12 The Measurement of the Critical Velocity
119
Here, v is an arbitrary velocity with |v| < c o . We therefore use the velocity −v of
the preferred frame o with respect to
in order to synchronise the clocks in the
reference system
. All other velocities observed from
have thus been fixed.
Summarising our reflections, we followed H. Poincaré, who in 1898 in his famous
essay ‘La Mesure du Temps’ called attention to the fundamental importance of a
definition of the simultaneity for two events needed in order to describe natural
processes. Poincaré [74] stated that ‘Il est difficile de séparer le probl` eme qualitatif
de la simultanéité du probl` eme quantitatif de la mesure du temps; soit qu’on se
serve d’un chronom` etre, soit qu’on ait `
a tenir compte d’une vitesse de transmission,
comme celle de la lumi` ere, car on ne saurait mesurer une pareille vitesse sans mesurer
un temps. · · · La simultanéité de deux événements, ou l’ordre de leur succession,
l’égalité de deux durées, doivent ˆ
etre définies de telle sorte que l’énoncé des lois
naturelles soit aussi simple que possible. En d’autres termes, toutes ces r` egles, toutes
ces définitions ne sont que le fruit d’un opportunisme inconscient.”
2
Indeed, our elementary principle is in fact nothing more than an arrangement
between the observers of both reference systems in order to be able to compare the
measured velocities of both systems of reference.
We now assume that the observer in
has distributed and set up his clocks so that
he can synchronise them according to our elementary principle of relativity. We will
do exactly this for our two clocks positioned at the end points of our length X, the
stationary rod in
, which possess the velocity v when observed from o . One of
the clocks U
o
v is positioned at the left end point of the length at the coordinate origin
of
. We have started this clock according to our initial condition (134). All other
clocks have to follow this clock. At the right-hand side of the length is the clock U
∗
v .
This clock has to be synchronised with U
o
v . In order to achieve this, we designate an
event as A, when the right end point of the length with the clock U
∗
v finds itself at
the coordinate origin of the reference system o .
Both clocks U
o
v and U
∗
v then run synchronically, in the sense of our elementary
principle of relativity, if and only if the hand of U
∗
v finds itself at the position t
A =
−x
/v when the event A takes place. When the coordinate origin of o moving,
according to our principle, from
with the velocity −v arrives at the left-hand
side of our measuring section, the coordinate origin of
, then the pointer of U
∗
v
has moved by x
/v and comes to hand setting 0 just as the pointer of U
o
v does in
accordance with our initial condition. Therefore, a synchronisation, in other words an
simultaneity in
has been defined for both clocks, according to which the velocity
−v for the reference system o was ascertained. This is exactly what we wanted to
achieve, see Fig. 12.4. If we take (113a) for x
into consideration, then the event A
is described by the observer in
in the following manner,
2 ‘It is difficult to separate the qualitative problem of simultaneity from the quantitative problem of
the measurement of time; no matter whether a chronometer is used, or whether account must be
taken of a velocity of transmission, as that of light, because such a velocity could not be measured
without measuring a time. · · · The simultaneity of two events or the order of their succession, the
equality of two durations, are to be so defined that the enunciation of the natural laws may be as
simple as possible. In other words, all rules, all these definitions are only the fruit of an unconscious
opportunism’. Quoted from the English translation in Poincaré [74].
119
Here, v is an arbitrary velocity with |v| < c o . We therefore use the velocity −v of
the preferred frame o with respect to
in order to synchronise the clocks in the
reference system
. All other velocities observed from
have thus been fixed.
Summarising our reflections, we followed H. Poincaré, who in 1898 in his famous
essay ‘La Mesure du Temps’ called attention to the fundamental importance of a
definition of the simultaneity for two events needed in order to describe natural
processes. Poincaré [74] stated that ‘Il est difficile de séparer le probl` eme qualitatif
de la simultanéité du probl` eme quantitatif de la mesure du temps; soit qu’on se
serve d’un chronom` etre, soit qu’on ait `
a tenir compte d’une vitesse de transmission,
comme celle de la lumi` ere, car on ne saurait mesurer une pareille vitesse sans mesurer
un temps. · · · La simultanéité de deux événements, ou l’ordre de leur succession,
l’égalité de deux durées, doivent ˆ
etre définies de telle sorte que l’énoncé des lois
naturelles soit aussi simple que possible. En d’autres termes, toutes ces r` egles, toutes
ces définitions ne sont que le fruit d’un opportunisme inconscient.”
2
Indeed, our elementary principle is in fact nothing more than an arrangement
between the observers of both reference systems in order to be able to compare the
measured velocities of both systems of reference.
We now assume that the observer in
has distributed and set up his clocks so that
he can synchronise them according to our elementary principle of relativity. We will
do exactly this for our two clocks positioned at the end points of our length X, the
stationary rod in
, which possess the velocity v when observed from o . One of
the clocks U
o
v is positioned at the left end point of the length at the coordinate origin
of
. We have started this clock according to our initial condition (134). All other
clocks have to follow this clock. At the right-hand side of the length is the clock U
∗
v .
This clock has to be synchronised with U
o
v . In order to achieve this, we designate an
event as A, when the right end point of the length with the clock U
∗
v finds itself at
the coordinate origin of the reference system o .
Both clocks U
o
v and U
∗
v then run synchronically, in the sense of our elementary
principle of relativity, if and only if the hand of U
∗
v finds itself at the position t
A =
−x
/v when the event A takes place. When the coordinate origin of o moving,
according to our principle, from
with the velocity −v arrives at the left-hand
side of our measuring section, the coordinate origin of
, then the pointer of U
∗
v
has moved by x
/v and comes to hand setting 0 just as the pointer of U
o
v does in
accordance with our initial condition. Therefore, a synchronisation, in other words an
simultaneity in
has been defined for both clocks, according to which the velocity
−v for the reference system o was ascertained. This is exactly what we wanted to
achieve, see Fig. 12.4. If we take (113a) for x
into consideration, then the event A
is described by the observer in
in the following manner,
2 ‘It is difficult to separate the qualitative problem of simultaneity from the quantitative problem of
the measurement of time; no matter whether a chronometer is used, or whether account must be
taken of a velocity of transmission, as that of light, because such a velocity could not be measured
without measuring a time. · · · The simultaneity of two events or the order of their succession, the
equality of two durations, are to be so defined that the enunciation of the natural laws may be as
simple as possible. In other words, all rules, all these definitions are only the fruit of an unconscious
opportunism’. Quoted from the English translation in Poincaré [74].