Classification of Petrov Homogeneous Spaces

Space-time manifolds $V_4$ admitting groups of motions occupy a special place in the theory of gravitation, since the vector Killing fields defining generators of these groups serve to construct the conserved physical quantities connected with the properties of the gravitational field. Therefore, these fields play an important role in describing the geometry of curved spacetime and physical phenomena occurring within it.

A large number of papers have been devoted to this role of the Killing vector fields (see, for example, [1,2,3,4]). The question of group motions in Riemannian spaces was first considered in the works of Bianchi [5,6], Fubini and Rimini [7,8] back in the late 19th and early 20th centuries.

These spaces are of particular interest in cosmology, which, as a modern science, in fact, began to develop after the appearance of works [9,10], in which special cases of homogeneous spaces were used for the first time. These spaces are still the foundation of cosmological models, including in alternative theories of gravitation (see [11,12]), as well as in the early stages of the Universe’s development.

The classification of space-time manifolds with groups of motions is based on the theory of continuous groups of transformations (see [13,14]).

A complete classification of four-dimensional pseudo-Riemannian spaces founded on the canonical real structures of the motion groups acting in the spaces was realized by A.Z. Petrov in [15,16,17]. Petrov constructed non-equivalent sets of operators for all groups of motions of four-dimensional pseudo-Riemannian spaces, integrated the Killing equations and found the metrics of all these spaces. When classifying isotropic spaces, Petrov used the approaches described in the papers by Kruchkovich [18,19].

The obtained results of this classification have found application in the theory of gravitation, first of all, when considering the problem of exact solutions of the field gravitational equations (see [17,20,21]). In more recent works, various aspects of using spaces with groups of motions were considered. Thus, the authors of [22,23,24] studied the intersections of homogeneous and Stackel spaces, as well as physical processes in homogeneous spaces. In [25,26], the problems of constructing integrals of motion for classical and quantum scalar charged particles moving in spaces with groups of motions and in admissible electromagnetic fields were considered. The classification of the exact solutions of Maxwell equations in homogeneous spaces in the presence of admissible electromagnetic fields was carried out in [27,28,29]. A special direction of application of homogeneous spaces is the construction of the theory of noncommutative integration of the equations of motion of charged quantum particles in external fields [30,31,32,33].

In this paper, the final stage of the Petrov classification is carried out. As it is known, the Killing vector fields specify infinitesimal transformations of the group of motions of space $V_4$. In the case where the group of motions $G_3$ acts in a simply transitive way in the homogeneous space $V_4$, the geometry of the non-isotropic hypersurface is determined by the geometry of the transitivity space $V_3$ of the group $G_3$. In this case, the metric tensor of the space $V_3$ can be given by a nonholonomic reper consisting of three independent vectors ${\ell }_{\left(a\right)}^{\alpha }$, which define the generators of the group $G_3$ of finite transformations in the space $V_3$. The representation of the metric tensor of $V_4$ spaces by means of vector fields ${\ell }_{\left(a\right)}^{\alpha }$ has a great physical meaning and makes it possible to substantially simplify the equations of mathematical physics in such spaces. Therefore, the Petrov classification should be complemented by the classification of vector fields ${\ell }_{\left(a\right)}^{\alpha }$ connected to Killing vector fields. For homogeneous spaces, this problem has been largely solved (see, e.g., [34,35,36]). A complete solution of this problem is presented in the present paper, where I refine the Petrov classification for homogeneous spaces in which the group $G_3$, which belongs to type $VIII$ according to the Petrov classification, acts simply transitively. In addition, this paper provides the complete classification of vector fields ${\ell }_{\left(a\right)}^{\alpha }$ for spaces $V_4$ in which the group $G_3$ acts simply transitively on isotropic hypersurfaces.

Consider a Riemannian space $V_4$ with signature $(-1,1,1,1)$, on the hypersurface $V_3$ of which the motion groups $G_3\left(N\right)$ act simply transitively (N corresponds to the number of the group $G_3$ in the Bianchi classification). In the case of a null hypersurface of transitivity $V_3^{*}\left(N\right)$, the space $V_4^{*}\left(N\right)$ will be called a null homogeneous Petrov space of type N. If the hypersurface of transitivity $V_3\left(N\right)$ is non-zero, then $V_4\left(N\right)$ will be called a homogeneous non-isotropic Petrov space of type N.

The geometry of a homogeneous Petrov space is connected with the geometry of the three-dimensional space of transitivity ${\tilde{V}}_3\left(N\right)$ of the group $G_3\left(N\right)$. For a non-null homogeneous Petrov space, this connection is direct, since in this case a hypersurface of transitivity $V_3\left(N\right)$ is in fact the space ${\tilde{V}}_3\left(N\right)$. For a null Petrov space, this connection is less obvious, but also exists, since the Killing vector fields of the space $V_4^{*}\left(N\right)$ are generally a linear combination of the Killing vector fields of the space ${\tilde{V}}_3\left(N\right)$ with coefficients depending on the wave variable $u^0$.

The contravariant components of the metric tensor in a non-null semi-geodesic coordinate system can be represented in the following form:The admissible transformations of the coordinates $u^i$, which do not violate the form of (1), are of the following form:It follows from the Killing equations that The following index designations are used here and hereafter:Zero indices number the fourth row and column. Greek letters denote the coordinate indices of the semi-geodetic coordinate system $u^i$. The small Latin letters a, b, and c denote the indices of the nonholonomic coordinate system associated with the group $G_3\left(N\right)$.

The classification of null homogeneous Petrov spaces was carried out in the isotropic semi-geodesic coordinate system (see [15], p. 158). In this coordinate system, the contravariant components of the metric tensor $g^{ij}$ have the form There and further, the indices denoted by $p,q$ change within $÷1,2$. In the present paper, the canonical form of the isotropic semi-geodesic coordinate system, in which is used:It follows from the Killing equations that in the canonical coordinate system the components of the Killing vector fields of the group $G_3\left(N\right)$ satisfy the following conditions (see [17]):The admissible coordinate transformations $\left\{u^i\right\}$, which do not violate the conditions (6), (7), are of the following form:The homogeneous space $V_3\left(N\right)$ has two equivalent definitions. According to the first one, $V_3\left(N\right)$ is the space of transitivity for the group $G_3\left(N\right)$, whose operators are constructed with Killing vector fields ${\xi }_{\left(a\right)}^{\alpha }$:According to the second definition, in the homogeneous space $V_3\left(N\right)$ the group $G_3\left(N\right)$ of non-infinitesimal shifts acts. The group operators have the form According to the first definition, two infinitely close points of space $V_3\left(N\right)$ can be combined using the motion group of space $G_3\left(N\right)$. According to the second definition, this can be done for any two points of the space. As is known (see, for example, [34]) p. 483), the metric tensor of the homogeneous space $V_3\left(N\right)$ is expressed through the vector fields ${\ell }_{\left(a\right)}^{\alpha }$ as follows:$\alpha ,\beta ,\gamma ÷1,2,3$; $a,b,c,÷1,2,3$; ${\eta }^{ab}=const$ (in the case where ${\eta }^{ab}$ is considered part of the metric tensor $g^{ij},{\eta }^{ab}$ are functions of the variable $u^0$). The sets of operators ${\mathrm{X}}_a,{\mathrm{Y}}_b$ obey the same structure equations:Hence, the sets of vector fields $\left\{{\xi }_{\left(a\right)}^{\alpha }\right\}$ and $\left\{{\ell }_{\left(a\right)}^{\alpha }\right\}$ are equivalent to each other with respect to coordinate transformations. Therefore, both fields can be simultaneously considered as Killing vector fields and as nonholonomic reper vector fields (see (11)) in the same space $V_3\left(N\right)$ but in different coordinate systems connected by admissible transformations of the form (2) or (8). In one coordinate system, these fields are connected by relations (see [34], p. 484): where These considerations apply to non-null homogeneous Petrov spaces. In the case of null Petrov spaces, they cannot be used directly because the covariant metric tensor of the space $V_3\left(N\right)$ is not the metric tensor of the isotropic hypersurface of the space $V_4^{*}$. Therefore, the proof of the (4) relations based on the above second definition of homogeneous space for null spaces is inapplicable. Nevertheless, the representation of (11), where the vector fields ${\ell }_{\left(a\right),\beta }^{\alpha }$ satisfy the system of Equation (14), holds for null homogeneous Petrov spaces as well. It can be shown that if vector fields satisfy a system of equations of the form (14), then ${\zeta }_a^{\alpha }$ are Killing vector fields for a space with metric tensor (11). Indeed, substituting the expressions and (11) into the Killing equations, we obtain the identity. As Equation (17) has three independent solutions, the vector fields ${\zeta }_{\left(a\right),\beta }^{\alpha }$ (also as ${\ell }_a^{\alpha }$) form the group $G_3$.

The classification has been carried out by Petrov in three stages.

1. Selection of a semi-geodesic coordinate system.

2. Solving structural Equation (4). In the second stage, all sets of Killing vector fields that are non-equivalent with respect to admissible transformations of variables have been listed. In this way, the choice of semi-geodesic coordinate systems in which the solutions of the systems of structural equations can be represented in elementary functions is also realized.

3. Solving the Killing equations. In the third step, using the solutions found in the second step, the Killing equations have been integrated: and the metric tensors of all the appropriate homogeneous Petrov spaces have been found.

Representation of the metric tensor of homogeneous space in the form (11) has a deep physical meaning, and it is demanded when solving problems of mathematical physics in the theory of gravitation. Therefore, in this paper, the classification of homogeneous Petrov spaces has been supplemented by the fourth stage.

4. Solving the systems of Equation (14). In the fourth stage, all non-equivalent solutions of the systems of Equation (14) have been found. When the components of the Killing vector fields do not depend on the wave variable $u^0$, the components of the metric $g^{\alpha \beta }$ are found directly from the (16) relations without directly using the Killing equations. Let us consider the peculiarities of the implementation of this step in the presence of such dependence.

Each of the groups $G_3\left(II\right),G_3\left(III\right),G_3\left(V\right)$ has several independent sets of group operators. One of these sets contains a linear combination of group operators of the following form (see [17]):Using the solutions of the system of Equation (15), one can find the contravariant components (7) of the metric tensor $g^{\alpha \beta }$ of the space $V_4^{*}$ admitting Killing vector fields ${\xi }_{\left(a\right)}^{\alpha }$. Then the contravariant components of the metric tensor ${\tilde{g}}^{\alpha \beta }$ can be represented as The function $\Delta g^{\alpha \beta }$ is found from the Killing equations:Here and hereafter,

When making the classification of homogeneous Petrov spaces, one can restrict oneself to the null case only, since the group of admissible coordinate transformations (8) is a subgroup of the group of admissible coordinate transformations (2). Therefore, all sets of the group operators $G_3\left(N\right)$, which act in the space $V_4^{*}\left(N\right)$, form equivalence classes with respect to the transformation group (2) for the set of operators of the same group, which acts in the space $V_4\left(N\right)$. One feature must be borne in mind when constructing the metric tensor $g^{\alpha \beta }$ according to the formula (11). In the non-null space $V_4\left(N\right)$, the number of independent functions ${\eta }^{ab}\left(u^0\right)$ cannot be reduced by admissible transformations of form (2).

The first step of the classification is to choose a coordinate system, in which one of the Killing fields has to be represented in the form Using admissible coordinate transformations (2), it can always be made for any vector. Moreover, it is possible to make the function $\mathrm{A}$ equal to unity. However, using admissible transformations (8) this can be carried out only for the vector that is already in this shape. Indeed, under the transformation (8), the operator ${\mathrm{X}}_a$ is transformed as follows:The following notations are used here and hereafter:The letters ${\Phi }^{\alpha },{\mathrm{A}}_a,{\mathrm{B}}_a,$ and ${\mathrm{R}}_a$ denote functions depending on the variables $u^0$ and $u^p$. Functions ${\phi }^a,a_a,b_a,r_a$ depend on $u^0$ and one of the variables $u^p$. Since the Jacobian of the transformation (8) does not equal to zero, it is possible to represent ${\tilde{X}}_1$ in the form (20) only if ${\mathrm{B}}_1={\mathrm{R}}_1=0$. In this case, any of the remaining operators, for example, ${\mathrm{X}}_2$, can be diagonalized using admissible transformations (8). Indeed, without restriction of generality, let us require that in a new coordinate system $\left\{\tilde{u}\right\}$ the operator ${\mathrm{X}}_2$ has the form For this purpose, the following condition must be fulfilled:The system (22) reduces to the form of the Cauchy–Kovalevskaya system and is consistent. Thus, operator ${\mathrm{X}}_2$ can be presented in the following form:The first option is received.

Obviously, this version is valid only for solvable groups. Hereinafter, operators ${\mathrm{X}}_1,{\mathrm{X}}_2$ are chosen as operators of the abelian subgroups of the groups $G_3\left(I\right)-G_3\left(VII\right)$. Therefore, from ${\mathrm{A}}_{,2}=0$ it follows that ${\mathrm{A}}_1=a_1(u^0,u^3).$

If $|{\mathrm{B}}_1|+|{\mathrm{R}}_1|\ne 0$, from (22) it follows that ${\mathrm{X}}_1$ can be presented in the form The admissible coordinate transformations that do not violate the condition (24) are as follows:Operator ${\mathrm{X}}_2$ commutes with operator ${\mathrm{X}}_1$. Therefore, after the admissible transformations (25) it takes the following form:Obviously, only the following options are possible:

In the last case, one can use the transformations (25) to simplify operator ${\mathrm{X}}_3$. Thus, these options together with versions for unsolvable groups have to be considered separately when making the classification.

When solving structural equations, admissible transformations (25) can be used to simplify the form of the operator ${\mathrm{X}}_3.$

For insoluble groups $G_3\left(VIII\right),G_3\left(IX\right)$ there is following version:

In this case, the admissible transformations have the form (25). They can be used to simplify the form for one of the operators ${\mathrm{X}}_p.$