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Albert Einstein
>
Relativity: The Special and General Theory
> Appendix 2.
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CONTENTS
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BIBLIOGRAPHIC RECORD
Albert Einstein
(1879–1955).
Relativity: The Special and General Theory.
1920.
Appendix II
Minkowski’s Four-Dimensional Space (“World”)
[S
UPPLEMENTARY TO
S
ECTION
XVII
]
W
E
can characterise the Lorentz transformation still more simply if we introduce the imaginary
ct
in place of
t,
as time-variable. If, in accordance with this, we insert
and similarly for the accented system
K',
then the condition which is identically satisfied by the transformation can be expressed thus:
1
That is, by the afore-mentioned choice of “co-ordinates” (11
a
) is transformed into this equation.
2
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}
.
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.”
4
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
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 imaginary time co-ordinate); the Lorentz transformation corresponds to a “rotation” of the co-ordinate system in the four-dimensional “world.”
5
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