Solving Linear Tensor Equations II: Including Parity Odd Terms in $4$-dimensions

TL;DR

This paper extends previous work on solving linear tensor equations in 4D by including parity odd terms, providing explicit solutions for unknown tensor components based on known sources.

Contribution

It introduces a method to explicitly solve 4D linear tensor equations with parity odd terms, expanding the scope of previous solutions.

Findings

Explicit solutions for tensor components in 4D equations

Inclusion of parity odd terms in tensor equations

Unique and exact solutions derived

Abstract

In this letter, focusing in $4$-dimensions, we extend our previous results of solving linear tensor equations. In particular, we consider a $30$ parameter linear tensor equation for the unknown tensor components $N_{\alpha\mu\nu}$ in terms of the known (sources) components $B_{\alpha\mu\nu}$. Our extension now consists in including also the parity even linear terms in $N_{\alpha\mu\nu}$ (and the associated traces) formed by contracting the latter with the $4$-dimensional Levi-Civita pseudo-tensor. Assuming a generic non-degeneracy condition and following a step by step procedure we show how one can solve explicitly for the unknown tensor field components $N_{\alpha\mu\nu}$ and consequently derive its unique and exact solution in terms of the components $B_{\alpha\mu\nu}$.