Mass–energy equivalence

In special relativity, the mass-energy equivalence usually expressed as E = mc² is the concept that there is an energy equivalence to any mass.

For an isolated body at rest, this energy, called the "rest energy", is defined absolutely, is proportional to the quantity of matter of the body, is always positive, and does not depend on the observer or the inertial frame of reference used to observe it.

Formula E = mc²

In 1905, Albert Einstein published the paper "Does the Inertia of a Body Depend Upon Its Energy Content?"^[1], containing one of the most famous equations in the field of physics: the mass-energy equivalence, relating energy and mass differences, as ΔE = Δmc².

The use of the energy difference ΔE by Einstein was consistent with classical mechanics, where energy is always expressed relative to a constant, arbitrary energy level, and can be positive or negative. ^[2]

Today, the mass-energy equivalence is written absolutely, as a completely definite energy that is always positive, observer-independent, and is directly proportional to the mass of a free (isolated) body, in the form ^[3]

Eo=mc^{2}

where

E₀ = the energy (also called "rest energy") equivalent to the mass (in joules),
m = mass (in kilograms), also called the invariant mass, and
c = the speed of light in a vacuum (celeritas) (in meters per second).

This equation applies to any closed system, for any composite body consisting of many particles, which is at rest. ^[4]

If the body is not at rest, then the full Energy-momentum relation must be used (L. Okun', op. cit.), as

E^{2}=m^{2}c^{4}+p^{2}c^{2},\;

where c is the speed of light, $E\;$ is total energy, $m\;$ is invariant mass, and $p\;$ is momentum.

History

The equivalence of energy and matter was first enunciated, in approximate form, in 1717 by Isaac Newton, in "Query 30" of the Opticks, where he states:

Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?

The mass-energy equivalence in terms of E = mc² was derived by Henri Poincare in 1900 ^[5] using the Newtonian relation p=mv and, more rigorously, by Albert Einstein (1905, op. cit.).

After 1906, the mass-energy equivalence was also used as E=Mc², in terms of the "relativistic mass"

M=\gamma m\!

where

M is the relativistic mass

m is the mass, and

\gamma ={1 \over {\sqrt {1-{\frac {|\mathbf {v} |^{2}}{c^{2}}}}}}\!

is the Lorentz factor,

v is the relative velocity between the observer and the object, and

c is the speed of light.

A difficulty with this approach is that since γ depends on velocity, observers in different inertial reference frames will calculate different values for the energy-mass equivalence. Other difficulties have been summarized by Okun^[6]. The current practice today, in scientific work, is to use the rest energy E₀ only. See Mass in special relativity for additional discussion.

Conservation of mass and energy

The concept of mass-energy equivalence complements but does not unite the concepts of conservation of mass and conservation of energy. While any mass can be converted to energy (such as kinetic energy, heat, or light), not every energy can be converted to mass. In relativity theory, in spite of popular philosophical discussions otherwise, mass and energy are not two forms of the same thing.

Binding energy and the mass defect

Max Planck first pointed out that the mass-energy equivalence formula implied that bound systems would have a mass less than the sum of their parts. The difference, called a mass defect, is a measure of the binding energy — the strength of the bond holding together the parts (in other words, the energy needed to break them apart). The greater the mass defect, the larger the binding energy.

Early experimenters realized that the very high binding energies of the atomic nuclei should allow calculation of their binding energies from mass differences. For example, the mass of a helium nucleus (that has two protons and two neutrons) is somewhat less than two times the proton mass plus two times the mass of a neutron. However it was not until the discovery of the neutron in 1932, and the measurement of the free neutron rest mass, that this calculation could actually be performed. Very shortly thereafter, the first transmutation reactions (such as ${}^{7}\mathrm {Li} +\mathrm {p} \rightarrow 2\,{}^{4}\mathrm {He}$ ) were able to verify the correctness of Einstein's mass-energy equivalence formula to an accuracy of 1%. See nuclear binding energy for example calculation.

The mass of a bound system is, thus, somewhat smaller than the sum of the masses of its constituent parts

(Mass of bound system) = (sum of masses of its parts) - (binding energy)/c².

Nuclear binding energies range from 1 to 9 MeV. The most stable nuclei in existence are iron-58 (iron with 58 nucleons) and nickel-62, which have the highest binding energies per nucleon of all nuclei.

Atomic energy

Contrary to popular belief, the strong nuclear binding energy is the main reason for the power of nuclear fission and nuclear fusion used in energy generation and atomic weapons, not E = mc². The main energy contribution is due to binding energy conversion to other forms of energy. This binding energy conversion is responsible for the mass defect, not the other way around. ^[7].

Practical examples

Physicists usually perform their calculations using the CGS measurement system (centimeters, grams, seconds, dynes, and ergs), because in these units the E and B fields of electromagnetism have the same unit (their unification is a consequence of special relativity). The formula can also use the SI system (with E in joules, m in kg, and c in meters per second). Using SI units, E=mc² is calculated as follows:

E = (1 kg) × (299,792,458 m/s)² = 89,875,517,873,681,764 J (≈90 × 10¹⁵ Joules)

Accordingly, one gram of mass — the mass of a U.S. dollar bill — is equivalent to the following amounts of energy:

≡ 89,875,517,873,681.764 J (≈90 terajoules), precisely by definition

≡ 24,965,421.631 578 267 777… kilowatt-hours (≈25 GW-hours)

= 21,466,398,651,400.058 278 398 777 1090 calories (≈21 Tcal)^[8]

= 21.466 398 651 400 058 278 398 777 1090 kilotons of TNT-equivalent energy (≈21 kt)^[8]

= 85,185,554,537.701 118 960 880 666 4808 BTUs (≈85 billion BTUs)^[8]

References

^ A. Einstein, "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?", Ann. d. Phys., 17, 891 (1905). Online English copy
^ Lev Davidovich Landau and Evgenii Mikhailovich Lifshits, (1987) Elsevier, ISBN 0750627689
^ Lev B. Okun', The Concept of Mass, Physics Today, June 1989.
^ L. D. Landau and E. M. Lifshits, op. cit., chapter 9 , p. 27
^ H. Poincare, Arch. Neerland, 5, 252 (1900)
^ Lev Borisovich Okunʹ, The Relations of Particles, (1991) World Scientific, ISBN 981020454X, p. 116-119.
^ Systematic trends in nuclear binding energies allow energy to be obtained by nuclear fission of heavy nuclei (heavier than iron or nickel) or nuclear fusion of light nuclei (lighter than iron or nickel). In nuclear fission, most of the energy released comes from the difference in binding energy when a heavier nucleus is split into lighter nuclei (that are much more strongly bound). In nuclear fusion, fusing lighter atomic nuclei to give heavier nuclei sets off energy because the binding energy of the end product is larger than the sum of binding energies of the initial nuclei.
^ ^a ^b ^c Conversions used: 1956 International (Steam) Table (IT) values where one calorie ≡ 4.1868 J and one BTU ≡ 1055.05585262 J. Weapons designers’ conversion value of one gram TNT ≡ 1000 calories used.

Bodanis, David (2001). E=mc²: A Biography of the World's Most Famous Equation. Berkley Trade. ISBN 0425181642.
Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0716743450.{{cite book}}: CS1 maint: multiple names: authors list (link)