2.3 The optical equations

Since this work concerns NGCs and, in particular, shear-free and asymptotically shear-free NGCs, it is necessary to first define them and then study their properties.

Given a Lorentzian manifold with local coordinates, a x, and an NGC, i.e., a foliation by a three parameter family of null geodesics,

a a w x = X (r,y ), (23 )
with r the affine parameterization and the (three) yw labeling the geodesics, the tangent vector field a a a l = DX ≡ ∂rX satisfies the geodesic equation
la∇alb = 0.

The two complex optical scalars (spin coefficients), ρ and σ, are defined by

1 ρ = -(− ∇ala + i curl la), (24 ) 2∘ ------------ curl la ≡ (∇ [alb]∇alb)
a b σ = ∇ (alb)m m

with ma an arbitrary complex (spacelike) vector satisfying mama = mala = mama- − 1 = 0. Equivalently σ can be defined by its norm,

( ) -- 1- ab 1- a 2 σσ = 2 ∇ (alb)∇ l − 2(∇al ) ,

with an arbitrary phase.

The ρ and σ satisfy the optical equations of Sachs [56], namely,

∂ρ- 2 ∂r = ρ + σ ¯σ + Φ00, (25 )
∂ σ --- = 2ρσ + ψ0, (26 ) ∂r
Φ00 = Rablalb, a bc d ψ0 = − Cabcdl m lm ,
where Φ00 and Ψ0 are, respectively, a Ricci and a Weyl tensor tetrad component (see below). In flat space, i.e., with Φ00 = Ψ0 = 0, excluding the degenerate case of ρρ-− σ σ-= 0, plane and cylindrical fronts, the general solution is
iΣ − r ρ = -2-----2----0-0, (27 ) r + Σ − σ σ
------σ0------- σ = r2 + Σ2 − σ0σ0. (28 )
The complex 0 σ (referred to as the asymptotic shear) and the real Σ (called the twist) are determined from the original congruence, Equation (23View Equation). Both are functions just of the parameters, yw. Their behavior for large r is given by
1 iΣ Σ2 σ0σ0 − 4 ρ = − --+ -2-+ --3 − --3--+ O(r ), (29 ) r0 r r r σ = σ--+ O (r −4). (30 ) r2
From this, σ0 gets its name as the asymptotic shear. In Section 3, we return to the issue of the explicit construction of NGCs in Minkowski space and in particular to the construction and detailed properties of regular shear-free congruences.

Note the important point that, in 𝕄, the vanishing of the asymptotic shear forces the shear to vanish. The same is not true for asymptotically-flat spacetimes. Specifically, for future null asymptotically-flat spaces described in a Bondi tetrad and coordinate system, we have, from other considerations, that

Φ00 = O (r−6), −5 ψ0 = O (r ), Σ = 0,
which leads to the asymptotic behavior of ρ and σ,
-- 1 σ0 σ0 −5 ρ = ρ = − r-+ --r3- + O (r ), σ0 σ = -2-+ O (r−4), r
with the two order symbols explicitly depending on the leading terms in Φ00 and Ψ0. The vanishing of σ0 does not, in this nonflat case, imply that σ vanishes. This case, referred to as asymptotically shear-free, plays the major role later. It will be returned to in greater detail in Section 4.
  Go to previous page Go up Go to next page