9.3 A specific construction for the Kerr spacetime

The angular momentum of Bergqvist and Ludvigsen [86] for the Kerr spacetime is based on their special flat, nonsymmetric but metric, connection explained briefly in Section 8.3. But their idea is not simply the use of the two &tidle;∇ e-constant spinor fields in Bramson’s superpotential. Rather, in the background of their approach there are twistor-theoretical ideas. (The twistor-theoretic aspects of the analogous flat connection for the general Kerr–Schild class are discussed in [234].)

The main idea is that, while the energy-momentum is a single four-vector in the dual of the Hermitian subspace of SA-⊗ ¯SB′, the angular momentum is not only an anti-symmetric tensor over the same space, but should depend on the ‘origin’, a point in a four-dimensional affine space M 0 as well, and should transform in a specific way under the translation of the ‘origin’. Bergqvist and Ludvigsen defined the affine space M0 to be the space of the solutions Xa of &tidle;∇aXb = gab − Hab, and showed that M0 is, in fact, a real, four-dimensional affine space. Then, for a given XAA ′, to each ∇&tidle;a-constant spinor field λA they associate a primed spinor field by μA ′ := XA ′AλA. This μA ′ turns out to satisfy the modified valence-one twistor equation &tidle; ′ ′ ′ ′ B ∇A (A μB ) = − HAA BB λ. Finally, they form the 2-form

( A- B) [ A- ( CD B-) A- ( CD B) A- B-] W X, λ ,λ ab := i λA ∇BB ′ XA ′C𝜀 λD − λB ∇AA ′ XB ′C 𝜀 λ D + 𝜀A′B′λ(AλB ) , (9.3 )
and define the angular momentum AB- J𝒮 (X ) with respect to the origin Xa as 1∕(8πG ) times the integral of W (X, λA, λB)ab on some closed, orientable spacelike two-surface 𝒮. Since this Wab is closed, ∇ [aWbc] = 0 (similar to the Nester–Witten 2-form in Section 8.3), the integral JA𝒮B-(X ) depends only on the homology class of 𝒮. Under the ‘translation’ Xe ↦→ Xe + ae of the ‘origin’ by a &tidle; ∇a-constant one-form ae, it transforms as AB- &tidle; AB-- (A- B-)B-′ J𝒮 (X ) = J𝒮 (X ) + a B′P𝒮, where the components aAB-′ are taken with respect to the basis {λAA} in the solution space. Unfortunately, no explicit expression for the angular momentum in terms of the Kerr parameters m and a is given.

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