Physics • Year 12 • Module 7 • Lesson 12
Time Dilation
Build HSC Band 5–6 extended-response technique on experimental design, data-driven analysis, and multi-step reasoning about relativistic time.
1. Data + scenario: the Hafele-Keating atomic clock experiment (Band 5–6)
8 marks Band 5–6
Scenario. In October 1972, J.C. Hafele and R.E. Keating flew four caesium atomic clocks on commercial aircraft eastward and westward around the world and compared them to identical clocks left at the US Naval Observatory (USNO) in Washington DC. The table below shows their simplified results. Special relativity predicts that the flying clocks (moving faster relative to the Earth’s centre) tick slower. General relativity predicts that clocks higher in Earth’s gravitational field tick faster.
| Clock journey | Special relativity prediction (ns) | GR gravitational prediction (ns) | Net predicted (ns) | Observed (ns) |
|---|---|---|---|---|
| Eastward flight (clocks faster, longer path) | −184 | +144 | −40 | −59 ± 10 |
| Westward flight (clocks slower, shorter path) | −96 | +179 | +83 | +273 ± 7 |
Adapted from Hafele & Keating, Science 177 (1972), 166–170. Positive = flying clock ran fast; negative = flying clock ran slow relative to USNO.
Q1. Analyse and evaluate the Hafele-Keating data to assess the experimental evidence for special relativistic time dilation. In your response you must:
- Explain what the special relativity column predicts and why both eastward and westward clocks show a negative special relativistic contribution.
- Use the data to assess the degree of agreement between prediction and observation for the eastward journey, showing a percentage discrepancy calculation.
- Explain why the westward journey shows a net positive result even though special relativity predicts a negative contribution.
- Evaluate whether the data provide convincing experimental support for special relativistic time dilation, distinguishing this from general relativistic effects.
- State one experimental limitation and one reason why the large uncertainty in the eastward result (±10 ns) does not invalidate the special relativistic prediction.
2. Multi-step calculation — the twin paradox scenario (Band 5–6)
7 marks Band 5–6
Scenario. In 2040, an astronaut (“Kai”) leaves Earth aboard a spacecraft travelling at 0.95c to reach the star Tau Ceti, 11.9 light-years from Earth. Kai immediately turns around and returns at the same speed. Kai’s Earth twin (“Aria”) remains on Earth throughout. Use c = 3.0 × 108 m s−1 and 1 light-year = 9.46 × 1015 m.
(a) Calculate the Lorentz factor γ for Kai’s spacecraft at 0.95c. (1 mark)
(b) Calculate the total round-trip time measured by Aria (Earth frame). Show your working. (2 marks)
(c) Calculate the total round-trip time measured by Kai (ship’s frame, proper time). (1 mark)
(d) State the age difference between Kai and Aria when Kai returns. Explain why this is not a paradox, with specific reference to the asymmetry between their situations. (2 marks)
(e) If the spacecraft could somehow travel at 0.9999c instead, state and explain (without calculation) whether Kai would be even younger upon return relative to Aria, and why. (1 mark)
Q1 — Sample Band 6 response (8 marks), annotated
SR prediction explanation: Special relativity predicts that any clock moving faster relative to a given inertial frame (here, Earth’s inertial centre) will tick more slowly — Δt = γΔt⊂0;. Both eastward and westward aircraft clocks move faster than the ground clock, so both show a negative SR contribution (they run slow relative to the USNO clock). The eastward jets move faster relative to Earth’s rotational frame than westward jets (they add orbital speed to the Earth’s rotation rather than subtracting it), explaining the larger negative SR prediction for the eastward journey (−184 ns vs −96 ns) [1].
Eastward percentage discrepancy: Predicted net = −40 ns. Observed = −59 ns. Discrepancy = |−59 − (−40)| = 19 ns. Percentage = 19 / 40 × 100 = 47.5%. This is relatively large, suggesting the eastward result is less precise [1].
Westward net positive result: For the westward journey, the GR gravitational contribution (+179 ns, clocks run faster at altitude) outweighs the SR velocity contribution (−96 ns, clocks run slower due to speed). The net prediction is +179 + (−96) = +83 ns, meaning the westward flying clocks run fast compared to the USNO. The observed +273 ns is larger than predicted, suggesting larger uncertainties or systematic effects [1].
Evaluation of evidence for SR: To isolate the SR prediction, one must disentangle it from the GR contribution. The data are broadly consistent with SR’s prediction that faster-moving clocks tick slower (both flights show negative SR contributions). However, the large discrepancy for the eastward journey (≈47.5%) weakens the quantitative case for SR alone. The westward result also shows a large discrepancy from the net prediction. While the sign and order of magnitude of the effects are consistent with special relativity, the experimental uncertainties and the need to disentangle SR from GR effects mean this experiment confirms SR qualitatively and approximately, not precisely in isolation [1 + 1].
Limitation: Aircraft vibration, temperature fluctuations, and unpredictable atmospheric pressure variations can affect the frequency of caesium atomic clocks, introducing noise beyond the stated ± uncertainties. The flight paths were not perfectly controlled inertial trajectories, meaning speed and altitude varied continuously [1].
Why uncertainty does not invalidate: The ±10 ns uncertainty means the true eastward value could lie between −49 and −69 ns. The predicted value of −40 ns lies just outside this range, but only marginally. Given that the experiment was conducted with 1972 technology on commercial aircraft (not in controlled conditions), the agreement is still significant. More importantly, the pattern (eastward slower, westward faster, matching SR + GR sign predictions) is consistent and has been confirmed by many subsequent, more precise experiments [1].
Marking criteria (8 marks): 1 = correct explanation of negative SR for both flights (faster speed = slower clock in Earth frame). 1 = percentage discrepancy calculated correctly for eastward (~47.5% or equivalent). 1 = westward net positive correctly explained (GR exceeds SR magnitude). 1 = evaluation identifies sign and magnitude consistency. 1 = evaluation correctly notes limitation of isolating SR from GR. 1 = valid experimental limitation stated (vibration/temperature/non-inertial flight). 1 = correct reasoning why ±10 ns uncertainty does not invalidate (marginal/pattern consistent/later experiments). 1 = precise use of relativistic terminology throughout.
Q2 — Sample Band 6 response (7 marks), annotated
(a) γ: γ = 1/√(1 − 0.95²) = 1/√(1 − 0.9025) = 1/√0.0975 = 1/0.3122 ≈ 3.20 [1 mark]
(b) Aria’s time: Round-trip distance = 2 × 11.9 = 23.8 light-years. At 0.95c: Δt = 23.8 / 0.95 = 25.05 years (Earth frame) [2 marks: 1 for distance, 1 for time]
(c) Kai’s proper time: Δt⊂0; = Δt / γ = 25.05 / 3.20 ≈ 7.83 years [1 mark]
(d) Age difference: Aria ages 25.05 years; Kai ages 7.83 years. Aria is approximately 17.2 years older than Kai upon reunion [0.5]. This is not a paradox because the situations are asymmetric: Kai must accelerate to turn around at Tau Ceti, changing inertial frames, while Aria remains in a single inertial frame throughout. General relativity resolves the accelerating phase, confirming Kai ages less. The acceleration breaks the symmetry that special relativity’s reciprocal time dilation would otherwise imply [1.5 → round to 2].
(e) Higher speed: At 0.9999c, γ would be much larger (~70.7). Kai’s proper time would be much smaller (Δt⊂0; = Δt / γ → 0 as v → c). Kai would be even younger relative to Aria: the greater the speed, the larger γ, the greater the time dilation, the more Kai’s clock runs slow in Aria’s frame [1].