12 The Measurement of the Critical Velocity
117
for our observer in
is isotropic, i.e. that the velocity has the same value for the
way there and back, then this would be a sensation. We have however not shown
this. The decisive question is the question about the signal velocity c
→ that the
observer in
measures for the first part of the measuring section. If we really could
prove that c
→ = c o
, then because of (133a) the following would immediately result,
c
→ = c
← = c o
and we would have reached our goal. However, without further
reflections, we cannot solve this problem.
In order to be able to measure a ‘true’ velocity (and not just an effective value
for way there and back), we need two clocks, one at the start and the second at the
end of the length to be measured. We have to synchronise these clocks in order to
be able to measure the time needed for the signal to travel. We need, in order to be
able to synchronise these clocks, a ‘true’ velocity and not just an effective value. We
intended to determine such a velocity, but such a velocity is not at our disposal. We
find ourselves in a loop. Let us recapitulate and examine our situation.
The moving observer can make pure geometrical measurements in his reference
system
without the help of any clocks. All he needs is a sufficient number of
measuring-rods L
o in an unbroken line, so that he can determine by counting these
measuring-rods, distances and lengths in his reference system
. Using this method,
he can determine x
according to (113a). All occurring events in the preferred frame
o can be attributed with the coordinates (x, t) and the observer in
can determine
the space coordinates x
for these events. He can thus study geometry. The observer
in
cannot however determine an order of time for the events that take place
somewhere and compare this order with the order of time that has been determined
for these events in the reference system o . The observer can ‘see’ that the objects
in
move, but he is not able to determine a well-defined velocity for these motions.
He can distribute his clocks throughout the system, but as long as he does not have a
well-defined rule of how to set up these clocks, any ‘measured’ velocity using these
clocks would be completely arbitrary.
We thus need a rule that informs us how we have to start a clock with a certain
hand setting at an arbitrary position in
. We will do this for a clock positioned at
the beginning of our length, the origin of our coordinates
. We will call this clock
U
o
v . Let us label the clocks resting in
generally with the subscript v. We want to
determine that U
o
v together with the clock U o at the coordinate origin of the reference
system, o both have the value 0 when these clocks meet. This coincidence of the
coordinate origins will be called the event O. Our (arbitrarily determined) initial
condition is thus, see Fig. 12.3,
o : O (x = 0 ; t = 0) ←→
: O (x
= 0 ; t
= 0) .
Initial condition
(134)
Then when all clocks in
have started they should all run in the same manner,
because they are all of the same construction. Thus, the only question is when to
start the other clocks in
(except U
o
v , which is started by the initial condition). As
harmless as this question may seem, this process determines the time procedure of
every event, of every physical law. We must also accomplish a way of setting clocks
for all possible systems of reference
, in other words for all possible velocities v,
117
for our observer in
is isotropic, i.e. that the velocity has the same value for the
way there and back, then this would be a sensation. We have however not shown
this. The decisive question is the question about the signal velocity c
→ that the
observer in
measures for the first part of the measuring section. If we really could
prove that c
→ = c o
, then because of (133a) the following would immediately result,
c
→ = c
← = c o
and we would have reached our goal. However, without further
reflections, we cannot solve this problem.
In order to be able to measure a ‘true’ velocity (and not just an effective value
for way there and back), we need two clocks, one at the start and the second at the
end of the length to be measured. We have to synchronise these clocks in order to
be able to measure the time needed for the signal to travel. We need, in order to be
able to synchronise these clocks, a ‘true’ velocity and not just an effective value. We
intended to determine such a velocity, but such a velocity is not at our disposal. We
find ourselves in a loop. Let us recapitulate and examine our situation.
The moving observer can make pure geometrical measurements in his reference
system
without the help of any clocks. All he needs is a sufficient number of
measuring-rods L
o in an unbroken line, so that he can determine by counting these
measuring-rods, distances and lengths in his reference system
. Using this method,
he can determine x
according to (113a). All occurring events in the preferred frame
o can be attributed with the coordinates (x, t) and the observer in
can determine
the space coordinates x
for these events. He can thus study geometry. The observer
in
cannot however determine an order of time for the events that take place
somewhere and compare this order with the order of time that has been determined
for these events in the reference system o . The observer can ‘see’ that the objects
in
move, but he is not able to determine a well-defined velocity for these motions.
He can distribute his clocks throughout the system, but as long as he does not have a
well-defined rule of how to set up these clocks, any ‘measured’ velocity using these
clocks would be completely arbitrary.
We thus need a rule that informs us how we have to start a clock with a certain
hand setting at an arbitrary position in
. We will do this for a clock positioned at
the beginning of our length, the origin of our coordinates
. We will call this clock
U
o
v . Let us label the clocks resting in
generally with the subscript v. We want to
determine that U
o
v together with the clock U o at the coordinate origin of the reference
system, o both have the value 0 when these clocks meet. This coincidence of the
coordinate origins will be called the event O. Our (arbitrarily determined) initial
condition is thus, see Fig. 12.3,
o : O (x = 0 ; t = 0) ←→
: O (x
= 0 ; t
= 0) .
Initial condition
(134)
Then when all clocks in
have started they should all run in the same manner,
because they are all of the same construction. Thus, the only question is when to
start the other clocks in
(except U
o
v , which is started by the initial condition). As
harmless as this question may seem, this process determines the time procedure of
every event, of every physical law. We must also accomplish a way of setting clocks
for all possible systems of reference
, in other words for all possible velocities v,