Stationary and axisymmetric solutions of higher-dimensional general relativity

By T. Harmark. 2004

212 citations recorded by the source corpus.

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 $D\ensuremath{-}2$ 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 $D\ensuremath{-}2$ by $D\ensuremath{-}2$ matrix field living in three-dimensional flat space (apart from a subclass of solutions that instead reduce to a set of equations on a $D\ensuremath{-}2$ by $D\ensuremath{-}2$ 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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/4bf09dc752ca4ac34198eac089cbd855234b3e4a

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Citations in the corpus

Black Saturn (2007, methodology): “For the analysis of axisymmetric solutions we make use of the results of [29, 30].”
The Phase Structure of Higher-Dimensional Black Rings and Black Holes (2007, background): “In [19] it is proven that five-dimensional stationary and axisymmetric solutions are unique given the conserved asymptotic charges and the rod structure as defined in [20, 21] (generalizing [22]).”
Thermodynamics of Asymptotically Locally AdS Spacetimes (2005, background): “More recently, this result was generalized for pure gravity in D dimensions and D − 2 orthogonal (non-orthogonal) commuting isometries in [42] ([43]).”
Uniqueness Theorem for 5-Dimensional Black Holes with Two Axial Killing Fields (2007, background): “In fact, what we will show is that the solution is uniquely det ermined in terms of its mass, the two angular momenta, and a datum called “rod struct ure” that has been introduced in a somewhat different form from a more local perspective by Harmark [14, 13], see [7] for a special case 2.”
Black Holes in Higher Dimensions (2008, background): “, n + 1 (with a0 = −∞, an+1 = +∞) [127].”