## D -Space Metric

In the following the derivation of the -space metric, is given.

We begin with the cut function, that satisfies the good cut eqation . The ( are (for the time being) completely independent of each other though is to be treated as being “close” the complex conjugate of .

(Later we will introduce instead of via

for the purpose of simplifying an integration.)

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 Equation (381) into the linearized GCE equation we have
which integrates immediately to

The are three independent , functions. By taking linear combinations they can be written as

where is our usual . The coefficients are functions only of the coordinates, .

Assuming that the monople 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 Equation (382), we obtain

which integrates to

The general solution to the linearized GCE is thus

We now demonstrate that

In the integral, (385), we replace the independent variables by

after some algebraical manipulation we obtain
and (surprisingly)
so that

Inserting Equations (387), (388) and (390) into (385) we obtain

Using the form Equation (389) the last integral can be easily evaluated (most easily done using and ) leading to
our sort for relationship.

We can go a step further. By taking the derivative of Equation (391) with respect to , we easily find the covariant form of , namely