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Mathematics for Physics

206 Total Energy

A relativistic energy equal to the Lorentz factor times rest energy that is used to express the full energy of a free particle including rest energy and kinetic energy.

Rest-mass energy. A particle has rest energy even when it is at rest. Rest energy is the energy equivalent of rest mass for a stationary particle. Rest mass is the mass measured in the particle’s rest frame. This principle is used to fix the energy baseline of any mass.

Motion-scaled total energy. In motion, the total energy is the rest energy scaled by the Lorentz factor. Total energy is the full energy of a free particle in a given frame. This principle is used to compute the energy of a high-speed particle.

The total energy is

\[ E = \gamma mc^{2} = E_{0} + K \]

where

  • \(E\) is the total energy.
  • \(\gamma\) is the Lorentz factor of the particle.
  • \(m\) is the rest mass.
  • \(c\) is the speed of light.
  • \(E_{0} = mc^{2}\) is the rest energy.
  • \(K\) is the kinetic energy.

Unification of rest and kinetic energy. Total energy is the sum of rest energy and relativistic kinetic energy. Relativistic kinetic energy is the excess of total energy over rest energy. This principle is used to recover Newtonian kinetic energy at low speed.

Energy as the time part of four-momentum. Total energy is the time component of the four-momentum and is not itself an invariant. Four-momentum is the four-vector \(\left(\dfrac{E}{c},\,\mathbf{p}\right)\). This principle is used to relate energy, momentum, and mass in collisions.

Conservation of total relativistic energy. In an isolated collision, the sum of the total energies is conserved, even when rest masses change. This principle is used to balance nuclear reactions and particle decays.

Note: Also called relativistic energy.

206.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — \(E=\gamma mc^{2}=E_{0}+K\); rest energy; energy conservation.
  2. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(E=\dfrac{mc^{2}}{\sqrt{1-\dfrac{u^{2}}{c^{2}}}}\); four-momentum; conservation.
  3. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — \(E=\gamma mc^{2}\); four-momentum.
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — rest energy; energy as the time component of four-momentum.
  5. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — energy-momentum four-vector.