Flat subspaces of the \(SL(n,\mathbb {R})\) chiral equations

Abstract

In this work, we introduce a method for finding exact solutions to the vacuum Einstein field equations in higher dimensions from a given solution to the chiral equation. When considering a \(n + 2\)-dimensional spacetime with n commutative Killing vectors, the metric tensor can take the form \(\hat{g} = f ( \rho , \zeta ) ( d \rho ^2 + d \zeta ^2 ) + g_{\mu \nu } ( \rho , \zeta ) d x^\mu d x^\nu \). Then, the Einstein field equations in vacuum reduce to a chiral equation, \(( \rho g_{, z} g ^{-1} )_{, \bar{z}} + ( \rho g_{, \bar{z}} g ^{-1} )_{, z} = 0\), and two differential equations, \(( \ln f \rho ^{1-1/n} )_{, Z} = \frac{\rho }{2} \operatorname {tr} ( g_{, _Z} g^{-1} )^2\), where \(g \in SL( n, \mathbb {R} )\) is the normalized matrix representation of \(g_{\mu \nu }\), \(z = \rho + i \zeta \) and \(Z = z, \bar{z}\). We use the ansatz \(g = g ( \xi ^a )\), where the parameters \(\xi ^a\) depend on z and \(\bar{z}\) and satisfy a generalized Laplace equation, \(( \rho \xi ^a _{, z} )_{, \bar{z}} + ( \rho \xi ^a _{, \bar{z}} )_{, z} = 0\). The chiral equation to the Killing equation, \(A_{a, \xi ^b} + A_{b, \xi ^a} = 0\), where \(A_a = g_{, \xi ^a} g^{-1}\). Furthermore, we assume that the matrices \(A_a\) commute with each other; in this way, they fulfill the Killing equation.