We know that we can obtain AdS$_2\times S^2$ by considering the near horizon limit of extremal RN black holes in $4d$ with various asymptotics, i.e., either Minkowski$_4$, or (A)dS$_4$. How about the effective geometry which appears in the near horizon limit of non-extremal charged black holes? I know that, in general, the coordinate singularity can be removed by choosing global Rindler coordinates. So for any black hole, would the non-extremal near horizon geometry simply be $R_2\times S^2$ where $R_2$ refers to a $2d$ Rinder geometry? Or are there certain exceptions or subtle cases?
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The near–horizon limit must still be a solution of Einstein(–Maxwell) equations. But $R_2\times S^2$ 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.
