Inner Cauchy Horizon of Axisymmetric and Stationary Black Holes with Surrounding Matter in Einstein-Maxwell Theory

We study the interior electrovacuum region of axisymmetric and stationary black holes with surrounding matter and find that there exists always a regular inner Cauchy horizon inside the black hole, provided the angular momentum $J$ and charge $Q$ of the black hole do not vanish simultaneously. In particular, we derive an explicit relation for the metric on the Cauchy horizon in terms of that on the event horizon. Moreover, our analysis reveals the remarkable universal relation $(8\pi J{)}^2+(4\pi Q^2{)}^2=A^{+}{A}^{-}$, where $A^{+}$ and $A^{-}$ denote the areas of event and Cauchy horizon, respectively.

Figure 1

Sketch of a part of a black hole spacetime in Weyl coordinates (left panel) and Boyer-Lindquist type coordinates (right panel). ${\mathcal{A}}^{+}$ and ${\mathcal{A}}^{-}$ denote upper and lower parts of the symmetry axis and ${\mathcal{H}}^{+}$ and ${\mathcal{H}}^{-}$ denote event and Cauchy horizons. In the Letter we will integrate the linear problem for the Einstein-Maxwell equations along the closed dashed line.