The Einstein Tensor and Its Generalizations

Abstract

The Einstein tensor G^ij is symmetric, divergence free, and a concomitant of the metric tensor g_ab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.