#### Relativistic Momentum and Energy
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The relativistic momentum \({\boldsymbol p}\) is defined as \[ {\boldsymbol p} \equiv m {\boldsymbol \eta} = \frac{m {\boldsymbol u}}{\sqrt{1 - u^2/c^2}}. \] The relativistic energy is defined as \[ E \equiv \frac{m c^2}{\sqrt{1 - u^2/c^2}} \equiv c p^0. \] These can be combined into the energy-momentum four-vector \[ p^\mu \equiv m \eta^\mu \]

When the object is stationary, its energy is the

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

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

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

For velocities much smaller than the speed of light, we can expand this to \[ E_{\mbox{kin}} = \frac{1}{2} mu^2 + \frac{3}{8} \frac{mu^4}{c^2} + ... \]

In a closed system,

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

N.B.: don't confuse an invariant quantity with a conserved quantity.

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42