Sectional curvature and the energy–momentum tensor

Abstract

Many years ago Ehlers and Kundt showed that a spacetime M is an Einstein space if and only if the sectional curvatures of any pair of orthogonal non-null 2-spaces at any point of M are equal. This paper generalizes this result by first showing a very straightforward relation between the sectional curvatures of such orthogonal pairs of 2-spaces and the trace-free part of the Ricci tensor and then by establishing for each algebraic (Segre) type of the energy–momentum tensor precisely which orthogonal pairs of non-null 2-spaces have the same sectional curvature. The results are described in a manifold theoretic sense and are tabulated for each Segre type.