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Mathematics for Physics

179 Mass-Energy Equivalence

The principle that mass and energy are interchangeable forms of the same quantity, expressed by Einstein’s relation between energy and rest mass, that is used to relate rest mass to rest energy.

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 rest mass.

The mass-energy relation is

\[ E = mc^{2} \]

where

  • \(E\) is the rest energy.
  • \(m\) is the rest mass.
  • \(c\) is the speed of light.

Rest-mass deficit. Rest mass of a closed system is not separately conserved. A drop in rest mass appears as kinetic energy or radiation. A rest-mass deficit is the decrease of total rest mass from reactants to products. This principle is used to compute the energy released in fission, fusion, and annihilation.

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

Unified conservation of mass-energy. Conservation of mass and conservation of energy become one conservation law for total relativistic energy. Total relativistic energy is the sum of rest energies and kinetic energies. This principle is used to balance collisions and decays in which mass and kinetic energy convert into each other.

179.1 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. — \(E=mc^{2}\); nuclear mass deficit; unified energy conservation.
  3. Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — rest energy; annihilation; inertia of internal energy.
  4. Shankar, R. Fundamentals of Physics I. Yale University Press, 2019. — rest-mass transmutation; unified conservation.
  5. Emam, M. H. Covariant Physics. Oxford University Press, 2021. — \(E=mc^{2}\); conservation of total energy.
  6. Carroll, S. M. Spacetime and Geometry. Cambridge University Press. — rest energy; inertia of stored energy.