160
16 Two Axiomatic Systems for Special Relativity
the principle of relativity, in other words he made both principles equal. Although
Poincaré [75] was wrong, when in 1909 he believed he had to postulate both, the
principle of relativity (which according to Einstein, cf. p.7, is formulated as two
hypotheses) and: ‘One needs to make still a third hypothesis, .... . A body in translational motion suffers a deformation in the direction in which it is displaced’.
3
Poincaré, however, oversaw that the Lorentz contraction was a deduction made from
the principle of relativity. We will just see however that one can as well start out from
the hypothesis of FitzGerald- Lorentz contraction, complete it to a closed axiomatic
system and then deduce the Einsteinian principle of relativity with its universal constancy of the speed of light from this. Therefore, Poincaré was correct in his evaluation
that both principles were equal. The main point is that the definition of simultaneity
in this approach to relativity is now strictly separated, what we are now going to
explain.
The completed Lorentzian axiomatic system of the Special Theory of Relativity from the outset is composed of two independent postulates. As we explained at
the beginning of this chapter, these are (A) a definition of simultaneity and (B) an
axiomatic statement about certain physical facts. These two postulates dissect the
Einsteinian, universal, abstract principle into two basic assumptions that are independent from each other. Let us start with the physical facts that we will demand
according to Lorentz:
1. In a preferred inertial system o , a moving rod suffers a length contraction as
this was proposed by Lorentz [62] in 1904, cf. also lorentz [60], according to
L
= L o
1 −
v 2
c
2
L
.
(112)
In addition, we have to postulate that a clock moving with respect to this preferred
system o goes behind according to Eq. (121) for time dilatation,
t
= t
1 −
v 2
c
2
L
.
(121)
In order to be able to measuring these facts, synchronised clocks should be available
in this preferred reference system o . For this, we demand the propagation of light
being isotropic in o . Then, we found the synchronisation of clocks in o on this
isotropy. Notice that even this synchronisation of the o -clocks is a definition: Clocks
are synchronised in such a way that light propagation is observed isotropically.
This postulate (121) was also proposed by Lorentz, although not in the sense
of an independent axiom. Both postulates (112) and (121) were seen by Lorentz
[63] as the elementary experimental statements of the Special Theory of Relativity.
In his lectures from the year 1906 published in 1909 in the book ‘The theory of
Electrons’, Lorentz [64] writes, ‘Then on account of the different rates of a moving
3 Quoted from the translation in Pais [71].
16 Two Axiomatic Systems for Special Relativity
the principle of relativity, in other words he made both principles equal. Although
Poincaré [75] was wrong, when in 1909 he believed he had to postulate both, the
principle of relativity (which according to Einstein, cf. p.7, is formulated as two
hypotheses) and: ‘One needs to make still a third hypothesis, .... . A body in translational motion suffers a deformation in the direction in which it is displaced’.
3
Poincaré, however, oversaw that the Lorentz contraction was a deduction made from
the principle of relativity. We will just see however that one can as well start out from
the hypothesis of FitzGerald- Lorentz contraction, complete it to a closed axiomatic
system and then deduce the Einsteinian principle of relativity with its universal constancy of the speed of light from this. Therefore, Poincaré was correct in his evaluation
that both principles were equal. The main point is that the definition of simultaneity
in this approach to relativity is now strictly separated, what we are now going to
explain.
The completed Lorentzian axiomatic system of the Special Theory of Relativity from the outset is composed of two independent postulates. As we explained at
the beginning of this chapter, these are (A) a definition of simultaneity and (B) an
axiomatic statement about certain physical facts. These two postulates dissect the
Einsteinian, universal, abstract principle into two basic assumptions that are independent from each other. Let us start with the physical facts that we will demand
according to Lorentz:
1. In a preferred inertial system o , a moving rod suffers a length contraction as
this was proposed by Lorentz [62] in 1904, cf. also lorentz [60], according to
L
= L o
1 −
v 2
c
2
L
.
(112)
In addition, we have to postulate that a clock moving with respect to this preferred
system o goes behind according to Eq. (121) for time dilatation,
t
= t
1 −
v 2
c
2
L
.
(121)
In order to be able to measuring these facts, synchronised clocks should be available
in this preferred reference system o . For this, we demand the propagation of light
being isotropic in o . Then, we found the synchronisation of clocks in o on this
isotropy. Notice that even this synchronisation of the o -clocks is a definition: Clocks
are synchronised in such a way that light propagation is observed isotropically.
This postulate (121) was also proposed by Lorentz, although not in the sense
of an independent axiom. Both postulates (112) and (121) were seen by Lorentz
[63] as the elementary experimental statements of the Special Theory of Relativity.
In his lectures from the year 1906 published in 1909 in the book ‘The theory of
Electrons’, Lorentz [64] writes, ‘Then on account of the different rates of a moving
3 Quoted from the translation in Pais [71].