Twin paradox — time dilation

What just happened?

The twin flew close to the speed of light and came back. Their clock — and with it the heart, the cells, the whole body — ticked slower than on Earth, so they aged less. It is not an illusion: it is real time dilation.

Lorentz factor:

— for every speed $v < c$ we have $\gamma \ge 1$. β = v/c is the speed as a fraction of the speed of light, c ≈ 3·10⁸ m/s.

The traveller’s proper time:

t — time measured on Earth, t′ — time experienced by the traveller. Since $\gamma \ge 1$, then $t' \le t$ — the traveller always comes back younger.

Why isn’t it symmetric?

The “paradox” disappears because the twins are not equivalent: it is the rocket that brakes and turns around — it accelerates. Only the traveller changes reference frame, so it is their time that runs shorter. The Earth twin stays in a single inertial frame the whole time.

Why you need this for the exam:

γ = 1/√(1−β²) and t′ = t/γ solve problems on time dilation and length contraction. The trick: compute γ from the speed, then simply divide (time) or multiply — and remember that γ is never less than 1.

Remember: the closer to the speed of light, the larger γ and the slower the traveller’s time runs — they come back younger. t′ = t/γ, and γ ≥ 1 always.