Generalized Weyl Solutions

TL;DR

Emparan and Reall generalize Weyl's class of static axisymmetric vacuum solutions to arbitrary dimensions, showing that the general D-dimensional spacetime with D-2 orthogonal commuting Killing vectors reduces to axisymmetric harmonic functions (either D-3 in 3D space or D-4 in 2D space). They classify solutions by rod-source configurations along the symmetry axis and construct several new higher-dimensional spacetimes, most notably a five-dimensional asymptotically flat black ring with horizon topology $S^1\times S^2$ balanced by a conical deficit. The work also explores interpretations via KK reductions and Wick rotations, yielding black holes and bubbles in KK vacua, black strings in KK throats, and multi-horizon configurations, thereby revealing rich horizon topologies beyond spherical. Together, these results provide a systematic framework for generating and analyzing higher-dimensional vacuum solutions with diverse topologies and asymptotics, expanding the landscape of exact solutions in General Relativity.

Abstract

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 $D\ge 4$. The general solution of the D-dimensional vacuum Einstein equations that admits D-2 orthogonal commuting non-null Killing vector fields is given either in terms of D-3 independent axisymmetric solutions of Laplace's equation in three-dimensional flat space or by D-4 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 x S^2 held in equilibrium by a conical singularity in the form of a disc.

Figures (10)

• Figure 1: Sources for the harmonic functions of one of the Weyl forms of flat space. The thin lines denote the $z$-axis and the thick lines denote thin rods along this axis. The left and right ends of the figure are to be interpreted as at $z=-\infty$ and $z=+\infty$ respectively. The sources for $U_1$ and $U_2$ are semi-infinite rods of mass $1/2$ per unit length. The $U_1$ source lies along $z \ge a$ and the $U_2$ source along $z \le a$ for some $a$. In the classification of Section \ref{['sec:classes']}, this is a class 0 solution.
• Figure 2: Sources for (a) the four dimensional and (b) five dimensional Schwarzschild solutions. The black hole interpretation requires that $x^1$ is the timelike coordinate. If in (a) $x^2$ is the timelike coordinate then this describes an expanding bubble in the $M^{1,2}\times S^1$ vacuum. If both $x^1$ and $x^2$ are spacelike then this describes a static KK $S^2$ bubble (when a trivial time direction is added). If in (b) $x^2$ (or $x^3$) corresponds to time, then it describes an expanding bubble in the $M^{1,3}\times S^1$ vacuum. If $x^1$, $x^2$ and $x^3$ are all spatial coordinates, then it describes an $S^3$ bubble. In the classification of Section \ref{['sec:classes']}, these solutions are class I.
• Figure 3: Sources for (a) C-metric; (b) black ring; (c) black hole plus KK bubble; (d) black string and KK bubble. Note that the sources for the $U_i$'s have to add up to an infinite rod. In the classification of Section \ref{['sec:classes']}, these solutions are class II.
• Figure 4: Spatial sections of the black ring metric. The coordinate $\phi$ is suppressed. The surfaces of constant $y$ are nested surfaces of topology $S^2 \times S^1$. The coordinate $\psi$ is the coordinate on $S^1$. The coordinates $x$ and $\phi$ are, respectively, the polar and azimuthal angles on $S^2$. The smallest constant $y$ surface corresponds to the horizon, at $y=-\infty$. The surface at $y=-1$ degenerates into an axis of rotation where the orbits of $\psi$ shrink to zero. The surfaces of constant $x$ are denoted by dotted lines. $x=-1$ points out of the ring and $x=+1$ points into the ring. The conical singularity may be chosen to lie inside the ring or, as in the case shown, outside the ring (so that it extends to infinity). Infinity is at $x=y=-1$.
• Figure 5: 1. Schematic depiction of the $R\theta$ plane of the KK bubble. The throat of the bubble (where the KK circle shrinks to zero size) is at $R=2M$. In the $(x,y)$ coordinates, this corresponds to $x=1$ or $y=-\infty$. The axes $\theta = 0,\pi$ correspond to $x=-1$ and $y=-1$ respectively. Solid and dashed lines denote curves of constant $x$ and $y$ respectively. 2. The $xy$ plane of the metric \ref{['eqn:holebubble']}. There is a horizon at $y=-\infty$ and the KK circle shrinks to zero size at $x=1$.