On the regularity of deformed extremal horizons

It has recently been argued that extremal black holes can act as amplifiers of new physics, due to horizon instabilities that enhance the effects of ultraviolet corrections. In this paper, we revisit some of these claims and investigate the viability of a class of non-spherical extremal black holes. In particular, we revisit the regularity of perturbed extremal Reissner–Nordström AdS black holes showing that, while some certain components of the scalar stress energy tensor diverge, the backreaction remains finite. We also study geodesic completeness, identifying a simple geometric constraint which, if satisfied, ensures that null geodesics cross the horizon smoothly. This analysis suggests the existence of a broad class of spacetimes with regular non-spherical horizons.

💡 Research Summary

The paper revisits the claim that extremal black holes, because of horizon instabilities, act as amplifiers of ultraviolet (UV) physics and that perturbations by a scalar field inevitably render the horizon singular. Focusing on extremal Reissner–Nordström anti‑de Sitter (RN‑AdS) black holes, the authors perform a detailed analytic study of a test massless scalar field on the background geometry, both in static coordinates and in ingoing Eddington–Finkelstein‑like coordinates.

In the static chart the Klein‑Gordon equation separates into radial and temporal parts. Near the outermost root (r_{+}) of the metric function (f(r)) the radial equation exhibits a regular singular point. For a non‑extremal (simple root) horizon the indicial equation gives (\gamma=0) for all angular modes (\ell), guaranteeing regularity. For an extremal (double root) horizon the indicial exponents become
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