11 A Clock Paradox
109
clocks resting in o , as well as inside of the crystal his known equation for time
dilatation containing the expression c L , the speed of light ˜
t = t
1 − v 2 /c
2
L . There
is thus no contradiction. Does this mean that our Eq. (121) for the time dilatation of
breather clocks is worthless? Naturally not.
We examine an infinitely extended crystal as well as the observers inside of this
crystal. This representation of ‘internal observers’ as we say goes back to Einstein
and was used by him in a demonstration of his ideas on the General Theory of
Relativity. Einstein argued using his ‘two-dimensional creatures’, his ‘flat beings’
on a sphere, see for example L. Infeld [42], A. Einstein [17]. Just as was used in the
case above, we will use such an ideal construction in order to show to what degree
the development of our theory depends on the possible methods of measurement.
2
For our internal observers, the crystal as a whole does not move. We could also
formulate it as follows. The small internal observers inside the crystal see this as an
infinitely large world. They do now ask themselves about the motion of their world.
Only the craziest and audacious philosophers amongst them would think of or even
mention the possibility—leading to the accusation of blasphemy and insanity and
probably ending up being burnt at the stake as a warlock—that their world is finite.
The outside observers of Einstein’s Special Relativity and the internal observers
inside the crystal work with two completely different classes of reference systems.
There is only one common element if they find themselves in the same preferred
frame o . This is the solution to the paradox.
Let us, however, now continue with the theoretical conceptions the observers
inside of the infinitely large widths of the crystal make. Here, we will consequently
use only those ‘mechanical instruments’ defined by the sine-Gordon equation and
will especially not use the speed of light. We will make sure that all the ‘instruments’
we use are of mechanical origin in our crystal. These are the many structures above
the ground state, the energetically distinguished state of an ideal lattice.
Using A. Seeger [86], we will not only understand this ground state (the vacuum)
as an ideal lattice, but we will also count the infinitely long dislocations passing as
straight lines through the lattice as the vacuum, cf. (97).
2 This Einsteinian concept of an internal observer was used in a different context in continuum
mechanics by E. Kröner [51, 52]. E. Kröner refers to the problems connected to the spatial
measurements inside of the crystal. The problem is thus completely different to our own. The
Krönerian internal observers should not be mistaken with our internal observers.
109
clocks resting in o , as well as inside of the crystal his known equation for time
dilatation containing the expression c L , the speed of light ˜
t = t
1 − v 2 /c
2
L . There
is thus no contradiction. Does this mean that our Eq. (121) for the time dilatation of
breather clocks is worthless? Naturally not.
We examine an infinitely extended crystal as well as the observers inside of this
crystal. This representation of ‘internal observers’ as we say goes back to Einstein
and was used by him in a demonstration of his ideas on the General Theory of
Relativity. Einstein argued using his ‘two-dimensional creatures’, his ‘flat beings’
on a sphere, see for example L. Infeld [42], A. Einstein [17]. Just as was used in the
case above, we will use such an ideal construction in order to show to what degree
the development of our theory depends on the possible methods of measurement.
2
For our internal observers, the crystal as a whole does not move. We could also
formulate it as follows. The small internal observers inside the crystal see this as an
infinitely large world. They do now ask themselves about the motion of their world.
Only the craziest and audacious philosophers amongst them would think of or even
mention the possibility—leading to the accusation of blasphemy and insanity and
probably ending up being burnt at the stake as a warlock—that their world is finite.
The outside observers of Einstein’s Special Relativity and the internal observers
inside the crystal work with two completely different classes of reference systems.
There is only one common element if they find themselves in the same preferred
frame o . This is the solution to the paradox.
Let us, however, now continue with the theoretical conceptions the observers
inside of the infinitely large widths of the crystal make. Here, we will consequently
use only those ‘mechanical instruments’ defined by the sine-Gordon equation and
will especially not use the speed of light. We will make sure that all the ‘instruments’
we use are of mechanical origin in our crystal. These are the many structures above
the ground state, the energetically distinguished state of an ideal lattice.
Using A. Seeger [86], we will not only understand this ground state (the vacuum)
as an ideal lattice, but we will also count the infinitely long dislocations passing as
straight lines through the lattice as the vacuum, cf. (97).
2 This Einsteinian concept of an internal observer was used in a different context in continuum
mechanics by E. Kröner [51, 52]. E. Kröner refers to the problems connected to the spatial
measurements inside of the crystal. The problem is thus completely different to our own. The
Krönerian internal observers should not be mistaken with our internal observers.