16
1 Congruences of World Lines
By differentiation, we then derive
a
y
=
Du y
dt
=
D
dt
d ˜
t
dt
u
y
=
d 2 ˜
t
dt 2 u
y
+
d ˜
t
dt
2
a
y ,
(1.82)
where
a
y
=
D
d ˜
t
u
y
=
D 2 Y y
d ˜
t 2 .
(1.83)
Analogously we obtain for the derivative of the deviation vector
D 2 η y
dt 2 =
d 2 ˜
t
dt 2
Dη y
d ˜
t
+
d ˜
t
dt
2 D 2 η y
d ˜
t 2 .
(1.84)
With this re-parametrisation the exact deviation equation (1.78) is recast into the
simpler form
D 2
d ˜
t 2 η
y 1 = −σ
y 1 y 2 a
y 2 − σ
y 1 x 2 ˜
a
x 2 − σ
y 1 y 2 y 3 u
y 2 u
y 3
−2σ
y 1 y 2 x 3 u
y 2
K
x 3 y 4 u
y 4 − H
x 3 y 4
Dσ y 4
d ˜
t
−σ
y 1 x 2 x 3
K
x 2 y 4 u
y 4 − H
x 2 y 4
Dσ y 4
d ˜
t
×
K
x 3 y 5 u
y 5 − H
x 3 y 5
Dσ y 5
d ˜
t
.
(1.85)
Now everything is synchronous in the sense that both curves are parametrized by ˜
t.
It should be stressed that the generalized deviation equation qualitatively reproduces
several other results for generalized deviation equations in the literature [12–18].
The generalized exact deviation equation (1.85) contains quantities which are
not defined along the reference curve Y , in particular the covariant derivatives of the
world function. In the next step we expand all quantities in powers of the deviation
around Y .
For a general bitensor B ... with a given index structure, we have the following
general expansion, up to the third order (in powers of σ y ):
B y 1 ...y n = A y 1 ...y n + A y 1 ...y n+1 σ
y n+1
+
1
2
A y 1 ...y n+1 y n+2 σ
y n+1 σ
y n+2 + O
σ
3
,
(1.86)