We briefly describe this procedure. One initially chooses as a background an exact solution of the Einstein equations; three cases were studied: flat Minkowski spacetime, the Schwarzschild spacetime with a ‘small’ mass and the Schwarzschild spacetime with a finite, ‘zero order’, mass. For such backgrounds, the set of spin coefficients is known and fixed. On this background the Maxwell equations were integrated to obtain the desired electromagnetic field that acts as the gravitational perturbation. Bramson has done this for a pure electric dipole solution [19, 2] on the Minkowski background. Recent work has used an electric and magnetic dipole field with a Coulomb charge . The resulting Maxwell field, in each case, is then inserted into the asymptotic Bianchi identities, which, in turn, determine the behavior of the perturbed asymptotic Weyl tensor, i.e., the Maxwell field induces nontrivial changes to the gravitational field. Treating the Maxwell field as first order, the calculations were done to second order, as was done earlier in this review.
Using the just obtained Weyl tensor terms, one can proceed to the integration of the spin-coefficient equations and the second-order metric tensor. For example, one finds that the dipole Maxwell field induces a second-order Bondi shear, . (This in principle would lead to a fourth-order gravitational energy loss, which in our approximation is ignored.)
Returning to the point of view of this section, the perturbed Weyl tensor can now be used to obtain the same physical identifications described earlier, i.e., by employing a null rotation to set , equations of motion and asymptotic physical quantities, (e.g., center of mass and charge, kinematic expressions for momentum and angular momentum, etc.) for the interior of the system could be found. Although we will not repeat these calculations here, we present a few of the results. Though the calculations are similar to the earlier ones, they differ in two ways: there is no first-order freely given Bondi shear and the perturbation term orders are different.
For instance, the perturbations induced by a Coulomb charge and general electromagnetic dipole Maxwell field in a Schwarzschild background lead to energy, momentum, and angular momentum flux relations :.
The familiarity of such results is an exhibit in favor of the physical identification methods described in this review, i.e., they are a confirmation of the consistency of the identification scheme.
Living Rev. Relativity 15, (2012), 1
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