Generalized Weyl solutions

By Roberto Emparan, Harvey Reall. 2001

281 citations recorded by the source corpus.

It was shown by Weyl that the general static axisymmetric solution of the vacuum Einstein equations in four dimensions is given in terms of a single axisymmetric solution of the Laplace equation in three-dimensional flat space. Weyl’s construction is generalized here to arbitrary dimension D>4. The general solution of the D-dimensional vacuum Einstein equations that admits D22 orthogonal commuting non-null Killing vector fields is given either in terms of D23 independent axisymmetric solutions of Laplace’s equation in threedimensional flat space or by D24 independent solutions of Laplace’s equation in two-dimensional flat space. Explicit examples of new solutions are given. These include a five-dimensional asymptotically flat ‘‘black ring’’ with an event horizon of topology S 1 3S 2 held in equilibrium by a conical singularity in the form of a disk.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/79a1b639324f7df1f949c9779d18217818902027

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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].”
Uniqueness Theorem for 5-Dimensional Black Holes with Two Axial Killing Fields (2007, background): “Even for a fixed set of of asymptotic charges m,J1,J2 the invariant lengths of the rods l1 = z1−z2, l2 = z2−z3 may be different for the different Black Ring solutions, cor responding to the fact that there exist non-isometric Black Ring solution s with equal asymptotic charges [6, 7].”
Stationary and axisymmetric solutions of higher-dimensional general relativity (2004, methodology): “For the special case when all the D − 2 Killing vector fields are orthogonal to each other, the canonical form of the metric that we find in this paper is equivalent to the form of the so-called generalized Weyl solutions of Emparan and Reall [16].”
Black Holes in Higher Dimensions (2008, methodology): “Then it is possible to prove that, under natural suitable conditions, the two-dimensional spaces orthogonal to all three Killing vectors are integrable [82].”
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]).”
Black rings in (Anti)-de Sitter space (2008, background): “Of all the solutions we have constructed approximately, the one that would seem more likely to admit a closed exact analytic solution is the static black ring in deSitter in five dimensions, generalizing the fairly simple metric for a vacuum static black ring [34].”
Supersymmetric black rings and three-charge supertubes (2004, background): “ters (m5) for branes carrying dipole charges. 3The action of D= 11 supergravity is given in equation (B.8). 6 x= −1 x= +1 x ψ y y= −1 x= const Figure 1: Coordinate system for black ring metrics (from [24, 5]). The diagram sketches a section at constant tand φ. Surfaces of constant yare ring-shaped, ”
Static axisymmetric vacuum solutions and non-uniform black strings (2002, methodology): “e radial direction, with no naked singularities, such as for the 5 dimensional black string, there is no known general analytic solution. Whilst the Weyl solution does generalise to higher dimensions [2,3,30], it does not describe axisymmetric geometries. Finding this general axisymmetric solution i”
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]).”
Bicycling black rings (2007, methodology): “We also refer the reader to [26, 27, 28] for literature on the inverse scattering technique and to [29] and [30] for Weyl solution techniques and rod diagram representations.”