It is worthwhile to discuss some of the issues related to these approximations. One important issue is
how to use the gauge freedom, Eq. (3.8),
, to simplify
and 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,
![]() |
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
Finally, from the reality condition on the , Eqs. (3.23
), (3.26
) and (3.25
) yield, with
and
treated as small,
![]() |
We then have, to linear order,
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
![]() This work is licensed under a Creative Commons License. E-mail us: |