Lorentzian homogeneous Ricci-flat metrics on almost abelian Lie groups
Yuichiro Sato, Takanao Tsuyuki
TL;DR
The paper analyzes Lorentzian left-invariant metrics on almost abelian Lie groups of dimension $n \ge 4$ to characterize homogeneous Ricci-flat but non-flat spacetimes. It derives explicit algebraic criteria for Ricci-flat and flat conditions across three metric forms determined by the structure matrix $A$, using decompositions into symmetric and skew-symmetric components. A central achievement is the higher-dimensional generalization of the Petrov solution, providing an explicit metric form $ds^{2} = e^{-2\alpha x_{n}}[ \cos(2\beta x_{n})(-dx_{1}^{2}+dx_{2}^{2}) - 2\sin(2\beta x_{n})dx_{1}dx_{2}] + \sum_{i=3}^{n-1} e^{-2\lambda_{i} x_{n}} dx_{i}^{2} + dx_{n}^{2}$ with constraints $\alpha \neq 0$, $\beta > 0$, $2\alpha + \sum_{i=3}^{n-1} \lambda_{i} = 0$, $2\alpha^{2} - 2\beta^{2} + \sum_{i=3}^{n-1} \lambda_{i}^{2} = 0$, $\prod_{i=3}^{n-1} \lambda_{i} \neq 0$, $\lambda_{3} > \cdots > \lambda_{n-1}$, which recovers the Petrov solution in four dimensions. The work further analyzes isometry actions, decomposability, and the geometry of the resulting spaces, showing that the general solution is not locally symmetric nor a plane wave in general, with special subcases recovering known plane-wave structures.
Abstract
We study Lorentzian left-invariant metrics on almost abelian Lie groups of dimensions larger than three. An almost abelian Lie group is a Lie group whose Lie algebra has a codimension one abelian ideal. Ricci-flat and non-flat conditions on the Lie algebra are derived. In particular, we generalize the four-dimensional Petrov solution to arbitrarily higher dimensions.