On the Electromagnetic Mass of a Moving Electron

by Ebenezer Cunningham

IN his discussion of the electromagnetic mass of a moving electron (Theorie der Elektrizität, ii. p. 205), Abraham raises an objection to the Lorentz conception of an electron as having, at rest, a spherical shape, but in motion the shape of an oblate spheroid the ratio of whose axes is ${\sqrt {1-v^{2}/c^{2}}}$, v being the velocity relative to the tether, and c the velocity of light. The present paper reconsiders Abraham's discussion and comes to the conclusion that the objection is not valid. The discussion was suggested by the fact that it has been proved that Maxwell's equations represent equally well the sequence of electromagnetic phenomena relative to a set of axes moving relative to the aether, as relative to a set of axes fixed in the aether. More explicitly this is stated as follows : —

If there are two sets of rectangular axes (A, A') coinciding at a certain instant, of which A' is moving relative to A with velocity v in the direction of the axis of x, which is conceived as at rest, and if x, y, z, t be space and time variables associated with A, and x', y' z', t' similar variables associated with A', then the equations

${\frac {1}{c}}{\frac {\partial E}{\partial t}}=curl\ H,\quad {\frac {1}{c}}{\frac {\partial H}{\partial t}}=-curl\ E$,

transform identically into the equations

${\frac {1}{c}}{\frac {\partial E'}{\partial t'}}=curl\ H',\quad {\frac {1}{c}}{\frac {\partial H'}{\partial t'}}=-curl\ E'$,

the accented and unaccented magnitudes being connected by the relations

${\begin{array}{ll}x'=\beta (x-vt],\\y'=y,\\z'=z,&\beta =\left(1-v^{2}/c^{2}\right)^{-{\frac {1}{2}}}\\t'=\beta \left(t-{\frac {vx}{c^{2}}}\right),\end{array}}$

${\begin{array}{l}E'=\beta \left({\frac {E_{x}}{\beta }},\ E_{y}-vH_{z},\ E_{z}+vH_{y}\right),\\\\H'=\beta \left({\frac {H_{x}}{\beta }},\ H_{y}-vE_{z},\ H_{z}+vE_{y}\right)\end{array}}$

Further, if $\rho ={\tfrac {1}{4\pi }}div\ E$, and $\rho '={\tfrac {1}{4\pi }}div\ E'$, the volume integrals taken through corresponding regions $\tau ,\ \tau ',\ \int _{\tau }\rho d\tau$ and $\int _{\tau '}\rho 'd\tau '\!$ are identically equal, giving an exact correspondence as regards distribution of electric charge.

Thus the above transformation renders the electromagnetic equations of a system independent of a uniform translation of the whole system through the aether.^[1]

According to this transformation, as a more geometrical correspondence, the length of a line in the direction of the axis of x moving with the axes A', as measured in the coordinates x', y', z', t', is greater than its length measured in the coordinates x, y, z, t in the ratio $1:\left(1-v^{2}/c^{2}\right){}^{\frac {1}{2}}$, so that Lorentz's hypothesis of the reduction in the dimensions of a body when it moves relatively to an observer is reduced by this geometrical correspondence to the assumption that in the variables associated with axes moving with it its shape is unaltered — an assumption suggested by the fact that the electromagnetic equations referred to those variables are independent of the motion through the aether, and by the attempt to form a purely electromagnetic theory of matter. Thus if the single electron at rest has a spherical configuration, and there are no other than electromagnetic forces, we should expect it in motion to have a spherical configuration when measured by the variables x' y' z' which means that as measured by the variables x, y, z it will have the spheroidal shape as suggested by Lorentz.

The electron as conceived by Abraham, on the other hand, is spherical always as regards the variables x, y, z, and a prolate spheroid as regards x' y' z' the ratio of the axes being $1:\left(1-v^{2}/c^{2}\right){}^{\frac {1}{2}}$.

In either case the electromagnetic mass is defined as the ratio of the external mechanical force on the electron to the acceleration of the centre of the electron, and Abraham develops two expressions for the longitudinal mass, i. e. for the ratio of the force in the direction of motion to the acceleration in the same direction, viz.: ${\tfrac {dG}{dv}}$ and ${\tfrac {1}{v}}{\frac {dW}{dv}}$ for the case of the so-called quasistationary motion, G being the electromagnetic momentum and W the electromagnetic energy. For the latter case these two expressions are proved identical, but for the Lorentz electron they are not equal, and Abraham deduces that W cannot be the whole energy of the electron. But the fact is that in this case the mass as above defined is not equal to ${\tfrac {1}{v}}{\frac {dW}{dv}}$, this expression being obtained on the assumption that the electron is "rigid" (Theorie der Elek. ii. p. 155). If the change in the shape of the electron with the change in velocity is taken into account, it will be found that the mass as obtained from the change in momentum is identical with the mass as obtained from the change in the energy, as it clearly must be, since a quantity defined in a perfectly definite manner cannot from consistent equations be shown to have two different values.

The assumption on which the theory is built is that the forces from sources exterior to the electron balance those due to the electron itself: this is the assumption that there is no inertia other than electromagnetic, and we deduce the equation

${\frac {dW}{dt}}=-{\frac {dA}{dt}}$,

where W is the electromagnetic energy, and A is the work done by the forces due to the electron itself.

If v₀ is the velocity of the centre of the electron, v=(v₀ + v₁) the velocity of the charge at any point of it, F the mechanical force per unit charge, we have

${\frac {dA}{dt}}=\int \rho (vF)d\tau$,

(vF) being the vector product of v and F.

If ξ η ζ are the coordinates relative to the centre of the electron of the element of charge whose velocity is v, and x y z of the same element when the electron is at rest, ξ=βx, η=y, ζ=z; so that the velocity v₁ of the charge relative to the centre is

${\frac {d\xi }{dt}}=x{\frac {d\beta }{dt}}=-{\frac {xv_{0}}{c^{2}\beta }}{\frac {dv_{0}}{dt}}$

in the direction of the axis of x.

Thus, if F_x is the component of F in that direction,

${\frac {dA}{dt}}=v_{0}\int \rho F_{x}d\tau +\int \rho x{\frac {d\beta }{dt}}F_{x}d\tau$, $=-v_{0}K-{\frac {v_{0}f}{c^{2}\beta }}\int \rho xF_{x}d\tau$,

K being the total mechanical force on the electron;

$=-v_{0}K-{\frac {v_{0}f}{c^{2}\beta }}\int \rho _{0}x\left(F_{x}\right)_{0}d\tau _{0}$;

where the suffix 0 in the last integral refers to the corresponding quantities when the electron is at rest, so that the region of integration is spherical.

For quasistationary motion W is a function of v₀ only, and therefore

${\frac {dW}{dt}}={\frac {dW}{dv_{0}}}{\frac {dv_{0}}{dt}}={\frac {dW}{dv_{0}}}f$.

Hence

${\frac {dW}{dv_{0}}}f=+v_{0}K+{\frac {v_{0}f}{c^{2}\beta }}\int \rho _{0}x\left(F_{x}\right)_{0}d\tau _{0}$

and therefore

$f\left\{{\frac {1}{v_{0}}}{\frac {dW}{dv_{0}}}-{\frac {1}{c^{2}\beta }}\int \rho _{0}x\left(F_{x}\right)_{0}d\tau _{0}\right\}=K$,

so that the longitudinal mass is equal to

${\frac {1}{v_{0}}}{\frac {dW}{dv_{0}}}-{\frac {1}{c^{2}\beta }}\int \rho _{0}x\left(F_{x}\right)_{0}d\tau _{0}$

It is the second term in this expression that is neglected by Abraham, and which he has to account for by assuming the energy of the electron to be made up of W together with a term not electromagnetic in origin.

We proceed to evaluate this expression in the two cases (i.) of a sphere with uniform volume density; (ii.) of a sphere with uniform surface charge.

(i.) Volume Distribution.

${\begin{array}{l}G={\frac {4}{5}}{\frac {e^{2}}{ac^{2}}}{\frac {v_{0}}{\beta }};\\\\W={\frac {3}{5}}{\frac {e^{2}}{a\beta }}\left(1+{\frac {v_{0}^{2}}{3c^{2}}}\right).\end{array}}$

Hence

${\begin{array}{rl}{\frac {dG}{dv_{0}}}&={\frac {4}{5}}{\frac {e^{2}}{ac^{2}}}\left\{{\frac {1}{\beta }}+{\frac {v_{0}^{2}/c^{2}}{\beta ^{3}}}\right\}\\\\&={\frac {4}{5}}{\frac {e^{2}}{ac^{2}\beta ^{3}}},\\\\{\frac {1}{v_{0}}}{\frac {dW}{dv_{0}}}&={\frac {3}{5}}{\frac {e^{2}}{a\beta }}\left\{{\frac {2}{3c^{2}}}+\left(1+{\frac {v_{0}^{2}}{3c^{2}}}\right){\frac {1}{c^{2}\beta ^{2}}}\right\}\\\\&={\frac {1}{5}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\left\{5-{\frac {v_{0}^{2}}{c^{2}}}\right\}.\\\\\int \rho _{0}x\left(F_{x}\right)_{0}d\tau _{0}&=\rho _{0}\int _{0}^{a}\int _{0}^{\pi }r\ \cos \theta \cdot {\frac {4}{3}}\pi r\rho _{0}\ \cos \theta \ 2\pi r^{2}\ \sin \theta \ d\theta \ dr\\\\&={\frac {8\pi ^{2}\rho _{0}^{2}}{3}}\cdot {\frac {a^{3}}{5}}\cdot {\frac {2}{3}}\\\\&={\frac {1}{5}}{\frac {e^{2}}{a}}.\end{array}}$.

$\therefore$

${\begin{array}{rl}{\frac {1}{v_{0}}}{\frac {dW}{dv_{0}}}&-{\frac {1}{c^{2}\beta }}\int \rho _{0}x\left(F_{x}\right)_{0}d\tau _{0}\\\\&={\frac {1}{5}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\left\{5-{\frac {v_{0}^{2}}{c^{2}}}-1+{\frac {v_{0}^{2}}{c^{2}}}\right\}\\\\&={\frac {4}{5}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\\\\&={\frac {dG}{dv_{0}}}.\end{array}}$.

(ii.) Surface Distribution.

${\begin{array}{rl}G&={\frac {2}{3}}{\frac {e^{2}}{ac^{2}}}{\frac {v_{0}}{\beta }}.\\\\W&={\frac {1}{2}}{\frac {e^{2}}{a\beta }}\left(1+{\frac {v_{0}^{2}}{3c^{2}}}\right),\\\\{\frac {dG}{dv_{0}}}&={\frac {2}{3}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}.\\\\{\frac {1}{v_{0}}}{\frac {dW}{dv_{0}}}&={\frac {1}{6}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\left\{5-{\frac {v_{0}^{2}}{c^{2}}}\right\}.\end{array}}$.

In this case the integral $\int \rho _{0}x\left(F_{x}\right)_{0}d\tau _{0}$ becomes $\int \sigma _{0}x\left(F_{x}\right)_{0}dS_{0}$, over the surface of the sphere of radius a, and $F_{x}=2\pi \sigma _{0}\cos \theta$.

Thus

${\begin{array}{rl}\int \sigma _{0}x\left(F_{x}\right)_{0}dS_{0}&=4\pi ^{2}a^{3}\sigma _{0}^{2}\int _{0}^{\pi }\cos ^{2}\theta \sin \theta d\theta \\\\&={\frac {8}{3}}\pi a^{3}\sigma _{0}^{2}={\frac {1}{6}}{\frac {e^{2}}{a}},\end{array}}$,

and

${\begin{array}{rl}{\frac {1}{v_{0}}}{\frac {dW}{dv_{0}}}&-{\frac {1}{c^{2}\beta }}\int \sigma _{0}x\left(F_{x}\right)_{0}dS_{0}\\\\&={\frac {1}{6}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\left\{5-{\frac {v_{0}^{2}}{c^{2}}}-1+{\frac {v_{0}^{2}}{c^{2}}}\right\}\\\\&={\frac {2}{3}}{\frac {e^{2}}{ac^{2}\beta ^{3}}}\\\\&={\frac {dG}{dv_{0}}}.\end{array}}$.

Thus in both cases the corrected expression for the longitudinal mass as derived from the energy gives the same result as that obtained from the momentum, and no other forces other than electromagnetic come into play.

On the other hand, since from what has been said above it appears that the electron will naturally retain the spherical shape as measured by the variables associated with the moving axes, it appears that some extraneous forces would be required to cause it to retain the spherical shape to an observer remaining at rest.

It is perhaps worth noting that "the principle of relativity" propounded by Bucherer in the Phil. Mag. of April 1907 is in essence identical with the statement made in the beginning of this paper. The principle referred to may be stated thus : that in the sequence of electromagnetic phenomena the giving of an additional uniform translational velocity v to the whole system of electric and magnetic bodies will not affect the phenomena observed if this velocity v is at the same time given to the observer. The transformation of space and time variables mentioned above shows a means of explaining this dependence of the electromagnetic phenomena on relative motion only; and conversely it is a comparatively simple matter to show that it is the only means. For it is required, among other things, to explain how a light-wave travelling outwards in all directions with velocity C relative to an observer A, may at the same time be travelling outwards in all directions with the same velocity relative to an observer B moving relative to A with velocity v. This can clearly not be done without some transformation of the space and time variables of the two observers.

Suppose two observers A, B, to be situated momentarily in the same spot, and let B be moving relatively to A with velocity v (measured by A in his own system of space and time). Let the direction of motion of B be A's axis of x, and let the instant of coincidence be A's time t=0.

Suppose axes of ξ η ζ to be B's system of coordinates moving with him with velocity v relative to A's axes of x y z, and coinciding with them at t=0.

Associated with a given point at a given time as marked by the values (x, y, z, t) will be unique values of (ξ, η, ζ, τ), τ being B's measure of the interval elapsed from the time of his coincidence with A, and conversely. There must therefore be a linear transformation from the variables (x, y, z, t) to (ξ, η, ζ, τ).

Consider now points on the axis of x (or ξ). Then the transformation must be of the form

$\xi ={\frac {a'x+b't+c'}{ax+bt+c}}$, $\tau ={\frac {a''x+b''t+c''}{ax+bt+c}}$,

Now ξ will not in general be infinite unless x is infinite, and also when x and t are zero ξ and τ are also zero. Hence the transformation must be of the simpler form

${\begin{array}{llll}\xi =a'x+b't&&{\frac {x=b''\xi -b'\tau }{\triangle }}\\&\mathrm {or} &&\triangle =(a'b''-b'a'')\\\tau =a''x+b''t&&t={\frac {-a''\xi +a'\tau }{\triangle }}\end{array}}$

the coefficients a', b', a", b" being functions of the relative velocity v.

Now if a point starts from A at time t=0 and travels with B, its coordinate ξ is always zero by virtue of the relation x=vt.

Hence

$b'=-a'v\,$

i.e.

$\xi =a'(x-vt)\,$

Now consider what is involved in saying that if a point moves along the axis of x relative to A with the velocity v of light, it also moves with velocity c relative to B. If a point moves from the position x at time t to the position x+δx at time t+δt let the corresponding changes in ξ and τ be δξ, δτ.

Then

$\delta \xi =a'\delta x+b'\delta t,\,$

$\delta \tau =a''\delta x+b''\delta t\,$

Hence

${\frac {\delta \xi }{\delta \tau }}={\frac {a'\delta x+b'\delta t}{a''\delta x+b''\delta t}}$.

Hence if the point has velocity n in A's system of coordinates and $\nu$ in that of B

$\nu ={\frac {a'n+b'}{a''n+b''}}$

In particular if $n=\pm c$, $\nu =\pm c$,

$a''c^{2}\pm c(b''-a')-b'=0$,

so that

$b''=a'\,$ and $a''={\frac {b'}{c^{2}}}=-{\frac {va'}{c^{2}}}$.

Thus

${\begin{array}{l}\xi =a'(x-vt),\\\\\tau =a'\left\{-{\frac {vx}{c^{2}}}+t\right\}.\end{array}}$.

If this transformation be reversed we have

${\begin{array}{l}x=\alpha '(\xi +v\tau )\\\\t=\alpha '\left(+{\frac {v\xi }{c^{2}}}+\tau \right)\ \mathrm {where} \ \alpha '={\frac {1}{a'\left(1-{\frac {v^{2}}{c^{2}}}\right)}}\end{array}}$

and α' will be the same Function of (-v) that a' is of v.

But the transformation shows that if x₁ x₂ be two points fixed relative to A and ξ₁ ξ₂ their coordinates in B at any time τ,

$x_{2}-x_{1}=\alpha '\left(\xi _{2}-\xi _{1}\right)$

i. e. a line of length l as seen by A appears to be of length ${\tfrac {l}{\alpha '}}$, as seen by B moving relatively to it. But this will be the same whichever be the direction of B's motion along the axis of x, so that if $\alpha '=f(v),\ f(v)=f(-v)$, i.e. $a'=\alpha '$.

Hence

$a'^{2}\left(1-{\frac {v^{2}}{c^{2}}}\right)=1$, i. e. $\alpha '=\left(1-{\frac {v^{2}}{c^{2}}}\right)^{-{\frac {1}{2}}}$.

Thus the transformation is finally

${\begin{array}{l}\xi =\beta (x-vt),\\\\\tau =\beta \left(-{\frac {vx}{c^{2}}}-t\right)\ \mathrm {where} \ \beta =\left(1-{\frac {v^{2}}{c^{2}}}\right)^{-{\frac {1}{2}}}\end{array}}$

Now let points not on the axis of x be considered. Since the axes of x and ξ coincide at all time, y and z always vanish when η and ζ vanish.

Hence $y=\lambda \eta$ and $z=\mu \zeta$, and λ and μ will not change if the velocity of motion of B be changed from v to -v ; thus if $\lambda =\phi (v)$; $\phi (v)=\phi (-v)$.

But since by reversing the transformation

$\eta =y/\lambda ,\ \phi (-v)={\frac {1}{\lambda }}$, and therefore λ=1.

Similarly μ=1.

The general transformation between x y z t and ξ η ζ τ is therefore

${\begin{array}{c}y=\eta ,\ z=\zeta ,\\\\\xi =\beta (x-vt)+c_{1}y+d_{1}z\,\\\\\tau =\beta \left(-{\frac {vx}{c^{2}}}+t\right)+c_{2}y+d_{2}z.\end{array}}$.

The two last equations reducing to those above for y=0 and z=0.

But our hypothesis requires that the equation

$\xi ^{2}+\eta ^{2}+\zeta ^{2}=c^{2}\tau ^{2}\,$

shall be a result of the equation

$x^{2}+y^{2}+z^{2}=c^{2}t^{2}.\,$

This is so if, and only if, c₁, c₂, d₁, d₂ are all zero.

Thus we have arrived exactly at the transformation as given above, and, as Einstein has shown (loc. cit.), in order that the electromagnetic equations may be invariant under this transformation the electric and magnetic vectors in the two systems must be correlated in the manner done in this paper.

Bucherer in the paper referred to does not take into account this necessary modification of coordinates, and therefore when in the latter part of it he evaluates the electromagnetic mass of the electron on the assumption that it is spherical he is in reality considering the Abraham electron, and so obtains Abraham's expression for its mass.

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1. The transformation in question is given by Einstein in a paper in the Annalen der Physik, xvii. (q. v.). It is in substance the same as that given by Larmor in 'Aether and Matter,' chap, xi., though the correlation is only proved to hold as far as the second power of v/c. Prof. Larmor tells me he has known for some time that it was exact. Vide also Lorentz, Amsterdam Proceedings, 1903-4.