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Equations of Motion in General Relativity$

Hideki Asada, Toshifumi Futamase, and Peter Hogan

Print publication date: 2010

Print ISBN-13: 9780199584109

Published to Oxford Scholarship Online: January 2011

DOI: 10.1093/acprof:oso/9780199584109.001.0001

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(p.133) Appendix C Null geodesic congruences

(p.133) Appendix C Null geodesic congruences

Source:
Equations of Motion in General Relativity
Publisher:
Oxford University Press

(p.133) Appendix C

Null geodesic congruences

Let k μ be the components of a null vector field, in coordinates z μ, in a space‐time with metric tensor components g μν. Take the integral curves of k μ to be geodesics with r (say) an affine parameter along them. Then we can write k μ = ∂z μ/∂r, g μν k μ k ν = k μ k μ = 0 and

k μ | ν k ν = 0 ,
(C.1)
with the stroke indicating covariant differentiation with respect to the Riemannian connection associated with the metric tensor g μν. The integral curves of the vector field k μ constitute a null geodesic congruence. Along any curve C of the congruence let ζ μ be an infinitesimal connecting vector joining each point of C to a point of any neighboring curve having the same value of the affine parameter r, then ζ μ is transported along C according to the transport law
ζ μ | ν k ν = k μ | ν ζ ν .
(C.2)

Multiplying this by k μ we see that it implies

r ( k μ ζ μ ) = 0 ,
(C.3)
so that the scalar product k μ ζ μ is constant along the null geodesic C. At any point P of C let e μ be a unit time‐like vector. Thus e μ e μ = 1. The physical significance of introducing e μ will emerge below. For the moment we note that we can extend e μ to a unit time‐like vector field along C by parallel transport and thus e | ν μ k ν = 0 . We can use e μ to normalize k μ by requiring k μ e μ = 1. Then we can define a parallel transported, null vector field along C by
l μ = e μ 1 2 k μ ,
(C.4)
satisfying
l μ l μ = 0 , l μ k μ = 1.
(C.5)

At P on C the pair of null vectors k μ, l μ can be completed to a null tetrad by the addition of a complex null vector m μ and its complex conjugate m̄μ, and so m μ m μ = 0 = m̄μ m̄μ, satisfying

m μ m ¯ μ = 1 , m μ k μ = m ¯ μ k μ = m μ l μ = m ¯ μ l μ = 0.
(C.6)

(p.134) The vectors m μ, m̄μ can be extended to vector fields on C by parallel transport. Thus m | ν μ k ν = m ¯ | ν μ k ν = 0 . The metric tensor can be written in terms of the tetrad as

g μ ν = m μ m ¯ ν m ¯ μ m ν + k μ l ν + l μ k ν .
(C.7)

Returning to the infinitesimal connecting vector ζ μ we place the following relations upon it:

ζ μ e μ = 0 and ( δ ν μ e μ e ν ) k ν ζ μ = 0.
(C.8)

Hence ζ μ is orthogonal to e μ and also orthogonal to the projection of k μ orthogonal to e μ. We will describe presently a physical interpretation of these relations. Clearly they imply that ζ μ is orthogonal to e μ and to k μ and thus by (C.4) is orthogonal to both l μ and k μ. Hence we can write

ζ μ = ζ ¯ m μ + ζ m ¯ μ ,
(C.9)
where ζ is a complex‐valued function of the coordinates z μ. Substituting this into (C.2) results in
ζ ¯ r m μ + ζ r m ¯ μ = ζ ¯ k μ | ν m ν + ζ k μ | ν m ¯ ν ,
(C.10)
from which we conclude, on multiplying by m μ, that
ζ r = σ ζ ¯ ρ ζ ,
(C.11)
where
σ = k μ | ν m μ m ν , ρ = k μ | ν m μ m ¯ ν .
(C.12)

Notwithstanding appearances the modulus of the complex variable σ and the real and imaginary parts of the complex variable ρ are independent of the choice of null tetrad. In other words they can be calculated from a knowledge only of g μν and k μ. Clearly the argument of the complex variable σ does depend upon the choice of null tetrad since a simple change of tetrad of the form m μe m μ, for ψ a real constant, does not leave σ invariant. Expanding the tensor with components k μǀν on the null tetrad, bearing in mind that k μ k μǀν = 0 = k ν k μǀν, k μǀν is real and the definitions of σ and ρ in (C.12), we find that

k μ | ν = a k μ k ν + b k μ m ν + b ¯ k μ m ¯ ν + c m μ k ν + c ¯ m ¯ μ k ν + σ ¯ m μ m ν + σ m ¯ μ m ¯ ν + ρ ¯ m μ m ¯ ν + ρ m ¯ μ m ν ,
(C.13)
where a is a real‐valued function of the coordinates and b, c are complex‐valued functions of the coordinates, with the bar as always denoting complex conjugation. We find by direct calculation, using the scalar products listed in (C.5) and (C.6), that
k μ | μ = ρ ρ ¯ ,
(C.14)
k μ | ν k μ | ν = 2 σ σ ¯ + 2 ρ ρ ¯ ,
(C.15)
k μ | ν k μ | ν = 2 σ σ ¯ + ρ 2 + ρ 2 ,
(C.16)

(p.135) from which we conclude that ρ = − θ with

θ = 1 2 k μ | μ and ω 2 = 1 2 k [ μ | ν ] k μ | ν 0 ,
(C.17)
where the square brackets denote skew‐symmetrization and
ǀ σ ǀ = 1 2 k ( μ | ν ) k μ | ν θ 2 ,
(C.18)
with the round brackets denoting symmetrization. Since ǀσǀ, θ, and ω are determined only by the metric and the tangent to the null geodesic congruence it makes sense to look for a physical/geometrical interpretation of these scalars which have the potential to distinguish one null geodesic congruence from another. For this we turn to (C.11) and the relations (C.8) involving the infinitesimal connecting vector ζ μ.

At an event P on the null geodesic C we consider e μ to be the 4‐velocity of a small plane circular opaque disk placed in the path of a small bundle of photons with null geodesic world‐lines in the neigborhood of C. The first condition involving ζ μ in (C.8) means that ζ μ, the position vector of points on the boundary of the disk, lies in the rest‐frame of the disk. The second condition in (C.8) requires the disk to be oriented relative to the photon paths so that the photons strike the disk at right angles when viewed in the rest‐frame of the disk. If Q is an event on C an affine parameter distance dr into the future from the event P then at Q we consider e μ to be the 4‐velocity of a small plane screen on which a shadow of the disk is projected. The position vector of points on the boundary of the shadow is given by ζ μ, which lies in the rest‐frame of the screen on account of the first of (C.8). On account of the second of (C.8) the disk is oriented relative to the photon paths so that the photons strike the screen at right angles when observed in the rest‐frame of the screen. In passing from the disk to the screen rr + dr and ζ in (C.11) is infinitesimally changed according to

ζ ζ = ( 1 ρ d r ) ζ σ ζ ¯ d r .
(C.19)

Since the disk is assumed circular we can take ζ = e , with 0 ≤ ϕ ≤ 2π, in (C.19). Thus the complex variable ζ′ specifies points on the boundary of the shadow relative to points on the boundary of the disk which are specified by ζ. Writing as above ρ = −θ we consider first the case ω = 0 = σ. In this case (C.19) reduces to ζ′ = (1 + θ dr) ζ and thus we see that the shadow is a circle whose radius is larger than the disk radius if θ 〉 0 and smaller than the disk radius if θ 〈 0. For this reason we interpret the scalar θ as the expansion (if positive) or contraction (if negative) of the congruence. Next we consider the case θ = 0 = σ. Now (C.19) becomes ζ′ = (1 + iω dr) ζ = e iω dr ζ, neglecting O(dr 2)–terms. Hence in this case the shadow is also a circle of the same radius as the disk but points on the boundary of the shadow are rotated relative to points on the boundary of the disk through a small angle ω dr. The scalar ω is referred to as the twist of the congruence for this reason. Finally the case θ = 0 = ω results in (C.19) becoming ζ′ = e σ e dr. Hence the shadow is elliptical with the semi‐ major and semi‐minor axes corresponding to ϕ given by e 2 = ±(σ/σ̄)½. The lengths of these axes are thus l ± = 1 ± ǀσǀ dr and their ratio is therefore l +/l = 1 + 2 ǀσǀ dr approximately. The area of the shadow is the same as the area of the circular disk (p.136) (neglecting O(dr 2)‐terms) and thus we call the scalar ǀσǀ the shear of the null geodesic congruence. In practice the complex scalar σ is usually referred to as the complex shear of the congruence.

In section 1 of Chapter 5 the generators of the future null cones with vertices on the time‐like world‐line r = 0 in Minkowskian space‐time are the null geodesic integral curves of the vector field k μ. The derivatives of k μ are given by formula (5.11). From these derivatives one readily verifies that this null geodesic congruence has vanishing twist and shear and has expansion θ = ( 1 \ 2 ) k , μ μ = r 1 . In section 2 of Chapter 5 a twist‐free, geodesic null congruence is defined via the 1‐form (5.37). This congruence has complex shear σ and expansion θ given by (5.50) and (5.51) respectively. The complex shear is calculated using the first of (C.12) with m μ and m̄μ given via the 1‐form 2 m μ d z μ = ϑ 1 + i ϑ 2 and its complex conjugate, with ϑ1 and ϑ2 given by (5.46) and (5.47).

The theory summarized in this appendix is commonly referred to in the literature as the Ehlers‐Sachs theorem which first appeared in Ehlers et al. (1961). Particularly elegant treatments of it can be found in Pirani (1965) and Chandrasekhar (1983).