Translation:The Reflection of Light at Moving Mirrors

by Vladimir Varićak

The Reflection of Light at Moving Mirrors.

By V. Varićak.

In the following I would like to give a non-Euclidean interpretation of Einstein's formulas for the reflection of light at moving mirrors.^[1] The ray of light incident at a reflecting coordinate plane $\xi =0$ shall be defined by the quantities $A$, $\cos \varphi$, $\nu$. These quantities are related to a stationary coordinate system. Mirror $\xi =0$ shall move with velocity $v$ in the direction of the positive abscissa axis of the stationary system. For the direction cosine of the reflected ray, one thus has the formula according to Einstein:

$\cos \varphi '''={\frac {\cos \varphi ''+{\frac {v}{c}}}{1+{\frac {v}{c}}\cos \varphi ''}}$

(1)

If one puts herein

$\varphi '''=\Pi \left(u_{1}'''\right),\ \varphi ''=\Pi \left(u_{1}''\right),\ {\frac {v}{c}}=\operatorname {th} \ u$

(2)

then it becomes

$\operatorname {th} \ u_{1}'''={\frac {\operatorname {th} \ u_{1}''+\operatorname {th} \ u}{1+\operatorname {th} \ u_{1}'''\operatorname {th} \ u}}$

or

$u_{1}'''=u_{1}''+u\,$.

(3)

Now it is furthermore

$\varphi ''=\pi -\varphi ',\ \varphi '=\Pi \left(u_{1}'\right)$

and therefore

$u_{1}''=-u_{1}'\,$

(5)

Einstein's formula for $\cos \varphi '$ was already transformed by me as the aberration equation (in my first report^[2]), into the form:

$u_{1}'=u_{1}-u\,$

(6)

and thus one has:

$u_{1}''=u-u_{1}\,$

and

$u_{1}'''=2u-u_{1}\,.$

(8)

Here, $u_{1}$ means the perpendicular belonging to the parallel angle $\varphi$. Equation (8) replaces Einstein's formula

$\cos \varphi '''={\frac {\left(1+\left({\frac {v}{c}}\right)^{2}\right)\cos \varphi -2{\frac {v}{c}}}{1-2{\frac {v}{c}}\cos \varphi +\left({\frac {v}{c}}\right)^{2}}}$

From Fig. 1, the construction of the reflected ray is easily to be seen by formula (8). In the construction it is advantageous, to use angle $\psi$ being supplementary to $\psi '''$

Fig. 1

It is $\psi =\Pi \left(u_{1}-2u\right)$. For $u_{1}-2u$ one has $\psi ={\tfrac {\pi }{2}}$. One ordinarily also considers $\psi$ as the reflection angle.

However, we can arrive at equation (8) in a still shorter way. Namely, the reflection angle at the moving mirror can be determined in the same way as in the stationary one, by means of construction on the basis of Huyghens' principle. I only mention the relevant explanations of W. M. Hicks^[3] and E. Kohl^[4], undertaken by them in the course of investigating the Michelson-Morley experiment.

Hicks assumes $v$ as being positive if the mirror is approaching the incident rays. In his formula (1) we thus have to assume $v$ as being negative, to bring it into accordance with our definitions. Then it reads in our notation

$\tan {\frac {1}{2}}\psi ={\frac {c+v}{c-v}}\tan {\frac {1}{2}}\varphi$

(9)

According to the relation that holds between the parallel angle and the corresponding perpendicular, we can write

$e^{-u_{1}'''}={\frac {1+\operatorname {th} \ u}{1-\operatorname {th} \ u}}e^{-u_{1}}$

or

$u_{1}'''=u_{1}-2u\,$

(10)

It's known that one has to take $u_{1}'''$ as being negative for angle $\varphi '''$, since it is supplementary to $\psi$. In which relation the magnitude of angle $\psi$ is with respect to angle $\varphi$, depends on the direction of motion of the mirror relative to the light source. In the case considered, angle $\psi$ is larger than $\varphi$, since it is related to the smaller perpendicular as parallel angle. For the ratio of amplitudes and frequencies, Einstein gives the following equation:

${\frac {A'''}{A''}}={\frac {\nu '''}{\nu ''}}={\frac {1+{\frac {v}{c}}\cos \varphi ''}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}$

(11)

Due to relation (2) we can write it in the form

${\frac {A'''}{A''}}={\frac {\nu '''}{\nu ''}}=\left(1+\operatorname {th} \ u\ \operatorname {th} \ u_{1}'\right)\operatorname {ch} \ u$

or also

${\frac {A'''}{A''}}={\frac {\nu '''}{\nu ''}}={\frac {\operatorname {ch} \left(u+u_{1}''\right)}{\operatorname {ch} \ u_{1}''}}$

(12)

If equation (7) is considered, then it becomes

${\frac {A'''}{A''}}={\frac {\nu '''}{\nu ''}}={\frac {\operatorname {ch} \left(2u+u_{1}\right)}{\operatorname {ch} \left(u+u_{1}\right)}}={\frac {\operatorname {ch} \left(u_{1}-2u\right)}{\operatorname {ch} \left(u_{1}-u\right)}}$

(13)

However, for the reflected ray it is

$A''=A',\ \nu ''=\nu '\,;$

(14)

according to formula (28) on p. 292 of this journal, one has

${\frac {A'}{A}}={\frac {\nu '}{\nu }}={\frac {\operatorname {ch} \left(u_{1}-u\right)}{\operatorname {ch} \ u_{1}}}$

(15)

and thus it becomes

${\frac {A'''}{A}}={\frac {\nu '''}{\nu }}={\frac {\operatorname {ch} \left(u_{1}-2u\right)}{\operatorname {ch} \ u_{1}}}$

(16)

The relations of the amplitudes and frequencies of the incident and reflected light, can be represented by the relation of the arcs of two distance lines between shared normals. Equation (16) replaces Einstein's equations

${\frac {A'''}{A}}={\frac {\nu '''}{\nu }}={\frac {1-2{\frac {v}{c}}\cos \varphi +\left({\frac {v}{c}}\right)^{2}}{1-\left({\frac {v}{c}}\right)^{2}}}$

(17)

Fig. 2

We have graphically represented formulas (15) and (16) in Fig. 2. It is easily seen, that one obtains $\nu '''(A''')$ by reflection of $\nu (A)$ upon $\nu '(A')$. In this way, also angle $\varphi '''$ can be determined by reflection of the incident ray at the aberrated ray. Formula (15) for Doppler's principle and formula (16) for the amplitude and frequency of the reflected light are of the same form; as well as aberration equation (6) and formula (10) for the reflection angle.

We denoted this velocity by $v$, which is represented by distance $u$ (for $c=1$), and by $v'$ we want to denote that velocity corresponding to the double distance $2a$. Then it follows from the previously mentioned equations, that the same light ray appears to an observer moving with velocity $v'$, as of the same constitution as it would appear for a resting observer after the reflection at a mirror moving with velocity $v$. In both cases the motion must be of the same direction.

Also the procedure by Bateman^[5] is in connection with this result, who derived the laws of reflection at moving mirrors on the basis of the presupposition: the image of an object shall emerge by the space-time transformation

${\begin{array}{l}x'=-x\ \operatorname {ch} \,2u+l\ \operatorname {sh} \,2u\\l'=l\ \operatorname {ch} \,2u-x\ \operatorname {sh} \,2u\end{array}}$

(18)

He writes this in another form.

For a light ray which is incident perpendicularly, we have $\varphi =0$, thus $u_{1}=\infty$, and formula (16) goes over into

$\nu '''=\nu e^{-2u}\,.$

(19)

The relation of frequencies and amplitudes can in this case be represented as the relation of two coaxial limiting arcs.

Agram, May 14, 1910

(Received May 23, 1910.)

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1. Ann. d. Phys. 17, 914, 1905.
2. This journal, 11, 95. 1910.
3. Phil. Mag. 3, 15, 1902
4. Ann. d. Phys. 28, 262, 1909.
5. H. Bateman, The reflection of light at an ideal plane mirror moving with a uniform velocity of translation. Phil. Mag. 18, 892, 1909.