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Electromagnetic mass was initially a concept of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, denoting as to how much the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
, or the
self-energy In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or effective mass, due to interactions between ...
, is contributing to the mass of charged particles. It was first derived by
J. J. Thomson Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered. In 1897, Thomson showed that ...
in 1881 and was for some time also considered as a dynamical explanation of
inertial mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
''per se''. Today, the relation of
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
,
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
,
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
, and all forms of energy – including electromagnetic energy – is analyzed on the basis of
Albert Einstein Albert Einstein ( ; ; 14 March 1879 â€“ 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
and
mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
. As to the cause of mass of
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, an ...
s, the
Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
in the framework of the relativistic
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
is currently used. However, some problems concerning the electromagnetic mass and self-energy of charged particles are still studied.


Charged particles


Rest mass and energy

It was recognized by
J. J. Thomson Sir Joseph John Thomson (18 December 1856 – 30 August 1940) was a British physicist and Nobel Laureate in Physics, credited with the discovery of the electron, the first subatomic particle to be discovered. In 1897, Thomson showed that ...
in 1881 that a charged sphere moving in a space filled with a medium of a specific inductive capacity (the electromagnetic aether of
James Clerk Maxwell James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
), is harder to set in motion than an uncharged body. (Similar considerations were already made by
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet, (; 13 August 1819 – 1 February 1903) was an Irish migration to Great Britain, Irish English physicist and mathematician. Born in County Sligo, Ireland, Stokes spent all of his career at the University ...
(1843) with respect to
hydrodynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
, who showed that the inertia of a body moving in an incompressible
perfect fluid In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density \rho_m and ''isotropic'' pressure ''p''. Real fluids are "sticky" and contain (and conduct) heat. Perfect fluids are idealized models in whi ...
is increased.) So due to this self-induction effect, electrostatic energy behaves as having some sort of
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
and "apparent" electromagnetic mass, which can increase the ordinary mechanical mass of the bodies, or in more modern terms, the increase should arise from their electromagnetic
self-energy In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or effective mass, due to interactions between ...
. This idea was worked out in more detail by
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
(1889), Thomson (1893), George Frederick Charles Searle (1897),
Max Abraham Max Abraham (; 26 March 1875 – 16 November 1922) was a German physicist known for his work on electromagnetism and his opposition to the theory of relativity. Biography Abraham was born in Danzig, Imperial Germany (now Gdańsk in Poland) t ...
(1902),
Hendrik Lorentz Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorentz t ...
(1892, 1904), and was directly applied to the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
by using the
Abraham–Lorentz force In the physics of electromagnetism, the Abraham–Lorentz force (also Lorentz–Abraham force) is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation by self-interaction. It is also called ...
. Now, the electrostatic energy E_\mathrm and mass m_\mathrm of an electron at rest was calculated to be :E_\mathrm=\frac\frac,\qquad m_\mathrm=\frac\frac where e is the charge, uniformly distributed on the surface of a sphere, and a is the
classical electron radius The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energ ...
, which must be nonzero to avoid infinite energy accumulation. Thus the formula for this electromagnetic energy–mass relation is :m_\mathrm=\frac\frac This was discussed in connection with the proposal of the electrical origin of matter, so
Wilhelm Wien Wilhelm Carl Werner Otto Fritz Franz Wien (; 13 January 1864 – 30 August 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to deduce Wien's displacement law, which calculates the emission of a blackbody ...
(1900), and Max Abraham (1902), came to the conclusion that the total mass of the bodies is identical to its electromagnetic mass. Wien stated, that if it is assumed that
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
is an electromagnetic effect too, then there has to be a proportionality between electromagnetic energy, inertial mass, and gravitational mass. When one body attracts another one, the electromagnetic energy store of gravitation is according to Wien diminished by the amount (where M is the attracted mass, G the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
, r the distance): :G=\frac
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 â€“ 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
in 1906 argued that when mass is in fact the product of the electromagnetic field in the aether – implying that no "real" mass exists – and because matter is inseparably connected with mass, then also
matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...
doesn't exist at all and electrons are only concavities in the aether.


Mass and speed


Thomson and Searle

Thomson (1893) noticed that electromagnetic momentum and energy of charged bodies, and therefore their masses, depend on the speed of the bodies as well. He wrote: In 1897, Searle gave a more precise formula for the electromagnetic energy of charged sphere in motion: :E_\mathrm^=E_\mathrm\left frac\ln\frac-1\right\qquad\beta=\frac, and like Thomson he concluded:


Longitudinal and transverse mass

From Searle's formula, Walter Kaufmann (1901) and
Max Abraham Max Abraham (; 26 March 1875 – 16 November 1922) was a German physicist known for his work on electromagnetism and his opposition to the theory of relativity. Biography Abraham was born in Danzig, Imperial Germany (now Gdańsk in Poland) t ...
(1902) derived the formula for the electromagnetic mass of moving bodies: :m_=\frac\cdot m_\mathrm\cdot\frac\left \frac\ln\left(\frac\right)+\frac\right/math> However, it was shown by Abraham (1902), that this value is only valid in the longitudinal direction ("longitudinal mass"), i.e., that the electromagnetic mass also depends on the direction of the moving bodies with respect to the aether. Thus Abraham also derived the "transverse mass": :m_=\frac\cdot m_\mathrm\cdot\frac\left left(\frac\right)\ln\left(\frac\right)-1\right/math> On the other hand, already in 1899 Lorentz assumed that the electrons undergo
length contraction Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
in the line of motion, which leads to results for the acceleration of moving electrons that differ from those given by Abraham. Lorentz obtained factors of k^3 \varepsilon parallel to the direction of motion and k\varepsilon perpendicular to the direction of motion, where k = \sqrt and \varepsilon is an undetermined factor. Lorentz expanded his 1899 ideas in his famous 1904 paper, where he set the factor \varepsilon to unity, thus: :m_=\frac,\quad m_=\frac , So, eventually Lorentz arrived at the same conclusion as Thomson in 1893: no body can reach the speed of light because the mass becomes infinitely large at this velocity. Additionally, a third electron model was developed by
Alfred Bucherer Alfred Heinrich Bucherer (* 9 July 1863 in Cologne; † 16 April 1927 in Bonn) was a German physicist, who is known for his experiments on relativistic mass. He also was the first who used the phrase "theory of relativity" for Einstein's theory of ...
and
Paul Langevin Paul Langevin (; ; 23 January 1872 â€“ 19 December 1946) was a French physicist who developed Langevin dynamics and the Langevin equation. He was one of the founders of the ''Comité de vigilance des intellectuels antifascistes'', an ant ...
, in which the electron contracts in the line of motion, and expands perpendicular to it, so that the volume remains constant. This gives: :m_=\frac,\quad m_=\frac


Kaufmann's experiments

The predictions of the theories of Abraham and Lorentz were supported by the experiments of Walter Kaufmann (1901), but the experiments were not precise enough to distinguish between them. In 1905 Kaufmann conducted another series of experiments (
Kaufmann–Bucherer–Neumann experiments The Kaufmann–Bucherer–Neumann experiments measured the dependence of the inertial mass (or momentum) of an object on its velocity. The historical importance of this series of experiments performed by various physicists between 1901 and 1915 is ...
) which confirmed Abraham's and Bucherer's predictions, but contradicted Lorentz's theory and the "fundamental assumption of Lorentz and Einstein", ''i.e.'', the relativity principle. In the following years experiments by
Alfred Bucherer Alfred Heinrich Bucherer (* 9 July 1863 in Cologne; † 16 April 1927 in Bonn) was a German physicist, who is known for his experiments on relativistic mass. He also was the first who used the phrase "theory of relativity" for Einstein's theory of ...
(1908), Gunther Neumann (1914) and others seemed to confirm Lorentz's mass formula. It was later pointed out that the Bucherer–Neumann experiments were also not precise enough to distinguish between the theories – it lasted until 1940 when the precision required was achieved to eventually prove Lorentz's formula and to refute Abraham's by these kinds of experiments. (However, other experiments of different kind already refuted Abraham's and Bucherer's formulas long before.)


Poincaré stresses and the problem

The idea of an electromagnetic nature of matter, however, had to be given up. Abraham (1904, 1905) argued that non-electromagnetic forces were necessary to prevent Lorentz's contractile electrons from exploding. He also showed that different results for the longitudinal electromagnetic mass can be obtained in Lorentz's theory, depending on whether the mass is calculated from its energy or its momentum, so a non-electromagnetic potential (corresponding to of the electron's electromagnetic energy) was necessary to render these masses equal. Abraham doubted whether it was possible to develop a model satisfying all of these properties. To solve those problems,
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 â€“ 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
in 1905 and 1906 introduced some sort of pressure ("Poincaré stresses") of non-electromagnetic nature. As required by Abraham, these stresses contribute non-electromagnetic energy to the electrons, amounting to of their total energy or to of their electromagnetic energy. So, the Poincaré stresses remove the contradiction in the derivation of the longitudinal electromagnetic mass, they prevent the electron from exploding, they remain unaltered by a
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
(''i.e.'' they are Lorentz invariant), and were also thought as a dynamical explanation of
length contraction Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
. However, Poincaré still assumed that only the electromagnetic energy contributes to the mass of the bodies. As it was later noted, the problem lies in the factor of electromagnetic rest mass – given above as m_\mathrm=\tfrac E_\mathrm/c^2 when derived from the Abraham–Lorentz equations. However, when it is derived from the electron's electrostatic energy alone, we have m_\mathrm=E_\mathrm/c^2 where the factor is missing. This can be solved by adding the non-electromagnetic energy E_\mathrm of the Poincaré stresses to E_\mathrm, the electron's total energy E_\mathrm now becomes: :\frac=\frac=\frac=\frac\frac=\fracm_\mathrm=m_\mathrm Thus the missing factor is restored when the mass is related to its electromagnetic energy, and it disappears when the total energy is considered.


Inertia of energy and radiation paradoxes


Radiation pressure

Another way of deriving some sort of electromagnetic mass was based on the concept of
radiation pressure Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is a ...
. These pressures or tensions in the electromagnetic field were derived by
James Clerk Maxwell James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
(1874) and
Adolfo Bartoli Adolfo Bartoli (19 March 1851 – 18 July 1896) was an Italian physicist, who is best known for introducing the concept of radiation pressure from thermodynamical considerations. Born in Florence, Bartoli studied physics and mathematics at the Un ...
(1876). Lorentz recognized in 1895 that those tensions also arise in his theory of the stationary aether. So if the electromagnetic field of the aether is able to set bodies in motion, the action / reaction principle demands that the aether must be set in motion by matter as well. However, Lorentz pointed out that any tension in the aether requires the mobility of the aether parts, which is not possible since in his theory the aether is immobile. (unlike contemporaries like Thomson who used fluid descriptions) This represents a violation of the reaction principle that was accepted by Lorentz consciously. He continued by saying, that one can only speak about ''fictitious'' tensions, since they are only mathematical models in his theory to ease the description of the electrodynamic interactions.


Mass of the fictitious electromagnetic fluid

In 1900 Poincaré studied the conflict between the action/reaction principle and Lorentz's theory. He tried to determine whether the
center of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weight function, weighted relative position (vector), position of the distributed mass sums to zero. Thi ...
still moves with a uniform velocity when electromagnetic fields and radiation are involved. He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum (such a momentum was also derived by Thomson in 1893 in a more complicated way). Poincaré concluded, the electromagnetic field energy behaves like a fictitious
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
(„fluide fictif“) with a mass density of E_/c^2 (in other words m_=E_/c^2). Now, if the center of mass frame (COM-frame) is defined by both the mass of matter ''and'' the mass of the fictitious fluid, and if the fictitious fluid is indestructible – it is neither created or destroyed – then the motion of the center of mass frame remains uniform. But this electromagnetic fluid is not indestructible, because it can be absorbed by matter (which according to Poincaré was the reason why he regarded the em-fluid as "fictitious" rather than "real"). Thus the COM-principle would be violated again. As it was later done by Einstein, an easy solution of this would be to assume that the mass of em-field is transferred to matter in the absorption process. But Poincaré created another solution: He assumed that there exists an immobile non-electromagnetic energy fluid at each point in space, also carrying a mass proportional to its energy. When the fictitious em-fluid is destroyed or absorbed, its electromagnetic energy and mass is not carried away by moving matter, but is transferred into the non-electromagnetic fluid and remains at exactly the same place in that fluid. (Poincaré added that one should not be too surprised by these assumptions, since they are only mathematical fictions.) In this way, the motion of the COM-frame, including matter, fictitious em-fluid, and fictitious non-em-fluid, at least ''theoretically'' remains uniform. However, since only matter and electromagnetic energy are directly observable by experiment (not the non-em-fluid), Poincaré's resolution still violates the reaction principle and the COM-theorem, when an emission/absorption process is ''practically'' considered. This leads to a paradox when changing frames: if waves are radiated in a certain direction, the device will suffer a
recoil Recoil (often called knockback, kickback or simply kick) is the rearward thrust generated when a gun is being discharged. In technical terms, the recoil is a result of conservation of momentum, as according to Newton's third law the force requ ...
from the momentum of the fictitious fluid. Then, Poincaré performed a
Lorentz boost In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
(to first order in ) to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow
perpetual motion Perpetual motion is the motion of bodies that continues forever in an unperturbed system. A perpetual motion machine is a hypothetical machine that can do work infinitely without an external energy source. This kind of machine is impossible, a ...
, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Therefore, he argued that also in this case there has to be another compensating mechanism in the ether. Poincaré came back to this topic in 1904. This time he rejected his own solution that motions in the ether can compensate the motion of matter, because any such motion is unobservable and therefore scientifically worthless. He also abandoned the concept that energy carries mass and wrote in connection to the above-mentioned recoil: These iterative developments culminated in his 1906 publication "The End of Matter" in which he notes that when applying the methodology of using an electric or magnetic field deviations to determine charge-to-mass ratios, one finds that the apparent mass added by charge makes up all of the apparent mass, thus the "real mass is equal to zero." Thus he goes on to postulate that electrons are only holes or motion effects in the aether while the aether itself is the only thing "endowed with inertia." He then goes on to address the possibility that all matter might share this same quality and thereby his position changes from viewing aether as a "fictitious fluid" to suggesting it might be the only thing that actually exists in the universe, finally stating "In this system there is no actual matter, there are only holes in the aether." Finally he repeats this exact problem of "Newton's principle" from 1904 again in 1908 publication in his section on "the principle of reaction" he notes that the actions of radiation pressure cannot be tied solely to matter in light of Fizeau's proof that the Hertz notion of total ether drag is untenable. This, he clarifies in the next section in his own explanation of
Mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
: Thus Poincaré's mass of a fictitious fluid led him to, instead, later find that the mass of matter itself was "fictitious." Einstein's own 1906 publication grants credit to Poincare for previously exploring the mass-energy equivalence and it is from these comments that it is commonly reported that
Lorentz ether theory What is now often called Lorentz ether theory (LET) has its roots in Hendrik Lorentz's "theory of electrons", which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th cen ...
is "mathematically equivalent."


Momentum and cavity radiation

However, Poincaré's idea of momentum and mass associated with radiation proved to be fruitful, when in 1903
Max Abraham Max Abraham (; 26 March 1875 – 16 November 1922) was a German physicist known for his work on electromagnetism and his opposition to the theory of relativity. Biography Abraham was born in Danzig, Imperial Germany (now Gdańsk in Poland) t ...
introduced the term „electromagnetic momentum“, having a field density of E_/c^2 per cm3 and E_/c per cm2. Contrary to Lorentz and Poincaré, who considered momentum as a fictitious force, he argued that it is a real physical entity, and therefore conservation of momentum is guaranteed. In 1904,
Friedrich Hasenöhrl Friedrich Hasenöhrl (; 30 November 1874 – 7 October 1915) was an Austrian physicist. Life Friedrich Hasenöhrl was born in Vienna, Austria-Hungary in 1874. His father was a lawyer and his mother belonged to a prominent aristocratic family. A ...
specifically associated inertia with ''radiation'' by studying the dynamics of a moving
cavity Cavity may refer to: Biology and healthcare *Body cavity, a fluid-filled space in many animals where organs typically develop **Gastrovascular cavity, the primary organ of digestion and circulation in cnidarians and flatworms *Dental cavity or too ...
. Hasenöhrl suggested that part of the mass of a body (which he called ''apparent mass'') can be thought of as radiation bouncing around a cavity. The apparent mass of radiation depends on the temperature (because every heated body emits radiation) and is proportional to its energy, and he first concluded that m=\tfracE/c^2. However, in 1905 Hasenöhrl published a summary of a letter, which was written by Abraham to him. Abraham concluded that Hasenöhrl's formula of the apparent mass of radiation is not correct, and on the basis of his definition of electromagnetic momentum and longitudinal electromagnetic mass Abraham changed it to m=\tfracE/c^2, the same value for the electromagnetic mass for a body at rest. Hasenöhrl recalculated his own derivation and verified Abraham's result. He also noticed the similarity between the apparent mass and the electromagnetic mass that Poincaré would comment on in 1906. However, Hasenöhrl stated that this energy-apparent-mass relation ''only'' holds as long a body radiates, i.e. if the temperature of a body is greater than 0 K.


Modern view


Mass–energy equivalence

The idea that the principal relations between mass, energy, momentum and velocity can only be considered on the basis of dynamical interactions of matter was superseded, when
Albert Einstein Albert Einstein ( ; ; 14 March 1879 â€“ 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
found out in 1905 that considerations based on the special
principle of relativity In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. For example, in the framework of special relativity the Maxwell equations have ...
require that all forms of energy (not only electromagnetic) contribute to the mass of bodies (
mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
). That is, the entire mass of a body is a measure of its energy content by E=mc^2, and Einstein's considerations were independent from assumptions about the constitution of matter. By this equivalence, Poincaré's radiation paradox can be solved without using "compensating forces", because the mass of matter itself (not the non-electromagnetic aether fluid as suggested by Poincaré) is increased or diminished by the mass of electromagnetic energy in the course of the emission/absorption process. Also the idea of an electromagnetic explanation of gravitation was superseded in the course of developing
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. So every theory dealing with the mass of a body must be formulated in a relativistic way from the outset. This is for example the case in the current
quantum field In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
explanation of mass of
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, an ...
s in the framework of the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
, the
Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other bein ...
. Because of this, the idea that any form of mass is ''completely'' caused by interactions with electromagnetic fields, is not relevant any more.


Relativistic mass

The concepts of longitudinal and transverse mass (equivalent to those of Lorentz) were also used by Einstein in his first papers on relativity. However, in special relativity they apply to the entire mass of matter, not only to the electromagnetic part. Later it was shown by physicists like
Richard Chace Tolman Richard Chace Tolman (March 4, 1881 â€“ September 5, 1948) was an American mathematical physicist and physical chemist who made many contributions to statistical mechanics. He also made important contributions to theoretical cosmology in ...
that expressing mass as the ratio of force and acceleration is not advantageous. Therefore, a similar concept without direction dependent terms, in which force is defined as \vec = \mathrm\vec/\mathrmt, was used as
relativistic mass The word "mass" has two meanings in special relativity: '' invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity o ...
:M=\frac,\qquad m_=\frac, This concept is sometimes still used in modern physics textbooks, although the term 'mass' is now considered by many to refer to
invariant mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
, see
mass in special relativity The word "mass" has two meanings in special relativity: ''invariant mass'' (also called rest mass) is an invariant quantity which is the same for all Observer (special relativity), observers in all reference frames, while the relativistic mass is ...
.


Self-energy

When the special case of the electromagnetic
self-energy In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or effective mass, due to interactions between ...
or self-force of charged particles is discussed, also in modern texts some sort of "effective" electromagnetic mass is sometimes introduced – not as an explanation of mass ''per se'', but in addition to the ordinary mass of bodies. Many different reformulations of the
Abraham–Lorentz force In the physics of electromagnetism, the Abraham–Lorentz force (also Lorentz–Abraham force) is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation by self-interaction. It is also called ...
have been derived – for instance, in order to deal with the -problem (see next section) and other problems that arose from this concept. Such questions are discussed in connection with
renormalization Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
, and on the basis of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, which must be applied when the
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
is considered physically point-like. At distances located in the classical domain, the classical concepts again come into play. A rigorous derivation of the electromagnetic self-force, including the contribution to the mass of the body, was published by Gralla et al. (2009).


problem

Max von Laue Max Theodor Felix von Laue (; 9 October 1879 – 24 April 1960) was a German physicist who received the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals. In addition to his scientific endeavors with cont ...
in 1911 also used the Abraham–Lorentz equations of motion in his development of special relativistic dynamics, so that also in special relativity the factor is present when the electromagnetic mass of a charged sphere is calculated. This contradicts the mass–energy equivalence formula, which requires the relation m_\mathrm=E_\mathrm/c^2 without the factor, or in other words, four-momentum doesn't properly transform like a
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
when the factor is present. Laue found a solution equivalent to Poincaré's introduction of a non-electromagnetic potential (Poincaré stresses), but Laue showed its deeper, relativistic meaning by employing and advancing
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
's
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
formalism. Laue's formalism required that there are additional components and forces, which guarantee that spatially extended systems (where both electromagnetic and non-electromagnetic energies are combined) are forming a stable or "closed system" and transform as a four-vector. That is, the factor arises only with respect to electromagnetic mass, while the closed system has total rest mass and energy of m_\mathrm=E_\mathrm/c^2. Another solution was found by authors such as
Enrico Fermi Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...
(1922),
Paul Dirac Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
(1938)
Fritz Rohrlich Fritz Rohrlich (May 12, 1921 – November 14, 2018) was an American theoretical physicist and educator who published in the fields of quantum electrodynamics, classical electrodynamics of charged particles, and the philosophy of science. Life and ...
(1960), or
Julian Schwinger Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
(1983), who pointed out that the electron's stability and the 4/3-problem are two different things. They showed that the preceding definitions of four-momentum are non-relativistic ''per se'', and by changing the definition into a relativistic form, the electromagnetic mass can simply be written as m_\mathrm=E_\mathrm/c^2 and thus the factor doesn't appear at all. So every part of the system, not only "closed" systems, properly transforms as a four-vector. However, binding forces like the Poincaré stresses are still necessary to prevent the electron from exploding due to Coulomb repulsion. But on the basis of the Fermi–Rohrlich definition, this is only a dynamical problem and has nothing to do with the transformation properties any more. Also other solutions have been proposed, for instance, Valery Morozov (2011) gave consideration to movement of an imponderable charged sphere. It turned out that a flux of nonelectromagnetic energy exists in the sphere body. This flux has an impulse exactly equal to of the sphere electromagnetic impulse regardless of a sphere internal structure or a material, it is made of. The problem was solved without attraction of any additional hypotheses. In this model, sphere tensions are not connected with its mass.


See also

*
History of special relativity The history of special relativity consists of many theoretical results and empirical findings obtained by Albert A. Michelson, Hendrik Lorentz, Henri Poincaré and others. It culminated in the theory of special relativity proposed by Albert Eins ...
*
Abraham–Lorentz force In the physics of electromagnetism, the Abraham–Lorentz force (also Lorentz–Abraham force) is the recoil force on an accelerating charged particle caused by the particle emitting electromagnetic radiation by self-interaction. It is also called ...
*
Wheeler–Feynman absorber theory The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is an interpretation of electrodynamics derived from the assump ...


Secondary sources ( references)


Primary sources

{{reflist, 25em, refs= {{cite journal , doi=10.1002/andp.19023150105 , author=Abraham, Max , year=1903 , title=Prinzipien der Dynamik des Elektrons , journal=Annalen der Physik , volume=315 , issue=1 , pages=105–179 , bibcode=1902AnP...315..105A , title-link=s:de:Prinzipien der Dynamik des Elektrons (1903) {{cite journal , author=Abraham, Max , year=1904 , title=Die Grundhypothesen der Elektronentheorie , trans-title= The Fundamental Hypotheses of the Theory of Electrons , journal=Physikalische Zeitschrift , volume=5 , pages=576–579 , title-link=s:de:Die Grundhypothesen der Elektronentheorie {{cite book , last=Abraham, M. , year=1905 , title= Theorie der Elektrizität: Elektromagnetische Theorie der Strahlung , publisher=Teubner , location=Leipzig , page
201
€“208 , url=https://archive.org/details/theoriederelekt04fpgoog , via=Internet Archive
{{cite book , author=Bucherer, A. H. , year=1904 , title=Mathematische Einführung in die Elektronentheorie , publisher=Teubner , location=Leipzig , url=https://archive.org/details/mathematischeei00buchgoog , via=Internet Archive {{cite journal , author=Dirac, Paul , year=1938 , title=Classical Theory of Radiating Electrons , journal=Proceedings of the Royal Society of London A , volume=167 , issue=929 , pages=148–169 , url=http://gallica.bnf.fr/ark:/12148/bpt6k56260v/f164 , via=gallica.bnf.fr , doi=10.1098/rspa.1938.0124 , bibcode=1938RSPSA.167..148D, doi-access=free {{cite journal , author=Einstein, Albert , year=1905a , title=Zur Elektrodynamik bewegter Körper , journal=Annalen der Physik , volume=322 , issue=10 , pages=891–921 , bibcode=1905AnP...322..891E , doi=10.1002/andp.19053221004 , url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_891-921.pdf, doi-access=free. See also
English translation
{{cite journal , author=Einstein, Albert , year=1905b , title=Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? , journal=Annalen der Physik , volume=323 , issue=13 , pages=639–643 , bibcode=1905AnP...323..639E , doi=10.1002/andp.19053231314 , url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_18_639-641.pdf, doi-access=free See also {{cite web , url=http://www.fourmilab.ch/etexts/einstein/E_mc2/www/ , title=the English translation {{cite journal , author=Einstein, Albert , year=1906 , title=Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie , journal=Annalen der Physik , volume=325 , issue=8 , pages=627–633 , bibcode=1906AnP...325..627E , doi=10.1002/andp.19063250814 , s2cid=120361282 , url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1906_20_627-633.pdf {{cite journal , author=Fermi, Enrico , year=1922 , title=Über einen Widerspruch zwischen der elektrodynamischen und relativistischen Theorie der elektromagnetischen Masse , trans-title= Concerning a Contradiction between the Electrodynamic and Relativistic Theory of Electromagnetic Mass , journal=Physikalische Zeitschrift , volume=23 , pages=340–344 {{cite journal , author1=Gralla, Samuel E. , author2=Harte, Abraham I. , author3=Wald, Robert M. , year=2009 , title=Rigorous derivation of electromagnetic self-force , journal=Physical Review D , volume=80 , issue=2 , pages=024031 , doi=10.1103/PhysRevD.80.024031 , arxiv=0905.2391 , bibcode=2009PhRvD..80b4031G , s2cid=118781808 {{cite journal , author=Hasenöhrl, Friedrich , year=1904 , title=Zur Theorie der Strahlung in bewegten Körpern , trans-title= On the Theory of Radiation in Moving Bodies , journal=Annalen der Physik , volume=320 , issue=12 , pages=344–370 , bibcode=1904AnP...320..344H , doi=10.1002/andp.19043201206 , title-link=s:de:Zur Theorie der Strahlung in bewegten Körpern {{cite journal , author=Hasenöhrl, Friedrich , year=1905 , title=Zur Theorie der Strahlung in bewegten Körpern. 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Berichtigung {{cite magazine , author=Heaviside, Oliver , year=1889 , title=On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric , magazine=Philosophical Magazine , series=5 , volume=27 , issue=167 , pages=324–339 , doi=10.1080/14786448908628362 , title-link=s:Motion of Electrification through a Dielectric {{cite journal , author=Kaufmann, Walter , year=1902 , title=Die elektromagnetische Masse des Elektrons , trans-title= The Electromagnetic Mass of the Electron , journal=Physikalische Zeitschrift , volume=4 , issue=1b , pages=54–56 {{cite journal , author=Kaufmann, Walter , year=1905 , title=Über die Konstitution des Elektrons , trans-title= On the Constitution of the Electron , journal=Sitzungsberichte der Königlich Preußische Akademie der Wissenschaften , volume=45 , pages=949–956 {{cite journal , author=Kaufmann, Walter , year=1906 , title=Über die Konstitution des Elektrons , trans-title= On the Constitution of the Electron , journal=Annalen der Physik , volume=324 , issue=3 , pages=487–553 , bibcode=1906AnP...324..487K , doi=10.1002/andp.19063240303, url=https://zenodo.org/record/1424061 {{cite book , author=Laue, Max von , year=1911 , title=Das Relativitätsprinzip , trans-title=The Principle of Relativity , url=https://archive.org/details/dasrelativittsp00lauegoog , via=Internet Archive , publisher=Vieweg , location=Braunschweig {{cite journal , author=Lorentz, Hendrik Antoon , year=1892 , title=La Théorie electromagnétique de Maxwell et son application aux corps mouvants , url=https://archive.org/details/lathorielectrom00loregoog , via=Internet Archive , journal=Archives Néerlandaises des Sciences Exactes et Naturelles , volume=25 , pages=363–552 {{cite book , author=Lorentz, Hendrik Antoon , year=1895 , title=Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern , trans-title= Attempt of a Theory of Electrical and Optical Phenomena in Moving Bodies , location=Leiden , publisher=E.J. Brill , title-link=s:de:Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern {{cite journal , author=Lorentz, Hendrik Antoon , year=1899 , title=Simplified Theory of Electrical and Optical Phenomena in Moving Systems , journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences , volume=1 , pages=427–442 , title-link=s:Simplified Theory of Electrical and Optical Phenomena in Moving Systems , bibcode=1898KNAB....1..427L {{cite journal , author=Lorentz, Hendrik Antoon , year=1904 , title=Electromagnetic phenomena in a system moving with any velocity smaller than that of light , journal=Proceedings of the Royal Netherlands Academy of Arts and Sciences , volume=6 , pages=809–831 , title-link=s:Electromagnetic phenomena , bibcode=1903KNAB....6..809L {{cite journal , author=Poincaré, Henri , year=1900 , title=La théorie de Lorentz et le principe de réaction , journal=Archives Néerlandaises des Sciences Exactes et Naturelles , volume=5 , pages=252–278 , title-link=s:fr:La théorie de Lorentz et le principe de réaction See also the {{cite web , url=http://www.physicsinsights.org/poincare-1900.pdf , title=English translation {{cite book , author=Poincaré, Henri , year=1904 , chapter=
The Principles of Mathematical Physics ''The'' () is a grammatical article in English, denoting persons or things already mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the ...
, title=Congress of arts and science, universal exposition, St. Louis, 1904 , volume=1 , pages=604–622 , publisher=Houghton, Mifflin and Company , location=Boston and New York
{{cite journal , author=Poincaré, Henri , year=1905 , title=Sur la dynamique de l'électron , trans-title= On the Dynamics of the Electron , journal=Comptes Rendus , volume=140 , pages=1504–1508 , title-link=s:fr:Sur la dynamique de l'électron (juin) {{cite journal , author=Poincaré, Henri , year=1906 , title=Sur la dynamique de l'électron , trans-title= On the Dynamics of the Electron , journal=Rendiconti del Circolo Matematico di Palermo , volume=21 , pages=129–176 , doi=10.1007/BF03013466 , bibcode=1906RCMP...21..129P , title-link=s:fr:Sur la dynamique de l'électron (juillet) , hdl=2027/uiug.30112063899089 , s2cid=120211823 , hdl-access=free {{cite journal , author=Poincaré, Henri , year=1906 , title=La fin de la matière , trans-title= The End of Matter , journal=Athenæum {{cite book , author=Poincaré, Henri , year=1908–1913 , title=The foundations of science (Science and Method) , chapter= The New Mechanics , location=New York , publisher=Science Press , pages=486–522 {{cite journal , author=Rohrlich, Fritz , year=1960 , title=Self-Energy and Stability of the Classical Electron , journal=American Journal of Physics , volume=28 , issue=7 , pages=639–643 , doi=10.1119/1.1935924 , bibcode=1960AmJPh..28..639R {{cite journal , author=Schwinger, Julian , year=1983 , title=Electromagnetic mass revisited , journal=
Foundations of Physics ''Foundations of Physics'' is a monthly journal "devoted to the conceptual bases and fundamental theories of modern physics and cosmology, emphasizing the logical, methodological, and philosophical premises of modern physical theories and procedur ...
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{{cite magazine , author=Searle, George Frederick Charles , year=1897 , title=On the Steady Motion of an Electrified Ellipsoid , magazine=Philosophical Magazine , series=5 , volume=44 , issue=269 , pages=329–341 , doi=10.1080/14786449708621072 , title-link=s:On the Steady Motion of an Electrified Ellipsoid {{cite journal , author=Stokes, George Gabriel , year=1844 , journal=Transactions of the Cambridge Philosophical Society , volume=8 , issue=1 , title=On some cases of fluid motion , url=https://archive.org/details/transactionsofca08camb , via=Internet Archive , pages=105–137 (Read May 29, 1843) {{cite magazine , author=Thomson, Joseph John , year=1881 , title=On the Electric and Magnetic Effects produced by the Motion of Electrified Bodies , magazine=Philosophical Magazine , series=5 , volume=11 , issue=68 , pages=229–249 , doi=10.1080/14786448108627008 , title-link=s:On the Electric and Magnetic Effects produced by the Motion of Electrified Bodies See also {{cite journal , url=https://zenodo.org/record/1431147 , title=via zenodo.org, doi=10.1080/14786448108627008, date=April 1881, last1=Thomson, first1=J. J., journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, volume=11, issue=68, pages=229–249 {{cite book , author=Thomson, Joseph John , year=1893 , title=Notes on recent researches in electricity and magnetism , url=https://archive.org/details/notesonrecentres00thom , via=Internet Archive , publisher=Clarendon Press , location=Oxford {{cite magazine , author=R. Tolman , year=1912 , title=Non-Newtonian Mechanics. The Mass of a Moving Body , magazine=Philosophical Magazine , volume=23 , issue=135 , pages=375–380 , doi=10.1080/14786440308637231 , title-link=s:The Mass of a Moving Body {{cite journal , author=Wien, Wilhelm , year=1900 , title=Über die Möglichkeit einer elektromagnetischen Begründung der Mechanik , trans-title= On the Possibility of an Electromagnetic Foundation of Mechanics , journal=Annalen der Physik , volume=310 , issue=7 , pages=501–513 , doi=10.1002/andp.19013100703 , bibcode=1901AnP...310..501W , title-link=s:de:Über die Möglichkeit einer elektromagnetischen Begründung der Mechanik {{cite journal , author=Morozov, Valery B. , year=2011 , title=On the question of the electromagnetic momentum of a charged body , journal=Physics Uspekhi , volume=54 , issue=4 , pages=371–374 , doi=10.3367/UFNe.0181.201104c.0389 , bibcode=2011PhyU...54..371M , arxiv=2007.03468, s2cid=120857631 Special relativity Obsolete theories in physics