A family of warped product semi-Riemannian Einstein metrics

We study warped product 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 (n−1)-dimensional 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 (n−1)-dimensional translation group and the fiber F is Ricci-flat. In particular, we obtain explicit solutions, in the case vacuum, for the Einstein field equation.