In numerous purely analytic applications outside of general
relativity, matching techniques have successfully cured
pathologies in perturbative expansions [93]. Matching is a strategy for obtaining a global solution by
patching together solutions obtained using different coordinate
systems for different regions. By adopting each coordinate system
to a length scale appropriate to its domain, a globally
convergent perturbation expansion is sometimes possible in cases
where a single coordinate system would fail. Burke showed that
matching could be used to eliminate some of the divergences
arising in perturbative calculations of gravitational radiation [94]. Kates and Kegles further showed that the use of a null
coordinate system for the exterior region is essential in the
perturbation expansions of curved space radiation fields [95]. They investigated the perturbative description of a scalar
field on a Schwarzschild background, in which case the asymptotic
behavior of the Schwarzschild light cones differs drastically
from that of the artificial Minkowski light cones used in the
perturbative expansions based upon the flat space Green function.
Use of the Minkowski light cones leads to
nonuniformities
in the expansion of the radiation fields which are eliminated by
the use of exterior coordinates based upon the true light cones.
Kates, Anderson, Kegles and Madonna extended this work to the
fully general relativistic case and reached the same conclusion [96].
Anderson later applied this approach to the slow motion
approximation of a binary system and obtained a derivation of the
radiation reaction effect on the orbital period which avoided
some objections concerning earlier approaches [97]. The use of the true light cones was also essential in
formulating as a mathematical theorem that the Bondi news
function satisfies the Einstein quadrupole formula to leading
order in a Newtonian limit [98]. Although questions of mathematical consistency still remain in
the perturbative treatment of gravitational radiation, it is
clear that the use of characteristic methods pushes these
problems to a higher perturbative order. Characteristic
techniques are used in present day codes for the Zerilli equation
[99].

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr19985
© MaxPlanckGesellschaft. ISSN 14338351
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