10 Measuring-Rods and Clocks in Motion
95
by us at
x = x(t) = v t .
Position of measuring-rod and clock
of the observer in
(109)
Let us first examine the measuring-rod. The measuring-rod moved at the uniform velocity v has to develop out of the displayed function q
I
o (x) in Fig. 9.2
by a displacement of v · t to the right, with the important condition that the
sine-Gordon equation maintains its validity. If one simply displaces the function
q
I
o (x) =
2a
π
arctan exp[π x/L o ] by v t to the right, we get according to (105) the
function
¯
q
I
o (x) =
2a
π
arctan exp
π(x − vt)
L o
.
(110)
Using this line form, we would receive a moving observer’s measuring-rod L o , and
this measuring-rod would be exactly identical to the observer’s measuring-rod L o
at b = v t in o according to Eq. (107). In this case, we could say that our anxiety
concerning the changing of length of a moving measuring-rod were unfounded.
The decisive point is however that the function (110) is not a possible line form
in the lattice. Function (110) does not fulfil the sine-Gordon equation which can
easily be checked by mathematical calculation. We thus make an important observation in our crystal:
It is impossible to rigidly displace the line defining our measuring-rod.
In other words, in the reality of our continuum defined by the crystalline background, there are no measuring-rods that stay unchanged during a motion! In Chap. 8,
we developed the sine-Gordon equation for this continuum, and this equation now
causes a line deformation dependent on the velocity v of the motion. Actually, the
line form has to be compressed in order to be moveable in order to be a corresponding solution q
I
= q
I
(x, t) of the sine-Gordon equation, compare A. Seeger [86],
H. G¨ unther [34], note the remark on p. 341,
q
I
= q
I
(x, t) =
2a
π
arctan exp
π(x − vt)
L o γ
,
γ =
1 −
v 2
c 2
o
.
⎫
⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
(111)
In fact, q
I
(x, t) is a line form localised at x = v t similar to the function q
I
o (x) that
moves with the velocity v to the right along the x-axis, see Fig. 10.2.
We now compare the functions (111) and (107). At time t = 0, the moving
measuring-rod L
only covers a fraction γ L o of the measuring-rod at rest L o at
b = 0. The same occurs for all other points in time with b = v t. The length L
that
we measure in o for the moving measuring-rod is L
= L o γ. This is already the
well-known Lorentz contraction that the sine-Gordon equation quite simply delivers:
95
by us at
x = x(t) = v t .
Position of measuring-rod and clock
of the observer in
(109)
Let us first examine the measuring-rod. The measuring-rod moved at the uniform velocity v has to develop out of the displayed function q
I
o (x) in Fig. 9.2
by a displacement of v · t to the right, with the important condition that the
sine-Gordon equation maintains its validity. If one simply displaces the function
q
I
o (x) =
2a
π
arctan exp[π x/L o ] by v t to the right, we get according to (105) the
function
¯
q
I
o (x) =
2a
π
arctan exp
π(x − vt)
L o
.
(110)
Using this line form, we would receive a moving observer’s measuring-rod L o , and
this measuring-rod would be exactly identical to the observer’s measuring-rod L o
at b = v t in o according to Eq. (107). In this case, we could say that our anxiety
concerning the changing of length of a moving measuring-rod were unfounded.
The decisive point is however that the function (110) is not a possible line form
in the lattice. Function (110) does not fulfil the sine-Gordon equation which can
easily be checked by mathematical calculation. We thus make an important observation in our crystal:
It is impossible to rigidly displace the line defining our measuring-rod.
In other words, in the reality of our continuum defined by the crystalline background, there are no measuring-rods that stay unchanged during a motion! In Chap. 8,
we developed the sine-Gordon equation for this continuum, and this equation now
causes a line deformation dependent on the velocity v of the motion. Actually, the
line form has to be compressed in order to be moveable in order to be a corresponding solution q
I
= q
I
(x, t) of the sine-Gordon equation, compare A. Seeger [86],
H. G¨ unther [34], note the remark on p. 341,
q
I
= q
I
(x, t) =
2a
π
arctan exp
π(x − vt)
L o γ
,
γ =
1 −
v 2
c 2
o
.
⎫
⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
(111)
In fact, q
I
(x, t) is a line form localised at x = v t similar to the function q
I
o (x) that
moves with the velocity v to the right along the x-axis, see Fig. 10.2.
We now compare the functions (111) and (107). At time t = 0, the moving
measuring-rod L
only covers a fraction γ L o of the measuring-rod at rest L o at
b = 0. The same occurs for all other points in time with b = v t. The length L
that
we measure in o for the moving measuring-rod is L
= L o γ. This is already the
well-known Lorentz contraction that the sine-Gordon equation quite simply delivers: