The Spacetime Interval

The Spacetime Interval

The Lorentz transformation mixes space and time — different observers disagree on $\Delta x$ and $\Delta t$ separately. But there is a quantity they all agree on: the spacetime interval:

$s^2 = c^2 \Delta t^2 - \Delta x^2$

(Using the $+---$ metric signature.) No matter which inertial frame you compute this in, you get the same number. It is the relativistic generalisation of the distance formula.

Classification of Intervals

Proper Time

For a timelike interval, the proper time $\Delta\tau$ is the time measured by a clock that travels directly between the two events:

$\Delta\tau = \frac{\sqrt{s^2}}{c}$

Invariance Example

Event: $\Delta t = 5$ s, $\Delta x = 3c$ m. Then $s^2 = 25c^2 - 9c^2 = 16c^2$. After a Lorentz boost to $v = 0.6c$ ($\gamma = 1.25$): $\Delta t' = 4$ s, $\Delta x' = 0$. So $s^2{}'= 16c^2 - 0 = 16c^2$. Invariant confirmed.

Your Task

Implement spacetime_interval_sq(delta_t, delta_x) returning $s^2 = c^2 \Delta t^2 - \Delta x^2$, and interval_type(s_sq) returning "timelike", "lightlike", or "spacelike". Use $c = 299792458.0$ m/s defined inside each function.