Rindler approximation to Kerr-Newman black hole

We show that the Rindler approximation to the time-radial part of the Kerr and Kerr-Newman metrics near their external $h_{+}$ and internal $h_{-}$ horizons only holds outside$h_{+}$ and inside$h_{-}$, so respectively inside and outside the external and internal ergospheres, regions where, in Boyer-Lindquist coordinates, both $g_{tt}$ and $g_{rr}$ are negative, but preserving the Lorentzian character of the metric, and $ r > 0$i.e. outside the region $ r < 0$ where closed timelike curves exist. At each point, the choice of Rindler coordinates is not trivial, but depends on the polar angle $\theta$. The approximation, as is known, automatically gives the absolute values of the surface gravities $\kappa_{\pm}$ as the corresponding proper accelerations, and therefore the Hawking temperatures $\tau_{\pm}$ at $h_{\pm}$ .