254
23 A Particle Solution—The Inertia of Energy
q = q
III
(x, t), with whose help we were able to define the clocks of the internal
observers in Chaps. 9 and 10. The breather not only possesses velocity as an object,
but it also possesses internal motion, namely its own oscillations. The calculations
for the verification of the particle property of a breather solution would therefore be
somewhat more complicated. We therefore refer the reader to the original paper, cf.
Günther [37].
Without actually delving deeper into the problem, we note that non-linear equations generally contain the laws of motion of their particle solutions. The sum of two
solutions of the sine-Gordon equation (88) in contrast to electrodynamics because of
the non-linear term sin(
2π
a
q) cannot simply just be another solution of this equation.
Two solutions can only be successfully combined to another solution if one determines their ‘correct’ motions (and the mutually caused deformations) with respect to
one another. Keeping this in mind, the sine-Gordon equation contains the dynamics
of its particles. The Maxwellian equations are linear, and one only has to supplementarily postulate the relativistically corrected Newtonian mechanics in order to be
able to calculate the electromagnetic field of two charged particles through time. The
relativistic, non-linear sine-Gordon equation with its critical velocity c o inside of a
crystal on the one hand corresponds to the system composed of Maxwellian equations and the relativistic, mechanical equations of motion with the speed of light c L
on the other hand. The equivalence, in the boundaries of our one-dimensional model
between Einstein’s Special Theory of Relativity of our physical spacetime and the
Special Theory of Relativity on a solid is complete.
23 A Particle Solution—The Inertia of Energy
q = q
III
(x, t), with whose help we were able to define the clocks of the internal
observers in Chaps. 9 and 10. The breather not only possesses velocity as an object,
but it also possesses internal motion, namely its own oscillations. The calculations
for the verification of the particle property of a breather solution would therefore be
somewhat more complicated. We therefore refer the reader to the original paper, cf.
Günther [37].
Without actually delving deeper into the problem, we note that non-linear equations generally contain the laws of motion of their particle solutions. The sum of two
solutions of the sine-Gordon equation (88) in contrast to electrodynamics because of
the non-linear term sin(
2π
a
q) cannot simply just be another solution of this equation.
Two solutions can only be successfully combined to another solution if one determines their ‘correct’ motions (and the mutually caused deformations) with respect to
one another. Keeping this in mind, the sine-Gordon equation contains the dynamics
of its particles. The Maxwellian equations are linear, and one only has to supplementarily postulate the relativistically corrected Newtonian mechanics in order to be
able to calculate the electromagnetic field of two charged particles through time. The
relativistic, non-linear sine-Gordon equation with its critical velocity c o inside of a
crystal on the one hand corresponds to the system composed of Maxwellian equations and the relativistic, mechanical equations of motion with the speed of light c L
on the other hand. The equivalence, in the boundaries of our one-dimensional model
between Einstein’s Special Theory of Relativity of our physical spacetime and the
Special Theory of Relativity on a solid is complete.