Proper Time and the Twin Paradox

Proper Time

Every clock measures its own proper time $\tau$ — the time elapsed along its own worldline. For a clock moving at constant velocity $v$ for coordinate time $\Delta t$:

$\Delta\tau = \frac{\Delta t}{\gamma} = \Delta t\sqrt{1 - \frac{v^2}{c^2}}$

Because $\gamma \geq 1$, proper time is always less than or equal to coordinate time. Moving clocks run slow — time dilation.

The Twin Paradox

Alice stays on Earth; Bob travels at $v = 0.8c$ for $T/2$ then returns. Earth's coordinate time for the round trip is $T$. Bob's proper time is:

$\tau_\text{Bob} = \frac{T}{\gamma}$

For $v = 0.8c$, $\gamma = 5/3 \approx 1.667$, so Bob ages only $0.6 T$ — he returns younger than Alice.

There is no paradox: Bob's worldline is not straight (he decelerates and turns around), so the situation is not symmetric. Alice follows the straight worldline between the two events — which is the worldline of maximum proper time (the spacetime geodesic).

Your Task

Implement proper_time(t, v) returning the proper time for coordinate time $t$ at constant speed $v$, and twin_age_difference(T, v) returning how much less the traveling twin ages compared to the stay-at-home twin. Use $c = 299792458.0$ m/s defined inside each function.