12 The Measurement of the Critical Velocity
113
∂
2
∂ ¯
x 2 ¯
q( ¯
x, ¯
t) −
1
c 2
o
∂
2
∂ ¯
t 2 ¯
q( ¯
x, ¯
t) =
D
σ
sin
2π
a
¯
q( ¯
x, ¯
t )
.
(128)
In other words, referred to our new coordinates, the sine-Gordon equation remains
invariant. The critical velocity has one and the same value c o for both space directions.
Our one-dimensional continuum is isotropic.
If we wish to measure the velocity c o in o , we can use our measuring-rods and
clocks. Due to the isotropy in o , we have to take the following steps: We consider a
length x L o in o . We determine the coordinates x 1 and x 2 of the end points out of
reasons of simplicity as x 1 = 0 and x 2 = x. In other words, x is the number that
informs us how many times our measuring-rod L o fits on to the length. The statement
of time t (the coordinate t in o ) also informs us how many times the pendulum
of our normal clock with the oscillation period of T o swung. At the time t o = 0 at
x 1 = 0, we send a sound signal to x 2 . At time t 1 , it is reflected and arrives, due to
the isotropy of signal propagation, at time t o = 2t 1 at x 1 = 0, so that we have
measured the critical velocity c o in o (see Fig. 12.1) according to
c o =
2x
t o
=
2x
2t 1
.
(129)
We now consider a measuring section, a ‘rod’ X (being a corresponding structure
inside of the crystal) moving with the uniform velocity v relative to the preferred
frame o . This rod is at rest in a reference system
. The internal observers in
can measure the length of this rod simply by using their measuring-rods L
and
determine the coefficient of measure x
. We can measure any sort of lengths in the
preferred frame o as was discussed in Chap. 10 after Eqs. (112) and (113), p. 96.
Fig. 12.1 The measuring of the critical velocity c o in the preferred frame o . At time t o = 0, a
‘sound signal’ is sent from x 1 = 0 to x 2 . This signal is reflected at time t 1 and arrives at time
t o = 2t 1 at x 1 . In the figure, the clock is gauged in such a way that the time t o is shown at 15,
in other words ‘quarter past’. Taking the fact as granted that the critical velocity in our preferred
frame o has the same value on its way in both directions (see the explanation in the text above), we
have measured using c o = 2x//t o the critical velocity c o in the reference system o , see (129)
113
∂
2
∂ ¯
x 2 ¯
q( ¯
x, ¯
t) −
1
c 2
o
∂
2
∂ ¯
t 2 ¯
q( ¯
x, ¯
t) =
D
σ
sin
2π
a
¯
q( ¯
x, ¯
t )
.
(128)
In other words, referred to our new coordinates, the sine-Gordon equation remains
invariant. The critical velocity has one and the same value c o for both space directions.
Our one-dimensional continuum is isotropic.
If we wish to measure the velocity c o in o , we can use our measuring-rods and
clocks. Due to the isotropy in o , we have to take the following steps: We consider a
length x L o in o . We determine the coordinates x 1 and x 2 of the end points out of
reasons of simplicity as x 1 = 0 and x 2 = x. In other words, x is the number that
informs us how many times our measuring-rod L o fits on to the length. The statement
of time t (the coordinate t in o ) also informs us how many times the pendulum
of our normal clock with the oscillation period of T o swung. At the time t o = 0 at
x 1 = 0, we send a sound signal to x 2 . At time t 1 , it is reflected and arrives, due to
the isotropy of signal propagation, at time t o = 2t 1 at x 1 = 0, so that we have
measured the critical velocity c o in o (see Fig. 12.1) according to
c o =
2x
t o
=
2x
2t 1
.
(129)
We now consider a measuring section, a ‘rod’ X (being a corresponding structure
inside of the crystal) moving with the uniform velocity v relative to the preferred
frame o . This rod is at rest in a reference system
. The internal observers in
can measure the length of this rod simply by using their measuring-rods L
and
determine the coefficient of measure x
. We can measure any sort of lengths in the
preferred frame o as was discussed in Chap. 10 after Eqs. (112) and (113), p. 96.
Fig. 12.1 The measuring of the critical velocity c o in the preferred frame o . At time t o = 0, a
‘sound signal’ is sent from x 1 = 0 to x 2 . This signal is reflected at time t 1 and arrives at time
t o = 2t 1 at x 1 . In the figure, the clock is gauged in such a way that the time t o is shown at 15,
in other words ‘quarter past’. Taking the fact as granted that the critical velocity in our preferred
frame o has the same value on its way in both directions (see the explanation in the text above), we
have measured using c o = 2x//t o the critical velocity c o in the reference system o , see (129)