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 $\hat{g}=f(\ddot{I},\hat{I}\P )(d\ddot{I}{}^2+d\hat{I}{\P }^2)+g_{\hat{I}¼\hat{I}½}(\ddot{I},\hat{I}\P )d{x}^{\hat{I}}¼d{x}^{\hat{I}}½$. Then, the Einstein field equations in vacuum reduce to a chiral equation, $(\ddot{I}g_{,z}{g}^{-1}{)}_{,\bar{z}}+(\ddot{I}g_{,\bar{z}}{g}^{-1}{)}_{,z}=0$, and two differential equations, $(\ln f\ddot{I}{}^{1-1/n}{)}_{,Z}=\frac{\ddot{I}}{}2\mathrm{tr}(g_{{,}_Z}{g}^{-1}{)}^2$, where $g\in SL(n,\mathbb{R})$ is the normalized matrix representation of $g_{\hat{I}¼\hat{I}½}$, $z=\ddot{I}+i\hat{I}\P$ and $Z=z,\bar{z}$. We use the ansatz $g=g(\hat{I}{¾}^a)$, where the parameters $\hat{I}{¾}^a$ depend on $z$ and $\bar{z}$ and satisfy a generalized Laplace equation, $(\ddot{I}\hat{I}{¾}_{,z}^a{)}_{,\bar{z}}+(\ddot{I}\hat{I}{¾}_{,\bar{z}}^a{)}_{,z}=0$. The chiral equation to the Killing equation, $A_{a,\hat{I}{¾}^b}+A_{b,\hat{I}{¾}^a}=0$, where $A_a=g_{,\hat{I}{¾}^a}{g}^{-1}$. Furthermore, we assume that the matrices $A_a$ commute with each other; in this way, they fulfill the Killing equation.

References in corpus (1)

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

Cited by in corpus (1)

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