Some curvature properties of perfect fluid spacetimes

Abstract

In this paper we assume that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations. With this new and innovative approach, we investigate some curvature properties of perfect fluid spacetimes. Firstly, we  provide a necessary condition for a Lorentzian manifold to be a perfect fluid spacetime. We prove that a conformally flat perfect fluid  spacetime is a spacetime of quasi constant sectional curvature. Also, we acquire that a conharmonically flat perfect fluid spacetime  represents the radiation era. Next, we show that if a stiff matter fluid obeys Yang’s equations, then the vorticity of the fluid vanishes.  Moreover, we find that if the Ricci tensor is Killing, then the perfect fluid spacetime is (i) expansion-free and shear-free, and its flow is  geodesic, however, not necessarily vorticity-free, and (ii) σ and p are constant. Besides, we establish that Ricci semi-symmetric or Ricci  symmetric perfect fluid spacetimes represent either dark matter era, or phantom era. Finally, we acquire that if a perfect fluid is Ricci  symmetric, then either the spacetime represents a dark matter era, or a static spacetime.