10 Measuring-Rods and Clocks in Motion
101
The Lorentz factor γ =
1 − v 2 /c 2
o is always smaller than 1. The period of oscillation T
of the moving clock is thus always larger than the period of oscillation T o
of the static clock. The unit of measure T
for time is increased by motion. Here we
have found the famous time dilatation, written down using the units of measure of
time, in other words the periods of oscillation of the moving and the static clocks.
This is not however that what we can immediately see and compare when we observe
a clock, the positions of the arms on the face of the clock.
The position of the arms on the face of the clock that are used to read the time count
the number of the oscillations. This number represents the coefficient of measure of
time. The arms of the moving clock always go behind the arms of a static clock.
When the moving clock U v with the hand setting t
arrives at the stationary clock
U 2 with the hand setting t, then the time T = t T o = t
T
has passed since it started
at the clock U 1 . Thus, because of (120), following is valid
t
= t
1 −
v 2
c 2
o
.
Time dilatation
of the moving clock
(121)
The moving clock goes behind.
Once again we wish to emphasise: In order to check (121) one has to compare
the readings of the single moving clock U v to the readings of at least two static
clocks. These two clocks have to have the distance v t from each other and have to
be stationary in the preferred frame o , see Fig. 10.6.
If the first static clock is not positioned at the origin of the coordinates, in other
words we start at an arbitrary clock in o , we have to use t and t
instead of the
readings t and t
as depicted by the arms of the clock. We thus write
t
= t
1 −
v 2
c 2
o
.
(121a)
In the Eq. (121) for the time dilatation, the coefficient of measure of time is used. The
reduced coefficient of measure of time t
is that what one can immediately read from
the face of the clock. Here we wish to remind the reader of the difference concerning
the formula (112) for Lorentz contraction. In (112) it is contraction of the moving
unit of measure which is primarily observed. For the unit of measurement of time,
the period of oscillation, time dilatation has to be formulated according to (120).
We will now prove that q
III
(x, t) is just as much a solution of the sine-Gordon
equation (88) as q
III
o (x, t). According to (117) it is
q
III
(x, t) = q
III
o
x − vt
γ
,
t − vx/c
2
o
γ
.
We put
101
The Lorentz factor γ =
1 − v 2 /c 2
o is always smaller than 1. The period of oscillation T
of the moving clock is thus always larger than the period of oscillation T o
of the static clock. The unit of measure T
for time is increased by motion. Here we
have found the famous time dilatation, written down using the units of measure of
time, in other words the periods of oscillation of the moving and the static clocks.
This is not however that what we can immediately see and compare when we observe
a clock, the positions of the arms on the face of the clock.
The position of the arms on the face of the clock that are used to read the time count
the number of the oscillations. This number represents the coefficient of measure of
time. The arms of the moving clock always go behind the arms of a static clock.
When the moving clock U v with the hand setting t
arrives at the stationary clock
U 2 with the hand setting t, then the time T = t T o = t
T
has passed since it started
at the clock U 1 . Thus, because of (120), following is valid
t
= t
1 −
v 2
c 2
o
.
Time dilatation
of the moving clock
(121)
The moving clock goes behind.
Once again we wish to emphasise: In order to check (121) one has to compare
the readings of the single moving clock U v to the readings of at least two static
clocks. These two clocks have to have the distance v t from each other and have to
be stationary in the preferred frame o , see Fig. 10.6.
If the first static clock is not positioned at the origin of the coordinates, in other
words we start at an arbitrary clock in o , we have to use t and t
instead of the
readings t and t
as depicted by the arms of the clock. We thus write
t
= t
1 −
v 2
c 2
o
.
(121a)
In the Eq. (121) for the time dilatation, the coefficient of measure of time is used. The
reduced coefficient of measure of time t
is that what one can immediately read from
the face of the clock. Here we wish to remind the reader of the difference concerning
the formula (112) for Lorentz contraction. In (112) it is contraction of the moving
unit of measure which is primarily observed. For the unit of measurement of time,
the period of oscillation, time dilatation has to be formulated according to (120).
We will now prove that q
III
(x, t) is just as much a solution of the sine-Gordon
equation (88) as q
III
o (x, t). According to (117) it is
q
III
(x, t) = q
III
o
x − vt
γ
,
t − vx/c
2
o
γ
.
We put