The Twin Paradox
Who comes home younger, and why.
Lesson at a glanceShow the key points
- Earth: 2D/v; traveller: 2D/(γv).
- The traveller changes frames; 'Earth now' jumps.
- Straight worldlines have the longest proper time.
Twins say goodbye; one flies to a star and back at high speed. Each sees the other’s clock running slow, so who ends up younger? Working it through carefully pulls together everything in this chapter.
The trip
Twins Asha and Bina live on Earth. Bina boards spaceship S₁ heading for planet P, which is 6 light-years away (in the Earth frame) and at rest relative to Earth. At P she jumps across to ship S₂ heading home. Both ships move at 0.6c relative to Earth, so γ = 1.25.
Asha’s view (Earth)
Each leg takes 6 ly / 0.6c = 10 years of Earth time, 20 years in all. Bina is moving the whole time, so her clocks, including her heartbeat and her ageing, run slow by γ: each leg lasts 10/1.25 = 8 years for her. Asha predicts Bina returns having aged 16 years while Asha aged 20. Bina will be 4 years younger.
This is a real effect: fast-moving unstable particles held in circular orbits by magnets live measurably longer than identical particles at rest.
Bina’s view, and the paradox
On S₁, Earth and P are moving, so their 6 ly separation is contracted to 6/1.25 = 4.8 ly. The planet approaches at 0.6c and arrives in 4.8/0.6 = 8 years by her clock. The return is the same: 16 years total. Bina agrees with Asha about her own age.
But Bina also sees Asha’s clock as moving, so running slow: in Bina’s 16 years Asha should age only 16/1.25 = 12.8 years. So Bina expects Asha to be the younger one. Each thinks the other is younger. Who is right?
The resolution
The situation isn’t symmetric. Asha stays in one inertial frame. Bina changes frames at the planet, from S₁ to S₂, and when you change frames you change which distant events count as “now” (the previous lesson). Track the clocks carefully:
| Moment | Bina's clock | Earth clock, as judged by Bina |
|---|---|---|
| Leaving Earth (on S₁) | 0 | 0 (the planet's clock reads 3.6 y ahead in S₁: rear clock leads by L₀v/c² = 6 × 0.6) |
| Arriving at P, still on S₁ | 8 y | 8 × 0.8 = 6.4 y (planet's clock: 10 y) |
| Just after boarding S₂ | 8 y | Earth is now the rear clock, leading the planet's by 3.6 y: 10 + 3.6 = 13.6 y |
| Arriving home | 16 y | 13.6 + 6.4 = 20 y |
The time-dilation argument covered only the two coasting legs (6.4 + 6.4 = 12.8 years). It missed the 7.2 years by which “Earth now” jumps forward when Bina switches ships. Add it in and both twins agree: Asha aged 20 years, Bina 16.
Why the turning twin is younger
On a spacetime diagram the stay-at-home twin follows a straight line and the traveller a bent one. In relativity a straight (inertial) path between two events always has the longest proper time; any detour is shorter. That makes Bina the younger.
- 1Each inertial observer sees the other's clocks run slow, but the traveller changes frames.
- 2Changing frames shifts which distant events are simultaneous: 'Earth now' jumps by 2L₀v/c².
- 3Earth: 2D/v; traveller: 2D/(γv). For 6 ly at 0.6c: 20 y versus 16 y.
- 4Inertial (straight) worldlines have the longest proper time.
Check yourself
Keep exploring
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