15 The Principle of Relativity: The Lost Crystal
145
oscillates slower than the breather solution resting with respect to the crystal, which
we have shown in Fig. 10.5. This is also, in principle, experimentally provable. From
this follows that the reference system o of the static crystal is distinguished and that
both the Lorentz contraction and time dilatation only take place, in reality, relative
to the crystal?
Such an evaluation of the situation always silently requires one thing that all
measurements are naturally carried out using the measuring instruments of the outside
observer, as are all other measurements in physics. We can indeed do it this way and
also arrive at the same evaluation as seen above but maybe different measuring
methods would be better suited for the observed phenomena.
What does ‘in reality’ mean? We do not intent to start a philosophical discussion
here, however we intend to concentrate alone on what happens inside of a crystal,
what functions inside of a crystal, what has its reality inside of a crystal. If we however solely concentrate on the crystal and the considered class of phenomena, then
the question concerning the symmetry of these phenomena is a problem of primary
importance. The mathematical methods of physical examination of such phenomena depend heavily on symmetry. Connected to this is the question concerning the
description of phenomena using reference systems moving relative to each other,
which is of elementary interest. If however we intend to describe these observed
phenomena using reference systems moving relative to the crystal lattice, then we
must first decide which measuring-rods and clocks are best suited for this purpose
and we must also ask for a meaningful definition of a velocity in these reference
systems? We have answered this question using our elementary principle of relativity, see Chap. 12, as well as Fig. 12.4: ‘If the observer in o registers that
moves
with the velocity v, then the observer in
should observe that o moves with the
velocity −v’. Everything else is deduction.
Let us once again make it clear that our measuring procedure, based purely on
mechanical phenomena in a crystal, tells us that the observer in
does in fact
register that a breather clock resting relative to the crystal oscillates slower than the
breather clock resting relative to
. We can derive these facts from Fig. 12.6.
In order to do this, we first have to consider the event B with t B = 0 for the
o clock U ∗ and with t
B =
−vvx
c 2
o γ
for the
clock U
∗
v . After time t = x/v, U ∗
with the hand setting t lies opposite the
clock U
o
v , whose hand then shows
the time t
S =
x
v
γ, because of the time dilatation of the clock which moves with
respect to o . Thus whilst the time t = x/v on one of the o clocks has run out,
the observer in
determines a time period t
(which he has to read from both
clocks U
∗
v and U
o
v ) according to t
= t
S − t
B =
x
v
γ −
−vvx
c 2
o γ
=
x
v
(γ +
v
2
c 2
o γ
) =
x
v
c
2
o γ
2 +v
2
c 2
o γ
=
x
v
c
2
o −v
2 +v
2
c 2
o γ
=
x
v
1
γ
, and thus
t = t
γ .
(162)
This is Eq. (121a) for the time dilatation of a moving clock, but with swapped positions as was determined by the observer in the reference system
moving relative to
145
oscillates slower than the breather solution resting with respect to the crystal, which
we have shown in Fig. 10.5. This is also, in principle, experimentally provable. From
this follows that the reference system o of the static crystal is distinguished and that
both the Lorentz contraction and time dilatation only take place, in reality, relative
to the crystal?
Such an evaluation of the situation always silently requires one thing that all
measurements are naturally carried out using the measuring instruments of the outside
observer, as are all other measurements in physics. We can indeed do it this way and
also arrive at the same evaluation as seen above but maybe different measuring
methods would be better suited for the observed phenomena.
What does ‘in reality’ mean? We do not intent to start a philosophical discussion
here, however we intend to concentrate alone on what happens inside of a crystal,
what functions inside of a crystal, what has its reality inside of a crystal. If we however solely concentrate on the crystal and the considered class of phenomena, then
the question concerning the symmetry of these phenomena is a problem of primary
importance. The mathematical methods of physical examination of such phenomena depend heavily on symmetry. Connected to this is the question concerning the
description of phenomena using reference systems moving relative to each other,
which is of elementary interest. If however we intend to describe these observed
phenomena using reference systems moving relative to the crystal lattice, then we
must first decide which measuring-rods and clocks are best suited for this purpose
and we must also ask for a meaningful definition of a velocity in these reference
systems? We have answered this question using our elementary principle of relativity, see Chap. 12, as well as Fig. 12.4: ‘If the observer in o registers that
moves
with the velocity v, then the observer in
should observe that o moves with the
velocity −v’. Everything else is deduction.
Let us once again make it clear that our measuring procedure, based purely on
mechanical phenomena in a crystal, tells us that the observer in
does in fact
register that a breather clock resting relative to the crystal oscillates slower than the
breather clock resting relative to
. We can derive these facts from Fig. 12.6.
In order to do this, we first have to consider the event B with t B = 0 for the
o clock U ∗ and with t
B =
−vvx
c 2
o γ
for the
clock U
∗
v . After time t = x/v, U ∗
with the hand setting t lies opposite the
clock U
o
v , whose hand then shows
the time t
S =
x
v
γ, because of the time dilatation of the clock which moves with
respect to o . Thus whilst the time t = x/v on one of the o clocks has run out,
the observer in
determines a time period t
(which he has to read from both
clocks U
∗
v and U
o
v ) according to t
= t
S − t
B =
x
v
γ −
−vvx
c 2
o γ
=
x
v
(γ +
v
2
c 2
o γ
) =
x
v
c
2
o γ
2 +v
2
c 2
o γ
=
x
v
c
2
o −v
2 +v
2
c 2
o γ
=
x
v
1
γ
, and thus
t = t
γ .
(162)
This is Eq. (121a) for the time dilatation of a moving clock, but with swapped positions as was determined by the observer in the reference system
moving relative to