D O C . 3 1 I D E A S A N D M E T H O D S 1 2 3
.
(3)
This transformation, which is called the “Galilei-transformation,” is in an im-
portant relation with the Newtonian equations of motion in mechanics. If one uses
in those equations the variables according to (3) instead of the variables
, one obtains in those new coordinates equations of exactly the same form.
One says: the equations of classical mechanics are covariant under Galilei transfor-
mations. This is the analytical expression of the principle of special relativity in
classical mechanics.
However, the Galilei transformation with its absolute time cannot, ac-
cording to previous considerations, do justice to the actual behavior of measuring
rods and clocks. For a light ray propagating along the positive -axis according to
,
one would get relative to , from (3)
,
in contradiction to the requirement that the principle of the constancy of the speed
of light must also hold relative to .
10. The Lorentz
Transformation[20]
The intuitive reasoning which led to the Galilei transformation (3) is, according to
our previous analysis, not sound. Because, contrary to the fourth equation in (3),
we already recognized the relativity of simultaneity. Furthermore, if we rigorously
want to interpret fig. 2, we have to add that the sketch has to hold for a specific time
value of the unprimed (“resting”) system . We cannot know if the coordinate
of length , as seen from also equals when seen from (relativity of
length). Consequently, the foundation of the first equation in (3) falls, and, in anal-
ogy, for the second and third equations as well.
Now, in order to get usable transformation equations in place of (3), one only has
to satisfy the condition that one and the same light ray has to have speed relative
x′ x vt – =
y′= y
z′= z
t′= t
⎭
⎪
⎪
⎬
⎪
⎪
⎫
x′, y′, z′, t′
x y z t , , ,
t t′) = (
x
x ct =
K′
x′ c v)t′ – ( =
K′
[p. 10]
t K
x′ K′, x′ K
c