Page:VaricakRel1910c.djvu/1

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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 shall be defined by the quantities , , . These quantities are related to a stationary coordinate system. Mirror shall move with velocity 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:

(1)

If one puts herein

(2)

then it becomes

or

. (3)

Now it is furthermore

and therefore

(5)

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

(6)

and thus one has:

and

(8)

Here, means the perpendicular belonging to the parallel angle . Equation (8) replaces Einstein's formula

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 being supplementary to

Fig. 1

It is . For one has . One ordinarily also considers 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 as being positive if the mirror is approaching the incident rays. In his formula (1) we thus have to assume as being negative, to bring it into accordance with our definitions. Then it reads in our notation

(9)

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

or

(10)

It's known that one has to take as being negative for angle , since it is supplementary to . In which relation the magnitude of angle is with respect to angle , depends on the direction of motion of the mirror relative to the light source. In the case considered, angle is larger than , since it is related to the smaller perpendicular as parallel angle.

  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.