2
1 C o n g r u e n c e s o f W o r l d L i n e s
Fig. 1.1 Sketch of the congruence x a = x a (s, ξ a ), the infinitesimal connecting vector ζ a , and
the orthogonal connecting vector η a . Here we denote fixed parameters by the “•” symbol
is the unit tangent vector field to the lines of the congruence and represents
the 4-velocity of the particle with world line ξ α . If ξ α and ξ α + δξ α , with
δξ α infinitesimal constants, label neighbouring lines of the congruence then the
infinitesimal connecting vector
ζ
a
=
∂x a
∂ξ α δξ
α ,
(1.3)
defined along ξ α , joins points on ξ α and ξ α + δξ α of equal parameter value s on
each line. We have
∂ζ a
∂s
=
∂ 2 x a
∂s ∂ξ α δξ
α
=
∂
∂ξ α
∂x a
∂s
δξ
α
=
∂u a
∂ξ α δξ
α .
(1.4)
But u a = u a (x(s, ξ )) and so, by the chain rule,
∂u a
∂ξ α =
∂u a
∂x b
∂x b
∂ξ α ,
(1.5)
giving us
∂ζ a
∂s
=
∂u a
∂x b
∂x b
∂ξ α δξ
α
=
∂u a
∂x b ζ
b ,
(1.6)