Electromagnetic mass

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This article deals with the electromagnetic mass, e.g. T. also called "effective mass". For the corresponding term in solid state physics see effective mass .

The electromagnetic mass , also apparent mass or effective mass , is a concept of classical mechanics or electrodynamics . It indicates to what extent the electromagnetic field or the self-energy contributes to the mass of a charged particle . The electromagnetic mass was first derived by JJ Thomson in 1881.

At the beginning of the 20th century, electromagnetic mass was considered to explain the origin of mass. However, this interpretation was rejected in favor of the statements of the theory of relativity . Mass, momentum, speed and all possible types of energy are related to each other through the equivalence of mass and energy . As far as the cause of the mass of elementary particles is concerned, the Higgs mechanism is used within the framework of the relativistic Standard Model . For the special case of the electromagnetic self-energy of charged particles, however, the presence of an “effective” electromagnetic mass can still be assumed.

Charged particles

Rest mass and energy

As early as 1843, George Gabriel Stokes showed in the context of hydrodynamics that the "effective" inertia of a moving body is increased in an incompressible perfect fluid. JJ Thomson (1881) made similar considerations . He recognized that an electrically charged sphere in a medium (the electromagnetic ether of James Clerk Maxwell ) , which has a specific inductive power, is more difficult to set in motion than an uncharged body. Through this self-induction effect, electrostatic energy behaves as if it had momentum and an "apparent" electromagnetic mass, which can increase the ordinary mechanical mass of a body. In other words: This mass comes from the electromagnetic self-energy of the particles. Thomson's idea was detailed developed by Oliver Heaviside (1889), Thomson (1893), George Frederick Charles Searle (1897), Max Abraham (1902), Hendrik Lorentz (1892, 1904), and was on the dynamics of the then discovered electron under Use of the Abraham-Lorentz equations applied. The electrostatic energy E _em and the mass m _{em of} an electron at rest were found with:

${\displaystyle E_{em}={\frac {1}{2}}{\frac {e^{2}}{a}},\qquad m_{em}={\frac {2}{3}}{\frac {e^{2}}{ac^{2}}}}$

where e is the uniformly distributed charge and a is the classical electron radius, which must be finite in order to avoid infinitely large energy values. This results in the electromagnetic energy-mass relationship with

${\displaystyle m_{em}={\frac {4}{3}}{\frac {E_{em}}{c^{2}}}}$

Some researchers like Wilhelm Wien (1900) and Abraham (1902) came to the conclusion that the total mass of a body is equal to its electromagnetic mass. Wien and others also assumed that gravitation is also of electromagnetic origin, and consequently that electromagnetic energy, inertial mass, and heavy mass must be proportional to each other. According to Vienna, when one body attracts another, the electromagnetic energy supply of gravitation is reduced by the amount (where M is the attracted mass, G is the gravitational constant , and r is the distance):

${\displaystyle G{\frac {{\frac {4}{3}}{\frac {E_{em}}{c^{2}}}M}{r}}}$

Henri Poincaré also said in 1906 that if the mass is actually the product of the electromagnetic field in the ether - according to which no “real” mass exists - and if it is assumed that the concept of matter is inextricably linked with that of mass, then consequently none exists Matter and electrons are only cavities in the ether.

Mass and speed

Thomson and Searle

Thomson (1893) noticed that the energy of charged bodies increases with increasing speed. It follows that more and more energy is required to accelerate the mass further, which was interpreted as an increase in mass with greater speed. He wrote (where v is the speed of the body and c is the speed of light):

" [S. 21] In the limit area v = c the increase in mass becomes infinitely great, consequently a charged sphere moving with the speed of light behaves as if its mass were infinitely great, and therefore its speed will remain the same; or in other words, it is impossible to increase the speed of a charged body moving through a dielectric above the speed of light. "

In 1897 Searle gave a more precise formula for the increase in the electromagnetic energy of a moving sphere:

${\displaystyle E_{em}^{v}=E_{em}\left[{\frac {1}{\beta }}\ln {\frac {1+\beta }{1-\beta }}-1\right],\qquad \beta ={\frac {v}{c}},}$

and like Thomson he concluded:

“ ... at v = c the energy becomes infinitely large, so that it seems impossible to get a charged body to move at a speed greater than the speed of light. . "

Longitudinal and transverse mass

Predictions on the velocity dependence of the transverse electromagnetic mass according to Abraham, Lorentz and Bucherer

Based on Searle's formula, Walter Kaufmann (1901) and Abraham (1902) derived the formula for the electromagnetic mass of moving bodies:

${\displaystyle m_{L}={\frac {3}{4}}\cdot m_{em}\cdot {\frac {1}{\beta ^{2}}}\left[-{\frac {1}{\beta ^{2}}}\ln \left({\frac {1+\beta }{1-\beta }}\right)+{\frac {2}{1-\beta ^{2}}}\right]}$

Abraham was able to show, however, that this formula is only correct in the longitudinal direction ("longitudinal mass"), i.e. H. the electromagnetic mass also depends on the direction of the moving electrons. Hence, Abraham derived the "transverse mass":

${\displaystyle m_{T}={\frac {3}{4}}\cdot m_{em}\cdot {\frac {1}{\beta ^{2}}}\left[\left({\frac {1+\beta ^{2}}{2\beta }}\right)\ln \left({\frac {1+\beta }{1-\beta }}\right)-1\right]}$

On the other hand, Lorentz had already assumed in 1899 that electrons are subject to a length contraction in the direction of motion , which is supposed to prevent observers from being able to measure their state of motion relative to the ether. As a result, the values ​​for the acceleration of the electrons deviated from Abraham's values. In 1899, and somewhat more precisely in 1904, Lorentz gave the following values ​​for the longitudinal and transverse mass (which agreed with the values derived from the theory of relativity by Albert Einstein in 1905 ):

${\displaystyle m_{L}={\frac {m_{em}}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{3}}},\quad m_{T}={\frac {m_{em}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$,

In addition, Alfred Bucherer and Paul Langevin (1904) developed a further electron model, according to which the electrons contract in the direction of movement, but expand perpendicular to it, whereby the volume remains constant. They received the following values:

${\displaystyle m_{L}={\frac {m_{em}\left(1-{\frac {1}{3}}{\frac {v^{2}}{c^{2}}}\right)}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{8/3}}},\quad m_{T}={\frac {m_{em}}{\left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{2/3}}}}$

Kaufmann's experiments

→ Main article : Kaufmann-Bucherer-Neumann experiments

The formulas for the transverse mass in the theories of Abraham and Lorentz were supported by the experiments of Kaufmann (1901–1903), but they were not precise enough to make a decision between the theories. For this reason, Kaufmann carried out further experiments in 1905 that were roughly in agreement with Abraham and Bucherer's formulas, but contradicted the Lorentz-Einstein formula. The subsequent experiments by Bucherer and others, however, showed a better agreement with the Lorentz-Einstein formula than with those of Abraham and Bucherer. In retrospect, however, these experiments were not accurate enough to make a decision between the alternatives, which could only be achieved in 1940. However, this only applied to this type of experiment; in others, the Lorentz-Einstein formula could be precisely confirmed much earlier (from 1917).

Poincaré tensions and the 4/3 problem

The idea of ​​an electromagnetic justification of matter, however, was incompatible with the Lorentz electron. Abraham (1904, 1905) showed that a non-electromagnetic force was needed to prevent the Lorentz electrons from simply exploding, due to the electrostatic repulsion of the individual sections of their field. He also showed that different values ​​follow for the longitudinal electromagnetic mass, depending on whether they are derived from its electromagnetic energy or its momentum. He calculated that a non-electromagnetic potential (corresponding to a ⅓ of the electromagnetic energy) was necessary in order to match the various results. Abraham doubted that it would be possible to develop a theory that would meet all of these requirements.

To solve this problem, Henri Poincaré (1905) introduced the Poincaré voltages named after him - exerting negative pressure - which represent a non-electromagnetic potential within the electrons. As requested by Abraham, they add non-electromagnetic energy to the electrons, which amounts to ¼ of their total energy or ⅓ of their electromagnetic energy. The Poincaré tensions thus resolve the contradiction in the derivation of the longitudinal mass, they prevent the explosion of the electrons, they remain unchanged due to a Lorentz transformation (they are therefore Lorentz invariant), and were also used by Poincaré as the dynamic cause for the Considered length contraction. Poincaré, however, stuck to the opinion that only electromagnetic energy contributes to the mass of the body.

As noted later, the root of the problem lies in the ⁴ ⁄ ₃ factor of the electromagnetic rest mass - i.e. m _em = (4/3) E _em / c ² if this is derived from the Abraham-Lorentz equations. However, if it is derived from the electrostatic energy of the electrons, the result is a mass of m _es = E _em / c ² without the factor. This contradiction is resolved by the non-electromagnetic energy E _{p of} the Poincaré voltages, resulting in the total energy of the electrons E _tot :

${\displaystyle {\frac {E_{tot}}{c^{2}}}={\frac {E_{em}+E_{p}}{c^{2}}}={\frac {E_{em}+{\frac {E_{em}}{3}}}{c^{2}}}={\frac {4}{3}}{\frac {E_{em}}{c^{2}}}={\frac {4}{3}}m_{es}=m_{em}}$

As a result, the missing ⁴ ⁄ ₃ factor is restored when the mass is related to its electromagnetic energy, which was common at the time, and it disappears when the total energy is taken into account.

Inertia of energy and radiation paradoxes

Radiation pressure

Another way that was used to create some kind of connection between electromagnetic energy and ground was based on the concept of radiation pressure . This pressure or the presence of voltages in the electromagnetic field was derived from James Clerk Maxwell (1874) and Adolfo Bartoli (1876). Lorentz (1895) was also able to show that these Maxwellian stresses follow from his theory of the ether at rest. However, there was the problem that bodies can be moved by these tensions, but they cannot work back on the resting ether, because the latter was absolutely immobile by definition. As a result, the principle of actio and reactio was violated in Lorentz's theory , which Lorentz accepted. He also explained that in an ether at rest one can only speak of "fictitious" voltages, and consequently they are only mathematical models to facilitate the description of electrodynamic interactions.

Mass of the fictitious electromagnetic fluid

In 1900 Poincaré examined this conflict between the reaction principle and the Lorentzian theory. He found out that the principle of maintaining the center of gravity movement of a material system, if electromagnetic fields or radiation are present, is no longer valid due to the violation of the reaction principle. In order to avoid this, he derived from Maxwell's voltages or the Poynting vector the presence of an electromagnetic pulse in the electromagnetic fields (such a pulse was already derived in 1893 by Thomson in a more complicated way, however.) From this he concluded that the electromagnetic field energy behaves like a “fictitious” fluid (“fluid fictif”) to which a mass of E _em / c ² (ie m _em = E _em / c ² ) can be assigned. If the center of gravity system is now considered to be composed of the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is regarded as indestructible (it is neither emitted nor absorbed), then the movement of the center of gravity remains uniform.

However, this solution was insufficient for the case when the electromagnetic energy is converted or absorbed into other forms of energy. This would result in the fluid connected to it being destroyed - which for Poincaré is the reason why this fluid or its momentum and mass is only to be regarded as fictitious. A simple solution to this problem would have been (as Einstein later did) to assume that the mass of the electromagnetic energy passes directly into matter during absorption, and that its mass consequently increases or decreases during the emission or absorption process. But this was not considered by Poincaré, but he invented another fictional, non-electromagnetic fluid. This is immovable in every place in the room and also has a fictitious mass proportional to its energy. If the fictitious electromagnetic fluid has now been destroyed, it transfers its energy and mass to the non-electromagnetic fluid, under the condition that this mass remains exactly in this place and is not carried along with the matter. (Poincaré added that one should not be too surprised about these assumptions, as they are only mathematical fictions.) If the mass of matter and the mass of the two fluids (electromagnetic and non-electromagnetic) together are now taken into account, it remains here too the movement of the center of gravity is uniform.

The resulting fact that in the case of emission or absorption the center of gravity movement of the system - consisting of the mass of matter and the electromagnetic fluid - is no longer uniform (because the effects of the non-electromagnetic fluid are not experimentally accessible), led Poincaré Regarding the following radiation paradox: When a beam is emitted in a certain direction, the body suffers a recoil due to the momentum of the beam. Poincaré now carried out a Lorentz transformation (for low speeds) into a system that moved relative to it. He noticed that the conservation of energy was maintained, but the law of conservation of momentum was violated, which made it possible to construct a perpetual motion machine , which Poincaré found very problematic. So he had to assume that there was an additional compensatory force that compensates for this effect. (If he had assumed like Einstein that the mass is taken up or given off by the matter itself, this problem would not exist, swu)

Poincaré took up this topic again in 1904. This time he rejected the solution that movements in the ether can compensate for the movements of matter, because these could not be determined experimentally and therefore scientifically useless. He also dismissed the concept that energy is related to mass and wrote in relation to the recoil during radiation emission:

"The apparatus will recoil as if it were a cannon and the energy it emitted was a sphere, and this contradicts Newton's principle because our projectile has no mass here, it is not matter, it is energy."

Momentum and cavity radiation

Poincaré's original idea of ​​a connection between momentum and mass with electromagnetic radiation turned out to be entirely correct. Abraham expanded Poincaré's formalism and introduced the concept of the “electromagnetic pulse”, the field density of which was E _em / c per cm ² and E _em / c ² per cm³. In contrast to Lorentz and Poincaré, he understood this as a real and not as a fictitious quantity, which guarantees conservation of momentum.

The work of Friedrich Hasenöhrl (1904) was also carried out in this context . He studied the effects of cavity radiation and calculated that it increases the mass of moving bodies. He derived the formula m _em = (8/3) E _em / c ² for the "apparent" mass due to radiation and temperature, i.e. H. electromagnetic radiation can transfer mass from one body to another. He and Abraham corrected this in 1905 by replacing the ⁸ ⁄ ₃ factor with the ⁴ ⁄ ₃ factor, giving them the same formula as for electromagnetic mass.

Modern view

Equivalence of mass and energy

The idea that the relationship between mass, energy, speed, and momentum must be determined by observing the dynamic structure of matter became obsolete when Albert Einstein derived the equivalence of mass and energy from the special theory of relativity in 1905 . It follows from it that all forms of energy contribute to the mass of a body according to E / c ² . In contrast to Poincaré's assumption, the mass of the absorbing body itself is increased or decreased through absorption or emission of energy, which resolves Poincaré's radiation paradox. Furthermore, the idea that gravity is of electromagnetic origin had to be abandoned with the development of general relativity .

Any theory that deals with the dynamic relationships between the mass of a body must therefore be formulated from the outset according to relativistic points of view. This is the case with the currently valid quantum field theoretical explanation of the mass of elementary particles within the framework of the standard model , the Higgs mechanism . Because of this mechanism, the assumption that the mass of all bodies is completely determined by dynamic interactions with electromagnetic fields is no longer relevant.

Relativistic mass

The concepts of longitudinal and transversal mass (equivalent to those of Lorentz) were also used by Einstein in his first work on the theory of relativity. Here, however, these apply to the entire mass, not just to the electromagnetic part. Tolman (1912) showed, however, that the related definition of mass as the quotient of force and acceleration is unfavorable. If it is used instead , the direction-dependent terms vanish and the relativistic mass results ${\displaystyle {\vec {F}}=\mathrm {d} {\vec {p}}/\mathrm {d} t}$

${\displaystyle M={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},\qquad m_{0}={\frac {E}{c^{2}}}}$.

This concept is used in some physics textbooks to this day. However, many call it obsolete and only speak of the "invariant mass", which corresponds to the older term rest mass. Instead, the effects of higher speeds are described in terms of relativistic energy and momentum.

Self-energy

In special cases when it comes to questions of the self-energy or self-power of charged particles, the use of an “effective” electromagnetic mass makes sense - no longer as an explanation for the entire mass of matter, but as a supplement to the ordinary mass. Variants and modifications of the Abraham-Lorentz equations are and have been proposed again and again (to solve the 4/3 problem, for example, see next section). This is also related to the renormalization in the context of quantum mechanics and quantum field theory. Quantum physics concepts must be taken into account if the electron is viewed as physically point-shaped. For larger distances, the classic concepts come back into play. A derivation of the electromagnetic self-forces was for example by Gralla et al. (2009), which also includes the contribution of self-strength to the mass of the body.

4/3 problem

Max von Laue (1911) also used the Abraham-Lorentz equations of motion in his further development of the special-relativistic dynamics, whereby the ⁴ ⁄ ₃ factor also occurs here when the electromagnetic mass is calculated from the self-field of a charged, spherical electron. This now contradicts the equivalence formula, which requires the relationship m _em = E _em / c ² without the ⁴ ⁄ ₃ factor, otherwise the four-pulse would no longer be correctly transformed as a four-vector . Laue now found a solution, which was equivalent to Poincaré's introduction of a non-electromagnetic potential (Poincaré tensions), but he was able to show its deeper, relativistic meaning because he further developed Minkowski's spacetime formalism. Laues formalism required that additional components and forces occur so that spatially extended systems always form a "closed system", where electromagnetic and non-electromagnetic energies are combined. Although a ⁴ ⁄ ₃ factor still occurs in the electromagnetic mass , it disappears when the entire system is taken into account, which ultimately results in the relationship m _tot = E _tot / c ² .

Alternative solutions were found by Enrico Fermi (1922), Paul Dirac (1938), Fritz Rohrlich (1960) and Julian Schwinger (1983). They showed that the foregoing definitions of the four-pulse in conjunction with the Abraham-Lorentz equations from the outset were not Lorentz covariant, and put in their place a formulation whereby the electromagnetic mass simply as m _em = e _em c / ² is written can and the ⁴ ⁄ ₃ factor does not appear at all. Every part of the system, whether closed or not, can be transformed as a four-vector. This showed that the stability of the electrons and the ⁴ ⁄ ₃ problem were not linked to one another, contrary to previous views. Nevertheless (if the electron is viewed as an extended, spherical object) similar mechanisms as the Poincaré voltages are necessary to maintain the stability of the electrons. In the Fermi-Rohrlich definition, however, this is only a dynamic problem and has nothing to do with the transformation properties of the system.

See also

• History of the special theory of relativity

Individual evidence

Secondary sources

1. ↑ Feynman, Ch. 28
2. ↑ ^{a b} Pais, pp. 155-159
3. ↑ Miller, pp. 45-47, 102-103
4. ↑ Miller (1981): 334-352
5. ↑ ^{a b} Janssen / Mecklenburg (2007)
6. ↑ ^{a b} Miller (1981), 382-383
7. ↑ ^{a b} Janssen / Mecklenburg (2007), pp. 32, 40
8. ↑ Miller (1981), 41ff
9. ↑ ^{a b} Darrigol (2005), 18-21
10. ↑ Miller (1981), 359-360
11. ↑ Rohrlich (1997)

• Darrigol, Olivier: The Genesis of the theory of relativity . In: Séminaire Poincaré . 1, 2005, pp. 1-22.

• Feynman, RP: Electromagnetic mass . In: The Feynman Lectures on Physics , Volume 2. Addison-Wesley Longman, Reading 1970, ISBN 0-201-02115-3 .

• Janssen, Michel & Mecklenburg, Matthew: From classical to relativistic mechanics: Electromagnetic models of the electron . In: VF Hendricks, et al. (Ed.): Interactions: Mathematics, Physics and Philosophy . Springer, Dordrecht 2007, pp. 65-134.

• Miller, Arthur I .: Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905-1911) . Addison-Wesley, Reading 1981, ISBN 0-201-04679-2 .

• Pais, Abraham : Electromagnetic Mass: The First Century . In: Subtle is the Lord: The Science and the Life of Albert Einstein . Oxford University Press, New York 1982, ISBN 0-19-520438-7 .

• Rohrlich, F .: The dynamics of a charged sphere and the electron . In: American Journal of Physics . 85, No. 11, 1997, pp. 1051-1056. doi : 10.1119 / 1.18719 .

• Rohrlich, F .: Classical charged particles , 3rd edition, World Scientific, Singapore 1964/2007, ISBN 9812700048 .

Primary sources

1. ↑ Stokes, George Gabriel: On some cases of fluid motion . In: Transactions of the Cambridge Philosophical Society . 8, No. 1, 1844, pp. 105-137.
2. ^ Thomson, Joseph John: On the Electric and Magnetic Effects produced by the Motion of Electrified Bodies . In: Philosophical Magazine . 11, No. 68, 1881, pp. 229-249.
3. ^ Heaviside, Oliver: On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric . In: Philosophical Magazine . 27, No. 167, 1889, pp. 324-339.
4. ↑ ^{a b c} Thomson, Joseph John: Notes on recent researches in electricity and magnetism . Clarendon Press, Oxford 1893.
5. ^ ^{A b} Searle, George Frederick Charles: On the Steady Motion of an Electrified Ellipsoid . In: Philosophical Magazine . 44, No. 269, 1897, pp. 329-341.
6. ↑ ^{a b c d e} Abraham, Max: Principles of the dynamics of the electron . In: Annals of Physics . 315, No. 1, 1903, pp. 105-179. bibcode : 1902AnP ... 315..105A . doi : 10.1002 / andp.19023150105 .
7. ^ Lorentz, Hendrik Antoon: La Théorie électromagnétique de Maxwell et son application aux corps mouvants . In: Archives néerlandaises des sciences exactes et naturelles . 25, 1892, pp. 363-552.
8. ↑ ^{a b} Lorentz, Hendrik Antoon: Electromagnetic phenomena in a system that moves with any speed that cannot be reached by light . In: Blumenthal, Otto & Sommerfeld, Arnold (Ed.): The principle of relativity. A collection of treatises 1904b / 1913, pp. 6-26.
9. ↑ ^{a b} Vienna, Wilhelm: About the possibility of electromagnetic justification of mechanics . In: Annals of Physics . 310, No. 7, 1900, pp. 501-513. doi : 10.1002 / andp.19013100703 .
10. ↑ Poincaré, Henri: The end of matter . In: Athenæum . 1906.
11. ↑ When in the limit v = c , the increase in mass is infinite, thus a charged sphere moving with the velocity of light behaves as if its mass were infinite, its velocity therefore will remain constant, in other words it is impossible to increase the velocity of a charged body moving through the dielectric beyond that of light.
12. ↑ ... when v = c the energy becomes infinite, so that it would seem to be impossible to make a charged body move at a greater speed than that of light
13. ↑ ^{a b} Kaufmann, Walter: The electromagnetic mass of the electron . In: Physikalische Zeitschrift . 4, No. 1b, 1902, pp. 54-56.
14. ^ Lorentz, Hendrik Antoon: Simplified Theory of Electrical and Optical Phenomena in Moving Systems . In: Proceedings of the Royal Netherlands Academy of Arts and Sciences . 1, 1899, pp. 427-442.
15. ↑ Bucherer, AH: Mathematical introduction to electron theory . Teubner, Leipzig 1904.
16. ↑ Kaufmann, Walter: About the constitution of the electron . In: Session reports of the Royal Prussian Academy of Sciences . 45, 1905, pp. 949-956.
17. ↑ Kaufmann, Walter: About the constitution of the electron . In: Annals of Physics . 324, No. 3, 1906, pp. 487-553.
18. ↑ Abraham, Max: The basic hypotheses of the electron theory . In: Physikalische Zeitschrift . 5, 1904, pp. 576-579.
19. ↑ Abraham, M .: Theory of Electricity: Electromagnetic Theory of Radiation . Teubner, Leipzig 1905, pp. 201-208.
20. ^ Poincaré, Henri: Sur la dynamique de l'électron . In: Comptes rendus hebdomadaires des séances de l'Académie des sciences . 140, 1905, pp. 1504-1508. See also German translationhttp: //vorlage_digitalisat.test/1%3D~GB%3D~IA%3DPoincareDynamikA~MDZ%3D%0A~SZ%3D~doppelseiten%3D~LT%3Ddeutsche%20%C3%9Cetzung~PUR%3D .
21. ^ Poincaré, Henri: Sur la dynamique de l'électron . In: Rendiconti del Circolo matematico di Palermo . 21, 1906, pp. 129-176. See also German translationhttp: //vorlage_digitalisat.test/1%3D~GB%3D~IA%3DPoincareDynamikB~MDZ%3D%0A~SZ%3D~doppelseiten%3D~LT%3Ddeutsche%20%C3%9Cetzung~PUR%3D .
22. ↑ Lorentz, Hendrik Antoon: Attempt of a theory of electrical and optical phenomena in moving bodies . EJ Brill, Leiden 1895.
23. ^ Poincaré, Henri: La théorie de Lorentz et le principe de réaction . In: Archives néerlandaises des sciences exactes et naturelles . 5, 1900, pp. 252-278. . See also German translationhttp: //vorlage_digitalisat.test/1%3D~GB%3D~IA%3DPoincareReaktion~MDZ%3D%0A~SZ%3D~doppelseiten%3D~LT%3Ddeutsche%20%C3%9Cetzung~PUR%3D .
24. ↑ Poincaré, Henri: The present state and the future of mathematical physics . In: The Value of Science (Chapters 7-9) . BG Teubner, Leipzig 1904/6, pp. 129-159.
25. ^ Poincaré, Henri: Science and method . Xenomos, Berlin 1908b / 2003, ISBN 3-936532-31-1 .
26. ↑ Hasenöhrl, Friedrich: On the theory of radiation in moving bodies . In: Annals of Physics . 320, No. 12, 1904, pp. 344-370. bibcode : 1904AnP ... 320..344H . doi : 10.1002 / andp.19043201206 .
27. ↑ Hasenöhrl, Friedrich: On the theory of radiation in moving bodies. Rectification . In: Annals of Physics . 321, No. 3, 1905, pp. 589-592. bibcode : 1905AnP ... 321..589H . doi : 10.1002 / andp.19053210312 .
28. ↑ ^{a b} Einstein, Albert: On the electrodynamics of moving bodies . In: Annals of Physics . 322, No. 10, 1905, pp. 891-921. bibcode : 1905AnP ... 322..891E . doi : 10.1002 / andp.19053221004 . .
29. ↑ Einstein, Albert: Does the inertia of a body depend on its energy content? . In: Annals of Physics . 323, No. 13, 1905, pp. 639-643. bibcode : 1905AnP ... 323..639E . doi : 10.1002 / andp.19053231314 . .
30. ↑ Einstein, Albert: The principle of the conservation of the center of gravity movement and the inertia of energy . In: Annals of Physics . 325, No. 8, 1906, pp. 627-633. bibcode : 1906AnP ... 325..627E . doi : 10.1002 / andp.19063250814 .
31. ^ R. Tolman: Non-Newtonian Mechanics. The mass of a moving body. . In: Philosophical Magazine . 23, 1912, pp. 375-380.
32. ↑ Gralla, Samuel E .; Harte, Abraham I .; Wald, Robert M .: Rigorous derivation of electromagnetic self-force . In: Physical Review D . 80, No. 2, 2009, p. 024031. arxiv : 0905.2391 . doi : 10.1103 / PhysRevD.80.024031 .
33. ↑ Laue, Max von: The principle of relativity . Vieweg, Braunschweig 1911a.
34. ↑ Fermi, Enrico: About a contradiction between the electrodynamic and relativistic theory of electromagnetic mass . In: Physikalische Zeitschrift . 23, 1922, pp. 340-344.
35. ^ Dirac, Paul: Classical Theory of Radiating Electrons . In: Proceedings of the Royal Society of London A . 167, No. 929, 1938, pp. 148-169. doi : 10.1098 / rspa.1938.0124 .
36. ^ Rohrlich, Fritz: Self-Energy and Stability of the Classical Electron . In: American Journal of Physics . 28, No. 7, 1960, pp. 639-643. doi : 10.1119 / 1.1935924 .
37. ^ Schwinger, Julian: Electromagnetic mass revisited . In: Foundations of Physics . 13, No. 3, 1983, pp. 373-383. doi : 10.1007 / BF01906185 .