1.3 The World Function and Deviation Equations
13
We collect the following useful identities for the world function σ :
[σ ] = [σ x ] = [σ y ] = 0,
(1.63)
[σ x 0 x 1 ] = [σ y 0 y 1 ] = g y 0 y 1 ,
(1.64)
[σ x 0 y 1 ] = [σ y 0 x 1 ] = −g y 0 y 1 ,
(1.65)
[σ x 0 x 1 x 2 ] = [σ x 0 x 1 y 2 ]
= [σ x 0 y 1 y 2 ] = [σ y 0 y 1 y 2 ] = 0,
(1.66)
[σ (n indices) ] ∼ ∇
n−4 R abcd for n ≥ 4.
(1.67)
Apart from the world function σ (x, y), another important bitensor is the parallel
propagator g y x (x, y)—not to be confused with the metric—that allows for the
parallel transportation of objects along the unique geodesic that links the points x
and y. For example, given a vector V x at x, the corresponding vector at y is obtained
by means of the parallel transport along the geodesic curve as V y = g y x (x, y)V x .
The coincidence limits of the parallel propagator and its first derivative are given
by:
g
x 0 y 1
= δ
y 0 y 1 ,
(1.68)
g
x 0 y 1 ;x 2
=
g
x 0 y 1 ;y 2
= 0,
(1.69)
g
x 0 y 1 ;x 2 x 3
= −
g
x 0 y 1 ;x 2 y 3
=
g
x 0 y 1 ;x 2 x 3
= −
g
x 0 y 1 ;y 2 y 3
=
1
2
R
y 0 y 1 y 2 y 3 ,
(1.70)
[g
x 0 x 1 ;(n indices) ] ∼ ∇
n−2 R abcd for n ≥ 2.
(1.71)
With these bitensor related concepts in mind we now return to the deviation of
the two curves X and Y depicted in Fig. 1.2. Conceptually the closest object to the
connecting vector between the two points x and y is the covariant derivative of the
world function σ y . Note that σ y is tangent to the geodesic Z at y (its length being
the geodesic length between y and x) and only in flat space-time it coincides with
the connecting vector. Keeping in mind such an interpretation, let us now work out
a propagation equation for this “generalized” connecting vector along the reference
curve, cf. Fig. 1.2. Following our conventions the reference curve will be Y (t) and
we define the generalized connecting vector to be:
η
y
:= −σ
y .
(1.72)
13
We collect the following useful identities for the world function σ :
[σ ] = [σ x ] = [σ y ] = 0,
(1.63)
[σ x 0 x 1 ] = [σ y 0 y 1 ] = g y 0 y 1 ,
(1.64)
[σ x 0 y 1 ] = [σ y 0 x 1 ] = −g y 0 y 1 ,
(1.65)
[σ x 0 x 1 x 2 ] = [σ x 0 x 1 y 2 ]
= [σ x 0 y 1 y 2 ] = [σ y 0 y 1 y 2 ] = 0,
(1.66)
[σ (n indices) ] ∼ ∇
n−4 R abcd for n ≥ 4.
(1.67)
Apart from the world function σ (x, y), another important bitensor is the parallel
propagator g y x (x, y)—not to be confused with the metric—that allows for the
parallel transportation of objects along the unique geodesic that links the points x
and y. For example, given a vector V x at x, the corresponding vector at y is obtained
by means of the parallel transport along the geodesic curve as V y = g y x (x, y)V x .
The coincidence limits of the parallel propagator and its first derivative are given
by:
g
x 0 y 1
= δ
y 0 y 1 ,
(1.68)
g
x 0 y 1 ;x 2
=
g
x 0 y 1 ;y 2
= 0,
(1.69)
g
x 0 y 1 ;x 2 x 3
= −
g
x 0 y 1 ;x 2 y 3
=
g
x 0 y 1 ;x 2 x 3
= −
g
x 0 y 1 ;y 2 y 3
=
1
2
R
y 0 y 1 y 2 y 3 ,
(1.70)
[g
x 0 x 1 ;(n indices) ] ∼ ∇
n−2 R abcd for n ≥ 2.
(1.71)
With these bitensor related concepts in mind we now return to the deviation of
the two curves X and Y depicted in Fig. 1.2. Conceptually the closest object to the
connecting vector between the two points x and y is the covariant derivative of the
world function σ y . Note that σ y is tangent to the geodesic Z at y (its length being
the geodesic length between y and x) and only in flat space-time it coincides with
the connecting vector. Keeping in mind such an interpretation, let us now work out
a propagation equation for this “generalized” connecting vector along the reference
curve, cf. Fig. 1.2. Following our conventions the reference curve will be Y (t) and
we define the generalized connecting vector to be:
η
y
:= −σ
y .
(1.72)