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,
By taking linear combinations they can be written as
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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
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where is a spin-wt
quantity. From Eq. (D.3
), we obtain
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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
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
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http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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