Time dilation

What just happened?

The rocket sped up to near the speed of light, and its light clock began to tick slower than ours. This is time dilation — moving clocks run slow.

Time dilation:

— $\Delta t_0$ is the proper time (measured by the moving clock), and $\Delta t$ is the time in our frame. Since $\gamma\ge 1$, more of it passes for us: the moving clock "falls behind".

When is the effect noticeable?

For everyday speeds $v\ll c$, so $\gamma\approx 1$ and the difference is immeasurable — time flows "classically". The effect becomes dramatic only when $v$ approaches $c$ (then $\gamma\to\infty$).

The twin paradox:

A twin who flies off in a fast rocket and returns will be younger than the one who stayed on Earth — because less proper time passed in their frame. This is not an illusion, but a real consequence of $\Delta t=\gamma\,\Delta t_0$.

Why you need this for the exam:

The formula $\Delta t=\gamma\Delta t_0$ plus computing γ from $v/c$ solves typical problems on time dilation and the twin paradox. The trick: identify which clock measures the proper time $\Delta t_0$ (the one that moves), and multiply by γ.

Remember: moving clocks run slow — Δt = γ·Δt₀. The closer to c, the larger γ; for v≪c γ≈1 and the effect vanishes.