a Lorentz boost

How do the coordinates of an event change when you switch to a frame moving at constant velocity? In Newtonian physics you just shift x by vt and leave time alone (a Galilean transformation). In relativity, because the speed of light is the same for everyone, time itself must transform too. A Lorentz boost is that correct rule — the rotation-like mixing of space and time between two inertial frames in relative motion.

For a boost of speed v along x, with beta = v/c and gamma = 1/sqrt(1 - beta^2), the transformation is: ct' = gamma(ct - beta x), x' = gamma(x - beta ct), y' = y, z' = z. In matrix form x'^mu = Lambda^mu_nu x^nu, where Lambda is defined by exactly this mixing. A boost is precisely the Lorentz transformation that preserves the interval ds^2 while changing the state of motion (as opposed to a spatial rotation). It reduces to the Galilean transformation x' = x - vt, t' = t in the limit v << c (gamma -> 1).

All of relativity's kinematic surprises are boosts read off in components: time dilation (dt' = gamma dt for a clock at rest), length contraction (a rod's length shrinks by 1/gamma), relativity of simultaneity (the beta x term mixing into t'), and the velocity-addition law. A boost is a 'rotation' in the (ct, x) plane, but through an imaginary/hyperbolic angle — the rapidity — which is why boosts along the same axis simply add their rapidities. Caveat: boosts along different axes do NOT commute, and composing two non-parallel boosts yields a boost plus a residual rotation (the Wigner rotation, source of Thomas precession) — boosts alone do not form a subgroup.