Page:Grundgleichungen (Minkowski).djvu/17

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.^[1] 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 $A(x_{0},\ y_{0},\ z_{0})$ at the time $t_{0}=0$ 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 AP, i.e. less than the time required for the propagation of light from A to P, and if $q={\frac {t-t_{0}}{AP}}<1$, then by a special Lorentz transformation, in which AP is taken as the axis, and which has the moment q, we can introduce a time parameter t which (see equation 11, 12, § 4) has got the same value $t'=0$ for both space-time points $A,t_{0}$, and P, t. So the two events can now be comprehended to be simultaneous.

Further, let us take at the same time $t_{0}=0$, two different space-points A, B, or three space-points A, B, C which are not in the same space-line, and compare therewith a space point P, which is outside the line AB, or the plane ABC at another time t, and let the time difference $t-t_{0}$ be less than the time which light requires for propagation from the line AB, or the plane ABC to P. Let q be the ratio of $t-t_{0}$ by the second time. Then if a Lorentz transformation is taken in which the perpendicular from P on AB, or from P on the plane ABC is the axis, and q is the moment, then all the three (or four) events $A,\ t_{0};\ B,\ t_{0};\ (C,\ t_{0})$ and P, t are simultaneous.

If four space-points, which do not lie in one plane are conceived to be at the same time to, then it is no longer permissible to make a change of the time parameter by a Lorentz-transformation, without at the same time destroying the character of the simultaneity of these four space points.

To the mathematician, accustomed on the one hand to the methods of treatment of the poly-dimensional manifold, and on the other hand to the conceptual figures of the so-called non-Euclidean Geometry, there can be no difficulty in adopting this concept

1. Just as beings which are confined within a narrow region surrounding a point on a spherical surface, may fall into the error that a sphere is a geometric figure in which one diameter is particularly distinguished from the rest.