Time Dilation

Time Dilation

One of the most astonishing predictions of special relativity is that moving clocks run slow. This is not a mechanical effect — it is a fundamental property of spacetime itself.

Proper Time and Coordinate Time

The proper time $\Delta\tau$ is the time measured by a clock that travels with the moving object (its own rest frame). The coordinate time $\Delta t$ is the time measured by a stationary observer watching the clock move past.

The relationship is:

$\Delta t = \gamma\,\Delta\tau$

Because $\gamma \geq 1$, the coordinate time is always at least as large as the proper time. A moving clock ticks fewer times than a stationary clock between two events: it is running slow as seen from outside.

Inverting the relation gives the proper time from a measured coordinate time:

$\Delta\tau = \frac{\Delta t}{\gamma}$

The Muon Experiment

Cosmic rays create muons at 15 km altitude moving at $v \approx 0.998c$ toward the ground. In their own rest frame, a muon lives only about $\tau = 2.2\,\mu\text{s}$ — classically it should decay after traveling just $2.2\,\mu\text{s} \times 0.998c \approx 660\text{ m}$. Yet muons reach sea level because, from our frame, their internal clock runs slow: $\gamma \approx 15$, so the coordinate lifetime is $\approx 33\,\mu\text{s}$ and they cover the full 15 km.

Your Task

Implement:

• time_dilation(tau, v) — returns coordinate time $\Delta t = \gamma\,\Delta\tau$
• proper_time(t, v) — returns proper time $\Delta\tau = \Delta t / \gamma$

Use $c = 299792458.0$ m/s, defined inside each function.