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 _dx , dy , dz , dt _. Then these
magnitudes always fulfill the condition.[21]
[21] Cf. Appendixes I and II. The relations which are derived there
for the co-ordinates themselves are valid also for co-ordinate
_differences_, and thus also for co-ordinate differentials
(indefinitely small differences).
_dx_2 + _dy_2 + _dz_2 _c_2_dt_2 = _dx _2 + _dy _2 + _dz _2
_c_2_dt _2.
The validity of the Lorentz transformation follows from this condition.
We can express this as follows: The magnitude
_ds_2 = _dx_2 + _dy_2 + _dz_2 _c_2 _dt_2,
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 _x, y, z_,
image034
by _x_1, _x_2, _x_3, _x_4, we also obtain the result that
_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.
is independent of the choice of the body of reference. We call the
magnitude _ds_ the distance apart of the two events or
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