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Lorentzian symmetric spaces which are Einstein-Yang-Mills with respect to invariant metric connections

Marco Castrillón López, Pedro M. Gadea, Eugenia Rosado Maria

TL;DR

The paper determines which four-dimensional Lorentzian symmetric spaces admit Einstein-Yang-Mills solutions with invariant connections and a diagonal holonomy metric. By leveraging Cahen–Wallach's classification and Komrakov’s Lorentzian pairs, it tests a broad set of cases, deriving explicit first EYM solutions (or proving nonexistence) and identifying ten spaces where nontrivial solutions exist. It further shows that the second EYM equation is automatically satisfied for these symmetric configurations, yielding a clean list of viable geometries along with their cosmological and gravitational constants. The results provide a complete classification within the chosen invariant-connection framework and connect the solutions to concrete geometric models, highlighting the role of symmetry in Einstein-Yang-Mills theory.

Abstract

We classify four-dimensional connected simply-connected indecomposable Lorentzian symmetric spaces $M$ with connected nontrivial isotropy group furnishing solutions of the Einstein-Yang-Mills equations. Those solutions with respect to some invariant metric connection $Λ$ in the bundle of orthonormal frames of $M$ and some diagonal metric on the holonomy algebra corresponding to $Λ$.

Lorentzian symmetric spaces which are Einstein-Yang-Mills with respect to invariant metric connections

TL;DR

The paper determines which four-dimensional Lorentzian symmetric spaces admit Einstein-Yang-Mills solutions with invariant connections and a diagonal holonomy metric. By leveraging Cahen–Wallach's classification and Komrakov’s Lorentzian pairs, it tests a broad set of cases, deriving explicit first EYM solutions (or proving nonexistence) and identifying ten spaces where nontrivial solutions exist. It further shows that the second EYM equation is automatically satisfied for these symmetric configurations, yielding a clean list of viable geometries along with their cosmological and gravitational constants. The results provide a complete classification within the chosen invariant-connection framework and connect the solutions to concrete geometric models, highlighting the role of symmetry in Einstein-Yang-Mills theory.

Abstract

We classify four-dimensional connected simply-connected indecomposable Lorentzian symmetric spaces with connected nontrivial isotropy group furnishing solutions of the Einstein-Yang-Mills equations. Those solutions with respect to some invariant metric connection in the bundle of orthonormal frames of and some diagonal metric on the holonomy algebra corresponding to .

Paper Structure

This paper contains 21 sections, 4 theorems, 56 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lorentzian symmetric pairs
  4. The first Einstein-Yang-Mills equation
  5. The cases ${\bf 1.1^1}(7)$; and ${\bf 1.1^1}(10)\;(t=0)$.
  6. The cases ${\bf 1.1^2}(9)$, ${\bf 1.1^2}(10)$; and ${\bf 1.1^2}(12)\; (t=0)$.
  7. The case ${\bf 1.1^3}(1)$.
  8. The case ${\bf 1.1^4}(1)$.
  9. The cases ${\bf 1.4^1}(24)$, ${\bf 1.4^1}(25)$ and ${\bf 1.4^1}(26)$.
  10. The cases ${\bf 2.1^2}(1)$, ${\bf 2.1^2}(2)$, ${\bf 2.1^2}(3)$
  11. The cases ${\bf 2.1^2}(4)$, ${\bf 2.1^2}(5)$ and ${\bf 2.1^2}(6)$.
  12. The case ${\bf 2.4^1}(3)$.
  13. The cases ${\bf 2.5^2}(4)$, ${\bf 2.5^2}(5)$, ${\bf 2.5^2}(6)$ and ${\bf 2.5^2}(7)$.
  14. The case ${\bf 3.2^2}(2)$.
  15. The cases ${\bf 3.3^2}(2)$, ${\bf 3.3^2}(3)$ and ${\bf 3.3^2}(4)$.
  16. ...and 6 more sections

Key Result

Proposition 3.1

The four-codimensional Lorentzian reductive pairs $(\mathfrak{k},\mathfrak{h})$ are the seventy-six pairs given in Table aaa. The four-codimensional Lorentzian symmetric pairs $(\mathfrak{k},\mathfrak{h})$ are the thirty-eigth pairs listed in Table bbb.

Theorems & Definitions (8)

  • Remark 2.1
  • Proposition 3.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Theorem 5.1
  • proof
  • Remark 5.2