26 Eigen Stresses and Dislocations
291
with respect to the lattice, an inertial mass m, that is connected to the energy E via
the Einsteinian equation E = m c
2
T . Even for the inertial mass m of the field created
by a screw dislocation the strict relativistic equation m = m o /
1 − v 2 /c
2
T applies,
if we designate the inertial mass of the field of the stationary dislocation as m o .
These relationships are analogue to those of the electron with its electromagnetic
field. One can for example relatively easily calculate that the strain velocity field
(401) of a dislocation can be summarised as a Lorentz tensor, just as the electromagnetic field of an electron can, cf. Günther [34]. We now have to add the field mass to
the bare mass of a dislocation that we numerically approximated and introduced in
Chap. 8, as well as in Eq. (295). If we once again accept a correspondence between the
characteristic velocity c o of the sine-Gordon equation and the transversal speed of
sound c T , c o = c T , then both mass portions cannot be differentiated because of their
identical velocity dependencies; these are facts that we exactly find confirmed when
dealing with electrons. The field Eqs. (397) and (398) thus describe a physical situation that is completely analogue to the conditions concerning electrons according
to the Maxwell-Lorentz theory with all of its mathematical consequences. Here,we
especially think of those consequences that arise, if quantum theory is taken into
account with the consequence of the secondary relativistic effects such as pair creation and vacuum polarisation. The creation of a electron-positron pairs in quantum
electrodynamics corresponds to the creation of elementary dislocation loops in the
theory of dislocations, cf. Günther [33, 35].
In Chaps. 9–21,we developed a complete Special Theory of Relativity for the
internal observers of our crystal on the basis of the sine-Gordon equation. We did
this by restricting the number of classes of competitive mechanical phenomena. We
restricted ourselves to the solutions of the sine-Gordon equation and ignored the
creation of elastic deformations by dislocations. We now saw that these deformations are determined by those equations according to (397) and (398) that fulfil all
demands made by a relativistic theory. Can we therefore include elastic deformations
caused by dislocations in our relativistic considerations inside of the crystal? Can the
internal observers of our crystal support their measurements and thoughts on elastic
deformations, as well as plastic deformations? Such a conclusion would be too far
fetched here. Equations (397) and (398) only apply to a strongly restricted class of
dislocations, the straight screw dislocations in an isotropic, unbounded continuum.
Not even the straight screw dislocation with a kink belong in this class!
But we can show that the part of elastic deformations bound structurally to dislocations, bound to any dislocations, is in fact determined by only relativistic wave
equations with one single signal velocity. These facts will be examined by us in the
next chapter. In the view of further illustrations to this problem,we must refer the
reader to Günther [32, 33].
It has been know for quite a whilst, here we refer the reader to a paper of Eshelby
[19] from 1949 that the eigen stresses of dislocations show certain relicts of relativistic
behaviour. These relicts gradually dissipate with the increasing complexity of Hooke’s
tensor. We only discover unrestricted relativistic behaviour for straight screw dislocations in an isotropic, unbounded medium, as seen above. It does not good putting
291
with respect to the lattice, an inertial mass m, that is connected to the energy E via
the Einsteinian equation E = m c
2
T . Even for the inertial mass m of the field created
by a screw dislocation the strict relativistic equation m = m o /
1 − v 2 /c
2
T applies,
if we designate the inertial mass of the field of the stationary dislocation as m o .
These relationships are analogue to those of the electron with its electromagnetic
field. One can for example relatively easily calculate that the strain velocity field
(401) of a dislocation can be summarised as a Lorentz tensor, just as the electromagnetic field of an electron can, cf. Günther [34]. We now have to add the field mass to
the bare mass of a dislocation that we numerically approximated and introduced in
Chap. 8, as well as in Eq. (295). If we once again accept a correspondence between the
characteristic velocity c o of the sine-Gordon equation and the transversal speed of
sound c T , c o = c T , then both mass portions cannot be differentiated because of their
identical velocity dependencies; these are facts that we exactly find confirmed when
dealing with electrons. The field Eqs. (397) and (398) thus describe a physical situation that is completely analogue to the conditions concerning electrons according
to the Maxwell-Lorentz theory with all of its mathematical consequences. Here,we
especially think of those consequences that arise, if quantum theory is taken into
account with the consequence of the secondary relativistic effects such as pair creation and vacuum polarisation. The creation of a electron-positron pairs in quantum
electrodynamics corresponds to the creation of elementary dislocation loops in the
theory of dislocations, cf. Günther [33, 35].
In Chaps. 9–21,we developed a complete Special Theory of Relativity for the
internal observers of our crystal on the basis of the sine-Gordon equation. We did
this by restricting the number of classes of competitive mechanical phenomena. We
restricted ourselves to the solutions of the sine-Gordon equation and ignored the
creation of elastic deformations by dislocations. We now saw that these deformations are determined by those equations according to (397) and (398) that fulfil all
demands made by a relativistic theory. Can we therefore include elastic deformations
caused by dislocations in our relativistic considerations inside of the crystal? Can the
internal observers of our crystal support their measurements and thoughts on elastic
deformations, as well as plastic deformations? Such a conclusion would be too far
fetched here. Equations (397) and (398) only apply to a strongly restricted class of
dislocations, the straight screw dislocations in an isotropic, unbounded continuum.
Not even the straight screw dislocation with a kink belong in this class!
But we can show that the part of elastic deformations bound structurally to dislocations, bound to any dislocations, is in fact determined by only relativistic wave
equations with one single signal velocity. These facts will be examined by us in the
next chapter. In the view of further illustrations to this problem,we must refer the
reader to Günther [32, 33].
It has been know for quite a whilst, here we refer the reader to a paper of Eshelby
[19] from 1949 that the eigen stresses of dislocations show certain relicts of relativistic
behaviour. These relicts gradually dissipate with the increasing complexity of Hooke’s
tensor. We only discover unrestricted relativistic behaviour for straight screw dislocations in an isotropic, unbounded medium, as seen above. It does not good putting