Local version of the no-hair theorem

TL;DR

This paper investigates the local geometric properties of non-extremal isolated horizons in 4D vacuum Einstein spacetimes with cosmological constant, extending the no-hair theorem to a local setting and classifying solutions by area and angular momentum.

Contribution

It provides a local version of the no-hair theorem for axisymmetric, Petrov type D isolated horizons with cosmological constant, generalizing previous global results.

Findings

Solutions form a 2D family parametrized by area and angular momentum.

Embeddability in Kerr-de Sitter, Kerr-anti de Sitter, and near-extremal horizons is established.

Uniqueness of axisymmetric type D isolated horizons is demonstrated.

Abstract

Non-extremal isolated horizons embeddable in 4-dimensional spacetimes satisfying the vacuum Einstein equations with cosmological constant are studied. The horizons are assumed to be stationary to the second order. The Weyl tensor at the horizon is assumed to be of the Petrov type D. The corresponding equation on the intrinsic horizon geometry is solved in the axisymmetric case. The family of the solutions is $2$-dimensional, it is parametrized by the area and the angular momentum. The embeddability in the Kerr - de Sitter, the Kerr - anti de Sitter and the Near extremal Horizon spacetimes obtained by the Horowitz limit from the extremal Kerr - de Sitter and extremal Kerr - anti de Sitter is discussed. This uniqueness of the axisymmetric type D isolated horizons is a generalization of the similar earlier result valid in the cosmological constant free case.