We will only look at a specific region of the spacetime – the “near region” – defined by
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 , 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
Living Rev. Relativity 15, (2012), 11
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