coordinate system S, in which observer M (as he thinks) is at rest. However, if we move the coordinate origin to O', i.e., if we take as the fundamental system , then and all points of the new ordinate axis will be transformed into the state of rest. However, also the time parameter is changed by that transformation.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/Varicak1912g.png/300px-Varicak1912g.png)
The unit time of the observer at a specific point shall be represented by the hyperbolic cosine of the Lobachevskian abscissa of that point. The unit time of the observer in O or in M is equal to "1" in the unprimed system, while the unit time of the observer in O'(M') becomes equal to , if we put OO' = u. For the observer resting in O it appears that the clock moving with velocity u stays behind in the ratio . When evaluating the duration of an event by means of the moving clock, the resting observer will find a smaller number. Thus there is the relation
(39) |
or
However, in the primed system the unit time of the observer in is equal to "1", while the unit time of the observer in O is equal to . Both systems are completely and equally justified. Thus we cannot speak of a time duration per se. Consequently it is not allowed to speak about the simultaneity of two events in an absolute sense.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Varicak1912h.png/300px-Varicak1912h.png)
At first let us consider an example.[1] From a material point A at rest at the coordinate origin in a valid reference frame, a short light signal propagates in all directions at time l = 1. At time l > 1 the points that receive the signal are located on a sphere, of which we only want to consider the intersection circle with the XY-plane. On this circle two other material Points B and C at rest in S, may be located; therefore they simultaneously receive the signal, that is, by the same value l.
- ↑ M. Laue, Das Relativitätsprinzip, 1911, 35. M. Planck, Acht Vorlesungen über theoretische Physik, 1910, 118.