Stationary axisymmetric solutions of five dimensional gravity

TL;DR

This work identifies a hidden $\mathrm{SL}(3,\mathbb{R})$ symmetry in five-dimensional vacuum gravity and isolates an $\mathrm{SO}(2,1)$ subgroup that preserves asymptotic flatness. By acting with this one-parameter group on static, axisymmetric seeds, the authors generate rotating, stationary solutions, analyze their mass and angular momenta, and show that the rod structure (sources of the Weyl potentials) is preserved while the rod orientations rotate. They explicitly derive the five-dimensional Myers–Perry black hole from Schwarzschild using a sequence of two $\mathrm{SO}(2,1)$ transformations (plus a flip and coordinate changes), illustrating the method's power. They conjecture a generalization: starting from a static seed with $N$ finite rods and applying $2N$ transformations in a flip–$\mathrm{SO}(2,1)$ sequence can generate the most general stationary axisymmetric solution with the same rod structure. The results provide a systematic, group-theoretic solution-generating framework for exploring the 5D solution space and its connection to 4D/5D boundary transitions via a $D$-transformation.

Abstract

We consider stationary axisymmetric solutions of general relativity that asymptote to five dimensional Minkowski space. It is known that this system has a hidden SL(3,R) symmetry. We identify an SO(2,1) subgroup of this symmetry group that preserves the asymptotic boundary conditions. We show that the action of this subgroup on a static solution generates a one-parameter family of stationary solutions carrying angular momentum. We conjecture that by repeated applications of this procedure one can generate all stationary axisymmetric solutions starting from static ones. As an example, we derive the Myers-Perry black hole starting from the Schwarzschild solution in five dimensions.