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

I. A. Sarmiento-Alvarado, P. Wiederhold, T. Matos

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 $\hat g = f ( ρ, ζ) ( d ρ^2 + d ζ^2 ) + g_{μν} ( ρ, ζ) d x^μd x^ν$. 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.

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 -dimensional spacetime with commutative Killing vectors, the metric tensor can take the form . Then, the Einstein field equations in vacuum reduce to a chiral equation, , and two differential equations, , where is the normalized matrix representation of , and . We use the ansatz , where the parameters depend on and and satisfy a generalized Laplace equation, . The chiral equation to the Killing equation, , where . Furthermore, we assume that the matrices commute with each other; in this way, they fulfill the Killing equation.
Paper Structure (11 sections, 33 theorems, 110 equations, 2 figures)

This paper contains 11 sections, 33 theorems, 110 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Chiral equation
  3. Kaluza-Klein metrics
  4. The centralizer of a set of matrices
  5. Commutative algebra
  6. Matrix $g_0$
  7. Exponential matrix
  8. Examples
  9. Conclusion
  10. Equivalence classes
  11. Commutative algebras

Key Result

Lemma 1

Let $\{ A_1, \ldots, A_r \}$ be a subset of pairwise commuting matrices in $\mathbf{M}_n$ and let $g \in \mathbf{Sym}_{n}$ be a matrix function of the variables $\xi^1, \ldots, \xi^r$. If $g_{, \xi^a} = A_a g$ for all $a \in \{ 1, \ldots, r \}$, then $g ( \xi^a ) = e ^{\xi^a A_a} g_0$, where $g_0$ i

Figures (2)

  • Figure 1: Behavior of $g_{tt} (r)$ with $p = 2$ and $q_1 = 0$.
  • Figure 2: Evolution of $g_{tt} (r)$ considering three values of $q_1/M$: 0, 0.5 and 1.5.

Theorems & Definitions (65)

  • Lemma 1
  • proof
  • Definition 1
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • proof
  • Definition 2
  • Definition 3
  • ...and 55 more