Symmetries of extremal horizons

TL;DR

This paper proves a generalization of Hawking's rigidity theorem for extremal horizons, showing that such horizons inherently possess symmetries and that these symmetries influence the behavior of matter fields and instabilities.

Contribution

It establishes an intrinsic symmetry property of extremal horizons in arbitrary dimensions, extending Hawking's rigidity theorem and analyzing implications for matter fields and instabilities.

Findings

Extremal horizons admit a Killing vector field on their cross-sections.

Enhanced near-horizon geometries have isometry groups containing SO(2,1) or R^2 ⋉ SO(1,1).

Matter fields inherit the symmetries of the horizons.

Abstract

We prove an intrinsic analogue of Hawking's rigidity theorem for extremal horizons in arbitrary dimensions: any compact cross-section of a rotating extremal horizon in a spacetime satisfying the null energy condition must admit a Killing vector field. If the dominant energy condition is satisfied for null vectors, it follows that an extension of the near-horizon geometry admits an enhanced isometry group containing $SO(2,1)$ or the 2D Poincar\'e group $\mathbb{R}^2 \rtimes SO(1,1)$. In the latter case, the associated Aretakis instability for a massless scalar field is shifted by one order in the derivatives of the field transverse to the horizon. We consider a broad class of examples including Einstein-Maxwell(-Chern-Simons) theory and Yang-Mills theory coupled to charged matter. In these examples we show that the symmetries are inherited by the matter fields.