2.2 Bondi coordinates and null tetrad

Proceeding with our examination of the properties of + ℑ, we introduce, in the neighborhood of + ℑ, what is known as a Bondi coordinate system: (uB,r,ζ, ¯ζ). In this system, uB, the Bondi time, labels the null surfaces, r is the affine parameter along the null geodesics of the constant uB surfaces and ζ = eiϕcot(𝜃∕2 ), the complex stereographic angle labeling the null geodesics of ℑ+. To reach + ℑ, we simply let r → ∞, so that + ℑ has coordinates ¯ (uB, ζ,ζ). The time coordinate uB, the topologically ℝ portion of + ℑ, labels “cuts” of + ℑ. The stereographic angle ζ accounts for the topological generators of the S2 portion of ℑ+, i.e., the null generators of ℑ+. The choice of a Bondi coordinate system is not unique, there being a variety of Bondi coordinate systems to choose from. The coordinate transformations between any two, known as Bondi–Metzner–Sachs (BMS) transformations or as the BMS group, are discussed later in this section.

Associated with the Bondi coordinates is a (Bondi) null tetrad system, (la,na, ma,ma). The first tetrad vector la is the tangent to the geodesics of the constant uB null surfaces given by [44Jump To The Next Citation Point]

a la = dx-- = gab∇buB, (5 ) dr la∇alb = 0, (6 ) ∂ ∂ la---- = ---. (7 ) ∂xa ∂r
The second null vector na is normalized so that:
a lan = 1. (8 )
In Bondi coordinates, we have [44Jump To The Next Citation Point]
∂ ∂ ∂ ∂ na---a = ---- + U ---+ XA ---A, (9 ) ∂x ∂uB ∂r ∂x
for functions U and XA to be determined, and A = 2,3 with x2 = ζ and x3 = ¯ζ. At ℑ+, na is tangent to the null generators of ℑ+.

The tetrad is completed with the choice of a complex null vector a m, a (m ma = 0) which is itself orthogonal to both la and na, initially tangent to the constant uB cuts at ℑ+ and parallel propagated inward on the null geodesics. It is normalized by

ma ¯ma = − 1. (10 )
Once more, in coordinates, we have [44Jump To The Next Citation Point]
a ∂ ∂ A ∂ m --a-= ω---+ ξ ---A, (11 ) ∂x ∂r ∂x
for some ω and ξA to be determined. All other scalar products in the tetrad are to vanish.

With the tetrad thus defined, the contravariant metric of the spacetime is given by

gab = lanb + lbna − ma ¯mb − mbm¯a. (12 )
In terms of the metric coefficients U, ω, A X, and A ξ, the metric can be written as:
( ) 0 1 0 gab = |( 1 g11 g1A |) , (13 ) 1A AB 0 g g g11 = 2(U − ωω¯), 1A A A ¯A g = X − (ω¯ξ + ωξ ), gAB = − (ξAξ¯B + ξB¯ξA).
Thus, the spacetime metric is determined from the metric coefficients.

There remains the issue of both coordinate and tetrad freedom, i.e., local Lorentz transformations. Most of the time we work in one arbitrary but fixed Bondi coordinate system, though for special situations more general coordinate systems are used. The more general transformations are given, essentially, by choosing an arbitrary slicing of ℑ+, written as uB = G (s,ζ, ¯ζ) with s labeling the slices. To keep conventional coordinate conditions unchanged requires a rescaling of r : r → r′ = (∂sG )−1r. It is also useful to be able to shift the origin of r by ′ ¯ r = r − r0(uB, ζ,ζ) with arbitrary ¯ r0(uB, ζ,ζ).

The tetrad freedom of null rotations around na, performed in the neighborhood of ℑ+, will later play a major role. For an arbitrary function L(uB, ζ, ¯ζ) on ℑ+, the null rotation about the vector na [44Jump To The Next Citation Point] is given by

¯L L la → l∗a = la − --ma − -m¯a + 0(r−2), (14 ) r r ma → m ∗a = ma − L-na + 0(r−2), (15 ) r na → n∗a = na. (16 )

Eventually, by the appropriate choice of the function L(uB, ζ, ¯ζ), the new null vector, l∗a, can be made into the tangent vector of an asymptotically shear-free NGC.

The second type of tetrad transformation is the rotation in the tangent (ma, ma ) plane, which keeps a l and a n fixed:

a iλ a m → e m , λ ∈ ℝ. (17 )
This latter transformation provides motivation for the concept of spin weight. A quantity η (ζ, ¯ζ) (s) is said to have spin-weight s if, under the transformation, Equation (17View Equation), it transforms as
η → η∗(s)(ζ, ¯ζ) = eisλη(s)(ζ, ¯ζ). (18 )

An example would be to take a vector on ℑ+, say ηa, and form the spin-weight-one quantity, η(1) = ηama.

Comment: For later use we note that L (u ,ζ,ζ¯) B has spin weight, s = 1.

For each s, spin-s functions can be expanded in a complete basis set, the spin-s harmonics, sYlm(ζ,ζ¯) or spin-s tensor harmonics, (s) Yli...j(ζ, ¯ζ) ⇔ sYlm(ζ,ζ¯).

A third tetrad transformation, the boosts, are given by

l#a = Kla, n#a = K − 1na. (19 )
These transformations induce the idea of conformal weight, an idea similar to spin weight. Under a boost transformation, a quantity, η(w), will have conformal weight w if
# w η(w) → η (w) = K η(w). (20 )

Sphere derivatives of spin-weighted functions ¯ η(s)(ζ,ζ ) are given by the action of the operators ∂ and its conjugate operator ¯∂, defined by [17Jump To The Next Citation Point]

s 1−s∂(P--η(s)) ∂η(s) = P ∂ ζ , (21 )
− s ¯∂η = P1+s ∂(P---η(s)), (22 ) (s) ∂ ¯ζ
where the function P is the conformal factor defining the sphere metric,
4d ζd¯ζ ds2 = ----2-, P

most often taken as

P = P0 ≡ 1 + ζζ¯.

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