D O C . 3 1 I D E A S A N D M E T H O D S 1 3 3
, ,
,
changes (6a)
into[30]
(10)
In a formal sense, this is the most simple condition to characterize the generalized
Lorentz transformation.
Comparing (10) to (9), one realizes the complete mathematical analogy existing
between the Lorentz transformation and the Euclidean transformation. The Lorentz
transformation is a Euclidean transformation in a four-dimensional space-time
continuum, provided one makes use of an imaginary time
coordinate.[31]
Long ago, physics and geometry derived in a graphically vivid manner, by
means of the theory of vectors, the formal qualities which the laws of nature must
show in order to be covariant under purely spatial transformations (9). An analo-
gous extension of these formal constructions into the four-dimensional provides us
with these structures and equations that are covariant under generalized Lorentz
transformations (10). By recognizing this, Minkowski considerably simplified the
application of the theory of special relativity. By his method one is able to judge
directly whether or not systems of equations do comply with the requirements of
relativity, without having to carry out a transformation of them.
However, it has to be emphasized that the equivalence of the time coordinate
with the spatial coordinates is only a formal and not a physical one,
which is obvious from what has been said above.
As is obvious from (9), the entire conceptual system of Euclidean geometry can
be built upon the fact that the square of the spatial distances
(11)
is a quantity independent of the choice of coordinates where is a quantity mea-
surable with a 〈unit〉 measuring rod. In a corresponding way, it is of fundamental
importance for the theory of special relativity that the quantity
x x1 = y x2 = z x3 =
–1ct x4 =
dx1
2
dx2
2
dx3
2
dx4
2
+ + +
dx12 ′ dx22 ′ dx32 ′ dx42. ′
+ + + =
x4
x1, x2, x3
ds2 dx1
2
dx2
2
dx3
2
+ + =
[p. 19]
ds