### 6.3 Near-region scalar-wave equation

Let us go back to the scalar wave equation around the Kerr–Newman black hole. We will now study a
particular range of parameters, where the wave equations simplify. We will assume that the wave has low
energy and low mass as compared to the black hole mass and low electric charge as compared to the black
hole charge,
where . 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

The solutions 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

where has been defined in (200).