15 The Principle of Relativity: The Lost Crystal
147
measure for the length of the rod X. In order to find out which length the observer in
determines for the rod X, we need the coordinates of its end points at one and the
same point of time in
, let us say at t
= 0. For the left-hand end point, we once
again have x
= 0 (this is event O). The right-hand end point finds itself at event B,
thus at time t
B =
−vvx
c 2
o γ
at x
= ˜
x
= x/γ. Thus, the right-hand end point finds
itself at time t
= 0 at the position x
= ˜
x
+ v · t
B =
x
γ
(1 − v
2
/c
2
o ). This x
is the coefficient of measure for the length of the resting rod X in o measured in
.
Compared to the coefficient of measure x of its length determined in the reference
system o , the following is valid
x
= γ · x .
(163)
For the rod X = x L o resting in the reference system o , the moving length x
L
is measured in
. In other words, the measuring-rod L
has to be used exactly γ x
times in
in order to fill out the distance in
corresponding to the moving length
in question. This situation has been shown in Fig. 15.1, using an arbitrary x.
Perhaps the statement in Eq. (163) becomes clearer if we consider the case x =
1. In other words, X = L o . Here, the length of the measuring-rod resting in the
reference system o is determined from the reference system
using the measuringrod L
resting in
. This situation has been shown and described in Fig. 15.2. The
observer in
registers that the measuring-rod L o that from his point of view is in
motion with the reference system o can be covered using the fraction γ L
of his
measuring-rod. As curious as this fact may seem from the outside—because from
the outside we always register that L
is smaller than L o and that it can become
arbitrarily small if the velocity v of
relative to o sufficiently approaches c o —for
the internal observer, who gauges his velocities using our elementary principle of
relativity, the world looks completely different (Fig. 15.1).
Equation (163) is Eq.(113a) for the Lorentz contraction of a moving rod, with
exchanged roles, so that the observer in the reference system
moving relative to
the reference system o measures the rod X resting in the reference system o :
The moving rod is contracted, 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.
The hasty conclusion of an internal observer in o , that the reference system
moves with respect to the crystal, would be disputed and reversed using exactly
the same argumentation. It thus remains a fact that the crystal cannot be proven by
anything in the world of the internal observers.
However, no one will want to conclude from this that we have proven the crystal
to be non-existent. Anyone who, against reality, still wishes to believe this only has
to take a chunk of crystal and bang it against his head until he has assured himself of
the opposite. It is important for the success of this reflective experiment that one uses
ones own head and not the head of another. We as outside observers do not have any
difficulties in accepting the existence of crystals. The internal observers however
need a more subtle line of thought. They have to answer the question concerning
147
measure for the length of the rod X. In order to find out which length the observer in
determines for the rod X, we need the coordinates of its end points at one and the
same point of time in
, let us say at t
= 0. For the left-hand end point, we once
again have x
= 0 (this is event O). The right-hand end point finds itself at event B,
thus at time t
B =
−vvx
c 2
o γ
at x
= ˜
x
= x/γ. Thus, the right-hand end point finds
itself at time t
= 0 at the position x
= ˜
x
+ v · t
B =
x
γ
(1 − v
2
/c
2
o ). This x
is the coefficient of measure for the length of the resting rod X in o measured in
.
Compared to the coefficient of measure x of its length determined in the reference
system o , the following is valid
x
= γ · x .
(163)
For the rod X = x L o resting in the reference system o , the moving length x
L
is measured in
. In other words, the measuring-rod L
has to be used exactly γ x
times in
in order to fill out the distance in
corresponding to the moving length
in question. This situation has been shown in Fig. 15.1, using an arbitrary x.
Perhaps the statement in Eq. (163) becomes clearer if we consider the case x =
1. In other words, X = L o . Here, the length of the measuring-rod resting in the
reference system o is determined from the reference system
using the measuringrod L
resting in
. This situation has been shown and described in Fig. 15.2. The
observer in
registers that the measuring-rod L o that from his point of view is in
motion with the reference system o can be covered using the fraction γ L
of his
measuring-rod. As curious as this fact may seem from the outside—because from
the outside we always register that L
is smaller than L o and that it can become
arbitrarily small if the velocity v of
relative to o sufficiently approaches c o —for
the internal observer, who gauges his velocities using our elementary principle of
relativity, the world looks completely different (Fig. 15.1).
Equation (163) is Eq.(113a) for the Lorentz contraction of a moving rod, with
exchanged roles, so that the observer in the reference system
moving relative to
the reference system o measures the rod X resting in the reference system o :
The moving rod is contracted, 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.
The hasty conclusion of an internal observer in o , that the reference system
moves with respect to the crystal, would be disputed and reversed using exactly
the same argumentation. It thus remains a fact that the crystal cannot be proven by
anything in the world of the internal observers.
However, no one will want to conclude from this that we have proven the crystal
to be non-existent. Anyone who, against reality, still wishes to believe this only has
to take a chunk of crystal and bang it against his head until he has assured himself of
the opposite. It is important for the success of this reflective experiment that one uses
ones own head and not the head of another. We as outside observers do not have any
difficulties in accepting the existence of crystals. The internal observers however
need a more subtle line of thought. They have to answer the question concerning