Spacetimes foliated by non-expanding and Killing horizons: higher dimension

TL;DR

This paper explores the geometry of spacetimes with foliations by non-expanding horizons in arbitrary dimensions, revealing conditions for extremal Killing horizons and deriving related metrics under Einstein's equations.

Contribution

It generalizes the study of NEH foliations to higher dimensions and arbitrary matter, establishing conditions for extremal Killing horizons and deriving new spacetime metrics.

Findings

Spacetimes foliated by NEHs satisfy specific geometric conditions.

Existence of a transversal extremal Killing horizon is linked to NEH foliation.

Derived all vacuum solutions with NEH foliations in arbitrary dimensions.

Abstract

The theory of non-expanding horizons (NEH) geometry and the theory of near horizon geometries (NHG) are two mathematical relativity frameworks generalizing the black hole theory. From the point of view of the NEHs theory, a NHG is just a very special case of a spacetime containing an NEH of many extra symmetries. It can be obtained as the Horowitz limit of a neighborhood of an arbitrary extremal Killing horizon. An unexpected relation between the two of them, was discovered in the study of spacetimes foliated by a family of NEHs. The class of 4-dimensional NHG solutions (either vacuum or coupled to a Maxwell field) was found as a family of examples of spacetimes admitting a NEH foliation. In the current paper we systematically investigate geometries of the NEHs foliating a spacetime for arbitrary matter content and in arbitrary spacetime dimension. We find that each horizon belonging to the foliation satisfies a condition that may be interpreted as an invitation for a transversal extremal Killing horizon to exist. Assuming the existence of a transversal extremal Killing horizon, we derive all the spacetime metrics satisfying the vacuum Einstein's equations.