Exact Solutions of Maxwell Equations in Homogeneous Spaces with the Group of Motions G₃(VIII)

Abstract

The problem of the classification of the exact solutions to Maxwell’s vacuum equations for admissible electromagnetic fields and homogeneous space-time with the group of motions $G_3\left(VIII\right)$ according to the Bianchi classification is considered. All non-equivalent solutions are found. The classification problem for the remaining groups of motion, $G_3\left(N\right)$, has already been solved in other papers. All non-equivalent solutions of empty Maxwell equations for all homogeneous spaces with admissible electromagnetic fields are now known.

1. Introduction

If the symmetry of space-time and physical fields is given by Killing fields whose number is no less than three, it is possible to reduce the field equations and the equations of motion of the tested charged particles to ordinary differential equations.

Spaces admitting complete sets of mutually commutative Killing tensor fields of rank no greater than two are of special interest in the theory of gravitation. Such spaces are called Steckel spaces. The theory of Steckel spaces was developed in [1,2,3,4,5,6,7] (see also [8,9,10,11] and the bibliographies given there). The equations of motion of test particles in Stackel spaces can be integrated using the commutative integration method (CIM) (or the method of complete separation of variables). Exact solutions to the gravitational equations are still actively used in the study of various aspects of gravitational theory and cosmology (see, for example, refs. [12,13,14,15,16,17,18,19,20,21,22,23]).

Another method for exact integration of the equations of motion for a test particle (the method of non-commutative integration (NCIM)) was proposed in [24]. This method is applied to spaces admitting non-commutative groups of motion $G_r\left(r\right),r\ge 3$ (see A. Petrov [25]). It allows for reducing the equations of motion to systems of ordinary differential equations. By analogy with Stackel spaces, we call them poststack spaces (PSS). PSS are also actively studied in gravitational theory and cosmology (see, e.g., [26,27,28,29,30,31,32,33,34] ). The classification of electromagnetic fields in which the Klein–Gordon–Fock equations and Hamilton–Jacobi equations admit non-commutative algebras of symmetry operators for a charged sample particle was carried out in [35,36,37,38].

Commutative and non-commutative integration methods have a similar classification problem, namely enumerating all non-equivalent metrics and electromagnetic potentials satisfying the requirements of the given symmetry. For Stackel spaces, the problem of classifying admissible external electromagnetic fields and electrovacuum solutions of the Einstein–Maxwell equations was solved in [39].

In previous works ([40,41,42]), the non-null PSS of all types were considered according to the Bianchi classification, except type $VIII$. In the present work, all non-equivalent exact solutions of Maxwell’s vacuum equations for non-null PSS of type $VIII$ are obtained. Thus, this classification is complete for all non-null PSS.

2. Admissible Electromagnetic Fields in Homogeneous Spaces

According to its definition (see [43]), the space-time $V_4$ is homogeneous if its metric can be represented in a semi-geodesic coordinate system as follows:

$$d{s}^2=-{d{u}^0}^2+{\eta }_{ab}{e}_{\alpha }^a{e}_{\beta }^{b}d{u}^{\alpha }d{u}^{\beta },g_{ij}=-{\delta }_i^0{\delta }_j^0+{\delta }_i^{\alpha }{\delta }_j^{\beta }{e}_{\alpha }^a{e}_{\beta }^b{\eta }_{ab}\left(u^0\right),det\left|{\eta }_{ab}\right|={\eta }^2>0,{\dot{e}}_{\alpha }^a=0,$$

(1)

where the condition

$$[Y_a,Y_b]=C_{ab}^c{Y}_c,Y_a=e_a^{\alpha }{\widehat{\partial }}_{\alpha }$$

(2)

is satisfied. Here, $e_a^{\alpha }$ are the triad of the dual vectors:

$$e_{\alpha }^b{e}_a^{\alpha }={\delta }_a^b$$

(3)

and $C_{bc}^a$ are structural constants of the group $G_3\left(N\right)$, which acts on $V_4$. The vectors of the frame $e_{\alpha }^a$ define a non-holonomic coordinate system in the hypersurface of transitivity $V_3$ of the group $G_3\left(N\right)$. Here and elsewhere, dots denote the derivatives of the variable $u^0$. The coordinate indices of the semi-geodesic coordinate system are denoted by the letters $i,j,k=0,1,\dots 3$. The variables of the local coordinate system on $V_3$ are provided with indices $\alpha ,\beta ,\gamma =1,\dots 3.$ Indices of a non-holonomic frame are provided with the indices $a,b,c=1,\dots 3$. The rule is used according to which of the the repeating upper and lower indices are summarized within the index range.

It has been proven in the paper [36] that for a charged test particle moving in the external electromagnetic field with potential $A_i$, the Hamilton–Jacobi equation:

$$g^{ij}(p_i+A_i)(p_j+A_j)=m^2(p_i+A_i=P_i),$$

(4)

and the Klein–Gordon–Fock equation:

$$\widehat{H}\phi =(g^{ij}(-i{\widehat{p}}_l+A_l)(-i{\widehat{p}}_j+A_j)=m^2\phi (-i{\widehat{p}}_j+A_j={\widehat{P}}_j)$$

(5)

admit the integrals of motion

$$X_{\alpha }={\xi }_{\alpha }^i{p}_i(or{\widehat{X}}_{\alpha }={\xi }_{\alpha }^i{\widehat{p}}_i),$$

(6)

if and only if the condition

$${\xi }_a^{\alpha }{\left({\xi }_b^{\beta }{A}_{\beta }\right)}_{,\alpha }=C_{ab}^c{\xi }_b^{\beta }{A}_{\beta }$$

(7)

is satisfied. Here, $p_i={\partial }_i\phi$, ${\widehat{p}}_k=-ı{\widehat{\nabla }}_k$ (${\widehat{\nabla }}_k$ is the covariant derivative operator corresponding to the partial derivative operator ${\widehat{\partial }}_i$ and $\phi$ is a scalar function of the particle with mass m), ${\xi }_{\alpha }^i$ is the Killing vector, and $C_{ab}^c$ are structural constants:

$$[{\widehat{X}}_a,{\widehat{X}}_b]=C_{ab}^c{\widehat{X}}_c.$$

If $A_i$ satisfies condition (7), the electromagnetic field is called admissible. All admissible electromagnetic fields for groups of motion $G_r\left(N\right)(r\ge 3)$ acting transitively on hypersurfaces of space-time have been found in [36,37,38].

Let us show that solutions of the system of Equation (7) for HPSS of type $VIII$ can be represented in the form:

$$A_{\alpha }={\alpha }_a\left(u^0\right)e_{\alpha }^a\Rightarrow {\mathbf{A}}_a=e_a^{\alpha }{A}_{\alpha }={\alpha }_a\left(u^0\right).$$

(8)

To prove this, let us find the frame vector using the metric tensor of Bianchi’s $VIII$-type space (see [25]).

$$\begin{array}{c}{ds}^2={d{u}^1}^2{a}_{11}+2d{u}^{1}d{u}^2(a_{11}{u^1}^2-2a_{13}{u}^1+a_{12})exp(-u^3)+2d{u}^{1}d{u}^3(a_{13}-a_{11}{u}^1)\\ +{d{u}^2}^2(a_{11}{u^1}^4-4a_{13}{u^1}^3+2(a_{12}+2a_{33}){u^1}^2-4a_{23}{u}^1+a_{22})exp(-2u^3)\\ 2d{u}^{2}d{u}^3(-a_{11}{u^1}^3+3a_{13}{u^1}^2-2(a_{12}+2a_{33})u^1+a_{23})exp(-u^2)+{d{u}^3}^2(a_{11}{u^1}^2-2a_{13}{u}^1+a_{33})+\epsilon {d{u}^0}^2.\end{array}$$

(9)

where $a_{ab}$ are arbitrary functions on $u^0,{\epsilon }^2=1$. To obtain the functions $e_a^{\alpha }$, it is sufficient to consider the components $g_{11},g_{12}$, and $g_{13}$ from system (1). The solution can be represented in the form:

$$e_{\alpha }^a=\left(\begin{array}{ccc}1& 0& 0\\ {u^1}^{2}exp(-u^1)& exp(-u^3)& -2u^{1}exp(-u^3)\\ -u^1& 0& 1\end{array}\right),e_a^{\alpha }=\left(\begin{array}{ccc}1& 0& 0\\ {u^1}^2& exp\left(u^3\right)& 2u^1\\ u^1& 0& 1\end{array}\right).$$

(10)

The lower index numbers the lines. The solution of the system of Equation (7) has been found in [36]. It has the form:

$$A_1={\alpha }_0\left(u^0\right),A_2=({\alpha }_0{u^1}^2+2{\beta }_0\left(u^0\right)u^1+{\gamma }_0\left(u^0\right)),A_3=-({\alpha }_0{u}^1+{\beta }_0).$$

By denoting: ${\alpha }_0={\alpha }_1,{\gamma }_0={\alpha }_2,{\beta }_0=-{\alpha }_3,$ we obtain (8).

3. Maxwell’s Equations

All exact solutions of empty Maxwell’s equations for solvable groups have been found in papers [40,41]. The present paper solves the problem for the group $G_3\left(VIII\right)$.

Consider empty Maxwell’s equations for an admissible electromagnetic field in homogeneous space with a group of motions $G_r$:

$$\frac{1}{\sqrt{-g}}{\left(\sqrt{-g}{F}^{ij}\right)}_{,j}=0.$$

(11)

The metric tensor and the electromagnetic potential are defined by relations (1) and (8). When $i=0$, from Equation (11) it follows:

$$\frac{1}{\sqrt{-g}}{\left(\sqrt{-g}{F}_{0.}^{.\alpha }\right)}_{\alpha }=\frac{1}{e}{\left(e_a^{\alpha }e{\eta }^{ab}{\dot{\alpha }}_b\right)}_{,\alpha }={\rho }_a\frac{\left({\eta }^{ab}\eta {\dot{\alpha }}_b\right)}{\eta }=0({\rho }_a=e_{a,\alpha }^{\alpha }+e_a^{\alpha }{e}_{,\alpha }/e).$$

(12)

Here, it is denoted:

$$g=-det\left|\right|g_{\alpha \beta }\left|\right|=-{\left(\eta e\right)}^2,where{\eta }^2=det\left|\right|{\eta }_{\alpha \beta }\left|\right|,e=det\left|\right|e_{\alpha }^a\left|\right|.$$

Let $i=\alpha$. Then, from Equation (11), it follows:

$$\frac{1}{\eta }{\left(\eta F_{0.}^{.\alpha }\right)}_{,0}=\frac{1}{e}{\left(e{F}^{\beta \alpha }\right)}_{,\beta }\Rightarrow \frac{1}{\eta }{\left(\eta {\eta }^{ab}{e}_a^{\alpha }{\dot{\alpha }}_b\right)}_{,0}=\frac{1}{e}{\left(e_b^{\beta }{\eta }^{ab}{e}_{\tilde{a}}^{\alpha }{e}_{\tilde{b}}^{\gamma }{\eta }^{\tilde{a}\tilde{b}}{F}_{\beta \gamma }e{e}_a^{\nu }\right)}_{,\nu }\Rightarrow$$

(13)

$$e{\left({\dot{\alpha }}_b\eta {\eta }^{ab}\right)}_{,0}=\eta e_{\alpha }^a{\left(e{e}_b^{\beta }{e}_{{\tilde{a}}_1}^{\alpha }{e}_{\tilde{b}}^{\gamma }{F}_{\beta \gamma }\right)}_{|a_1}{\eta }^{a_{1}b}{\eta }^{\tilde{a}\tilde{b}}.$$

(14)

Let us find components of $F_{\alpha \beta }$ using relation (8).

$$F_{\alpha \beta }=(e_{\beta ,\alpha }^a-e_{\beta ,\alpha }^a){\alpha }_a=e_{\beta }^c{e}_c^{\gamma }{e}_{\alpha }^d{e}_d^{\nu }(e_{\gamma ,\nu }^a-e_{\nu ,\gamma }^a){\alpha }_a=e_{\beta }^b{e}_{\alpha }^a{e}_{\gamma }^c(e_{a|b}^{\gamma }-e_{b|a}^{\gamma }){\alpha }_c=e_{\beta }^b{e}_{\alpha }^a{C}_{ba}^c{\alpha }_c.$$

(15)

Then,

$${\left(e{F}^{\alpha \beta }\right)}_{,\beta }={\eta }^{ab}{\eta }^{\tilde{a}\tilde{b}}{C}_{\tilde{b}b}^d{\alpha }_d({\left(e{e}_a^{\alpha }\right)}_{|\tilde{a}}+e{e}_a^{\alpha }{e}_{\tilde{a},\gamma }^{\gamma }).$$

(16)

We present the structural constants of a group $G_3$ in the form:

$$C_{ab}^c=C_{12}^c{\epsilon }_{\tilde{a}\tilde{b}}^{12}+C_{13}^c{\epsilon }_{\tilde{a}\tilde{b}}^{13}+C_{23}^c{\epsilon }_{\tilde{a}\tilde{b}}^{23},$$

(17)

where

$${\epsilon }_{ab}^{AB}={\delta }_a^A{\delta }_b^B-{\delta }_b^A{\delta }_a^B.$$

Let us denote:

$${\sigma }_1=C_{23}^a{\alpha }_a,{\sigma }_2=C_{31}^a{\alpha }_a,{\sigma }_3=C_{12}^a{\alpha }_a;$$

$$\left\{\begin{array}{c}\hfill {\gamma }_1={\sigma }_1{\eta }_{11}+{\sigma }_2{\eta }_{12}+{\sigma }_3{\eta }_{13},\\ \hfill {\gamma }_2={\sigma }_1{\eta }_{12}+{\sigma }_2{\eta }_{22}+{\sigma }_3{\eta }_{23},\\ \hfill {\gamma }_3={\sigma }_1{\eta }_{13}+{\sigma }_2{\eta }_{23}+{\sigma }_3{\eta }_{33}.\end{array}\right.$$

Equation (16) will take the form:

$$\begin{array}{c}\eta {\left(\eta {\eta }^{ab}{\dot{\alpha }}_b\right)}_{,0}={\delta }_1^a({\gamma }_1\left(C_{32}^1\right)-{\gamma }_2(C_{31}^1+{\rho }_3)+{\gamma }_3(C_{21}^1+{\rho }_2))+{\delta }_2^a({\gamma }_1(C_{32}^2+{\rho }_3)+\\ {\gamma }_2{C}_{13}^2-{\gamma }_3(C_{12}^2+{\rho }_1))+{\delta }_3^a(-{\gamma }_1(C_{23}^3+{\rho }_2)+{\gamma }_2(C_{13}^3+{\rho }_1)+{\gamma }_3{C}_{21}^3),\end{array}$$

(18)

$${\rho }_a{\eta }^{ab}{\dot{\alpha }}_b=0.$$

(19)

To decrease the order of Equation (18), we introduce new independent functions:

$$b^a={\delta }_a^c{b}_c=\eta {\eta }^{ab}{\dot{\alpha }}_b\Rightarrow \eta {\dot{\alpha }}_a={\eta }_{ab}{b}^b.$$

(20)

Let us introduce the function:

$$n_{ab}=n_{ab}\left(u^0\right)=\frac{{\eta }_{ab}}{\eta }\Rightarrow det{n}_{ab}=n=\frac{1}{\eta }.$$

(21)

Then, Maxwell’s Equations (18) and (21) take the form of a system of linear algebraic equations on the unknown functions $n_{ab}$:

$$\begin{array}{c}{\dot{b}}^a={\delta }_1^a({\tilde{\gamma }}_1\left(C_{32}^1\right)-{\tilde{\gamma }}_2(C_{31}^1+{\rho }_3)+{\tilde{\gamma }}_3(C_{21}^1+{\rho }_2))+{\delta }_2^a({\tilde{\gamma }}_1(C_{32}^2+{\rho }_3)+\\ {\tilde{\gamma }}_2{C}_{13}^2-{\tilde{\gamma }}_3(C_{12}^2+{\rho }_1))+{\delta }_3^a(-{\tilde{\gamma }}_1(C_{23}^3+{\rho }_2)+{\tilde{\gamma }}_2(C_{13}^3+{\rho }_1)+{\tilde{\gamma }}_3{C}_{21}^3)({\tilde{\gamma }}_a=n{\gamma }_a),\end{array}$$

(22)

$${\dot{\alpha }}_a=n_{ab}{b}^b.$$

(23)

Equation (19):

$${\rho }_a{b}^a=0$$

(24)

is a restriction on the function $b^a$ (if ${\rho }_a\ne 0$). Let us obtain the Maxwell’s equations for the group $G_3\left(VIII\right)$. Non-zero structural constants, in this case, have the form:

$$C_{12}^3=2,C_{13}^1=1,C_{32}^2=1\Rightarrow$$

(25)

From here, it follows that

$${\sigma }_1=-{\alpha }_2,{\sigma }_2=-{\alpha }_1,{\sigma }_3=2{\alpha }_3.$$

Using these relations, we obtain Maxwell’s Equation (18) in the form:

$$\widehat{B}\widehat{n}=\widehat{\omega },$$

(26)

where

$$\widehat{B}=\left(\begin{array}{cccccc}{a}_1& a_2& a_3& 0& 0& 0\\ b_1& b_2& b_3& 0& 0& 0\\ 0& a_1& 0& a_2& a_3& 0\\ 0& b_1& 0& b_2& b_3& 0\\ 0& 0& a_1& 0& a_2& a_3\\ 0& 0& b_1& 0& b_2& b_3\end{array}\right),$$

(27)

$${\widehat{n}}^T=(n_{11},n_{12},n_{13},n_{22},n_{23},n_{33});{\widehat{\omega }}^T=(-{\dot{b}}_2,{\dot{a}}_2,-{\dot{b}}_1,{\dot{a}}_1,\frac{{\dot{b}}_3}{2},-\frac{{\dot{a}}_3}{2}).$$

Hereafter, the following notations are used:

$${\alpha }_1=a_2,{\alpha }_2=a_1,{\alpha }_3=-\frac{a_3}{2}.$$

(28)

Let us find the algebraic complement of the matrix $\widehat{B}:$

$$\widehat{V}=\left(\begin{array}{cccccc}{b}_1{v}_1^2& -a_1{v}_1^2& b_2{v}_1^2& -a_2{v}_1^2& b_3{v}_1^2& -a_3{V}_1^2\\ b_1{v}_1{v}_2& -a_1{v}_1{v}_2& b_2{v}_1{v}_2& -a_2{v}_1{v}_2& b_3{v}_1{v}_2& -a_3{v}_1{v}_2\\ b_1{v}_1{v}_3& -a_1{v}_1{v}_3& b_2{v}_{1}V{v}_3& -a_2{v}_1{v}_3& b_3{v}_1{v}_3& -a_3{v}_1{v}_3\\ b_1{v}_2^2& -a_1{v}_2^2& b_2{v}_2^2& -a_2{v}_2^2& b_3{v}_2^2& -a_3{v}_2^2\\ b_1{v}_2{v}_3& -a_1{v}_2{v}_3& b_2{v}_2{v}_3& -a_2{v}_2{v}_3& b_3{v}_2{v}_3& -a_3{v}_2{v}_3\\ b_1{v}_3^2& -a_1{v}_3^2& b_2{v}_3^2& -a_2{v}_3^2& b_3{v}_3^2& -a_3{V}_3^2\end{array}\right)$$

(29)

$$v_1=a_2{b}_3-a_3{b}_2,v_2=a_3{b}_2-a_2{b}_3,v_3=a_1{b}_2-a_2{b}_1.$$

As $\widehat{B}$ is a singular matrix, $\widehat{V}$ is the annulling matrix for $\widehat{B}$:

$$\widehat{V}\widehat{B}=0.$$

(30)

Therefore, when ${v_1}^2+{v_2}^2+{v_3}^2\ne 0$, one of the equations from system (26) can be replaced by the equation:

$${a_3}^2+{b_3}^2=4(a_1{a}_2+b_1{b}_2+c)(c=const).$$

(31)

Depending on the rank of the matrix $\widehat{B}$, one or more functions $n_{ab}\left(u^0\right)$ are independent. It is possible to express the remaining functions $n_{ab}$ through the functions $a_a,b_a$. To find non-equivalent solutions of the system (26), one should consider the following variants:

$1.a_1\ne 0;2.a_1=0,a_2\ne 0;3.a_1=a_2=0,a_3\ne 0.$ Taking this observation into account, let us consider all non-equivalent options.

4. Solutions of Maxwell Equations

Since the functions $a_a$ satisfy the condition:

$$a_1^2+a_2^2+a_3^2\ne 0,$$

the $rank$ of matrix (29) cannot be less than three if

$$v_1^2+v_2^2+v_3^2\ne 0\Rightarrow rank\left|\right|\widehat{B}\left|\right|=5.$$

In order to obtain a complete solution to the classification problem, it is necessary:

(I) To consider all non-equivalent variants with non-zero minors of $rank=5$ of the matrix $\widehat{B}$;

(II) To consider all non-equivalent variants under the condition: $v_a=0(rank\le 3).$

The components of the matrix $\widehat{\eta }$ and the functions ${\alpha }_a$ are given by formulae (21) and (28). In view of these circumstances, let us list all exact solutions of empty Maxwell equations for PSS of type VIII.

I. $rank\left|\right|\widehat{B}\left|\right|=5$.

1. $a_1{v}_1\ne 0\Rightarrow$ the minor ${\widehat{B}}_{12}$ and its inverse matrix $\widehat{P}={\widehat{B}}_{12}^{-1}$ have the form:

$${\widehat{B}}_{12}=\left(\begin{array}{ccccc}{a}_2& a_3& 0& 0& 0\\ a_1& 0& a_2& a_3& 0\\ b_1& 0& b_2& b_3& 0\\ 0& a_1& 0& a_2& a_3\\ 0& b_1& 0& b_2& b_3\end{array}\right),$$

(32)

$$\widehat{P}=\left(\begin{array}{ccccc}-\frac{v_2}{a_1{v}_1}& -\frac{a_3{b}_2}{{\alpha }_1{v}_1}& \frac{a_2{a}_3}{a_1{v}_1}& -\frac{a_3{b}_3}{a_1{v}_1}& \frac{a_3^2}{a_1{v}_1}\\ -\frac{V_3}{a_1{v}_1}& \frac{a_2{b}_2}{a_1{v}_1}& -\frac{a_2^2}{a_1{v}_1}& \frac{a_2{b}_3}{a_1{v}_1}& -\frac{a_2{a}_3}{a_1{v}_1}\\ -\frac{V_2^2}{a_1{v}_1^2}& \frac{(a_3{b}_1{v}_1-a_2{b}_3{v}_3)}{a_1{v}_1^2}& \frac{a_3(a_2{v}_2-a_1{v}_1)}{a_1{v}_1^2}& -\frac{a_3{b}_3{v}_2}{a_1{v}_1^2}& \frac{a_2^2{v}_2}{a_1{v}_1^2}\\ -\frac{v_2{v}_3}{a_1{v}_1^2}& \frac{a_2{b}_2{v}_2}{a_1{v}_1^2}& -\frac{a_2^2{V}_2}{a_1{v}_1^2}& -\frac{a_3{b}_3{v}_3}{a_1{v}_1^2}& \frac{a_3^2{v}_3}{a_1{v}_1^2}\\ -\frac{v_3^2}{a_1{v}_1^2}& \frac{a_2{b}_2{v}_3}{a_1{v}_1^2}& -\frac{a_2^2{v}_3^2}{a_1{v}_1^2}& \frac{(a_3{b}_2{v}_3-a_2{b}_1{v}_1)}{a_1{v}_1^2}& \frac{a_2(a_1{v}_1-a_3{v}_3)}{a_1{v}_1^2}\end{array}\right)$$

(33)

Then, the solution to Equation (26) is as follows:

$${\widehat{n}}_1={\widehat{P}}_1{\widehat{\omega }}_1,$$

(34)

where

$${\widehat{n}}_1^T=(n_{12},n_{13},n_{22},n_{23},n_{33});$$

$${\widehat{\omega }}_1^T=(-({\dot{b}}_2+a_1{n}_{11}),-{\dot{b}}_1,{\dot{a}}_1,\frac{{\dot{b}}_3}{2},-\frac{{\dot{a}}_3}{2}).$$

Functions $n_{11},$$a_a$, and $b_a$ are arbitrary functions of $u^0$ that obey condition (31).

2. $a_2{v}_1\ne 0$. Obviously, we obtain a non-equivalent solution to the previous one only if $a_1=0$. In order to implement the classification, a similar choice should be made for all other variants. The matrix ${\widehat{B}}_{14}$ and its inverse matrix ${\widehat{P}}_2={\widehat{B}}_{14}^{-1}$ have the form:

$${\widehat{B}}_{14}=\left(\begin{array}{ccccc}{a}_2& {\alpha }_3& 0& 0& 0\\ b_2& b_3& 0& 0& 0\\ 0& 0& a_2& a_3& 0\\ 0& 0& 0& a_2& a_3\\ 0& b_1& 0& b_2& b_3\end{array}\right),{\widehat{P}}_2=\left(\begin{array}{ccccc}\frac{b_3}{v_1}& -\frac{a_3}{v_1}& 0& 0& 0\\ -\frac{b_2}{v_1}& \frac{{\alpha }_2}{v_1}& 0& 0& 0\\ \frac{a_3^2{b}_1{b}_2}{a_2{v}_1^2}& -\frac{a_3^2{b}_1}{v_1^2}& \frac{1}{a_2}& -\frac{a_3{b}_3}{a_2{v}_1}& \frac{a_3^2}{a_2{v}_1}\\ -\frac{a_3{b}_1{b}_2}{v_1^2}& \frac{a_2{a}_3{b}_1}{v_1^2}& 0& \frac{b_3}{v_1}& -\frac{a_3}{v_1}\\ \frac{a_2{b}_1{b}_2}{v_1^2}& -\frac{a_2^2{b}_1}{v_1^2}& 0& -\frac{b_2}{v_1}& \frac{a_2}{v_1}\end{array}\right)$$

(35)

Then, the solution to Equation (26) is as follows:

$${\widehat{n}}_2={\widehat{P}}_2{\widehat{\omega }}_2,$$

(36)

where

$${\widehat{n}}_2^T=(n_{12},n_{13},n_{22},n_{23},n_{33});$$

$${\widehat{\omega }}_2=(-{\dot{b}}_2,({\dot{a}}_2-b_1{n}_{11}),-{\dot{b}}_1,\frac{{\dot{b}}_3}{2},-\frac{{\dot{a}}_3}{2}).$$

Functions $n_{11}$, $a_a$, and ${\beta }_a$ are arbitrary functions of $u^0$ that obey condition (31).

3. $a_3{v}_1\ne 0\Rightarrow a_1=a_2=0\Rightarrow$ the minor ${\widehat{B}}_{16}^{-1}$ and its inverse matrix ${\widehat{P}}_3={\widehat{B}}_{16}^{-1}$ have the form:

$${\widehat{B}}_{16}=\left(\begin{array}{ccccc}0& a_3& 0& 0& 0\\ b_2& b_3& 0& 0& 0\\ 0& 0& 0& a_3& 0\\ b_1& 0& b_2& b_3& 0\\ 0& 0& 0& 0& a_3\end{array}\right),{\widehat{P}}_3=\left(\begin{array}{ccccc}-\frac{b_3}{a_3{b}_2}& \frac{1}{b_3}& 0& 0& 0\\ \frac{1}{a_3}& 0& 0& 0& 0\\ \frac{b_1{b}_3}{a_3{b}_2^2}& -\frac{b_1}{b_2^2}& -\frac{b_3}{b_2{a}_3}& \frac{1}{b_2}& 0\\ 0& 0& \frac{1}{a_3}& 0& 0\\ 0& 0& 0& 0& \frac{1}{a_3}\end{array}\right)$$

(37)

Then, the solution to Equation (26) is as follows:

$${\widehat{n}}_3={\widehat{P}}_3{\widehat{\omega }}_3,$$

(38)

where

$${\widehat{n}}_3^T=(n_{12},n_{13},n_{22},n_{23},n_{33}),{\widehat{\omega }}_3=(-{\dot{b}}_2,({\dot{a}}_2-b_1{n}_{11}),-{\dot{b}}_1,0,\frac{{\dot{b}}_3}{2}).$$

(39)

4. ${\alpha }_1{v}_2\ne 0,\Rightarrow v_1=0\Rightarrow$ the minor ${\widehat{B}}_{24}^{-1}$ and its inverse matrix ${\widehat{P}}_4={\widehat{B}}_{24}^{-1}$ have the form:

$${\widehat{B}}_{24}=\left(\begin{array}{ccccc}{a}_1& a_2& a_3& 0& 0\\ 0& a_1& 0& a_3& 0\\ 0& b_1& 0& b_3& 0\\ 0& 0& a_1& a_2& a_3\\ 0& 0& b_1& b_2& b_3\end{array}\right),{\widehat{P}}_4=\left(\begin{array}{ccccc}\frac{1}{{\alpha }_1}& \frac{a_2{b}_3}{a_1{v}_2}& -\frac{a_2{a}_3}{a_1{v}_2}& \frac{a_3{b}_3}{a_1{v}_2}& -\frac{a_3^2}{a_1{v}_2}\\ 0& -\frac{b_3}{v_2}& \frac{a_3}{v_2}& 0& 0\\ 0& 0& 0& -\frac{b_3}{v_2}& \frac{a_3}{v_2}\\ 0& \frac{b_1}{v_2}& -\frac{a_1}{v_2}& 0& 0\\ 0& \frac{b_1{v}_3}{{v_2}^2}& -\frac{a_1{v}_3}{v_2}& \frac{b_1}{v_2}& -\frac{a_1}{v_2}\end{array}\right)$$

(40)

Then, the solution to Equation (26) is as follows:

$${\widehat{n}}_4={\widehat{P}}_4{\widehat{\omega }}_4,$$

(41)

where

$${\widehat{n}}_4^T=(n_{11},n_{12},n_{13},n_{23},n_{33}),{\widehat{\omega }}_4=(-{\dot{b}}_2,-({\dot{b}}_1+a_2{n}_{22}),({\dot{a}}_1-b_2{n}_{22}),\frac{{\dot{b}}_3}{2},-\frac{{\dot{a}}_3}{2})$$

(42)

Functions $n_{22},$$a_a$, and ${\beta }_a$ are arbitrary functions of $u^0$ that obey condition (31) and $a_2{\beta }_3=a_3{\beta }_2.$

5. ${\alpha }_2{V}_2\ne 0,\Rightarrow a_1=V_1=0\Rightarrow$ the minor ${\widehat{B}}_{44}^{-1}$ and its inverse matrix ${\widehat{P}}_5={\widehat{W}}_{44}^{-1}$ have the form:

$${\widehat{B}}_{44}=\left(\begin{array}{ccccc}0& a_2& a_3& 0& 0\\ b_1& b_2& b_3& 0& 0\\ 0& 0& 0& a_3& 0\\ 0& 0& 0& a_2& a_3\\ 0& 0& b_1& b_2& b_3\end{array}\right),\widehat{P_5}=\left(\begin{array}{ccccc}-\frac{b_2}{b_1{a}_2}& \frac{1}{b_1}& 0& 0& 0\\ \frac{1}{a_2}& 0& 0& \frac{b_3}{a_2{b}_1}& -\frac{a_3}{a_2{b}_1}\\ 0& 0& 0& -\frac{b_3}{a_3{b}_1}& \frac{1}{b_1}\\ 0& 0& \frac{1}{a_3}& 0& 0\\ 0& 0& -\frac{a_2}{a_3^2}& \frac{1}{a_3}& 0\end{array}\right)$$

(43)

Then, the solution to Equation (26) is as follows:

$${\widehat{n}}_5={\widehat{P}}_2{\widehat{\omega }}_5,$$

(44)

where

$${\widehat{n}}_5^T=(n_{11},n_{12},n_{13},n_{23},n_{33});{\widehat{\omega }}_5=(-{\dot{b}}_2,{\dot{\alpha }}_2,-({\dot{b}}_1+a_2{n}_{22}),\frac{{\dot{b}}_3}{2},-\frac{{\dot{\alpha }}_3}{2})$$

(45)

Functions $n_{22},$$a_a$, and $b_a$ are arbitrary functions of $u^0$ that obey condition (31) and $a_2{b}_3=a_3{b}_2.$

6. $a_3{v}_2\ne 0,v_1=0,\Rightarrow a_1=a_2=b_2=0$. From condition (31) it follows:

$$a_3=ccos2\phi ,b_3=csin2\phi ,$$

where $\phi$ is an arbitrary function of $u^0$. The minor ${\widehat{B}}_{46}^{-1}$ and its inverse matrix ${\widehat{P}}_6={\widehat{B}}_{46}^{-1}$ have the form:

$${\widehat{B}}_{64}=\left(\begin{array}{ccccc}0& 0& ccos\phi & 0& 0\\ b_1& 0& & 0& 0\\ 0& 0& 0& ccos\phi & 0\\ 0& b_1& 0& csin\phi & 0\\ 0& 0& 0& 0& ccos\phi \end{array}\right),{\widehat{\mathsf{\Omega }}}_6=\left(\begin{array}{ccccc}-\frac{sin\phi }{b_{1}cos\phi }& \frac{1}{b_1}& 0& 0& 0\\ 0& 0& -\frac{sin\phi }{b_{1}cos\phi }& \frac{1}{b_1}& 0\\ \frac{1}{ccos\phi }& 0& 0& 0& 0\\ 0& 0& \frac{1}{ccos\phi }& 0& 0\\ 0& 0& 0& 0& \frac{1}{ccos\phi }\end{array}\right).$$

(46)

Then, the solution to Equation (26) is as follows:

$${\widehat{n}}_6={\widehat{P}}_6{\widehat{\omega }}_6,$$

(47)

where

$${\widehat{n}}_6^T=(n_{11},n_{12},n_{13},n_{23},n_{33});{\widehat{\omega }}_6=(0,0,-{\dot{b}}_1,0,c\dot{\phi }cos\phi ).$$

Functions $n_{22}$, $b_1$, and $\phi$ are arbitrary functions of $u^0$.

7. $a_1{v}_3\ne 0$$\Rightarrow v_1=v_2=0,$ otherwise, we obtain a solution equivalent to the previous ones. As $v_3\ne 0\Rightarrow$$a_3=b_3=0$, the minor ${\widehat{B}}_{26}$ and its inverse matrix ${\widehat{P}}_7={\widehat{B}}_{26}^{-1}$ have the form:

$${\widehat{B}}_{26}=\left(\begin{array}{ccccc}{\alpha }_1& {\alpha }_2& 0& 0& 0\\ 0& {\alpha }_1& 0& a_2& 0\\ 0& b_1& 0& b_2& 0\\ 0& 0& {\alpha }_1& 0& {\alpha }_2\\ 0& 0& b_1& 0& b_2\end{array}\right),{\widehat{P}}_7=\left(\begin{array}{ccccc}\frac{1}{{\alpha }_1}& -\frac{{\alpha }_2{b}_2}{{\alpha }_1{v}_3}& \frac{{\alpha }_2^2}{{\alpha }_1{v}_3}& 0& 0\\ 0& \frac{b_2}{v_3}& -\frac{{\alpha }_2}{v_3}& 0& 0\\ 0& 0& 0& \frac{b_2}{v_3}& -\frac{{\alpha }_2}{v_3}\\ 0& -\frac{b_1}{v_3}& \frac{{\alpha }_1}{v_3}& 0& 0\\ 0& 0& 0& -\frac{b_1}{v_3}& \frac{{\alpha }_1}{v_3}\end{array}\right).$$

(48)

Then, the solution to Equation (26) is as follows:

$${\widehat{n}}_{3a}={\widehat{P}}_7{\widehat{\omega }}_7.$$

(49)

where

$${\widehat{n}}_7^T=(n_{11},n_{12},n_{13},n_{22},n_{23});$$

$${\widehat{\omega }}_7^T=(-{\dot{b}}_2,-{\dot{b}}_1,{\dot{a}}_1,0,0).$$

8. $a_2{v}_3\ne 0$$\Rightarrow a_1=v_1=v_2=0,$ otherwise, we obtain a solution equivalent to the previous ones. As $v_3\ne 0\Rightarrow$${\alpha }_3=b_3=0,$ the minor ${\widehat{B}}_{64}$ and its inverse matrix ${\widehat{P}}_8={\widehat{B}}_{64}^{-1}$ have the form:

$${\widehat{B}}_{64}=\left(\begin{array}{ccccc}0& {\alpha }_2& 0& 0& 0\\ b_1& {\alpha }_2& 0& 0& 0\\ 0& 0& 0& a_2& 0\\ 0& 0& 0& 0& {\alpha }_2\\ 0& 0& b_1& 0& b_2\end{array}\right),{\widehat{P}}_8=\left(\begin{array}{ccccc}-\frac{b_2}{a_2{b}_1}& -\frac{1}{b_1}& 0& 0& 0\\ \frac{1}{a_2}& 0& 0& 0& 0\\ 0& 0& 0& -\frac{b_2}{b_1{a}_2}& \frac{1}{b_1}\\ 0& 0& \frac{1}{a_2}& 0& 0\\ 0& 0& 0& \frac{1}{a_2}& 0\end{array}\right).$$

(50)

Then, the solution to Equation (26) is as follows:

$${\widehat{n}}_8={\widehat{P}}_8{\widehat{\omega }}_8.$$

(51)

where

$${\widehat{n}}_8^T=(n_{11},n_{12},n_{13},n_{22},n_{23}),{\widehat{\omega }}_8^T=(-{\dot{b}}_2,-{\dot{b}}_1,0,0,0).$$

Functions $n_{33},$$a_2{\beta }_1$, and ${\beta }_2$ are arbitrary functions of $u^0$ that obey condition (31).

II. $rank\left|\right|\widehat{B}\left|\right|<5$

9. $v_a=0$. Let us represent the system of Maxwell’s equations in the form:

$$\widehat{Q}{\widehat{n}}_I={\widehat{\omega }}_I,$$

(52)

where

$$\widehat{Q}=\left(\begin{array}{cccccc}{a}_1& a_2& a_3& 0& 0& 0\\ 0& a_1& 0& a_2& a_3& 0\\ 0& 0& a_1& 0& a_2& a_3\\ b_1& b_2& b_3& 0& 0& 0\\ 0& b_1& 0& b_2& b_3& 0\\ 0& 0& b_1& 0& b_2& b_3\end{array}\right),$$

$${\widehat{\omega }}_I=({\widehat{\omega }}_{\beta },{\widehat{\omega }}_{\alpha });{\widehat{\omega }}_{\beta }^T=(-{\dot{b}}_2,-{\dot{b}}_1,\frac{{\dot{b}}_3}{2}),{\widehat{\omega }}_{\alpha }^T=({\dot{a}}_2,{\dot{a}}_1,-\frac{{\dot{a}}_3}{2})$$

$${\widehat{n}}_I=({\widehat{n}}_{\alpha },{\widehat{b}}_{\alpha });{\widehat{n}}_{\alpha }^T=(n_{11},n_{12},n_{13}),{\widehat{n}}_{\beta }^T=(n_{22},n_{23},n_{33})$$

Consider all possible options.

(a) $a_1\ne 0\Rightarrow b_a=\frac{{\alpha }_a{b}_1}{{\alpha }_1}$. Maxwell’s Equation (52) take the form:

$${\widehat{B}}_I{\widehat{n}}_{\alpha }=({\widehat{\omega }}_{\beta }-{\widehat{B}}_{II}{\widehat{n}}_{\beta })\Rightarrow {\widehat{n}}_{\alpha }={\widehat{B}}_I^{-1}({\widehat{\omega }}_{\beta }-{\widehat{B}}_{II}{\widehat{n}}_{\beta }),$$

$$b_1{\widehat{B}}_I{\widehat{n}}_{\alpha }=a_1{\widehat{\omega }}_{\alpha }-b_1{\widehat{B}}_{II}{\widehat{n}}_{\beta }\Rightarrow b_1{\widehat{\omega }}_{\beta }-a_1{\widehat{\omega }}_{\alpha }=0\Rightarrow$$

$$\left\{\begin{array}{c}\hfill a_1{\dot{a}}_2+b_1{\dot{b}}_2=0,\\ \hfill a_1{\dot{a}}_3+b_1{\dot{b}}_3=0,\\ \hfill a_1{\dot{a}}_1+b_1{\dot{b}}_1=0.\end{array}\right.$$

(53)

Here,

$${\widehat{B}}_I=\left(\begin{array}{cccc}{a}_1& a_2& a_3& \\ 0& a_1& 0\\ 0& 0& a_1\end{array}\right),{\widehat{B}}_I^{-1}=\left(\begin{array}{cccc}\frac{1}{a_1}& -\frac{a_2}{a_1^2}& -\frac{a_3}{a_1^2}& \\ 0& \frac{1}{a_1}& 0\\ 0& 0& \frac{1}{a_1}\end{array}\right),{\widehat{B}}_{II}=\left(\begin{array}{cccc}0& 0& 0& \\ a_2& a_3& 0\\ 0& a_2& a_3,\end{array}\right)$$

From the last equation of system (53) it follows that

$$a_1=e_{0}sin\phi ,b_1=e_{0}cos\phi ,e_0=const.$$

Thus, $b_2=a_2\frac{cos\phi }{sin\phi }$ and $b_3=a_3\frac{cos\phi }{sin\phi }$, and from the previous equations, it follows that

$$a_a=e_0{q}_{a}sin\phi ,b_a=e_0{q}_{a}cos\phi ,q_a=const,q_1=1.$$

Then, matrices ${\widehat{B}}_I,{\widehat{B}}_I^{-1}$, and ${\widehat{B}}_{II}$ and line ${\widehat{\omega }}^T$ take the form:

$${\widehat{B}}_I={\widehat{w}}_{1}sin\phi ,{\widehat{B}}_I^{-1}=\frac{1}{sin\phi }{\widehat{w}}_1^{-1},{\widehat{B}}_{II}={\widehat{w}}_{2}sin\phi .$$

$${\widehat{w}}_1=\left(\begin{array}{cccc}1& q_2& q_3& \\ 0& 1& 0\\ 0& 0& 1\end{array}\right),{\widehat{w}}_1^{-1}=\left(\begin{array}{ccc}1& -q_2& -q_3\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),{\widehat{w}}_2=\left(\begin{array}{cccc}0& 0& 0& \\ q_2& q_3& 0\\ 0& q_2& q_3,\end{array}\right)$$

$${\widehat{\omega }}_{\beta }^T=\dot{\phi }{\widehat{c}}^T=\dot{\phi }sin\phi (q_2,1,-\frac{q_3}{2})$$

Then, the solution to Equation (26) is as follows:

$${\widehat{n}}_{\alpha }={\widehat{w}}^{-1}(\dot{\phi }\widehat{c}-\widehat{q}{\widehat{n}}_{\beta })$$

(b) $a_1=0\Rightarrow a_2\ne 0$. Let us use the previous results, in which the indices 1 and 2 are reversed: $1\iff 2$. The solution of Maxwell’s equation has the form:

$${\widehat{n}}_{\alpha }={\widehat{w}}^{-1}(\dot{\phi }\widehat{c}-\widehat{q}{\widehat{n}}_{\beta })$$

$${\widehat{n}}_{\alpha }^T=(n_{22},n_{12},n_{23}),{\widehat{n}}_{\beta }^T=(n_{11},n_{13},n_{33}),$$

$${\widehat{w}}^{-1}=\left(\begin{array}{ccc}1& 0& -q\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\widehat{q}=\left(\begin{array}{cccc}0& 0& 0& \\ 0& q& 0\\ 0& 0& q,\end{array}\right),{\widehat{c}}^T=(0,1,-\frac{q}{2}).$$

$$a_2=e_{0}sin\phi ,b_2=e_{0}cos\phi ,a_3=e_{0}qsin\phi ,b_3=e_{0}qcos\phi ,q=const,\phi =\phi \left(u^0\right).$$

(c) $a_3\ne 0.$ The solutions, which are not equivalent to the previous ones, can be obtained under the conditions $a_1=a_2=0\Rightarrow b_1=b_2=0$. From Maxwell’s equations it follows that

$$a_3{n}_{13}=a_3{n}_{23}=0,a_3{n}_{33}=\frac{{\dot{b}}_3}{2},b_3{n}_{33}=-\frac{{\dot{a}}_3}{2}\Rightarrow a_3{\dot{a}}_3+b_3{\dot{b}}_3=0.$$

The solution has the form

$$n_{33}=\dot{\phi },n_{13}=n_{23}=a_1=a_2=b_1=b_2=0,a_3=qcos2\phi ,b_3=qsin2\phi .$$

Functions $\phi ,n_{11},n_{12}$, and $n_{22}$ are arbitrary functions on $u^0$.

5. Conclusions

In previous works [40,41,42], all non-equivalent solutions of Maxwell’s empty equations for admissible electromagnetic fields in homogeneous space-time metrics of all types according to Bianchi’s classification (except type $VIII$) were found. The present work completes the first stage of the classification problem formulated in the introduction. The next step is the classification of the corresponding exact solutions of the Einstein–Maxwell equations. All solutions obtained in the completed classification have a form suitable for further use and have sufficient arbitrariness so that the Einstein–Maxwell equations have nontrivial solutions. The use of the triad of frame vectors (see [43]) allows us to reduce the Einstein–Maxwell equations with the energy-momentum tensor of the admissible electromagnetic field to an overcrowded system of ordinary differential equations. To perform the classification, we need to study the coexistence conditions of these systems of equations. It is possible to use additional symmetries of homogeneous spaces and admissible electromagnetic fields (see [38]). In the future, we will begin to solve this classification problem.