Pre-Quantum Electrodynamics
Relativistic Momentum and Energyred.rm.rme
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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Created: 2024-02-27 Tue 10:31