Abstract
In this paper, we study an
n
-dimensional space-time (
M
,
g
) with Einstein field equation and nondegenerate Ricci tensor. It is shown that taking the Ricci tensor
as another semi-Riemannian metric on
M
, the identity map
is a harmonic map. We also obtain a characterization of a vacuum using a differential equation satisfied by the electromagnetic stress tensor on space-time. In addition, we also show that if the Ricci tensor of (
M
,
g
) is parallel and the signatures of
g
and
are same, then the semi-Riemannian manifold
is an Einstein manifold.