Electromagnetic mass

**Electromagnetic mass** was initially a concept of classical mechanics, denoting as to how much the electromagnetic field, or the self-energy, is contributing to the mass of charged particles. It was first derived by J. J. Thomson in 1881 and was for some time also considered as a dynamical explanation of inertial mass *per se*.

Today, the relation of mass, momentum, velocity and all forms of energy, including electromagnetic energy, is analyzed on the basis of Albert Einstein's special relativity and mass–energy equivalence. As to the cause of mass of elementary particles, the Higgs mechanism in the framework of the relativistic Standard Model is currently used. In addition, 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 in 1881 that a charged sphere moving in space is harder to set in motion than an uncharged body.

* Similar considerations were already made by George Gabriel Stokes (1843) with respect to hydrodynamics, who showed that the inertia of a body moving in an incompressible perfect fluid is increased.)

So due to this self-induction effect, electrostatic energy behaves as having some sort of momentum 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.

* This idea was worked out in more detail by Oliver Heaviside (1889), Thomson (1893), George Frederick Charles Searle (1897), Max Abraham (1902), Hendrik Lorentz (1892, 1904), and was directly applied to the electron by using the Abraham–Lorentz force.

Now, the electrostatic energy

E

e
m

{\displaystyle E\_{em}}

 and mass

m

e
m

{\displaystyle m\_{em}}

 of an electron at rest was calculated to be

E

e
m

=

1
2

e

2

r

,

m

e
m

=

2
3

e

2

r

c

2

{\displaystyle E\_{em}=\frac{1}{2}\frac{e^{2}}{r},\qquad m\_{em}=\frac{2}{3}\frac{e^{2}}{rc^{2}}}

where

e

{\displaystyle e}

 is the charge, uniformly distributed on the surface of a sphere with radius r (classical electron radius), which must be nonzero to avoid infinite energy accumulation. Thus the formula for this electromagnetic energy–mass relation is

m

e
m

=

4
3

E

e
m

c

2

{\displaystyle m\_{em}=\frac{4}{3}\frac{E\_{em}}{c^{2}}}

This was discussed in connection with the proposal of the electrical origin of matter, so Wilhelm Wien (1900), and Max Abraham (1902), came to the conclusion that the total mass of the bodies is identical to its electromagnetic mass.

Henri Poincaré 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 doesn't exist at all and electrons are only concavities in the aether.

#### Possible explanation of gravity

Wien stated, that if it is assumed that gravitation 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

{\displaystyle M}

 is the attracted mass,

G

{\displaystyle G}

 the gravitational constant,

r

{\displaystyle r}

 the distance):

G

4
3

E

e
m

c

2

M

r

{\displaystyle G\frac{\frac{4}{3}\frac{E\_{em}}{c^{2}}M}{r}}

### 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:

> [p. 21] 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.
>
>
>

In 1897, Searle gave a more precise formula for the electromagnetic energy of charged sphere in motion:

E

e
m

v

=

E

e
m

[

1
β

ln
⁡

1
β

1
−
β

−
1

]

,

# β

v
c

,

{\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:

> ... 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.
>
>
>

#### Longitudinal and transverse mass

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

m

L

=

3
4

⋅

m

e
m

⋅

1

β

2

[

−

1

β

2

ln
⁡

(

1
β

1
−
β

)

2

1
−

β

2

]

{\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]}

However, it was shown by Abraham (1902), that this value is only valid when measured parallel to the direction of motion ("longitudinal mass"). Thus Abraham also derived the "transverse mass":

m

T

=

3
4

⋅

m

e
m

⋅

1

β

2

[

(

1

β

2

2
β

)

ln
⁡

(

1
β

1
−
β

)

−
1

]

{\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, already in 1899 Lorentz assumed that the electrons undergo length contraction 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

ε

{\displaystyle k^3 \varepsilon}

 parallel to the direction of motion and

k
ε

{\displaystyle k\varepsilon}

 perpendicular to the direction of motion, where

# k

1
−

v
2

/

c
2

{\displaystyle k = \sqrt{1- v^2 / c^2}}

 and

ε

{\displaystyle \varepsilon}

 is an undetermined factor. Lorentz expanded his 1899 ideas in his famous 1904 paper, where he set the factor

ε

{\displaystyle \varepsilon}

 to unity, thus:

m

L

=

m

e
m

(

1
−

v

2

c

2

)

3

,

m

T

=

m

e
m

1
−

v

2

c

2

{\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}}}} }

,
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 and Paul Langevin, in which the electron contracts in the line of motion, and expands perpendicular to it, so that the volume remains constant. This gives:

m

L

=

m

e
m

(

1
−

1
3

v

2

c

2

)

(

1
−

v

2

c

2

)

8

/

3

,

m

T

=

m

e
m

(

1
−

v

2

c

2

)

2

/

3

{\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

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) 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 (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.

This 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 4/3 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 1/3 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 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 1/4 of their total energy or to 1/3 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 (*i.e.* they are Lorentz invariant), and were also thought as a dynamical explanation of length contraction. 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 4/3 factor of electromagnetic rest mass – given above as

m

e
m

=
(
4

/

3
)

E

e
m

/

c
2

{\displaystyle m\_{em}=(4/3)E\_{em}/c^2}

 when derived from the Abraham–Lorentz equations. However, when it is derived from the electron's electrostatic energy alone, we have

m

e
s

=

E

e
m

/

c
2

{\displaystyle m\_{es}=E\_{em}/c^2}

 where the 4/3 factor is missing. This can be solved by adding the non-electromagnetic energy

E
p

{\displaystyle E\_p}

 of the Poincaré stresses to

E

e
m

{\displaystyle E\_{em}}

, the electron's total energy

E

t
o
t

{\displaystyle E\_{tot}}

 now becomes:

E

t
o
t

c

2

=

E

e
m

E

p

c

2

=

E

e
m

E

e
m

3

c

2

=

4
3

E

e
m

c

2

=

4
3

m

e
s

=

m

e
m

{\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}}

Thus the missing 4/3 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. These pressures or tensions in the electromagnetic field were derived by James Clerk Maxwell (1874) and Adolfo Bartoli (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. 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 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 („fluide fictif“) with a mass density of

E

e
m

/

c
2

{\displaystyle E\_{em}/c^2}

 (in other words

m

e
m

=

E

e
m

/

c
2

{\displaystyle m\_{em}=E\_{em}/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, incl. 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 from the momentum of the fictitious fluid. Then, Poincaré performed a Lorentz boost (to first order in v/c) 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, 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:

> The apparatus will recoil as if it were a cannon and the projected energy a ball, and that contradicts the principle of Newton, since our present projectile has no mass; it is not matter, it is energy.
>
>
>

### Momentum and cavity radiation

However, Poincaré's idea of momentum and mass associated with radiation proved to be fruitful, when Max Abraham introduced the term „electromagnetic momentum“, having a field density of

E

e
m

/

c
2

{\displaystyle E\_{em}/c^2}

 per cm3 and

E

e
m

/

c

{\displaystyle E\_{em}/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 specifically associated inertia with *radiation* by studying the dynamics of a moving cavity. 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
(
8

/

3
)
E

/

c
2

{\displaystyle m=(8/3)E/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
(
4

/

3
)
E

/

c
2

{\displaystyle m=(4/3)E/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. 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

### 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 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

F
→

=

d

p
→

/

d

t

{\displaystyle \vec{F} = \mathrm{d}\vec{p}/\mathrm{d}t}

, was used as relativistic mass

# M

m

0

1
−

v

2

c

2

,

m

0

=

E

c
2

,

{\displaystyle M=\frac{m\_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},\qquad m\_{0}=\frac{E}{c^2},}

This concept is sometimes still used in modern physics textbooks, although the term 'mass' is now considered by many to refer to invariant mass, see mass in special relativity.

### 4/3 problem

Max von Laue in 1911 also used the Abraham–Lorentz equations of motion in his development of special relativistic dynamics, so that also in special relativity the 4/3-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

e
m

=

E

e
m

/

c
2

{\displaystyle m\_{em}=E\_{em}/c^2}

 without the 4/3 factor, or in other words, four-momentum doesn't properly transform like a four-vector when the 4/3 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's spacetime 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 4/3 factor arises only with respect to electromagnetic mass, while the closed system has total rest mass and energy of

m

t
o
t

=

E

t
o
t

/

c
2

{\displaystyle m\_{tot}=E\_{tot}/c^2}

.

Another solution was found by authors such as Enrico Fermi (1922), Paul Dirac (1938) Fritz Rohrlich (1960), or Julian Schwinger (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

e
m

=

E

e
m

/

c
2

{\displaystyle m\_{em}=E\_{em}/c^2}

 and thus the 4/3 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 1/3 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.

The 4/3 problem for electromagnetic field becomes clearer when the generalized Poynting theorem is used in a physical system for all active fields.
In this case, it is shown that the cause of the 4/3 problem is the difference between the four-vector and four-tensor.

* Indeed, the energy and momentum of the system form a four-momentum.
* However, the energy and momentum densities of electromagnetic field are temporary components of the stress-energy tensor and do not form a four-vector. The same applies to integrals over the volume of these components.