Page:The principle of relativity (1920).djvu/92

 If then, by referring back to equations (9), we carry out the transformation (1) through the angle ψ and a subsequent rotation round the Z-axis through the angle φ, we perform a Lorentz-transformation at the end of which m_{y} = 0, e_{y} = 0, and therefore m and e shall both coincide with the new Z-axis. Then by means of the invariants m^2 - e^2, (me) the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled.

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By the Lorentz transformation, we are allowed to effect certain changes of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three. Since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves. In fact, the following considerations will prove conclusive.

Let a reference system (x, y, z, t) for space time points (events) be somehow known. Now if a space point A (x_{0}, y_{0}, z_{0}) the time t_{0} be compared with a space point P (x, y, z) at the time t, and if the difference of time t - t_{0}, (let t > t_{0}) be less than the length A P i.e. less than the time required for the propogation of light from