Static Spherically Symmetric Spacetime as a Function of Scalar Curvature

Abstract

In this paper, we constructed a complete solution of four-dimensional static spherically symmetric spacetime as a function of scalar curvature in spherical coordinates. It is shown that static spherically symmetric spacetimes which satisfies $R_{ij}\,\,=\,\alpha (r)\,g_{ij}$ for all $i,j\,=\,0,1,2,3$ are necessarily Einstein manifolds with non-zero cosmological constant. Also, static spherically symmetric metrics with vanishing covariant derivative of Ricci tensor are deduced. Some important solutions of these metrics with scalar curvatures which are arbitrary functions of r (non-constant, constant and zero) are introduced. Some of these solutions are well known solutions of Einstein field equations.