Chapter 21
Violation of Relativity—The
Rediscovered Crystal
For an internal observer of a crystal, all of the phenomena observed here, such as
plastic and certain elastic deformations of a crystal, represent motions in a relativistic
spacetime continuum. We discovered this in Chap. 15 and confirmed this in Chap. 18
where we discussed the Doppler effect. For the outside observer, the crystal is just
the place where all these phenomena occur, and he can understand these phenomena
with the help of the crystal. Furthermore, the crystal makes him clear the internal
observer’s explanations and descriptions for these phenomena. This crystal has, from
the internal observer’s point of view, no state of motion and is principally nothing
other than a relativistic spacetime continuum. This is valid as long as all assumptions
that are needed for this ideal model description are fulfilled. These assumptions are
as follows:
1. The primary discrete distribution of the elastically coupled masses is replaced with
a continuous distribution according to (84).
2. We calculate using the linearised theory of elasticity according to (85) and also
using the equation c T = c o if we include transversal sound waves.
3. The principal dependence m = m o /
1 − v 2 /c 2
o of the lattice atom’s mass m on
their velocity v according to the Einstein’s Special Theory of Relativity is ignored.
The velocity v of the oscillating atoms does not become larger than the speed of
sound c T , of which we assume that it is many times smaller than the speed of light
c L , so that v
2
/c
2
L < c
2
T /c
2
L 1 .
We thus work with non-relativistic mechanics for the lattice atoms. Structures
on this lattice, kinks for example, show a relativistic behaviour, however, where the
sound velocity is the limit, but not the light velocity.
The law of motion (79) for dislocations together with Eqs. (80) and (81) for interaction forces results in the sine-Gordon equations (88) and (95) with the critical
velocity c o .
We will stay here and consider for a whilst the area, defined by the strict model
assumption 1–3. We will alternatively change our position from that of an internal
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_21
231
Violation of Relativity—The
Rediscovered Crystal
For an internal observer of a crystal, all of the phenomena observed here, such as
plastic and certain elastic deformations of a crystal, represent motions in a relativistic
spacetime continuum. We discovered this in Chap. 15 and confirmed this in Chap. 18
where we discussed the Doppler effect. For the outside observer, the crystal is just
the place where all these phenomena occur, and he can understand these phenomena
with the help of the crystal. Furthermore, the crystal makes him clear the internal
observer’s explanations and descriptions for these phenomena. This crystal has, from
the internal observer’s point of view, no state of motion and is principally nothing
other than a relativistic spacetime continuum. This is valid as long as all assumptions
that are needed for this ideal model description are fulfilled. These assumptions are
as follows:
1. The primary discrete distribution of the elastically coupled masses is replaced with
a continuous distribution according to (84).
2. We calculate using the linearised theory of elasticity according to (85) and also
using the equation c T = c o if we include transversal sound waves.
3. The principal dependence m = m o /
1 − v 2 /c 2
o of the lattice atom’s mass m on
their velocity v according to the Einstein’s Special Theory of Relativity is ignored.
The velocity v of the oscillating atoms does not become larger than the speed of
sound c T , of which we assume that it is many times smaller than the speed of light
c L , so that v
2
/c
2
L < c
2
T /c
2
L 1 .
We thus work with non-relativistic mechanics for the lattice atoms. Structures
on this lattice, kinks for example, show a relativistic behaviour, however, where the
sound velocity is the limit, but not the light velocity.
The law of motion (79) for dislocations together with Eqs. (80) and (81) for interaction forces results in the sine-Gordon equations (88) and (95) with the critical
velocity c o .
We will stay here and consider for a whilst the area, defined by the strict model
assumption 1–3. We will alternatively change our position from that of an internal
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_21
231