Solving Linear Tensor Equations II: Including Parity Odd Terms in Four Dimensions

Abstract

In this paper, focusing on 4-dimensional space, we extend our previous results of solving linear tensor equations. In particular, we consider a 30-parameter linear tensor equation for the unknown tensor component Nαμν in terms of the known component (source) Bαμν. The extension also included the parity even linear terms in Nαμν (and the associated traces), which are formed by contracting the latter with the 4-dimensional Levi-Civita pseudotensor. Assuming generic non-degeneracy conditions and following a step-by-step procedure, we show how to explicitly solve for the unknown tensor field component Nαμν and, consequently, derive its unique and exact solution in terms of the component Bαμν.