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Stationary and Axisymmetric Solutions of Higher-Dimensional General Relativity

Troels Harmark

TL;DR

The work develops a canonical framework for pure-gravity solutions in $D$ dimensions with $D-2$ commuting Killing fields, reducing the Einstein equations to a matrix equation for $G_{ij}(r,z)$ on flat 3-space and coupling it to a solvable $ u$-system. It introduces a rod-structure along the axis $r=0$ that encodes horizons and compact directions, providing a geometric classification tool for higher-dimensional spacetimes. The methodology is validated through detailed treatments of the four- and five-dimensional asymptotics and explicit canonical forms for the Kerr, Myers–Perry, and black ring solutions, including regularity and horizon data derived from rod directions. The framework offers a path to constructing new stationary axisymmetric solutions in higher dimensions and deepens understanding of the phase structure and uniqueness properties of higher-dimensional black holes.

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.

Stationary and Axisymmetric Solutions of Higher-Dimensional General Relativity

TL;DR

The work develops a canonical framework for pure-gravity solutions in dimensions with commuting Killing fields, reducing the Einstein equations to a matrix equation for on flat 3-space and coupling it to a solvable -system. It introduces a rod-structure along the axis that encodes horizons and compact directions, providing a geometric classification tool for higher-dimensional spacetimes. The methodology is validated through detailed treatments of the four- and five-dimensional asymptotics and explicit canonical forms for the Kerr, Myers–Perry, and black ring solutions, including regularity and horizon data derived from rod directions. The framework offers a path to constructing new stationary axisymmetric solutions in higher dimensions and deepens understanding of the phase structure and uniqueness properties of higher-dimensional black holes.

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.

Paper Structure

This paper contains 27 sections, 10 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Stationary and axisymmetric solutions
  3. Deriving canonical form of metric and the Einstein equations
  4. Compact notation for the equations for $G_{ij}(r,z)$
  5. Behavior of solutions near $r=0$
  6. Behavior of $G(r,z)$ near $r=0$
  7. The rod-structure of a solution
  8. Asymptotically flat space-times
  9. Perturbation of diagonal metric
  10. Four-dimensional asymptotic Minkowski-space
  11. Five-dimensional asymptotic Minkowski-space
  12. Rotating black hole solutions
  13. Kerr solution
  14. Five-dimensional Myers-Perry solution
  15. Myers-Perry solution with one angular momentum
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $V_{(i)}$, $i=1,...,D-2$, be $D-2$ commuting Killing vector fields such that: Then the two-planes orthogonal to the Killing vector fields $V_{(i)}$, $i=1,...,D-2$, are integrable. $\square$

Theorems & Definitions (10)

  • Theorem 2.1
  • Theorem 3.1
  • Lemma E.1
  • Corollary E.2
  • Lemma E.3
  • Corollary E.4
  • Lemma E.5
  • Corollary E.6
  • Lemma E.7
  • Theorem E.8