## D -Space Metric

In the following, the derivation of the -space metric of (4.24) is given. We begin with the cut function, that satisfies the good-cut equation . The ( are (for the time being) completely independent of each other, though is to be treated as being ‘close’ the complex conjugate of . Taking the gradient of , multiplied by an arbitrary four vector (i.e., ), we see that it satisfies the linear good cut equation,

Let be a particular solution, and assume for the moment that the general solution can be written as
with the four components of to be determined. Substituting Eq. (D.2) into the linearized good-cut equation, we have
which integrates immediately to
where the are three independent , functions.

By taking linear combinations they can be written as

where is our usual , and the coefficients are functions only of the coordinates . Assuming that the monopole term in is sufficiently large so that it has no zeros and then by rescaling we can write as a monopole plus higher harmonics in the form

where is a spin-wt quantity. From Eq. (D.3), we obtain

which integrates to

The general solution to the linearized good-cut equation is thus

We now demonstrate that

In the integral of (D.6), we replace the independent variables by

After some algebraic manipulation we obtain
and (surprisingly)
so that

Inserting Eqs. (D.8), (D.9) and (D.11) into (D.6) we obtain

Using the form Eq. (D.10) the last integral can be easily evaluated (most easily done using and ) leading to

In particular, note that when (i.e., the case of an everywhere shear-free NGC) and the metric on -space reduces to the complex Minkowski metric, as claimed throughout the text. We can go a step further by taking the derivative of Eq. (D.12) with respect to to find the covariant form of , namely