Chapter 15
The Principle of Relativity: The Lost
Crystal
We now make a discovery. We have already written down the Lorentz transformation
(151) without having given the matter any great thought, in fact we only used it in
mathematical calculations. In Chap. 10 we introduced a new variable u according
to (114) for a moving kink q
I
(x, t), so that the moving kink went over, according
to (115) into the function q
I
o (u), so that q
I
o (u) fulfils in the variable u the static
sine-Gordon equation,
∂
2
∂u 2 q
I
o (u) =
D
σ
sin
2π
a
q
I
o (u)
.
(157)
We had, for the moving breather q
III
(x, t) according to (122), introduced new
variables u and w instead of x and t so that the function q
III
(x, t) changed into
the breather function q
III
(u, w), so that q
III
o (u, w) fulfils the sine-Gordon equation
in the variables u, w,
∂
2
∂u 2 q
III
o (u, w) −
1
c 2
o
∂
2
∂w 2 q
III
o (u, w) =
D
σ
sin
2π
a
q
III
o (u, w)
.
(158)
However, (114) is already half of the transformation (151) if we write x
in place
of u, and Eq. (122) is completely identical to the Lorentz transformation (151) if
one replaces u with x
and w with t
. These facts are generally applicable. We can,
using exactly the same mathematical steps as with variables u and w in Chap. 10,
calculate, using the variables x
and t
, that if the function q(x, t) fulfils the sineGordon equation
∂
2
∂x 2 q(x, t) −
1
c 2
o
∂
2
∂t 2 q(x, t) =
D
σ
sin
2π
a
q(x, t)
,
(159)
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license to Springer Nature Singapore Pte Ltd. 2020
H. Günther, Elementary Approach to Special Relativity,
https://doi.org/10.1007/978-981-15-3168-2_15
141