Relativistic Energy & Momentum Reference Cheat Sheet

Key Facts

• The Lorentz factor is $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, where $v$ is speed and $c$ is the speed of light.
• Relativistic momentum is $p = \gamma mv$, which approaches classical momentum $p = mv$ when $v \ll c$.
• Total relativistic energy is $E = \gamma mc^2$, including both rest energy and kinetic energy.
• Rest energy is $E_0 = mc^2$, which is the energy an object has even when $v = 0$.
• Relativistic kinetic energy is $K = (\gamma - 1)mc^2$, not $K = \frac{1}{2}mv^2$ at high speeds.
• The energy-momentum relation is $E^2 = (pc)^2 + (mc^2)^2$, which works for massive particles and simplifies for photons.
• For a photon with $m = 0$, the energy-momentum relation becomes $E = pc$.
• No object with nonzero rest mass can reach $c$ because $\gamma$ increases without bound as $v$ approaches $c$.

Vocabulary

Lorentz factor

The factor $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ that measures how strongly relativistic effects appear at speed $v$.

Rest energy

The energy $E_0 = mc^2$ stored in an object's mass when it is not moving relative to the observer.

Total energy

The full relativistic energy $E = \gamma mc^2$, equal to rest energy plus kinetic energy.

Relativistic momentum

The momentum $p = \gamma mv$ of an object moving at speed $v$, including the effect of the Lorentz factor.

Invariant mass

The rest mass $m$ of an object, which is the same in all inertial reference frames.

Photon

A massless particle of light that travels at speed $c$ and has energy $E = pc$.