It is worthwhile to discuss some of the issues related to these approximations. One important issue is
how to use the gauge freedom, Equation (79),
, to simplify the ‘velocity vector’,
A Notational issue: Given a complex analytic function (or vector) of the complex variable , say
, then
can be decomposed uniquely into two parts,
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where all the coefficients in the Taylor series for and
are real. With but a slight extension
of conventional notation we refer to them as real analytic functions.
With this notation, we also write
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By using the reparametrization of the world line, via , with the decomposition
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the ‘velocity’ transforms as
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One can easily check that by the appropriate choice of we can make
and
orthogonal, i.e.,
The remaining freedom in the choice of is simply an additive complex constant, which is used
shortly for further simplification.
We now write , which, with the slow motion approximation, yields, from the
normalization,
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i.e., is second order. Since
is second order,
is a constant plus a second-order
term. Using the remaining complex constant freedom in
, the constant can be set to zero:
Finally, from the reality condition on the , Equations (94
), (97
) and (96
) yield, with
and
treated as small,
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We then have, to linear order,
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