1
Congruences of World Lines
Abstract
Material particles and photons are of paramount importance in constructing
and analysing models of gravitational fields in general relativity. Their histories
in space-time are congruences of time-like and light-like (or null) world lines
respectively. Material particles and photons therefore play a central role in this
book and we begin by describing the standard theory of congruences of timelike world lines and of null congruences. In the light-like case we shall only
require a knowledge of null geodesic congruences. For use later we describe
geodesic deviation equations with the help of Synge’s world function and parallel
propagator.
1.1
Time-Like Conguences
A three parameter family of time-like world lines, with one line passing through
each point of space-time, constitutes a time-like congruence. Such a congruence
is the history of a material continuum. A space-like hypersurface intersects the
lines of the congruence and the points of , with intrinsic local coordinates ξ α , α =
1, 2, 3 provide the three parameters necessary to label the lines of the congruence.
The parametric equations of the congruence take the form
x
a
= x
a (s, ξ
α ) ,
(1.1)
with ξ α = constants on each line of the congruence (and so, henceforth, we shall
refer to “the line ξ α ” for convenience) and s is taken as arc length or proper time
along each line, c.f. Fig. 1.1. Thus
u
a
=
∂x a
∂s
with u
a u a = 1 ,
(1.2)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1_1
1
Congruences of World Lines
Abstract
Material particles and photons are of paramount importance in constructing
and analysing models of gravitational fields in general relativity. Their histories
in space-time are congruences of time-like and light-like (or null) world lines
respectively. Material particles and photons therefore play a central role in this
book and we begin by describing the standard theory of congruences of timelike world lines and of null congruences. In the light-like case we shall only
require a knowledge of null geodesic congruences. For use later we describe
geodesic deviation equations with the help of Synge’s world function and parallel
propagator.
1.1
Time-Like Conguences
A three parameter family of time-like world lines, with one line passing through
each point of space-time, constitutes a time-like congruence. Such a congruence
is the history of a material continuum. A space-like hypersurface intersects the
lines of the congruence and the points of , with intrinsic local coordinates ξ α , α =
1, 2, 3 provide the three parameters necessary to label the lines of the congruence.
The parametric equations of the congruence take the form
x
a
= x
a (s, ξ
α ) ,
(1.1)
with ξ α = constants on each line of the congruence (and so, henceforth, we shall
refer to “the line ξ α ” for convenience) and s is taken as arc length or proper time
along each line, c.f. Fig. 1.1. Thus
u
a
=
∂x a
∂s
with u
a u a = 1 ,
(1.2)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
P. A. Hogan, D. Puetzfeld, Frontiers in General Relativity, Lecture Notes
in Physics 984, https://doi.org/10.1007/978-3-030-69370-1_1
1