Stationary and axisymmetric solutions of higher-dimensional general relativity

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.

An inverse scattering formalism for STU supergravity

“…For this computation, group rotation is sufficient, but we expect that in more complicated situations, in particular for configurations involving three or more poles, one needs to develop some other algorithmic techniques to find appropriate vectors. In this regard, ideas from the interval structure [26][27][28] of gravitational solutions can be useful, but at the moment this remains an open challenging problem.…”

How to stop (worrying and love) the bubble: boundary changing solutions

“…This solution has symmetry group R × U(1) and hence qualifies as Weyl-Papapetrou (a stationary axisymmetric vacuum solution) [35,30,36]. Setting…”

“…3 Spherical prolate coordinates are a special case of C-metric coordinates; see [29,30] and references therein. Our spherical prolate diagrams are analogs of C-metric diagrams in [31].…”

Black Holes in Higher-Dimensional Gravity

“…As an example we note the generalized Weyl ansatz [24,25] for static and stationary solutions with D − 2 commuting Killing vectors, in which the Einstein equations simplify considerably. For the static case, this ansatz is for example relevant for bubble-black hole sequences [49] in five and six-dimensional KK space.…”

“…Here, it has been found that in addition to the Myers-Perry (MP) black holes [15], there exist rotating black rings [16,2] and multi-black hole solutions like black-Saturns and multi-black rings [17,18,19,20] including those with two independent angular momenta [21,22,23]. All of these are exact solutions which have been obtained with the aid of special ansätze [24,25] based on symmetries and inverse-scattering techniques [26,27,28,29]. We refer in particular to the review [2] for further details on the black ring in five dimensions and Ref.…”