15 The Principle of Relativity: The Lost Crystal
143
theory. We also discovered the second Einsteinian postulate, the constancy of critical
velocity, in the same manner in Chap. 12. Let us once more summarise this:
From law (79) for the motion of dislocations relative to the lattice, we can derive
the sine-Gordon equation (88) with its critical velocity c o defined by the parameters
of the lattice, in the same way as we derived the wave equation (61) from Eq. (31)
for a linear chain. The periodical potential of the lattice, in which the linear chain
of dislocation masses is embedded, delivers the sine term to us. Due to this nonlinearity, the sine-Gordon equation possesses soliton solutions, with which we can
define natural measuring-rods and clocks. We thus arrive at a remarkable conclusion:
The contraction of length and the time dilatation of moving measuring-rods or clocks, respectively, are reduced to elastic interactions in the linear chains of dislocations in the lattice.
Thus, the circle is closed. The mechanical oscillations, with which we started
in Chap. 4, give us a physical mechanism for explaining the elementary effects of
Special Relativity: Length contraction and time dilatation as registered by the internal
observer in a crystal can be explained using Hooke’s law for the constituents of this
crystal,, i.e. the atoms that remain invisible for the internal observers—a curious
connection.
We will repeat this so that no misunderstanding can arise: The effects of Einstein’s
Special Relativity of our physical spacetime caused by the speed of light have nothing
at all to do with elastic interactions of the atoms in a crystalline solid. We will come
back to this point later on.
Using our elementary principle of relativity, we are now able to define simultaneity
in the ‘moving reference system’, that makes the Lorentz invariance of the sineGordon equation visible. Here, we note together with Poincaré [73, 74] that this
definition of simultaneity is not altogether necessary; however, using it easily allows
for a special symmetrical formulation of the physical laws. Once this has been done,
we derive the constancy of the critical velocity for all observers moving uniformly
towards each other, which then leads to the whole Einsteinian principle of relativity,
the complete equivalence of all reference systems moving uniformly towards each
other.
Here, one could protest, saying that we are always only talking about the sineGordon equation, whereas the physics inside of a crystal deals with several other
states that are not described by this equation. Correct is that we do in fact strictly
limit the range of states to those structures described by the sine-Gordon equation, that
as we know, describe the deviation from the ‘vacuum state’ of the lattice (compare to
Chap. 8). The number of these states is nevertheless immensely large. The non-linear
character of the sine-Gordon equation belongs to the comprehensive area of soliton
physics, where new solutions are being made using new methods. For the first view
of these, see the paper of A. Seeger [86]. In the light of this, limiting our states does
not seem so narrow.
Dislocation structures belong to the area of plasticity or micro-plasticity, respectively, that always also lead to elastic deformations of the lattice, that we ignored
here. This coupling of dislocations with the elastic deformations of the lattice is
determined by the totality of the parameters of Hooke’s tensor. This means for the
143
theory. We also discovered the second Einsteinian postulate, the constancy of critical
velocity, in the same manner in Chap. 12. Let us once more summarise this:
From law (79) for the motion of dislocations relative to the lattice, we can derive
the sine-Gordon equation (88) with its critical velocity c o defined by the parameters
of the lattice, in the same way as we derived the wave equation (61) from Eq. (31)
for a linear chain. The periodical potential of the lattice, in which the linear chain
of dislocation masses is embedded, delivers the sine term to us. Due to this nonlinearity, the sine-Gordon equation possesses soliton solutions, with which we can
define natural measuring-rods and clocks. We thus arrive at a remarkable conclusion:
The contraction of length and the time dilatation of moving measuring-rods or clocks, respectively, are reduced to elastic interactions in the linear chains of dislocations in the lattice.
Thus, the circle is closed. The mechanical oscillations, with which we started
in Chap. 4, give us a physical mechanism for explaining the elementary effects of
Special Relativity: Length contraction and time dilatation as registered by the internal
observer in a crystal can be explained using Hooke’s law for the constituents of this
crystal,, i.e. the atoms that remain invisible for the internal observers—a curious
connection.
We will repeat this so that no misunderstanding can arise: The effects of Einstein’s
Special Relativity of our physical spacetime caused by the speed of light have nothing
at all to do with elastic interactions of the atoms in a crystalline solid. We will come
back to this point later on.
Using our elementary principle of relativity, we are now able to define simultaneity
in the ‘moving reference system’, that makes the Lorentz invariance of the sineGordon equation visible. Here, we note together with Poincaré [73, 74] that this
definition of simultaneity is not altogether necessary; however, using it easily allows
for a special symmetrical formulation of the physical laws. Once this has been done,
we derive the constancy of the critical velocity for all observers moving uniformly
towards each other, which then leads to the whole Einsteinian principle of relativity,
the complete equivalence of all reference systems moving uniformly towards each
other.
Here, one could protest, saying that we are always only talking about the sineGordon equation, whereas the physics inside of a crystal deals with several other
states that are not described by this equation. Correct is that we do in fact strictly
limit the range of states to those structures described by the sine-Gordon equation, that
as we know, describe the deviation from the ‘vacuum state’ of the lattice (compare to
Chap. 8). The number of these states is nevertheless immensely large. The non-linear
character of the sine-Gordon equation belongs to the comprehensive area of soliton
physics, where new solutions are being made using new methods. For the first view
of these, see the paper of A. Seeger [86]. In the light of this, limiting our states does
not seem so narrow.
Dislocation structures belong to the area of plasticity or micro-plasticity, respectively, that always also lead to elastic deformations of the lattice, that we ignored
here. This coupling of dislocations with the elastic deformations of the lattice is
determined by the totality of the parameters of Hooke’s tensor. This means for the