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

Abstract

This paper classifies the exact solutions of the Maxwell vacuum equations for the case when the electromagnetic fields and metrics of homogeneous spaces are invariant with respect to the motion group $G_3\left(IX\right)$. All the appropriate non-equivalent exact solutions of the Maxwell vacuum equations are found.

1. Introduction

All the known methods of integration of the main differential equations of mathematical physics are based on the complete reduction of these equations to a system of ordinary differential equations. The reduction is carried out using symmetry operators. For the equations of motion of a classical or quantum sample particle in external electromagnetic and gravitational fields, the symmetry operators are integrals of motion. It is known that a necessary condition for the existence of integrals of motion is the existence of the spacetime symmetry given by the Killing fields.

Thus, the problem of exact integration is closely related to the study of the spacetime symmetry. At present, two methods of the exact integration of equations of motion are known. These are the methods of commutative (CIM) and noncommutative (NCIM) integration. The first method is based on the theory of the complete separation of variables, and it is applicable in Stackel spaces. Stackel spaces admit complete sets consisting of mutually commuting Killing fields. The theory of Stackel spaces was developed in [1,2,3,4,5,6,7]. A description of the theory and a detailed bibliography can be found in [8,9,10] (see also, [11]). Solutions of the field equations, which are still used widely in the theory of gravitation, have been constructed on the basis of Stackel spaces. These solutions are often used in the study of various effects in gravitational fields (see, for example, [12,13,14,15,16,17,18,19,20,21,22,23,24,25]).

The second method (NCI method) is based on the use of noncommutative algebras of symmetry operators that are linear in moments and constructed using vector Killing fields. This method was proposed in [26]. The development of the method and its application to gravity theory can be found in [27,28,29,30].

As in Stackel spaces, in the spaces with a noncommutative group of motions, the equations of motion of a test particle admit the complete reduction to a system of ordinary differential equations. Therefore, we call the spacetime manifolds admitting noncommutative groups $G_r,r\ge 3$ post-Stackel spaces (PSS).

By analogy with Stackel spaces, we call the PSS non-isotropic, if a group $G_r$ (or its subgroup of rank 3) acts transitively on a non-isotropic hypersurface of spacetime, or isotropic, if the hypersurface is isotropic. For non-isotropic post-Stackel spaces, we also use the term “homogeneous post-Stackel spaces (HPSS)”.

The same classification problems can be considered for the PSS as for the Stackel spaces. For example, in the papers [9,10], a complete classification was given for the case when the Hamilton–Jacobi equation for a charged test particle admits the complete separation of variables in the external electromagnetic field. A similar classification problem was solved for the PSS as well. In [31], PSS with transitive four-parameter groups of motions were considered; in [32], HPSS were considered (see also, [33]); in [34], PSS with groups acting on isotropic hypersurfaces of transitivity were considered. PSS with four-parameter groups of motions were considered in [35], provided that these groups have transitive three-parameter subgroups. Thus, the potentials of all admissible electromagnetic fields have been found, for which the Hamilton–Jacobi and Klein–Gordon–Fock equations have algebras of symmetry operators given by the groups of motions of post-Stackel spaces. It was proved that the Klein–Gordon–Fock equation admits the algebra of symmetry operators given by groups of motions of PSS, if and only if the Hamilton–Jacobi equations admit the appropriate algebra of the integrals of motion.

The next classification problem is the classification of the electrovacuum solutions of the Einstein–Maxwell equations for the case when the CIM and NCIM methods are applicable. During the century-long history of general relativity, many exact solutions of the vacuum and electrovacuum Einstein equations have been found (see, for example, [36]). Nevertheless, this problem remains relevant. The main purpose of the classification is not so much to find new exact solutions but to list all the gravitational and electromagnetic fields in which the equations of motion of test particles can be exactly integrated or at least reduced to systems of ordinary differential equations. This problem is divided into two stages.

In the first stage, all the non-equivalent classes of solutions of the vacuum Maxwell equations for the potentials of admissible electromagnetic fields are found. In the second stage, the obtained classification is used to classify the corresponding electrovacuum spaces. Historically, for Stackel spaces this problem was solved before the problem of the first stage (see the bibliography given in [9,10,37]). The present article is devoted to solving the first stage of this classification problem. All the non-equivalent solutions of empty Maxwell equations in homogeneous spaces of type IX, according to Bianchi’s classification, are found.

2. Admissible Electromagnetic Fields in Homogeneous Spaces

There are two definitions of homogeneous spaces. According to the first, a spacetime $V_4$ is homogeneous if its subspace $V_3$, endowed with the Euclidean space signature, admits coordinate transformations (forming the group $G_3\left(N\right)$ of motions of spaces $V_4$) that enable the connection of any two points in $V_3$ (see [38]). This definition directly implies that the metric of the $V_4$ in the semi-geodesic coordinate system $\left[u^i\right]$ can be represented as follows:

$$d{s}^2=-{d{u}^0}^2+{\eta }_{ab}{l}_{\alpha }^a{l}_{\beta }^{b}d{u}^{\alpha }d{u}^{\beta },g_{ij}=-{\delta }_i^0{\delta }_j^0+{\delta }_i^a{\delta }_j^b,$$

(1)

$$e_{\alpha }^a{e}_{\beta }^b{\eta }_{ab}\left(u^0\right),det\left|{\eta }_{ab}\right|>0e_{\alpha ,0}^a=0.$$

The coordinate indices of the variables of the semi-geodesic coordinate system are denoted by lower case Latin letters: $i,j,k=0,1\dots 3$. The coordinate indices of the variables of the local coordinate system on the hypersurface $V_3$ are denoted by lower case Greek letters: $\alpha ,\beta ,\gamma =1,\dots 3.$ The time variable is denoted by a 0 index. The group indices and indices of nonholonomic frame are denoted by $a,b,c=1,\dots 3$. The summation is performed over the repeated upper and lower indices within the index range.

The 1-form $e_{\alpha }^{a}d{u}^{\alpha }$ is invariant under the acting of the group $G_3\left(N\right)$. The vectors of the frame $e_{\alpha }^a$ define a nonholonomic coordinate system in $V_3$. The dual triplet of vectors

$$e_a^{\alpha },e_a^{\alpha }{e}_{\alpha }^b={\delta }_a^b,e_a^{\alpha }{e}_{\beta }^a={\delta }_{\beta }^{\alpha }$$

defines the operators of the group algebra:

$${\widehat{Y}}_a=e_a^{\alpha }{\partial }_a,[{\widehat{Y}}_a,{\widehat{Y}}_b]=C_{ab}^c{\widehat{Y}}_c.$$

(2)

According to another definition, space–time $V_4$ is homogeneous if it admits a three-parameter group of motions $G_3\left(N\right)$, whose hypersurface $V_3$ of transitivity has the Euclidean space signature. The Killing vector fields ${\xi }_a^{\alpha }$ and their dual vector fields ${\xi }_{\alpha }^a$ form another frame of the space $V_3$ and another representation of the algebra of the group $G_3$. The vector fields ${\xi }_a^{\alpha }$ satisfy the Killing equations,

$$g_{,\gamma }^{\alpha \beta }{\xi }_a^{\gamma }=g^{\alpha \gamma }{\xi }_{a,\gamma }^{\beta }+g^{\beta \gamma }{\xi }_{a,\gamma }^{\alpha },$$

(3)

and set the infinitesimal group operators of the algebra $G_3$ as

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

(4)

Let us consider an electromagnetic field with potential $A_i$. For a charged test particle, moving in this external electromagnetic field, it has been proved that the Hamilton–Jacobi equation

$$g^{ij}{P}_i{P}_j=m,P_i=p_i+A_i$$

(5)

and the Klein–Gordon–Fock equation

$$\widehat{H}\phi =\left(g^{ij}{\widehat{P}}_i{\widehat{P}}_j\right)\phi =m^2\phi ,{\widehat{P}}_k={\widehat{p}}_k+A_k$$

(6)

admit the integrals of motion, which are given by the Killing vectors

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

if and only if the conditions

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

(7)

are satisfied (see paper [32]). Here, $p_i={\partial }_i\phi$; ${\widehat{p}}_k=-l{\widehat{\nabla }}_k;$ (${\widehat{\nabla }}_k$ is the covariant derivative operator corresponding to the partial derivative operator-${\widehat{\partial }}_i$ in the coordinate field $u^i),$ and $\phi$ is a scalar function of a particle with mass m; ${\tilde{\mathbf{A}}}_a={\xi }_a^{\alpha }{A}_{\alpha }.$

The electromagnetic field, whose potential satisfies condition (7), is called admissible. All admissible electromagnetic fields for the groups of motion $G_r\left(N\right)(r\le 4)$ acting transitively on the hypersurfaces of the spacetime have been found in [32,33,34,35].

Solutions of the set of equations (7) for HPSS of type $IX$ have the form:

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

(8)

To prove this, let us find the frame vector. We use the metric tensor of $IX$-type space by Bianchi, found in Petrov’s book [39]. As is known, the Bianchi type IX metric contains as a special case the space of constant positive curvature and, therefore, is of special interest for cosmology.

$${ds}^2={d{u}^1}^2[a_{11}-(a_{12}cos2u^3+a_{22}sin2u^3)]+2d{u}^{1}d{u}^3((a_{13}cos{u}^3-a_{23}sin{u}^3)+$$

(9)

$$+2d{u}^{1}d{u}^2[(a_{13}cos{u}^3-a_{23}sin{u}^3)cos{u}^1+(a_{12}cos2u^3-a_{22}sin2u^3)sin{u}^1]$$

$$+{d{u}^2}^2[a_{33}{cos{u}^1}^2+(a_{23}cos{u}^3+a_{13}sin{u}^3)sin2u^1+(a_{12}sin2u^3+a_{22}cos2u^3+a_{11}){sin{u}^1}^2]$$

$$2d{u}^{2}d{u}^3(a_{33}cos{u}_1+(a_{23}cos{u}^3+a_{13}sin{u}^3)sin{u}^1)+{d{u}^3}^2{a}_{33}+e{d{u}^0}^2.$$

$a_{ab}$ are arbitrary functions on $u^0$.

To obtain the functions $l_a^{\alpha }$, it is sufficient to consider the components $g_{13},g_{23}$ from the system (3). The solution has the form:

$$l_{\alpha }^a={\delta }_{\alpha }^p{l}_p^a(u^1,u^3)+{\delta }_{\alpha }^3{\delta }_3^a.$$

From the equations

$$g_{13}=a_{13}cos{u}^3-a_{23}sin{u}^3={\eta }_{3a}{l}_1^a,g_{23}=a_{33}cos{u}_1+(a_{23}cos{u}^3+a_{13}sin{u}^3)sin{u}^1={\eta }_{3a}{l}_1^a,$$

it follows that

$$l_{\alpha }^a=\left(\begin{array}{ccc}cos{u}^3& -sin{u}^3& 0\\ sin{u}^{1}sin{u}^3& sin{u}^{1}cos{u}^3& cos{u}^1\\ 0& 0& 1\end{array}\right),l_a^{\alpha }=\left(\begin{array}{ccc}cos{u}^3& \frac{sin{u}^3}{sin{u}^1}& -\frac{cos{u}^{1}sin{u}^3}{sin{u}^1}\\ -sin{u}^3& \frac{cos{u}^3}{sin{u}^1}& -\frac{cos{u}^{1}cos{u}^3}{sin{u}^1}\\ 0& 0& 1\end{array}\right),$$

(10)

$l_{\alpha }^a{l}_b^{\alpha }={\delta }_b^a.$

The lower index numbers are lines. One can show that the vector fields (10) satisfy Equations (1) and (2). We present the components of the vectors ${\xi }_a^{\alpha }$ in the form of a matrix:

$$\left|\right|{\xi }_a^{\alpha }\left|\right|=\left(\begin{array}{ccc}0& 1& 0\\ cos{u}^2& -\frac{cos{u}^{1}sin{u}^2}{sin{u}^1}& \frac{sin{u}^2}{sin{u}^1}\\ -sin{u}^2& -\frac{cos{u}^{1}cos{u}^2}{sin{u}^1}& \frac{cos{u}^2}{sin{u}^1}\end{array}\right).$$

The components ${\tilde{\mathbf{A}}}_{\alpha }$ can be expressed through ${\mathbf{A}}_{\alpha }$ as follows:

$${\tilde{\mathbf{A}}}_a=Z_a^b{\mathbf{A}}_b,$$

where

$$\left|\right|Z_a^b={\xi }_a^{\alpha }{l}_{\alpha }^b\left|\right|=\left(\begin{array}{ccc}sin{u}^{1}sin{u}^3& sin{u}^{1}cos{u}^3& cos{u}^1\\ (cos{u}^{2}cos{u}^3-sin{u}^{2}sin{u}^{3}cos{u}^1)& -(cos{u}^{2}sin{u}^3+sin{u}^{2}cos{u}^{3}cos{u}^1)& sin{u}^{1}sin{u}^2\\ -(sin{u}^{2}cos{u}^3+cos{u}^{2}sin{u}^{3}cos{u}^1)& (sin{u}^{2}sin{u}^3-cos{u}^{2}cos{u}^{3}cos{u}^1)& cos{u}^{2}sin{u}^1\end{array}\right).$$

It can be shown by direct calculation that the elements of the matrix $Z_a^b$ satisfy the equation:

$$Z_{a|c}^b=C_{ca}^{a_1}{Z}_{a_1}^b,|a=l_a^{\alpha }{\partial }_{\alpha }.$$

(11)

Therefore, Equation (7) can be reduced to the form:

$${\xi }_a^{\alpha }{\mathbf{A}}_{b,\alpha }=0\Rightarrow {\mathbf{A}}_a={\alpha }_a\left(u^0\right).$$

(12)

3. Maxwell’s Equations with Zero Electromagnetic Field Sources in a Homogeneous Spacetime

All the exact solutions of vacuum Maxwell equations for solvable groups have been found in papers [40,41]. In the present paper, the problem is solved for the group $G_3\left(IX\right)$.

We use the first definition of homogeneous spaces. Note, that for the spacetime with the groups of motions $G_3\left(I\right)-G_3\left(VI\right),G_3\left(IX\right)$, both definitions are equivalent.

Consider the Maxwell equations with zero electromagnetic field sources in homogeneous space in the presence of an electromagnetic field invariant with respect to the group $G_r$:

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

(13)

The metric tensor is defined by relations (1), and the electromagnetic potential is defined by relations (7). When $i=0$, from the set of Equation (13), it follows that

$$\frac{1}{\sqrt{-g}}{\left(\sqrt{-g}{g}^{\alpha \beta }{F}_{0\beta }\right)}_{\alpha }=\frac{1}{l}{\left(l{l}_a^{\alpha }{\eta }^{ab}{\dot{\alpha }}_b\right)}_{,\alpha }={\eta }^{ab}{\rho }_a{\dot{\alpha }}_b=0.$$

(14)

Here, we denote $g=-det\left|\right|g_{\alpha \beta }\left|\right|=-{\left(\eta l\right)}^2,$ where ${\eta }^2=det\left|\right|{\eta }_{\alpha \beta }\left|\right|,l=det\left|\right|l_{\alpha }^a\left|\right|,$ and ${\rho }_a=l_{a,\alpha }^{\alpha }+l_{|a}/l;$ the dots mean the time derivatives. Let $i=\alpha$. Then, from Equation (13), it follows:

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

(15)

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

(16)

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

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

(17)

Then,

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

(18)

The structural constants of a group $G_3$ can be present in the form:

$$C_{ab}^c=C_{12}^c{\epsilon }_{\tilde{a}\tilde{b}}^{12}+C_{13}^c{\epsilon }_{\tilde{a}\tilde{b}}^{13}+C_{23}^c{\epsilon }_{\tilde{a}\tilde{b}}^{23},{\epsilon }_{ab}^{AB}={\delta }_a^A{\delta }_b^B-{\delta }_b^A{\delta }_a^B.$$

(19)

Using the notations

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

$${\gamma }_1={\sigma }_1{\eta }_{11}+{\sigma }_2{\eta }_{12}+{\sigma }_3{\eta }_{13},{\gamma }_2={\sigma }_1{\eta }_{12}+{\sigma }_2{\eta }_{22}+{\sigma }_3{\eta }_{23},{\gamma }_3={\sigma }_1{\eta }_{13}+{\sigma }_2{\eta }_{23}+{\sigma }_3{\eta }_{33},$$

let us reduce Maxwell’s equations (13) to the form:

$$\eta {\left({\eta }^{ab}{\dot{\alpha }}_b\right)}_{,0}={\delta }_1^a({\gamma }_1\left(C_{32}^1\right)-{\gamma }_2(C_{31}^1+{\rho }_3)+{\gamma }_3(C_{21}^1+{\rho }_2))+{\delta }_2^a({\gamma }_1(C_{32}^2+{\rho }_3)+$$

(20)

$${\gamma }_2{C}_{13}^2-{\gamma }_3\left(C_{12}^2{\rho }_1\right))+{\delta }_3^a(-{\gamma }_1(C_{23}^3+{\rho }_2)+{\gamma }_2(C_{13}^3+{\rho }_1)+{\gamma }_3{C}_{21}^3).$$

The order of Equation (20) can be decreased by introducing a new independent function:

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

(21)

Let us consider the Maxwell equations for the group $G_3\left(IX\right)$. In this case, the nonzero structural constants are as follows:

$$C_{12}^3=C_{31}^2=C_{23}^1=1.$$

The functions ${\sigma }_a$ and ${\gamma }_1$ have the form

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

$${\gamma }_1={\alpha }_1{\eta }_{11}+{\alpha }_2{\eta }_{12}+{\alpha }_3{\eta }_{13},{\gamma }_2={\alpha }_1{\eta }_{12}+{\alpha }_2{\eta }_{22}+{\alpha }_3{\eta }_{23},{\gamma }_1={\alpha }_1{\eta }_{13}+{\alpha }_2{\eta }_{23}+{\alpha }_3{\eta }_{33}.$$

Using these relations, we obtain Maxwell’s Equations (14) and (20) as a system of linear algebraic equations on the unknown functions $n_{ab}$:

$$n_{ab}=\frac{{\eta }_{ab}}{\eta }\Rightarrow \eta =\frac{1}{det{n}_{ab}}.$$

(22)

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

(23)

where

$$\widehat{W}=\left(\begin{array}{cccccc}{\alpha }_1& {\alpha }_2& {\alpha }_3& 0& 0& 0\\ {\beta }_1& {\beta }_2& {\beta }_3& 0& 0& 0\\ 0& {\alpha }_1& 0& {\alpha }_2& {\alpha }_3& 0\\ 0& {\beta }_1& 0& {\beta }_2& {\beta }_3& 0\\ 0& 0& {\alpha }_1& 0& {\alpha }_2& {\alpha }_3\\ 0& 0& {\beta }_1& 0& {\beta }_2& {\beta }_3\end{array}\right),$$

(24)

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

Index T means the transposition of a matrix. Let us find the algebraic complement of the matrix $\widehat{W}$:

$$\widehat{V}=\left(\begin{array}{cccccc}{\beta }_1{V}_1^2& -{\alpha }_1{V}_1^2& {\beta }_2{V}_1^2& -{\alpha }_2{V}_1^2& {\beta }_3{V}_1^2& -{\alpha }_3{V}_1^2\\ {\beta }_1{V}_1{V}_2& -{\alpha }_1{V}_1{V}_2& {\beta }_2{V}_1{V}_2& -{\alpha }_2{V}_1{V}_2& {\beta }_3{V}_1{V}_2& -{\alpha }_3{V}_1{V}_2\\ {\beta }_1{V}_1{V}_3& -{\alpha }_1{V}_1{V}_3& {\beta }_2{V}_1{V}_3& -{\alpha }_2{V}_1{V}_3& {\beta }_3{V}_1{V}_3& -{\alpha }_3{V}_1{V}_3\\ {\beta }_1{V}_2^2& -{\alpha }_1{V}_2^2& {\beta }_2{V}_2^2& -{\alpha }_2{V}_2^2& {\beta }_3{V}_2^2& -{\alpha }_3{V}_2^2\\ {\beta }_1{V}_2{V}_3& -{\alpha }_1{V}_2{V}_3& {\beta }_2{V}_2{V}_3& -{\alpha }_2{V}_2{V}_3& {\beta }_3{V}_2{V}_3& -{\alpha }_3{V}_2{V}_3\\ {\beta }_1{V}_3^2& -{\alpha }_1{V}_3^2& {\beta }_2{V}_3^2& -{\alpha }_2{V}_3^2& {\beta }_3{V}_3^2& -{\alpha }_3{V}_3^2\end{array}\right).$$

(25)

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

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

(26)

Therefore, one of the equations from the system (23) can be replaced by the equation:

$${\delta }^{ab}({\dot{\alpha }}_a{\dot{\alpha }}_b+{\dot{\beta }}_a{\dot{\beta }}_b)\Rightarrow {\delta }^{ab}({\alpha }_a{\alpha }_b+{\beta }_a{\beta }_b)=c^2=const.$$

(27)

Depending on the rank of the matrix $\widehat{W}$, one or more functions $n_{ab}$ are independent. The remaining functions $n_{ab}$ can be expressed through them and through the functions ${\alpha }_a$ and ${\beta }_a$. For classification, it is necessary to find non-equivalent solutions of the system (23). Obviously, this system is symmetric with respect to the transposition $l_1^{\alpha }\leftrightarrow l_2^{\alpha }$. Therefore, the reference indices $a=1$ and $a=2$ can be interchanged. Taking this observation into account, let us consider all the non-equivalent options.

4. Solutions of Maxwell Equations

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

$${\widehat{W}}_{12}=\left(\begin{array}{ccccc}{\alpha }_2& {\alpha }_3& 0& 0& 0\\ {\alpha }_1& 0& {\alpha }_2& {\alpha }_3& 0\\ {\beta }_1& 0& {\beta }_2& {\beta }_3& 0\\ 0& {\alpha }_1& 0& {\alpha }_2& {\alpha }_3\\ 0& {\beta }_1& 0& {\beta }_2& {\beta }_3\end{array}\right),$$

(28)

$${\widehat{\Omega }}_1=\left(\begin{array}{ccccc}-\frac{V_2}{{\alpha }_1{V}_1}& -\frac{{\alpha }_3{\beta }_2}{{\alpha }_1{V}_1}& \frac{{\alpha }_2{\alpha }_3}{{\alpha }_1{V}_1}& -\frac{{\alpha }_3{\beta }_3}{{\alpha }_1{V}_1}& \frac{{\alpha }_3^2}{{\alpha }_1{V}_1}\\ -\frac{V_3}{{\alpha }_1{V}_1}& \frac{{\alpha }_2{\beta }_2}{{\alpha }_1{V}_1}& -\frac{{\alpha }_2^2}{{\alpha }_1{V}_1}& \frac{{\alpha }_2{\beta }_3}{{\alpha }_1{V}_1}& -\frac{{\alpha }_2{\alpha }_3}{{\alpha }_1{V}_1}\\ -\frac{V_2^2}{{\alpha }_1{V}_1^2}& \frac{({\alpha }_3{\beta }_1{V}_1-{\alpha }_2{\beta }_3{V}_3)}{{\alpha }_1{V}_1^2}& \frac{{\alpha }_3({\alpha }_2{V}_2-{\alpha }_1{V}_1)}{{\alpha }_1{V}_1^2}& -\frac{{\alpha }_3{\beta }_3{V}_2}{{\alpha }_1{V}_1^2}& \frac{{\alpha }_2^2{V}_2}{{\alpha }_1{V}_1^2}\\ -\frac{V_2{V}_3}{{\alpha }_1{V}_1^2}& \frac{{\alpha }_2{\beta }_2{V}_2}{{\alpha }_1{V}_1^2}& -\frac{{\alpha }_2^2{V}_2}{{\alpha }_1{V}_1^2}& -\frac{{\alpha }_3{\beta }_3{V}_3}{{\alpha }_1{V}_1^2}& \frac{{\alpha }_3^2{V}_3}{{\alpha }_1{V}_1^2}\\ -\frac{V_3^2}{{\alpha }_1{V}_1^2}& \frac{{\alpha }_2{\beta }_2{V}_3}{{\alpha }_1{V}_1^2}& -\frac{{\alpha }_2^2{V}_3}{{\alpha }_1{V}_1^2}& \frac{({\alpha }_3{\beta }_2{V}_3-{\alpha }_2{\beta }_1{V}_1)}{{\alpha }_1{V}_1^2}& \frac{{\alpha }_2({\alpha }_1{V}_1-{\alpha }_3{V}_3)}{{\alpha }_1{V}_1^2}\end{array}\right).$$

(29)

Then, the solution of Equation (23) can be represented as:

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

(30)

where

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

Function $n_{11},$ as well as the functions ${\alpha }_a$ and ${\beta }_a$ are arbitrary functions of $u^0$ that obey condition (27).

2. ${\alpha }_2{V}_1\ne 0,\Rightarrow {\alpha }_1=0\Rightarrow$ the minor ${\widehat{W}}_{14}^{-1}$ and its inverse matrix ${\widehat{\Omega }}_2={\widehat{W}}_{14}^{-1}$ have the form:

$${\widehat{W}}_{14}=\left(\begin{array}{ccccc}{\alpha }_2& {\alpha }_3& 0& 0& 0\\ {\beta }_2& {\beta }_3& 0& 0& 0\\ 0& 0& {\alpha }_2& {\alpha }_3& 0\\ 0& 0& 0& {\alpha }_2& {\alpha }_3\\ 0& {\beta }_1& 0& {\beta }_2& {\beta }_3\end{array}\right),{\widehat{\Omega }}_2=\left(\begin{array}{ccccc}\frac{{\beta }_3}{V_1}& -\frac{{\alpha }_3}{V_1}& 0& 0& 0\\ -\frac{{\beta }_2}{V_1}& \frac{{\alpha }_2}{V_1}& 0& 0& 0\\ \frac{a_3^2{\beta }_1{\beta }_2}{{\alpha }_2{V}_1^2}& -\frac{{\alpha }_3^2{\beta }_1}{V_1^2}& \frac{1}{{\alpha }_2}& -\frac{{\alpha }_3{\beta }_3}{{\alpha }_2{V}_1}& \frac{a_3^2}{{\alpha }_2{V}_1}\\ -\frac{a_3{\beta }_1{\beta }_2}{V_1^2}& \frac{{\alpha }_2{\alpha }_3{\beta }_1}{V_1^2}& 0& \frac{{\beta }_3}{V_1}& -\frac{a_3}{V_1}\\ \frac{a_2{\beta }_1{\beta }_2}{V_1^2}& -\frac{{\alpha }_2^2{\beta }_1}{V_1^2}& 0& -\frac{{\beta }_2}{V_1}& \frac{a_2}{V_1}\end{array}\right).$$

(31)

The solution of Equation (23) can be represented as:

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

(32)

where

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

$${\widehat{\omega }}_2=(-{\dot{\beta }}_1,-{\beta }_1{n}_{11},-{\dot{\beta }}_2,-{\dot{\beta }}_3,{\dot{\alpha }}_3).$$

Function $n_{11},$ as well as the functions ${\alpha }_a$ and ${\beta }_a$ are arbitrary functions of $u^0$ that obey condition (27).

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

$${\widehat{W}}_{16}=\left(\begin{array}{ccccc}0& a_3& 0& 0& 0\\ {\beta }_2& {\beta }_3& 0& 0& 0\\ 0& 0& 0& a_3& 0\\ {\beta }_1& 0& {\beta }_2& {\beta }_3& 0\\ 0& 0& 0& 0& a_3\end{array}\right),{\widehat{\Omega }}_3=\left(\begin{array}{ccccc}-\frac{{\beta }_3}{a_3{\beta }_2}& \frac{1}{{\beta }_3}& 0& 0& 0\\ \frac{1}{a_3}& 0& 0& 0& 0\\ \frac{{\beta }_1{\beta }_3}{a_3{\beta }_2^2}& -\frac{{\beta }_1}{{\beta }_2^2}& -\frac{{\beta }_3}{{\beta }_2{a}_3}& \frac{1}{{\beta }_2}& 0\\ 0& 0& \frac{1}{a_3}& 0& 0\\ 0& 0& 0& 0& \frac{1}{a_3}\end{array}\right).$$

(33)

Then, the solution of Equation (23) can be represented as:

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

(34)

where

$${\widehat{n}}_3^T=(n_{12},n_{13},n_{22},n_{23},n_{33});$$

$${\widehat{\omega }}_3^T=(-{\dot{\beta }}_1,-{\beta }_1{n}_{11},-{\dot{\beta }}_2,0,-{\dot{\beta }}_3).$$

Function $n_{11},$ as well as the functions ${\alpha }_3$ and ${\beta }_a$ are arbitrary functions of $u^0$ that obey condition (27).

4. $a_1{V}_3\ne 0$$\Rightarrow V_1=V_2=0;$ otherwise, we obtain a solution equivalent to the previous ones. As $V_3\ne 0\Rightarrow$${\alpha }_3={\beta }_3=0.$ The minor ${\widehat{W}}_{62}$ and its inverse matrix ${\widehat{\Omega }}_4={\widehat{W}}_{62}^{-1}$ have the form:

$${\widehat{W}}_{26}=\left(\begin{array}{ccccc}{\alpha }_1& {\alpha }_2& 0& 0& 0\\ 0& {\alpha }_1& 0& a_2& 0\\ 0& {\beta }_1& 0& {\beta }_2& 0\\ 0& 0& {\alpha }_1& 0& {\alpha }_2\\ 0& 0& {\beta }_1& 0& {\beta }_2\end{array}\right),{\widehat{\Omega }}_4=\left(\begin{array}{ccccc}\frac{1}{{\alpha }_1}& -\frac{{\alpha }_2{\beta }_2}{{\alpha }_1{V}_3}& \frac{{\alpha }_2^2}{{\alpha }_1{V}_3}& 0& 0\\ 0& \frac{{\beta }_2}{V_3}& -\frac{{\alpha }_2}{V_3}& 0& 0\\ 0& 0& 0& \frac{{\beta }_2}{V_3}& -\frac{{\alpha }_2}{V_3}\\ 0& -\frac{{\beta }_1}{V_3}& \frac{{\alpha }_1}{V_3}& 0& 0\\ 0& 0& 0& -\frac{{\beta }_1}{V_3}& \frac{{\alpha }_1}{V_3}\end{array}\right).$$

(35)

Then, the solution of Equation (23) can be represented as:

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

(36)

where

$${\widehat{n}}_4^T=(n_{11},n_{12},n_{13},n_{22},n_{23});$$

$${\widehat{\omega }}_4^T=(-{\dot{\beta }}_1,-{\dot{\beta }}_2,{\dot{\alpha }}_2,0,0).$$

Function $n_{33},$ as well as the functions ${\alpha }_1,{\alpha }_2,$ and ${\beta }_a$, are arbitrary functions of $u^0$ that obey the condition (27).

5. $V_a=0$. Let us represent the system of Maxwell equations in the form:

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

were

$$\widehat{Q}=\left(\begin{array}{cccccc}{a}_1& a_2& a_3& 0& 0& 0\\ 0& a_1& 0& a_2& a_3& 0\\ 0& 0& {\alpha }_1& 0& a_2& a_3\\ {\beta }_1& {\beta }_2& {\beta }_3& 0& 0& 0\\ 0& {\beta }_1& 0& {\beta }_2& {\beta }_3& 0\\ 0& 0& {\beta }_1& 0& {\beta }_2& {\beta }_3\end{array}\right),$$

(37)

$${\widehat{\omega }}_I=({\widehat{\omega }}_{\beta },{\widehat{\omega }}_{\alpha });{\widehat{\omega }}_{\beta }=({\omega }_1,{\omega }_3,{\omega }_5),{\widehat{\omega }}_{\alpha }=({\omega }_2,{\omega }_4,{\omega }_6)$$

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

Let us consider all possible options.

(a) $a_1\ne 0\Rightarrow {\beta }_a=\frac{{\alpha }_a{\beta }_1}{{\alpha }_1}$. Maxwell’s equations will take the form:

$${\widehat{W}}_I{\widehat{n}}_{\alpha }=({\widehat{\omega }}_{\beta }-{\widehat{Q}}_1{\widehat{n}}_{\beta })\Rightarrow {\widehat{n}}_{\alpha }={\widehat{W}}_I^{-1}({\widehat{\omega }}_{\beta }-{\widehat{Q}}_1{\widehat{n}}_{\beta }),$$

$${\beta }_1{\widehat{W}}_I{\widehat{n}}_{\alpha }={\alpha }_1{\widehat{\omega }}_{\alpha }-{\beta }_1{\widehat{Q}}_1{\widehat{n}}_{\beta }\Rightarrow {\beta }_1{\widehat{\omega }}_{\beta }-{\alpha }_1{\widehat{\omega }}_{\alpha }=0\Rightarrow$$

$$\left\{\begin{array}{c}\hfill {\alpha }_1{\dot{\alpha }}_2+{\beta }_1{\dot{\beta }}_2=0,\\ \hfill {\alpha }_1{\dot{\alpha }}_3+{\beta }_1{\dot{\beta }}_3=0,\\ \hfill {\alpha }_1{\dot{\alpha }}_1+{\beta }_1{\dot{\beta }}_1=0.\end{array}\right.$$

(38)

Here:

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

From the last equation of the system (38) it follows:

$$a_1=esin\phi ,{\beta }_1=ecos\phi ,e=const,$$

Thus, ${\beta }_2={\alpha }_2\frac{cos\phi }{sin\phi },{\beta }_3={\alpha }_3\frac{cos\phi }{sin\phi }$, and from the previous equations it follows:

$${\alpha }_a=e{c}_{a}sin\phi ,{\beta }_a=e{c}_{a}cos\phi ,e,c_a=const,c_1=1.$$

Then matrices ${\widehat{W}}_I,{\widehat{W}}_I^{-1},{\widehat{Q}}_I$ and lines ${\widehat{\omega }}^T$ take the form:

$${\widehat{W}}_I=sin\phi \widehat{w},{\widehat{W}}_I^{-1}=\frac{1}{sin\phi }{\widehat{w}}^{-1},{\widehat{Q}}_I=sin\phi \widehat{q}.$$

$$\widehat{w}=\left(\begin{array}{cccc}1& c_2& c_3& \\ 0& 1& 0\\ 0& 0& 1\end{array}\right),{\widehat{w}}^{-1}=\left(\begin{array}{ccc}1& -c_2& -c_3\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),\widehat{q}=\left(\begin{array}{cccc}0& 0& 0& \\ c_2& c_3& 0\\ 0& c_2& c_3,\end{array}\right)$$

$${\widehat{\omega }}_{\beta }^T=\dot{\phi }{\widehat{C}}^T=\dot{\phi }sin\phi (c_2,1,c_3),$$

The solution of Maxwell’s equations can be represented as:

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

Functions $n_{22},n_{23},$ and $n_{33},$ as well as the function $\phi$, are arbitrary functions of $u^0$.

(b) $V_a=0,{\alpha }_3\ne 0.$ The solutions, which are not equivalent to the previous ones, can be obtained under the conditions $a_1=a_2=0\Rightarrow {\beta }_1={\beta }_2=0$. From Maxwell’s equations it follows:

$${\alpha }_3{n}_{13}={\alpha }_3{n}_{23}=0,a_3{n}_{33}=-{\dot{\beta }}_3,{\beta }_3{n}_{33}={\dot{\alpha }}_3\Rightarrow a_3{\dot{a}}_3+{\beta }_3{\dot{\beta }}_3=0.$$

The solution has the form:

$$n_{33}=\dot{\phi },n_{13}=n_{23}={\alpha }_1={\alpha }_2={\beta }_1={\beta }_2=0,a_3=ccos\phi ,{\beta }_3=csin\phi .$$

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

5. Conclusions

It is known that homogeneous spaces of $IV$ and $IX$ types according to the Bianchi classification include as special cases the spaces of constant curvature. Hence, they receive special interest in cosmology. In the universe with the metric of homogeneous space, all physical fields are invariant with respect to the group of motions of the spacetime. Therefore, these exact fields should be considered first when solving the self-consistent Einstein equations, in particular, the Einstein–Maxwell equations. The final goal of the classification of PSS with admissible electromagnetic fields is to enumerate all the electrovacuum solutions of the Einstein–Maxwell equations. In [40,41], the complete classification of the vacuum solutions of the Maxwell equations for homogeneous spaces with solvable groups of motions was carried out. In the present paper, the same problem was solved for HPSS of $IX$-type. For the final decision of the first stage of the classification problem, it remains to consider the HPSS $VIII$-type, which will be detailed in the next paper. The results obtained will be used in the second stage for integration of the corresponding Einstein–Maxwell equations.