Page:The principle of relativity (1920).djvu/146



resolved in such world-lines, and I may just deviate from my point if I say that according to my opinion the physical laws would find their fullest expression as mutual relations among these lines.

By this conception of time and space, the (x, y, z) manifoldness t = 0 and its two sides t < 0 and t > 0 falls asunder. If for the sake of simplicity, we keep the null-point of time and space fixed, then the first named group of mechanics signifies that at t = 0 we can give the x, y, and z-axes any possible rotation about the null-point corresponding to the homogeneous linear transformation of the expression

x^2 + y^2 + z^2.

The second group denotes that without changing the expression for the mechanical laws, we can substitute (x - αt, y - βt, z - yt?] for (x, y, z) where (α, β, γ) are any constants. According to this we can give the time-axis any possible direction in the upper half of the world t > 0. Now what have the demands of orthogonality in space to do with this perfect freedom of the time-axis towards the upper half ?

To establish this connection, let us take a positive parameter c, and let us consider the figure

c^2t^2 - x^2 - y^2 - z^2 = 1

According to the analogy of the hyperboloid of two sheets, this consists of two sheets separated by t = 0. Let us consider the sheet, in the region of t > 0, and let us now conceive the transformation of x, y, z, t in the new system of variables; (x, y, z, t) by means of which the form of the expression will remain unaltered. Clearly the rotation of space round the null-point belongs to this group of transformations. Now we can have a full idea of the transformations which we picture to ourselves from a particular