Four-momentum

Four-momentum is the relativistic four-vector whose time component represents energy and whose spatial components represent three-momentum. It transforms between inertial frames under Lorentz transformations and has an invariant magnitude determined by rest mass.

Lorentz transformation: A linear transformation relating spacetime coordinates measured in inertial frames moving at constant relative velocity. It mixes four-momentum’s energy and spatial components when changing inertial frames.
Special relativity: The theory of space, time, and motion in inertial frames, with the speed of light invariant. Four-momentum is the energy–momentum quantity formulated to obey special relativity.
Conservation of four-momentum: The conservation law requiring total four-momentum to remain unchanged in an isolated interaction. It supplies frame-independent energy and momentum bookkeeping for particle reactions.
Newtonian momentum: The product of an object's mass and velocity in classical mechanics. Unlike four-momentum, it does not combine energy with spatial momentum or transform relativistically.
Energy–momentum relation: The relativistic equation relating a particle’s energy, momentum magnitude, and rest mass. It expresses the invariant norm of four-momentum as the particle’s rest mass.
Three-momentum: The vector quantity equal to a body's mass times its velocity in Newtonian mechanics. These three spatial components appear alongside energy in the relativistic four-vector.
Invariant mass in particle physics: The mass reconstructed from the combined four-momenta of a particle’s decay products. Adding measured decay-product four-momenta reveals the mass of an unstable parent particle.
Energy: A scalar physical quantity measuring a system’s capacity to produce changes or perform work. Energy alone is frame-dependent; four-momentum packages it with momentum into a Lorentz-covariant object.
Invariant mass: The mass of a system defined by the Lorentz-invariant magnitude of its total four-momentum. A particle’s four-momentum magnitude determines its invariant mass in every inertial frame.
Relativistic energy: The total energy of a body in special relativity, including its rest energy. It is the time component paired with three-momentum.

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