# Extremal horizons stationary to the second order: new constraints

## TL;DR

This paper derives new geometric constraints for null surfaces in Einstein vacuum spacetimes, providing a complete characterization of those embeddable in extremal Kerr black holes, including cases with a cosmological constant.

## Contribution

It introduces a new second-order constraint for NE-SF null surfaces and characterizes their embeddability in extremal Kerr spacetimes.

## Key findings

- Derivation of a new second-order geometric constraint.
- Complete characterization of embeddable NE-SF null geometries.
- Applicability to spacetimes with a cosmological constant.

## Abstract

We consider non-expanding shear free (NE-SF) null surface geometries embeddable as extremal Killing horizons to the second order in Einstein vacuum spacetimes. A NE-SF null surface geometry consists of a degenerate metric tensor and a consistent torsion free covariant derivative. We derive the constraints implied by the existence of an embedding. The first constraint is well known as the near horizon geometry equation. The second constraint we find is new. The constraints lead to a complete characterization of those NE-SF null geometries that are embeddable in the extremal Kerr spacetime. Our results are also valid for spacetimes with a cosmological constant.