Equivalence of mass and energy

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The sculpture Relativity Theory in the Berlin Walk of Ideas for the 2006 FIFA World Cup in Germany

The equivalence of mass and energy, or E = mc² for short, is a law of nature discovered by Albert Einstein in 1905 as part of the special theory of relativity . In today's formulation, it says that the mass and the rest energy of an object are proportional to one another:

Therein is the speed of light .

A change in the internal energy of a system therefore also means a change in its mass. Due to the large constant conversion factor , energy turnover, as is typical in everyday life, is accompanied by only small, barely measurable changes in mass. The mass of a typical car battery increases by only 40 ng due to the electrical energy stored in it  .

In nuclear physics , elementary particle physics and astrophysics , the equivalence of mass and energy is much more evident. Due to the binding energy released during their formation, the mass of atomic nuclei is almost one percent smaller than the sum of the masses of their unbound core components . By annihilating a particle with its antiparticle , the entire energy contained in the mass of the particle can even be converted into radiation energy.

The validity of the equivalence of mass and energy has been experimentally checked in many tests of the relativistic energy-momentum relationship and confirmed with high accuracy.

Overview and examples

The fact that the equivalence of mass and energy went unnoticed in classical physics as in everyday life can be understood from the size of the factor . According to the energy turnover of normal size (for example in chemical reactions such as combustion or the generation of heat through mechanical work) only extremely small changes in mass, which are hardly measurable with the scales even today. As a result, two separate conservation laws were established for closed systems : conservation of the total mass , conservation of all energy . However, since conversions between kinetic energy and rest energy are possible (examples: inelastic collision , radioactive decay ) and only the rest energy is equivalent to the mass, the conservation of mass is not generally valid. The change in the mass of an object associated with an energy transfer is also referred to as a mass increase or mass defect, depending on the sign. Instead of two conservation laws, you only have one, the energy conservation law .

In the combustion of coal energy is in the form of heat and radiation -free, the composition of the resulting carbon dioxide is only immeasurably smaller than the sum of the masses of the raw materials of carbon and oxygen . In general, the increase in energy associated with an increase in temperature only makes a negligible contribution to the mass. The sun, for example, would only be around 0.0001 percent less massive if it were cold.

In everyday situations the rest energy of a body exceeds its kinetic energy by many orders of magnitude. Even at the speed of a satellite in earth orbit (approx. 8 km / s), its kinetic energy is on the one hand less than a billionth of its rest energy:

On the other hand, it is so large that a satellite burns up if the energy is converted into an equal amount of heat when it re-enters the atmosphere.

A hydrogen - atom - consisting of an electron and a proton - has about 1 / 70,000,000 less mass than the two free particles together. This mass difference was released as binding energy during the formation of the atom . For atomic nuclei, however , this mass defect is quite large: at 12 C, for example, around 0.8%.

Well-known examples for the equivalence of mass and energy are:

  • Annihilation radiation: A particle pair electron - positron , which together have a mass of approx. , Can dissolve in radiation, in two massless gamma quanta of 511 keV energy each. The rest energy of the system before mutual annihilation is exactly as great as the energy of the radiation that is generated.
  • Nuclear fission : an atomic nucleus of the element uranium can burst into several fragments, the masses of which together are approx. 0.1% smaller than the original uranium nucleus. The energy released in the process corresponds exactly to this decrease in mass and can (if a correspondingly large amount of substance is split) u. a. appear as an explosion ( atomic bomb ) or heat source ( nuclear power plant ).
  • Nuclear fusion : When helium is formed from hydrogen, about 0.8% of its mass disappears, which is the main energy source of many stars (see stellar nuclear fusion ). The sun loses around 4 million tons of mass every second just because of the light it emits. Compared to the total mass of the sun of around , however, this effect is negligible. Even after several billion years, the sun has lost far less than a per thousand of its mass in this way.


Modern physics formulates the terms mass and energy with the help of the energy-momentum relation of the special theory of relativity: According to this, every closed physical system (hereinafter referred to as "body") has a total energy and an impulse as well as a mass . Energy and momentum have different values depending on the chosen reference system (on which the speed of the body depends), whereas the mass always has the same value. The quantities form the four components of the body's energy-momentum four-vector . The norm of this four-vector is determined (except for a constant factor ) by the mass :

Adjusted according to the energy:

In the center of gravity system ( ) there is again for the energy , also often referred to as rest energy .

Viewed from a different frame of reference, the same body has different values ​​for the four components. These values ​​can be obtained by applying the Lorentz transformation . If the body moves with speed relative to the chosen reference system , its energy and momentum are determined accordingly

whereby .

The norm of the four-vector is retained (see above), so the mass is a Lorentz invariant .

If one expands the equation to powers of into a Taylor series , one gets:

The "zeroth" link in this series is again the rest energy of the body. All higher members together form the kinetic energy . In the first of these members stands out and the classical kinetic energy results


This is a good approximation if in the non-relativistic case (i.e. ) all further terms can be neglected because they contain powers of . At very high speeds, these higher terms cannot be neglected. They then represent the disproportionate increase in kinetic energy for relativistic speeds.


In 1907 Einstein extended his considerations to include gravity . The equivalence principle , i.e. the equality of inert and heavy mass, led him to the conclusion that an increase in the rest energy of a system also results in an increase in the heavy mass. In the continuation of this idea within the framework of the general theory of relativity , it emerged that not only the mass, but the energy-momentum tensor is to be regarded as the source of the gravitational field.

One example is gravitational collapse . If the nuclear heat generation in the interior of a star with a sufficiently large total mass is extinguished, its matter is concentrated in such a small space that the gravitational field, which is getting stronger and stronger, contributes to further attraction and contraction itself through its field energy. The result is a black hole .



The relationship between mass, energy and the speed of light was considered by several authors as early as 1880 in the context of Maxwell's electrodynamics. Joseph John Thomson (1881), George Searle (1897), Wilhelm Wien (1900), Max Abraham (1902) and Hendrik Lorentz (1904) discovered that the electromagnetic energy adds an " electromagnetic mass " to the body according to the formula (in modern Notation)


Friedrich Hasenöhrl (1904/05) arrived at the same formula by looking at the electromagnetic cavity radiation of a body, whereby he also determined the dependence of the mass on the temperature. Henri Poincaré (1900), on the other hand, concluded from considerations on the reaction principle that electromagnetic energy has a "fictitious" mass of

corresponds. The electromagnetic mass was also referred to as the "apparent" mass, as it was initially distinguished from the "true" mechanical mass of Newton.

In 1905 Albert Einstein derived from the special theory of relativity he had developed shortly before that the mass of a body must change when the body absorbs or releases the energy . He won this result for the case that the energy conversion involves electromagnetic radiation. But he was the first to recognize the general validity: This equivalence must also apply to all other possible forms of energy turnover, and also to the entire rest energy and the entire mass accordingly


The equivalence of mass and energy was thus embedded in a comprehensive theory, the special theory of relativity.

Albert Einstein also called this equivalence "inertia of energy".

This was followed by a series of further theoretical derivations of the statement that under the most varied of conditions a change in the rest energy corresponds to the change in the mass in the shape (see the time table below). Einstein himself published 18 such derivations, the last one in 1946. It was regularly emphasized that this did not prove full equivalence in the form , but only in the form or equivalent to any constant summand. Since such a summand is always freely selectable, because the zero point is a matter of convention when specifying a total energy, it can be set equal to zero (as a “far more natural” choice (Einstein 1907)). In this form, the equivalence of mass and rest energy became an integral part of theoretical physics before it could be verified by measurements.

Experimentally, the equivalence of the changes in mass and energy in the form from 1920 onwards based on the mass defect of the nuclear masses became accessible. From the 1930s onwards, this equivalence was confirmed quantitatively in nuclear reactions, in which both the energy conversions and the difference in the masses of the reactants before and after the reaction were measurable. Initially, however, the error limits were 20%.

An experimental test of the equivalence in the form is possible by measuring the energy conversion in the creation or destruction of particles . Fermi was the first to adopt such a process in 1934 when beta radiation was created . He treated the newly generated and emitted electrons with the help of the quantum mechanical Dirac equation , which is based on the energy-momentum relationship of the special theory of relativity and thus ascribes the energy consumption to the generation of an electron at rest ( ) . This was confirmed by measuring the maximum kinetic energy of the electrons and comparing it with the energy balance of the nuclear transformation.

Today the validity of the equivalence of mass and energy has been experimentally confirmed with high accuracy:


Starting with 1905, the interpretation and meaning of the equivalence of mass and energy were gradually developed and deepened.

  • 1905: Einstein derives from the principle of relativity and electrodynamics that during the emission of radiation the mass of a body decreases by, whereby the given energy is. Einstein concludes that "the inertia of a body depends on its energy content", that is, the mass is a measure of its energy content.
  • 1906: Einstein shows with the help of a simple circular process that a change in the energy of a system must result in a change in its mass in order for the movement of the center of gravity to remain uniform. The form in which the energy is present is irrelevant. Einstein refers to Poincaré, who drew a similar conclusion in 1900 , albeit limited to purely electromagnetic energy.
  • May 1907: Einstein explains that the expression for the energy of a moving mass point of the mass takes on the simplest form if the expression (without additional additive constant) is chosen for its energy in the resting state . In a footnote, he uses the expression principle of equivalence of mass and energy for the first time . In addition, for a system of moving mass points, Einstein uses the formula (where the energy is in the center of gravity system) to describe the increase in mass when the kinetic energy of the mass points is increased.
  • June 1907: Max Planck introduces thermodynamic considerations and the principle of the smallest effect , and uses the formula (where the pressure and volume is) to represent the relationship between mass, its latent energy and thermodynamic energy in bodies. Following this, Johannes Stark uses the formula in October and applies it in connection with the quantum hypothesis .
  • December 1907: Einstein derives the formula in which the mass of the body is before and the mass is after the transfer of energy . He concludes that “the inertial mass and energy of a physical system appear as like things. In terms of inertia, a mass is equivalent to an energy content of magnitude . [...] Far more natural [than to distinguish between "true" and "apparent" mass] it appears to regard any inert mass as a store of energy. "
  • 1909: Gilbert N. Lewis and Richard C. Tolman use two variations of the formula: and , where the energy of a moving body, the rest energy, m is the relativistic mass, and the invariant mass. Analogous expressions are also used by Hendrik Antoon Lorentz in 1913 , although he writes the energy on the left: and , where the energy of a moving mass point is the rest energy, M is the relativistic mass and m is the invariant mass.
  • For a more detailed justification of the equivalence relationship, the relationship to the energy-momentum tensor is worked out. This is done for the first time by Max von Laue (1911). However, he limits his investigation to "static closed systems" in which, for example, electromagnetic forces and mechanical stresses are in balance. Felix Klein generalized this proof in 1918, according to which the restriction to static systems is not necessary.
  • In 1932 Cockroft and Walton succeeded in the first direct experimental demonstration of the equation in nuclear reactions . The gain in kinetic energy corresponds (within the error limits of 20% at that time) to the decrease in the total mass of the reactants.
  • In 1933, the positron , the antiparticle to the electron, discovered a few months earlier, is created as a pair with the electron, for which the energy is required. When they were destroyed together, discovered in 1934, it was precisely this energy that was re-emitted as destruction radiation. Both processes are initially not interpreted as a conversion between energy and mass, but rather as the excitation of an electron previously hidden with negative energy in the Dirac Sea into the visible world of positive energy, whereby the hole created in the Dirac Sea appears as a positron.
  • In 1934 Enrico Fermi for the first time accepted the possibility that massive particles could be produced. For the formation process, the β-radioactivity, he applies the law of conservation of energy and the relativistic formula for the energy of the particles . The energy is therefore consumed for the creation of a resting particle . With this, Fermi succeeds in the first quantitatively correct theory of β-radiation and - incidentally - the first confirmation of the full equivalence of mass and energy.
  • In 1935 Einstein gives a new derivation of , solely from the conservation of momentum during impact and without reference to electromagnetic radiation. By referring to the fact that in the case of energy, from the concept of the term, an additive constant is arbitrary, he chooses it in such a way that it holds.
  • In 1965 Roger Penrose , Wolfgang Rindler and Jürgen Ehlers show that the special theory of relativity cannot, in principle, exclude an additive constant in an equation , whereby it stands for the assumed (Lorentz invariant) part of the mass that cannot be undershot by removing energy. However, they conclude from the experimental observations on particle formation and annihilation that it is true. Mitchell J. Feigenbaum and David Mermin confirm and deepen this result in 1988.

Einstein's derivation

In 1905, Einstein discovered the connection between mass and energy through the following thought experiment. Poincaré had developed a similar thought experiment in 1900, but was unable to clarify it satisfactorily.

A body rests in a frame of reference and has a certain rest energy about which we do not need to know anything more. It sends out two equal flashes of the same energy in opposite directions. Then the impulses of the light flashes are also the same, but opposite, so that the body remains at rest because of the maintenance of the overall impulse. Because of the conservation of energy, the body now has the energy


We consider the same process from a second frame of reference that moves relative to the first at speed in the emission direction of one of the light flashes. The values ​​of all energies calculated in the second system are denoted by ... It could be that the energy scales of both reference systems have different zero points that differ by a constant . Since the conservation of energy in the second frame of reference applies just as well as in the first (principle of relativity), it follows


Since the body remains at rest in the first system, it moves at the same speed in the second system after the emission as before. Its energy in the second frame of reference is therefore greater by the kinetic energy than in the first. Therefore:

By subtracting the sides of these two equations in pairs, the unknown rest energies and the constant drop out and we get:

The decisive point is now: The two flashes of light, which have opposite directions and the same energies in the rest system of the body, are also opposed in the second reference system (due to the choice of the direction of movement), but have different energies. One shows redshift, the other blueshift. After the Lorentz transformation of the electrodynamic fields, their energies are respectively , where . Together, their energy is greater than in the first frame of reference:

The two values ​​for the kinetic energy before and after the emission are therefore also different according to the above equation. The kinetic energy decreases due to the emission


Since the speed of the body remains the same during emission, but afterwards it has a lower kinetic energy than before, its mass must have decreased. To determine this change, we use the formula that is valid in the borderline case and develop the right-hand side of the last equation according to powers up to the term . It turns out . So the release of energy leads to a decrease in mass .

Einstein concluded this thought, published in 1905, with the words (symbols modernized):

“If a body emits the energy in the form of radiation, its mass is reduced by . [...] The mass of a body is a measure of its energy content. [...] It cannot be ruled out that a test of the theory will succeed in bodies whose energy is highly variable (e.g. in the case of radium salts). "

- Albert Einstein

Einstein circumvents the problem of the unknown rest energy by eliminating this quantity from the equations in his thought experiment. For the energy output, he chooses electromagnetic radiation and derives the change in mass from it. In 1905 he added the statement, without proof, that this applied to any type of energy loss. From 1907/08 he suggests, "since we can dispose of the zero point [...], [...] to regard any inert mass as a store of energy", that is .

E = mc² and the atomic bomb

From 1897 Henri Becquerel , Marie and Pierre Curie and Ernest Rutherford had researched ionizing rays and concluded from their inexplicably high energy at the time that the underlying nuclear reactions are a million times more energetic than chemical reactions. Rutherford and Frederick Soddy (1903) assumed that the energy source was an enormous reservoir of latent energy in the body, which must also be present in normal matter. Rutherford (1904) speculated that perhaps one day one could control the decay of radioactive elements and release an enormous amount of energy from a small amount of matter. With Einstein's equation (1905) this energy could be read from the different nuclear masses, which could actually be proven in the 1930s.

However, the equation does not say how to start the fission of heavy atomic nuclei. The observation of the induced nuclear fission by Otto Hahn and Fritz Straßmann was decisive, as was the fact that the neutrons released in the process can trigger a chain reaction in enriched uranium. Contrary to what popular scientific reports claim, the connection between rest energy and mass did not play a special role in the development of the atomic bomb (“ Manhattan Project ” in the USA from 1942). Albert Einstein influenced the development of the atomic bomb less through his physical knowledge, but at most politically. He wrote a letter to President Roosevelt in which he advocated the development of the atomic bomb in the United States.

Web links

Commons : Einstein formula  - collection of images, videos and audio files

Individual evidence

  1. a b c d Albert Einstein: Does the inertia of a body depend on its energy content? In: Annals of Physics . tape 323 , no. 13 , 1905, pp. 639–643 ( physik.uni-augsburg.de [PDF]).
  2. D. Mermin : It's About Time: Understanding Einstein's Relativity. Princeton University Press, 2005, pp. 160 ff. ( Google Books ).
  3. a b c Albert Einstein: About the principle of relativity and the conclusions drawn from it . In: Yearbook of radioactivity and electronics . tape 4 , 1908, pp. 411-462 ( soso.ch [PDF]).
  4. ^ ET Whittaker: A History of the theories of aether and electricity . 2nd Edition. 1: The classical theories. 1951, volume 2 : The modern theories 1900–1926. 1953. Nelson, London.
  5. M. Jannsen, M. Mecklenburg: From classical to relativistic mechanics. Electromagnetic models of the electron . In: VF Hendricks et al. (Ed.): Interactions: Mathematics, Physics and Philosophy . Springer, Dordrecht 2007, p. 65-134 ( netfiles.umn.edu [PDF]).
  6. Max Born: Einstein's theory of relativity . Ed .: Jürgen Ehlers, Markus Pössel. 7th edition. Springer, Berlin / Heidelberg / New York and others 2003, ISBN 3-540-00470-X (first edition: 1964).
  7. a b Max Jammer: The concept of mass in physics. Wissenschaftliche Buchgesellschaft, Darmstadt 1964, English original: Concepts of Mass in Classical and Modern Physics. Harvard UP, Cambridge (Mass) 1961; Harper, New York 1964; Dover, New York 1997. ISBN 0-486-29998-8 .
  8. a b O. Darrigol: The Genesis of the theory of relativity . In: Séminaire Poincaré . tape 1 , 2005, p. 1–22 ( bourbaphy.fr [PDF; 526 kB ]).
  9. a b c Albert Einstein: About the inertia of energy required by the principle of relativity . In: Annals of Physics . tape 328 , no. 7 , 1907, pp. 371–384 , doi : 10.1002 / andp.19073280713 , bibcode : 1907AnP ... 328..371E ( physik.uni-augsburg.de [PDF]).
  10. a b Albert Einstein: The principle of the maintenance of the center of gravity movement and the inertia of energy . In: Annals of Physics . tape 325 , no. 8 , 1906, pp. 627–633 , doi : 10.1002 / andp.19063250814 , bibcode : 1906AnP ... 325..627E ( physik.uni-augsburg.de [PDF]).
  11. a b J.D. Cockcroft, ETS Walton: Experiments with High Velocity Positive Ions. II. The Disintegration of Elements by High Velocity Protons . In: Proc.Royal Soc. A 137, no. 1 , 1933, pp. 229-242 , JSTOR : 95941 .
  12. R. Stuewer: Mass Energy and the neutron in the early Thirties. In: Einstein in Context: A Special Issue of Science in Context. Science in Context, Vol 6 (1993), pp. 195 ff. Excerpt in Google books
  13. ^ KT Bainbridge: The Equivalence of Mass and Energy. Phys. Rev. 44 (1933), p. 123.
  14. Simon Rainville, James K. Thompson, Edmund G. Myers, John M. Brown, Maynard S. Dewey, Ernest G. Kessler, Richard D. Deslattes, Hans G. Börner, Michael Jentschel, Paolo Mutti, David E. Pritchard: World Year of Physics: A direct test of E = mc2 . In: Nature . tape 438 , no. 7071 , December 22, 2005, p. 1096-1097 , doi : 10.1038 / 4381096a .
  15. a b Eugene Hecht: How Einstein confirmed E 0 = mc 2 . In: American Journal of Physics . tape 79 , no. 6 , 2011, p. 591-600 , doi : 10.1119 / 1.3549223 .
  16. Max Planck: On the dynamics of moving systems . In: Session reports of the Royal Prussian Academy of Sciences, Berlin . First half volume, no. 29 , 1907, pp. 542-570 ( archive.org ).
  17. J. Stark: Elementary quantum of energy, model of negative and positive electricity . In: Physikalische Zeitschrift . tape 24 , no. 8 , 1907, pp. 881 ( archive.org ).
  18. ^ Gilbert N. Lewis, Richard C. Tolman: The Principle of Relativity, and Non-Newtonian Mechanics . In: Proceedings of the American Academy of Arts and Sciences . tape 44 , 1909, pp. 709-726 ( Wikisource ).
  19. Hendrik Antoon Lorentz: The principle of relativity. Three lectures given at Teyler's Foundation in Haarlem (1913) . BG Teubner, Leipzig and Berlin 1914 ( Wikisource ).
  20. Michel Janssen: The Trouton Experiment, E = MC 2, and a Slice of Minkowski Space-Time . In: Lindy Divarci, Jürgen Renn, Petra Schröter, John J Stachel (eds.): Revisiting the foundations of relativistic physics: festschrift in honor of John Stachel . Kluwer Academic, Dordrecht / Boston [a. a.] 2003, ISBN 1-4020-1284-5 , pp. 27-54 .
  21. Max von Laue: On the dynamics of the theory of relativity . In: Annals of Physics . tape 340 , no. 8 , 1911, pp. 524–542 , doi : 10.1002 / andp.19113400808 , bibcode : 1911AnP ... 340..524L ( gallica.bnf.fr ).
  22. Felix Klein: About the integral form of the conservation laws and the theory of the spatially closed world . In: Göttinger Nachrichten . 1918, p. 394-423 ( archive.org ). - Comment: In the archive picture 605 ff. Of 634.
  23. ^ Val L. Fitch: Elementary Particle Physics. Pp. 43-55. In: Benjamin Bederson (Ed.): More Things in Heaven and Earth: A Celebration of Physics at the Millennium. Vol. II, Springer, 1999, ISBN 978-1-4612-7174-1 . limited preview in Google Book search.
  24. ^ M. Laurie Brown, F. Donald Moyer: Lady or tiger? The Meitner-Hupfeld Effect and Heisenberg's neutron theory . In: Amer. Journ. of Physics . tape 52 , 1984, pp. 130-136 . And publications specified there.
  25. Enrico Fermi: Attempt a theory of beta rays . In: Journal of Physics . tape 88 , 1934, pp. 161 .
  26. Albert Einstein: Elementary Derivation of the Equivalence of Mass and Energy . In: Bull. Of the American Mathematical Society . tape 41 , 1935, pp. 223-230 .
  27. ^ R. Penrose, W. Rindler, J. Ehlers: Energy Conservation as the Basis of Relativistic Mechanics I and II . In: Amer. Journ of Physics . tape 33 , 1965, pp. 55-59 and 995-997 .
  28. MJ Feigenbaum, D. Mermin: E = mc2 . In: Amer. Journ of Physics . tape 56 , 1988, pp. 18-21 , doi : 10.1119 / 1.15422 .
  29. ^ Ernest Rutherford: Radioactivity . University Press, Cambridge 1904, pp. 336-338 ( archive.org ).
  30. Werner Heisenberg: Physics And Philosophy: The Revolution In Modern Science . Harper & Brothers, New York 1958, pp. 118-119 ( archive.org ).
  31. ^ Cover picture of Time Magazine July 1946. On: content.time.com.
  32. Markus Pössel, Max Planck Institute for Gravitational Physics: From E = mc-square to the atomic bomb. ( Memento of April 30, 2008 in the Internet Archive ) And: Is the whole the sum of its parts? ( Memento of the original from April 20, 2008 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / www.einstein-online.info


  1. a b Einstein used the letter L for the energy difference in his publication .
  2. a b c d e The formula is given here with the symbol c for the speed of light; Einstein used the letter V in his publication .