# Pre-Quantum Electrodynamics
Author: Jean-Sébastien Caux

#### Relativistic Momentum and Energy
red.rm.rme

The {\bf relativistic momentum} \({\boldsymbol p}\) is defined as \[ {\boldsymbol p} \equiv m {\boldsymbol \eta} = \frac{m {\boldsymbol u}}{1 - u^2/c^2}. \] The {\bf relativistic energy} is defined as \[ E \equiv \frac{m c^2}{\sqrt{1 - u^2/c^2}}. \] These can be combined into the {\bf energy-momentum four-vector} \[ p^\mu \equiv m \eta^\mu. \]

When the object is stationary, its energy is the

{\bf Rest energy} \[ E_{\mbox{\tiny rest}} \equiv m c^2. \]

When moving, the difference between relativistic and rest energies is the

{\bf Kinetic energy} \[ E_{\mbox{\tiny kin}} \equiv E - mc^2 = mc^2 \left( \frac{1}{\sqrt{1-u^2/c^2}} - 1 \right). \]

For velocities much smaller than the speed of light, we can expand this to \[ E_{\mbox{\tiny kin}} = \frac{1}{2} mu^2 + \frac{3}{8} \frac{mu^2}{c^2} + ... \]

In a closed system,

{\bf Total relativistic energy and momentum is conserved} \[ E^2 - c^2 p^2 = m^2 c^4 \]

N.B.: don't confuse an {\bf invariant} quantity with a {\bf conserved} quantity.

Except where otherwise noted, all content is licensed under a Creative Commons Attribution 4.0 International License.

Author: Jean-Sébastien Caux

Created: 2022-03-07 Mon 20:38