1 2 2 D O C . 3 1 I D E A S A N D M E T H O D S
measuring rod that is at rest relative to
.2[19]
It is obvious that the results and of the two entirely different processes of
measurement do not need to be the same. (Relativity of lengths.)
It is now easily seen that the relativity of times and lengths does not justify the
consequence drawn at section 5b. The dilemma expounded in sec. 5 is herewith re-
solved.
9. Galilei Transformation
Description in physics always uses a coordinate system (body of reference) upon
which all processes are referred. Whatever occurs is composed of “pointlike
events,” each one determined chronologically and spatially relative to the coordi-
nate system through three spatial coordinates and a time value .
On the other hand, we know that in classical mechanics there is already an infi-
nite manifold of admissible coordinate systems (inertial systems), all completely
equivalent for the description of nature. If are the space-time coordinates
of a pointlike event in reference to system , and are the coordinates
of the same event in reference to a system that moves with velocity v relative
to , then it is clear that, with a given orientation and location of relative to ,
the primed coordinates must be completely determined by the unprimed ones (co-
ordinate transformation).
Classical mechanics implicitly as-
sumed the absolute character of times
and lengths. Consequently, for the two
coordinate systems whose relative posi-
tion and orientation is sketched in the at-
tached figure, one had to assume the
transformation:
2
The unit measuring rods in (a) and (b) shall be equal to each other when compared at
relative rest to each other. It shall be noted here that this condition of equality (also for
clocks) is a lasting one, independent of the past history of motion and an essential precon-
dition of the entire theory.
K
l′ l
[p. 9]
x y z , , t
x y z t , , ,
K x′, y′, z′, t′
K′
K K′ K
⎧ ⎪ ⎨ ⎪ ⎩
x′
y′
x
x′
vt
⎧ ⎨ ⎩
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
v
x
y y′ =
y
Fig. 2