We are now in a position
to formulate more exactly the idea of
Minkowski, which was only vaguely indicated in Section 17. In
accordance with the special theory of relativity, certain co-ordinate
systems are given preference for the description of the
four-dimensional, space-time continuum. We called these “Galileian
co-ordinate systems." For these systems, the four co-ordinates
, which determine an event or--in other words, a point of the
four-dimensional continuum--are defined physically in a simple
manner, as set forth in detail in the first part of this book. For the
transition from one Galileian system to another, which is moving
uniformly with reference to the first, the equations of the Lorentz
transformation are valid. These last form the basis for the derivation
of deductions from the special theory of relativity, and in themselves
they are nothing more than the expression of the universal validity of
the law of transmission of light for all Galileian systems of
reference.
Minkowski found that the Lorentz transformations satisfy the following
simple conditions. Let us consider two neighboring events, the
relative position of which in the four-dimensional continuum is given
with respect to a Galileian reference-body by the space co-ordinate
differences
and the time-difference
. With reference to a
second Galileian system we shall suppose that the corresponding
differences for these two events are
. Then these
magnitudes always fulfill the condition 1.
The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude
which belongs to two adjacent points of the four-dimensional
space-time continuum, has the same value for all selected (Galileian)
reference-bodies. If we replace ,
, by
, we also obtain the result that
is independent of the choice of the body of reference. We call the magnitude ds the “distance” apart of the two events or four-dimensional points.
Thus, if we choose as time-variable the imaginary variable
instead of the real quantity
, we can regard the space-time
continuum--accordance with the special theory of relativity--as a
“Euclidean” four-dimensional continuum, a result which follows from
the considerations of the preceding section.
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