Integrable and non-integrable structures in Einstein - Maxwell equations with Abelian isometry group \mathcal{G}_2
aa r X i v : . [ g r- q c ] F e b Integrable and non-integrable structuresin Einstein - Maxwell equations withAbelian isometry group G ∗ Steklov Mathematical Institute of Russian Academy of Sciences
Abstract
The classes of electrovacuum Einstein - Maxwell fields (with a cosmological con-stant), which metrics admit an Abelian two-dimensional isometry group G with non-null orbits and electromagnetic fields possess the same symmetry, are considered. Forthe fields with such symmetries, we describe the structures of the so called ”nondy-namical degrees of freedom” which presence as well as the presence of a cosmologicalconstant change (in a strikingly similar ways) the vacuum and electrovacuum dynami-cal equations and destroy their well known integrable structures. The modifications ofthe known reduced forms of Einstein - Maxwell equations – the Ernst equations andself-dual Kinnersley equations which take into account the presence of non-dynamicaldegrees of freedom are found and the subclasses of fields with different non-dynamicaldegrees of freedom are considered. These are: (I) vacuum metrics with cosmologicalconstant, (II) space-time geometries in vacuum with isometry groups G which orbitsdo not admit the orthogonal 2-surfaces (none-orthogonally-transitive isometry groups)and (III) electrovacuum fields with more general structures of electromagnetic fieldsthan in the known integrable cases. For each of these classes of fields, in the case ofdiagonal metrics, all field equations can be reduced to the only nonlinear equation ofthe fourth order for one real function α ( x , x ) which characterise the element of areaon the orbits of the isometry group G . Simple examples of solutions for each of theseclasses are presented. It is pointed out that if for some two-dimensional reduction ofEinstein’s field equations in four or higher dimensions, the function α ( x , x ) possess a”harmonic” structure, instead of being (together with other field variables) a solutionof some nonlinear equations, this can be an indication of possible complete integrabilityof these reduced dynamical equations for the fields with vanishing of all non-dynamicaldegrees of freedom. Keywords : gravitational and electromagnetic fields, Einstein - Maxwell equations, cosmologicalconstant, isometry group, integrability, exact solutions ∗ e-mail: [email protected] ontents G . . . . 31.2 G -symmetry of fields and integrability of Einstein - Maxwell equations . . . 31.3 G -symmetry and non-dynamical degrees of freedom of fields . . . . . . . . . 6 G G -symmetry reduced Einstein - Maxwell field equations . . . . . . . . . . . 112.2.1 Non-dynamical degrees of freedom of electromagnetic field . . . . . . 122.2.2 Self-dual form of symmetry-reduced Maxwell equations . . . . . . . . 122.2.3 Non-dynamical degrees of freedom of gravitational field . . . . . . . . 122.2.4 Self-dual form of symmetry-reduced Einstein - Maxwell equations . . 142.2.5 Dynamical degrees of freedom and dynamical equations . . . . . . . . 172.3 Classes of fields with non-dynamical degrees of freedom . . . . . . . . . . . . 182.3.1 Vacuum fields with cosmological constant (Λ = 0) . . . . . . . . . . . 182.3.2 Vacuum fields with non-ortogonally-transitive isometry group G ( ω aµ =0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Electrovacuum fields with non-dynamical degrees of freedom ( e o = 0) 222.4 Examples of solutions with (I) Λ = 0, (II) ω aµ = 0, (III) e o = 0 . . . . . . . . 232.4.1 Construction of solutions for the classes of fields (I), (II), (III) . . . . 232.4.2 Simplest examples of solutions for classes of fields (I), (II), (III) . . . 262.5 Non-dynamical degrees of freedom and integrability . . . . . . . . . . . . . . 27 For deeper understanding of various mathematical structures which may arise in the studiesof some classes of objects or particular models in different areas of mechanics, mathematicaland theoretical physics, it may be useful to consider these classes of objects and models withinmore general ones because such kind of analysis can lead to some useful generalizations of theidentified structures or can show explicitly the obstacles which do not allow these structuresto be extended to these more general cases. In the present paper, this concerns the studiesof various features of the structures of vacuum Einstein’s and electrovacuum Einstein -Maxwell equations for the fields which are described by the known integrable reductions ofthese equations and which constitute some subclass in the whole class of gravitational andelecromagnetic fields which space-times admit the Abelian isometry group G with non-nullorbits and which electromagnetic fields possess the same symmetries. The history of discovery of integrable reductions of Einstein’s field equations in General Relativity – theEinstein equations for vacuum, electrovacuum Einstein - Maxwell equations, the Einstein - Maxwell - Weylequations and others, discussion of some related questions and corresponding references can be found in [1] .1 Integrability of vacuum Einstein equations with the isometrygroup G In the modern theory of gravity a behaviour of strong gravitational fields and their inter-action with electromagnetic fields is described respectively by vacuum Einstein equationsand by electrovacuum Einstein - Maxwell equations. Because of rather complicate nonlinearstructure of these equations, the development of effective methods for analytical investiga-tion of these equations and construction of rather rich classes of their exact solutions becamepossible only after a discovery of existence of integrable reductions of these equations forthe fields with certain space-time symmetries. Though the integrability of these equationswas conjectured much earlier [2] and different signs and features of this integrability (as weunderstand these now) could be recognized in the results of later considerations in differentmathematical contexts presented by different authors [3]–[6],[7], the actual discovery of in-tegrability of vacuum Einstein equations was made in the paper of Belinski and Zakharov[8] for the class of metrics which depend on time and one spatial coordinate and which pos-sess 2 × . In [8] the authors presented a spectral problem whichwas equivalent to the symmetry reduced vacuum Einstein equations and which consistedof an overdetermined linear system with a spectral parameter, which compatibility condi-tions are satisfied for solutions of vacuum Einstein equations from the mentioned above classof metrics, and of a set of supplement conditions for the choice of solutions of this linearsystem providing the equivalence of the whole spectral problem to the nonlinear system offield equations. In [8] a method was developed for generating infinite hierarchies of vacuummulti-soliton solutions, starting from arbitrary chosen already known vacuum solution of thesame equations which plays the role of background for solitons. Besides that, in [8] it wasshown that for this type of vacuum metrics the other (”non-soliton”) part of the space ofsolutions can be described in terms of the Riemann-Hilbert problem on the spectral planeand the corresponding system of linear singular integral equations. G -symmetry of fields and integrability of Einstein - Maxwellequations To the time of discovery of vacuum Einstein equations it was found that the similar reductionsof electrovacuum Einstein - Maxwell equations also possess the features which indicate to The similar results for stationary axisymmetric vacuum fields were published a bit later [9]. In a bit earlier papers, Kinnersley and Chitre [6] Maison [7] constructed some linear systems with complexparameters which compatibility conditions were satisfied by the solutions of vacuum Einstein equations.However, a formulation of a complete spectral problem which would be equivalent to the dynamical equationshad not been given in these papers. Besides that, the linear system constructed in [6] was used in anothercontext: it was considered there as the system for a generating function for an infinite hierarchy of matrixpotentials which was associated with each stationary axisymmetric solution of the field equations and whichwas used for a description of the action of the infinite-dimensional algebra of internal symmetries on thesolutions of Einstein equations. The linear system constructed in [7] servde as indication that vacuumEinstein equations for such fields can occur to be integrable ”in the spirit of Lax”. There was a remarkable technical step also made in [8], which is important for explicit construction andanalysis of vacuum soliton solutions. This is a very simple and explicit general ”determinant” expressionfor calculation of the conformal factor in the conformally flat part of the metric found in [8]. A compactdeterminant forms for all components of Belinski and Zakharov vacuum soliton solutions were found in [10]. Soon after the papers [8, 9] it became clear that the formulation of the inverse scatteringtransform and construction of vacuum soliton solutions suggested by Belinski and Zakharovdo not admit a straightforward generalization to the case of interacting gravitational andelectromagnetic fields. A bit later, the integrable structure of of Einstein - Maxwell equa-tions for space-times with the isometry groups G , with block-diagonal form of metric andwith some specific structure of electromagnetic field components was found in the author’spapers [15, 16], where a spectral problem equivalent to another (complex selfdual) formof the (symmetry reduced) Einstein - Maxwell equations was constructed. This spectralproblem consists of (a) the linear system which compatibility conditions are satisfied forsolutions of Einstein - Maxwell equations and (b) the supplementary conditions of existencefor this linear system of the matrix integral of a special structure providing the equivalenceof the complete spectral problem to electrovacuum field equations. Besides that, in the pa-pers [15, 16] a method for generating of electrovacuum soliton solutions was developed. Thismethod allows to generate the electrovacuum soliton solutions starting from arbitrary chosenknown electrovacuum solution (with the same symmetry) which plays the role of backgroundfor solitons, but technically, this method differs essentially from its vacuum analogue con-structed by Belinski and Zakharov in [8, 9]. More detail description of electrovacuum solitonsconstructing was presented in [17]. On the relation between the spectral problems as well asbetween Belinski and Zakharov vacuum solitons and vacuum limit of electrovacuum solitonssee [18].The most general approach to a description of the spaces of solutions of integrable reduc-tions of vacuum Einstein equations and electrovacuum Einstein - Maxwell equations (calledas the ”monodromy transform approach”) was suggested in [19], where a system of linearsingular integral equations equivalent to the nonlinear dynamical equations was constructedwithout any supplementary restrictions on the class of vacuum or electrovacuum fields, such Later, a matrix system of linear singular integral equations of Hauser and Ernst was reduced to muchsimpler form of a scalar integral equation by Sibgatullin [13], who restricted in these integral equationsthe choice of the seed solution by the Minkowski space-time and expressed (following Hauser and Ernst[14]) the Geroch group parameters in the kernel of these integral equations in terms of the values of theErnst potentials on the axis of symmetry for generating solution. This reduced form of Hauser and Ernstintegral equation was used many times by Sibgatullin with co-authors for explicit calculations of manyparticular examples of asymptotically flat stationary axisymmetric solutions of Einstein - Maxwell equationswith the Ernst potentials which boundary values on the axis are rational functions of the Weyl coordinate z along the axis. However, one can notice that all asymptotically flat solutions constructed in this way,were some particular cases of already known vacuum or electrovacuum soliton solutions generated on theMinkowski background or represented some analytcal continuations of these solitons in the spaces of theirfree parameters.
4s regularity of the axis of symmetry or the absence of initial singularities in time-dependentsolutions. In this approach, every local solution of integrable reductions of vacuum Einsteinequations or electrovacuum Einstein - Maxwell equations is characterized by a set of coor-dinate independent holomorphic functions of the spectral parameter – the monodromy dataof the normalized fundamental solution of the corresponding spectral problem which enterexplicitly into the kernel of the mentioned above system of linear singular integral equations.This system of integral equations admits a unique solution for any choice of the monodromydata as functions of the spectral parameter. For a special class of arbitrary rational and”analytically matched” monodromy data functions, this system of linear integral equationsadmits an explicit solution [17, 20, 21]. This allows to construct infinite hierarchies of so-lutions of the field equations which extend essentially the classes of soliton solutions andinclude the solutions for some new types of field configurations.Thus, discovery of integrable reductions of Einstein equations and Einstein - Maxwellequations and development of vacuum and electrovacuum soliton generating transforma-tions as well as constructing of new types of field configurations solving the correspondinglinear singular integral equations provide us with effective methods for constructing themulti-parametric families of exact solutions and finding many physically interesting exam-ples. In particular, these include such non-trivial field configurations as nonlinear superpo-sitions of two charged rotating massive sources of the Kerr-Newman type [17], the solutionfor a Schwarzschild black hole immersed into the external gravitational and electromagneticfields of Bertotti-Robinson magnetic universe [21], equilibrium configurations of two massivecharged sources of Reissner-Nordstrom type [22] and others. Applications of electrovacuumsoliton generating transformations was considered in [17]. Besides that, in the recent pa-per [23] a large class of solutions was constructed. These solutions describe a nonlinearinteraction of electrovacuum soliton waves with a strong (non-soliton) pure electromagneticwaves of arbitrary amplitudes and profiles propagating in and interacting with the curvatureof background space-time of homogeneously and anisotropically expanding universe whichmetric is described by a symmetric Kasner solution. For solutions of this class, the role ofthe background for solitons is played by the class of such pure electromagnetic travellingwaves propagating in the symmetric Kasner background which was found in [24].The examples of solutions given above show that the existence of integrable reductions ofthe field equations in the Einstein theory of gravity open the ways for development of variouseffctive methods for constructing of solutions. Therefore, it may be of a large interest tostudy the possibilities of generalization of these methods to much wider classes of fields.In this way, it may be interesting to consider those features of the structure of integrablereductions of Einstein’s field equations which are retained for some more general classes offields as well as those obstacles which destroy this integrability in more general situations.As far as for all known cases, the fields described by the integrable reductions of Einstein’sfield equations belong to the classes for which the space-time metric admits two-dimensionalAbelian group of isometries G , generated by two commuting non-null Killing vector fields,an obvious next step of our considerations is to compare the structures of the Einstein’s fieldequations in the integrable cases with the structure of these equations for the whole classesof fields which possess the same G -symmetry.5 .3 G -symmetry and non-dynamical degrees of freedom of fields The space-time geometries and electromagnetic field configurations, for which the Einstein -Maxwell equations were found to be integrable, admit the Abelian two-dimensional isometrygroups. However, for this integrability it is not enough to assume such symmetry. The inte-grable structures of these field equations were found only for some subclasses of space-timeswith such isometry groups. In the literature, further restrictions on classes of fields with G -symmetry which lead eventually to integrability of the dynamical parts of this symmetryreduced field equations had arose already in different contexts and were used in different,but equivalent forms. In particular, in some cases it was assumed that two constants con-structed from two commuting Killing vector fields vanish, Another (equivalent) assumptionwas that the orbits of the isometry group G admit the orthogonal 2-surfaces and herefore,in appropriate coordinates metric takes 2 × G . For these classes of fields we show, thatbesides the dynamical variables these electromagnetic and gravitation fields possess some”non-dynamical degrees of freedom” which we use further in order to derive the necessaryconditions for the mentioned above integrability in more physical terms. The term ”non-dynamical degrees of freedom” means here those metric and electromagnetic field compo-nents for which we do not have the dynamical equations but instead of these, the part ofEinstein equations, which represent the constraints, admit the explicit solutions which allowto express these field components in terms of the other (dynamical) variables and a set ofarbitrary constant parameters which arise from the constraint equations as the constantsof integration. These non-dynamical degrees of freedom can possess an important phys- There are different classes of space-times which admit the Abelian two-dimensional isometry groups.The difference between these classes may be in the topology and metric signature of the isometry grouporbits. In particular, the orbits of Killing vector fields can be compact or not (as in the cases of axial andcylindrical symmetries or in the case of plane waves respectively) and one of two commuting Killing vectorfields can be time-like and the other one – the space-like (as for stationary axisymmetric fields) or bothKilling vector fields can be space-like (as for waves and cosmological models). However, it is useful to notethat this separation of solution into different classes is local: the same solution in different space-time regionscan belong to different classes described above. G -symmetries, can enteralso into the symmetry reduced dynamical parts of vacuum Einstein equations and elec-trovacuum Einstein - Maxwell equations as free parameters and change crucially the struc-ture of these dynamical equations. It is interesting to note that the form in which theseparameters enter the dynamical part of the field equations is very similar to the case ofpresence of cosmological constant. That is why in this paper, we include the cosmologicalconstant in our considerations as one more non-dynamical degree of freedom of the gravi-tational field. The presence of these arbitrary constants, similarly to the case of presenceof the cosmological constant, destroy the well known today integrable structures of vacuumEinstein equations and electrovacuum Einstein - Maxwell equations, however, it is necessaryto appreciate that the question of integrability (probably, in some other form) of these moregeneral equations remains to be opened.Nonetheless, an important property of arbitrary cosntants which determine the non-dynamical degrees of freedom of gravitational and electromagnetic fields in a general classesof vacuum and electrovacuum spacetimes with G -symmetries is that for some particular val-ues of these constants the corresponding field components, identified with the non-dynamicaldegrees of freedom, become pure gauge and threfore, these components can vanish for ap-propriate choice of coordinates and gauges. In this case, the dynamical equations reduceto those which were found earlier to be integrable. Thus, we show that the necessary andsufficient condition providing the integrability of G -symmetry reduced vacuum Einsteinequations and electrovacuum Einstein - Maxwell equations can be presented in an equiva-lent but more physical form as the condition of vanishing of all non-dynamical degrees offreedom of gravitational and electromagnetic fields.Further, we consider different subclasses of fields with non-dynamical degrees of freedom:(I) vacuum metrics with non-vanishing cosmological constant, (II) space-time geometries inwhich the orbits of the isometry group G do not admit the ortogonal 2-surfaces, i.e. theisometry group G is not orthogonally-transitive and (III) electrovacuum fields more generalstructures of electromagnetic fields than in the known integrable cases. For each of thesesubclasses, in the case of diagonal metrics all field equations can be reduced to one nonlinearequation for one real function α ( x , x ), which possesses a geometrical interpretation asdescribing the element of the area on the orbits of the isometry group G . The simpleexamples of solutions are geven here for each of these subclasses of fields with non-vanishingnon-dynamical degrees of freedom.In the Concluding remarks, we summarized the results of this paper concerning the”status” of fields described by integrable reductions of vacuum Einstein equations and elec-trovacuum Einstein - Maxwell equations within the corresponding general classes of vacuumor electrovacuum fields with a cosmological constant in space-times which metrics admittwo-dimensional Abelian isometry group and electromagnetic fields possess the same sym-metry. 7 Electrovacuum space-times with two-dimensionalAbelian isometry group G γ = c = 1): R ik − Rg ik + Λ g ik = 8 πT (M) ik , T (M) ik = 14 π ( F il F kl − F lm F lm g ik ) ∇ k F ik = 0 , ∇ [ i F jk ] = 0 , (1)where i, j, k, . . . = 0 , , ,
3, metric signature is ( − + ++) and Λ is a cosmological constant.Most of the known today exact solutions of these equations possess the two-dimensional sym-metries. We consider here the whole class of solutions of (1) which admit a two-dimensionalAbelian isometry group which action is generated by two commuting non-null Killing vectorfields, denoted further as the isometry group G . Orbit space and the choice of coordinates.
If the space-time admits two-dimensional Abelianisometry group G , its action foliates this spacetime by two-dimensional non-null surfaces –the orbits of this group. A factor of this space-time manifold with respect to the action of G is a two-dimensional manifold called as the orbit space of this isometry group. In suchspace-times, the coordinate systems can be chosen so that the coordinates x and x are thecoordinates on the orbit space and another pair of coordinates x and x are the coordinateson the orbits. It is convenient to choose the coordinates x and x as natural parameterson the lines of two commuting Killing vector fields which generate the action of G and aretangent to its orbits. In these coordinates the components of a chosen pair of Killing vectorfields take the forms ξ (3) = ∂∂x , ξ (4) = ∂∂x or ξ k ( a ) = δ ka (2)where the index in parenthesis a = 3 , x µ = { x , x } and these are independent of x a = { x , x } . Metric and electromagnetic field components.
In the space-time with two commuting isome-tries (2), in the coordinates described above, the components of metric and the Maxwelltensor of electromagnetic field can be presented in a general form ds = g µν dx µ dx ν + g ab ( dx a + ω aµ dx µ )( dx b + ω bν dx ν ) ,F ik = F µν F µb F aν F ab ! , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) µ, ν, . . . = 1 , a, b, . . . = 3 , g µν , g ab , ω aµ , F µν , F µa = − F aµ and F ab depend on x µ = { x , x } only. It is importantthat we do not preset in advance if the time coordinate is among ( x , x ) or it is among8 x , x ) and therefore, we consider in a unified manner both possible classes of fields –the stationary fields with one spatial isometry and time-dependent fields with two spatialisometries. Conformal coordinates on the orbit space of G . In (3), the components g ab determine themetric on the orbits of the isometry group G and the components g µν – the metric on theorbit space. It is well known that on a two-dimensional surface the local coordinates canbe chosen so that its metric takes a conformally flat form. We use this convenient choice oflocal coordinates on the orbit space x µ = { x , x } for which g µν = f η µν , η µν = (cid:18) ǫ ǫ (cid:19) , ǫ = ± ,ǫ = ± , ǫ ≡ − ǫ ǫ (4)where, by definition, f >
0, and we use the sign symbols ǫ and ǫ to consider in a unifiedmanner all possible cases of the signature of metric on the orbits. Namely, for stationary fieldsfor which the time coordinate is among the coordinates x a = { x , x } and both coordinates x µ are space-like, the values of the symbols ǫ and ǫ coincide, but for other cases, in whichthe time coordinate is among the coordinates x µ = { x , x } , the symbols ǫ and ǫ take theopposite values. Therefore, for the sign symbol ǫ introduced in (4) and for its ”square root” j which will be used further we have ǫ = (cid:26) − j = (cid:26) , for ǫ = 1 ,i, for ǫ = − . We call also the case ǫ = 1 as the hyperbolic case and ǫ = − − + ++), we have ǫ = − ǫ = 1 for ǫ = 1 and ǫ = ǫ = 1 for ǫ = −
1. For Levi-Civitasymbol determined on the orbit space x µ = { x , x } we have: ε µν = (cid:18) − (cid:19) , ε µν = η µγ ε γν = (cid:18) ǫ − ǫ (cid:19) , ε µν = ε νγ η γν = − ǫ (cid:18) − (cid:19) . The conditions (4) determine the choice of coordinates x µ up to arbitrary transformationsof the form ( x + jx ) → A ( x + jx ) and ( x − jx ) → B ( x − jx ) which transform theconformal factor f defined in (4) as f → A ′ B ′ f . This choice of coordinates can be specifiedmore, using for these coordinates the functions determined by space-time geometry itself. Metric on the orbits of G . The 2 × g ab of the metric (3) de-scribes the metric on the orbits of the isometry group G . In the symmetry reduced Einstein- Maxwell equations, these components will play the role of dynamical variables for gravi-tational field. However, it is more convenient to use instead of these variables, a function α > k g ab k ≡ ǫα (5)and a 2 × k h k which components with lower indices coincide with those for k g k : h ab ≡ g ab , but following Kinnersley [3], its indices will be rised and lowered like the two-component spinor indeces.Then, the components of k h k with lower, mixed and upper indices9ill possess the expressions h ab ≡ g ab , h ab = − h ac ǫ cb ,h ab = ǫ ac h cb , h ab = h ac ǫ cb , where ǫ ab = ǫ ab = (cid:18) − (cid:19) (6)It is worth to mention here also some useful relations:det k h ab k = − ǫα , h cc = 0 , h ac h cb = − ǫα δ ab , g ab = ǫα − h ab . (7) Components of the Ricci tensor for metrics (3).
In our notations, the components of theRicci tensor for metrics (3) possess the expressions R µν = 1 f (cid:2) − ∂ δ (cid:0) f δ f (cid:1) η µγ + f γ α µ + f µ α γ − α δ f δ η µγ f α − α µγ α − ǫ α ∂ µ h cd ∂ γ h dc (cid:3) η γν + ǫ f (cid:0) T c h cd T d (cid:1) δ µν + ω cµ R cν ,R aµ = − ǫ f α ε µγ ∂ γ (cid:18) αf h ac T c (cid:19) , where T a ≡ ε µν ∂ µ ω aν ,R ab = − ǫ f α η γδ ∂ γ (cid:2) α∂ δ g ac g cb (cid:3) − ǫ f g ac T c T b − R aγ ω bγ , (8)where f µ = ∂ µ f , α µ = ∂ µ α , α µν = ∂ µ ∂ ν α and f γ = η γδ f δ ; the functions f > α > Self-dual Maxwell tensor and its complex vector potential.
The Maxwell tensor and its dualconjugate can be combined in the self-dual Maxwell tensor + F ik = F ik + i ∗ F ik , ∗ F ik = 12 ε iklm F lm , + F ik = i ε iklm + F lm , (9)where ε iklm = e iklm / √− g is the four-dimensional Levi-Civita tensor and e iklm is a four-dimensional antisymmetric Levi-Civita symbol such that e = 1. For this tensor, a pairof Maxwell equations in the second line of (1) can be combined in a complex equation whichis equivalent to the existence of a complex vector potential ε iklm ∇ k + F lm = 0 , ⇒ + F ik = ∂ i Φ k − ∂ k Φ i . (10)For spacetimes which admit the isometry group G , provided the electromagnetic fieldsshare this symmetry, the Lie drivatives of the Maxwell tensor along the Killing vector fields ξ ( a ) vanish. This means that in the coordinates (3) the components of self-dual Maxwelltensor are independent of the coordinates x a , but in general, this should not be necessarytrue for the components of its vector potential Φ i . In the coordinates (3), the componentsof Φ i split in two parts Φ i = { Φ µ , Φ a } so that + F µν = ∂ µ Φ ν − ∂ ν Φ µ , + F µb = ∂ µ Φ b − ∂ b Φ µ , + F bc = ∂ b Φ c − ∂ c Φ b . (11)10or later convenience, we present here also the general expressions for the components ofself-dual Maxwell tensor with upper indices + F µν = 1 f η µγ (cid:18) + F γδ + ω cγ + F δc − + F γc ω cδ + ω cγ + F cd ω dδ (cid:19) η δν + F µa = − + F µγ ω aγ + 1 ǫα f η µγ ( + F γd − ω cγ + F cd ) h da + F ab = 1 α h ac + F cd h db − ǫα f h ac ( + F cγ − + F cd ω dγ ) η γδ ω bδ − ǫα f ω aγ η γδ ( + F δc − ω dδ + F dc ) h cb + ω aγ + F γδ ω bδ . (12)To write the equations (1) for spacetimes with metric and electrpmagnetic fields (3), weneed now the expressions for the components of the energy-momentum tensor. The energy-momentum tensor for electromagnetic fields.
It is easy to check that in generalthe energy-momentum tensor for electromagnetic field (1) can be expressed in terms of self-dual Maxwell tensor and its complex conjugate: T (M) ik = − π + F il + F lk (13)In the coordinates (3) the components of this tensor possess the expressions T (M) µν = i πf α (cid:2) e o ( ε γδ ∂ γ Φ δ ) δ µν − ε νγ ∂ γ Φ c ∂ µ Φ c (cid:3) ,T (M) aµ = i πf α ε µγ ∂ γ (cid:0) e o Φ a − e o Φ a (cid:1) ,T (M) ab = i πf α ε γδ (cid:2) ∂ γ Φ a ∂ δ Φ b − e o ( ∂ γ Φ δ ) δ ab (cid:3) . (14)Now, the Einstein - Maxwell field equations (1) can be written, using the above notationsand the expressions (8) and (14). G -symmetry reduced Einstein - Maxwell field equations In this section we consider a complete set of Einstein - Maxwell equations for electrovaccumfields (with a cosmological constant Λ) which admit an Abelian isometry group G . Wedescribe the structure of these equations and determine their constraint and dynamicalparts. The general solution of constraint equations presented below in this section allows todetermine the non-dynamical degrees of freedom of gravitational and electromagnetic fieldsand find the general form of G -symmetry reduced dynamical equations. Finally, we showin this section that for some special values of integration constants and Λ = 0, the non-dynamical degrees of freedom become pure gauge and vanishing for appropriate choice ofgauges and coordinates. In this case, it occurs that the physical condition of vanishing ofnon-dynamical degrees of freedom of gravitational and electromagnetic fields is equivalent topure geometrical condition of 2 × G -symmetries to the corresponding subclasses whichdynamical equations are completely integrable.11 .2.1 Non-dynamical degrees of freedom of electromagnetic field It is easy to see that the independence of the components of self-dual Maxwell tensor on thecoordinates x a implies a specific structure of some of the components of this tensor: + F ab = e o ǫ ab , e o = const, (15)where e o is an arbitrary complex constant. This means that the corresponding componentsof complex vector potential Φ i are the linear functions of the coordinates x a . Up to arbitrarygauge transformation, these components possess the structureΦ µ = Φ µ ( x γ ) , Φ a = 12 e o ǫ ca x c + φ a ( x γ ) . (16)In this class of fields, the real and imaginary parts of arbirary complex constant e can beconsidered as determining two non-dynamical degrees of freedom of electromagnetic field.The corresponding components of electromagnetic potential representing these degrees offreedom can be calculated from the constraint part of self-dual Maxwell equations. In terms of a complex vector potential the Maxwell equations (10) are equivalent to the firstorder duality equations + F ik = i ε iklm + F lm , + F ik = ∂ i Φ k − ∂ k Φ i . (17)In the coordinates (3), using the expressions (12) and (16), we can split the duality equations(17) in two parts. One of them is the so called constraint equation ε µν ∂ µ Φ ν = ε µν ∂ µ Φ c ω cν − e o (cid:2) ε µν ǫ cd ω cµ ω dν + iǫα − f (cid:3) , (18)One can see here that the components Φ µ of a complex electromagnetic potential representthe non-dynamical degrees of freedom of electromagnetic field because these components aredetermined (up to arbitrary gauge transformation Φ µ → Φ µ + ∂ µ χ ) by the equation (18),provided the solution for Φ a , α , ω aµ and f is known. Another part of (17) in the metrics (3)reduce to a system of duality equations (similarly to the Kinnersley equations [3]) ∂ µ Φ a − e o ǫ ca ω cµ = iα − ε µγ h ac ( ∂ γ Φ c − e o ǫ dc ω dγ ) , (19)where the components Φ a of a complex potential play the role of dynamical variables. The elecrovacuum Einstein - Maxwell field equations (1) for spacetimes with G -symmetriescan be written using the expressions (8), (13), (14). In this form, these equations consist ofthree subsystems coupled to each other, but possessing different structures.12 hree subsystems of Einstein-Maxwell equations. For electrovacuum fields the trace T (M) kk of the energy-momentum tensor vanishes and therefore, the curvature scalar R = 4Λ andelectrovacuum Einstein - Maxwell equations can be written in the form R µν = 8 πT (M) µν + Λ δ µν , R aν = 8 πT (M) aν , R ab = 8 πT (M) ab + Λ δ ab , (20)In general, for spacetimes with the isometry group G , the three subsystems of equations (20)are coupled to each other and possess rather complicate structure. However, these structurescan be simplified when one will observe that the second subsystem in (20) represents aconstraint equations which can be solved explicitly and its solution allows to identify some”non-dynamical” degrees of freedom for gravitational field. Constraint equations for ω aµ . The second subsystem in (20), in accordance with theexpressions (8) and (14), takes the form of constraint equations ∂ µ h αf h ac T c + 2 iǫ ( e o Φ a − e o Φ a ) i = 0 (21)which can be solved explicitly and the general solution of these equations includes arbitraryreal two-dimensional vector constant of integration: αf h ac T c = − iǫ ( e o Φ a − e o Φ a ) + ℓ a , where ℓ a = const. (22)This relation allows to exclude T a from the other equations (20) and besides that, using thedefinition of T a given in (8) we obtain the equation for ω aµ of the form ε µν ∂ µ ω aν = ǫα − f h ac L c , L a = ℓ a − iǫ ( e o Φ a − e o Φ a ) (23)which allows to calculate the metric functions ω aµ (up to an arbitrary gauge transformation ω aµ → ω aµ + ∂ µ ω a ) in terms of h ab , Φ a , f and the constant vector ℓ a . Thus, we can considerthe metric functions ω aµ , together with the components Φ µ of complex electromagneticpotential, as the non-dynamical degrees of fredom for gravitational and electromagneticfields determined (up to the gauge transformations) by the constraint equations (23) and(18) respectively with the arbitrarily chosen constants ℓ a and e o . Constraint equations for the conformal factor f With the constraint equations (18) and(23), the first subsystem in (20) reduces to f µ α ν + f ν α µ f = F µν + Xη µν (24)where the components of F µν are independent of f and the function X includes the secondderivatives of f . These functions possess rather long expressions: F µν ≡ α µν + ǫ α ∂ µ h cd ∂ ν h dc + iε νγ h M γ c ǫ cd M µd − M γc ǫ cd M µd i ,X ≡ ∂ δ (cid:0) αf δ f (cid:1) + 2 f α Λ − ǫe o e o fα − fα ( L c h cd L d ) . (25)13here F µν is symmetric, X and F µν are real and the components M are defined as M µa ≡ ∂ µ Φ a − e o ω aµ , M µa = M µc ǫ ca . (26)Instead of the equations (24), it is more convenient to consider an equivalent system whichconsists of the non-diagonal component ( . . . ) , the combination ( . . . ) + ǫ ( . . . ) and acontraction of (24) with η µν , i.e. the equation ( . . . ) − ǫ ( . . . ) . It is easy to see that thefirst two of these equations do not include the second derivatives of f and these equationscan be presented in the form ∂ ff = F ( h ab , ω aγ , Φ a ) , ∂ ff = F ( h ab , ω aγ , Φ a ) , (27)where the explicit expressions for F and F in terms of the components of F µν defined in(25), with the notations α = ∂ α and α = ∂ α , are F = α ( F + ǫ F ) − ǫα F α − ǫα ) , F = − α ( F + ǫ F ) + 2 α F α − ǫα ) . Instead of the third of the mentioned above equations, it is more convenient to consider theequation for a Ricci scalar R −
4Λ = 0 which, multiplied by ( − f α ), reads as α∂ γ (cid:16) f γ f (cid:17) + 2 η γδ ∂ γ ∂ δ α − f α ( L c h cd L d ) + ǫ α η γδ ∂ γ h cd ∂ δ h dc + 4 αf Λ = 0It is remarkable that after a very long calculations it can be shown that this equationas well as the compatibility condition ∂ F − ∂ F = 0 of a pair of equations (27) aresatisfied identically, provided all field variables satisfy the constraint equations (18), (23)and (27) as well as the self-dual Maxwell equations (17) and the last subsystem in (20), i.e. R ab = 8 πT ab + Λ δ ab which structure is considered just below. Similarly to the self-dual Maxwell equations (17), following Kinnersley [32], one can expressthe projections of the Ricci tensor R i ( a ) in terms of self-dual bivectors + K ij ( a ) : R i ( a ) = i ε ijkl ∇ j + K kl ( a ) , + K ij ( a ) ≡ K ij ( a ) + i ∗ K ij ( a ) , K ij ( a ) ≡ ∇ i ξ j ( a ) − ∇ j ξ i ( a ) (28)where the suffix in parenthesis numerates the Killing vector fields (2), and the dual bivectors ∗ K ij ( a ) are defined similarly to a dual Maxwell tensor (10). For the corresponding projectionsof the energy-momentum tensor we obtain T i ( a ) = ε ijkl ∇ j + S kl ( a ) + 18 π + F ij ( L ξ ( a ) Φ j ) , + S ij ( a ) = − i π Φ ( a ) + F ij , Φ ( a ) ≡ Φ k ξ k ( a ) , (29)where a bar means complex conjugation, bivectors + S kl ( a ) with a = 3 , L ξ ( a ) means the Lie derivatives with respect to the Killing vector fields ξ ( a ) . Therefore, the secondand third subsystems of Einstein - Maxwell equations (20) take the form ε ijkl ∇ j + H kl ( a ) = − i + F ij ( L ξ ( a ) Φ j ) − i Λ δ i ( a ) , where + H ik ( a ) ≡ + K ik ( a ) + 32 iπ + S ik ( a ) . (30)14or each Killing vector field, + H ij ( a ) are self-dual bivectors: + H ij ( a ) = i ε ijkl + H kl ( a ) (31)In contrast to the case of self-dual form of Maxwell equations (17), the selfdual bivectors + H ij ( a ) do not possess the potentials because the right hand side of the equation (30) doesnot vanish, besides the special cases (Λ = 0, e o = 0) or (Λ = 0, Φ µ = 0). However, in generalsome matrix potentials associated with self-dual bivectors + H ij ( a ) can be constructed andthis leads to some (more complicate) self-dual form of dynamical equations for gravitationalfield. To obtain these equations, we consider the components of self-dual bivectors definedabove, in the coordinates adapted for G -symmetry of space-time geometry and fields. Inthese coordinates and our notations described earlier in this paper, the expressions (28) forRicci tensor components read as R µa = i f α ε µγ ǫ cd ∂ γ + K cda , R ba = i f α ε γδ ǫ bc ∂ γ + K δca , (32)where we omit the parenthesis in the last suffix which numerates the Killing vector fields.The components of the bivectors + K ij ( a ) possess the expressions + K µa ( b ) = ∂ µ h ab + iα − ε µν h ac ∂ ν h cb + iǫǫ ca ω cµ L b + K ab ( c ) = iǫǫ ab L c , L a ≡ αf h ac T c (33)The Einstein - Maxwell equations corresponding to the first part of Ricci tensor components(32) were considered in the previous section where it was shown that these equations leadto constraint equations for metric functions ω aµ and the explicit expressions (23) for thefunctions L a . Therefore, we have to consider now the Einstein - Maxwell equations corre-sponding to the second part of equations (32). Using the explicit expressions for componentsof the energy-momentum tensor of electromagnetic field (14), the constraint equations (18)and self-dual Maxwell equations (19), we can write these equations as ε γδ ∂ γ (cid:0) + K δab + 2Φ b ∂ δ Φ a + 2 e o Φ δ δ ab (cid:1) = 2 if α Λ δ ab (34)where the index b was rised as a spinor index: + K δab = − + K δac ǫ cb . Going further, we split thismatrix equation into its ”trace” and ”traceless” parts. The first of them leads to a dynamicalequation for α : η γδ ∂ γ ∂ δ α + f α ( L c h cd L d ) + 2 f (cid:0) α Λ + ǫe o e o α (cid:1) = 0 . (35)while the traceless part of (34) can be presented in the form ε γδ ∂ γ (cid:2) N δab + iǫǫ ca ω cµ ǫ bd ( ℓ d − iǫe o Φ d ) + Z δ δ ab (cid:3) = 0 , (36)15here we introduced the notations ( M µa = ǫ ca M µc and M µa were defined in (26)): N µab = ∂ µ h ab + iα − ε µν h ac ∂ ν h cb + 2Φ b M µa Z µ ≡ iǫε µγ ∂ γ α + i ǫω cµ L c − Φ c ∂ µ Φ c . (37)Taking into account the self-dual form of Maxwell equations (19), it is easy to see that thecomponents N µab and M µa satisfy the ”two-dimensional” duality relations: N µab = iα − ε µν h ac N νcb M µa = iα − ε µν h ac M νc . (38)These equations would coincide with the duality equations which were constructed by Kin-nersley [3] for electrovacuum, provided N µab and M µa would have the potentials, but this isnot so in a more general case considered here. However, the equations (36) mean that theexpression in square brackets possess some matrix potential b H ab so that ∂ µ b H ab = N µab + iǫω aµ ( ℓ b − iǫe o Φ b ) + Z µ δ ab . (39)where ω aµ = ǫ ca ω cµ and ℓ b = ǫ bc ℓ c . Therefore, we can express the components of N µab interms of the components of this potential as N µab = ∂ µ b H ab − iǫω aµ ( ℓ b − iǫe o Φ b ) − Z µ δ ab (40)Substitution of these expressions into the duality relations (38) leads to some modificationof Kinnesrley self-dual equations for gravitational field in our more general case in which thenon-dynamical degrees of freedom do not vanish. These modified Kinnersley’s equationsfor the dynamical variables h ab and Φ a takes the form ∂ µ b H ab − Z µab = iα − ε µν h ac ( ∂ ν b H cb − Z νcb ) , Z µab ≡ iǫω aµ ( ℓ b − iǫe o Φ b ) + Z µ δ ab ,∂ µ Φ a − e o ω aµ = iα − ε µν h ac ( ∂ ν Φ c − e o ω cν ) ,∂ µ b H ab = ∂ µ h ab + iα − ε µν h ac ∂ ν h cb + 2Φ b ( ∂ µ Φ a − e o ω aµ ) + Z µab . (41)Unfortunately, these duality equations and the dynamical equation (35) for the function α do not represent a closed system of dynamical equations for the dynamical variables h ab andΦ a . The non-dynamical degrees of freedom and the corresponding constants – real ℓ a andcomplex e o also enter the equations (35) and (41). In this general case, the decoupling ofthe equations does not take place and the complete set of Einstein - Maxwell field equationsincludes, besides the dynamical equation (41) for h ab and Φ a and the equation (35) for thefunction α , also the constraint equations (23) for ω aµ and (27) for the conformal factor f . As we shall see below, the matrix potential b H ab in the limit of vanishing of non-dynamcal degrees offreedom of gravitational and electromagnetic fields, does not coincide with the known Kinnersley’s matrixpotential H ab but differes from this by some additional terms. Therefore, we say here not about a generalizedpotential, but about a modified one. .2.5 Dynamical degrees of freedom and dynamical equations In this subsection we consider the second order equations which arise from G -symmetryreduced Einstein - Maxwell equations for the components of metric h ab and complex electro-magnetic potential Φ a . Though in general case considered in this paper the dynamicalequations for these field components do not decouple from the equations for other fieldcomponents, we call here these field variables as the dynamical degrees of freedom becausein the most important subcases, in which the other degrees of freedom of fields vanish andthe field equations become integrable, just these field components play the role of dynamicalvariables for gravitational and electromagnetic fields. The equations (41) imply the secondorder dynamical equations for h ab and Φ a which can be presented in a symmetric form η µν ∂ µ (cid:2) α − ( ∂ ν h ac ) h cb (cid:3) + iǫε µν (cid:0) M µa M ν b − M µa M νb (cid:1) + ǫfα L a L c h cb + 2 f (cid:0) ǫα Λ + e o e o α (cid:1) δ ab = 0 ,η µν ∂ µ (cid:2) α − M νc h cb (cid:3) − ie o α − f h bc L c = 0 . (42)where L a and M γa were defined in (23) and (26) respectively. The trace of the left hand sideof the first equation in (42), leads again to dynamical equation (35) for α : η γδ ∂ γ ∂ δ α + f α ( L c h cd L d ) + 2 f (cid:0) α Λ + ǫe o e o α (cid:1) = 0 . Using this equation and introducing the new matrix variable k ab such that k ab ≡ α − h ab , k cc = 0 , k ac k cb = − ǫδ ab , det k k ab k = − ǫ, (43)the equations (42) can be simplified and presented in the ”tracelrss” form η γδ ∂ γ (cid:2) α ( ∂ δ k ac ) k cb (cid:3) + iǫε γδ (cid:0) M γa M δb − M γa M δb (cid:1) − ǫfα (cid:2) L a L c k cb −
12 ( L c k cd L d ) δ ba (cid:3) = 0 ,η µν ∂ µ (cid:2) M νc k cb (cid:3) − ie o α − f k bc L c = 0 . (44)It is interesting to note here that the equations (42) or the equations (44) do not representa closed systems for dynamical variables h ab and Φ a or k ab and Φ a respectively becausethe non-dynamical variables f and ω aµ (through the structure of M µa ) also enter theseequations. Having the aim to decouple the dynamical equations for h ab and Φ a from the other(constraint) equations for other (non-dynamical) degrees of freedom, we could solve easily theequation (35) with respect to f and thus to express it in terms of h ab and Φ a only. However,substitution of this expression for f into the dynamical equations does not solve completelythe problem of decoupling of these equations. Besides that, we have to understand, if someconstraints on the field variables can arise if we substitute this expression for f into theconstraint equations (27). Nontheless, despite of the absence in general of decoupling of theequations (42) as the equations for h ab and Φ a from the constraint equations for ω aµ and f , we call here the field variables h ab and Φ a as dynamical variables and the equations (42)as dynamical equations because in the most important subcase in which the constants ℓ a e o vanish, such decoupling take place and leads to the integrability of the dynamicalequations (42). To understand what happens in the other cases in which such decouplingdo not take place, we restrict our considerations by some most interesting particular caseswhich illustrate this complicated structure of the equations. As it was shown in the previous section, in general class of electrovacuum spacetimes whichmetrics admit the Abelian isometry group G , gravitational and electromagnetic fields canpossess (besides the well known dynamical degrees of freedom) some non-dynamical degreesof freedom. The gravitational and electromagnetic field components corresponding to thesenon-dynamical degrees of freedom can be determined completely as the solutions of the con-straint equations which include a set of arbitrary constants which characterise these degreesof freedom. This set of constants includes the components of constant real two-dimensionalvector ℓ a which non-zero components lead to such property of space-time geometry thatthe orbits of the isometry group G are not 2-surface orthogonal and therefore, the space-time metric can not possess a block-diagonal structure, and the complex constant e o whichpossess the electromagnetic nature. Its non-zero value leads to more complicate structureof electromagnetic field. A striking similarity of the structure of the field equations forthese electrovacuum fields with non-dynamical degrees of freedom and of elecrovacuum fieldequations with G isometry group and non-vanishing cosmological constant made it reason-able to include in our considerations the corresponding class of fields with non-vanishingcosmological constant as one more non-dynamical degree of freedom of gravitational field. Λ = 0 ) In this subsection we consider physically important subclass of the described above class ofspacetimes. This is a class of pure vacuum metrics with cosmological constant and with theAbelian isometry group G . For this subclass, we choose for simplicity ℓ a = 0 , e o = 0 , Φ a = 0 , Λ = 0 . (45)With these conditions, the equation (42) can be presented in the form η µν ∂ µ (cid:2) α − h ac ∂ ν h cb (cid:3) + 2 ǫαf Λ δ ab = 0 (46)and, taking the trace of (46) or using the condition (45) in (35), we obtain f = − η γδ ∂ γ ∂ δ α α Λ (47)Substituting this expression into (46) we obtain a closed system of dynamical equations fora matrix k ab ≡ α − h ab which algebraic properties was already mentioned in (43): η µν ∂ µ (cid:2) α k ac ∂ ν k cb (cid:3) = 0 , k cc = 0 , det k k ab k = ǫ, k ac k cb = − ǫδ ab . (48)18 ynamical equations in terms of scalar functions. Using (47) and parametrization k g ab k ≡ k h ab k = ǫ (cid:18) H H Ω H Ω H Ω + ǫα H − (cid:19) , (49)where ǫ = ± g and the functions α > H > g /g , theequations (46) can be reduced to a pair of equations for two scalar functions H and Ω: η µν (cid:0) H µν + α µ α H ν − α µν α H − H µ H ν H − ǫ H α Ω µ Ω ν (cid:1) = 0 ,η µν (cid:0) Ω µν − α µ α Ω ν + 2 H µ H Ω ν (cid:1) = 0 . (50)Here and below, the greek letters as well as the numbers 1 and 2 in the suffices of scalarfunctions mean the partial derivatives with respect to the coordinates x µ = { x , x } . Generalization of the Ernst equations.
Sometimes it is convenient, similarly to pure vac-uum case [33], to rewrite the second of the equations (50) in a different form which allowsto introduce the potential φ : η µν ∂ µ ( α − H ∂ ν Ω) = 0 ⇒ ∂ µ Ω = αH − ε µν ∂ ν φ. Using this potential, the equations (50) can be presented in the form η µν (cid:0) H µν + α µ α H ν − α µν α H − H µ H ν H + 1 H φ µ φ ν (cid:1) = 0 ,η µν (cid:0) φ µν + α µ α φ ν − H µ H φ ν (cid:1) = 0 . (51)Following again pure vacuum procedure [33] we can combine these equations into the Ernst-like equation for a complex potential E ≡ H + iφ which takes the form(Re E ) η µν (cid:0) E µν + α µ α E ν − α µν α Re E (cid:1) − η µν E µ E ν = 0 (52)where the only difference with vacuum Ernst equation is the presence of the last term in theparenthesis, while for vacuum vith Λ = 0 this term vanishes because in this case η µν α µν = 0. Complete system of field equations.
Another compact form of the equations (51) arises ifwe use a new unknown function ψ such that H = αe ψ . Then we obtain η µν ( ψ µν + α µ α ψ ν + α − e − ψ φ µ φ ν ) = 0 ,η µν ( φ µν − α µ α φ ν − ψ µ φ ν ) = 0 . (53)Besides the equations (53) or, equivalently, the equation (52), some supplement equationsfor the functions ψ , φ and α arise from the equations (27) and (47). These supplementequations possess the structures ( ψ + ǫψ + α − e − ψ ( φ + ǫφ ) = K ψ ψ + 2 α − e − ψ φ φ = K (54)19here the right hand sides K , K are determined completely by the function α ( x , x ): K ( I )1 = 2 B √ αA (cid:20) α (cid:0) A √ αB (cid:1) + ǫα (cid:0) A √ αB (cid:1) (cid:21) + 2( α + ǫα ) AαB , K ( I )2 = 2 B √ αA (cid:20) α (cid:0) A √ αB (cid:1) + α (cid:0) A √ αB (cid:1) (cid:21) + 4 α α AαB , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A = α − ǫα ,B = α − ǫα . (55)It is worth noting here that a complete system of field equations which we obtain here, i.e.the equations (53) and (54) together with (55) does not admit a direct passage to the limitΛ = 0 and to the corresponding condition A = 0, because we use the expression (47) for theconformal factor where it was assumed that Λ = 0.A complete system of field equations (53) and (54) together with (55) possess rathercomplicate structure which does not allow, for example, to analyse the general structure ofits space of solutions. However, we can clarify the situation a bit more, if we restrict ourconsiderations by a class of fields with diagonal metrics – static fields ( ǫ = −
1) or, e.g., planewaves with linear polarization ( ǫ = 1). The fields with diagonal metrics.
For the class of fields with diagonal metrics we haveΩ = 0 , φ = 0and the equations (53), (54) and (55) reduces by the substitution H = αe ψ to the equations ψ − ǫψ + α α ψ − ǫ α α ψ = 0 ,ψ + ǫψ = K ψ ψ = K (56)where the functions K and K are determined in terms of the function α by the expressions(55), i.e. K = K ( I )1 and K = K ( I )2 . The last two equations in (56) can be solved directlywith respect to ψ and ψ . This leads to the following solutions: ψ = ± √ r K + q K − ǫ K , ψ = ± √ K q K + p K − ǫ K . (57)where one should choose only the upper signs or only the lower signs. (Another solutionarise if we interchange the expressions for ψ and ψ .) Besides that, as it is easy to see, anaditional condition arises from (56). This condition is K > ǫ = 1 . The expressions (57) for ψ and ψ give rise also to a pair of equations which should besatisfied by the functions K and K . These equations arise from the first equation in (56)and the compatibility condition ∂ ( ψ ) − ǫ∂ ( ψ ) + α α ψ − ǫ α α ψ = 0 , ∂ ψ − ∂ ψ = 0 . K and K : ∂ K − ∂ K + α α K + α α ( −K + p K − ǫ K ) = 0 ,∂ K − ǫ∂ K + α α K − ǫα α ( K + p K − ǫ K ) = 0 . (58)where K and K were expressed in (55) in terms of the function α and its derivetives.However, it is easy to show that in (58) we have only one equation for α , because certainlinear combination of the equations (58), which does not include the square root, reduces toidentity in view of (55). Therefore, we can consider only one of the equations (58), e.g. thefirst one, which can be presented in the form ∂ ( α K ) − ∂ ( α K ) = α q K − ǫ K (59)where K = K ( I )1 and K = K ( I )2 which were defined in (55). Taking ”square” of this equationleads to a polynomial equation for α and its derivatives up to the fourth order. This polynomcan be reduced a bit because it can be factorized and the multiplier α ( α − ǫα ) canbe canceled. As a result, we obtain still rather complicate, highly nonlinear differentialequation for α which left hand side is a polynom with respect to α and its derivatives ofall orders up to the fourth one. This polynomial is homogeneous with respect to α andits degree of homogeneity is equal to 7. We do not present here the explicit form of thispolynomial differential equation because it is rather long. Though a complicate structure ofthis equation does not allow to expect that this equation for α may occur to be completelyintegrable, its existence shows the structure of the space of solutions of the class of diagonalvacuum metrics with cosmological constant which possess the Abelian isometry group G .Namely, the derived nonlinear fourth-order differential equation for α actually representthe only dynamical equation, which solutions determine completely all corresponding metriccomponents using the relations derived above. However, it is necessary to mention, that thereare some restrictions on the choice of the solutions of dynamical equation for α . Besides anobvious condition α >
0, in the hyperbolic case ( ǫ = 1) the condition K > K is determined in terms of α in (55). Another condition for the choice ofthe solution of the mentioned above squared (polynomial) differential equation is that thissolution should satisfy also the original equation (59) which space of solutions includes only”half” of the solution space of the squared equation. G ( ω aµ = 0 ) To consider the simplest subcase of general class of metrics with non-dynamical degrees offreedom discussed in this paper, in which the orbits of the isometry group G are not 2-surface-orthogonal and therefore, the metrics do not possess the 2 × ω aµ = 0), we chose the case of vacuum fields with G -symmetry for which we putΛ = 0 , Φ a = 0 , e o = 0 , ℓ a = 0 . Using a freedom of SL (2 , R ) linear transformations of the coordinates x a = ( x , x ), wecan reduce, without any loss of generality, the components of the constant vector ℓ a to the21implest form and assume for simplicity that in the transformed coordinates the matrix h ab of metric components is diagonal. Thus, we put ℓ a = { ℓ o , } , Ω = 0 . In this case, similarly to the previous case of vacuum fields with cosmological constant, wecan find again the conformal factor f from the equation (35): f = − ǫǫ ǫ ℓ o αHA where the sign ǫ and the function H arise from the parametrization (49) for h ab . Thefunctions A and B were defined in (55). It is very surprising that in the present casethe general equations (42) after a different sabstitution for H = α e ψ reduce to the sameequations (56), but the expressions for K and K differ from the expressions (55): K ( II )1 = K ( I )1 + 7 α + ǫα α , K ( II )2 = K ( I )2 + 14 α α α . (60)Further, the basic equations for this class can be constructed using the same expressions(56) – (59), where we have to use K = K ( II )1 and K = K ( II )2 . As a result, the polynomialequation for α , which we obtain ”squaring” (59), will differ from (59) only by numericalcoefficients. e o = 0 ) For simplicity, we consider here the class of elecromagnetic fields with non-dynamical electro-magnetic degrees of freedom for which the metric h ab on the orbits of the isometry groupis diagonal and besides that, the projections of complex electromagnetic potential on theorbits, the metric components which make the orbits to be not 2-surface orthogonal and thecosmological constant vanish, i.e. for this class of fieldsΩ = 0 , Φ a = 0 , ℓ a = 0 , Λ = 0 , e o = 0 . In this case, similarly to the previous cases, we find the conformal factor f from (35): f = − ǫǫ α e o e o A It is remarkable that in this case the same substitution as in the case of presence of cosmo-logical constant, i.e. H = αe ψ leads to the same dynamical equations (56) for ψ and α , inwhich, however, the expressions for K K , denoted here as K ( III )1 and K ( III )2 , differ from(58) and are determined by the expressions K ( III )1 = K ( I )1 + 4 α + ǫα α , K ( III )2 = K ( I )2 + 8 α α α . (61)Further reduction of these dynamical equations follows the same equations (56) – (59), where K = K ( III )1 and K = K ( III )2 . This leads again to the only dynamical equation for α whichdiffers from the similar equations derived in two previous cases only by numerical coefficients.22 .4 Examples of solutions with (I) Λ = 0 , (II) ω aµ = 0 , (III) e o = 0 In this section, we describe a construction of (presumably) new families of solutions of allthree types (I), (II) and (III) and give explicitly the simplest examples.The particular solutions for all three classes of fields considered above can be constructedin a unified way, if we begin with the system of equations for the functions ψ and α ψ − ǫψ + α α ψ − ǫ α α ψ = 0 ,ψ + ǫψ = K , ψ ψ = K , (62)which arises in the same form for all three classes of fields, but with different values of K and K , defined respectively in (55), (60) or (61). However, introducing the parameter n ,which takes the values n = 1 for the class (I), n = − n = − K and K can be presented in a convenient general form K = 2 B √ αA (cid:20) α (cid:0) A √ αB (cid:1) + ǫα (cid:0) A √ αB (cid:1) (cid:21) + ( α + ǫα ) α (cid:16) AB + 1 − nα (cid:17) , K = 2 B √ αA (cid:20) α (cid:0) A √ αB (cid:1) + α (cid:0) A √ αB (cid:1) (cid:21) + 2 α α α (cid:16) AB + + 1 − nα (cid:17) , (63)where, as before, we use the notations A = α − ǫα B = α − ǫα . To solve the equations derived above, we use a simple ansatz in which the unknown functions ψ and α are assumed to be depending on a one unknown function only: ψ = ψ ( w ) , α = α ( w ) , w = w ( x , x ) . (64)Substitution of this ansatz into the first of the equations (62) solves this equation providedthe function w satisfies the following linear equation and the functions ψ ( w ) and α ( w ) satisfythe easily solvable relation w − ǫw = 0 , ψ ′′ + α ′ α ψ ′ = 0 ⇒ ψ ′ ( w ) = k o α ( w ) . (65)where k o is an arbitrary real constant. Then, we also obtain from (62) K = k o α ( w ) ( w + ǫw ) , K = 2 k o α ( w ) w w . (66)Taking into account the linear equation (65) for w , one finds that α ( w ) should satisfy anordinary differential equation2 α (cid:2) α ′ α ′′′ − ( α ′′ ) (cid:3) − (cid:2) k o + n ( α ′ ) (cid:3) α ′′ = 0 (67)23here the prime denotes a differentiation with respect to w . A standard sabstitution (thedot denotes a differentiation with respect to α ) α ′ = P ( α ) , α ′′ = ˙ P P, α ′′′ = ¨
P P + ˙ P P, (68)after a change of the independent variable α = e x transforms the equation (67) into P xx − (cid:0) k o P + n + 2 (cid:1) P x = 0 . (69)If we introduce the notations for auxiliary constants (where g o is an arbitrary real constant) h ± = P ± ( n + 2) R o , P ± = g o ± R o , R o = s g o + k o n + 2 , h o = k o ( n + 2) R o (70)and set in the expressions for P ± and h ± respectively( I ) : n = 1 , ( II ) : n = − , ( III ) : n = − , the solution for the above equations can be presented in a parametric form α = α o ( P − P + ) h + ( P − P − ) h − , e ψ = e ψ o (cid:18) P − P + P − P − (cid:19) h o , w = 2 α o n + 2 Z ( P − P + ) − h + ( P − P − ) h − dP, (71)where α o and ψ o are arbitrary real constants and the integral for w can be expressed in termsof the hypergeometric function. For the conformal factor f we have different expressions fordifferent cases: f ( I ) = − ǫ α o Λ ( w − ǫw )( P − P + ) − h + ( P − P − ) h − ,f ( II ) = 4 ǫǫ ǫ α o e ψ o ℓ o ( w − ǫw )( P − P + ) h o +2 h + ( P − P − ) − h o − h − ,f ( III ) = ǫǫ e o e o ( w − ǫw )( P − P + )( P − P − ) . (72)where we have to choose in the expressions for P ± and h ± the corresponding values n = 1, n = − n = −
3. The choice of the signs ǫ , ǫ , ǫ and of the solution w of the linearequation (65) should satisfy the only condition f >
0. Besides that, in the case (II) theequations for calculation of metric functions ω aµ (which presence is an obstacle for theexistence of 2-surfaces orthogonal to the isometry group orbits) take the form ∂ ω a − ∂ ω a = 4 ǫǫ α o ℓ o ( w − ǫw )( P − P + ) − h + ( P − P − ) h − δ a , (73)and in the case (III) we obtain the similar equation for calculation of the components Φ µ ofcomplex electromagnetic potential ∂ Φ − ∂ Φ = − iǫ α o e o ( w − ǫw )( P − P + ) − h + ( P − P − ) h − , (74)24here in the expressions for P ± and h ± in (73) we should choose n = −
6, and in (74)we should choose n = −
3. Each of these two equations can be solved if we use a gaugetransformations ω aµ → ω aµ + ∂ µ k a and Φ µ → Φ µ + ∂ µ φ k a φ are arbitrary functions.This allows to simplify the components ω aµ and Φ µ , choosing one of the components ω µ and one of the components Φ µ , as well as well as both components ω µ equal to zero.Then, nonvanishing components among ω µ and Φ µ can be expressed in quadratures (seethe examples given below).It is important to note here that though the constructed above solutions depend on arbi-trary choosing solution w ( x , x ) of the linear equation (65) and the constants of integration k o and g o , these solutions represent only finite-parametric families, while the arbitrary func-tion can be excluded from the solutions by an appropriate coordinate transformation. In the hyperbolic case ( ǫ = 1 ), we may choose ǫ = − x , x ) = ( t, x ) and introducethe null coordinates u = t − x and v = t + x , for which the metric on the orbit space takes theform g µν dx µ dx ν = − f dudv . Then the local transformations of these coordinates u → p ( u ), v → q ( v ), where p ( u ) and q ( v ) are arbitrary real functions with p ′ > q ′ >
0, leavesthis form of metric unchanged. The corresponding new time-like and space-like coordinatesare e x = ( p ( u ) + q ( v )) and e x = ( − p ( u ) + q ( v )) respectively. On the other hand, thefunction w ( x , x ) satifies the linear equation (65). In the null coordinates u and v the linearequation for w takes the form w uv = 0 and therefore, its solutions possess the structures w = e p ( u ) + e q ( v ) with some functions e p ( u ) and e q ( v ). If we choose in the mentioned justabove coordinate transformation p ( u ) = ± e p ( u ) and q ( v ) = ± e q ( v ), where the sign is chosenappropriately to have p ′ > q ′ >
0, we obtain that in new coordinates ( e x , e x ) thefunction w ( x , x ) takes one of the simplest forms w = x or w = x (where we omit the e onthe transformed coordinates). In these cases, we obtane physically different types of fields– e.g., w = t may correspond to time-dependent cosmological solutions, while w = x maycorrespond to spatially inhomogeneous static fields. In the elliptic case ( ǫ = − ), we may choose ǫ = 1 and ( x , x ) = ( x, y ) and introduce twocomplex conjugated to each other coordinates ξ = x + iy and η = x − iy , for which the metricon the orbit space takes the form g µν dx µ dx ν = f dξdη . Then the local transformations of thesecoordinates ξ → p ( ξ ), η → q ( η ), where p ( ξ ) and q ( η ) are arbitrary holomorphic functions,leaves this form of metric unchanged. The corresponding new space-like coordinates are e x = ( p ( ξ ) + q ( η )) and e x = i ( − p ( ξ ) + q ( η )). On the other hand, w ( x , x ) satisfiesthe linear equation (65). In the coordinates ξ and η the linear equation for w takes theform w ξη = 0 and therefore, its solutions is w = e p ( ξ ) + e q ( η ) with some functions e p ( ξ )and its complex conjugate e q ( η ). After the coordinate transformation with p ( ξ ) = e p ( ξ ) and q ( η ) = e q ( η ), the function w ( x , x ) takes one of the simplest forms w = x or w = x (asbefore, we also omit e on the transformed coordinates). From physical point of view, thismay correspond to static fields with plane, or cylindrical, or, may be, some other spatialsymmetry. 25 .4.2 Simplest examples of solutions for classes of fields (I), (II), (III) To construct really simplest examples of solutions for these classes of fields, we use in the ex-pressions derived previously one more ansatz k o = 0, that leads immediately to the followingvery strong restrictions k o = 0 ⇒ ψ = 0 ⇒ K = K = 0 . In this case, the equation (67) admits a solution with an arbitrary constant c o α = c o w − /n , (75)where, as before, the parameter n for different cases takes the values n = 1 for the class (I), n = − n = − k o = 0 and take the limit R o →
0. Besides that, toobtain a correct (Lorentzian) signature of the metric, it is necessary to make an appropriatechoice of the signs ǫ , ǫ and ǫ , as well as of the simplest form of w such as w = x or w = x (in the metrics given below, we denote the coordinates ( x , x , x , x ) respectively as( t, x, y, z )). Class (I): Vacuum metrics with a cosmological constant.
In this case, for Λ > ǫ = 1, ǫ = 1, ǫ = − w = x = t and, after some rescalings, we obtain ds = t − (cid:2)
3Λ ( − dt + dx ) + dy + dz (cid:3) , and for Λ < ǫ = 1, ǫ = 1, ǫ = − w = x = x , that leads to the metric ds = x − (cid:2) − Λ) ( − dt + dx ) + dy + dz (cid:3) . It is easy to see that these metrics represent respectively de-Sitter and anti-de Sitter metricswhich cover certain parts of these space-times.
Class (II): Vacuum metrics with non-orthogonally-transitive isometry groups G . In the simplest case of metrics, for which the orbits of the isometry group G do not admitthe existence of 2-surfaces orthogonal to the orbits , we can choose ǫ = 1, ǫ = 1, ǫ = − w = x = x ω = ω = ω = 0. the corresponding solution is ds = s o x − / ( − dt + dx ) + x / ( dy + s o x − / dt ) + dz , where s o is an arbitrary constant. After an obvious coordinate transformation { t, x, y, z } →{ T, X, Y, Z } , this metric can be reduced to the following form ds = 2( dT + XdY ) dY + dX + dZ . The isometry groups G , which orbits admit the existence of 2-surfaces orthogonal to the orbits are calledsometimes as orthogonally-transitive ones. k o = 0, the Riemann tensor vanishes and therefore, these metrics are flat.This means that we obtained here the examples of non-orthogonally-transitive subgroups ofthe isometry group of the Minkowski space-time. A bit more complicate calculations for themetrics of the class (II) with k o = 0, described by the expressions (70)–(73) with n = − Class (III): Solution with electromagnetic non-dynamical degrees of freedom.
Choosing ǫ = 1, ǫ = 1, ǫ = − w = x = x and Φ = 0 we obtain the solution ( ds = q o q o x − / ( − dt + dx ) + x / ( dy + dz )Φ t = iq o x − / . An interesting property of this solution is that in this space-time, in contrast to electrovac-uum fields described by integrable reductions of Einstein - Maxwell equations, the electricand magnetic fields possess the components orthogonal to the orbits of the isometry group G . Thus, even the simplest examples of vacuum and electrovacuum solutions given aboveshow that the presence of non-dynamical degrees of freedom of gravitational and electro-magnetic fields can give rise to some non-trivial solutions which may be interesting fromphysical as well as from geometrical points of view. Here we show that vanishing of non-dynamical degrees of freedom of gravitational and elec-tromagnetic fields is a sufficient condition for Einstein - Maxwell equations for electrovacuumspace-times with Abelian isometry group G to reduce to completely integrable system.In the previous sections, the non-dynamical degrees of freedom of gravitational and elec-tromagnetic fields were identified respectively with the metric functions ω aµ and the com-ponents Φ µ of a complex electromagnetic vector potential and characterised by a set ofconstant parameters which consists of the components of a constant two-dimensional vector ℓ a , a complex constant e o and a cosmological constant Λ. In general, these parametersenter the symmetry reduced dynamical equations and make the structure of these equationsrather complicate. However, for a special choice of the values of these constant parametersΛ = 0 , ℓ a = 0 , e o = 0 , (76)all the mentioned above non-dynamical degrees of freedom become pure gauge and vanishafter appropriate coordinate and gauge transformations. Indeed, as one can see from the It is wondering that the ”deformation” of the structure of G -symmetry reduced Einstein - Maxwellequations due to the presence of non-dynamical degrees of freedom of fields is very similar to that causedby the presence of a cosmological constant. This allows us to consider (formally, at least) the cosmologicalconstant as one more non-dynamical degree of freedom of gravitational field. ε µν ∂ µ ω aν = 0 , ε µν ∂ µ Φ ν = ε µν ∂ µ Φ c ω cν . The first of these equations means that locally the functions ω aµ possess the structure ω aµ = ∂ µ ω a and therefore, there exists a coordinate transformation x a → x a − ω a ( x γ ) which leads tothe condition ω aµ = 0. Then the second of the equations given just above implies ε µν ∂ µ Φ ν = 0and therefore, the functions Φ µ possess locally the structure Φ µ = ∂ µ φ ( x γ ) and after thegauge transformation of electromagnetic field potential Φ µ → Φ µ − ∂ µ φ the correspondingΦ µ vanish. Thus, for the parameters (76) all non-dynamical degrees of freedom vanish, ω aµ = 0 , Φ µ = 0 , (77)and the space-time metric and complex vector electromagnetic potential (3) take the forms ds = f η µν dx µ dx ν + g ab dx a dx b , Φ i = { , , Φ a } , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) µ, ν, . . . = 1 , a, b, . . . = 3 , f , g ab and Φ a depend on x µ = { x , x } only.This class of metrics and electromagnetic fields do not describe some physical and ge-ometrical features of gravitational and electromagnetic fields which can take place withina general class of electrovacuum space-times which admit the Abelian isometry group G .These restrictions include the grevitational fields with non-orthoginally-transitive groups ofisometries G and some more complicate structures of electromagnetif fields. However, thewell known remarkable property of this class of fields is that the Einstein - Maxwell equa-tions in this case simplify considerably and become completely integrable. Just this classof electrovacuum fields and integrability of the corresponding Einstein - Maxwell equationswere used by different authors in numerous studies of recent decades of many aspects ofbehaviour of strong gravitational and electromagnetic fields and their nonlinear interactions.
In this paper a structure of field equations for the general classes of vacuum and electrovac-uum fields for which the space-time admits two-dimensional Abelian isometry group and theelectromagnetic field possess the same symmetry. These classes of fields include all fieldsdescribed by the known integrable reductions of vacuum Einstein equations and electrovac-uum Einstein - Maxwell equations. However, the fields in these classes can include also thecomponents, which are called here as the non-dynamical degrees of freedom of gravitational Such structure of complex electromagnetic potential in the hyperbolic case, i.e. for time-dependentfields give rise to electric and magnetic fields which directions are tangent to the orbits of G , while theelectromagnetic non-dynamical degrees of freedom, if not vanish, correspond to electric and magnetic fieldswith the components in the spatial direction orthogonal to the orbits of G . In the elliptic case, i.e. forstationary fields with one spatial symmetry, this structures of metric and complex electromagnetic potentialgive rise to electric and magnetic fields with the components in the directions orthogonal to the orbits of G ,while the non-dynamical electromagnetic degrees of freedom, if not vanish, give rise to electric and magneticfields with the components in the directions tangent to the orbits of G . G -symmetry reduced dynamical equations modified appropriately for the case ofpresence of non-dynamical degrees of frredom. These are the modified Ernst equations forcomplex scalar potentials and matrix sel-dual Kinnersley equatyions. The simplest examplesare presented for solutions with different non-vanishing non-dynamical degrees of freedom.It is worth mentioning here once more a surprising analogy between all considered abovecases of a presence of non-dynamical degrees of freedom of gravitational and electromagneticfields (from one side) and the case of presence of a cosmological constant (from the otherside) which change the structures of G -symmetry reduced Einstein - Maxwell equations (incomparison with the integrable reductions of these equations) in a very similar ways which donot allow to generalize for these cases the well known methods used for solution of integrablereductions of these equations arising in the absence of the mentioned above factors. Thesimplest examples of solutions with non-dynamical degrees of freedom of fields were given.It was shown also that for a special choice of values of the constant parameters whichcharacterise non-dynamical degrees of freedom, these degrees of freedom occur to be puregauge and these can vanish for appropriate choice of coordinates and gauge transformationsof fields. Thus, it occurs that in the case of existence of two-dimensional Abelian groupof isometries, the vanishing of all non-dynamical degrees of freedom of electrovacuum fieldsprovide the sufficient condition for integrability of this symmetry reduced vacuum Einsteinequations and electrovacuum Einstein - Maxwell equations. We note also that the similarnon-dynamical degrees of freedom of fields exist also in the other cases of Einstein’s fieldequations which admit the integrable two-dimensional reductions. In this cases, the similarassumptions about the space-time symmetry and vanishing of all non-dynamical degreesof freedom also occur to be the sufficient conditions for integrability of the correspondingsymmetry reduced dynamical equations. In particular, these are e.g., the Einstein - Maxwell- Weyl equations which describe the nonlinear interaction of gravitational, electromagneticand massless two-component Weyl spinor field [34], as well as two-dimensional reductions ofEinstein equations for some string gravity models in space-times of four or higher dimensions[35].It is interesting to note also that for all known integrable reductions of Einstein’s fieldequations the condition of ”harmonical” structure is satisfied for the function α ( x , x ), whichdetermines the element of area on the orbits of the isometry group G (in four-dimensionalspace-times) or G D − (for gravity models in D -dimensional space-times with D > α is a dynamicalvariable which should satisfy a complicate nonlinear equations. Therefore, if for some two-dimensional reduction of Einstein’s field equations the function α is a ”harmonic” function,this can be an indication of possible complete integrability of these reduced dynamical equa-tions for the fields with vanishing of all non-dynamical degrees of freedom.29 cknowledgments This work is supported by the Russian Science Foundation under grant 14-50-00005.
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