12 The Measurement of the Critical Velocity
121
Fig. 12.5 At event A the clocks U ∗
v and U o meet. The positions of the hand settings can be directly
compared. The coefficient of measure t A for the time shown by U o is calculated using (135) and
(136) with t
A = 15, 6 as t A = γ t
A = −9, 4. Once again, the dotted line combines both points
describing event A. Both of the left end points cannot be combined by the dotted line because we
have illustrated two different events which will be described below
The hand of the moving clock U
∗
v with respect to o , moves more slowly due to the
time dilatation (121a) namely by the value t
A ,
t
A = t A
1 −
v 2
c 2
o
=
x
v
1 −
v 2
c 2
o
.
(138)
The hand of U
∗
v finds itself for the coincidence of both coordinate origins at the
position t
B = t
A + t
A , so that with (135) and (138)
t
B =
−x
v
1 − v 2 /c 2
o
+
x
v
1 −
v 2
c 2
o
=
−x
v
1
1 − v 2 /c 2
o
−
1 −
v 2
c 2
o
and thus
t
B =
−x v/c
2
o
1 − v 2 /c 2
o
.
(139)
The event B is described by both reference systems as follows,
: B
x
2 = x
, t
B =
−x v
c 2
o
1 − v 2 /c 2
o
,
(140)
o : B (x 2 = x , t B = 0 ) .
(141)