On a Special Class of Type-I Gravitational Fields

Source abstract

This paper contains an investigation of the algebraic structure and the analytic properties of a class of normal hyperbolic Riemannian 4-spaces restricted by the following condition: There exists a timelike unit vector ua such that the Riemann tensor satisfies *Rabcdubud = 0. This condition is shown to be equivalent to the statement that the conform tensor is Petrov type I with real eigenvalues, ua being a principal vector and an eigenvector of the Ricci tensor. This means that there is no flux of nongravitational energy relative to an observer travelling with 4-velocity ua. The eigen null directions (Debever vectors) of the conform tensor lie in a timelike hyperplane spanned by ua and the two eigenvectors of εac ≡ −Cabcdubud belonging to the eigenvalues with largest absolute value. The conform tensor is degenerate (type D) if and only if a Debever direction projected into the rest space of an observer ua is an eigendirection of εab. The complete set of Bianchi identities is examined. It yields an expression for the covariant eigentime derivative of εab and an algebraic relation linking the rotation and shear of ua to the curvature tensor of the Riemannian space. The general results are applied to special Einstein spaces (Rab = 0) admitting a congruence of timelike curves without shear and rotation. We get a new simplified proof of the theorem that in the case of nondegeneracy such spaces are static, the curves of the congruence being paths of an isometric motion.