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
Taking the gradient of , multiplied by an arbitrary four vector , (i.e., ), we see that it satisfies the linear Good Cut equation,
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
The general solution to the linearized GCE is thus
We now demonstrate that
In the integral, (385), we replace the independent variables by
Inserting Equations (387), (388) and (390) into (385) we obtain
We can go a step further. By taking the derivative of Equation (391) with respect to , we easily find the covariant form of , namely
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