17 The Twin Paradox
195
founding the Special Theory of Relativity, came to the conclusion of the inertia
of energy, a result known to virtually everyone as the famous equation E = m c
2
(cf. also Chap. 3). We will discuss the connected processes, including those less
spectacular processes in our solid objects, based on the foundations of our general
equation of motion for dislocations (79) in Chap. 23.
The simple reason that the effects, being plastic deformations, resulting from the
energy mass equivalence inside of a solid are so relatively harmless, is that the sound
velocity, or our critical velocity c o of the sine-Gordon equation is, at least with respect
to the accessible material atomic lattices, minute in comparison with the speed of light
c L . However, under different physical conditions there may be other types of atomic
lattices—and such atomic lattices are indeed favoured by astrophysicists—where the
difference between the critical velocity c o and the speed of light c L is only very small,
therefore, taking Einstein’s Special Theory of Relativity into consideration, c o < c L ,
but c o ≈ c L . In such an atomic lattice, but not in our everyday earthly crystals, where
the whole processes are harmless, the mutual destruction of two dislocations with
opposite signs would lead to an inferno, which when compared to nuclear fusion
would make the latter look like a little blaze. The enormous dimension of such a
process simply lies in the explanation that dislocations traverse a very large part of
the crystal, in other words their space, whereas our elementary particles are confined
to space points. A dislocation mass therefore exceeds the masses of the point particles
by many powers of ten.
We should be happy that the sound velocity is so small. On the other hand, it
would be attractive to consider a crystal with a critical velocity only slightly less
that of speed of light. Here, we must of course admit that when considering such
possibilities, we overstep the limits of our self-restricted calculations and now only
argue qualitatively, whereby we inevitably accept incalculabilities and uncertainties
concerning the conclusions. For atoms moving inside of the lattice with a velocity comparable to that of light c L , their inertia m would no longer be a constant, a
constant that we took as granted, but according to Special Relativity (see Chap. 3)
would be dependent on its own velocity. Our accepted assumptions (68) of linear
theory of elasticity could no longer be upheld. However, even in this case we would
qualitatively await such states of excitation of a lattice which would be very similar
to the solutions of the wave equation and of the sine-Gordon equation discussed by
us, which originally brought us into a position where we were able to discuss the
sound velocity c T and the critical velocity c o for plastic deformations, and subsequently comparing these to the speed of light c L . If we however assume a critical
velocity c o ≈ c L , c o < c L , then the clocks of the internal observer moving with the
velocity v inside of the crystal, when observed from outside, almost go behind by the
speed of light, thus they behave almost like the outside observer’s clocks, as Special
Relativity demands. Yet those breather clocks moving inside of the crystal and those
resting relative to the crystal are, when observed from outside, physically completely
different clocks, and their difference in pace therefore cannot result in a conclusion
being made by an outside observer, on the behaviour of a moving clock since outside
Special Relativity is not applicable in this case, as discussed in Chap. 11. The numerical value of the sound velocity is for this set of circumstances of no importance.
195
founding the Special Theory of Relativity, came to the conclusion of the inertia
of energy, a result known to virtually everyone as the famous equation E = m c
2
(cf. also Chap. 3). We will discuss the connected processes, including those less
spectacular processes in our solid objects, based on the foundations of our general
equation of motion for dislocations (79) in Chap. 23.
The simple reason that the effects, being plastic deformations, resulting from the
energy mass equivalence inside of a solid are so relatively harmless, is that the sound
velocity, or our critical velocity c o of the sine-Gordon equation is, at least with respect
to the accessible material atomic lattices, minute in comparison with the speed of light
c L . However, under different physical conditions there may be other types of atomic
lattices—and such atomic lattices are indeed favoured by astrophysicists—where the
difference between the critical velocity c o and the speed of light c L is only very small,
therefore, taking Einstein’s Special Theory of Relativity into consideration, c o < c L ,
but c o ≈ c L . In such an atomic lattice, but not in our everyday earthly crystals, where
the whole processes are harmless, the mutual destruction of two dislocations with
opposite signs would lead to an inferno, which when compared to nuclear fusion
would make the latter look like a little blaze. The enormous dimension of such a
process simply lies in the explanation that dislocations traverse a very large part of
the crystal, in other words their space, whereas our elementary particles are confined
to space points. A dislocation mass therefore exceeds the masses of the point particles
by many powers of ten.
We should be happy that the sound velocity is so small. On the other hand, it
would be attractive to consider a crystal with a critical velocity only slightly less
that of speed of light. Here, we must of course admit that when considering such
possibilities, we overstep the limits of our self-restricted calculations and now only
argue qualitatively, whereby we inevitably accept incalculabilities and uncertainties
concerning the conclusions. For atoms moving inside of the lattice with a velocity comparable to that of light c L , their inertia m would no longer be a constant, a
constant that we took as granted, but according to Special Relativity (see Chap. 3)
would be dependent on its own velocity. Our accepted assumptions (68) of linear
theory of elasticity could no longer be upheld. However, even in this case we would
qualitatively await such states of excitation of a lattice which would be very similar
to the solutions of the wave equation and of the sine-Gordon equation discussed by
us, which originally brought us into a position where we were able to discuss the
sound velocity c T and the critical velocity c o for plastic deformations, and subsequently comparing these to the speed of light c L . If we however assume a critical
velocity c o ≈ c L , c o < c L , then the clocks of the internal observer moving with the
velocity v inside of the crystal, when observed from outside, almost go behind by the
speed of light, thus they behave almost like the outside observer’s clocks, as Special
Relativity demands. Yet those breather clocks moving inside of the crystal and those
resting relative to the crystal are, when observed from outside, physically completely
different clocks, and their difference in pace therefore cannot result in a conclusion
being made by an outside observer, on the behaviour of a moving clock since outside
Special Relativity is not applicable in this case, as discussed in Chap. 11. The numerical value of the sound velocity is for this set of circumstances of no importance.