On higher dimensional Einstein spacetimes with a warped extra dimension

TL;DR

This paper investigates higher dimensional warped Einstein spacetimes with one extra dimension, analyzing their algebraic properties, singularities, and generating new solutions like spinning black strings in AdS space.

Contribution

It clarifies the relation between Brinkmann metrics and other coordinate systems, and demonstrates how to generate new solutions such as spinning black strings.

Findings

Relation between Weyl aligned null directions of slices and full spacetime

Identification of spacetime singularities via curvature invariants

Construction of spinning and accelerating black strings in AdS space

Abstract

We study a class of higher dimensional warped Einstein spacetimes with one extra dimension. These were originally identified by Brinkmann as those Einstein spacetimes that can be mapped conformally on other Einstein spacetimes, and have subsequently appeared in various contexts to describe, e.g., different braneworld models or warped black strings. After clarifying the relation between the general Brinkmann metric and other more specific coordinate systems, we analyze the algebraic type of the Weyl tensor of the solutions. In particular, we describe the relation between Weyl aligned null directions (WANDs) of the lower dimensional Einstein slices and of the full spacetime, which in some cases can be algebraically more special. Possible spacetime singularities introduced by the warp factor are determined via a study of scalar curvature invariants and of Weyl components measured by geodetic observers. Finally, we illustrate how Brinkmann's metric can be employed to generate new solutions by presenting the metric of spinning and accelerating black strings in five dimensional anti-de Sitter space.