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

I. A. Sarmiento-Alvarado, Petra Wiederhold, Tonatiuh Matos

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 $\ (αg_{, \bar{z}} g^{-1})_{, z} + \ (αg_{, z} g^{-1})_{, \bar{z}} = 0$ with $ g\in SL(n,\mathbb{R})$. The other block reduces to the differential equation $(\ln f α^{1-1/n})_{, z} = 1/2 αtr( g_{, z} g^{-1})^2$ and its complex conjugate. We use the ansatz $g = g(ξ) $, where $ξ$ 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.

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

Abstract

In this work we find solutions of the ()-dimensional Einstein Field Equations (EFE) with commuting Killing vectors in vacuum. In the presence of Killing vectors, the EFE can be separated into blocks of equations. The main part can be summarized in the chiral equation with . The other block reduces to the differential equation and its complex conjugate. We use the ansatz , where 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.
Paper Structure (9 sections, 27 theorems, 101 equations)

This paper contains 9 sections, 27 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Field equations
  3. One-dimensional subspaces
  4. The subspace $\mathcal{I} ( A )$
  5. Jordan matrices
  6. Computing one-dimensional subspaces
  7. Equivalence classes for the matrix $A$
  8. Example: one-dimensional $SL(5,\mathbb{R})$-subspaces
  9. Conclusions

Key Result

Theorem 1

For any non-zero matrix $A \in \mathbf{M}_{n}$, $\mathcal{I}( A )$ is a subspace of the vector space $\mathbf{M}_{n}$.

Theorems & Definitions (39)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Lemma 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 2: from horn_johnson_1985
  • ...and 29 more