A family of warped product semi-Riemannian Einstein metrics

TL;DR

This paper classifies and constructs explicit warped product semi-Riemannian Einstein metrics, especially Ricci flat solutions, with bases conformal to pseudo-Euclidean spaces and invariant under translation groups, relevant for vacuum Einstein equations.

Contribution

It provides a complete classification of Ricci flat warped product Einstein metrics with specific symmetry and conformal properties, including explicit solutions for vacuum Einstein equations.

Findings

Explicit Ricci flat solutions obtained

Classification of Einstein metrics with translation invariance

Solutions applicable to vacuum Einstein field equations

Abstract

We study warped products semi-Riemannian Einstein manifolds. We consider the case in that the base is conformal to an n-dimensional pseudo Euclidean space and invariant under the action of an translation group. We provide all such solutions in the case Ricci flat when the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an translation group and the fiber F is Ricci flat. In particular, we obtain explicit solutions, in the case vacuum, for the Einstein field equation.