Electromagnetic field of a charge asymptotically approaching spherically symmetric black hole

We consider a test charged particle falling onto a Schwarzschild black hole and evaluate its electromagnetic field. The Regge-Wheeler equation is solved analytically by approximating the potential barrier with Dirac delta function and rectangular barrier. We show that for asymptotically large time measured by a distant observer the electromagnetic field approaches the spherically symmetric electrostatic field exponentially fast. This implies that in the region accessible to a distant observer the initial state of separated charge and Schwarzschild black hole becomes asymptotically indistinguishable from the Reisnner-Nordstr\"om solution. Implications of this result for models with plasma accretion on black holes are discussed.7 a