We saw in Sections 3 and 4 how shear-free and asymptotically shear-free NGCs determine arbitrary complex analytic world lines in the auxiliary complex -space (or complex Minkowski space). In the examples from Sections 3 and 5, we saw how, in each of the cases, one could pick out a special GCF, referred to as the UCF, and the associated complex world line by a transformation to the complex center of mass or charge by requiring that the complex dipoles vanish. In the present section we consider the same problem, but now perturbatively for the general situation of asymptotically-flat spacetimes satisfying either the vacuum Einstein or the Einstein–Maxwell equations in the neighborhood of future null infinity. Since the calculations are relatively long and complicated, we give the basic details only for the vacuum case, but then present the final results for the Einstein–Maxwell case without an argument.
We begin with the Reissner–Nordström metric, considering both the mass and the charge as zeroth-order quantities, and perturb from it. The perturbation data is considered to be first order and the perturbations themselves are general in the class of analytic asymptotically-flat spacetimes. Though our considerations are for arbitrary mass and charge distributions in the interior, we look at the fields in the neighborhood of . The calculations are carried to second order in the perturbation data. Throughout we use expansions in spherical harmonics and their tensor harmonic versions, but terminate the expansions after . Clebsch–Gordon expansions are frequently used. See Appendix C.
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