126
12 The Measurement of the Critical Velocity
Hence, a composition of velocities, which is based on the Reichenbach transformation
(143) is valid,
4
u = u
γ
2
+ v −→ u
=
u − v
γ 2 .
Reichenbach’s composition
of velocities
(144)
Whilst the observer in o can only determine velocities smaller than c o (one should
think about the moving measuring rods and clocks), the observer in
registers
velocities u
that, because of the factor 1/γ
2 , can become arbitrarily large if his
velocity v in comparison to o is sufficiently near to c o . If one however wishes to
accept these peculiarities as an expression for the different nature of these reference
systems and also the distinction of our reference system o , then one cannot sustain
this statement. We will at once see this: Using (143), we would have introduced
a description into the reference system
that blocks our view of an important
symmetry of the observed phenomena in solids. Nevertheless, this description is
correct and reflects the reality.
Notice that as well as our clock paradox of the last chapter is nothing but an
illustration for Poincaré’s ideas. We here as well refer to H. Reichenbach [83], Chap. 2,
‘We could arrange the definition of simultaneity of a system K in such a manner that
it leads to the same results as that of another system K
which is in motion relative
to K ;…’, which is exactly what we have done in (143). We will come back to this
question and its consequences, e.g. concerning the contraction of moving length’s in
Chap. 16.
We will now leave absolute simultaneity and deal with the determination of the
velocity c o
in
using that synchronisation procedure, based on our elementary
principle of relativity for the
clocks. In connection to this, we will be able to show
in the next two chapters, that this reference system (and with it all other systems
moving with a uniform velocity with respect to o ) is completely equivalent to o .
Here, the important point is that we can prove the existence of such an instruction
for the synchronisation, with which we receive such a symmetry of the reference
systems. In the appendix, Chap. 26, we will give a further example for the abovequoted ideas of Poincaré and Reichenbach. There we will explain that the so-called
structural eigen strains, which will be generated by arbitrary dislocations in a solid
with arbitrary symmetry, fulfil one and the same d’Alembertian wave equation in all
reference systems
. This is, in its generality, a surprisingly simple result.
Let us now move on to the velocity c o
that the observer measures in
.
For this purpose, we send the signal attributed with the velocity c o by o on to
the measuring section with the intention to determine the value c o
in
. We start
the signal at the mutual coordinate origin, our event O. The observers in o and
register the same starting time t o = t o
= 0 at the starting point of the measuring
section X = x L o = x
L being at rest in
. The travel time t
∗ of the signal
in the reference system
up to the end point of the measuring section can be read
4 Notice that the single difference of (144) to the famous Einsteinian composition of velocities
(194) is the definition of simultaneity so that no contradiction can arise with experiments.
12 The Measurement of the Critical Velocity
Hence, a composition of velocities, which is based on the Reichenbach transformation
(143) is valid,
4
u = u
γ
2
+ v −→ u
=
u − v
γ 2 .
Reichenbach’s composition
of velocities
(144)
Whilst the observer in o can only determine velocities smaller than c o (one should
think about the moving measuring rods and clocks), the observer in
registers
velocities u
that, because of the factor 1/γ
2 , can become arbitrarily large if his
velocity v in comparison to o is sufficiently near to c o . If one however wishes to
accept these peculiarities as an expression for the different nature of these reference
systems and also the distinction of our reference system o , then one cannot sustain
this statement. We will at once see this: Using (143), we would have introduced
a description into the reference system
that blocks our view of an important
symmetry of the observed phenomena in solids. Nevertheless, this description is
correct and reflects the reality.
Notice that as well as our clock paradox of the last chapter is nothing but an
illustration for Poincaré’s ideas. We here as well refer to H. Reichenbach [83], Chap. 2,
‘We could arrange the definition of simultaneity of a system K in such a manner that
it leads to the same results as that of another system K
which is in motion relative
to K ;…’, which is exactly what we have done in (143). We will come back to this
question and its consequences, e.g. concerning the contraction of moving length’s in
Chap. 16.
We will now leave absolute simultaneity and deal with the determination of the
velocity c o
in
using that synchronisation procedure, based on our elementary
principle of relativity for the
clocks. In connection to this, we will be able to show
in the next two chapters, that this reference system (and with it all other systems
moving with a uniform velocity with respect to o ) is completely equivalent to o .
Here, the important point is that we can prove the existence of such an instruction
for the synchronisation, with which we receive such a symmetry of the reference
systems. In the appendix, Chap. 26, we will give a further example for the abovequoted ideas of Poincaré and Reichenbach. There we will explain that the so-called
structural eigen strains, which will be generated by arbitrary dislocations in a solid
with arbitrary symmetry, fulfil one and the same d’Alembertian wave equation in all
reference systems
. This is, in its generality, a surprisingly simple result.
Let us now move on to the velocity c o
that the observer measures in
.
For this purpose, we send the signal attributed with the velocity c o by o on to
the measuring section with the intention to determine the value c o
in
. We start
the signal at the mutual coordinate origin, our event O. The observers in o and
register the same starting time t o = t o
= 0 at the starting point of the measuring
section X = x L o = x
L being at rest in
. The travel time t
∗ of the signal
in the reference system
up to the end point of the measuring section can be read
4 Notice that the single difference of (144) to the famous Einsteinian composition of velocities
(194) is the definition of simultaneity so that no contradiction can arise with experiments.