146
15 The Principle of Relativity: The Lost Crystal
the reference system o . In other words, this means that the observer in
registers
that the breather clock resting relative to the crystal goes behind:
The moving clock goes behind, even if it rests by chance with respect to the crystal and only
moves relative to the observer, who himself moves relative to the crystal.
This only seemingly contradicts the information in Fig. 10.6. The relativity of time
dilatation for the internal observers in connection with the discussion concerning the
twin paradox in Chap. 16 is once again depicted in Fig. 17.2.
We also want to see how it can be that the static kink resting relative to the crystal
is registered by the observer in
as shorter than the kink resting relative to
by the Lorentz factor γ, which is once again apparently contradictory to what we
have shown in Fig. 10.2. Here, one has to keep in mind that we have only shown
momentary snapshots, as registered by the internal observer in the reference system
o . These observations are also made by us ’outside’ observers. We can peer into
a crystal from the outside and measure, using the measuring-rods and clocks that
we constructed using our ’outside’ physics. Important symmetrical properties of the
observed phenomena as measured by the internal observers remain invisible to us
when we use these outside measuring-rods and clocks. Momentary snapshots as
registered by the internal observer in the reference system
have been depicted in
Fig. 12.4 and the lower picture of Fig. 17.1.
How can it be possible for the observer in the reference system
(that moves
relative to o with the velocity v) to measure a Lorentz contracted length of a rod
being at rest in o (thus being at rest relative to the crystal)? Let us once again make
this clear: If we measure the length of a rod in the reference system in which it
rests, then it plays no role when we note the coordinates of its end points, because
these do not change. Due to the fact that the rod from Fig. 12.8 rests in the reference
system
, x
is the coefficient of measure of its length, even though both
clocks
show different time readings. However, if the rod moves, we need the coordinates
of both end points at one and the same point of time, so that we can calculate the
coefficient of measure of the moving length from the coordinate differences, also see
our explanations about Lorentz contractions at the end of Chap. 10.
Let us consider a rod X resting in o , as we have shown in Fig. 12.1, with the end
points x 1 = 0 and x 2 = x. The coordinate difference x is thus the coefficient of
measure of its length with the measuring-rod L o and it is X = x L o = ˜
x
L
, so
that it has to be ˜
x
= x/γ because of L
= γ L o . What is ˜
x
? We consider the
end points of the rod at time t = 0 in o . These are the events O and B with their
coordinates (x = 0, t = 0), or (x 2 = x, t B = 0) in o . We have already described
the synchronisation of clocks of the reference system
for this case in Fig. 12.6. In
, O has the coordinates (t
= 0, x
= 0) and event B has the coordinates, as seen
in Fig. 12.6 and using Eq. (140), x
= x/γ, t
B =
−vvx
c 2
o γ
. What is ˜
x
?
Although in this case the number ˜
x
also states how often the measuring-rod
L
fits into the positions, which are defined by the events O and B in
. Both of
these events however are the end points of a moving rod in
at two different points
of time. The quantity ˜
x
is thus not the length of the moving rod X with respect
to
anymore, but only the distance between the events O and B, that are only
simultaneous in o and whose coordinate difference makes up the coefficient of
15 The Principle of Relativity: The Lost Crystal
the reference system o . In other words, this means that the observer in
registers
that the breather clock resting relative to the crystal goes behind:
The moving clock goes behind, even if it rests by chance with respect to the crystal and only
moves relative to the observer, who himself moves relative to the crystal.
This only seemingly contradicts the information in Fig. 10.6. The relativity of time
dilatation for the internal observers in connection with the discussion concerning the
twin paradox in Chap. 16 is once again depicted in Fig. 17.2.
We also want to see how it can be that the static kink resting relative to the crystal
is registered by the observer in
as shorter than the kink resting relative to
by the Lorentz factor γ, which is once again apparently contradictory to what we
have shown in Fig. 10.2. Here, one has to keep in mind that we have only shown
momentary snapshots, as registered by the internal observer in the reference system
o . These observations are also made by us ’outside’ observers. We can peer into
a crystal from the outside and measure, using the measuring-rods and clocks that
we constructed using our ’outside’ physics. Important symmetrical properties of the
observed phenomena as measured by the internal observers remain invisible to us
when we use these outside measuring-rods and clocks. Momentary snapshots as
registered by the internal observer in the reference system
have been depicted in
Fig. 12.4 and the lower picture of Fig. 17.1.
How can it be possible for the observer in the reference system
(that moves
relative to o with the velocity v) to measure a Lorentz contracted length of a rod
being at rest in o (thus being at rest relative to the crystal)? Let us once again make
this clear: If we measure the length of a rod in the reference system in which it
rests, then it plays no role when we note the coordinates of its end points, because
these do not change. Due to the fact that the rod from Fig. 12.8 rests in the reference
system
, x
is the coefficient of measure of its length, even though both
clocks
show different time readings. However, if the rod moves, we need the coordinates
of both end points at one and the same point of time, so that we can calculate the
coefficient of measure of the moving length from the coordinate differences, also see
our explanations about Lorentz contractions at the end of Chap. 10.
Let us consider a rod X resting in o , as we have shown in Fig. 12.1, with the end
points x 1 = 0 and x 2 = x. The coordinate difference x is thus the coefficient of
measure of its length with the measuring-rod L o and it is X = x L o = ˜
x
L
, so
that it has to be ˜
x
= x/γ because of L
= γ L o . What is ˜
x
? We consider the
end points of the rod at time t = 0 in o . These are the events O and B with their
coordinates (x = 0, t = 0), or (x 2 = x, t B = 0) in o . We have already described
the synchronisation of clocks of the reference system
for this case in Fig. 12.6. In
, O has the coordinates (t
= 0, x
= 0) and event B has the coordinates, as seen
in Fig. 12.6 and using Eq. (140), x
= x/γ, t
B =
−vvx
c 2
o γ
. What is ˜
x
?
Although in this case the number ˜
x
also states how often the measuring-rod
L
fits into the positions, which are defined by the events O and B in
. Both of
these events however are the end points of a moving rod in
at two different points
of time. The quantity ˜
x
is thus not the length of the moving rod X with respect
to
anymore, but only the distance between the events O and B, that are only
simultaneous in o and whose coordinate difference makes up the coefficient of