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

TL;DR

This paper derives exact solutions to Einstein's equations with multiple symmetries by reducing complex differential equations to algebraic problems using a specific ansatz, facilitating the construction of physically relevant spacetime models.

Contribution

It introduces a novel algebraic method to solve the chiral equations in Einstein's field equations with multiple Killing vectors, simplifying the search for exact solutions.

Findings

Derived explicit solutions for Einstein's equations with n commuting Killing vectors.

Reduced complex differential equations to algebraic problems using a specific ansatz.

Enabled easy customization of solutions with desired physical properties.

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.