Chapter 12
The Measurement of the Critical Velocity
We now arrive at the most sensitive point of the Special Theory of Relativity: The
measurement of the critical signal velocity. The testing of the speed of light, the
critical velocity of electromagnetic phenomena was the main reason why the violent
dispute concerning the ether had arose. The unexpected result of this dispute was
Einstein’s postulate in his paper from 1905, cf. Einstein [14, 15], the principle of the
universal constancy of this signal velocity, which led as a consequence to a revised
formulation of the whole of physics.
Here, we will understand the quantity c o in the sine-Gordon equation by the
critical signal velocity. Primarily, we have no real picture of this velocity inside of
a mechanical continuum. We are far more used to dealing with sound velocities.
The sound velocities do give us an orientation of the magnitude of c o . In order to see
this, we consider the approximation of the underlying crystal replaced by an isotropic
continuum. In many cases, it is exactly this isotropic continuum that delivers a useful
concept in describing mechanical processes inside of a solid. There are two different
sound velocities inside of an isotropic continuum (see Chap. 25). The first is the
transversal polarised sound wave, where the oscillating vector s is orthogonal with
the direction of propagation k of the wave. The second is the longitudinal wave,
where the oscillating vector s is parallel with the direction of propagation k. Let
us consider transversal sound waves, those waves that move through the medium
with the transversal sound velocity c T . This velocity c T is closely related to the
critical velocity c o in the sine-Gordon equation. Here, we remind ourselves that
an excitation propagating along the dislocation line according to the sine-Gordon
equation is just as much a transversal displacement out of this dislocation direction
(see Chap. 7) as is the transversal displacement of the linear chain propagating with
the transversal sound velocitiy (see Chap. 6). Furthermore, the linear terms of the
Frenkel-Kontorova equation (77), which approximate the sine-Gordon equation,
describe the linear chain of the lattice atoms in the vicinity of the dislocation line.
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_12
111
The Measurement of the Critical Velocity
We now arrive at the most sensitive point of the Special Theory of Relativity: The
measurement of the critical signal velocity. The testing of the speed of light, the
critical velocity of electromagnetic phenomena was the main reason why the violent
dispute concerning the ether had arose. The unexpected result of this dispute was
Einstein’s postulate in his paper from 1905, cf. Einstein [14, 15], the principle of the
universal constancy of this signal velocity, which led as a consequence to a revised
formulation of the whole of physics.
Here, we will understand the quantity c o in the sine-Gordon equation by the
critical signal velocity. Primarily, we have no real picture of this velocity inside of
a mechanical continuum. We are far more used to dealing with sound velocities.
The sound velocities do give us an orientation of the magnitude of c o . In order to see
this, we consider the approximation of the underlying crystal replaced by an isotropic
continuum. In many cases, it is exactly this isotropic continuum that delivers a useful
concept in describing mechanical processes inside of a solid. There are two different
sound velocities inside of an isotropic continuum (see Chap. 25). The first is the
transversal polarised sound wave, where the oscillating vector s is orthogonal with
the direction of propagation k of the wave. The second is the longitudinal wave,
where the oscillating vector s is parallel with the direction of propagation k. Let
us consider transversal sound waves, those waves that move through the medium
with the transversal sound velocity c T . This velocity c T is closely related to the
critical velocity c o in the sine-Gordon equation. Here, we remind ourselves that
an excitation propagating along the dislocation line according to the sine-Gordon
equation is just as much a transversal displacement out of this dislocation direction
(see Chap. 7) as is the transversal displacement of the linear chain propagating with
the transversal sound velocitiy (see Chap. 6). Furthermore, the linear terms of the
Frenkel-Kontorova equation (77), which approximate the sine-Gordon equation,
describe the linear chain of the lattice atoms in the vicinity of the dislocation line.
© The Editor(s) (if applicable) and The Author(s), under exclusive
license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_12
111