Rigidity of the extremal Kerr-Newman horizon

TL;DR

The paper addresses the intrinsic rigidity of extremal horizons in four-dimensional Einstein-Maxwell theory by analyzing the quasi-Einstein-Maxwell structure on the horizon cross-section $M$. It derives a divergence identity for the 2D equations, defines $\rho = \sqrt{\psi^2+\beta^2}$ with $\beta = \star B$, and proves the existence of a positive function $\Gamma$ such that $K = \Gamma X + \nabla\Gamma$ is a Killing vector preserving the Maxwell data. Consequently, $\mathcal{L}_K g = 0$ and $\Gamma\rho$ is constant, forcing $M$ to be topologically $S^2$ and, under axial symmetry, identifying the non-gradient solutions with the extremal Kerr-Newman horizon (possibly with a cosmological constant $\lambda$). This completes the intrinsic classification of near-horizon geometries with compact cross-sections in 4D Einstein-Maxwell theory and clarifies rigidity phenomena for extremal horizons. The results extend to include a nonzero $\lambda$ and provide a framework for higher-dimensional generalizations in future work.

Abstract

We prove that the intrinsic geometry of compact cross-sections of an extremal horizon in four-dimensional Einstein-Maxwell theory must admit a Killing vector field or is static. This implies that any such horizon must be an extremal Kerr-Newman horizon and completes the classification of the associated near-horizon geometries. The same results hold with a cosmological constant.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Remark 2.4
• Corollary 2.5