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,
ωM = O (πœ–), μM = O (πœ–), qeQ = O (πœ–), (205 )
where πœ– β‰ͺ 1. From these approximations, we deduce that ωa,ωr+, ωQ and μa = O (πœ–) as well.

We will only look at a specific region of the spacetime – the “near region” – defined by

ωr = O(πœ–), μr = O (πœ–). (206 )
Note that the near region is a distinct concept from the near-horizon region r − r+ β‰ͺ M. Indeed, for sufficiently small ω and μ, the value of r defined by the near region can be arbitrarily large.

Using the approximations (205View Equation), the wave equation greatly simplifies. It can be solved both in the near region and in the far region r ≫ M 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 M β‰ͺ r β‰ͺ ω− 1. As noted in [68Jump To The Next Citation Point], conformal invariance results from the freedom to locally choose the radius of the matching surface within the matching region.

More precisely, using (205View Equation), the angular equation (198View Equation) reduces to the standard Laplacian on the two-sphere

[ 1 m2 ] -----∂πœƒ(sin πœƒ∂πœƒ) − ---2--+ Kl S (πœƒ) = O (πœ–2). (207 ) sinπœƒ sin πœƒ
The solutions eimΟ•S (πœƒ) are spherical harmonics and the separation constants are
2 Kl = l(l + 1) + O(πœ– ). (208 )

In the near region, the function V (r) defined in (201View Equation) is very small, V (r) = O (πœ–2). The near region scalar-wave equation can then be written as

[ ] α (r+)2 α(r− )2 ∂r(Δ ∂r) + ------------------− ------------------− l(l + 1) R (r ) = 0, (209 ) (r − r+)(r+ − r− ) (r − r− )(r+ − r− )
where α(r) has been defined in (200View Equation).
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