1.1 Time-Like Conguences
3
or, since ζ a = ζ a (x(s, ξ )),
∂ζ a
∂x b u
b
=
∂u a
∂x b ζ
b
⇔ ζ
a
,b u
b
= u
a
,b ζ
b ,
(1.7)
with the comma, as always, denoting partial differentiation. Since the Riemannian
connection is symmetric, this can be written in the manifestly covariant form
˙
ζ
a
= u
a ;b ζ
b ,
(1.8)
with ˙
ζ a = ζ a ;b u b with the semicolon denoting, as always, covariant differentiation
with respect to the Riemannian connection calculated with the metric g ab of the
space-time. In general we will use a dot to indicate the covariant derivative in the
direction u a of any quantity defined along ξ α . Hence in particular ˙
u a = u a ;b u b is
the 4-acceleration of the particle with world line ξ α . This is orthogonal to u a , that is
to say ˙
u a u a = 0, on account of the second of (1.2).
The infinitesimal connecting vector ζ a is not invariantly defined. It clearly
depends on the choice of origin of s on each world line ξ α . To translate the
origin of s by differing amounts on each world line we make the transformation
s → s = s + f (ξ α ), for some function f (ξ α ) which, of course, is constant on each
world line ξ α . The reader can show that under this transformation
ζ
a
→ ζ
= ζ
a
+
∂f
∂ξ α δξ
α u
a ,
(1.9)
so that the connecting vector acquires a component in the direction of the tangent
u a . Hence to work with a connecting vector which is invariant under (1.9) we define
the orthogonal connecting vector
η
a
= h
a
b ζ
b ,
(1.10)
with h a b = δ a
b − u a u b the projection tensor. The projection tensor projects
quantities orthogonal to u a (and thus in particular η a u a = 0) and satisfies h a b u b =
0, h a b h b c = h a c and h a a = 3. The second equation here is characteristic of
a projection, namely, the effect of applying the projection twice is the same as
applying it once! Using (1.8) and (1.10) we can deduce the transport law for η a
along ξ α :
h
a
b ˙
η
b
= A
a
b η
b with A
a
b = u
a ;c h
c
b .
(1.11)
The connecting vector η a is the position vector of ξ α + δξ α relative to ξ α for any
value of s (in other words at all points along ξ α ). Let (s) be the 3-space orthogonal
to u a at x a (s, ξ ). Then η a lies in (s) and at any s = s 0 (say) we have (s 0 ) as the
instantaneous rest frame of the particle with world line ξ α . The transport law (1.11)
means that if we move along ξ α infinitesimally by replacing s by s + ds then η a
3
or, since ζ a = ζ a (x(s, ξ )),
∂ζ a
∂x b u
b
=
∂u a
∂x b ζ
b
⇔ ζ
a
,b u
b
= u
a
,b ζ
b ,
(1.7)
with the comma, as always, denoting partial differentiation. Since the Riemannian
connection is symmetric, this can be written in the manifestly covariant form
˙
ζ
a
= u
a ;b ζ
b ,
(1.8)
with ˙
ζ a = ζ a ;b u b with the semicolon denoting, as always, covariant differentiation
with respect to the Riemannian connection calculated with the metric g ab of the
space-time. In general we will use a dot to indicate the covariant derivative in the
direction u a of any quantity defined along ξ α . Hence in particular ˙
u a = u a ;b u b is
the 4-acceleration of the particle with world line ξ α . This is orthogonal to u a , that is
to say ˙
u a u a = 0, on account of the second of (1.2).
The infinitesimal connecting vector ζ a is not invariantly defined. It clearly
depends on the choice of origin of s on each world line ξ α . To translate the
origin of s by differing amounts on each world line we make the transformation
s → s = s + f (ξ α ), for some function f (ξ α ) which, of course, is constant on each
world line ξ α . The reader can show that under this transformation
ζ
a
→ ζ
= ζ
a
+
∂f
∂ξ α δξ
α u
a ,
(1.9)
so that the connecting vector acquires a component in the direction of the tangent
u a . Hence to work with a connecting vector which is invariant under (1.9) we define
the orthogonal connecting vector
η
a
= h
a
b ζ
b ,
(1.10)
with h a b = δ a
b − u a u b the projection tensor. The projection tensor projects
quantities orthogonal to u a (and thus in particular η a u a = 0) and satisfies h a b u b =
0, h a b h b c = h a c and h a a = 3. The second equation here is characteristic of
a projection, namely, the effect of applying the projection twice is the same as
applying it once! Using (1.8) and (1.10) we can deduce the transport law for η a
along ξ α :
h
a
b ˙
η
b
= A
a
b η
b with A
a
b = u
a ;c h
c
b .
(1.11)
The connecting vector η a is the position vector of ξ α + δξ α relative to ξ α for any
value of s (in other words at all points along ξ α ). Let (s) be the 3-space orthogonal
to u a at x a (s, ξ ). Then η a lies in (s) and at any s = s 0 (say) we have (s 0 ) as the
instantaneous rest frame of the particle with world line ξ α . The transport law (1.11)
means that if we move along ξ α infinitesimally by replacing s by s + ds then η a