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 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
In the near region, the function defined in (201
) is very small,
. The near region
scalar-wave equation can then be written as
http://www.livingreviews.org/lrr-2012-11 |
Living Rev. Relativity 15, (2012), 11
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