Appendix II. Minkowski’s Four-Dimensional Space (“World”) [Supplementary to Section XVII]

Relativity: The Special and General TheoryAlbert Einstein

We can characterise the Lorentz transformation still more simply if we introduce the imaginary Ict\sqrt{-I} \cdot ct in place of tt, as time-variable. If, in accordance with this, we insert

x1=xx_1 = x x2=yx_2 = y x3=zx_3 = z x4=Ictx_4 = \sqrt{-I} \cdot ct

and similarly for the accented system K1K^1, then the condition which is identically satisfied by the transformation can be expressed thus:

x12+x22+x32+x42=x12+x22+x32+x42(12)\tag{12} {x'_1}^2 + {x'}_2^2 + {x'}_3^2 + {x'}_4^2 = x_1^2 + x_2^2 + x_3^2 + x_4^2

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 x4x_4, enters into the condition of transformation in exactly the same way as the space co-ordinates x1,x2,x3x_1, x_2, x_3. It is due to this fact that, according to the theory of relativity, the “time” x4x_4, enters into natural laws in the same form as the space co ordinates x1,x2,x3x_1, x_2, x_3.

A four-dimensional continuum described by the “co-ordinates” x1,x2,x3,x4x_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 (x1,x2,x3x'_1, x'_2, x'_3) with the same origin, then x1,x2,x3x'_1, x'_2, x'_3, are linear homogeneous functions of x1,x2,x3x_1, x_2, x_3 which identically satisfy the equation

x12+x22+x32=x12+x22+x32{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 four-dimensional “world.”

Appendix III. The Experimental Confirmation of the General Theory of Relativity

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