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We investigate the interior hyperbolic region of axisymmetric and stationaryblack holes surrounded by a matter distribution. First, we treat thecorresponding initial value problem of the hyperbolic Einstein equationsnumerically in terms of a single-domain fully pseudo-spectral scheme.Thereafter, a rigorous mathematical approach is given, in which soliton methodsare utilized to derive an explicit relation between the event horizon and aninner Cauchy horizon. This horizon arises as the boundary of the future domainof dependence of the event horizon. Our numerical studies provide strongevidence for the validity of the universal relation $\Ap\Am = (8\pi J)^2$ where$\Ap$ and $\Am$ are the areas of event and inner Cauchy horizon respectively,and $J$ denotes the angular momentum. With our analytical considerations we areable to prove this relation rigorously.

The interior of axisymmetric and stationary black holes: Numerical and analytical studies