164
16 Two Axiomatic Systems for Special Relativity
We therefore have to firstly decide what we wish to understand when we use the
term, ‘length of a moving rod’. Here, we use the generally accepted definition: The
length of a moving rod is equal to the difference of its coordinate end points at one
and the same point of time.
This automatically takes a further definition as granted: the definition of simultaneity.
Let us start with a primary preferred frame o , where a simultaneity is defined
on the basis of an isotropic propagation of light. Let us take Lorentz contraction of a
moving length with respect to o as granted.
are moving frames with respect to o .
A moving length, with respect to
, has a well-defined meaning if a simultaneity is
defined in
. Consider a rof X = x L o at rest in o with its coefficient of measure
x for its length in o . We will envisage two possibilities.
1. Firstly, we follow Poincaré [73, 74], (cf. also Chap. 12): ‘The simultaneity of
two events or the order of their succession, as well as the equality of two time intervals,
must be defined in such a way that the statements of the natural laws be as simple as
possible’. We can realise Poincaré’s demand with the help of our elementary principle
of relativity. In the last chapter, cf. the Figs. 12.9 and 15.1, we have seen this. Once
the definition of simultaneity is based on that principle the observer in the reference
system
moving relative to the reference system o measures a length contraction
for the rod resting in the reference system o , which is identical with the Lorentz
contraction measured for a moving rod in o . The coefficient of measure x
for
the length of the rod X = x L o = x
L
as determined in the reference system
is x
= γ x with γ =
1 − v 2 /c 2 . The moving rod is observed shortened,
even if it rests, by chance, with respect to the crystal and only moves relative to the
observer, who himself moves relative to the crystal. According to Reichenbach, the
contraction of the rod moving relative to
is called Einstein contraction. However,
this Einstein contraction is dependent on the definition of simultaneity in
, as we
will see now.
2. Let us suppose that the clocks in
are synchronised according to the ‘absolute
simultaneity’ as described by (143),
x
=
x − vt
γ
, t
= γ t .
Consider again the rod X = x L o at rest in o with the coefficient of measure x
for its length in o . The measuring-rods L
at rest in
are shortened, if their length
is measured in o , L
= γ L o . Hence, the coefficient of measure x
for the length of
the rod X as measured with L
, X = x
L
, is x
= x/γ. Here, we made use of
the fact that the coordinates of the end points have one and the same time coordinate
t in o if they have one and the same time coordinate t
= γ t in
. Reichenbach’s
Einstein ‘contraction’ in truth now is an extension. The moving length x
of the rod
X as observed from the reference system
is extended in this case of synchronisation. This means, due to the ‘absolute simultaneity’ the statement concerning the
lengths of moving and resting rods has an absolute meaning. Nevertheless, there is
no contradiction between the two statements ‘the rod is contracted’ or ‘the rod is
16 Two Axiomatic Systems for Special Relativity
We therefore have to firstly decide what we wish to understand when we use the
term, ‘length of a moving rod’. Here, we use the generally accepted definition: The
length of a moving rod is equal to the difference of its coordinate end points at one
and the same point of time.
This automatically takes a further definition as granted: the definition of simultaneity.
Let us start with a primary preferred frame o , where a simultaneity is defined
on the basis of an isotropic propagation of light. Let us take Lorentz contraction of a
moving length with respect to o as granted.
are moving frames with respect to o .
A moving length, with respect to
, has a well-defined meaning if a simultaneity is
defined in
. Consider a rof X = x L o at rest in o with its coefficient of measure
x for its length in o . We will envisage two possibilities.
1. Firstly, we follow Poincaré [73, 74], (cf. also Chap. 12): ‘The simultaneity of
two events or the order of their succession, as well as the equality of two time intervals,
must be defined in such a way that the statements of the natural laws be as simple as
possible’. We can realise Poincaré’s demand with the help of our elementary principle
of relativity. In the last chapter, cf. the Figs. 12.9 and 15.1, we have seen this. Once
the definition of simultaneity is based on that principle the observer in the reference
system
moving relative to the reference system o measures a length contraction
for the rod resting in the reference system o , which is identical with the Lorentz
contraction measured for a moving rod in o . The coefficient of measure x
for
the length of the rod X = x L o = x
L
as determined in the reference system
is x
= γ x with γ =
1 − v 2 /c 2 . The moving rod is observed shortened,
even if it rests, by chance, with respect to the crystal and only moves relative to the
observer, who himself moves relative to the crystal. According to Reichenbach, the
contraction of the rod moving relative to
is called Einstein contraction. However,
this Einstein contraction is dependent on the definition of simultaneity in
, as we
will see now.
2. Let us suppose that the clocks in
are synchronised according to the ‘absolute
simultaneity’ as described by (143),
x
=
x − vt
γ
, t
= γ t .
Consider again the rod X = x L o at rest in o with the coefficient of measure x
for its length in o . The measuring-rods L
at rest in
are shortened, if their length
is measured in o , L
= γ L o . Hence, the coefficient of measure x
for the length of
the rod X as measured with L
, X = x
L
, is x
= x/γ. Here, we made use of
the fact that the coordinates of the end points have one and the same time coordinate
t in o if they have one and the same time coordinate t
= γ t in
. Reichenbach’s
Einstein ‘contraction’ in truth now is an extension. The moving length x
of the rod
X as observed from the reference system
is extended in this case of synchronisation. This means, due to the ‘absolute simultaneity’ the statement concerning the
lengths of moving and resting rods has an absolute meaning. Nevertheless, there is
no contradiction between the two statements ‘the rod is contracted’ or ‘the rod is