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9.3 A specific construction for the Kerr spacetime

The angular momentum of Bergqvist and Ludvigsen [68] for the Kerr spacetime is based on their special flat, non-symmetric but metric connection explained briefly in Section 8.3, but their idea is not simply the use of the two ~ \~/ 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 [170].)

The main idea is that while the energy-momentum is a single four-vector in the dual of the Hermitian subspace of SA-- ox 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 4-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 ~ \~/ aXb = gab- Hab, and showed that M0 is, in fact, a real, four dimensional affine space. Then, for a given XAA', to each \~/ ~a-constant spinor field cA they associate a primed spinor field by mA':= XA'AcA. This mA' turns out to satisfy the modified valence 1 twistor equation ~ ' ' ' ' B \~/ A(A mB ) = - HAA BB c. Finally, they form the 2-form

( A- B) [ A- ( CD B-) A- ( CD B) A- B-] W X, c ,c ab:= i c A \~/ BB' XA'C e cD - cB \~/ AA' XB'C e c D + eA'B'c(Ac B) , (64)
and define the angular momentum A-B- JS (X) with respect to the origin Xa as 1/(8pG) times the integral of W (X, cA-,cB-)ab on some closed, orientable spacelike 2-surface S. Since this Wab is closed, \~/ [aWbc] = 0 (similarly to the Nester-Witten 2-form in Section 8.3), the integral J ASB-(X) depends only on the homology class of S. Under the ‘translation’ Xe '--> Xe + ae of the ‘origin’ by a ~ \~/ a-constant 1-form ae it transforms as AB- ~ A-B- (A- B-)B-' JS (X) = JS (X) + a B-'P S, where the components aA-B' are taken with respect to the basis {cAA} 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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