Caterina towards Alpha Centauri

Problem

Caterina leaves Earth at time $t=0$ and travels towards Alpha Centauri, which is $L=4$ light-years (ly) away, at speed $v=0{,}9\,c$. Having arrived at Alpha Centauri, she reverses course and returns to Earth at the same speed. Her twin brother Simone waits for her on Earth. When Caterina returns, how many years have passed for her? How many for Simone?

Solution

We work in units where $c=1$ ly/year.

Time for Simone (Earth frame, SRT). In SRT, Caterina moves with uniform straight-line motion. The outbound leg lasts: $\Delta t_\text{outbound}=\frac{L}{v}=\frac{4}{0{,}9}\approx 4{,}44\ \text{years}$ The return lasts the same, so for Simone: $t_C^\text{Simone}=2\cdot 4{,}44\approx 8{,}88\ \text{years}$

Time for Caterina (proper time). We use the invariant between the “departure” event ($E_A$) and the “arrival at Alpha Centauri” event ($E_B$): $(\Delta\tau_{AB})^2=(\Delta t_{AB})^2-(\Delta s_{AB})^2/c^2=4{,}44^2-4^2=19{,}71-16=3{,}71$ $\Delta\tau_\text{outbound}=\sqrt{3{,}71}\approx 1{,}93\ \text{years}$ By symmetry the return lasts the same amount, so the total for Caterina is: $\ev{t_C^\text{Caterina}\approx 2\cdot 1{,}93 = 3{,}86\ \text{years}}$

Conclusion. When Caterina returns to Earth, she has aged about $3{,}9$ years, while Simone has aged about $8{,}9$: the difference is about 5 years. If they were identical twins at the start, on her return Caterina is younger. It is not an optical illusion: she has truly lived less time than Simone.

Why it is not really a "paradox" not symmetric. Simone always stays in a single inertial frame; Caterina, on the other hand, changes reference frame halfway through the trip (she reverses course, experiencing an acceleration). The invariant formula $(\Delta\tau)^2=(\Delta t)^2-(\Delta s/c)^2$ applies separately to the outbound and return legs, and the sum of the two proper times is less than Simone's total time. Golden rule: between two events connected by motion, the proper time is maximal for the uniform straight-line trajectory (a single SR) and decreases if the traveller changes frame — "whoever stays still ages more".

It cannot be said, symmetrically, that Simone is the younger one: the two roles are

Links

Topics: Special relativity Concepts: Time dilation · Spacetime invariant Skills: Changing reference frame Methods: Minkowski diagram · Natural units