Stationary and Axisymmetric Solutions of Higher-Dimensional General Relativity

TL;DR

This paper generalizes the canonical form of stationary, axisymmetric solutions in higher-dimensional General Relativity, deriving a matrix differential equation framework and analyzing specific solutions like rotating black holes and rings.

Contribution

It introduces a canonical metric form for higher-dimensional stationary, axisymmetric solutions, extending previous four-dimensional results and simplifying the Einstein equations into matrix differential equations.

Findings

Derived a canonical metric form for D-dimensional solutions.

Reduced Einstein equations to matrix differential equations in flat space.

Analyzed rod sources for black hole and black ring solutions.

Abstract

We study stationary and axisymmetric solutions of General Relativity, i.e. pure gravity, in four or higher dimensions. D-dimensional stationary and axisymmetric solutions are defined as having D-2 commuting Killing vector fields. We derive a canonical form of the metric for such solutions that effectively reduces the Einstein equations to a differential equation on an axisymmetric D-2 by D-2 matrix field living in three-dimensional flat space (apart from a subclass of solutions that instead reduce to a set of equations on a D-2 by D-2 matrix field living in two-dimensional flat space). This generalizes the Papapetrou form of the metric for stationary and axisymmetric solutions in four dimensions, and furthermore generalizes the work on Weyl solutions in four and higher dimensions. We analyze then the sources for the solutions, which are in the form of thin rods along a line in the three-dimensional flat space that the matrix field can be seen to live in. As examples of stationary and axisymmetric solutions, we study the five-dimensional rotating black hole and the rotating black ring, write the metrics in the canonical form and analyze the structure of the rods for each solution.