Four-momentum
The Lorentz four-vector p^μ = (E/c, p_x, p_y, p_z) combining a particle's energy and three-momentum into a single object that transforms covariantly under boosts. Its invariant norm-squared p^μ p_μ = (E/c)² − |p|² = m²c² is the energy-momentum-mass relation; total four-momentum is conserved in any collision.
Definition
Four-momentum is the canonical dynamical object of special relativity: a Lorentz four-vector p^μ = (E/c, p_x, p_y, p_z) whose time component is the energy divided by c and whose spatial components are the relativistic three-momentum p = γmv. Under a Lorentz boost, the four components mix exactly as the (ct, x, y, z) coordinates do, so p^μ transforms covariantly — the same boost matrix that rotates spacetime coordinates rotates four-momentum components. This Lorentz-covariance is precisely why four-momentum is the right object to use: conservation of energy and conservation of three-momentum in one frame translate to conservation of all four components of total p^μ in every frame, automatically.
The Minkowski-norm-squared of the four-momentum is the energy-momentum-mass relation: p^μ p_μ = (E/c)² − |p|² = m²c², the most famous Lorentz invariant in physics after the speed of light itself. It says energy and momentum are not independent: a particle's rest mass m is the invariant length of its four-momentum vector (in natural units), and given any two of {E, |p|, m} the third is fixed. For a photon (m = 0) the relation collapses to E = |p|c; for a particle at rest (|p| = 0) it gives Einstein's E = mc²; for relativistic particles in general it is E² = (pc)² + (mc²)². Total four-momentum is conserved in any collision, and the invariance of p^μ p_μ provides the cleanest way to compute thresholds, decay kinematics, and scattering observables: pick the simplest frame, evaluate the scalar, then export the result to any other frame.