8.3 A specific construction for the Kerr spacetime

Logically, this specific construction should be presented in Section 12, but the technique that it is based on justifies its placement here.

By investigating the propagation law, Eqs. (8.3View Equation) and (8.4View Equation) of Ludvigsen and Vickers for the Kerr spacetimes, Bergqvist and Ludvigsen constructed a natural flat, (but nonsymmetric) metric connection [85]. Writing the new covariant derivative in the form C ∇&tidle;AA ′λB = ∇AA ′λB + Γ AA′B λC, the ‘correction’ term Γ AA′BC could be given explicitly in terms of the GHP spinor dyad (adapted to the two principal null directions), the spin coefficients ρ, τ and τ′, and the curvature component ψ2. Γ AA ′BC admits a potential [86Jump To The Next Citation Point]: Γ ′ = − ∇ B′H ′ ′ AA BC (C B )AA B, where H ′ ′ := 1ρ− 3(ρ + ρ¯)ψ o o ¯o ′¯o ′ ABA B 2 2 A B A B. However, this potential has the structure Hab = flalb appearing in the form of the metric gab = g0ab + f lalb for the Kerr–Schild spacetimes, where g0ab is the flat metric. In fact, the flat connection ∇&tidle;e above could be introduced for general Kerr–Schild metrics [234Jump To The Next Citation Point], and the corresponding ‘correction term’ Γ AA′BC could be used to easily find the Lánczos potential for the Weyl curvature [18].

Since the connection &tidle; ∇AA ′ is flat and annihilates the spinor metric 𝜀AB, there are precisely two linearly-independent spinor fields, say λ0A and λ1A, that are constant with respect to &tidle;∇AA ′ and form a normalized spinor dyad. These spinor fields are asymptotically constant. Thus, it is natural to choose the spin space (SA, 𝜀AB) --- to be the space of the &tidle;∇a-constant spinor fields, irrespectively of the two-surface 𝒮.

A remarkable property of these spinor fields is that the Nester–Witten 2-form built from them is closed: ′ du (λA,¯λB-) = 0. This implies that the quasi-local energy-momentum depends only on the homology class of 𝒮, i.e., if 𝒮1 and 𝒮2 are two-surfaces, such that they form the boundary of some hypersurface in M, then P AB-′= P AB-′ 𝒮1 𝒮2, and if 𝒮 is the boundary of some hypersurface, then P AB′= 0 𝒮. In particular, for two-spheres that can be shrunk to a point, the energy-momentum is zero, but for those that can be deformed to a cut of the future null infinity, the energy-momentum is that of Bondi and Sachs.

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