56
6 Lattice and the Continuum
critical vacuum speed of light c L . Those who know the Special Theory of Relativity,
immediately know that two critical velocities for Special Relativity are impossible.
There is no Lorentz transformation with its characteristic consequences of time
dilatation and Lorentz contractions for two critical velocities. In the light of this,
sound is not somewhat like light, at least not inside of an atomic lattice. One could at
best think of liquid or gaseous substances that also only possess one critical velocity.
These however, do not find our interests, because they have no internal mechanical
structures. These mediums remain empty for further questions concerning Special
Relativity.
In our case, it will not be the theory of elasticity, but plasticity on which the emphasis lies, and this is determined by the properties of one-dimensional dislocations. As
we will see in the following chapter, the dislocations inside of a crystal actually,
in a sense, define their own one-dimensional lattice, from which we consider the
transversal deflections, which then leads to one single defined critical velocity.
Of course, there is a relation between dislocations and elastic deformations, but
these we can neglect in the first approximation. In Chaps. 26–27, we will also discuss
the relations between the dislocations and the elasticity of three- dimensional lattices,
including those that have more than two critical velocities. There we will prove that
we can separate a well-defined part, the so-called structural eigen stresses, from the
elastic deformations caused by dislocations. These structural eigen stresses are, in
comparison to all elastic deformations, dependent only upon one single critical velocity, which has the surprising consequence that in the area of plasticity including the
‘structural elasticity’, Lorentz symmetry of a solid is valid. In other words, expressed
in the language of physics: In a solid, we discover a symmetry group which is isomorphic with the symmetry group of our physical spacetime. Here, we wish to express
that the terms ‘Lorentz transformations’, ‘Lorentz symmetry’, ‘Lorentz factor’, etc.
are in fact reserved in literature for corresponding terms that possess the parameter
of the speed of light, and we should therefore use a terminus ‘proper Minkowski
rotation’, compare to J. A. Schouten [85], Chap. I. H. F. Goenner [27], Chap. 1, has
pointed to the isomorphism of these correlations with arbitrary changes in the finite
critical velocity. We will keep the terms ‘Lorentz transformation’, ‘Lorentz symmetry’, etc. even for critical velocities occurring inside a solid, because here confusion
cannot occur, (cf. also Chap. 13) and thus use A. Seeger’s terminology, see for example Seeger [86]. For the relations between Special Relativity and classical field theory
see H. F. Goenner [28].
The various sound velocities for the complete theory of elasticity, however, lead
once again to a breaking of the Lorentz symmetry for the complete system of dislocations and elastic deformations. The valid Lorentz symmetry of a solid will thus
not always be easy to recognise. It is thus of interest to pick out such a class of physical phenomena for solids, for which this Lorentz symmetry applies in its complete
clearness. To this end, we will next examine crystals and their dislocations more
closely in the following chapter.
6 Lattice and the Continuum
critical vacuum speed of light c L . Those who know the Special Theory of Relativity,
immediately know that two critical velocities for Special Relativity are impossible.
There is no Lorentz transformation with its characteristic consequences of time
dilatation and Lorentz contractions for two critical velocities. In the light of this,
sound is not somewhat like light, at least not inside of an atomic lattice. One could at
best think of liquid or gaseous substances that also only possess one critical velocity.
These however, do not find our interests, because they have no internal mechanical
structures. These mediums remain empty for further questions concerning Special
Relativity.
In our case, it will not be the theory of elasticity, but plasticity on which the emphasis lies, and this is determined by the properties of one-dimensional dislocations. As
we will see in the following chapter, the dislocations inside of a crystal actually,
in a sense, define their own one-dimensional lattice, from which we consider the
transversal deflections, which then leads to one single defined critical velocity.
Of course, there is a relation between dislocations and elastic deformations, but
these we can neglect in the first approximation. In Chaps. 26–27, we will also discuss
the relations between the dislocations and the elasticity of three- dimensional lattices,
including those that have more than two critical velocities. There we will prove that
we can separate a well-defined part, the so-called structural eigen stresses, from the
elastic deformations caused by dislocations. These structural eigen stresses are, in
comparison to all elastic deformations, dependent only upon one single critical velocity, which has the surprising consequence that in the area of plasticity including the
‘structural elasticity’, Lorentz symmetry of a solid is valid. In other words, expressed
in the language of physics: In a solid, we discover a symmetry group which is isomorphic with the symmetry group of our physical spacetime. Here, we wish to express
that the terms ‘Lorentz transformations’, ‘Lorentz symmetry’, ‘Lorentz factor’, etc.
are in fact reserved in literature for corresponding terms that possess the parameter
of the speed of light, and we should therefore use a terminus ‘proper Minkowski
rotation’, compare to J. A. Schouten [85], Chap. I. H. F. Goenner [27], Chap. 1, has
pointed to the isomorphism of these correlations with arbitrary changes in the finite
critical velocity. We will keep the terms ‘Lorentz transformation’, ‘Lorentz symmetry’, etc. even for critical velocities occurring inside a solid, because here confusion
cannot occur, (cf. also Chap. 13) and thus use A. Seeger’s terminology, see for example Seeger [86]. For the relations between Special Relativity and classical field theory
see H. F. Goenner [28].
The various sound velocities for the complete theory of elasticity, however, lead
once again to a breaking of the Lorentz symmetry for the complete system of dislocations and elastic deformations. The valid Lorentz symmetry of a solid will thus
not always be easy to recognise. It is thus of interest to pick out such a class of physical phenomena for solids, for which this Lorentz symmetry applies in its complete
clearness. To this end, we will next examine crystals and their dislocations more
closely in the following chapter.