Effective Geometry of Non-Extremal Near Horizon RN Black Holes

The near–horizon limit must still be a solution of Einstein(–Maxwell) equations. But $R_2times S^2$ is not, since it has non-zero scalar curvature.

If in order to obtain the limit we simply “zoom in” on the event horizon of non–extreme Reissner–Nordström metric, while keeping the ratio $Q/M$ fixed, then the only consistent limit could be the flat Minkowski spacetime written in Rindler coordinates, with Maxwell field being too weak to cause curvature (“test field”).

The only nontrivial curved near-horizon limit could be obtained if, together with zooming in on the horizon, we also vary the charge of black hole so that the solution approaches extremality $|Q|to M$ in this limit. The limiting spacetime would still be the Bertotti–Robinson solution $mathrm{AdS}_2 times S^2$, but properly choosing approach to extremality we can ensure non-zero limit for surface gravity of the horizon (and thus temperature) via the preferred choice of time. This corresponds to the experiences of an accelerated observer on Bertotti–Robinson spacetime for whom an acceleration horizon would exist. Coordinate system for the patch of AdS factor seen by such observer is known as Rindler–AdS${}_2$ chart.

References:

A general theory for limits of spacetimes:

• Geroch, R. (1969). Limits of Spacetimes. Communications in Mathematical Physics, 13(3), 180-193. doi:10.1007/BF01645486, Project Euclid.

Another paper about such limits, with emphasis on near-horizon extreme RN limit and interesting illustrations:

• Bengtsson, I., Holst, S., & Jakobsson, E. (2014). Classics Illustrated: Limits of Spacetimes. Classical and Quantum Gravity, 31(20), 205008, doi:10.1088/0264-9381/31/20/205008, arXiv:1406.4326.

More about AdS${}_2$ as a black hole:

• Spradlin, M., & Strominger, A. (1999). Vacuum States for $mathrm{AdS}_2$ Black Holes. Journal of High Energy Physics, 1999(11), 021, doi:10.1088/1126-6708/1999/11/021, arXiv:hep-th/9812073.