Page:The principle of relativity (1920).djvu/93

 A to P, and if q = (t - t_{0})/A P < 1, then by a special Lorentz transformation, in which A P 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 A B, or the plane A B C, at another time t, and let the time difference t - t_{0} (t > t_{0}) be less than the time which light requires for propogation from the line A B, or the plane A B C) to P. Let q be the quotient of (t - t_{0}) by the second time. Then if a Lorentz transformation is taken in which the perpendicular from P on A B, or from P on the plane A B C 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 t_{0}, 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 of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.