Almost all extremal black holes in AdS are singular

We investigate the geometry near the horizon of a generic, four-dimensional extremal black hole. When the cosmological constant is negative, we show that (in almost all cases) tidal forces diverge as one crosses the horizon, and this singularity is stronger for larger black holes. In particular, this applies to generic nonspherical black holes, such as those satisfying inhomogeneous boundary conditions. Nevertheless, all scalar curvature invariants remain finite. Moreover, we show that nonextremal black holes have tidal forces that diverge in the extremal limit. Holographically, this singularity is reflected in anomalous scaling of the specific heat with temperature. Similar (albeit weaker) effects are present when the cosmological constant is positive, but not when it vanishes.

Automated summary

We studied the geometry near the horizon of a four-dimensional extremal black hole. When the cosmological constant is negative, we found that tidal forces diverge as one crosses the horizon, and this singularity is stronger for larger black holes. This applies to generic nonspherical black holes, and all scalar curvature invariants remain finite. Furthermore, nonextremal black holes have tidal forces that diverge in the extremal limit, reflecting in anomalous scaling of the specific heat with temperature holographically. Similar, albeit weaker, effects are present when the cosmological constant is positive, but not when it vanishes.