3. Mass and energy in relativity

Mass, momentum and units #

Invariant mass is the same for every inertial observer. Momentum and total energy change under a change of frame. Write E₀=mc² for rest energy and K=E−E₀ for kinetic energy. In natural units one often writes a mass in GeV; in SI the corresponding mass unit is GeV/c². For a composite system, add both energies and momenta before calculating its invariant mass.

3.1 The full energy-momentum relation #

The famous expression $E=mc^2$ is the rest-frame special case of

E^2=p^2c^2+m^2c^4.

For a particle at rest, $p=0$ and therefore

E_0=mc^2.

For a massless particle, $m=0$ , so

E=pc.

This is the form relevant to photons. Modern particle physics usually avoids the older language “relativistic mass” or “moving mass.” The invariant mass $m$ is frame-independent; energy and momentum transform as components of a four-vector. Zur Elektrodynamik bewegter K"orper (1905) · Review of Particle Physics (2026)

3.2 Natural units #

High-energy physics commonly sets

\hbar=c=1.

Then energy, momentum, and mass are expressed in the same units, often eV, MeV, or GeV. To convert back, the extremely useful CODATA identity is

\hbar c=197.3269804\ldots\ \mathrm{MeV\,fm},

which is exact once the defining SI constants are used. CODATA recommended values of the fundamental physical constants: 2022 (2024)

For example a characteristic length scale associated with momentum transfer $Q$ is roughly

\Delta x\sim\frac{\hbar c}{Q}.

Thus $Q=1\,\mathrm{GeV}$ probes distances of order $0.2\,\mathrm{fm}$ , while $Q=100\,\mathrm{GeV}$ probes around $2\times10^{-3}\,\mathrm{fm}$ .

3.3 Composite mass is a rest-frame energy eigenvalue #

For a closed composite system in its center-of-momentum frame,

M c^2 = E_{\mathrm{total}}(\mathbf{P}=0).

This equation is conceptually more reliable than “add the constituent masses.” A hydrogen atom weighs slightly less than a free proton plus free electron because its electromagnetic binding energy is negative. A bound nucleus weighs less than the corresponding free nucleons by its nuclear binding energy. A proton, however, is a relativistic strongly coupled QCD state, so its mass must be calculated from the QCD Hamiltonian or from correlation functions in lattice QCD rather than from a nonrelativistic constituent sum. Revisiting the proton mass decomposition (2020) · Lattice-QCD Validation of Hadron Mass and Trace-Anomaly Decomposition Sum Rules (2026)

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