rapidity

Velocities in relativity are awkward: they refuse to simply add (two 0.8c boosts do not give 1.6c), and they hit a wall at c. Rapidity is a repackaging of velocity that fixes both problems — a measure of 'how boosted' a frame is that adds up straightforwardly and runs all the way to infinity as speed approaches c.

The rapidity phi of a boost is defined by tanh(phi) = beta = v/c, equivalently gamma = cosh(phi) and gamma beta = sinh(phi). In this variable a Lorentz boost along x is literally a hyperbolic rotation: ct' = ct cosh(phi) - x sinh(phi), x' = x cosh(phi) - ct sinh(phi) — the same form as an ordinary rotation but with cos -> cosh and sin -> sinh. The key payoff: collinear boosts add their rapidities, phi_total = phi_1 + phi_2, and this reproduces the relativistic velocity-addition formula tanh(phi_1 + phi_2) = (tanh phi_1 + tanh phi_2)/(1 + tanh phi_1 tanh phi_2).

Rapidity makes the geometry transparent: a boost is a rotation through hyperbolic angle phi in the (ct, x) plane, which is why the Lorentz group's boost part looks like SO(1,1) with hyperbolic functions. As v -> c, phi -> infinity, so the light barrier is the statement that no finite sum of rapidities reaches infinity. In particle physics, differences in (longitudinal) rapidity are invariant under boosts along the beam, which is why detectors bin particles by rapidity (or its massless approximation, pseudorapidity). Caveat: rapidities add only for boosts along a single line; for non-collinear boosts you again pick up a Wigner rotation.