102
10 Measuring-Rods and Clocks in Motion
u :=
x − vt
γ
,
w :=
t − vx/c
2
o
γ
.
⎫
⎪ ⎬
⎪ ⎭
(122)
The partial differentiation with respect to x and t can be expressed using the partial
differentiation with respect to u and w. After simple rearrangements we receive
∂u
∂x
=
1
γ
,
∂u
∂t
=
−v
γ
,
∂w
∂x
=
−v
c 2
o γ
,
∂w
∂t
=
1
γ
,
∂
∂x
=
∂u
∂x
∂
∂u
+
∂w
∂x
∂
∂w
,
∂
∂t
=
∂u
∂t
∂
∂u
+
∂w
∂t
∂
∂w
,
∂
2
∂x 2 =
∂u
∂x
2 ∂
2
∂u 2 + 2
∂u
∂x
∂w
∂x
∂
2
∂u∂w
+
∂w
∂x
2 ∂
2
∂w 2 ,
∂
2
∂t 2 =
∂u
∂t
2 ∂
2
∂u 2 + 2
∂u
∂t
∂w
∂t
∂
2
∂u∂w
+
∂w
∂t
2 ∂
2
∂w 2 .
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(123)
With
q
III
(x, t) = q
III
o (u, w)
(124)
Fig. 10.5 The choice of the
initial condition. When the
clock U v moving with the
velocity v moves past the
static clock U 1 at x = 0,
both clocks should have their
hands at zero. The clock U 1
counts the oscillations of the
thin drawn line described by
the function q III
o (0, t), see
Fig. 9.4. The clock U v counts
the oscillations of the thick
curve described according to
(119) by q III (vt, t), see
Fig. 10.4
U v
U 1
`
`
`
` ` `
`
6
6
`
`
`
` ` `
`
6
v
q
t
x = vt
x
a
2
T o
T
=
5
3 T o