A specific construction for the Kerr spacetime

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, Equations (8.3 ) and (8.4 ) of Ludvigsen and Vickers for the Kerr spacetimes, Bergqvist and Ludvigsen constructed a natural flat, (but nonsymmetric) metric connection [77 ]. 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 [78 ]: $Γ ′ = − ∇ B′H ′ ′ AA BC (C B )AA B$ , 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 [212 ], and the corresponding ‘correction term’ $Γ AA′BC$ could be used to easily find the Lánczos potential for the Weyl curvature [11 ].

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, independent of the two-surface $𝒮$ .

A remarkable property of these spinor fields is that the Nester–Witten two-form built from them is closed: $′ du(λA,λ¯B--) = 0$ . 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 $AB ′ P 𝒮-- = 0$ . 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.