Neighbourhoods of Isolated Horizons and their stationarity

TL;DR

This paper develops a coordinate system near isolated horizons using geometric invariants, enabling the explicit expansion of spacetime metrics and identifying conditions for stationarity in four-dimensional electro-vacuum cases.

Contribution

It introduces a Bondi-like coordinate system based on invariants for non-expanding horizons and derives metric expansions and stationarity conditions in 4D.

Findings

Defined a Bondi-like coordinate system near horizons

Derived metric expansion formulas in 4D spacetime

Identified conditions for the existence of a Killing field near horizons

Abstract

A distinguished (invariant) Bondi-like coordinate system is defined in the spacetime neighbourhood of a non-expanding horizon of arbitrary dimension via geometry invariants of the horizon. With its use, the radial expansion of a spacetime metric about the horizon is provided and the free data needed to specify it up to given order are determined in spacetime dimension $4$. For the case of an electro-vacuum horizon in $4$-dimensional spacetime the necessary and sufficient conditions for the existence of a Killing field at its neighbourhood are identified as differential conditions on the horizon data and data on null surface transversal to the horizon.