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75
Relativity: The Special and General Theory
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Albert Einstein: Relativity
Appendix
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
in place of t, as time?variable. If, in accordance with this, we insert
x1
= x
x2
= y
x3
= z
x
4
=
and similarly for the accented system K¹, then the condition which is identically satisfied by the
transformation can be expressed thus :
x1'² + x2'² + x3'² + x
4
'² = x 2
+ x 2
+ x 2 + x 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 x1, x2, x3. 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 x1, x2, x3.
A four?dimensional continuum described by the "co?ordinates" x1, x2, x3, 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 x1, x2,
x3 which identically satisfy the equation
x'
2
+ x' 2 + x' 2 = x 2 + x 2 + x 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."
Next: The Experimental Confirmation of the General Theory of Relativity