On a type of spacetime (Q2815293)

The authors study the spacetime of general relativity regarded as a connected four-dimensional semi-Riemannian manifold \((M^4,g)\), where \(g\) is a Lorentz metric, i.e., the metric of signature \((-+++)\). The main focus is on a special type of spacetimes -- the conformally flat weakly Ricci symmetric spacetimes.NEWLINENEWLINE A nonflat Riemannian manifold \((M^n,g)\), \(n>2\), is said to be ``weakly Ricci symmetric'' if its Ricci tensor \(S\) is of type \((0,2)\) and satisfies the condition NEWLINE\[NEWLINE (\nabla_X S)(Y,Z)=A(X)S(Y,Z)+B(Y)S(X,Z)+D(Z)S(Y,X), NEWLINE\]NEWLINE where \(A,B\) and \(D\) are three nonzero 1-forms. Such an \(n\)-dimensional manifold is denoted by \((\mathrm{WRS})_n\).NEWLINENEWLINE The authors prove that the conformally flat spacetime \((\mathrm{WRS})_4\) with a nonzero scalar curvature is the Robertson-Walker spacetime, meaning that it is homogeneous and isotropic. Furthermore, a vector field \(\rho\) defined by \(E(X)=g(X,\rho)\) is irrotational (satisfying the condition \(g(\nabla_X\rho,Z)=g(\nabla_Z\rho,X)\)) and its integral curves are geodesics.NEWLINENEWLINE It is shown that a perfect fluid spacetime \((\mathrm{WRS})_4\) with a nonzero scalar curvature has vanishing vorticity and vanishing shear. Therefore, if the velocity vector field is always hypersurface orthogonal, then the possible local cosmological structures of the spacetime are of Petrov type \(I, D\) or \(O\).NEWLINENEWLINE At the end, a nontrivial example of a conformally flat \((\mathrm{WRS})_4\) spacetime with nonzero nonconstant scalar curvature is constructed.