Rindler approximation to Kerr black hole

TL;DR

This paper investigates the applicability of the Rindler approximation to Kerr and Kerr-Newman black hole metrics near their horizons, clarifying the regions where it holds and its implications for surface gravity and Hawking temperature.

Contribution

It demonstrates that the Rindler approximation is valid only outside the external horizon and inside the internal horizon, refining previous assumptions about its applicability.

Findings

Rindler approximation holds outside h_+ and inside h_- horizons.

Provides explicit dependence of Rindler coordinates on polar angle θ.

Automatically yields surface gravities and Hawking temperatures from the approximation.

Abstract

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 {\bf only} holds {\bf outside} $h_+$ and {\bf 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 $h_\pm$.