Flat subspaces of the $SL(n,\mathbb{R})$ chiral equations
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arXiv
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2026
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| author | Sarmiento-Alvarado, I. A. Wiederhold, P. Matos, T. |
| author_facet | Sarmiento-Alvarado, I. A. Wiederhold, P. Matos, T. |
| contents | 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 $\hat g = f ( ρ, ζ) ( d ρ^2 + d ζ^2 ) + g_{μν} ( ρ, ζ) d x^μd x^ν$. Then, the Einstein field equations in vacuum reduce to a chiral equation, $( ρg_{, z} g ^{-1} )_{, \bar z} + ( ρg_{, \bar z} g ^{-1} )_{, z} = 0$, and two differential equations, $( \ln f ρ^{1-1/n} )_{, Z} = \fracρ{2} \operatorname{tr} ( g_{, _Z} g^{-1} )^2$, where $g \in SL( n, \mathbb{R} )$ is the normalized matrix representation of $g_{μν}$, $z = ρ+ i ζ$ and $Z = z, \bar z$. We use the ansatz $g = g ( ξ^a )$, where the parameters $ξ^a$ depend on $z$ and $\bar z$ and satisfy a generalized Laplace equation, $( ρξ^a _{, z} )_{, \bar z} + ( ρξ^a _{, \bar z} )_{, z} = 0$. The chiral equation to the Killing equation, $A_{a , ξ^b} + A_{b , ξ^a} = 0$, where $A_a = g_{, ξ^a} g^{-1}$. Furthermore, we assume that the matrices $A_a$ commute with each other; in this way, they fulfill the Killing equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_07385 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Flat subspaces of the $SL(n,\mathbb{R})$ chiral equations Sarmiento-Alvarado, I. A. Wiederhold, P. Matos, T. General Relativity and Quantum Cosmology 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 $\hat g = f ( ρ, ζ) ( d ρ^2 + d ζ^2 ) + g_{μν} ( ρ, ζ) d x^μd x^ν$. Then, the Einstein field equations in vacuum reduce to a chiral equation, $( ρg_{, z} g ^{-1} )_{, \bar z} + ( ρg_{, \bar z} g ^{-1} )_{, z} = 0$, and two differential equations, $( \ln f ρ^{1-1/n} )_{, Z} = \fracρ{2} \operatorname{tr} ( g_{, _Z} g^{-1} )^2$, where $g \in SL( n, \mathbb{R} )$ is the normalized matrix representation of $g_{μν}$, $z = ρ+ i ζ$ and $Z = z, \bar z$. We use the ansatz $g = g ( ξ^a )$, where the parameters $ξ^a$ depend on $z$ and $\bar z$ and satisfy a generalized Laplace equation, $( ρξ^a _{, z} )_{, \bar z} + ( ρξ^a _{, \bar z} )_{, z} = 0$. The chiral equation to the Killing equation, $A_{a , ξ^b} + A_{b , ξ^a} = 0$, where $A_a = g_{, ξ^a} g^{-1}$. Furthermore, we assume that the matrices $A_a$ commute with each other; in this way, they fulfill the Killing equation. |
| title | Flat subspaces of the $SL(n,\mathbb{R})$ chiral equations |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2603.07385 |