We can characterise the Lorentz transformation still more simply if we
introduce the imaginary
in place of
, as time-variable. If, in
accordance with this, we insert
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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 , enters into
the condition of transformation in exactly the same way as the space
co-ordinates
. It is due to this fact that, according
to the theory of relativity, the “time”
, enters into natural
laws in the same form as the space co ordinates
.
A four-dimensional continuum described by the “co-ordinates"
, 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 (
) with the same origin, then
, are
linear homogeneous functions of
which identically
satisfy the equation
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."
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