6.4 Other related results

The ideas involved in the identification, at future null infinity, of interior physical quantities that were developed in the proceeding sections can also be applied to a variety of different perturbation schemes. Bramson, Adamo and Newman [9Jump To The Next Citation Point1Jump To The Next Citation Point3Jump To The Next Citation Point] have investigated how gravitational perturbations originating solely from a Maxwell radiation field can be carried through again using the asymptotic Bianchi identities to obtain, in a different context, the same identifications: a complex center-of-mass/charge world line, energy and momentum loss, as well as an angular momentum flux law that agrees exactly with the predictions of classical electromagnetic field theory. This scheme yields (up to the order of the perturbation) an approximation for the metric in the interior of the perturbed spacetime.

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 [91] on the Minkowski background. Recent work has used an electric and magnetic dipole field with a Coulomb charge [3Jump To The Next Citation Point]. 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, σ0. (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 0∗i ψ 1 = 0, 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 [3]:

′ 2 ( i′′ i′′ i′′ i′′) M B = − 3c5 D ED E + DM D M , (331 ) 1 P i′ =--- Dk′E′ DjM′′𝜖kji, 3c4 J k′ =-2- (Di ′′Dj′+ Di′′Dj ′)𝜖 , 3c3 E E M M ijk
all of which agree exactly with predictions from classical field theory [30Jump To The Next Citation Point].

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.

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