What is Relativistic momentum and energy? — Physics

Definition 5.2 University Physics — Year 3 · Chapter 5 — Relativistic Dynamics

A particle of mass $m$ and velocity $\vect v$ ( $\gamma = 1/\sqrt{1 - v^2/c^2}$ ) carries the momentum and the energy

\vect p = \gamma m\vect v , \qquad E = \gamma mc^2 .

At rest, $E_0 = mc^2$ : the rest energy. The kinetic energy is what motion adds, $E_k = E - mc^2 = (\gamma - 1)mc^2$ , which for $v \ll c$ reduces to the familiar $\tfrac12 mv^2$ (expand $\gamma$ ). In every isolated process, total $\vect p$ and total $E$ — rest energies included — are conserved.

Left: Bertozzi’s “ultimate speed” experiment — measured electron speeds (dots) follow the relativistic curve and saturate at c while Newton’s line sails past it. Right: the energy–momentum hyperbola; its offset at p = 0 is the rest energy, its asymptote the photon’s E = pc. — Left: Bertozzi’s “ultimate speed” experiment — measured electron speeds (dots) follow the relativistic curve and saturate at $c$ while Newton’s line sails past it. Right: the energy–momentum hyperbola; its offset at $p = 0$ is the rest energy, its asymptote the photon’s $E = pc$ .

Examples

Example 5.5 (Orders of magnitude)

Particle physics counts energy in electronvolts and masses in $\mathrm{MeV}/c^2$ : electron $0.511$ , proton $938.3$ , muon $105.7$ , pion $139.6$ . An electron of kinetic energy $1\,\mathrm{MeV}$ has $\gamma = 2.96$ and $v = 0.94c$ — already fully relativistic, which is why electronics stays classical ( $\mathrm{eV}$ ) and nuclear physics does not ( $\mathrm{MeV}$ ). A $6.8\,\mathrm{TeV}$ LHC proton has $\gamma = 7250$ and trails a photon by only $2.9\,\mathrm{m}/\mathrm{s}$ .

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