234
21 Violation of Relativity—The Rediscovered Crystal
fangs of the Laplacian demon: The former knows the state of the world at an arbitrary
point of time, because he exactly knows his initial state. We however now make an
astounding discovery concerning the world of the internal observers. The particles
of the internal observers are structures, structures built up from the positions of
lattice atoms. These lattice atoms themselves lay, as we know, outside of the internal
observers empirical world. They make up their space—their vacuum—the ‘ether’.
As a consequence, all the statistical fluctuations of the lattice atom positions, which
can even vary at different crystal positions depending on temperature, completely
fall out of the strict causal laws of motion that the internal observers register. The
Laplacian demon has been overpowered. The exact from its law of motion calculated
position of a kink cannot be accepted due to the statistical fluctuations of the lattice
atom positions. This calculated value only corresponds with the statistical mean
of many measurements. The internal observer registers for a single measurement a
statistical uncertainty, which was principally not reckoned with. Taken aback by this
the internal observer states, God plays dice’. The more we heat up the crystal, the
more the internal observer’s God licentiously plays dice.
What if the internal observers learn to make far more precise measurements and
are able to move to far greater frequencies using ever-increasing energies
1 ? The
internal observers would quickly arrive at the boundaries of validity of the model
assumptions formulated at the beginning of this chapter.
Even if we drop all three assumptions, we would stay to find, for those structures
on the lattice that find our interest, an equation, whose solutions are comparable to
those of the sine-Gordon equation—with one important difference: A dispersion is
created. We will now discuss this for the first of our three assumptions.
Equation (39) for the dispersion free linear chain, is bound to the condition that
many atoms are involved in a single oscillation, so that we can replace the deflection
of individual atoms with a continuous wave. If however the frequency is very large,
resulting in the wavelength becoming very small, so small that it moves into the
region of becoming the size of a distance between two atoms in a linear chain, then
only a few atoms are used in the construction of a wavelength. The approximation
(36), n N , is not fulfilled anymore, and we cannot replace the cosine term in the
strict Eq. (35) with the first two terms of its Taylor series. The angular velocity ω n is
not proportional to the wave number k n = n 2π/L anymore, and we must calculate
the signal velocity on the chain using Eq. (41), with the result (44). The wave’s
velocity of propagation c
(n)
K now depends on its frequency ω n . A dispersion is the
result. We can no longer speak of a uniform signal velocity, especially if we include
the non-linear theory of elasticity, as well as the relativistic (with the speed of light
c L ) change of mass of the lattice atoms.
The sine-Gordon equation, the central item of all our considerations, begins to
lose cohesion with increasing energies and frequencies. Without a uniform critical
velocity c o for all frequencies in the sine-Gordon equation, we would not receive the
1 The energy can only be increased by the velocity of motion and thus by the number of passages
across the zero position per second, in other words the frequency, because the amplitude of an
oscillation is restricted by the crystal’s geometry.
21 Violation of Relativity—The Rediscovered Crystal
fangs of the Laplacian demon: The former knows the state of the world at an arbitrary
point of time, because he exactly knows his initial state. We however now make an
astounding discovery concerning the world of the internal observers. The particles
of the internal observers are structures, structures built up from the positions of
lattice atoms. These lattice atoms themselves lay, as we know, outside of the internal
observers empirical world. They make up their space—their vacuum—the ‘ether’.
As a consequence, all the statistical fluctuations of the lattice atom positions, which
can even vary at different crystal positions depending on temperature, completely
fall out of the strict causal laws of motion that the internal observers register. The
Laplacian demon has been overpowered. The exact from its law of motion calculated
position of a kink cannot be accepted due to the statistical fluctuations of the lattice
atom positions. This calculated value only corresponds with the statistical mean
of many measurements. The internal observer registers for a single measurement a
statistical uncertainty, which was principally not reckoned with. Taken aback by this
the internal observer states, God plays dice’. The more we heat up the crystal, the
more the internal observer’s God licentiously plays dice.
What if the internal observers learn to make far more precise measurements and
are able to move to far greater frequencies using ever-increasing energies
1 ? The
internal observers would quickly arrive at the boundaries of validity of the model
assumptions formulated at the beginning of this chapter.
Even if we drop all three assumptions, we would stay to find, for those structures
on the lattice that find our interest, an equation, whose solutions are comparable to
those of the sine-Gordon equation—with one important difference: A dispersion is
created. We will now discuss this for the first of our three assumptions.
Equation (39) for the dispersion free linear chain, is bound to the condition that
many atoms are involved in a single oscillation, so that we can replace the deflection
of individual atoms with a continuous wave. If however the frequency is very large,
resulting in the wavelength becoming very small, so small that it moves into the
region of becoming the size of a distance between two atoms in a linear chain, then
only a few atoms are used in the construction of a wavelength. The approximation
(36), n N , is not fulfilled anymore, and we cannot replace the cosine term in the
strict Eq. (35) with the first two terms of its Taylor series. The angular velocity ω n is
not proportional to the wave number k n = n 2π/L anymore, and we must calculate
the signal velocity on the chain using Eq. (41), with the result (44). The wave’s
velocity of propagation c
(n)
K now depends on its frequency ω n . A dispersion is the
result. We can no longer speak of a uniform signal velocity, especially if we include
the non-linear theory of elasticity, as well as the relativistic (with the speed of light
c L ) change of mass of the lattice atoms.
The sine-Gordon equation, the central item of all our considerations, begins to
lose cohesion with increasing energies and frequencies. Without a uniform critical
velocity c o for all frequencies in the sine-Gordon equation, we would not receive the
1 The energy can only be increased by the velocity of motion and thus by the number of passages
across the zero position per second, in other words the frequency, because the amplitude of an
oscillation is restricted by the crystal’s geometry.