One-dimensional subspaces of the $SL(n,\mathbb{R})$ Chiral Equations

Abstract

In this work we find solutions of the ($n+2$)-dimensional Einstein Field Equations (EFE) with $n$ commuting Killing vectors in vacuum. In the presence of $n$ Killing vectors, the EFE can be separated into blocks of equations. The main part can be summarized in the chiral equation $\ (\alpha g_{, \bar{z}} g^{-1})_{, z} + \ (\alpha g_{, z} g^{-1})_{, \bar{z}} = 0$ with $ g\in SL(n,\mathbb{R})$. The other block reduces to the differential equation $(\ln f \alpha ^{1-1/n})_{, z} = 1/2 \alpha tr( g_{, z} g^{-1})^2$ and its complex conjugate. We use the ansatz $g = g(\xi ) $, where $\xi $ satisfies a generalized Laplace equation, so the chiral equation reduces to a matrix equation that can be solved using algebraic methods, turning the problem of obtaining exact solutions for these complicated differential equations into an algebraic problem. The different EFE solutions can be chosen with desired physical properties in a simple way.