Extremal horizons stationary to the second order: new constraints

Understand

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.