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8.3 A specific construction for the Kerr spacetime

Logically, this specific construction perhaps would have to be presented only in Section 12, but the technique that it is based on may justify its placing here.

By investigating the propagation law (59View Equation, 60View Equation) of Ludvigsen and Vickers, for the Kerr spacetimes Bergqvist and Ludvigsen constructed a natural flat, (but non-symmetric) metric connection [67]. Writing the new covariant derivative in the form C \~/ ~AA'cB = \~/ AA'cB + GAA'B cC, the ‘correction’ term C GAA'B could be given explicitly in terms of the GHP spinor dyad (adapted to the two principal null directions), the spin coefficients r, t and t', and the curvature component y2. GAA'BC admits a potential [68Jump To The Next Citation Point]: G ' = - \~/ B'H ' ' AA BC (C B)AA B, where H ' ':= 1r- 3(r + r)y 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 \~/ ~e above could be introduced for general Kerr-Schild metrics [170Jump To The Next Citation Point], and the corresponding ‘correction term’ GAA'BC could be used to find easily the Lánczos potential for the Weyl curvature [10].

Since the connection ~ \~/ AA' is flat and annihilates the spinor metric eAB, there are precisely two linearly independent spinor fields, say c0A and c1A, that are constant with respect to ~ \~/ AA' and form a normalized spinor dyad. These spinor fields are asymptotically constant. Thus it is natural to choose the spin space (SA-,eAB ) --- to be the space of the ~ \~/ a-constant spinor fields, independently of the 2-surface S.

A remarkable property of these spinor fields is that the Nester-Witten 2-form built from them is closed: ' du(cA-,cB--) = 0. This implies that the quasi-local energy-momentum depends only on the homology class of S, i.e. if S1 and S2 are 2-surfaces such that they form the boundary of some hypersurface in M, then P A-B'= PA-B-' S1 S2, and if S is the boundary of some hypersurface, then P A-B'= 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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