Spacetimes with a vanishing second Ricci invariant

Abstract

Spacetimes with a vanishing second Ricci invariant, but which are not necessarily Ricci - flat, though common in general relativity, are seldom studied in a coordinate and symmetry independent way by actually using their Ricci invariants. Yet, and as an example, it can be shown by use of Ricci invariants that no such spacetimes can actually represent a perfect fluid by way of Einstein's equations. The widely studied Kiselev black hole is a particularly simple example of such a spacetime. Yet it is frequently, and erroneously, referred to as a perfect fluid. This paper gathers together information relevant to the study of spacetimes with vanishing a second Ricci invariant. It is shown that such spacetimes need not be stationary, a point relevant to their possible physical significance.