Interior Reissner-Nordström metric and the scalar wave equation

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Abstract
We approximate the effective potential appearing in the radial part of the massless Klein-Gordon equation for the interior of a charged black hole by a second-order polynomial. This approximation allows an analytic treatment of the equation. Further we recover all the relevant properties in the interior region. For the interior normal modes the comparison between the analytical and numerical results yields an excellent agreement when the ratio of charge to mass for the black hole is close to unity. We use two theorems to bound the difference between the exact and approximate eigenvalues. A complementary scheme using both the analytical and numerical methods is proposed to solve related problems in the Kerr metric.