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Mathematics for Physics

201 Rest Energy

The energy an object has by virtue of its rest mass alone when stationary, equal to rest mass times the speed of light squared.

Inherent rest energy. A stationary body has rest energy equal to its rest mass times \(c^{2}\). Rest energy is the energy equivalent of rest mass when the body is at rest. Rest mass is the mass measured in the body’s rest frame. This principle is used to fix the energy stored in existence alone.

The rest energy is

\[ E_{0} = mc^{2} \]

where

  • \(E_{0}\) is the rest energy.
  • \(m\) is the rest mass.
  • \(c\) is the speed of light.

Rest mass as an energy reservoir. Rest mass is a reservoir of energy that everyday processes leave unused. This principle is used to compute the energy released when rest energy converts to kinetic energy and radiation.

Inertia of stored internal energy. Internal energy stored in a closed system increases that system’s rest mass and rest energy. Internal energy is the kinetic and potential energy of the constituents. This principle is used to show that a hotter or more compressed body has more rest energy than a colder or relaxed one.

Invariance of rest energy. Total energy and momentum change from frame to frame, while rest energy is the same in every inertial frame because it is measured in the unique rest frame. An invariant is a quantity all inertial observers assign the same value. This principle is used to separate the energy of motion from the energy of existence.

Rest-mass transmutation. Rest mass of a closed system is not separately conserved. A rest-mass deficit is the drop in total rest mass from reactants to products, and that drop appears as kinetic energy. This principle is used to compute fragment energies in decays and collisions.

201.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — source for the heading explanation.
  2. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — \(E_{0}=mc^{2}\); stored internal energy; invariant rest energy.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — rest energy; inertia of internal energy; rest-mass deficit.
  4. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — rest energy as a reservoir; \(E_{0}=mc^{2}\).
  5. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — rest-mass transmutation; fusion energy.
  6. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — rest energy as an invariant.