5.3 Type II twisting metrics

It was pointed out in the previous section that the RT metrics are the general relativistic analogues of the (real) Liénard–Wiechert Maxwell fields. The type II algebraically-special twisting metrics are the gravitational analogues of the complex Liénard–Wiechert Maxwell fields described earlier. Unfortunately they are far more complicated than the RT metrics. In spite of the large literature and much effort there are very few known solutions and much still to be learned [41, 58, 46]. We give a very brief description of them, emphasizing only the items of relevance to us.

A null tetrad system (and null geodesic coordinates) can be adopted for the type II metrics so that the Weyl tetrad components are

ψ0 = ψ1 = 0, ψ2 ⁄= 0.
It follows from the Goldberg–Sachs theorem that the degenerate principal null congruence is geodesic and shear-free. Thus, from the earlier discussions it follows that there is a unique angle field, -- L (uB,ζ,ζ). As with the complex Liénard–Wiechert Maxwell fields, the type II metrics and Weyl tensors are given in terms of the angle field, -- L (uB,ζ, ζ). In fact, the entire metric and the field equations (the asymptotic Bianchi identities) can be written in terms of L and a Weyl tensor component (essentially the Bondi mass). Since L (u ,ζ,ζ) B describes a unique shear-free NGC, it can be written parametrically in terms of a unique GCF, namely the UCF -- X (type II)(τ,ζ,ζ ). So, we have that
L (u ,ζ,ζ) = ∂ X , B (τ) (type II)-- uB = X (type II)(τ,ζ,ζ ).
Since -- X (type II)(τ,ζ,ζ) can be expanded in spherical harmonics, the l = (0,1) harmonics can be identified with a (unique) complex world line in ℋ-space. The asymptotic Bianchi identities then yield both kinematic equations (for angular momentum and the Bondi linear momentum) and equations of motion for the world line, analogous to those obtained for the Schwarzschild perturbation and the RT metrics. As a kinematic example, the imaginary part of the world line is identified as the intrinsic spin, the same identification as in the Kerr metric,
Si = MBc ξiI. (5.33 )
In Section 6, a version of these results will be derived in a far more general context.

Recently, the type II Einstein–Maxwell equations were studied using a slow-motion perturbation expansion around the Reissner–Nördstrom metric, keeping spherical harmonic contributions up to l = 2. It was found that the above-mentioned world line coincides in this case with that given by the Abraham–Lorentz–Dirac equation, prompting us to consider such spacetimes as ‘type II particles’ in the same way that one can refer to Reissner–Nördstrom–Schwarzschild or Kerr–Newman ‘particles’ [52].

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