Rapidity

Rapidity

Velocity is a poor coordinate for relativistic motion — it is bounded by $c$ and the relativistic addition formula is unwieldy. Rapidity $\phi$ is the natural alternative:

$\tanh\phi = \beta = \frac{v}{c}, \quad \phi = \text{arctanh}(\beta)$

Rapidity ranges from $-\infty$ to $+\infty$, has no upper bound, and makes velocity addition trivially additive:

$\phi_\text{total} = \phi_1 + \phi_2$

Converting back: $v = c\tanh\phi$.

Why It Works

The Lorentz transformation in rapidity form is a hyperbolic rotation — exactly like a Euclidean rotation but with $\cosh$ and $\sinh$ instead of $\cos$ and $\sin$:

$\gamma = \cosh\phi, \quad \gamma\beta = \sinh\phi$

Velocity Addition Example

Two ships each travel at $0.5c$ in the same direction. Classical result: $v = c$. Relativistic result using rapidity:

$\phi_\text{total} = 2\,\text{arctanh}(0.5) \approx 1.0986$

$v = c\tanh(1.0986) = 0.8c$

No matter how many $0.5c$ boosts you stack, you never reach $c$.

Your Task

Implement rapidity(v) returning $\phi = \text{arctanh}(v/c)$, velocity_from_rapidity(phi) returning $v = c\tanh\phi$, and add_rapidities(phi1, phi2) returning the resulting velocity $c\tanh(\phi_1 + \phi_2)$. Use $c = 299792458.0$ m/s defined inside each function.