15 The Principle of Relativity: The Lost Crystal
151
as the internal observers inside of our crystal are bound to their space. Thus we find
ourselves together with the most clever and refined mathematicians, physicists and
engineers in experimental atomic research centres and try with the utmost effort and
utmost energy to poke our vacuum a little in order to, with a bit of luck knock a bit
out of this unbelievingly large source.
Physically, on the basis of our crystal lattice, the theory of relativity reaches one of
the limits of its validity when it comes to length contractions and their corresponding
Lorentz factors γ =
1 − v 2 /c 2
o leading to distances lower than the lattice parameter of the crystal. Principally, such distances cannot be measured using kinks on
dislocations, because the field theoretical description of the kink with the help of the
sine-Gordon equation due to the used continuum approximation (85) is simply based
on the geometrical structures of these lines built up of a large number of lattice atoms.
We are thus in a position to be able to predict the moment when Special Relativity
loses its validity on the crystal lattice. We will return to this in Chap. 21.
Here in this connection, it is remarkable that we know of limiting lengths in our
‘outside physics’, so that geometric distances smaller than these lengths become
problematic, or cannot physically be meaningfully determined at all. An absolute
lowest value for space distances according to today’s science was already introduced
in 1906 by M. Planck. This elementary length l o named after him is made up of
three fundamental parameters, Planck’s constant , the gravitational constant f and
the speed of light c L . Together, these three parameters define Planck’s elementary
length l o =
f /c
3
L = 1, 6 · 10
−33 cm. W. Heisenberg even proposed the Compton
wavelength λ c of a nucleus with the mass m, λ c =
2π
mc L
= 1, 32 · 10
−13 cm, a value
twenty times larger than Planck’s length, as the lowest possible limit past which no
meaningful geometric measurements could be made, cf. also H. Treder [94, 95].
A space lattice made up of physical constituents whatever type they may be,
whose next neighbours are positioned at a distance equivalent to the Heisenberg, or
even the Planck length, could indeed according to our considerations supply us with
the physical background needed not only for Special Relativity but also for further
elementary properties of matter. Here, we explicitly repeat that the hypothetical lattice
‘constituents’ of our physical space is under no circumstances inertial masses of our
physical space such as atoms or elementary particles. The masses of our physical
space would be structures on such a lattice, so that the basic kinematic properties
of our masses, being that masses move relative to each other with a certain inertia,
could in principle not be attributed to the constituents of such a lattice. Nevertheless,
such a lattice would be physical real and would assert fundamental influence on the
physics of our matter—this would correspond to the statement made by A. Einstein
[12, 13] in 1920 concerning the ether problem, ‘To deny the ether is ultimately to
assume that empty space has no physical qualities whatever. The fundamental facts
of mechanics do not harmonise with this view’. The hypothetical discussion using a
lattice for our physical space has nothing to do with the introduction of a classical
ether. It is just a vague idea of a possible physical background of our relativistic
spacetime structure. In order to become more involved in this subject, we would
need to include the quantum theory and maybe even the theory of gravity.
151
as the internal observers inside of our crystal are bound to their space. Thus we find
ourselves together with the most clever and refined mathematicians, physicists and
engineers in experimental atomic research centres and try with the utmost effort and
utmost energy to poke our vacuum a little in order to, with a bit of luck knock a bit
out of this unbelievingly large source.
Physically, on the basis of our crystal lattice, the theory of relativity reaches one of
the limits of its validity when it comes to length contractions and their corresponding
Lorentz factors γ =
1 − v 2 /c 2
o leading to distances lower than the lattice parameter of the crystal. Principally, such distances cannot be measured using kinks on
dislocations, because the field theoretical description of the kink with the help of the
sine-Gordon equation due to the used continuum approximation (85) is simply based
on the geometrical structures of these lines built up of a large number of lattice atoms.
We are thus in a position to be able to predict the moment when Special Relativity
loses its validity on the crystal lattice. We will return to this in Chap. 21.
Here in this connection, it is remarkable that we know of limiting lengths in our
‘outside physics’, so that geometric distances smaller than these lengths become
problematic, or cannot physically be meaningfully determined at all. An absolute
lowest value for space distances according to today’s science was already introduced
in 1906 by M. Planck. This elementary length l o named after him is made up of
three fundamental parameters, Planck’s constant , the gravitational constant f and
the speed of light c L . Together, these three parameters define Planck’s elementary
length l o =
f /c
3
L = 1, 6 · 10
−33 cm. W. Heisenberg even proposed the Compton
wavelength λ c of a nucleus with the mass m, λ c =
2π
mc L
= 1, 32 · 10
−13 cm, a value
twenty times larger than Planck’s length, as the lowest possible limit past which no
meaningful geometric measurements could be made, cf. also H. Treder [94, 95].
A space lattice made up of physical constituents whatever type they may be,
whose next neighbours are positioned at a distance equivalent to the Heisenberg, or
even the Planck length, could indeed according to our considerations supply us with
the physical background needed not only for Special Relativity but also for further
elementary properties of matter. Here, we explicitly repeat that the hypothetical lattice
‘constituents’ of our physical space is under no circumstances inertial masses of our
physical space such as atoms or elementary particles. The masses of our physical
space would be structures on such a lattice, so that the basic kinematic properties
of our masses, being that masses move relative to each other with a certain inertia,
could in principle not be attributed to the constituents of such a lattice. Nevertheless,
such a lattice would be physical real and would assert fundamental influence on the
physics of our matter—this would correspond to the statement made by A. Einstein
[12, 13] in 1920 concerning the ether problem, ‘To deny the ether is ultimately to
assume that empty space has no physical qualities whatever. The fundamental facts
of mechanics do not harmonise with this view’. The hypothetical discussion using a
lattice for our physical space has nothing to do with the introduction of a classical
ether. It is just a vague idea of a possible physical background of our relativistic
spacetime structure. In order to become more involved in this subject, we would
need to include the quantum theory and maybe even the theory of gravity.