. From these approximations, we deduce that 
and 
as
well.
We will only look at a specific region of the spacetime – the “near region” – defined by
Note that the near region is a distinct concept from the near-horizon region
. Indeed, for
sufficiently small 
and 
, the value of 
defined by the near region can be arbitrarily
large.
Using the approximations (205
), the wave equation greatly simplifies. It can be solved both in the near
region and in the far region 
in terms of special functions. A complete solution can then be
obtained by matching near and far solutions together along a surface in the matching region
. As noted in [68
], conformal invariance results from the freedom to locally choose the
radius of the matching surface within the matching region.
More precisely, using (205), the angular equation (198) reduces to the standard Laplacian on the
two-sphere
are spherical harmonics and the separation constants are
In the near region, the function 
defined in (201) is very small, 
. The near region
scalar-wave equation can then be written as
has been defined in (200).
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Living Rev. Relativity 15, (2012), 11
http://www.livingreviews.org/lrr-2012-11 |
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![[ 1 m2 ] -----∂π(sin π∂π) − ---2--+ Kl S (π) = O (π2). (207 ) sinπ sin π](article1064x.gif){kind=small}
![[ ] α (r+)2 α(r− )2 ∂r(Δ ∂r) + ------------------− ------------------− l(l + 1) R (r ) = 0, (209 ) (r − r+)(r+ − r− ) (r − r− )(r+ − r− )](article1069x.gif){kind=small}