10 Measuring-Rods and Clocks in Motion
97
Fig. 10.3 Equation (113) of the Lorentz contraction, x = x/
1 − v 2 /c 2
o . Shown is v = 0, 8 c o ;
thus, γ =
1 − v 2 /c 2
o = 0, 6 and therefore x = x/0, 6 ; L = 0, 6 L
In order to avoid irritations, we want to call the reader’s attention to the fact that
there is another situation which formally leads to the same Eq. (113), but which
describes another physical situation. In fact, it is only necessary for the validity of
Eq. (113) that both the starting point and the end point of the length X are measured at
the same point of time t of o . If both the starting point with its coordinate x 1 and the
end point with its coordinate x 2 of a length X move in o with the same constant
velocity v, so that x 1 = x o + vt, x 2 = x o + x + vt, then our measuring-rod L o
should fit, according to the assumptions x times in the length X. However, the
moving measuring-rod L
fits x
times, X = x L o = x
L
, so that we only
have to exchange the coordinates with the coordinate differences in (113). Thus, the
following is again valid
x
=
x
1 − v 2 /c 2
o
.
(113a)
This physically describes the following: It is supposed of a static rod X referred
to
which on its part moves with a velocity v with respect to o , that its coefficient
of measure of its length in o is exactly x. The coefficient of measure x
of its
length in
is, due to the Lorentz contraction, determined by Eq. (113). We will
return to this case in Chap. 12.
There are certain details in the measuring of lengths that play an important role
later on. We have to distinguish these different situations:
1. We determine the spatial distance between two events, let us say the distance
between the event O and the event B that we are observing in an arbitrary inertial
system : In order to measure this distance, we would mark both positions where the
events occurred and we would then count how often our measuring-rod, available
to us in this reference system, fits between both points. This would give us our
coefficient of measure for the distance. We are not confronted with any problems
using this method of measurement.
2. The next object to have its length measured is a rod. The objective is to determine
the length of this rod in a reference system . Here, two situations have to be
differentiated. (a) The rod is stationary in the reference system . I can therefore mark
both end points without rushing, at any time and once again count how many times
my measuring-rod fits in between. (b) The following situation creates completely
new problems. The rod is moving with the uniform velocity v through our reference
97
Fig. 10.3 Equation (113) of the Lorentz contraction, x = x/
1 − v 2 /c 2
o . Shown is v = 0, 8 c o ;
thus, γ =
1 − v 2 /c 2
o = 0, 6 and therefore x = x/0, 6 ; L = 0, 6 L
In order to avoid irritations, we want to call the reader’s attention to the fact that
there is another situation which formally leads to the same Eq. (113), but which
describes another physical situation. In fact, it is only necessary for the validity of
Eq. (113) that both the starting point and the end point of the length X are measured at
the same point of time t of o . If both the starting point with its coordinate x 1 and the
end point with its coordinate x 2 of a length X move in o with the same constant
velocity v, so that x 1 = x o + vt, x 2 = x o + x + vt, then our measuring-rod L o
should fit, according to the assumptions x times in the length X. However, the
moving measuring-rod L
fits x
times, X = x L o = x
L
, so that we only
have to exchange the coordinates with the coordinate differences in (113). Thus, the
following is again valid
x
=
x
1 − v 2 /c 2
o
.
(113a)
This physically describes the following: It is supposed of a static rod X referred
to
which on its part moves with a velocity v with respect to o , that its coefficient
of measure of its length in o is exactly x. The coefficient of measure x
of its
length in
is, due to the Lorentz contraction, determined by Eq. (113). We will
return to this case in Chap. 12.
There are certain details in the measuring of lengths that play an important role
later on. We have to distinguish these different situations:
1. We determine the spatial distance between two events, let us say the distance
between the event O and the event B that we are observing in an arbitrary inertial
system : In order to measure this distance, we would mark both positions where the
events occurred and we would then count how often our measuring-rod, available
to us in this reference system, fits between both points. This would give us our
coefficient of measure for the distance. We are not confronted with any problems
using this method of measurement.
2. The next object to have its length measured is a rod. The objective is to determine
the length of this rod in a reference system . Here, two situations have to be
differentiated. (a) The rod is stationary in the reference system . I can therefore mark
both end points without rushing, at any time and once again count how many times
my measuring-rod fits in between. (b) The following situation creates completely
new problems. The rod is moving with the uniform velocity v through our reference