Intrinsic rigidity of extremal horizons

TL;DR

This paper proves that the geometry of extremal horizons inherently admits symmetries, classifies near-horizon geometries, and derives explicit solutions for certain cases, advancing understanding of black hole horizon structures.

Contribution

It establishes the intrinsic symmetry of extremal horizon cross-sections, classifies near-horizon geometries, and provides explicit solutions for the horizon Einstein equation on spheres.

Findings

Intrinsic geometry admits a Killing vector field.

Classification of extremal Kerr horizon as the general solution.

Explicit solvability of the horizon Einstein equation on spheres.

Abstract

We prove that the intrinsic geometry of compact cross-sections of any vacuum extremal horizon must admit a Killing vector field. If the cross-sections are two-dimensional spheres, this implies that the most general solution is the extremal Kerr horizon and completes the classification of the associated near-horizon geometries. The same results hold with a cosmological constant. Furthermore, we also deduce that any non-trivial vacuum near-horizon geometry, with a non-positive cosmological constant, must have a Lie algebra of Killing vector fields that contains $\mathfrak{sl}(2)\times \mathfrak{u}(1)$ in all dimensions under no symmetry assumptions. We also show that, if the cross-sections are two-dimensional, the horizon Einstein equation is equivalent to a single fourth order PDE for the K\"ahler potential, and that this equation is explicitly solvable on the sphere if the corresponding metric admits a Killing vector.