(p.133) Appendix C Null geodesic congruences
(p.133) Appendix C Null geodesic congruences
(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
Multiplying this by k μ we see that it implies
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
(p.134) The vectors m μ, m̄μ can be extended to vector fields on C by parallel transport. Thus. The metric tensor can be written in terms of the tetrad as
Returning to the infinitesimal connecting vector ζ μ we place the following relations upon it:
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
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 μ → e iψ 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
(p.135) from which we conclude that ρ = − θ − iω with
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 r → r + dr and ζ in (C.11) is infinitesimally changed according to
Since the disk is assumed circular we can take ζ = e iϕ, 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 − iω 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 iϕ − σ e −iϕ dr. Hence the shadow is elliptical with the semi‐ major and semi‐minor axes corresponding to ϕ given by e 2iϕ = ±(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. 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 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).