Chapter 3: Energy-Momentum Relation
The energy-momentum relation is the fundamental equation connecting energy, momentum, and mass. It's the relativistic version of the kinetic energy formula and reveals the deep connection between these quantities.
The Energy-Momentum Relation
\( E^2 = (pc)^2 + (mc^2)^2 \)
The "mass shell" condition—true for all free particles
At Rest (p = 0)
E = mc² (rest energy)
Massless (m = 0)
E = pc (photons)
General
\( E = \gamma mc^2 \)
Relativistic Kinetic Energy
\( K = E - mc^2 = (\gamma - 1)mc^2 \)
Total energy minus rest energy
At low speeds (v ≪ c), expanding γ - 1 ≈ v²/2c² gives:
\( K \approx \frac{1}{2}mv^2 \) (classical result)
The Dispersion Relation
In quantum mechanics, E = ℏω and p = ℏk. The energy-momentum relation becomes:
\( \omega^2 = c^2k^2 + \frac{m^2c^4}{\hbar^2} \)
This is the Klein-Gordon dispersion relation, fundamental to relativistic quantum field theory.