Godel Type Metrics in Einstein-Aether Theory II: Nonflat Background in Arbitrary Dimensions

TL;DR

This paper extends the class of G"odel-type solutions in Einstein-Aether theory to nonflat backgrounds in arbitrary dimensions, providing a method to generate exact solutions from these metrics.

Contribution

It generalizes previous flat-background results to nonflat backgrounds in any dimension, linking solutions to Ricci-flat and Maxwell equations.

Findings

Exact solutions in nonflat backgrounds are constructed.

The reduced field equations relate to Ricci-flat and Maxwell equations.

Explicit solutions are provided for specific background geometries.

Abstract

It was previously proved that the G\"{o}del-type metrics with flat three-dimensional background metric solve exactly the field equations of the Einstein-Aether theory in four dimensions. We generalize this result by showing that the stationary G\"{o}del-type metrics with nonflat background in $D$ dimensions solve exactly the field equations of the Einstein-Aether theory. The reduced field equations are the $(D-1)$-dimensional Euclidean Ricci-flat and the $(D-1)$-dimensional source-free Maxwell equations, and the parameters of the theory are left free except $c_{1}-c_{3}=1$. We give a method to produce exact solutions of the Einstein-Aether theory from the G\"{o}del-type metrics in $D$ dimensions. By using this method, we present explicit exact solutions to the theory by considering the particular cases: ($D-1$)-dimensional Euclidean flat, conformally flat, and Tangherlini backgrounds.