Time Dilation

Deriving the most famous equation in relativity

Imagine a clock consisting of two mirrors facing each other, distance $d$ apart. A photon bounces between them. One "tick" of the clock is the time for the photon to make a round trip: $\Delta t_0 = 2d/c$.

Now imagine this clock moving horizontally at speed $v$ past a stationary observer. From the observer's perspective, the photon must travel a longer diagonal path to hit the moving upper mirror and return. Since the speed of light is $c$ for all observers (second postulate), the photon takes longer to complete its round trip.

The geometry gives:

Substituting $d = c\Delta t_0/2$ and solving:

This is time dilation: a moving clock ticks slower by factor $\gamma$. Since $\gamma \geq 1$ always, the dilated time is always greater than or equal to the proper time. The effect is reciprocal — each observer sees the other's clock running slow. This is not a paradox because the situations are not symmetric: only one clock changes inertial frames (if it turns around).

Time dilation: $\Delta t = \gamma\Delta t_0$ where $\gamma \geq 1$. Proper time $\Delta t_0$ is measured by the clock present at both events (moving clock); it is always the shortest time. Dilated time $\Delta t$ is measured by any observer moving relative to that clock. The light clock derivation uses Pythagoras: the photon travels a longer diagonal path, so the tick takes longer.

Write $\Delta t = \gamma\Delta t_0$, the definition of proper time, and a sketch of the light clock diagonal path.

When theory meets experiment

We just saw that time dilation follows from Einstein's second postulate and Pythagoras applied to a light clock. That raises a question: is time dilation a real physical effect or just a theoretical curiosity? This card answers it → three experiments at different scales that confirm time dilation really happens.

Time dilation is not science fiction. Three experiments confirm it across very different scales: cosmic ray muons, atomic clocks on aircraft, and GPS satellites orbiting Earth every 12 hours.

Cosmic ray muons: Muons are unstable particles created when cosmic rays strike the upper atmosphere (~15 km). Their proper lifetime is $\tau_0 = 2.2\ \mu\text{s}$. At $v \approx 0.995c$, the Lorentz factor $\gamma \approx 10$. Classical physics predicts they travel only $0.995c \times 2.2\ \mu\text{s} \approx 650$ m before decaying. Yet muons reach sea level in abundance. In Earth's frame, time dilation extends their lifetime to $\gamma \tau_0 \approx 22\ \mu\text{s}$, allowing them to travel ~14 km. In the muon's own frame, the atmosphere is length-contracted (see Lesson 13), shrinking the 15 km to ~650 m.

Hafele-Keating experiment (1972): Atomic clocks were flown on commercial aircraft around the world. When compared to Earth-based reference clocks, the flying clocks showed time differences consistent with special and general relativistic predictions. The moving clocks lost time due to speed (special relativity) but gained time from being higher in Earth's gravitational field (general relativity).

GPS satellites: GPS satellites orbit at ~20,200 km altitude at ~14,000 km/h. Special relativistic time dilation makes their clocks tick slower by ~7 $\mu$s/day. General relativistic gravitational time dilation (clocks higher in a gravity well run faster) adds ~45 $\mu$s/day. The net effect (~38 $\mu$s/day faster) must be corrected, or GPS position errors would accumulate at ~10 km/day.

Three experimental confirmations: (1) Cosmic ray muons — proper lifetime $2.2\,\mu$s, but at $v \approx 0.995c$ ($\gamma \approx 10$) they reach sea level: dilated lifetime $\approx 22\,\mu$s. (2) Hafele-Keating (1972): atomic clocks on aircraft matched relativistic predictions. (3) GPS satellites: SR gives $-7\,\mu$s/day, GR gives $+45\,\mu$s/day; net $+38\,\mu$s/day must be corrected or GPS drifts ~10 km/day.

Record the three experiments and the GPS correction values in your book.

The twin paradox and how to resolve it

We just saw three experiments confirming time dilation is real. That raises a question: if a spaceship travels out and back, how do we calculate both the Earth-frame time and the ship's proper time, and why does the "paradox" not actually contradict relativity? This card answers it → a four-step worked example with the twin paradox resolved.

Twin paradox worked example at $v = 0.8c$: Earth-frame round trip $= 30$ years ($2\times12\,\text{ly}\div 0.8c$). Ship proper time: $\gamma = 1.667$, so $\Delta t_0 = 30/1.667 = 18$ years. Age difference $= 12$ years (Earth twin older). Paradox resolved: astronaut accelerates (changes frame) so situations are not symmetric.

Write the four steps: Earth distance/speed, $\gamma$, proper time, age difference.

Avoiding the traps that cost marks

We just saw how to solve twin-paradox problems and why the astronaut ages less. That raises a question: what are the most common errors students make about time dilation that lose marks in the HSC? This card answers it → three misconceptions with corrections and a four-step exam strategy.

Misconception 1: Time dilation only occurs near the speed of light.
Wrong. $\gamma > 1$ for all $v > 0$. At 900 km/h, $\gamma \approx 1 + 3\times10^{-13}$ — tiny but non-zero and measurable with atomic clocks (Hafele-Keating, 1972).

Misconception 2: "Proper" time is the Earth-frame time or the "correct" time.
Wrong. Proper time is the time measured by a clock at rest relative to the events — usually the moving observer's clock. For a spaceship journey, the ship clock gives proper time; Earth clocks give dilated time. Proper time is always the shortest.

Misconception 3: Each observer sees the other's clock running fast.
Wrong. Each inertial observer sees the other's clock running slow. This symmetry appears contradictory but is consistent because simultaneity is relative. The asymmetry only becomes apparent when one observer changes frames (accelerates).

Three traps: (1) "Proper" does not mean Earth-frame — it is the time in the clock's own rest frame (always shortest). (2) Moving clocks run slow, not fast — each observer sees the other's clock slow. (3) HSC strategy: identify proper time first → calculate $\gamma$ → apply $\Delta t = \gamma\Delta t_0$ → check $\Delta t > \Delta t_0$.

Write the three misconception corrections and the four-step exam strategy.