Characterizations of -flat curvature tensor on spacetimes and f(r,T) -gravity

Abstract

Abstract The prime goal of this article is to characterize spacetimes endowed with  -flat curvature tensor. It is demonstrated that a 4-dimensional  -flat perfect fluid spacetime is either a de-Sitter spacetime or locally isometric to Minkowski spacetime under certain energy condition. Moreover, it is established that a Ricci simple  -flat spacetime becomes a Robertson-Walker spacetime. In addition, we shown that a Ricci simple spacetime with harmonic  -curvature tensor is a generalized Robertson-Walker spacetime. Also, we explain several conclusions based on  -flat generalized Robertson-Walker spacetimes. Finally, we investigate  -flat spacetimes that satisfy the f r ,  -gravity and f r ,  m -gravity.