94
10 Measuring-Rods and Clocks in Motion
q
I
o (x)
=
2a
π
arctan exp
πx
L o
−→ L o ,
o :
q
III
o (x, t) =
2a
π
arctan
sin
2πt
T o
cosh
πx
L o
√
2
−→ T o .
(105)
A system of coordinates x and t belongs to such a reference system o with a
measuring-rod L o and an oscillation period T o . This means the following: We first
determine a starting position, the origin of the coordinates. This is an arbitrary ‘initial
event’ at the position 0 at time 0 and thus has the coordinates x = 0, t = 0. An
arbitrary event that takes place at a distance X = x L o from the ‘initial event’ and
after a time T = t T o has the coordinates x and t belonging to the reference system
o . We can also write o (x, t) and can thus say:
The coordinates x and t are the coefficients of measure for the time and length measurements
in the reference system o (x, t) with the units of measure L o and T o .
(The shortened versions of time t and space x can be used instead of the coordinates
x and t.)
The moving lengths and clocks also have to be solutions of the sine-Gordon
equation. Does this even exist? We firstly remind ourselves: If we displace a function
y = f (x) along the x-axis by b to the right, then this new function y = ¯
f (x) has to
be described according to (compare with Fig. 2.1)
y = ¯
f (x) = f (x − b) .
Displacement of y = f (x)
to the right by b
(106)
A sufficiently large number of measuring-rods L o and clocks preferred frame o . A
measuring-rod that has its central point lying at x = b is described by the line form
q
I
ob (x) =
2a
π
arctan exp
π(x − b)
L o
−→
Measuring-rod L o
at b in o
(107)
and a clock found at the position x = b is described by the line form
q
III
ob (x, t) =
2a
π
arctan
sin
2πt
T o
cosh
π(x−b)
L o
√
2
−→
Period T o of a clock
at x = b in o .
(108)
Because of
∂
∂x
f (x) =
∂
∂x
f (x − b) the functions in (107) and in (108) obviously
fulfil the sine-Gordon equation just as the functions in (105) do. We can in fact thus
position our measuring-rods and clocks wherever we want.
Now an observer has the constant velocity v against the preferred frame o . He
is then at rest in his own system of reference
. If we see this observer at our time
t = 0 at the position x = 0 as we take it, then we see him at any time t at the position
x = v t. The measuring-rod and the clock that rest next to the observer are registered
10 Measuring-Rods and Clocks in Motion
q
I
o (x)
=
2a
π
arctan exp
πx
L o
−→ L o ,
o :
q
III
o (x, t) =
2a
π
arctan
sin
2πt
T o
cosh
πx
L o
√
2
−→ T o .
(105)
A system of coordinates x and t belongs to such a reference system o with a
measuring-rod L o and an oscillation period T o . This means the following: We first
determine a starting position, the origin of the coordinates. This is an arbitrary ‘initial
event’ at the position 0 at time 0 and thus has the coordinates x = 0, t = 0. An
arbitrary event that takes place at a distance X = x L o from the ‘initial event’ and
after a time T = t T o has the coordinates x and t belonging to the reference system
o . We can also write o (x, t) and can thus say:
The coordinates x and t are the coefficients of measure for the time and length measurements
in the reference system o (x, t) with the units of measure L o and T o .
(The shortened versions of time t and space x can be used instead of the coordinates
x and t.)
The moving lengths and clocks also have to be solutions of the sine-Gordon
equation. Does this even exist? We firstly remind ourselves: If we displace a function
y = f (x) along the x-axis by b to the right, then this new function y = ¯
f (x) has to
be described according to (compare with Fig. 2.1)
y = ¯
f (x) = f (x − b) .
Displacement of y = f (x)
to the right by b
(106)
A sufficiently large number of measuring-rods L o and clocks preferred frame o . A
measuring-rod that has its central point lying at x = b is described by the line form
q
I
ob (x) =
2a
π
arctan exp
π(x − b)
L o
−→
Measuring-rod L o
at b in o
(107)
and a clock found at the position x = b is described by the line form
q
III
ob (x, t) =
2a
π
arctan
sin
2πt
T o
cosh
π(x−b)
L o
√
2
−→
Period T o of a clock
at x = b in o .
(108)
Because of
∂
∂x
f (x) =
∂
∂x
f (x − b) the functions in (107) and in (108) obviously
fulfil the sine-Gordon equation just as the functions in (105) do. We can in fact thus
position our measuring-rods and clocks wherever we want.
Now an observer has the constant velocity v against the preferred frame o . He
is then at rest in his own system of reference
. If we see this observer at our time
t = 0 at the position x = 0 as we take it, then we see him at any time t at the position
x = v t. The measuring-rod and the clock that rest next to the observer are registered