoring either term can predict the wrong sign for the final clock shift.
Part 6 — Why the Reference Frame Must Be Chosen Carefully
Airplane speed relative to the ground is not the same as speed relative to an Earth-centred inertial frame.
Because the ground itself rotates, the inertial speed contains Earth’s rotational velocity plus or minus the aircraft’s velocity component.
A naive calculation using only “900 km/h airplane speed” for both directions misses the core directional effect.
Part 7 — Real Atomic Clocks Drift
Portable cesium clocks are extremely good, but not perfect. Their rates can drift slightly with time and environmental conditions.
The experiment therefore required pre-flight and post-flight comparisons, clock-rate models, uncertainty estimates and multiple clocks.
A predicted 100-ns relativistic shift means little if clock drift is unknown at the microsecond scale. Metrology is part of the physics.
Part 8 — Why Flight Path and Altitude Matter
The aircraft did not remain at one altitude and one speed. Real routes included climbs, descents, layovers, latitude changes and varying ground speeds.
Proper time therefore must be integrated along the actual path rather than estimated from one average number if high accuracy is required.
The simplified formulas teach the mechanism. The historical comparison requires the route history.
RFE Stress Test — Relativity or Clock Drift?
- multiple clocks: do independent portable clocks show consistent directional shifts?
- pre/post calibration: is each clock’s drift measured around the flight interval?
- route reconstruction: are altitude, latitude and speed histories included?
- direction reversal: do eastward and westward predictions differ in the observed way?
- ground reference: are reference clocks stable and continuously compared?
- model decomposition: can velocity-only or gravity-only calculations explain both flights simultaneously?
The combined relativity model wins because one set of spacetime rules predicts opposite directional behaviour and the altitude contribution together.
Observation vs Inference
Observation: transported clocks return with measurable time offsets relative to ground references.
Metrological inference: the offsets exceed or follow calibrated clock drift and uncertainty in the expected directional pattern.
Relativistic inference: accumulated proper time depends on both motion and gravitational potential as predicted by special and general relativity.
Common Misconceptions and How to Repair Them
- “All airplane clocks lose time because they move.” Repair: altitude makes them gain gravitationally, and westward inertial speed can be lower than the ground’s.
- “Ground clocks are stationary.” Repair: Earth rotates, so ground clocks move in an inertial frame.
- “The experiment used ordinary wristwatches.” Repair: portable cesium atomic clocks were required.
- “One clock disagreement proves relativity.” Repair: drift, route and uncertainty must be modelled.
- “Hafele–Keating was the final precision test.” Repair: later atomic-clock experiments improved accuracy enormously.
Checkpoint Questions
- What is proper time?
- What does increased speed do to clock rate?
- What does increased gravitational potential do?
- Why is eastward flight different from westward flight?
- Why is ground-relative airspeed insufficient?
- What experimental problem does clock drift create?
- Why must route history be included?
Apply It — Same Plane, Higher Altitude
If the aircraft follows the same velocity path but flies higher, the gravitational contribution becomes more positive: the airborne clock tends to accumulate more proper time relative to the lower reference. Whether the total shift is positive or negative still depends on the velocity term.
Unfamiliar Transfer — GPS Is a Continuous Relativity Experiment
GPS satellites carry atomic clocks moving rapidly at high gravitational potential. Their clock rates differ from ground clocks because of both special and general relativity.
Navigation works only because those timing effects are modelled and corrected. A phone position is therefore connected to the same architecture:
clock path through spacetime → accumulated proper time → signal timing → distance estimate → position.
Answer Key
1. Time measured along one clock’s own worldline. 2. Greater inertial speed reduces proper time. 3. Higher potential makes a stationary clock run faster relative to a lower one. 4. Earth rotates, so east/west flights have different inertial speeds. 5. The ground itself moves. 6. Drift can imitate nanosecond-scale offsets. 7. Relativistic rate depends continuously on altitude and velocity.
Can You Explain WHY?
Explain why “moving clocks run slow” cannot by itself predict the Hafele–Keating results. A strong answer should connect Earth rotation → inertial speed → special-relativistic term → aircraft altitude → gravitational term → east/west asymmetry → accumulated proper time.
Singapore JC Science Bridge
JC Physics supplies motion, gravitation and relativity. Hafele–Keating adds measurement integration: a real clock samples changing speed, altitude and latitude over hours, so the final answer is an accumulated path effect rather than one instantaneous formula.
Deep Science Windows
- Chronometric geodesy: modern optical clocks can measure gravitational-potential differences through frequency.
- Sagnac effect: rotation creates direction-dependent timing around a closed path and is essential in Earth-based navigation.
- GPS relativity: satellite clocks combine orbital velocity and weaker gravity.
- Twin paradox: reunited clocks follow different worldlines and can accumulate different proper times without contradiction.
- Optical clocks: laboratory comparisons now test relativity far beyond 1970s portable cesium precision.
Evidence Boundaries
The simple weak-field equation is an approximation. Precise airborne-clock calculations require Earth’s rotation, actual trajectories, gravitational potential and clock calibration. Hafele–Keating agreed with relativity within the substantial uncertainty of its era; later experiments provide far tighter confirmation. The ground is a rotating reference, not a universal inertial rest frame.
Manual Summary — KNOW → CONNECT → EXPLAIN → APPLY → CHECK
- KNOW: proper time depends on velocity and gravitational potential.
- CONNECT: aircraft motion and altitude contribute with different signs.
- EXPLAIN: Earth rotation makes east/west inertial velocities different.
- APPLY: estimate nanosecond-scale clock shifts.
- CHECK: clock drift, route, altitude, reference frame and uncertainty.
Teaching Guide for Parents, Tutors and Teachers
Why this opening works: ordinary air travel becomes a direct spacetime experiment; the surprise is measurable clock disagreement, not abstract science-fiction speed.
- Central reasoning model: synchronised clocks → different worldlines → velocity + potential rate shifts → reunion → compare.
- Teaching sequence: proper time → SR velocity term → gravitational term → Earth rotation → east/west prediction → clock metrology → GPS.
- Diagnostic question: “Why can a westbound airplane clock gain time even though the airplane is moving?”
- If stuck: separate the two effects into a plus/minus ledger before combining them.
- Ready for more: integrate proper time in a rotating Earth metric and analyse GPS timing.
Quiet Teaching Standard: never reduce the result to “Einstein was right.” Require the learner to calculate which contribution changes sign with flight direction and which does not.