# The Entrainment of Light by Moving Bodies According to the Principle of Relativity

 The Entrainment of Light by Moving Bodies According to the Principle of Relativity  (1907)  by Max von Laue, translated by Wikisource
 In German: Die Mitführung des Lichtes durch bewegte Körper nach dem Relativitätsprinzip, Annalen der Physik 328 (10): 989–990, Online (Published September 24, 1907.)

The Entrainment of Light by Moving Bodies According to the Principle of Relativity

by M. Laue.

Since Einstein's electrodynamics,[1] which is based on the principle of relativity, is equivalent with the older theory of Lorentz as long one restricts himself to the first power of the relations of all body velocities to the speed of light, it is obvious that it allows to calculate Fresnel's dragging coefficient as a first approximation. But no reference is made in the literature as to how much easier this problem is resolved by the relativity principle than by the other theory, even with the simplification recently given by Lorentz.[2]

Namely, this is only an example of Einstein's addition theorem of velocities. There are two coordinate systems with parallel axes, the "primed" and the "unprimed", moving against each other along the direction of X with velocity v. A velocity w with respect to the primed system, whose direction forms the angle θ with the X'-axis, corresponds to a velocity in the unprimed system

 $w=\frac{\sqrt{v^{2}+w'^{2}+2vw'\cos\vartheta'-\frac{1}{c^{2}}v^{2}w'^{2}\sin^{2}\vartheta'}}{1+\frac{1}{c^{2}}vw'\cos\vartheta'}$.[3]

Now, if a body of refractive index n is at rest in the primed system, then the phase velocity of light in the primed system is:

 $w'=\frac{c}{n}$.

[ 990 ] The corresponding velocity in the unprimed system is therefore

 $w=\frac{\sqrt{v^{2}+\frac{c^{2}}{n^{2}}+2v\frac{c}{n}\cos\vartheta'-\frac{v^{2}}{n^{2}}\sin^{2}\vartheta'}}{1+\frac{v}{cn}\cos\vartheta'}$.

If the directions of the velocities v and c/n coincide, as in the experiment of Fresnel, then it is $\cos\vartheta'=\pm1$, and

 $w=\frac{\frac{c}{n}\pm v}{1\pm\frac{v}{cn}}$ $=\frac{c}{n}+\left(1-\frac{1}{n^{2}}\right)\left\{ \pm v-\frac{v^{2}}{cn}\pm\frac{v^{3}}{(cn)^{2}}-\frac{v^{4}}{(cn)^{3}}\pm\dots\right\}$.

If, however, for example $\vartheta'=\pm\pi/2$, it is

 $w=\sqrt{\frac{c^{2}}{n^{2}}+v^{2}\left(1-\frac{1}{n^{2}}\right)}=\frac{c}{n}+\frac{1}{2}\frac{v^{2}}{nc}\left(n^{2}-1\right)$ $-\frac{1}{2}\cdot\frac{1}{4}\frac{v^{4}}{nc^{3}}(n^{2}-1)^{2}+\frac{1}{2}\cdot\frac{1}{4}\cdot\frac{3}{6}\frac{v^{6}}{nc^{5}}(n^{2}-1)^{3}\dots$

In dispersive substances we have to fill in the value for n corresponding to the frequency in the primed system.

For the group velocity it is exactly the same, if we replace the refractive index n by the expression n + v(dn/dv) (v is frequency).

So, according to the relativity principle, light is completely carried by the body, however, just because of this its velocity relative to an observer (who does not participate in the motion of the body) is not the same as the vector sum of its velocity relative to the body and that of the body relative to an observer. In this way we are relieved of the need to introduce into optics an "aether", which penetrates the bodies without sharing their motion.

Berlin, July 1907.