Page:Eddington A. Space Time and Gravitation. 1920.djvu/64

For the moment the difficulty of thinking in terms of an unfamiliar geometry may be evaded by a dodge. Instead of real time $t$, consider imaginary time $\tau$; that is to say, let $t=\tau {\sqrt {-1}}$. Then $(t_{2}-t_{1})^{2}=-(\tau _{2}-\tau _{2})^{2}$, so that $s^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}+(\tau _{1}-\tau _{2})^{2}$. Everything is now symmetrical and there is no distinction between $\tau$ and the other variables. The continuum formed of space and imaginary time is completely isotropic for all measurements; no direction can be picked out in it as fundamentally distinct from any other.

The observer's separation of this continuum into space and time consists in slicing it in some direction, viz. that perpendicular to the path along which he is himself travelling. The section gives three-dimensional space at some moment, and the perpendicular dimension is (imaginary) time. Clearly the slice may be taken in any direction; there is no question of a true separation and a fictitious separation. There is no conspiracy of the forces of nature to conceal our absolute motion—because, looked at from this broader point of view, there is nothing to conceal. The observer is at liberty to orient his rectangular axes of $x$, $y$, $z$ and $\tau$ arbitrarily, just as in three-dimensions he can orient his axes of $x$, $y$, $z$ arbitrarily.

It can be shown that the different space and time used by the aviator in Chapter i correspond to an orientation of the time-axis along his own course in the four-dimensional world, whereas the ordinary time and space are given when the time-axis is oriented along the course of a terrestrial observer. The FitzGerald contraction and the change of time-measurement are given exactly by the usual formulae for rotation of rectangular axes^[1].

It is not very profitable to speculate on the implication of the mysterious factor ${\sqrt {-1}}$, which seems to have the property of turning time into space. It can scarcely be regarded as more than an analytical device. To follow out the theory of the four-

1. Appendix, Note 3.