162
16 Two Axiomatic Systems for Special Relativity
However, from where do we take the properties of our measuring-rods and clocks
in a preferred inertial frame postulated in the first postulate? The easiest method
would be if we could simply refer to precision experiments. In fact today, it is possible to do just this, because the Lorentz contraction of bodies, as well as the time
dilatation of clocks moving with respect to our laboratory can be proven with breathtaking accuracy. Such a starting point however remains theoretically unsatisfying.
It would be better to have an exact mathematical theory, whose solutions can be
identified as particles and using these, we could calculate and measure those postulated properties of a Lorentz contraction and of time dilatation. In the frame of
the classical field theory, a non-linear theory gives us stabile particle configurations,
so-called solitons, as solutions. Every non-linear system of equations that retains its
mathematical form after its variables were changed according to the Lorentz transformation (151) are Lorentz invariant as one would say allows soliton solutions. These
equations suffice as a model for particles with those properties demanded by the
first postulate. A mathematically simple example for this would be the sine-Gordon
equation for a single space dimension which we discussed in detail. In a solid, the
speed of light is replaced with a critical velocity c o , whose dimension lies in the
regions of sound velocities. For Special Relativity of our physical spacetime with the
speed of light, the one-dimensional sine-Gordon equation does not achieve much,
because here, as in opposition to a solid, we do not know of any physical objects that
could be identified using the solutions of this one-dimensional equation. Today, the
sine-Gordon equation for three-dimensional space with three-dimensional solutions
corresponding to the kink and breather solutions discussed by us increasingly occupies the thoughts of physicists. Here, we wish to refer to the papers of G. Leibbrandt
[55, 56]. If such solutions of the three-dimensional sine-Gordon equation can be
identified using physical objects, then one could, according to the Lorentz method,
also construct a three-dimensional Special Relativity with the help of this equation
in the same fashion as we did this for one space dimension in Chaps. 9–15.
In our case, where we have a continuum approximation of the crystal lattice, we
can traverse one step further. We have shown what the physical background for the
sine-Gordon equation that constitutes the fundamental of the explanation of Special
Relativity in this continuum according to the Lorentz method looks like. The elastic
coupled components of a periodical lattice structure constitute the background, which
underlies the laws of Newtonian motion. With the ideal lattice as a vacuum, the objects
of Special Relativity are realised by localised structural imperfections of this lattice.
2.
∗ On the other hand, after having started with 1. we may replace our postulat 2.
of the elementary principle of relativity with another postulate for synchronisation of
clocks in the systems
. We replace the postulate 2. with an ‘absolute simultaneity’
according to
t
(x, 0) = 0
(153)
as depicted in Fig. 12.8. The result of 1. and 2.
∗ is now Reichenbach transformation
(143), which replaces Lorentz transformation (177) resulting from 1. and 2.,
16 Two Axiomatic Systems for Special Relativity
However, from where do we take the properties of our measuring-rods and clocks
in a preferred inertial frame postulated in the first postulate? The easiest method
would be if we could simply refer to precision experiments. In fact today, it is possible to do just this, because the Lorentz contraction of bodies, as well as the time
dilatation of clocks moving with respect to our laboratory can be proven with breathtaking accuracy. Such a starting point however remains theoretically unsatisfying.
It would be better to have an exact mathematical theory, whose solutions can be
identified as particles and using these, we could calculate and measure those postulated properties of a Lorentz contraction and of time dilatation. In the frame of
the classical field theory, a non-linear theory gives us stabile particle configurations,
so-called solitons, as solutions. Every non-linear system of equations that retains its
mathematical form after its variables were changed according to the Lorentz transformation (151) are Lorentz invariant as one would say allows soliton solutions. These
equations suffice as a model for particles with those properties demanded by the
first postulate. A mathematically simple example for this would be the sine-Gordon
equation for a single space dimension which we discussed in detail. In a solid, the
speed of light is replaced with a critical velocity c o , whose dimension lies in the
regions of sound velocities. For Special Relativity of our physical spacetime with the
speed of light, the one-dimensional sine-Gordon equation does not achieve much,
because here, as in opposition to a solid, we do not know of any physical objects that
could be identified using the solutions of this one-dimensional equation. Today, the
sine-Gordon equation for three-dimensional space with three-dimensional solutions
corresponding to the kink and breather solutions discussed by us increasingly occupies the thoughts of physicists. Here, we wish to refer to the papers of G. Leibbrandt
[55, 56]. If such solutions of the three-dimensional sine-Gordon equation can be
identified using physical objects, then one could, according to the Lorentz method,
also construct a three-dimensional Special Relativity with the help of this equation
in the same fashion as we did this for one space dimension in Chaps. 9–15.
In our case, where we have a continuum approximation of the crystal lattice, we
can traverse one step further. We have shown what the physical background for the
sine-Gordon equation that constitutes the fundamental of the explanation of Special
Relativity in this continuum according to the Lorentz method looks like. The elastic
coupled components of a periodical lattice structure constitute the background, which
underlies the laws of Newtonian motion. With the ideal lattice as a vacuum, the objects
of Special Relativity are realised by localised structural imperfections of this lattice.
2.
∗ On the other hand, after having started with 1. we may replace our postulat 2.
of the elementary principle of relativity with another postulate for synchronisation of
clocks in the systems
. We replace the postulate 2. with an ‘absolute simultaneity’
according to
t
(x, 0) = 0
(153)
as depicted in Fig. 12.8. The result of 1. and 2.
∗ is now Reichenbach transformation
(143), which replaces Lorentz transformation (177) resulting from 1. and 2.,