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Solution for the Einstein-Maxwell equations invariant under an $(n - 1)$-dimensional group of dilations

Benedito Leandro, Ilton Menezes, Rafael Novais

TL;DR

This work investigates static Einstein--Maxwell–$\\Lambda$ spacetimes that are invariant under an $(n-1)$-dimensional group of dilations. By formulating the problem as an electrostatic system on a conformally related spatial metric, the authors reduce the field equations to a set of ordinary differential equations using a single invariant function $\xi$. They show a dichotomy: with $\Lambda\neq 0$ the solutions must be rotationally or translationally symmetric, while with $\Lambda=0$ solutions lie in the Majumdar--Papapetrou class, admitting a dilation symmetry and yielding new explicit dilation-invariant MP solutions. A central result provides a complete description of the admissible $\xi$-dependent ansatz, including a dilation-invariant MP family with a closed-form lapse involving an arctangent, and recovers the classical MP multi-centered solution as a corollary. The findings broaden the known symmetry structures of electrovacuum spacetimes and supply explicit, non-asymptotically-flat MP solutions with novel invariance properties.

Abstract

We consider an electrostatic system whose spatial factor is conformal to an $n$-dimensional Euclidean space. We provide a complete characterization of the most general ansatz, thereby reducing the associated electrostatic system of partial differential equations to an ordinary differential equation system. We prove that there are only two possibilities: either the cosmological constant is nonzero, in which case the solutions are necessarily invariant under rotations or translations, or the cosmological constant vanishes, and the solutions belong to the Majumdar-Papapetrou class with a degree of freedom associated with an invariant $(n-1)$-dimensional subgroup. As a result, we introduce a new solution to the electrovacuum system in the Majumdar-Papapetrou class that is invariant under an $(n-1)$-dimensional group of dilations.

Solution for the Einstein-Maxwell equations invariant under an $(n - 1)$-dimensional group of dilations

TL;DR

This work investigates static Einstein--Maxwell– spacetimes that are invariant under an -dimensional group of dilations. By formulating the problem as an electrostatic system on a conformally related spatial metric, the authors reduce the field equations to a set of ordinary differential equations using a single invariant function . They show a dichotomy: with the solutions must be rotationally or translationally symmetric, while with solutions lie in the Majumdar--Papapetrou class, admitting a dilation symmetry and yielding new explicit dilation-invariant MP solutions. A central result provides a complete description of the admissible -dependent ansatz, including a dilation-invariant MP family with a closed-form lapse involving an arctangent, and recovers the classical MP multi-centered solution as a corollary. The findings broaden the known symmetry structures of electrovacuum spacetimes and supply explicit, non-asymptotically-flat MP solutions with novel invariance properties.

Abstract

We consider an electrostatic system whose spatial factor is conformal to an -dimensional Euclidean space. We provide a complete characterization of the most general ansatz, thereby reducing the associated electrostatic system of partial differential equations to an ordinary differential equation system. We prove that there are only two possibilities: either the cosmological constant is nonzero, in which case the solutions are necessarily invariant under rotations or translations, or the cosmological constant vanishes, and the solutions belong to the Majumdar-Papapetrou class with a degree of freedom associated with an invariant -dimensional subgroup. As a result, we introduce a new solution to the electrovacuum system in the Majumdar-Papapetrou class that is invariant under an -dimensional group of dilations.
Paper Structure (3 sections, 6 theorems, 141 equations)

This paper contains 3 sections, 6 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Proof of the Main Result

Key Result

Theorem 1

Let $(\mathbb{R}^{n},g)$ be the Euclidean space with Cartesian coordinates $(x_1,\ldots,x_n)$ and Euclidean metric $g$. Then there exists a smooth function $\xi=\xi(x_1,\ldots,x_n)$ such that $(\Omega,\overline g,N(\xi),\psi(\xi))$, where $\overline g=g/\varphi^{2}(\xi)$, is a solution of the electr Here $\tau$, $\gamma_k$, and $\theta_k$ are real constants, and $\Gamma$ is a smooth real function.

Theorems & Definitions (14)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Remark 1: The domain $\Omega$ of the lapse function
  • Remark 2: Uniform equivalence and completeness of Theorem \ref{['thm3']}
  • Lemma 1
  • proof : Proof of Lemma \ref{['cont:lem1']}
  • Theorem 3
  • proof : Proof of Theorem \ref{['theo1']}
  • proof : Proof of Theorem \ref{['ansatz']}
  • ...and 4 more