Super-energy tensor for space–times with vanishing scalar curvature

Abstract

A four-index tensor is constructed with terms both quadratic in the Riemann tensor and linear in its second derivatives, which has zero divergence for space–times with vanishing scalar curvature. This tensor reduces in vacuum to the Bel–Robinson tensor. Furthermore, the completely timelike component referred to any observer is positive, and zero if and only if the space–time is flat (excluding some unphysical space–times). We also show that this tensor is the unique one that can be constructed with these properties. Such a tensor does not exist for general gravitational fields. Finally, we study this tensor in several examples: the Friedmann–Lemaı̂tre–Robertson–Walker space–times filled with radiation, the plane–fronted gravitational waves, and the Vaidya radiating metric.