144
15 The Principle of Relativity: The Lost Crystal
general case, 21 material parameters and generally lead to a very complicated coupled system of differential equations for elastic deformations under the influence
of general dislocation structures. One can however show that these deformations,
originally derived from arbitrary dislocation distributions, finally containing only
one single parameter can be determined using simple wave equations
1 and also fulfil the Lorentz symmetry (cf. Günther [35]). We therefore have another immensely
large class of states in a crystal that also underlie the special principal of relativity
with the velocity c o or sound velocity. A few further explanations will be made in
the appendix, in Chaps. 26 and 27. We can now also reasonably say that the class
of states at the disposal of our internal observers is a cosmos of its own, for which
Special Relativity with the critical velocity c o of the sine-Gordon equation applies
(or respectively, cum grano salis also with the transversal sound velocity c T ).
The un-hoped derivation of the famous Einsteinian principle of relativity from
our humble considerations of Newtonian mechanics has made us oversee something,
we lost our crystal during this process! The crystal has disappeared! There is now
nothing in the world of our internal observers capable of detecting or finding our
crystal. Our internal observer is not capable of detecting the slightest trace of the
crystal, even with the help of the most clever and complicated experiments. Every
single of these internal observers in any single reference system measures one and
the same velocity c o . Using his measuring-rods and clocks does not help him any
further. The observer in o still states that the clock resting in
goes behind and
that measuring-rod resting in
is shorter. Thus he is compelled to state that only
o is distinguished as the absolute resting reference system and that
possesses
the velocity v.
This conclusion however is null and void, since we know that the same sineGordon equation with the same physical parameters applies for the observer in
as it does for the observer in o . Thus, the observer in
discovers the kink and the
breather solutions with the length L o and the oscillation period T o . The length that
was observed from o as being shorter is registered in
as the normal length L o ,
and the oscillation period that we observed as dilated is registered in
as the normal
period T o . The observer in
registers that the measuring-rods and clocks at rest in
o possess the velocity −v. He thus discovers that the clock resting in o goes behind
by the Lorentz factor and that the measuring-rod is shortened correspondingly. The
sign of the velocity plays no role, because all of these effects only depend on the
ratio v
2
/c
2
o .
Here, we stop, reconsider and ask, because of the consequences of these conclusions: Is this in fact really so? Have we not shown explicitly in Chap. 10 that
the kink moving with respect to the crystal is shortened, shortened in comparison
with the kink resting with respect to the crystal. It is obviously the resting kink that
possesses the largest geometrical extension. The sketch in Fig. 10.2 is, in principle,
experimentally provable. The breather solution moving with respect to the crystal
1 Here, we wish to note that these equations go over to the Lorentz invariant equations with the
transversal sound velocity c T , in the special case of straight screw dislocations in an isotropic
medium, cf. Günther [34, 35].
15 The Principle of Relativity: The Lost Crystal
general case, 21 material parameters and generally lead to a very complicated coupled system of differential equations for elastic deformations under the influence
of general dislocation structures. One can however show that these deformations,
originally derived from arbitrary dislocation distributions, finally containing only
one single parameter can be determined using simple wave equations
1 and also fulfil the Lorentz symmetry (cf. Günther [35]). We therefore have another immensely
large class of states in a crystal that also underlie the special principal of relativity
with the velocity c o or sound velocity. A few further explanations will be made in
the appendix, in Chaps. 26 and 27. We can now also reasonably say that the class
of states at the disposal of our internal observers is a cosmos of its own, for which
Special Relativity with the critical velocity c o of the sine-Gordon equation applies
(or respectively, cum grano salis also with the transversal sound velocity c T ).
The un-hoped derivation of the famous Einsteinian principle of relativity from
our humble considerations of Newtonian mechanics has made us oversee something,
we lost our crystal during this process! The crystal has disappeared! There is now
nothing in the world of our internal observers capable of detecting or finding our
crystal. Our internal observer is not capable of detecting the slightest trace of the
crystal, even with the help of the most clever and complicated experiments. Every
single of these internal observers in any single reference system measures one and
the same velocity c o . Using his measuring-rods and clocks does not help him any
further. The observer in o still states that the clock resting in
goes behind and
that measuring-rod resting in
is shorter. Thus he is compelled to state that only
o is distinguished as the absolute resting reference system and that
possesses
the velocity v.
This conclusion however is null and void, since we know that the same sineGordon equation with the same physical parameters applies for the observer in
as it does for the observer in o . Thus, the observer in
discovers the kink and the
breather solutions with the length L o and the oscillation period T o . The length that
was observed from o as being shorter is registered in
as the normal length L o ,
and the oscillation period that we observed as dilated is registered in
as the normal
period T o . The observer in
registers that the measuring-rods and clocks at rest in
o possess the velocity −v. He thus discovers that the clock resting in o goes behind
by the Lorentz factor and that the measuring-rod is shortened correspondingly. The
sign of the velocity plays no role, because all of these effects only depend on the
ratio v
2
/c
2
o .
Here, we stop, reconsider and ask, because of the consequences of these conclusions: Is this in fact really so? Have we not shown explicitly in Chap. 10 that
the kink moving with respect to the crystal is shortened, shortened in comparison
with the kink resting with respect to the crystal. It is obviously the resting kink that
possesses the largest geometrical extension. The sketch in Fig. 10.2 is, in principle,
experimentally provable. The breather solution moving with respect to the crystal
1 Here, we wish to note that these equations go over to the Lorentz invariant equations with the
transversal sound velocity c T , in the special case of straight screw dislocations in an isotropic
medium, cf. Günther [34, 35].