### 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 [28, 42, 33]. 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 such that

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, . As
with the complex Liénard–Wiechert Maxwell fields, the type II metrics and Weyl tensors are given in
terms of the angle field, . In fact, the entire metric and the field equations (the asymptotic
Bianchi identities) can be written in terms of and a Weyl tensor component (essentially
the Bondi mass). Since describes a unique shear-free NGC, it can be written,
parametrically, in terms of a unique GCF, namely a UCF, , i.e., we have that
Since can be expanded in spherical harmonics, the harmonics
can be identified with a (unique) complex world line in -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,
In Section 6, a version of these results will be derived in a far more general context.