Relativity: The Special and General Theory | ||
2. APPENDIX II
MINKOWSKI'S FOUR-DIMENSIONAL SPACE ("WORLD")
(SUPPLEMENTARY TO SECTION XVII)
WE can characterise the Lorentz transformation still more simply if we introduce the imaginary [Description: Equation] in place of t, as time-variable. If, in accordance with this, we insert x1 = x x2 = y x3 = z x4 = [Description: Equation] and similarly for the accented system K', then the condition which is identically satisfied by the transformation can be expressed thus: x1'2 + x2'2 + x3'2 + x4'2 = x12 + x22 + x32 + x42 (12).
That is, by the afore-mentioned choice of "coordinates," (11a) is transformed into this equation.
We see from (12) that the imaginary time co-ordinate x4, enters into the condition of transformation in exactly the same way as the space co-ordinates x1, x2, x3. It is due to this fact that, according to the theory of relativity, the "time"
A four-dimensional continuum described by the "co-ordinates" x1, x2, x3, x4, 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 x1, x2, x3 which identically satisfy the equation x'12 + x'22 + x'32 = x12 + x22 + x32 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."
Relativity: The Special and General Theory | ||