Four-Vectors

Four-Vectors

In special relativity, space and time mix under Lorentz transformations. The natural language for this is 4-vectors — objects with one time component and three space components that transform covariantly.

Common 4-Vectors

Minkowski Norm

Using the $+{-}{-}{-}$ signature, the Minkowski norm squared is:

$|A|^2 = A_t^2 - A_x^2 - A_y^2 - A_z^2$

This quantity is Lorentz-invariant — all observers agree on it.

Invariant Mass

For the 4-momentum, the norm gives the invariant mass:

$|p|^2 = (E/c)^2 - |\vec{p}|^2 = (mc)^2$

A lightlike vector (photon) has $|A|^2 = 0$. A timelike vector has $|A|^2 > 0$.

Your Task

Implement:

• minkowski_norm_sq(at, ax, ay, az) — computes $A_t^2 - A_x^2 - A_y^2 - A_z^2$
• invariant_mass(E, px, py, pz) — returns mass in kg given $E$ in J and momentum in kg⋅m/s

Use $c = 299792458.0$ m/s defined inside each function that needs it.