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 neighbouring events, the
relative position of which in the four-dimensional continuum is given
with respect to a Galileian reference-body K by the space co-ordinate
differences dx, dy, dz and the time-difference dt. With reference to a
second Galileian system we shall suppose that the corresponding
differences for these two events are dx1, dy1, dz1, dt1. Then these
magnitudes always fulfil the condition*
dx2 + dy2 + dz2 - c^2dt2 = dx1 2 + dy1 2 + dz1 2 - c^2dt1 2.
The validity of the Lorentz transformation follows from this
condition. We can express this as follows: The magnitude
ds2 = dx2 + dy2 + dz2 - c^2dt2,
which belongs to two adjacent points of the four-dimensional
space-time continuum, has the same value for all selected (Galileian)
reference-bodies.
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