Translation:The Wave Radiation of a Moving Point Charge in Accordance with the Principle of Relativity

by Max von Laue

The Wave Radiation of a Moving Point Charge According to the Principle of Relativity

by M. Laue.

The wave radiation of a moving point charge was already investigated a number of years ago by Heaviside^[1] and especially detailed by Abraham^[2] on the basis of the electron theory. The relativity theory was already used by Lorentz^[3] for calculating the field of an oscillating dipole, but without specifying the radiated energy and momentum, as well as the back-reaction of radiation on the dipole. Now, both theories agree as regards the electromagnetic equations, as well as in the expression for the ponderomotive force. The difference lies solely in the form which they attribute to the moving charges; because one theory assumes that they are not affected by motion, and in the other theory they contract in the direction of velocity. But this difference may not come into question at distances great compared to the dimensions of the charged parts of space - both theories must give the same electromagnetic field, thus the same radiated energy and momentum, and the same force exerted on the electron. If we nevertheless consider the question here again, it happens to show how much easier the theory of relativity solves it. There is also an important difference between the two theories as regards the effect of this force, which give the clearest illustration of the inertia of energy required by the principle of relativity.

a) The back-reaction of wave radiation on a point charge.

Lorentz gives, as an approximation for the force ${\mathfrak {K}}$ on the electron as the result of radiation, the equation^[4]

${\mathfrak {K}}={\frac {2}{3}}{\frac {e^{2}}{c^{3}}}{\mathfrak {\ddot {q}}}$

Where e is the charge, c the speed of light, and ${\mathfrak {\ddot {q}}}$ the second derivative of the velocity ${\mathfrak {q}}$ with respect to time. In the derivation, all terms in ${\mathfrak {q}},{\mathfrak {\dot {q}}},{\mathfrak {\ddot {q}}}$ etc. which are quadratic or of even higher order, are neglected, but it's easy to convince himself that no part independent of the dimensions can occur except of the specified one, which does not contain ${\mathfrak {q}}$ as a factor. For ${\mathfrak {q}}=0$ and point charges, therefore, this equation is strictly valid, however great ${\mathfrak {\dot {q}}},{\mathfrak {\ddot {q}}}$ etc. may be.

We now, as usual, introduce two right angled coordinate systems, corresponding in the direction of axis, and uniformly moving against each other parallel to the x-axis with velocity v; the primed (x', y', z', plus time t') and the unprimed (x, y, z, t) system. One can always arrange that the electron (at the moment we choose) has the speed 0 in the primed system. It is then subjected to the force

${\mathfrak {K'}}={\frac {2}{3}}{\frac {e^{2}}{c^{3}}}{\mathfrak {{\ddot {q}}'}}$

which we want to transform to the unprimed system.

Einstein^[5] has given the transformation formulas for the velocity (the z-coordinate is treated equally as the y-coordinate):

${\mathfrak {q}}'_{x}=({\mathfrak {q}}_{x}-v)\cdot {\frac {c^{2}}{c^{2}-v{\mathfrak {q}}_{x}}},\ {\mathfrak {q}}'_{y}={\mathfrak {q}}_{y}{\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\mathfrak {q}}_{x}}}$

By differentiating with respect to t' and t, and by taking into account the equation

${\frac {dt'}{dt}}={\frac {c^{2}-v{\mathfrak {q}}_{x}}{c{\sqrt {c^{2}-v^{2}}}}}$^[6]

it follows:

${\mathfrak {\dot {q}}}'_{x}=\left({\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\mathfrak {q}}_{x}}}\right)^{2}{\mathfrak {\dot {q}}}_{x};\ {\mathfrak {q}}'_{y}=\left({\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\mathfrak {q}}_{x}}}\right)^{2}\left({\mathfrak {\dot {q}}}_{y}+{\frac {v{\mathfrak {q}}_{y}{\mathfrak {\dot {q}}}_{x}}{c^{2}-v{\mathfrak {q}}_{x}}}\right)$

${\mathfrak {\ddot {q}}}'_{x}=\left({\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\mathfrak {q}}_{x}}}\right)^{4}\left({\mathfrak {\ddot {q}}}_{x}+{\frac {3v{\mathfrak {\dot {q}}}_{x}^{2}}{c^{2}-v{\mathfrak {q}}_{x}}}\right)$;

${\mathfrak {\ddot {q}}}'_{y}=\left({\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\mathfrak {q}}_{x}}}\right)^{3}\left({\mathfrak {\ddot {q}}}_{y}+v{\frac {3{\mathfrak {\dot {q}}}_{y}{\mathfrak {\dot {q}}}_{x}+{\mathfrak {q}}_{y}{\mathfrak {\ddot {q}}}_{x}}{c^{2}-v{\mathfrak {q}}_{x}}}+{\frac {3v^{2}{\mathfrak {q}}_{y}{\mathfrak {\dot {q}}}_{x}^{2}}{(c^{2}-v{\mathfrak {q}}_{x})^{2}}}\right)$.

In this case, where ${\mathfrak {q}}'=0$, it is ${\mathfrak {q}}'_{x}=v,\ {\mathfrak {q}}'_{y}={\mathfrak {q}}_{x}=0$. If we add, that under the same condition

${\mathfrak {K}}_{x}={\mathfrak {K}}'_{x},\ {\mathfrak {K}}_{y}={\mathfrak {K}}'_{y}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$^[7]

we find:

${\mathfrak {K}}_{x}={\frac {2e^{2}c}{3(c^{2}-{\mathfrak {q}}^{2})^{2}}}\left({\mathfrak {\ddot {q}}}_{x}+{\frac {3\left|{\mathfrak {q}}\right|}{c^{2}-{\mathfrak {q}}^{2}}}{\mathfrak {\dot {q}}}_{x}^{2}\right)$

${\mathfrak {K}}_{y}={\frac {2e^{2}}{3c(c^{2}-{\mathfrak {q}}^{2})}}\left({\mathfrak {\ddot {q}}}_{y}+{\frac {3\left|{\mathfrak {q}}\right|}{c^{2}-{\mathfrak {q}}^{2}}}{\mathfrak {\dot {q}}}_{y}{\mathfrak {\dot {q}}}_{x}\right)$

${\mathfrak {K}}_{z}={\frac {2e^{2}}{3c(c^{2}-{\mathfrak {q}}^{2})}}\left({\mathfrak {\ddot {q}}}_{z}+{\frac {3\left|{\mathfrak {q}}\right|}{c^{2}-{\mathfrak {q}}^{2}}}{\mathfrak {\dot {q}}}_{z}{\mathfrak {\dot {q}}}_{x}\right)$.

One can rewrite the result into the vector equation:

${\mathfrak {K}}={\frac {2e^{2}}{3c(c^{2}-{\mathfrak {q}}^{2})}}\left[{\mathfrak {\ddot {q}}}+{\mathfrak {q}}{\frac {3({\mathfrak {q{\dot {q}}}})}{c^{2}-{\mathfrak {q}}^{2}}}+{\frac {\mathfrak {q}}{c^{2}-{\mathfrak {q}}^{2}}}\left(({\mathfrak {q{\ddot {q}}}})+{\frac {3({\mathfrak {q{\dot {q}}}})^{2}}{c^{2}-{\mathfrak {q}}^{2}}}\right)\right]$,

which is completely consistent with Abraham's result^[8].

The motion of an electron is quasi-stationary, as long as this force is negligible against the acceleration proportional to the inertia of its field. Since they are of the same order in both theories, also the limits of the quasi-stationary motion agree.

b) The radiation of uniformly moving light sources.

The simplest model of a light source is a molecular dipole of rapidly changing moment, consisting of two equal but oppositely charged ions in vacuum. We presuppose that its dimensions are small compared to the wavelength of the emitted radiation, or in the case of non-periodic processes, they are small compared with the wavelengths of all those vibrations that come into consideration by a Fourier decomposition of the radiation. For simplicity, we still assume that only one ion is involved in the oscillations. The generalization follows easily, given that identical shifts of the two charges in the opposite sense are equivalent.

To this dipole we want to apply some of the equations developed by Planck ^[9] for the dynamics of moving bodies. It must be remembered, however, that molecular structures have neither temperature nor volume in the sense of thermodynamics, thus their kinetic potential H is independent of these variables. The result is that we always have to set the pressure

$p={\frac {\partial H}{\partial V}}$

and the entropy

$S={\frac {\partial H}{\partial T}}$^[10]

to zero.

Now, this dipole should rest in the primed system. According to Formula 33) of the mentioned paper, its energy in relation to the unprimed System is (since the momentum ${\mathfrak {G}}'$ in the primed system disappears):

$E={\frac {cE'}{\sqrt {c^{2}-v^{2}}}}$.

The radiated energy per unit time is thus:

$-{\frac {dE}{dt}}=-{\frac {c}{\sqrt {c^{2}-v^{2}}}}{\frac {dE'}{dt'}}{\frac {dt'}{dt}}$,

but because of ${\mathfrak {q}}_{x}=v$

${\frac {dt'}{dt}}={\frac {\sqrt {c^{2}-v^{2}}}{c}}$,

it follows

$-{\frac {dE}{dt}}=-{\frac {dE'}{dt'}}$,

the emitted quantum of energy per unit time in both systems is the same.

Now it is well known that radiation from a stationary dipole is calculated from the acceleration ${\mathfrak {\dot {q}}}$ of the moving ion by the equation:^[11],

$-{\frac {dE'}{dt'}}={\frac {2}{3}}{\frac {e^{2}}{c^{2}}}{\mathfrak {\dot {q}}}^{'2}={\frac {2}{3}}{\frac {e^{2}}{c^{2}}}({\mathfrak {\dot {q}}}_{x}^{'2}+{\mathfrak {\dot {q}}}_{y}^{'2}+{\mathfrak {\dot {q}}}_{z}^{'2})$;

if we substitute in the above transformation equations ${\mathfrak {q}}_{x}=v,\ {\mathfrak {q}}_{y}={\mathfrak {q}}_{x}=0$ for ${\mathfrak {\dot {q}}}$, it follows:

$-{\frac {dE}{dt}}={\frac {2e^{2}c^{2}}{3(c^{2}-{\mathfrak {q}}^{2})^{2}}}\left({\frac {{\mathfrak {\dot {q}}}_{x}^{2}}{c^{2}-{\mathfrak {q}}^{2}}}+{\frac {{\mathfrak {\dot {q}}}_{y}^{2}+{\mathfrak {\dot {q}}}_{z}^{2}}{c^{2}}}\right)={\frac {2e^{2}c^{2}}{3(c^{2}-{\mathfrak {q}}^{2})^{2}}}\left({\mathfrak {\dot {q}}}^{2}+{\frac {({\mathfrak {q{\dot {q}}}})^{2}}{c^{2}-{\mathfrak {q}}^{2}}}\right)$,

which is also consistent with Abraham's result^[12].

By equation 46) of that paper, the momentum in the unprimed system has the value

${\mathfrak {G}}={\mathfrak {q}}\cdot {\frac {E}{c^{2}}}$.

A resting dipole can not experience any further motion, because the opposite one is equal in every direction. In the case considered the dipole is at rest in the primed system. Any such acceleration would be a sign of absolute motion of this system, which would be contrary to the principle of relativity. Therefore its speed ${\mathfrak {q}}$ relative to the unprimed system has to remain constant. And by radiation per unit time it therefore loses momentum

$-{\frac {d{\mathfrak {G}}}{dt}}=-{\mathfrak {q}}\cdot {\frac {1}{c^{2}}}{\frac {dE}{dt}}={\mathfrak {q}}{\frac {2e^{2}}{3c(c^{2}-{\mathfrak {q}}^{2})^{2}}}\left({\mathfrak {\dot {q}}}^{2}+{\frac {({\mathfrak {q{\dot {q}}}})^{2}}{c^{2}-{\mathfrak {q}}^{2}}}\right)$.

Again, also this result can be derived from the earlier theory^[13] and this implies that the dipole experiences a force ${\mathfrak {K}}$ directed opposite to the motion. The theory of relativity must agree with this conclusion. Also in that theory we have for the momentum:

${\mathfrak {K}}={\frac {d{\mathfrak {G}}}{dt}}$.

But while in the older theory the momentum ${\mathfrak {G}}$ is a function of the velocity alone and all forces have to change the latter, in the theory of relativity it is possible that a force alters the energy (instead of the velocity) and thus the inertia associated with it. This case occurs at the moving source. The older theory needs a force whose work covers a portion of the energy radiation to maintain uniform motion. According to the theory of relativity, however, it completely stems from the energy of the light source.

The given values for the energy and momentum radiation are valid, since they depend only on the acceleration of the moving electron, even if it doesn't oscillate together with a dipole, but is moving in any other way. It is already proven by Abraham, that the force ${\mathfrak {K}}$ calculated in a) gives the same values for energy- and momentum radiation, if we form the mean value

${\frac {1}{T}}{\overset {t+T}{\underset {t}{\int }}}({\mathfrak {Kq}})dt$ und ${\frac {1}{T}}{\overset {t+T}{\underset {t}{\int }}}{\mathfrak {K}}dt$

For an extended radiation source we may be able to draw a similar conclusion. In general, however, the exerted radiation of a resting body has a resulting momentum and accordingly exercises a motion drive, as is can be most easily seen by the example of a cavity (of perfectly reflecting substance) filled with radiation, having a small opening only at a single point. But if the body is isotropic, homogeneous and limited by a convex surface everywhere, it suffers a normal pressure from the radiation which is everywhere the same, not resulting in a net force. If one such body is at rest in the primed system, then it has a constant speed ${\mathfrak {q}}$ in relation to the unprimed system. Its momentum is according to equation 46) of Planck's treatise

${\mathfrak {G}}={\mathfrak {q}}\cdot {\frac {E+pV}{c^{2}}}$.

The radiation takes from it the energy per unit time

$W=-\left({\frac {dE}{dt}}+p{\frac {dV}{dt}}\right)$

so that the momentum decreases by

$-{\frac {d{\mathfrak {G}}}{dt}}={\frac {\mathfrak {q}}{c^{2}}}\left(W-V{\frac {dp}{dt}}\right)$.

If the radiation intensity remains stable, then this is also valid for the radiation pressure and we find with Lorentz^[14] a force directed against the velocity

${\mathfrak {K}}=-{\mathfrak {q}}\cdot {\frac {W}{c^{2}}}$

But unlike the older theory this causes no hindrance of motion, but only a decrease of inertia.

It remains to explain how it is possible, that in the same operation a body in the unprimed system experiences a force, while this is not the case in the primed system. Initially, this seems to contradict the transformation equations of the force components. But the above force ${\mathfrak {K}}$ is not an instantaneous electrodynamic force on a point charge 1, but is generated by integration of such forces over the whole body, so far it is filled with electrons, and then by forming the mean value for durations that are long against light periods. The limits of time integration are of course independent of the location. But t' is not only a function of time t now, but also of coordinate x. Integrals (formed in an analogous way) of this type in the primed and unprimed system are never related to integration areas (which correspond to each other) of the variables x', y', z', t' and x, y, z, t. Therefore, we can not conclude from one directly to the other.

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1. O. Heaviside, Nature 67, 6, 1902.
2. M. Abraham, Ann. d. Phys. (4) 14, 236, 1904. Theorie der Elektrizität II, Leipzig 1905, in particular § 13 to 15.
3. H. A. Lorentz, Proc. Amsterdam 1904, p. 803.
4. H. A. Lorentz, Enzykl. d. math. Wissensch. V, 1, p. 188, 1903.
5. A. Einstein, Ann. d. Phys. (4) 17, 891, 1905, § 5 and Jahrb. d. Radioakt. u. Elektronik 4, 411, 1907, equation 3.
6. See M. Planck, Ann d. Phys. (4) 26, 1, 1908, equation 13)
7. M. Planck, l. c., equations 21); A. Einstein, l. c, eqation 21).
8. See. Abraham's textbook, Equation 85).
9. M. Planck, l. c.
10. Equation 7) in that very place.
11. See. Abraham's textbook, equation 55) or M. Planck, Ann. d. Phys. (4) 9, 619, 1902.
12. There, equation 82 b).
13. There, equation 83).
14. H. A. Lorentz, Enzykl. d. math. Wissenschaften V, 1, S. 270, 1903.