APPENDIX II

Minkowski’s four-dimensional space ("World”)
(Supplementary to section 17)

We can characterise the Lorentz transformation still more simply if we introduce the imaginary eq. 25 in place of t, as time-variable.  If, in accordance with this, we insert

x[1] = x x[2] = y x[3] = z x[4] = eq. 25

and similarly for the accented system K1, then the condition which is identically satisfied by the transformation can be expressed thus : 

x[1]’2 + x[2]’2 + x[3]’2 + x[4]’2 = x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2
    (12).

That is, by the afore-mentioned choice of " coordinates,” (11a) [see the end of Appendix II] is transformed into this equation.

We see from (12) that the imaginary time co-ordinate x[4], enters into the condition of transformation in exactly the same way as the space co-ordinates x[1], x[2], x[3].  It is due to this fact that, according to the theory of relativity, the " time “x[4], enters into natural laws in the same form as the space co ordinates x[1], x[2], x[3].

A four-dimensional continuum described by the “co-ordinates” x[1], x[2], x[3], x[4], was called “world” by Minkowski, who also termed a point-event a " world-point.”  From a “happening” in three-dimensional space, physics becomes, as it were, an " existence " in the four-dimensional " world.”

This four-dimensional " world " bears a close similarity to the three-dimensional " space " of (Euclidean) analytical geometry.  If we introduce into the latter a new Cartesian co-ordinate system (x’[1], x’[2], x’[3]) with the same origin, then x’[1], x’[2], x’[3], are linear homogeneous functions of x[1], x[2], x[3] which identically satisfy the equation

        x’[1]^2 + x’[2]^2 + x’[3]^2 = x[1]^2 + x[2]^2 + x[3]^2

The analogy with (12) is a complete one.  We can regard Minkowski’s " world " in a formal manner as a four-dimensional Euclidean space (with an imaginary time coordinate) ; the Lorentz transformation corresponds to a " rotation " of the co-ordinate system in the fourdimensional " world.”

APPENDIX III

THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY

From a systematic theoretical point of view, we may imagine the process of evolution of an empirical science to be a continuous process of induction.  Theories are evolved and are expressed in short compass as statements of a large number of individual observations in the form of empirical laws, from which the general laws can be ascertained by comparison.  Regarded in this way, the development of a science bears some resemblance to the compilation of a classified catalogue.  It is, as it were, a purely empirical enterprise.