Theory of relativity

Relativistic momentum and energy, E = mc²

Energy and momentum form the four-momentum; rest mass is invariant and satisfies the energy–momentum relation.

An object of rest mass m moving at speed v has momentum p=γmv and total energy E=γmc². Its rest energy E₀=mc² remains when it is at rest.

Physical meaning

Avoid calling γmc² “relativistic mass” when m denotes invariant rest mass. Keeping mass and energy distinct makes conservation laws consistent across frames.

The four-momentum $P^\mu=(E/c,\mathbf p)$ has Minkowski norm $P_\mu P^\mu=-m^2c^2$ in the time-negative signature. This yields $E^2=p^2c^2+m^2c^4$, valid for massive particles and for photons when $m=0$. The mass $m$ is invariant; it does not increase with speed, so the phrase “relativistic mass” can obscure the distinction between mass and energy. In a center-of-momentum frame, total momentum vanishes, yet the system’s energy can still contribute to its invariant mass.

In a collision, each particle’s kinetic energy need not be conserved, but the total four-momentum of an isolated system is. The final products’ rest energy can differ from the initial total rest energy because initial kinetic energy may become new rest mass. For a photon, $E=pc$ because its invariant mass is zero, yet it carries energy and momentum. Two counter-propagating photons can form a system with nonzero invariant mass.

Quick check