Notes on a Cotton-type tensor for electrostatic systems

Abstract

We introduce a natural (0,3)-tensor associated with electrostatic systems in arbitrary dimensions. This tensor arises from the comparison between the Cotton and Weyl decompositions of the Riemann curvature tensor and extends several identities previously known in dimensions three and four. We prove that it is totally trace-free and satisfies the same algebraic symmetries as the Cotton tensor. We further investigate its behavior under the natural assumption that the electric field is collinear with the gradient of the lapse function, obtaining a simplified expression. Along the way, we extend a boundary collinearity result to arbitrary dimensions and derive several identities that may be useful in the study of electrostatic systems.