1.3 The World Function and Deviation Equations
15
in terms of the quantities which are defined at Y and then “propagated” by K x y and
H x y .
Multiplication of (1.76) with dt/d ˜
t yields
˜
u
x 3 = K
x 3 y 2 u
y 2
dt
d ˜
t
− H
x 3 y 1
Dσ y 1
dt
dt
d ˜
t
,
(1.77)
and insertion into (1.74) results in:
D 2
dt 2 η
y 1 = −σ
y 1 y 2 y 3 u
y 2 u
y 3 − σ
y 1 y 2 a
y 2 − σ
y 1 x 2 ˜
a
x 2
d ˜
t
dt
2
−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
dt
−σ
y 1 x 2 x 3
K
x 2 y 4 u
y 4 − H
x 2 y 4
Dσ y 4
dt
×
K
x 3 y 5 u
y 5 − H
x 3 y 5
Dσ y 5
dt
−σ
y 1 x 2
dt
d ˜
t
d 2 ˜
t
dt 2
K
x 2 y 3 u
y 3 − H
x 2 y 3
Dσ y 3
dt
.
(1.78)
Note that we may determine the factor d ˜
t/dt by requiring that the velocity along
the curve X is normalized, i.e. ˜
u x ˜
u x = 1, in which case (1.76) yields
d ˜
t
dt
= ˜
u x 1 K
x 1 y 2 u
y 2 − ˜
u x 1 H
x 1 y 2
Dσ y 2
dt
.
(1.79)
Up to this point the derivation has been completely general, i.e. (1.78) is the exact
form of the deviation equation for two general arbitrarily parametrized curves Y (t)
and X(˜ t). In particular, it allows for a comparison of two general, i.e. not necessarily
geodetic, world lines in space-time. Before we consider any approximate version of
(1.78), we switch to a synchronous parametrization by rewriting the velocity as
u
y
=
dY y
dt
=
d ˜
t
dt
dY y
d ˜
t
.
(1.80)
Thus we parametrize the first curve with the same parameter ˜
t that is used on the
second curve. Accordingly, we denote
u
y
=
dY y
d ˜
t
.
(1.81)