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Equivalence Between Self-energy and Self-mass in Classical Electron Model

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A cornerstone of physics, Maxwell‘s theory of electromagnetism, apparently contains a fatal flaw. The standard expressions for the electromagnetic field energy and the self-mass of an electron of finite extension do not obey Einstein‘s famous equation, \(E=mc^2\), but instead fulfill this relation with a factor 4/3 on the left-hand side. Furthermore, the energy and momentum of the electromagnetic field associated with the charge fail to transform as a four-vector. Many famous physicists have contributed to the debate of this so-called 4/3-problem without arriving at a complete solution. It has generally been assumed that, as originally suggested by Poincaré, the problems are connected to the question of stability of the charge distribution, and that relativistic equivalence between energy and self-mass can only be restored by inclusion of stabilizing forces. Alternative solutions to the problems have also been proposed. Nearly a century ago Fermi suggested a covariant definition of the electromagnetic energy and momentum, and sixty years later Kalckar et al. argued that the 4/3 problem is caused by omission of a relativistic correction in the standard evaluation of the self-force from Coulomb self-interaction. However, the relation between these suggestions has not been clear. We show that the relativistic correction implies that the mechanical momentum of an accelerated rigid body must be defined as the sum of the momenta of its parts for fixed time in the momentary rest frame of the body. For the total momentum of particles and field to be conserved, the total energy–momentum tensor must be divergence free, and this then requires that the momentum of the associated electromagnetic field be defined in the same way, consistent with the suggestion by Fermi. This comprehensive solution of the 4/3-problem demonstrates that there is no conflict of Maxwell‘s theory with special relativity and the questions of equivalence of electromagnetic energy and self-mass and of stability of a classical charge distribution are independent. In appendices we discuss the relations of our treatment with Fermi‘s seminal paper and with a classic paper by Dirac where he evaluated the damping self-force on a point electron from transport of energy and momentum in the electromagnetic field.

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Appendix A: Expansion of Electromagnetic Fields Near Accelerated Charge

The retarded electromagnetic field at a space-time point \((\mathbf{r}, t)\), produced by a point charge q carrying out an assigned motion \(\mathbf{r}_0 (t')\), is determined by the state of motion of the charge at an earlier time \(t'\) (see [4], formulas (14.13) and (14.14), or [22] formula (63.8)]),

$$\begin{aligned} \mathbf{E}_{ret} (\mathbf{r}, t)= & {} \frac{q}{R'^2} \frac{(1-\beta '^2) (\mathbf{n}' - {\pmb \beta }')}{(1-\mathbf{n}' \!\! \cdot \! {\pmb \beta }')^3} + \nonumber \\&+ \frac{q}{c R'} \frac{\mathbf{n}' \times [ (\mathbf{n}' - {\pmb \beta }') \times {\dot{{\pmb \beta }}}']}{(1-\mathbf{n}' \!\! \cdot \! {\pmb \beta }')^3}, \end{aligned}$$
$$\begin{aligned} \mathbf{H}_{ret} (\mathbf{r}, t)= & {} \mathbf{n}' \times \mathbf{E}_{ret} (\mathbf{r}, t), \end{aligned}$$

where \({\pmb \beta }' \equiv {\pmb \beta }(t') = \mathbf{v}(t')/c\) is the velocity of the charge (relative to the speed of light) and \({\dot{{\pmb \beta }}}'=d {\pmb \beta }' /dt'\) is the acceleration (divided by c), \(\mathbf{n}' \equiv \mathbf{n} (t')\) is a unit vector in the direction towards the observation point, \(\mathbf {r}\), from the electron position at time \(t'\) (i.e., in the direction of \(\mathbf{r} - \mathbf{r}_0 (t')\) ) and \(R' \equiv R (t')\) = \(\mid \! \mathbf{R} (t') \! \mid \equiv \mid \! \mathbf{r} - \mathbf{r}_0 (t') \! \mid \). The primed quantities refer to the time \(t'\) defined as

$$\begin{aligned} t' = t - \frac{1}{c} \mid \mathbf{r} - \mathbf{r}_0 (t') \mid . \end{aligned}$$

The expression for the advanced field, \(\mathbf{E}_{adv}\), can be obtained from Eq. (A1) by a change of variables: \({\pmb \beta }' \rightarrow - {\pmb \beta }'\) and modification of Eq. (A3) to \(t' = t + R'/c\). After the following expansions, leading to formulas expressed in variables related to the electron motion at time t, the corresponding formulas for the advanced field are obtained simply by a change of sign of the velocity and its second derivative.

Fig. 2
figure 2

Illustration of the geometry for calculation of the retarded field in the vicinity of an electron

We are interested in the retarded electric field (A1) in the neighborhood of a moving charge q. The notation for the calculation of the field at time t at a point P with coordinate vector \(\mathbf {r}\) is illustrated in Fig. 2. Our aim is to express the field as a function of the radius vector of the point relative to the position of the charge at the same time t, \( \mathbf{R} (t) = \mathbf{r} - \mathbf{r}_0 (t) \equiv \mathbf{n} \varepsilon \), the acceleration, \(c {\dot{{\pmb \beta }}} (t)\), and the derivative of the acceleration, \(c \ddot{{\pmb \beta }} (t)\). For simplicity, we perform the calculations in the rest frame of the charge at time t,

$$\begin{aligned} {\pmb \beta }(t) =0. \end{aligned}$$

We consider a variation of the distance \(\varepsilon \) of the point P from the charge for fixed \(\mathbf{n}\). The position of the charge at time t is fixed but the position at the earlier time \(t' =t - R'/c \) is a function of \(\varepsilon \) through the delay \(\tau \equiv R' /c\). We want to expand the field (A1) in the parameter \(\varepsilon \). The vector \(\mathbf{R}(t) - \mathbf{R} (t') = \mathbf{r}_0 (t') - \mathbf{r}_0 (t) \) depends on \(\varepsilon \) only through the parameter \(\tau \) and we may therefore first expand this vector in \(\tau \) and we obtain


To convert this expression into an expansion of \(R' = \mid \! \mathbf{R}(t') \! \mid \) in \(\varepsilon \) we note that since \(\beta = 0\) also the dependence on \(\varepsilon \) of the vector \(\mathbf{R}(t) - \mathbf{R} (t') \) must be of second and higher order. Hence the series expansion of \(R'\) has the form

$$\begin{aligned} R' \approx \varepsilon + a_1 \varepsilon ^2 + a_2 \varepsilon ^3. \end{aligned}$$

Inserting this into Eq. (A5) and keeping terms of up to third order in \(\varepsilon \) in the norm of the vector \(\mathbf{R} (t')\) we obtain for the coefficients \(a_1\) and \(a_2\) in Eq. (A6)

$$\begin{aligned} a_1= & {} - \frac{1}{2c} \ ({\dot{{\pmb \beta }}} \mathbf{n}) \end{aligned}$$
$$\begin{aligned} a_2= & {} \frac{3}{8c^2} \ ({\dot{{\pmb \beta }}} \mathbf{n})^2 + \frac{1}{6c^2} \ (\ddot{{\pmb \beta }} \mathbf{n}) + \frac{1}{8c^2} \ |{\dot{{\pmb \beta }}}|^2, \end{aligned}$$

where all quantities on the right hand side are taken at the time t. Also the velocity \(c {\pmb \beta }(t')\) is a function of \(\varepsilon \) only through \(\tau \) and may first be expanded in this parameter. To second order in \(\varepsilon \) this leads to

$$\begin{aligned} {\pmb \beta }(t') \approx - \frac{\varepsilon }{c} {\dot{{\pmb \beta }}} + \frac{\varepsilon ^2}{2c^2} \left( (\mathbf{n} {\dot{{\pmb \beta }}}) {\dot{{\pmb \beta }}} + \ddot{{\pmb \beta }} \right) . \end{aligned}$$

For the derivative, we need only include the first-order term \({\dot{{\pmb \beta }}}(t')\approx {\dot{{\pmb \beta }}} - (\varepsilon /c) \ddot{{\pmb \beta }} \).

The expansion of the unit vector \(\mathbf{n} (t')\) may be obtained from the ratio of the expressions in Eqs. (A5) and (A6), and to second order we obtain

$$\begin{aligned} \mathbf{n} (t')\approx & {} \mathbf{n} + \frac{\varepsilon }{2c} \left( (\mathbf{n} {\dot{{\pmb \beta }}}) \mathbf{n} - {\dot{{\pmb \beta }}} \right) \nonumber \\&- \frac{\varepsilon ^2}{2c^2} \left( \frac{1}{4} (\mathbf{n} {\dot{{\pmb \beta }}})^2 \mathbf{n} + \frac{1}{3} (\mathbf{n} \ddot{{\pmb \beta }}) \mathbf{n} + \frac{1}{4} |{\dot{{\pmb \beta }}}|^2 \mathbf{n} - \frac{1}{2} (\mathbf{n} {\dot{{\pmb \beta }}}) {\dot{{\pmb \beta }}} - \frac{1}{3} \ddot{{\pmb \beta }} \right) . \end{aligned}$$

Inserting these expansions into Eq. (A1) we obtain

$$\begin{aligned} \mathbf{E}_{ret}\approx & {} \frac{q}{\varepsilon ^2 } \, \mathbf{n} - \frac{q}{2c\varepsilon } \left[ \mathbf{n} (\mathbf{n} {\dot{{\pmb \beta }}}) + {\dot{{\pmb \beta }}} \right] \nonumber \\&+ \frac{q}{c^2} \left[ \, \frac{3}{8} \, (\mathbf{n} {\dot{{\pmb \beta }}})^2 \mathbf{n} + \frac{3}{4} \, (\mathbf{n} {\dot{{\pmb \beta }}}) {\dot{{\pmb \beta }}} - \frac{3}{8} \, |{\dot{{\pmb \beta }}}|^2 \mathbf{n} + \frac{2}{3} \, \ddot{{\pmb \beta }} \, \right] , \end{aligned}$$

and from insertion of Eqs. (A10) and (A11) into Eq. (A2),

$$\begin{aligned} \mathbf{H}_{ret} \approx \frac{q}{2c^2 } \, \mathbf{n} \times \ddot{{\pmb \beta }}. \end{aligned}$$

Expansions of this type were first performed by Page (see formulas (21)–(24) in [32]). Dirac did the same calculations in covariant form [2] and for \(\beta =0\) formula (A11) is the same as the expression (60) in [2]. Heitler also considered the expansion (A11), retaining terms of even order in \(\mathbf {n}\) only (see Eq. (14) in [25], §4]). The first two terms in Eq. (A11) were applied in [18] and characterized as a first-order expansion in the acceleration.

Appendix B: Equivalence from Flux of Field Momentum

To prove complete consistency of the two methods for calculation of the electromagnetic mass, from the internal forces between charge elements and from the flux of field momentum, we need to show that the result in Eq. (36) also holds for \(\varepsilon \rightarrow R\) and hence agrees with Eq. (14) in this limit. This is a little more complicated because we must now distinguish between the unit vector \(\mathbf{n}\) in the direction from a charge element dq to a point on the surface and the surface normal \(\mathbf{n}_{s}\) (see Fig. 3). The momentum flux may be written as a double integral over charges \(dq_1\) and \( dq_2\) at distances \(\varepsilon _1\) and \(\varepsilon _2\) from a point on the sphere and with unit vectors \(\mathbf{n}_1\) and \(\mathbf{n}_2\) towards this point. For simplicity we include here only the terms in Eq. (6) proportional to \(\varepsilon ^{-2}\) and \(\varepsilon ^{-1}\) which determine the reactive self-force and hence the electromagnetic electron mass. Using the symmetry between \(dq_1\) and \(dq_2\) we obtain

$$\begin{aligned} \mathbf{k}_s\approx & {} \frac{1}{4\pi } \int \!\! \! \int dq_1 dq_2 \left\{ \frac{ ( \mathbf{n}_2 \mathbf{n}_s) \mathbf{n}_1 }{\varepsilon _1^2 \varepsilon _2^2} - \frac{ \mathbf{n}_1 }{2c \varepsilon _1^2 \varepsilon _2} [( \mathbf{n}_2 {\dot{{\pmb \beta }}})(\mathbf{n}_2 { \mathbf n}_s) \right. \nonumber \\&+ (\mathbf{n}_s {\dot{{\pmb \beta }}})] - \frac{ (\mathbf{n}_2 \mathbf{n}_s) }{2c \varepsilon _2^2 \varepsilon _1} [( \mathbf{n}_1 {\dot{{\pmb \beta }}}) \mathbf{n}_1 + {\dot{{\pmb \beta }}}] - \frac{ ( \mathbf{n}_1 { \mathbf n}_2 ) \mathbf{n}_s }{2 \varepsilon _1^2 \varepsilon _2^2} \nonumber \\&\left. + \frac{\mathbf{n}_s }{2c \varepsilon _1^2 \varepsilon _2} [(\mathbf{n}_2 {\dot{{\pmb \beta }}})( \mathbf{n}_1 \mathbf{n}_2) \!+\! (\mathbf{n}_1 {\dot{{\pmb \beta }}})] \right\} \! \!. \end{aligned}$$
Fig. 3
figure 3

Geometry for calculation of the momentum flux into the sphere S with radius \(\varepsilon \). The electric field is generated by a uniformly distributed charge q on a spherical surface with radius \(R<\varepsilon \)

This is the momentum flux into the sphere which should be integrated over the surface as in Eq. (33). For the two terms proportional to \((\varepsilon _1^2\varepsilon _2^2)^{-1}\) this gives zero. Consider then the second and last terms, both with the pre-factor \((2c\varepsilon _1^2\varepsilon _2)^{-1}\) and with the factors

$$\begin{aligned} - \mathbf{n}_1 [(\mathbf{n}_2 {\dot{{\pmb \beta }}})(\mathbf{n}_2 \mathbf{n}_s ) + (\mathbf{n}_s {\dot{{\pmb \beta }}})] + \mathbf{n}_s [(\mathbf{n}_2 {\dot{{\pmb \beta }}})( \mathbf{n}_1 \mathbf{n}_2) + (\mathbf{n}_1 {\dot{{\pmb \beta }}})]. \end{aligned}$$

We rearrange to

$$\begin{aligned} (\mathbf{n}_2 {\dot{{\pmb \beta }}}) [(\mathbf{n}_1 { \mathbf n}_2) \mathbf{n}_s - (\mathbf{n}_2 \mathbf{n}_s) \mathbf{n}_1] + [ \mathbf{n}_s ( \mathbf{n}_1 {\dot{{\pmb \beta }}}) - \mathbf{n}_1 (\mathbf{n}_s {\dot{{\pmb \beta }}})]. \end{aligned}$$

In the expression (B1) the distances \(\varepsilon _1\) and \(\varepsilon _2\) are fixed for fixed values of \((\mathbf{n}_1 { \mathbf n}_s)\) and \((\mathbf{n}_2 \mathbf{n}_s)\). We keep the position of \(dq_2\) fixed but average over the position of \(dq_1\) on a circle around \(\mathbf{n}_s\), i.e., for fixed \( (\mathbf{n}_1 \mathbf{n}_s )\). This leads to \( \mathbf{n}_1 \rightarrow (\mathbf{n}_1 \mathbf{n}_s ) \mathbf{n}_s \). Introducing this replacement into the two expressions above we see that they both become equal to zero. This leaves

$$\begin{aligned} \mathbf{k}_s \approx \frac{1}{4\pi } \int \!\! \! \int dq_1 dq_2 \frac{ - ( \mathbf{n}_2 \mathbf{n}_s) }{2c \varepsilon _2^2 \varepsilon _1} [( \mathbf{n}_1 {\dot{{\pmb \beta }}}) \mathbf{n}_1 + {\dot{{\pmb \beta }}}]. \end{aligned}$$

First integrate over \( dq_2\). Only the Coulomb part of the field has survived and from electrostatics we know that the Coulomb field from a uniformly charged spherical shell is the same outside the shell as from the total charge at the center of the sphere. (The radius \(\varepsilon \) must remain infinitesimally larger than R, \(\varepsilon \rightarrow R_{+}\). Within the charged surface the field is only half as large). So the integral becomes

$$\begin{aligned} \mathbf{k}_s \approx \frac{-q}{4\pi \varepsilon ^2} \int dq_1 \frac{ 1 }{2c \varepsilon _1} [( \mathbf{n}_1 {\dot{{\pmb \beta }}}) \mathbf{n}_1 + {\dot{{\pmb \beta }}}]. \end{aligned}$$

This momentum flux should then be integrated over the sphere with radius \(\varepsilon \),

$$\begin{aligned} \mathbf{K}_{s} (\varepsilon )= & {} \varepsilon ^2 \!\! \int \! \!\! d\Omega _s \mathbf{k}_s \approx \! \\&- \frac{q^2}{8\pi c} \int \!\! \frac{d\Omega _s}{4\pi } \int _0^{2\pi } \! \! d\varphi \! \! \int _0^\pi \! \! d\theta \sin \theta \frac{1}{\varepsilon _1} \! \left[ (\mathbf{n}_1 {\dot{{\pmb \beta }}}) \mathbf{n}_1 \! + {\dot{{\pmb \beta }}} \right] , \end{aligned}$$

where the angles \(\theta \), \(\varphi \) define the direction towards \(dq_1\) from the center of the sphere relative to the direction of \(\mathbf{n}_s\). We may perform the integration over solid angles first. For fixed values of \(\theta \) and \(\varphi \) the distance \(\varepsilon _1\) is constant. The directions \(\mathbf{n}_s\) and \(\mathbf{n}_1\) rotate together, covering the \(4\pi \) solid angle, so the average over \(\mathbf{n}_s\) corresponds to an average over \(\mathbf{n}_1\). According to Eq. (13) we therefore obtain

$$\begin{aligned} \mathbf{K}_{s} (\varepsilon )\approx & {} \frac{-q^2}{8\pi c} \!\! \int _0^{2\pi } \!\! d \varphi \int _0^\pi \!\! \frac{d \theta \sin \theta }{ \sqrt{ (\varepsilon ^2 +R^2 - 2 \varepsilon R \cos \theta ) } } \, \frac{4}{3} {\dot{{\pmb \beta }}}= - \frac{4}{3} \frac{q^2}{2c \varepsilon } {\dot{{\pmb \beta }}}. \end{aligned}$$

For \(\varepsilon = R \) this result is identical to the reactive self-force in Eq. (3).

In analogy to Eq. (37), we must introduce the KLU-correction in the integral over forces, now with the expression (B1) for the momentum flux. Again the relativistic correction factor is only important for the two Coulomb terms. We can apply the same trick as before and make the replacements \(\mathbf{n}_1 \rightarrow (\mathbf{n}_1 \mathbf{n}_s) \mathbf{n}_s\) and then the two terms can be combined. Since the scalar multiplication of \(\mathbf{n}_1\) (or \(\mathbf{n}_2\)) by \(\mathbf{n}_s\) can be carried out after the integration over \(dq_1\) (or \(dq_2\)) we can use the fact that the field from the charged shell is the same as that from the total charge placed at the center, and we once again obtain the correction in Eq. (37).

Appendix C: Fermi Solution of 4/3-Paradox

Around 1920 Enrico Fermi wrote several papers related to the problems encountered in calculations of the electromagnetic mass of an electron [19]. However, they have remained relatively unknown to most of the physics community, probably because the papers were published in an Italian journal. The concluding paper was also published in German and it has recently become accessible on the Internet, translated into English [19]. In the words of Jantzen and Ruffini [30], though often quoted, it has rarely been appreciated nor understood for its actual content. These authors give a detailed account of Fermi’s work but like [18] their paper is published in a journal with a limited readership, and both papers have received very few citations. We shall here give a brief account of Fermi’s approach to the problem. As seen below, there are both similarities and interesting differences to the treatments we have discussed.

There is no doubt that Fermi’s view of the source of the problem was very similar to that expressed in the later paper by Kalckar, Lindhard and Ulfbeck [18], as demonstrated by the following quotes from the introduction. After introducing the two conflicting values or the electromagnetic mass, with and without the factor 4/3, Fermi writes: “Especially we will prove: The difference between the two values stems from the fact, that in ordinary electrodynamic theory of electromagnetic mass (though not explicitly) a relativistically forbidden concept of rigid bodies is applied. Contrary to that, the relativistically most natural and most appropriate concept of rigid bodies leads to the value \(U/c^2\) for the electromagnetic mass.” And further below: “In this paper, HAMILTON’s principle will serve as a basis, being most useful for the treatment of a problem subjected to very complicated conditions of a different nature than those considered in ordinary mechanics, because our system must contract in the direction of motion according to relativity theory. However, we notice that although this contraction is of order of magnitude \(v^2/c^2\), it changes the most important terms of electromagnetic mass, i.e, the rest mass.”

Fermi’s paper is not easy to read and understand, partly because he uses a description of relativistic kinematics with an imaginary time axis (and there are a number of confusing misprints). The Lorentz transformation between reference frames in relative motion can then formally be described as a simple rotation of the 4-dimensional coordinate system, with a complex angle of rotation. However, we shall keep the notation applied in the main part of this paper.

The electromagnetic self-force can be derived from the principle of least action [22]. Fermi distinguishes between two cases, A and B. In case A we disregard the relativistic effects and consider the time t to be a common parameter for all elements of the rigid body. The part of the action responsible for the interaction of the charges with the electromagnetic field is then

$$\begin{aligned} S_{int} = - \frac{1}{c^2} \int A_\nu j^\nu d^4 x = - \frac{1}{c} \int dq \int _{t_1}^{t_2} A_\nu \frac{dx^\nu }{dt}dt, \end{aligned}$$

where \(A^\nu =(\varphi , \mathbf{A})\) is the four-potential and \(j^\nu =(c\rho , \mathbf{j})\) the four-current, with \(\nu = 0,1,2,3\). The differential is \( d^4 x = cdtdV\), where dV is a differential spatial volume. In the last expression \(x^\nu (t)\) is the world line of the charge element dq, \(x^\nu =(ct, \mathbf{r})\), and \(A_\nu \) is the four-potential at this line.

According to the variational principle, the action should remain stationary for variations of the motion. In this connection, the definition of the integration region in Eq. (C1) is important. For a point particle, the initial and final coordinates are to be kept fixed and this constrains the variations of the world line. Similarly, we must require that the charge elements dq in Eq. (C1) have fixed coordinates at the limits of integration over t in the variation of the action integral,

$$\begin{aligned} \delta S_{int} = - \frac{1}{c} \int dq \int _{t_1}^{t_2} \left( \frac{ \partial A_\nu }{\partial x^\mu } \delta x^\mu \frac{dx^\nu }{dt} + A_\nu \frac{d}{dt} \delta x^\nu \right) dt , \end{aligned}$$

where \(\delta x^0 =0\) while \(\delta x^k\) for \(k=1-3\) are arbitrary functions of t except for the condition that they vanish at the limits of integration. The last term in Eq. (C2) can be integrated by parts,

$$\begin{aligned} \int _{t_1}^{t_2} dt A_\nu \frac{d}{dt} \delta x^\nu = - \int _{t_1}^{t_2} dt \frac{d A_\nu }{dt}\delta x^\nu =- \int _{t_1}^{t_2} dt \frac{ \partial A_\nu }{\partial x^\mu } \frac{d x^\mu }{dt} \delta x^\nu . \end{aligned}$$

Switching the symbols \(\mu \) and \(\nu \) in this last term we then obtain

$$\begin{aligned} \delta S_{int}= & {} - \frac{1}{c} \int dq \int _{t_1}^{t_2} dt \left( \frac{ \partial A_\nu }{\partial x^\mu } \delta x^\mu \frac{dx^\nu }{dt} - \frac{ \partial A_\mu }{\partial x^\nu } \frac{dx^\nu }{dt} \delta x^\mu \right) \nonumber \\= & {} \frac{1}{c} \int dq \int _{t_1}^{t_2} dt F_{\nu \mu } \frac{dx^\nu }{dt} \delta x^\mu = 0, \end{aligned}$$

where \(F_{\nu \mu }\) is the electromagnetic field tensor defined in Eq. (38). When the mechanical action for the particle is included in the variation (C2), the expression (C3) provides the Lorentz force in the equation of motion for the particle. Here we assume the mechanical mass to be zero and consider instead the balance between the force from an external field \( \mathbf{E}^{(e)}\) and that from the internal field \( \mathbf{E}^{(i)}\) given by Eq. (6) (first two terms). In the rest frame all the velocities vanish and only the term with \(\nu = 0\) remains in Eq. (C3). The four-vector \(F_{0\mu }\) is given by \((0,\mathbf {E})\) and the variational principle leads to the relation

$$\begin{aligned} \int \mathbf{E} dq = \int \left( \mathbf{E}^{(e)} + \mathbf{E}^{(i)} \right) dq =0. \end{aligned}$$

Fermi notes that “we would have arrived at these equations without further ado, when we (as it ordinarily happens in the derivation of the electromagnetic mass and as it was essentially done by M. Born as well) would have assumed from the outset, that the total force on the system is equal to zero. However, we have derived Eq. (C4) from HAMILTON’s principle, to demonstrate the source of the error” [19].

As we have seen in Sect. 2, evaluation of the internal contribution leads to the 4/3 coefficient in the self-force,

$$\begin{aligned} \int \mathbf{E}^{(i)} dq = - \frac{4}{3} \frac{U}{c^2} \mathbf{g}. \end{aligned}$$

Here U is the electromagnetic self-energy,

$$\begin{aligned} U= \int \!\!\! \int \frac{dqdq'}{2\varepsilon } , \end{aligned}$$

where \(\varepsilon \) is the distance between the two charge elements. The external force is obtained as

$$\begin{aligned} \mathbf{K} = \int \mathbf{E}^{(e)} dq , \end{aligned}$$

and we obtain from Eqs. (C4) and (C5) the force

$$\begin{aligned} \mathbf{K} = \frac{4}{3} \frac{U}{c^2} \mathbf{g}. \end{aligned}$$

Fermi concludes that comparison of this equation with the basic law of point dynamics, \(\mathbf{K}=m \mathbf{g}\), finally gives us

$$\begin{aligned} m= \frac{4}{3} \frac{U}{c^2}. \end{aligned}$$

Fermi then argues that this procedure cannot be correct. Instead we must introduce the coordinates and the field in the momentary rest frames, i.e., in 3-dimensional planes perpendicular to the four-velocity, and specify the integration region as a section between two such planes of the world tube traveled by the body. In Eq. (C1), both the product of the two four-vectors and the differential are invariant under Lorentz transformation. We may imagine the integration region split into differential slices between two such planes. The width of the slices is given by the differential time dt. From geometrical considerations of rigid acceleration of a body momentarily at rest, Fermi derived a relation corresponding to Eq. (19) and (D9) for \(\gamma =1\),

$$\begin{aligned} dt = \left( 1 + \mathbf{g \cdot R}/{c^2} \right) d t_0 , \end{aligned}$$

where dt and \(dt_0\) are the incremental times at \(\mathbf {r}\) and at a reference point on the body, \(\mathbf{r}_0\), respectively, \(\mathbf {g}\) is the acceleration at \(\mathbf{r}_0\), and \( \mathbf{R} = \mathbf{r} - \mathbf{r}_0\). The expression for the action integral then becomes

$$\begin{aligned} S_{int} = - \int dq \int \varphi \left( 1 + \mathbf{g \cdot R}/{c^2} \right) d \tau _0 , \end{aligned}$$

where \(\tau _0\) is the proper time for the reference point \(\mathbf{r}_0\) on the body.

This defines his case B. In his own words: “Now it can be immediately seen, that variation A is in contradiction with relativity theory, because it has no invariant characteristics against the world transformation, and is based on the arbitrary space xyz. On the other hand, variation B has the desired invariant characteristics, and is always based on the proper space, i.e., the space perpendicular to the world tube. Thus it is without doubt to be preferred before the previous one.”

Instead of Eq. (C3) we then obtain

$$\begin{aligned} \delta S_{int}= & {} - \int dq \int \delta \mathbf{r} \cdot \nabla \varphi \, \left( 1 + \mathbf{g \cdot R}/{c^2} \right) d \tau _0 \nonumber \\= & {} \int dq \int \delta \mathbf{r} \cdot \mathbf{E} \, \left( 1 + \mathbf{g \cdot R}/{c^2} \right) d \tau _0 = 0. \end{aligned}$$

Since the displacement \(\delta \mathbf{r}\) is arbitrary, this leads to the relation in Eq. (C4) with the additional factor in parenthesis. This factor can be neglected for the external field if we choose the reference point \(\mathbf{r}_0\) as the center of charge (it is a very small correction in any case) and we obtain

$$\begin{aligned} \mathbf{K} = \frac{4}{3} \frac{U}{c^2} \, \mathbf{g} - \frac{1}{c^2} \int \mathbf{E}^{(i)} (\mathbf{g R}) dq. \end{aligned}$$

Let us consider the last term in Eq. (C13). It is only important for the leading Coulomb term in \(\mathbf{E}^{(i)}\) and we obtain

$$\begin{aligned} - \frac{1}{c^2} \int \!\!\! \int \frac{\mathbf{r} - \mathbf{r}'}{|\mathbf{r} - \mathbf{r}'|^3} \, \, \mathbf{g} \! \cdot \! (\mathbf{r} - \mathbf{r}_0) dqdq'. \end{aligned}$$

Switching notation, \( \mathbf{r} \longleftrightarrow \mathbf{r}'\), we obtain the same expression with the last parenthesis replaced by \((\mathbf{r}_0 - \mathbf {r}')\). Averaging the two expressions we eliminate the reference coordinates and obtain for a spherically symmetric charge distribution

$$\begin{aligned} - \frac{1}{2 c^2} \int \!\!\! \int \frac{\mathbf{r} - \mathbf{r}'}{|\mathbf{r} - \mathbf{r}'|^3} \, \, \mathbf{g} \! \cdot \! (\mathbf{r} - \mathbf{r}') dqdq' = - \frac{1}{3} \frac{U}{c^2} \, \mathbf{g}. \end{aligned}$$

With this expression for the last term in Eq. (C13) we obtain the proper relativistic equivalence between electromagnetic mass and energy.

Did Fermi’s 1922-paper then present a satisfactory solution of the 4/3-problem, which was overlooked, not appreciated, or forgotten? In our view it fell well short of this. Fermi’s argument for case B to be preferred did not identify the key problem with the calculation in case A. This calculation is not as claimed in contradiction with relativity theory and there is nothing wrong with the result in Eq. (C4), except that it is not directly relevant to the description of an accelerated rigid body. It expresses the condition for conserved total momentum as a function of laboratory time. However, as pointed out in [18], the electromagnetic mass of the rigid body is not determined by the sum of forces in Eq. (C7) through the analogy to point dynamics, leading to Eq. (C9), but by the forces on the individual parts of the body through the relation in Eq. (23). As expressed in Eq. (26), this corresponds to a definition of the total momentum of the body as the sum of the momenta of its parts for fixed time in the momentary rest frame.

This relation is obtained more directly in case B because here the formulation of the variational calculation is consistent with the relativistic concept of rigid body motion. The origin of Fermi’s basic formula (C10) for time differentials at different positions is the Eq. (4) in [33], which links the proper time intervals in the small spatial region in the vicinity of the world line in Riemannian space. In that paper Fermi also introduced the so-called ‘Fermi coordinates’ applied here in case B (see §10, Chap. 2 in [34]).

Fermi’s paper was not cited in [18] but the authors’ view on this and later related papers is indicated by comments to a list of references in a note for a lecture series on ‘Surprises in Theoretical Physics’, given by Jens Lindhard at Aarhus University in 1988 [35]: “E. Fermi [19] started out from general relativity and suggested a covariant definition of self-mass, whereby \(4/3\rightarrow 1\). His suggestion was forgotten, even by himself. Similar attempts were tried by Wilson [20] and Kwal [21]. Rohrlich took it up again in his textbook [17], using a formal definition of a covariant classical electron. Dirac [2] formulated a classical theory of the electron, where he sidestepped the problem.”

Appendix D: Solution for Point Electron with Born–Dirac Tube

Following Dirac [2], let us surround the singular point-electron world line in space-time with a thin tube with constant radius \(\varepsilon \) in the electron’s rest frame for any instant of the electron time-coordinate \(\tau \) in the laboratory frame. The self-force is then to be calculated from the transport of momentum across the surface of this tube. Consider the four-vector \(\eta ^\mu = x^\mu - z^\mu (s)\), where \(z^\mu (s) = (c\tau , \mathbf{r}_0 (\tau ))\) is the world line of the electron and \(x^\mu \) is some point in the vicinity of this world line. The parameter s is the proper time multiplied by the velocity of light \(ds = cd\tau /\gamma \), where \(\gamma (\tau ) = (1 - \beta ^2)^{-1/2}\) is the Lorentz factor and \(c {\pmb \beta }= d \mathbf{r}_0 /d\tau \). We assume that the points \( x^\mu \) are such that the vector \(\eta ^\mu \) is perpendicular to the four-velocity of the electron, \(u^\mu = dz^\mu /ds \), and the surface of the Born–Dirac tube is then defined by two equations [2, 26]

$$\begin{aligned} \eta ^\mu \eta _\mu= & {} - \varepsilon ^2, \end{aligned}$$
$$\begin{aligned} \eta ^\mu u_\mu= & {} 0. \end{aligned}$$

These equations define a 2-dimensional structure (a sphere) in 4-dimensional space for fixed value of s. When the electron moves, this structure forms a 3-dimensional surface f of a tube. Equation (D2) defines a 3-dimensional plane which is perpendicular to the four-velocity and intersects the four-sphere defined by Eq. (D1). An analogue of the tube in three dimensions is illustrated in Fig. 4.

Following again Dirac, let us make a variation of the point \(x^\mu \) on the surface f to the point \(x^\mu + dx^\mu \), also on this surface. Let us suppose that this point is on the 3-dimensional plane corresponding to \( s+ds\). Differentiating the Eqs. (D1) and (D2) we obtain

$$\begin{aligned} (x^\mu - z^\mu )(dx_\mu - u_\mu ds )= & {} 0, \end{aligned}$$
$$\begin{aligned} (dx^\mu - u^\mu ds ) u_\mu +(x^\mu - z^\mu ) \frac{d}{ds} u_\mu ds= & {} 0. \end{aligned}$$

Using Eq. (D2) and the relation \(u^\mu u_\mu = 1\) we obtain from these equations the following relations

$$\begin{aligned} \eta ^{\mu } dx_{\mu }= & {} 0, \end{aligned}$$
$$\begin{aligned} u^{\mu } dx_{\mu }= & {} \left( 1-\eta ^{\mu } \frac{d}{ds} u_\mu \right) ds. \end{aligned}$$

Let us split up the four-space variation on the tube surface f into a part, \(dx^\mu _\perp \), orthogonal to the four-velocity \(u^\mu \) and a part, \(dx^\mu _\parallel \), parallel to \(u^\mu \). The latter can be written as \(dx_\parallel ^\mu =cdt(1,{\pmb \beta }(\tau ))\), i.e., the velocity is the same as that of the electron but the laboratory times t and \(\tau \) are different, as we also found from the analysis in Sect. 3. These differentials can be visualized in the 3-dimensional analogue in Fig. 4. Here the surface f is two-dimensional and \(dx^\mu \) can be split into a component along the circle, which is an intersection of the tube surface with the \(x'y'\)-plane, and a component parallel to the \(t'\) -axis, i.e., parallel to the electron three-velocity in the laboratory frame at the time \(\tau \).

Fig. 4
figure 4

Minkowski diagram of a 3-dimensional analogue of the Born–Dirac tube around the world line of an electron (dashed red line) accelerated in the x-direction. Here \(\tau \) is the time coordinate of the electron in the laboratory frame (txy) where it is at rest for \(\tau =0\), \(\eta ^2 \equiv \eta ^i\eta _i\) and \(\eta '^2 \equiv \eta '^i\eta '_i\). The coordinate system \((t',x',y')\) corresponds to the rest frame at a later time \(\tau \). The two circles with radius \(\varepsilon \) in the (xy) and \((x',y')\) planes indicate the cuts of the tube surface with these planes and \(\eta ^i\) with \(i = 1-3\) are the laboratory coordinates of a radius vector in one of the two circles (Color figure online)

Let us find the connection between the two times. According to Eq. (D2) the 4-plane intersecting the world-tube is defined by

$$\begin{aligned} c(t-\tau ) = {\pmb \beta }\cdot (\mathbf{r} - \mathbf{r}_0 (\tau )). \end{aligned}$$

The time variations of this equation gives,

$$\begin{aligned} c(dt - d\tau ) = ( \mathbf{R} {\dot{{\pmb \beta }}} )d\tau + {\pmb \beta }\cdot (d\mathbf{r} - c {\pmb \beta }d\tau ), \end{aligned}$$

where \(\mathbf{R} = \mathbf{r} - \mathbf{r}_0 (\tau )\). We found above that if we choose \(dx^\mu \) to be parallel to the electron velocity then \(d\mathbf{r} = c{\pmb \beta }dt\). Insertion of this into Eq. (D8) leads to

$$\begin{aligned} dt = d\tau \left( 1+ \frac{1}{c} \gamma ^2 ( \mathbf{R} {\dot{{\pmb \beta }}} ) \right) , \end{aligned}$$

which agrees with Eq. (19).

The 3-dimensional surface element of the tube is equal to \(d^3 f = |dx^\mu _\parallel | dS\), where dS is a surface element of the sphere defined in Eqs. (D1) and (D2), and using the relation (D6) we find (see also the expression (66) in [2])

$$\begin{aligned} d^3 f = \left( 1 - \eta ^\mu \frac{du_\mu }{ds} \right) ds dS. \end{aligned}$$

For a calculation of the momentum transport in the rest frame this reduces to

$$\begin{aligned} d^3 f = \left( 1 + \frac{\varepsilon }{c} \, \, \mathbf{n} \cdot {\dot{{\pmb \beta }}} \right) c d\tau \varepsilon ^2 \mathbf{n} d\Omega . \end{aligned}$$

We see that the factor in the parenthesis originates in the dependence of the time differential on the spatial coordinate in Eq. (D9), associated with the spatial variation of the acceleration. This in turn originates in the Lorentz contraction of the rigid sphere upon acceleration.

We now obtain for the momentum transport across a section of the tube corresponding to \(d\tau \), i.e., the transport through the rigid sphere surrounding the electron corresponding to this time interval,

$$\begin{aligned} d\mathbf{P}= d\tau \int \!\! \int \mathbf{k}_s \left( 1 + \frac{\varepsilon }{c} \, \, \mathbf{n}_s \cdot {\dot{{\pmb \beta }}} \right) \varepsilon ^2 d\Omega , \end{aligned}$$

with \(\mathbf{k}_s\) given in Eq. (34). As we have seen in Sect. 4 this leads to complete equivalence between the electromagnetic energy and mass outside the sphere.

Dirac calculated the energy–momentum transport through the tube for the retarded field from an accelerated point charge, including both terms in Eq. (4). However, to obtain an equation of motion he replaced the divergent inertial self-force (first term in Eq. (36) but without the factor 4/3!) by a term, \(- mc {\dot{{\pmb \beta }}}\), corresponding to a finite mass m. He applied an expansion similar to the one discussed in Sect. 2 but more general, avoiding the assumption \(\beta = 0\), and obtained a generalization of the formula (8) for the damping force,

$$\begin{aligned} F^\mu = \frac{2e^2}{3c} \left( \frac{d^2}{ds^2} u^\mu + \left( \frac{d}{ds} u^\nu \right) ^2 u^\mu \right) , \end{aligned}$$

with the four-force defined as the derivative of the four-momentum with respect to s ([2], last two terms on the left-hand side of Eq. (24)). Dirac discussed the \(0'\)th component of the four-force, the power term,

$$\begin{aligned} F^0 = \frac{2e^2}{3c} \left( \frac{d^2}{ds^2} u^0 + \left( \frac{d}{ds} u^\nu \right) ^2 u^0 \right) . \end{aligned}$$

The second term corresponds to the power of irreversible emission of radiation and, according to Dirac, gives the effect of radiation damping on the motion of the electron. The first term is a perfect differential of a so-called acceleration energy [1] and corresponds to reversible exchange of energy with the near field (see also [3]). However, it should be noted that the other terms of the four-force do not separate so neatly and are mixed under Lorentz transformations.

An interesting derivation of the formula (D13) is given in [22] (see also [36], §32]). The first term is an obvious relativistic generalization of Eq. (8) but it does not have the property required by any four-force that it be perpendicular to the four-velocity. The second term is then added as a plausible extension remedying this deficiency. And it is this term that now accounts for the radiation reaction!

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Khokonov, M.K., Andersen, J.U. Equivalence Between Self-energy and Self-mass in Classical Electron Model. Found Phys 49, 750–782 (2019).

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