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 $N_{\alpha \mu \nu }$ in terms of the known component (source) $B_{\alpha \mu \nu }$. The extension also included the parity even linear terms in $N_{\alpha \mu \nu }$ (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 $N_{\alpha \mu \nu }$ and, consequently, derive its unique and exact solution in terms of the component $B_{\alpha \mu \nu }$.

1. Introduction

Extending the results we obtained in [1], we consider the most general linear tensor equation in four dimensions, in which all of the possible parity odd (i.e., contractions with the Levi-Civita pseudotensor) combinations of a third-rank tensor1 are included on top of the even combinations. As we show in the proceeding discussion, this general linear tensor equation involved 30 parameters and consisted of 15 distinct combinations of the unknown third-rank tensor field $N_{\alpha \mu \nu }$. The task is to prescribe a method for solving the aforementioned tensor field in terms of a given (known) component2. These linear tensor equations appear in quadratic metric-affine gravity theories [6,7] and their solutions produce a distortion tensor3 in terms of a hypermomentum tensor. Having solved the distortion, spacetime torsion and non-metricity can then be readily computed.

By solving the distortion in terms of the connection, we can then see exactly how the microscopic properties of matter (encoded in the hypermomentum tensor) produce the non-Riemannian degrees of freedom of torsion and non-metricity. Therefore, it is of great importance to have a prescribed method for solving equations of this kind, since direct physical implications can be drawn from the solutions4.

Here, we only present a simple application of the results, which could possibly be applied to other branches of physics as well. We start with some basic definitions and, subsequently, state and prove our theorem.

2. Definitions

We now present some basic definitions that are used throughout this paper. We consider a 4-dimensional differentiable manifold endowed with a metric g and an affine connection ∇, namely $(g,\nabla ,\mathcal{M})$. The signature of the metric is denoted by s. In addition, $N_{\alpha \mu \nu }$ denotes the component of an arbitrary third-rank tensor field and ${\epsilon }^{\alpha \beta \kappa \lambda }$ is the component of the 4-dimensional Levi-Civita pseudotensor.

Definition 1.

We define the first, second and third contractions of $N_{\alpha \mu \nu }$, respectively, according to:

$$N_{\mu }^{\left(1\right)}:=N_{\alpha \beta \mu }{g}^{\alpha \beta },N_{\mu }^{\left(2\right)}:=N_{\alpha \mu \beta }{g}^{\alpha \beta },N_{\mu }^{\left(3\right)}:=N_{\mu \alpha \beta }{g}^{\alpha \beta }$$

(1)

Definition 2.

By contracting $N_{\alpha \mu \nu }$ with the Levi-Civita pseudotensor, three parity odd combinations are formed, as follows:

$$M_{\lambda \alpha \beta }^{\left(1\right)}:=N_{\mu \nu \lambda }{\epsilon }_{\alpha \beta }^{\mu \nu },M_{\nu \alpha \beta }^{\left(2\right)}:=N_{\mu \nu \lambda }{\epsilon }_{\alpha \beta }^{\mu \lambda },M_{\mu \alpha \beta }^{\left(3\right)}:=N_{\mu \nu \lambda }{\epsilon }_{\alpha \beta }^{\nu \lambda }$$

(2)

Note that by this construction, last pair of indices of each $M^{\left(i\right)}$ value was antisymmetric. In addition, a fourth pseudotrace could be constructed as:

$$M^{\alpha }=N^{\left(4\right)\alpha }:={\epsilon }^{\alpha \mu \nu \lambda }{N}_{\mu \nu \lambda }$$

(3)

Note also that further contractions of the $M^{\left(i\right)}$ values did not produce any new traces since $g^{\mu \nu }{M}_{\mu \nu \alpha }^{\left(1\right)}=-g^{\mu \nu }{M}_{\mu \nu \alpha }^{\left(2\right)}=g^{\mu \nu }{M}_{\mu \nu \alpha }^{\left(3\right)}=-M_{\alpha }$ and the rest are either identically vanishing or proportional to the latter. Using the above definitions, we now state and prove the main results of this paper.

3. The Theorem

Theorem 1.

In a 4-dimensional space of signature s, we consider the 30-parameter tensor equation:

$$\begin{array}{c}{a}_1{N}_{\alpha \mu \nu }+a_2{N}_{\nu \alpha \mu }+a_3{N}_{\mu \nu \alpha }+a_4{N}_{\alpha \nu \mu }+a_5{N}_{\nu \mu \alpha }+a_6{N}_{\mu \alpha \nu }+\sum _{i=1}^3\left(a_{7i}{N}_{\mu }^{\left(i\right)}{g}_{\alpha \nu }+a_{8i}{N}_{\nu }^{\left(i\right)}{g}_{\alpha \mu }+a_{9i}{N}_{\alpha }^{\left(i\right)}{g}_{\mu \nu }\right)\\ +b_{11}{M}_{\alpha \mu \nu }^{\left(1\right)}+b_{12}{M}_{\nu \alpha \mu }^{\left(1\right)}+b_{13}{M}_{\mu \nu \alpha }^{\left(1\right)}+b_{21}{M}_{\alpha \mu \nu }^{\left(2\right)}+b_{22}{M}_{\nu \alpha \mu }^{\left(2\right)}+b_{23}{M}_{\mu \nu \alpha }^{\left(2\right)}+b_{31}{M}_{\alpha \mu \nu }^{\left(3\right)}+b_{32}{M}_{\nu \alpha \mu }^{\left(3\right)}+b_{33}{M}_{\mu \nu \alpha }^{\left(3\right)}\\ +{\epsilon }_{\rho \alpha \mu \nu }\left(b_1{N}^{\left(1\right)\rho }+b_2{N}^{\left(2\right)\rho }+b_3{N}^{\left(3\right)\rho }\right)+c_1{M}_{\mu }{g}_{\alpha \nu }+c_2{M}_{\nu }{g}_{\alpha \mu }+c_3{M}_{\alpha }{g}_{\mu \nu }=B_{\alpha \mu \nu }\end{array}$$

(4)

where $a_i$, $a_{ji}$ $i=1,2,\dots ,6$, $j=7,8,9$ are scalars, $b_{kl}$, $c_m$ are pseudoscalars, $B_{\alpha \mu \nu }$ is a given (known) tensor and $N_{\alpha \mu \nu }$ is the component of the unknown tensor5 N. Define the matrices:

$$A:=\left(\begin{array}{ccccccccccccccc}{\alpha }_{11}& {\alpha }_{12}& {\alpha }_{13}& {\alpha }_{14}& {\alpha }_{15}& {\alpha }_{16}& {\alpha }_{17}& {\alpha }_{18}& {\alpha }_{19}& {\alpha }_{1,10}& {\alpha }_{1,11}& {\alpha }_{1,12}& {\alpha }_{1,13}& {\alpha }_{1,14}& {\alpha }_{1,15}\\ {\alpha }_{21}& {\alpha }_{22}& {\alpha }_{23}& {\alpha }_{24}& {\alpha }_{25}& {\alpha }_{26}& {\alpha }_{27}& {\alpha }_{28}& {\alpha }_{29}& {\alpha }_{2,10}& {\alpha }_{2,11}& {\alpha }_{1,12}& {\alpha }_{2,13}& {\alpha }_{2,14}& {\alpha }_{2,15}\\ {\alpha }_{31}& {\alpha }_{32}& {\alpha }_{33}& {\alpha }_{34}& {\alpha }_{35}& {\alpha }_{36}& {\alpha }_{37}& {\alpha }_{38}& {\alpha }_{39}& {\alpha }_{3,10}& {\alpha }_{3,11}& {\alpha }_{1,12}& {\alpha }_{1,13}& {\alpha }_{1,14}& {\alpha }_{1,15}\\ {\alpha }_{41}& {\alpha }_{42}& {\alpha }_{43}& {\alpha }_{44}& {\alpha }_{45}& {\alpha }_{46}& {\alpha }_{47}& {\alpha }_{48}& {\alpha }_{49}& {\alpha }_{4,10}& {\alpha }_{4,11}& {\alpha }_{4,12}& {\alpha }_{4,13}& {\alpha }_{4,14}& {\alpha }_{4,15}\\ {\alpha }_{51}& {\alpha }_{52}& {\alpha }_{53}& {\alpha }_{54}& {\alpha }_{55}& {\alpha }_{56}& {\alpha }_{57}& {\alpha }_{58}& {\alpha }_{59}& {\alpha }_{5,10}& {\alpha }_{5,11}& {\alpha }_{5,12}& {\alpha }_{5,13}& {\alpha }_{5,14}& {\alpha }_{5,15}\\ {\alpha }_{61}& {\alpha }_{62}& {\alpha }_{63}& {\alpha }_{64}& {\alpha }_{65}& {\alpha }_{66}& {\alpha }_{67}& {\alpha }_{68}& {\alpha }_{69}& {\alpha }_{6,10}& {\alpha }_{6,11}& {\alpha }_{6,12}& {\alpha }_{6,13}& {\alpha }_{6,14}& {\alpha }_{6,15}\\ {\alpha }_{71}& {\alpha }_{72}& {\alpha }_{73}& {\alpha }_{74}& {\alpha }_{75}& {\alpha }_{76}& {\alpha }_{77}& {\alpha }_{78}& {\alpha }_{79}& {\alpha }_{7,10}& {\alpha }_{7,11}& {\alpha }_{7,12}& {\alpha }_{7,13}& {\alpha }_{7,14}& {\alpha }_{7,15}\\ {\alpha }_{81}& {\alpha }_{82}& {\alpha }_{83}& {\alpha }_{84}& {\alpha }_{85}& {\alpha }_{86}& {\alpha }_{87}& {\alpha }_{88}& {\alpha }_{89}& {\alpha }_{8,10}& {\alpha }_{8,11}& {\alpha }_{8,12}& {\alpha }_{8,13}& {\alpha }_{8,14}& {\alpha }_{8,15}\\ {\alpha }_{91}& {\alpha }_{92}& {\alpha }_{93}& {\alpha }_{94}& {\alpha }_{95}& {\alpha }_{96}& {\alpha }_{97}& {\alpha }_{98}& {\alpha }_{99}& {\alpha }_{9,10}& {\alpha }_{9,11}& {\alpha }_{9,12}& {\alpha }_{9,13}& {\alpha }_{9,14}& {\alpha }_{9,15}\\ {\alpha }_{10,1}& {\alpha }_{10,2}& {\alpha }_{10,3}& {\alpha }_{10,4}& {\alpha }_{10,5}& {\alpha }_{10,6}& {\alpha }_{10,7}& {\alpha }_{10,8}& {\alpha }_{10,9}& {\alpha }_{10,10}& {\alpha }_{10,11}& {\alpha }_{10,12}& {\alpha }_{10,13}& {\alpha }_{10,14}& {\alpha }_{10,15}\\ {\alpha }_{11,1}& {\alpha }_{11,2}& {\alpha }_{11,3}& {\alpha }_{11,4}& {\alpha }_{11,5}& {\alpha }_{11,6}& {\alpha }_{11,7}& {\alpha }_{11,8}& {\alpha }_{11,9}& {\alpha }_{11,10}& {\alpha }_{11,11}& {\alpha }_{11,12}& {\alpha }_{11,13}& {\alpha }_{11,14}& {\alpha }_{11,15}\\ {\alpha }_{12,1}& {\alpha }_{12,2}& {\alpha }_{12,3}& {\alpha }_{12,4}& {\alpha }_{12,5}& {\alpha }_{12,6}& {\alpha }_{12,7}& {\alpha }_{12,8}& {\alpha }_{12,9}& {\alpha }_{12,10}& {\alpha }_{12,11}& {\alpha }_{12,12}& {\alpha }_{12,13}& {\alpha }_{12,14}& {\alpha }_{12,15}\\ {\alpha }_{13,1}& {\alpha }_{13,2}& {\alpha }_{13,3}& {\alpha }_{14,4}& {\alpha }_{14,5}& {\alpha }_{14,6}& {\alpha }_{14,7}& {\alpha }_{14,8}& {\alpha }_{14,9}& {\alpha }_{1,10}& {\alpha }_{1,11}& {\alpha }_{1,12}& {\alpha }_{1,13}& {\alpha }_{1,14}& {\alpha }_{1,15}\\ {\alpha }_{11}& {\alpha }_{12}& {\alpha }_{13}& {\alpha }_{14}& {\alpha }_{15}& {\alpha }_{16}& {\alpha }_{17}& {\alpha }_{18}& {\alpha }_{19}& {\alpha }_{1,10}& {\alpha }_{14,11}& {\alpha }_{14,12}& {\alpha }_{14,13}& {\alpha }_{14,14}& {\alpha }_{14,15}\\ {\alpha }_{15,1}& {\alpha }_{15,2}& {\alpha }_{15,3}& {\alpha }_{15,4}& {\alpha }_{15,5}& {\alpha }_{15,6}& {\alpha }_{15,7}& {\alpha }_{15,8}& {\alpha }_{15,9}& {\alpha }_{15,10}& {\alpha }_{15,11}& {\alpha }_{15,12}& {\alpha }_{15,13}& {\alpha }_{15,14}& {\alpha }_{15,15}\end{array}\right)$$

(5)

and

$$\Gamma :=\left(\begin{array}{cccc}{\gamma }_{11}& {\gamma }_{12}& {\gamma }_{13}& {\gamma }_{14}\\ {\gamma }_{21}& {\gamma }_{22}& {\gamma }_{23}& {\gamma }_{24}\\ {\gamma }_{31}& {\gamma }_{32}& {\gamma }_{33}& {\gamma }_{34}\\ {\gamma }_{41}& {\gamma }_{42}& {\gamma }_{43}& {\gamma }_{44}\end{array}\right)$$

(6)

where the ${\alpha }_{ij}$ and ${\gamma }_{ij}$ values are linear combinations of the original $a_i$, $a_{ji}$, $b_j$ and $c_k$ values (see Appendix B). Then, given that both of these matrices are non-singular, i.e., when

$$det\left(A\right)\ne 0anddet(\Gamma )\ne 0$$

(7)

holds true, then the general and unique solution of (4) reads:

$$\begin{array}{c}{N}_{\alpha \mu \nu }={\tilde{\alpha }}_{11}{\widehat{B}}_{\alpha \mu \nu }+{\tilde{\alpha }}_{12}{\widehat{B}}_{\nu \alpha \mu }+{\tilde{\alpha }}_{13}{\widehat{B}}_{\mu \nu \alpha }+{\tilde{\alpha }}_{14}{\widehat{B}}_{\alpha \nu \mu }+{\tilde{\alpha }}_{15}{\widehat{B}}_{\nu \mu \alpha }+{\tilde{\alpha }}_{16}{\widehat{B}}_{\mu \alpha \nu }+{\tilde{\alpha }}_{17}{\breve{B}}_{\alpha \mu \nu }+{\tilde{\alpha }}_{18}{\breve{B}}_{\mu \nu \alpha }+{\tilde{\alpha }}_{19}{\breve{B}}_{\nu \alpha \mu }\\ +{\tilde{\alpha }}_{1,10}{\overline{B}}_{\alpha \mu \nu }+{\tilde{\alpha }}_{1,11}{\overline{B}}_{\mu \nu \alpha }+{\tilde{\alpha }}_{1,12}{\overline{B}}_{\nu \alpha \mu }+{\tilde{\alpha }}_{1,13}{\mathring{B}}_{\alpha \mu \nu }+{\tilde{\alpha }}_{1,14}{\mathring{B}}_{\mu \nu \alpha }+{\tilde{\alpha }}_{1,15}{\mathring{B}}_{\nu \alpha \mu }\end{array}$$

(8)

where the ${\tilde{\alpha }}_{1i}$ values are the first-row elements of the inverse matrix $A^{-1}$ and

$${\widehat{B}}_{\alpha \mu \nu }=B_{\alpha \mu \nu }-\sum _{i=1}^4\sum _{j=1}^4\left(a_{7i}{\tilde{\gamma }}_{ij}{B}_{\mu }^{\left(j\right)}{g}_{\alpha \nu }+a_{8i}{\tilde{\gamma }}_{ij}{B}_{\nu }^{\left(j\right)}{g}_{\alpha \mu }+a_{9i}{\tilde{\gamma }}_{ij}{B}_{\alpha }^{\left(j\right)}{g}_{\mu \nu }\right)-{\epsilon }_{\alpha \mu \nu }^{\rho }\sum _{i=1}^3\sum _{j=1}^4{b}_i{\tilde{\gamma }}_{ij}{B}_{\rho }^{\left(j\right)}$$

(9)

$${\breve{B}}_{\alpha \mu \nu }={\epsilon }_{\alpha \mu }^{\beta \gamma }{\widehat{B}}_{\beta \gamma \nu }-2{(-1)}^s\sum _{j=1}^4\left[(b_{21}+b_{23}+b_{31}+b_{33}){\tilde{\gamma }}_{1j}+(b_{11}+b_{13}-b_{31}-b_{33}){\tilde{\gamma }}_{2j}-(b_{11}+b_{13}+b_{21}+b_{23}){\tilde{\gamma }}_{3j}\right]B_{[\alpha }^{\left(j\right)}{g}_{\mu ]\nu }$$

(10)

$${\overline{B}}_{\alpha \mu \nu }:={\epsilon }_{\alpha \nu }^{\beta \gamma }{\widehat{B}}_{\beta \mu \gamma }-2{(-1)}^s\sum _{j=1}^4\left[(b_{21}+b_{22}+b_{31}+b_{32}){\tilde{\gamma }}_{1j}+(b_{11}+b_{12}-b_{31}-b_{32}){\tilde{\gamma }}_{2j}-(b_{11}+b_{12}+b_{21}+b_{22}){\tilde{\gamma }}_{3j}\right]B_{[\alpha }^{\left(j\right)}{g}_{\mu ]\nu }$$

(11)

$${\mathring{B}}_{\alpha \mu \nu }:={\epsilon }_{\mu \nu }^{\beta \gamma }{\mathring{B}}_{\alpha \beta \gamma }-2{(-1)}^{s+1}\sum _{j=1}^4\left[(b_{22}+b_{23}+b_{32}+b_{33}){\tilde{\gamma }}_{1j}+(b_{12}+b_{13}-b_{32}-b_{33}){\tilde{\gamma }}_{2j}-(b_{22}+b_{23}+b_{12}+b_{13}){\tilde{\gamma }}_{3j}\right]B_{[\mu }^{\left(j\right)}{g}_{\nu ]\alpha }$$

(12)

where ${\tilde{\gamma }}_{ij}$ values are the elements of the inverse matrix ${\Gamma }^{-1}$.

Proof. 

The first step consists of removing the traces (and pseudotraces) of N and expressing them in terms of the corresponding traces of the known tensor B. To this end, we perform four distinct operations on (4), i.e., we contract the latter with $g^{\alpha \mu }$, $g^{\alpha \nu }$, $g^{\mu \nu }$ and ${\epsilon }^{\lambda \alpha \mu \nu }$, which (after some renaming of the indices) produces the system:

$$\begin{array}{c}{\gamma }_{11}{N}_{\mu }^{\left(1\right)}+{\gamma }_{12}{N}_{\mu }^{\left(2\right)}+{\gamma }_{13}{N}_{\mu }^{\left(3\right)}+{\gamma }_{14}{N}_{\mu }^{\left(4\right)}=B_{\mu }^{\left(1\right)}\end{array}$$

(13)

$$\begin{array}{c}{\gamma }_{21}{N}_{\mu }^{\left(1\right)}+{\gamma }_{22}{N}_{\mu }^{\left(2\right)}+{\gamma }_{23}{N}_{\mu }^{\left(3\right)}+{\gamma }_{24}{N}_{\mu }^{\left(4\right)}=B_{\mu }^{\left(2\right)}\end{array}$$

(14)

$$\begin{array}{c}{\gamma }_{31}{N}_{\mu }^{\left(1\right)}+{\gamma }_{32}{N}_{\mu }^{\left(2\right)}+{\gamma }_{33}{N}_{\mu }^{\left(3\right)}+{\gamma }_{34}{N}_{\mu }^{\left(4\right)}=B_{\mu }^{\left(3\right)}\end{array}$$

(15)

$$\begin{array}{c}{\gamma }_{41}{N}_{\mu }^{\left(1\right)}+{\gamma }_{42}{N}_{\mu }^{\left(2\right)}+{\gamma }_{43}{N}_{\mu }^{\left(3\right)}+{\gamma }_{44}{N}_{\mu }^{\left(4\right)}=B_{\mu }^{\left(4\right)}\end{array}$$

(16)

where ${\gamma }_{ij}$ values are linear combinations of the initial 27 parameters, the exact form of which we present in the Appendix A. We also used the notation $M_{\mu }=N_{\mu }^{\left(4\right)}$. The above is a system of four equations with four unknowns, which we could express in matrix form as:

$$\Gamma X=Y$$

(17)

where we defined the columns $X:={(N_{\mu }^{\left(1\right)},N_{\mu }^{\left(2\right)},N_{\mu }^{\left(3\right)},N_{\mu }^{\left(4\right)})}^T$ and $Y:=(B_{\mu }^{\left(1\right)},B_{\mu }^{\left(2\right)},B_{\mu }^{\left(3\right)},$$B_{\mu }^{\left(4\right)}{)}^T$ along with the matrix:

$$\Gamma :=\left(\begin{array}{cccc}{\gamma }_{11}& {\gamma }_{12}& {\gamma }_{13}& {\gamma }_{14}\\ {\gamma }_{21}& {\gamma }_{22}& {\gamma }_{23}& {\gamma }_{24}\\ {\gamma }_{31}& {\gamma }_{32}& {\gamma }_{33}& {\gamma }_{34}\\ {\gamma }_{41}& {\gamma }_{42}& {\gamma }_{43}& {\gamma }_{44}\end{array}\right)$$

(18)

Now, given that the matrix $\Gamma$ is not degenerate, i.e., when

$$det(\Gamma )\ne 0$$

(19)

then its inverse ${\Gamma }^{-1}$ exists. Then, multiplying (17) with the latter, we obtain:

$$X={\Gamma }^{-1}Y$$

(20)

which relate the unknown traces $N_{\mu }^{\left(i\right)}$ to the known traces $\left(B_{\mu }^{\left(i\right)}\right)$. More specifically, the component form of the above matrix equation produces the relationship:

$$N_{\mu }^{\left(i\right)}=\sum _{j=1}^4{\tilde{\gamma }}_{ij}{B}_{\mu }^{\left(j\right)}$$

(21)

where ${\tilde{\gamma }}_{ij}$ values are the elements of the inverse matrix ${\Gamma }^{-1}$. As a result, we have fully eliminated the traces of N in terms of the traces of B. We could then substitute the last equation back into (4) to obtain:

$$\begin{array}{c}{a}_1{N}_{\alpha \mu \nu }+a_2{N}_{\nu \alpha \mu }+a_3{N}_{\mu \nu \alpha }+a_4{N}_{\alpha \nu \mu }+a_5{N}_{\nu \mu \alpha }+a_6{N}_{\mu \alpha \nu }\\ +b_{11}{M}_{\alpha \mu \nu }^{\left(1\right)}+b_{12}{M}_{\nu \alpha \mu }^{\left(1\right)}+b_{13}{M}_{\mu \nu \alpha }^{\left(1\right)}+b_{21}{M}_{\alpha \mu \nu }^{\left(2\right)}+b_{22}{M}_{\nu \alpha \mu }^{\left(2\right)}+b_{23}{M}_{\mu \nu \alpha }^{\left(2\right)}+b_{31}{M}_{\alpha \mu \nu }^{\left(3\right)}+b_{32}{M}_{\nu \alpha \mu }^{\left(3\right)}+b_{33}{M}_{\mu \nu \alpha }^{\left(3\right)}={\widehat{B}}_{\alpha \mu \nu }\end{array}$$

(22)

where

$${\widehat{B}}_{\alpha \mu \nu }=B_{\alpha \mu \nu }-\sum _{i=1}^4\sum _{j=1}^4\left(a_{7i}{\tilde{\gamma }}_{ij}{B}_{\mu }^{\left(j\right)}{g}_{\alpha \nu }+a_{8i}{\tilde{\gamma }}_{ij}{B}_{\nu }^{\left(j\right)}{g}_{\alpha \mu }+a_{9i}{\tilde{\gamma }}_{ij}{B}_{\alpha }^{\left(j\right)}{g}_{\mu \nu }\right)-{\epsilon }_{\alpha \mu \nu }^{\rho }\sum _{i=1}^3\sum _{j=1}^4{b}_i{\tilde{\gamma }}_{ij}{B}_{\rho }^{\left(j\right)}$$

(23)

and we renamed $c_1=a_{74}$, $c_2=a_{84}$ and $c_3=a_{94}$ in order to obtain a more compact form. However, this only solved 4 of the 19 unknown combinations as they appear in (4), with 15 more remaining; thus, we need 15 more equations in order to fully solve for N. To this end, we then consider the five possible independent permutations of (22), which (including (22) itself and using shorthand sum notation)6 read:

$$\begin{array}{c}{a}_1{N}_{\alpha \mu \nu }+a_2{N}_{\nu \alpha \mu }+a_3{N}_{\mu \nu \alpha }+a_4{N}_{\alpha \nu \mu }+a_5{N}_{\nu \mu \alpha }+a_6{N}_{\mu \alpha \nu }+\sum _{i=1}^3\left(b_{i1}{M}_{\alpha \mu \nu }^{\left(i\right)}+b_{i2}{M}_{\nu \alpha \mu }^{\left(i\right)}+b_{i3}{M}_{\mu \nu \alpha }^{\left(i\right)}\right)={\widehat{B}}_{\alpha \mu \nu }\end{array}$$

(24)

$$\begin{array}{c}{a}_1{N}_{\nu \alpha \mu }+a_2{N}_{\mu \nu \alpha }+a_3{N}_{\alpha \mu \nu }+a_4{N}_{\mu \alpha \nu }+a_5{N}_{\alpha \nu \mu }+a_6{N}_{\nu \mu \alpha }+\sum _{i=1}^3\left(b_{i1}{M}_{\nu \alpha \mu }^{\left(i\right)}+b_{i2}{M}_{\mu \nu \alpha }^{\left(i\right)}+b_{i3}{M}_{\alpha \mu \nu }^{\left(i\right)}\right)={\widehat{B}}_{\nu \alpha \mu }\end{array}$$

(25)

$$\begin{array}{c}{a}_1{N}_{\mu \nu \alpha }+a_2{N}_{\alpha \mu \nu }+a_3{N}_{\nu \alpha \mu }+a_4{N}_{\nu \mu \alpha }+a_5{N}_{\mu \alpha \nu }+a_6{N}_{\alpha \nu \mu }+\sum _{i=1}^3\left(b_{i1}{M}_{\mu \nu \alpha }^{\left(i\right)}+b_{i2}{M}_{\alpha \mu \nu }^{\left(i\right)}+b_{i3}{M}_{\nu \alpha \mu }^{\left(i\right)}\right)={\widehat{B}}_{\mu \nu \alpha }\end{array}$$

(26)

$$\begin{array}{c}{a}_1{N}_{\alpha \nu \mu }+a_2{N}_{\mu \alpha \nu }+a_3{N}_{\nu \mu \alpha }+a_4{N}_{\alpha \mu \nu }+a_5{N}_{\mu \nu \alpha }+a_6{N}_{\nu \alpha \mu }+\sum _{i=1}^3\left(b_{i1}{M}_{\alpha \nu \mu }^{\left(i\right)}+b_{i2}{M}_{\mu \alpha \nu }^{\left(i\right)}+b_{i3}{M}_{\nu \mu \alpha }^{\left(i\right)}\right)={\widehat{B}}_{\alpha \nu \mu }\end{array}$$

(27)

$$\begin{array}{c}{a}_1{N}_{\nu \mu \alpha }+a_2{N}_{\alpha \nu \mu }+a_3{N}_{\mu \alpha \nu }+a_4{N}_{\nu \alpha \mu }+a_5{N}_{\alpha \mu \nu }+a_6{N}_{\mu \nu \alpha }+\sum _{i=1}^3\left(b_{i1}{M}_{\nu \mu \alpha }^{\left(i\right)}+b_{i2}{M}_{\alpha \nu \mu }^{\left(i\right)}+b_{i3}{M}_{\mu \alpha \nu }^{\left(i\right)}\right)={\widehat{B}}_{\nu \mu \alpha }\end{array}$$

(28)

$$\begin{array}{c}{a}_1{N}_{\mu \alpha \nu }+a_2{N}_{\nu \mu \alpha }+a_3{N}_{\alpha \nu \mu }+a_4{N}_{\mu \nu \alpha }+a_5{N}_{\nu \alpha \mu }+a_6{N}_{\alpha \mu \nu }+\sum _{i=1}^3\left(b_{i1}{M}_{\mu \alpha \nu }^{\left(i\right)}+b_{i2}{M}_{\nu \mu \alpha }^{\left(i\right)}+b_{i3}{M}_{\alpha \nu \mu }^{\left(i\right)}\right)={\widehat{B}}_{\mu \alpha \nu }\end{array}$$

(29)

In this way, we gather six equations but still have nine more to go. Continuing, we then contract (22) with ${\epsilon }_{\beta \gamma }^{\alpha \mu }$ and by using the identity ${\epsilon }_{\alpha \mu \beta \gamma }{\epsilon }^{\kappa \lambda \rho \sigma }={(-1)}^{s}4!{\delta }_{[\alpha }^{\kappa }{\delta }_{\mu }^{\lambda }{\delta }_{\beta }^{\rho }{\delta }_{\gamma ]}^{\sigma }$, its contractions and some long calculations, we finally arrive at (some useful identities are given in Appendix C):

$$\begin{array}{c}(a_1-a_6)M_{\nu \beta \gamma }^{\left(1\right)}+(a_4-a_3)M_{\nu \beta \gamma }^{\left(2\right)}+(a_2-a_5)M_{\nu \beta \gamma }^{\left(3\right)}+2{(-1)}^{s+1}(b_{21}+b_{23}+b_{31}+b_{33})N_{[\beta }^{\left(1\right)}{g}_{\gamma ]\nu }\\ +2{(-1)}^{s+1}(b_{11}+b_{13}-b_{31}-b_{33})N_{[\beta }^{\left(2\right)}{g}_{\gamma ]\nu }-2{(-1)}^{s+1}(b_{11}+b_{13}+b_{21}+b_{23})N_{[\beta }^{\left(3\right)}{g}_{\gamma ]\nu }\\ +2{(-1)}^s\left[-(b_{11}+b_{13})+2b_{12}\right]N_{\left[\beta \gamma \right]\nu }+2{(-1)}^s\left[-(b_{21}+b_{23})+2b_{22}\right]N_{\left[\beta \right|\nu \left|\gamma \right]}+2{(-1)}^s\left[-(b_{31}+b_{33})+2b_{32}\right]N_{\nu \left[\beta \gamma \right]}\\ ={\epsilon }_{\beta \gamma }^{\alpha \mu }{\widehat{B}}_{\alpha \mu \nu }\end{array}$$

(30)

We then rename the indices $\beta \to \alpha$, $\gamma \to \mu$ and use (21) to remove the traces of N. After some rearranging, we end up with:

$$\begin{array}{c}{(-1)}^s(2b_{12}-b_{11}-b_{13})N_{\alpha \mu \nu }+{(-1)}^s(2b_{32}-b_{31}-b_{33})N_{\nu \alpha \mu }-{(-1)}^s(2b_{22}-b_{21}-b_{23})N_{\mu \nu \alpha }\\ +{(-1)}^s(2b_{22}-b_{21}-b_{23})N_{\alpha \nu \mu }-{(-1)}^s(2b_{32}-b_{31}-b_{33})N_{\nu \mu \alpha }-{(-1)}^s(2b_{12}-b_{11}-b_{13})N_{\mu \alpha \nu }\\ +(a_1-a_6)M_{\nu \alpha \mu }^{\left(1\right)}+(a_4-a_3)M_{\nu \alpha \mu }^{\left(2\right)}+(a_2-a_5)M_{\nu \alpha \mu }^{\left(3\right)}={\breve{B}}_{\alpha \mu \nu }\end{array}$$

(31)

where we set

$${\breve{B}}_{\alpha \mu \nu }={\epsilon }_{\alpha \mu }^{\beta \gamma }{\widehat{B}}_{\beta \gamma \nu }-2{(-1)}^s\sum _{j=1}^4\left[(b_{21}+b_{23}+b_{31}+b_{33}){\tilde{\gamma }}_{1j}+(b_{11}+b_{13}-b_{31}-b_{33}){\tilde{\gamma }}_{2j}-(b_{11}+b_{13}+b_{21}+b_{23}){\tilde{\gamma }}_{3j}\right]B_{[\alpha }^{\left(j\right)}{g}_{\mu ]\nu }$$

(32)

Next, we consider the index permutation $\alpha \to \mu \to \nu \to \alpha$ in (31), firstly once and then two successive times to obtain two more equations:

$$\begin{array}{c}{(-1)}^s(2b_{12}-b_{11}-b_{13})N_{\mu \nu \alpha }+{(-1)}^s(2b_{32}-b_{31}-b_{33})N_{\alpha \mu \nu }-{(-1)}^s(2b_{22}-b_{21}-b_{23})N_{\nu \alpha \mu }\\ +{(-1)}^s(2b_{22}-b_{21}-b_{23})N_{\mu \alpha \nu }-{(-1)}^s(2b_{32}-b_{31}-b_{33})N_{\alpha \nu \mu }-{(-1)}^s(2b_{12}-b_{11}-b_{13})N_{\nu \mu \alpha }\\ +(a_1-a_6)M_{\alpha \mu \nu }^{\left(1\right)}+(a_4-a_3)M_{\alpha \mu \nu }^{\left(2\right)}+(a_2-a_5)M_{\alpha \mu \nu }^{\left(3\right)}={\breve{B}}_{\mu \nu \alpha }\end{array}$$

(33)

$$\begin{array}{c}{(-1)}^s(2b_{12}-b_{11}-b_{13})N_{\nu \alpha \mu }+{(-1)}^s(2b_{32}-b_{31}-b_{33})N_{\mu \nu \alpha }-{(-1)}^s(2b_{22}-b_{21}-b_{23})N_{\alpha \mu \nu }\\ +{(-1)}^s(2b_{22}-b_{21}-b_{23})N_{\nu \mu \alpha }-{(-1)}^s(2b_{32}-b_{31}-b_{33})N_{\mu \alpha \nu }-{(-1)}^s(2b_{12}-b_{11}-b_{13})N_{\alpha \nu \mu }\\ +(a_1-a_6)M_{\mu \nu \alpha }^{\left(1\right)}+(a_4-a_3)M_{\mu \nu \alpha }^{\left(2\right)}+(a_2-a_5)M_{\mu \nu \alpha }^{\left(3\right)}={\breve{B}}_{\nu \alpha \mu }\end{array}$$

(34)

In the same manner, we contract (22) once with ${\epsilon }_{\beta \gamma }^{\alpha \nu }$ and another time with ${\epsilon }_{\beta \gamma }^{\mu \nu }$ and again, after performing one and two successive permutations of the indices for each case, we gather six more equations:

$$\begin{array}{c}{(-1)}^s(b_{21}+b_{22}-2b_{23})N_{\alpha \mu \nu }-{(-1)}^s(b_{11}+b_{12}-2b_{13})N_{\nu \alpha \mu }-{(-1)}^s(b_{31}+b_{32}-2b_{33})N_{\mu \nu \alpha }\\ +{(-1)}^s(b_{11}+b_{12}-2b_{13})N_{\alpha \nu \mu }-{(-1)}^s(b_{21}+b_{22}-2b_{23})N_{\nu \mu \alpha }+{(-1)}^s(b_{31}+b_{32}-2b_{33})N_{\mu \alpha \nu }\\ +(a_4-a_2)M_{\mu \alpha \nu }^{\left(1\right)}+(a_1-a_5)M_{\mu \alpha \nu }^{\left(2\right)}+(a_6-a_3)M_{\mu \alpha \nu }^{\left(3\right)}={\overline{B}}_{\alpha \mu \nu }\end{array}$$

(35)

$$\begin{array}{c}{(-1)}^s(b_{21}+b_{22}-2b_{23})N_{\mu \nu \alpha }-{(-1)}^s(b_{11}+b_{12}-2b_{13})N_{\alpha \mu \nu }-{(-1)}^s(b_{31}+b_{32}-2b_{33})N_{\nu \alpha \mu }\\ +{(-1)}^s(b_{11}+b_{12}-2b_{13})N_{\mu \alpha \nu }-{(-1)}^s(b_{21}+b_{22}-2b_{23})N_{\alpha \nu \mu }+{(-1)}^s(b_{31}+b_{32}-2b_{33})N_{\nu \mu \alpha }\\ +(a_4-a_2)M_{\nu \mu \alpha }^{\left(1\right)}+(a_1-a_5)M_{\nu \mu \alpha }^{\left(2\right)}+(a_6-a_3)M_{\nu \mu \alpha }^{\left(3\right)}={\overline{B}}_{\mu \nu \alpha }\end{array}$$

(36)

$$\begin{array}{c}{(-1)}^s(b_{21}+b_{22}-2b_{23})N_{\nu \alpha \mu }-{(-1)}^s(b_{11}+b_{12}-2b_{13})N_{\mu \nu \alpha }-{(-1)}^s(b_{31}+b_{32}-2b_{33})N_{\alpha \mu \nu }\\ +{(-1)}^s(b_{11}+b_{12}-2b_{13})N_{\nu \mu \alpha }-{(-1)}^s(b_{21}+b_{22}-2b_{23})N_{\mu \alpha \nu }+{(-1)}^s(b_{31}+b_{32}-2b_{33})N_{\alpha \nu \mu }\\ +(a_4-a_2)M_{\alpha \nu \mu }^{\left(1\right)}+(a_1-a_5)M_{\alpha \nu \mu }^{\left(2\right)}+(a_6-a_3)M_{\alpha \nu \mu }^{\left(3\right)}={\overline{B}}_{\nu \alpha \mu }\end{array}$$

(37)

$$\begin{array}{c}{(-1)}^s(2b_{31}-b_{32}-b_{33})N_{\alpha \mu \nu }-{(-1)}^s(2b_{21}-b_{22}-b_{23})N_{\nu \alpha \mu }+{(-1)}^s(2b_{11}-b_{12}-b_{13})N_{\mu \nu \alpha }\\ -{(-1)}^s(2b_{31}-b_{32}-b_{33})N_{\alpha \nu \mu }-{(-1)}^s(2b_{11}-b_{12}-b_{13})N_{\nu \mu \alpha }+{(-1)}^s(2b_{21}-b_{22}-b_{23})N_{\mu \alpha \nu }\\ +(a_3-a_5)M_{\alpha \mu \nu }^{\left(1\right)}+(a_6-a_2)M_{\alpha \mu \nu }^{\left(2\right)}+(a_1-a_4)M_{\alpha \mu \nu }^{\left(3\right)}={\mathring{B}}_{\alpha \mu \nu }\end{array}$$

(38)

$$\begin{array}{c}{(-1)}^s(2b_{31}-b_{32}-b_{33})N_{\mu \nu \alpha }-{(-1)}^s(2b_{21}-b_{22}-b_{23})N_{\alpha \mu \nu }+{(-1)}^s(2b_{11}-b_{12}-b_{13})N_{\nu \alpha \mu }\\ -{(-1)}^s(2b_{31}-b_{32}-b_{33})N_{\mu \alpha \nu }-{(-1)}^s(2b_{11}-b_{12}-b_{13})N_{\alpha \nu \mu }+{(-1)}^s(2b_{21}-b_{22}-b_{23})N_{\nu \mu \alpha }\\ +(a_3-a_5)M_{\mu \nu \alpha }^{\left(1\right)}+(a_6-a_2)M_{\mu \nu \alpha }^{\left(2\right)}+(a_1-a_4)M_{\mu \nu \alpha }^{\left(3\right)}={\mathring{B}}_{\mu \nu \alpha }\end{array}$$

(39)

$$\begin{array}{c}{(-1)}^s(2b_{31}-b_{32}-b_{33})N_{\nu \alpha \mu }-{(-1)}^s(2b_{21}-b_{22}-b_{23})N_{\mu \nu \alpha }+{(-1)}^s(2b_{11}-b_{12}-b_{13})N_{\alpha \mu \nu }\\ -{(-1)}^s(2b_{31}-b_{32}-b_{33})N_{\nu \mu \alpha }-{(-1)}^s(2b_{11}-b_{12}-b_{13})N_{\mu \alpha \nu }+{(-1)}^s(2b_{21}-b_{22}-b_{23})N_{\alpha \nu \mu }\\ +(a_3-a_5)M_{\nu \alpha \mu }^{\left(1\right)}+(a_6-a_2)M_{\nu \alpha \mu }^{\left(2\right)}+(a_1-a_4)M_{\nu \alpha \mu }^{\left(3\right)}={\mathring{B}}_{\nu \alpha \mu }\end{array}$$

(40)

where we set

$${\overline{B}}_{\alpha \mu \nu }:={\epsilon }_{\alpha \nu }^{\beta \gamma }{\widehat{B}}_{\beta \mu \gamma }-2{(-1)}^s\sum _{j=1}^4\left[(b_{21}+b_{22}+b_{31}+b_{32}){\tilde{\gamma }}_{1j}+(b_{11}+b_{12}-b_{31}-b_{32}){\tilde{\gamma }}_{2j}-(b_{11}+b_{12}+b_{21}+b_{22}){\tilde{\gamma }}_{3j}\right]B_{[\alpha }^{\left(j\right)}{g}_{\mu ]\nu }$$

(41)

and

$${\mathring{B}}_{\alpha \mu \nu }:={\epsilon }_{\mu \nu }^{\beta \gamma }{\mathring{B}}_{\alpha \beta \gamma }-2{(-1)}^{s+1}\sum _{j=1}^4\left[(b_{22}+b_{23}+b_{32}+b_{33}){\tilde{\gamma }}_{1j}+(b_{12}+b_{13}-b_{32}-b_{33}){\tilde{\gamma }}_{2j}-(b_{22}+b_{23}+b_{12}+b_{13}){\tilde{\gamma }}_{3j}\right]B_{[\mu }^{\left(j\right)}{g}_{\nu ]\alpha }$$

(42)

We then place Equations (24)–(29) and (31)–(40) into that exact order and express the system of the above 15 equations in matrix form as:

$$A\mathcal{N}=\mathcal{B}$$

(43)

where A is the $15\times 15$ matrix of coefficients (see Appendix B). We also define the columns:

$$\mathcal{N}={\left(N_{\alpha \mu \nu },N_{\nu \alpha \mu },N_{\mu \nu \alpha },N_{\alpha \nu \mu },N_{\nu \mu \alpha },N_{\mu \alpha \nu },M_{\alpha \mu \nu }^{\left(1\right)},M_{\nu \alpha \mu }^{\left(1\right)},M_{\mu \nu \alpha }^{\left(1\right)},M_{\alpha \mu \nu }^{\left(2\right)},M_{\nu \alpha \mu }^{\left(2\right)},M_{\mu \nu \alpha }^{\left(2\right)},M_{\alpha \mu \nu }^{\left(3\right)},M_{\nu \alpha \mu }^{\left(3\right)},M_{\mu \nu \alpha }^{\left(3\right)}\right)}^T$$

(44)

as well as

$$\mathcal{B}={\left({\widehat{B}}_{\alpha \mu \nu },{\widehat{B}}_{\nu \alpha \mu },{\widehat{B}}_{\mu \nu \alpha },{\widehat{B}}_{\alpha \nu \mu },{\widehat{B}}_{\nu \mu \alpha },{\widehat{B}}_{\mu \alpha \nu },{\breve{B}}_{\alpha \mu \nu },{\breve{B}}_{\mu \nu \alpha },{\breve{B}}_{\nu \alpha \mu },{\overline{B}}_{\alpha \mu \nu },{\overline{B}}_{\mu \nu \alpha },{\overline{B}}_{\nu \alpha \mu },{\mathring{B}}_{\alpha \mu \nu },{\mathring{B}}_{\mu \nu \alpha },{\mathring{B}}_{\nu \alpha \mu }\right)}^T$$

(45)

Then, given that the matrix A is non-singular, namely

$$det\left(A\right)\ne 0$$

(46)

we can formally multiply the above matrix equation with $A^{-1}$ to obtain:

$$\mathcal{N}=A^{-1}\mathcal{B}$$

(47)

Finally, by equating the first elements of the latter column equation, we arrive at the stated result:

$$\begin{array}{c}{N}_{\alpha \mu \nu }={\tilde{\alpha }}_{11}{\widehat{B}}_{\alpha \mu \nu }+{\tilde{\alpha }}_{12}{\widehat{B}}_{\nu \alpha \mu }+{\tilde{\alpha }}_{13}{\widehat{B}}_{\mu \nu \alpha }+{\tilde{\alpha }}_{14}{\widehat{B}}_{\alpha \nu \mu }+{\tilde{\alpha }}_{15}{\widehat{B}}_{\nu \mu \alpha }+{\tilde{\alpha }}_{16}{\widehat{B}}_{\mu \alpha \nu }+{\tilde{\alpha }}_{17}{\breve{B}}_{\alpha \mu \nu }+{\tilde{\alpha }}_{18}{\breve{B}}_{\mu \nu \alpha }+{\tilde{\alpha }}_{19}{\breve{B}}_{\nu \alpha \mu }\\ +{\tilde{\alpha }}_{1,10}{\overline{B}}_{\alpha \mu \nu }+{\tilde{\alpha }}_{1,11}{\overline{B}}_{\mu \nu \alpha }+{\tilde{\alpha }}_{1,12}{\overline{B}}_{\nu \alpha \mu }+{\tilde{\alpha }}_{1,13}{\mathring{B}}_{\alpha \mu \nu }+{\tilde{\alpha }}_{1,14}{\mathring{B}}_{\mu \nu \alpha }+{\tilde{\alpha }}_{1,15}{\mathring{B}}_{\nu \alpha \mu }\end{array}$$

(48)

where ${\tilde{\alpha }}_{ij}$ values are the elements of the inverse matrix $A^{-1}$. The matrix is the exact and unique solution of (4), provided that the non-degeneracy conditions (19) and (46) were satisfied.  □

Evidently, when $B_{\alpha }=0$ and $det\left(A\right)\ne 0$, along with $det(\Gamma )\ne 0$, the unique solution of (4) is always $N_{\alpha \mu \nu }=0$. More precisely, we showed the following.

Corollary 1.

When $B_{\alpha \mu \nu }=0$ and both matrices A and $\Gamma$ are non-singular, then the unique solution of (4) is $N_{\alpha \mu \nu }=0$.

4. Conclusions

We considered the most general linear tensor equation of a third-rank tensor N in terms of a given source B in four dimensions. The latter was a 30-parameter tensor equation, as given by (4). By following a step-by-step procedure and given two rather general non-degeneracy conditions among the parameters, we provided the unique and exact solution of the component $N_{\alpha \mu \nu }$ in terms of the known component $B_{\alpha \mu \nu }$, its dualizations and its contractions. The solution is given by the expression (48). An immediate conclusion is that, provided the non-degeneracy conditions hold and in the absence of sources (i.e., when $B_{\alpha \mu \nu }=0$), $N_{\alpha \mu \nu }=0$ is the only (unique) solution of the 30-parameter tensor equation. As a final remark, let us note that our results have a natural application in metric-affine theories of gravity (see Appendix D) but could also be applied to other physical situations just as well.