194
17 The Twin Paradox
Fig. 17.9 End of the story of the twins. The twins meet at event S(v). There departure occurred
at event O. We once again calculate using v = 0, 8 c o , γ = 0, 6 and gauge all clocks according
to x P /v = 15 scale parts. The reference system has the velocity v with respect to o , and the
reference system has the velocity v with respect to . The hand of twin A’s personal clock U A
reads according to (222), as we discovered many times over, t
S(v) = 2 · 15/0, 6 = 50 scale parts
at event S(v), whilst the hand of brother A o ’s personal clock U A
o goes behind, as we have also
discovered many times over, by the factor of γ and only reads, according to (214) ˜
t S(v) = 30 scale
parts, assuming of course that brother A o did not secretly reset his clock (as he did in our story
when he changed reference systems)
observer are alone responsible for the dynamic processes in the interior of a crystal.
The energy E of a kink q
I moving with the velocity v,
q
I
= q
I
(x, t) =
2π
a
arctan exp
π(x − vt)
L o
1 − v 2 /c 2
o
,
has as an example nothing to do with the speed of light c L , however this energy
increases with the approximation of the parameter v to the critical velocity c o of
the sine-Gordon equation. We will show for this energy in Chap. 23, Eq.(297) on
p. 251 that E =
E o
√
1−v 2 /c 2
o
, where E o represents the energy of the kink at rest; see
also Seeger [].
From the dependence of an object’s energy to its velocity, where for our ‘outside
objects’ the speed of light is valid, Einstein [; ], three months after his famous paper