Locally Homogeneous Spaces, Induced Killing Vector Fields and Applications to Bianchi Prototypes
aa r X i v : . [ m a t h . DG ] J a n Locally Homogeneous Spaces, Induced KillingVector Fields and Applications to BianchiPrototypes
G.O. Papadopoulos ∗ National & Kapodistrian University of Athens, Physics DepartmentNuclear & Particle Physics SectionPanepistimioupolis, Ilisia GR 157–71, Athens, Greece
Th. Grammenos † University of Thessaly, Department of Civil EngineeringGR 383–34, Volos, Greece
Dedication
G.O. Papadopoulos wishes to dedicate this work to Gerasimos . Abstract
An answer to the question:
Can, in general, the adoption of a givensymmetry induce a further symmetry, which might be hidden at a firstlevel? has been attempted in the context of differential geometry of locallyhomogeneous spaces. Based on É. Cartan’s theory of moving frames,a methodology for finding all symmetries for any n -dimensional locallyhomogeneous space is provided. The analysis is applied to 3-dimensionalspaces, whereby the embedding of them into a 4-dimensional Lorentzianmanifold is examined and special solutions to Einstein’s field equationsare recovered.The analysis is mainly of local character, since the interest is focused onlocal structures based on differential equations (and their symmetries),rather than on the implications of, e.g., the analytic continuation of theirsolution(s) and their dynamics in the large. MSC-Class (2010): 83C05, 83C15, 83C20, 53B20, 53C30, 58J70, 22E65,22E70
Keywords : locally homogeneous spaces, Killing equations & vector fields,(local) isometric embedding, moving frames, gauge freedom
A long time ago, a not well known discovery was made (for details, see: [1]): inthe Bianchi Type III cosmological prototype, a fourth Killing vector field of the3-dimensional locally homogeneous space has been discovered and the adoptionof that, through proper prolongation, has led to a special (of the Kinnersley ∗ e-mail: [email protected] † e-mail: [email protected] Can, in general, the adoption of a given symmetryinduce a further symmetry, which might be hidden at a first level?
At least to the best of the authors’ knowledge, the only work discussing thehomogeneous case in 3 dimensions is [3]; but the treatment is not systematic(i.e., no general method, applicable to n dimensions, is presented), and from avery different point of view . Also, another similar work is to be found in [4],but the topological issues put severe limitations on the results.The idea of symmetry, as realised through the context and the applicationsof group theory, has been proven to be extremely fruitful in many aspects andin many fields of research. From the realm of differential equations to theareas of geometry and topology, and from simple (or even complex) technicalproblems to modern physical theories, symmetry has been a most significantcomponent in the effort towards understanding the nature of versatile problemsand, consequently, providing a solving process –to some extent, at least.Of course, although the interest in the present parer is focused on both theRiemannian geometry of locally homogeneous spaces (LHS), from a pragmaticpoint of view, and the EFEs within the context of the former, the followingshort thoughts seem to be quite general.In a very broad sense, one could say that there are two cases where the idea ofsymmetry is implemented: C In the first case, symmetry can be considered as an exact assumptionwhich can lead to a simplified, compared to the initial, problem with thehope of an equally simple and exact, as a statement, solution. A typicalexample for this case is the adoption of some symmetry (as an accurateassumption) upon solving the EFEs. Indeed, the existence of some Killingvector field(s) (KVF(s)) leads (in many instances) to analytic, closed inform, solutions to the EFEs –see [5] for a panoramic view on this issue. C In the second case, symmetry is a (usually “hidden”) feature characterisinga given system the implementation of which can clarify as well as simplifythe problem to be solved. Here, the archetype is the discovery (usingvarious standard methods) of, e.g., some Lie-point symmetries admittedby a system of differential equations; their existence signals a reduction(again, using standard methods) of the initial system to a simpler finalsystem. At a conceptual level, one could say that the initial differentialequations are given modulo some kind of redundancy, and that symmetry—through the reduction process— results in an irreducible, yet equivalent,system of differential equations, unveiling —at the same time— the truedegrees of freedom.A synthesis of these two cases (perhaps, best imagined through the merging oftheir corresponding examples –which are not chosen arbitrarily) can serve as the It is deemed appropriate to quote the abstract of reference [3]:
We derive necessary andsufficient tensor conditions for the existence of a four parameter isometry group G whichacts multiply transitively on a Riemannian V . We then apply these results to determinewhich spatially homogeneous cosmological models have induced 3-metrics which are invariantunder such a four parameter group . It is well known that Sophus Lie gave to the notion of continuous symmetry a rigorousmeaning by inventing the theory of continuous groups of transformations and applying hisideas to the theory of differential equations in order to generalise Evarist Galois’ theory onalgebraic equations. Hence, the word archetype is the most appropriate. whether the adoption of a given symmetry, and no other, mightlead to the discovery of another, extra symmetry .Returning to the thematics of interest, which is the Riemannian geometryof the LHS in n dimensions, the aforementioned considerations can be restatedas follows :Generally, a (pseudo)Riemannian geometry ( M , g ) , where M is a smooth n -dimensional manifold, and g a smooth metric tensor field, defined throughout M , defines a LHS in n dimensions if the Lie algebra of KVFs spans the tangentspace at all points of M . Then, locally, the space is completely described bythe structure constants of the underlying Lie algebra. These constants satisfyalgebraic constraints coming from the Jacobi identity. Of course, the KVFssatisfy the conditions: £ ξ A g = 0 (1.0.1)Now, it is easy to adapt the previous general comments on symmetries. Inprinciple, there are two ways to consider a symmetry expression like £ ζ g = 0 : HC Either as a system of partial differential equations (PDEs) of the 1st orderwhich, given some KVF(s) { ζ } as an initial symmetry assumption, is tobe solved in terms of the metric tensor field components. HC Or as a system of PDEs of the 1st order which, given a smooth metrictensor field, is to be solved in terms of the unknown(s) { ζ } .One could say that for the case of LHS, and since all the KVFs are given eachtime (by virtue of the very definition for a locally homogeneous Riemanniangeometry), the first possibility describes exactly the state of affairs, while thesecond is empty (or meaningless). But, on the other hand, the question posedearlier still remains: Let an n -fold { ξ A } of KVFs, corresponding to a given Lie algebra g n , be the onlyinitial symmetry assumption for a locally homogeneous Riemannian geometry in n dimensions. Let also a metric tensor field g red. which solves (1.0.1) . Undersuch a setting, are there other, non trivial solutions to the system £ ζ g red. = 0 ,in terms of some KVF(s) { ζ } ? The paradigm of Bianchi Type III (see, e.g., [1]), surprisingly enough, renderedthis question not only non trivial —because the answer there is affirmative—but also justified –since the discovery of the extra KVF has led to a specialsolution to EFEs for that prototype [1, 2]).Another perspective leading to this question is provided by the followinggeneral comments:Suppose that one wants to solve a symmetry condition of the form £ ζ g = 0 byimplementing formal solving processes for systems of PDEs. Then, in principle,two equally formal and associated problems will appear.The first problem, which is twofold in nature, is that (after a formal solvingprocess) redundant degrees of freedom might emerge –as the paradigm of theSchwarzschild space time shows. Indeed, substitution of all the three KVFs(defining the Lie algebra of the group SO (3 , R ) acting multiply transitively on The following discussion is not rigorous; it rather offers a flavour, on general grounds, ofwhat is going to be the central scope of this paper. Therefore, a naive approach in both theformalism and definitions/terminology is adopted. n dimensions (which is the objective of Section3), have important implications: I One realises that it is necessary to know all the initial symmetries in agiven setting, especially when the existence of some of them is a mereconsequence of the adoption of the rest. I Knowledge of the full symmetry group is tantamount to the knowledge ofnot only the full gauge freedom but also of the true degrees of freedom–see [6]. I Points I and I remain valid even when global/topological considerationsenter the analysis –see [7, 8]Section 2 provides some, necessary for the development in Sections 3 and4, mathematical preliminaries; essentially, elements from É. Cartan’s theory ofmoving frames are presented.The objective of Section 3 is to answer in detail the previous question (in itslast form) for any LHS in n dimensions and to discuss in full the implicationsof the extra symmetry (when present) along the lines described above.In Section 4 two simple applications are given: (a) the methodology is applied inthe special case when n = 3 (i.e., Bianchi prototypes), and (b) the embeddingof the 3-dimensional LHS into a 4-dimensional Lorentzian manifold (Bianchicosmological models) is considered. Then, the extra symmetry (when present,and by means of proper prolongations) is used to lead to special, vacuum (i.e.,Ricci flat) solutions to EFEs, which might be of interest.It should be stressed that the character of the present study is local because thedesired result is to communicate the basic ideas in the simplest possible form,and thus to avoid more technical, subtle matters related to topological issues.On the other hand, many results are independent of the topology. Of course,again, the methodology is susceptible of proper modifications in order to remainvalid when global considerations are to be taken into account, and this mightbe the goal of a future work. For the sake of both completeness and logical continuity, a short collection ofelements (i.e., basic notions, definitions, and results) from É. Cartan’s theoryof moving frames will be given. It is meant, of course, that the purpose of thisshort presentation is not to give a detailed account on the theory; it ratherserves as a reference collection. For extended and versatile treatments on the4ubject, see e.g., the last two from references [9]. The original reference is, ofcourse, [10].
Conventions adopted.
Lower case Latin indices are used for any coordinate spacein n dimensions, while capital Latin indices for either any (co)tangent space in n dimensions or any n parametric group space. Both classes of indices have astheir domain of definition the set { , , . . . n } .Let R be a (semi)Riemannian space described by the pair ( M , g ) , where M is an n dimensional, simply connected , Hausdorff and C ∞ manifold and g is a C m metric tensor field on it that is a non degenerate, covariant tensor field oforder 2, with the property that at each point of M one can choose a frame of n real vectors { e , . . . , e n } , such that g ( e A , e B ) = η AB where η (called framemetric ) is a (possibly constant) symmetric matrix with prescribed signature.The totality of the sets { e A } (i.e., the sets for every point on the manifold)determines the GL ( n, R ) frame bundle over M and defines the tangent bundle T ( M ) of M . Thus, the matrix η simply reflects the inner products of thevectors in the tangent bundle.Another fundamental notion is that of the cotangent bundle T ∗ ( M ) of M which,as a linear vector space, is the dual to T ( M ) . Indeed, if { θ A } denotes the basisof the cotangent space at a point on the manifold, then in a similar manner,the totality of the sets { θ A } (i.e., the sets for every point on the manifold)determines the GL ( n, R ) coframe bundle over M and defines the cotangentbundle T ∗ ( M ) of M .The duality relation is realised through a linear operation called contraction ( y ): e A y θ B = δ BA (2.0.2)where δ BA is the Kronecker delta.Élie Cartan has given a formulation of Riemannian geometry in terms ofsome basic p -forms (which are totally antisymmetric (cid:0) p (cid:1) tensors; by definition,the coframe vectors are 1-forms) and the four basic operations acting upon them:the wedge product ( ∧ ), the exterior differentiation ( d ), the contraction ( y ) witha frame vector (field) e and the Lie derivative ( £ e ) with respect to a framevector (field) e .Some very important properties are (for any p -form α , any q -form β , and anyframe vector(s) e ): e y ( α ∧ β ) = ( e y α ) ∧ β + ( − p α ∧ ( e y β ) (2.0.3a) d ( α ∧ β ) = ( d α ) ∧ β + ( − p α ∧ ( d β ) (2.0.3b) £ e d α = d £ eα (2.0.3c) £ eα = e y d α + d ( e y α ) (2.0.3d) £ e ( α ∧ β ) = ( £ eα ) ∧ β + α ∧ ( £ eβ ) (2.0.3e) £ e ( e y α ) = ( £ e e ) y α + e y ( £ e α ) (2.0.3f)Returning to the description of Riemannian geometry, the necessary equationsare provided by É. Cartan’s structure equations (CSEs): The adoption of this assumption is prompted by a potential implementation of
Poincaré’sLemma . Alternatively, this constraint can be replaced by another one, less restrictive, byconsidering simply connected neighbourhoods of a given point on the manifold instead. torsion vanishes, reads: d θ A + Γ AB ∧ θ B = 0 (2.0.4a)where the Γ AB define the connexion Γ AB = γ ABM θ M (2.0.4b)and the quantities γ ABM are called
Ricci rotation coefficients (or spinconnexion coefficients within the context of Newman-Penrose formalism).(B) The second set reads: Ω AB = d Γ AB + Γ AM ∧ Γ MB (2.0.5a)where the Ω AB define the curvature Ω AB = 12 R ABMN θ M ∧ θ N (2.0.5b)and the quantities R ABMN define the Riemann tensor.(C) Taking into account the exterior differential calculus (EDC), the action ofa d upon the first set results in the third set: ( d Γ AB + Γ AN ∧ Γ NB ) ∧ θ B = 0 (2.0.6a)or, by virtue of the second set: R ABMN θ B ∧ θ M ∧ θ N = 0 (2.0.6b)which are nothing but the Jacobi identities .(D) Taking into account the EDC, the action of a d upon the second set resultsin, by virtue of the second set itself, the fourth set: d Ω AB = Ω AM ∧ Γ MB − Γ AM ∧ Ω MB (2.0.7)which are nothing but the Bianchi identities .(E) The frame metric η , which is used to raise and lower (co)frame indices,and is subject to the metricity condition : dη AB = Γ AB + Γ BA (2.0.8)It is also used to define the first fundamental form , g : g = η AB θ A ⊗ θ B (2.0.9) Or algebraic Bianchi identities , or cyclic identities .
6t this point, a comment is deemed necessary. If: £ e A e B ≡ [ e A , e B ] = D MAB e M (2.0.10a)where the quantities D ABM are called structure functions , then, by using thebasic properties given a few lines back, it can easily be proven that: d θ A = − D AMN θ M ∧ θ N = 0 (2.0.10b)i.e., γ A [ BM ] ≡ ( γ ABM − γ AMB ) = − D ABM
This formalism provides a very elegant, powerful, and simple formulationnot only for the isometry condition but also for its integrability conditions. Infact, this formulation becomes even simpler when the frame metric is constanton its entire domain of definition ( rigid frame metric ). Indeed, following [11],the isometry condition: £ ζ g = 0 (2.0.11)is completely equivalent to the statement: £ ζ ( η AB θ A ⊗ θ B ) = η AB (cid:0) ( £ ζ θ A ) ⊗ θ B + θ A ⊗ ( £ ζ θ B ) (cid:1) = 0 (2.0.12)since dη AB = 0 ; i.e., the Lie dragging, with respect to a KVF ζ , of the firstfundamental form vanishes. Because the set of coframe vectors { θ A } constitutesa basis in the cotangent space, it follows that: £ ζ θ A = F AB θ B , in general: dF AB = 0 (2.0.13)for some non constant quantities F AB ; essentialy these are related to the KVF ζ ,and thus are to be determined . Upon substitution of this allocation to (2.0.12),it is: £ ζ g = 0 ⇔ ( £ ζ θ A = F AB θ B η AM F MB + η MB F MA = 0 (2.0.14)Once again, taking into account the EDC and the CSEs, the action of a d upon(2.0.14) results in the primary integrability condition: £ ζ Γ AB = F AM Γ MB − Γ AM F MB − dF AB (2.0.15)In a similar manner, taking into account the EDC and the CSEs, the action ofa d upon (2.0.15) results in the secondary integrability condition: £ ζ Ω AB = F AM Ω MB − Ω AM F MB (2.0.16)The integrability conditions of (2.0.16) are empty, by virtue of the Bianchiidentities.In summary: £ ζ g = 0 plusIntegrabilityConditions ⇔ g = η AB θ A ⊗ θ B dη AB = 0 £ ζ θ A = F AB θ B , in general: dF AB = 0 η AM F MB + η MB F MA = 0 £ ζ Γ AB = F AM Γ MB − Γ AM F MB − dF AB £ ζ Ω AB = F AM Ω MB − Ω AM F MB plusCSEs (2.0.17)7 LHS and induced KVFs: the full symmetrygroup
The starting poing is the following:
Definition 3.0.1.
Let a structure ( M , g , G r ) be such that: H M is an n dimensional, simply connected, Hausdorff and C ∞ manifold, H g is a C m metric tensor field, defined on the entire manifold that is a nondegenerate, covariant tensor field of order 2, with the property that ateach point of M one can choose a frame of n real vectors { e , . . . , e n } ,such that g ( e A , e B ) = h AB where h AB is a (possibly constant) symmetricmatrix of Euclidean signature, H G r is a local Lie group of transformations (associated with a Lie algebra g r )acting simply transitively on M . Therefore, n = r and a bijective mappingbetween the set of group parameters and (at least) some neighbourhoodsof points on the manifold can be established.Then this structure defines an n dimensional locally homogeneous space (forrigorous accounts see [9], and for generalisations like curvature homogeneousspaces , see [12]).First, let U p be an open neighbourhood of a given, albeit arbitrary, point p ∈ M , such that two conditions are met: (a) the set of group parameterscorresponding to the identity group element is a proper subset of U p , and (b)if the Lie algebra g n , as a (real) linear vector space, is spanned by the set ofKVFs { ξ A } , then none of its members has singular points on U p .The set { ξ A } can serve as the frame vectors on S q ∈ U p T q ( M ) ; consequently, theset comprised of their duals, say, { φ A } : ξ A y φ B = δ BA (3.0.18)can also serve as the coframe vectors on S q ∈ U p T ∗ q ( M ) , and it can be used todescribe the local Riemannian geometry along the lines of É. Cartan’s theory.Now, since: ∀ ξ A , ξ B ∈ g n : [ ξ A , ξ B ] = C MAB ξ M (3.0.19)where C MAB are the strcuture constants corresponding to the Lie algebra g n , aLie differentiation with respect to ξ A of the duality relation (3.0.18) results in: £ ξ A φ B = − C BAM φ M , ∀ A, B ∈ { , . . . , n } (3.0.20)and thus, a first fundamental form like: g = H AB φ A ⊗ φ B (3.0.21)would not trivially solve the symmetry condition: £ ξ A g = 0 (3.0.22)but, rather, there would be differential constraints upon the (non constant) H AB . Therefore, such an adoption would not offer much towards the search for8n irreducible (i.e., without spurious degrees of freedom) metric tensor field andwith all its symmetries known, simply because one would face a similar, to theinitial, problem: that of solving a system of PDEs for H AB .A simplification towards the solution to this problem is provided by the basisof the algebra e g n of the reciprocal —to G n — local Lie group of transformations e G n , spanned by, say, { X A } and having the defining property : ( ∀ ξ A ∈ g n ) ∧ ( ∀ X B ∈ e g n ) : £ ξ A X B ≡ [ ξ A , X B ] = 0 (3.0.23a) ∀ X A , X B ∈ e g n : [ X A , X B ] = − C MAB X M (3.0.23b)where (3.0.23b) reflect the initial conditions (say { X A | q ∈ U p = X A } ) needed fora solution to the system of PDEs (3.0.23a).At this point, an assumption is needed; the domain of definition S X , determinedby the solution to the system of PDEs (3.0.23a), is supposed to be such that: (a)no member of the set { X A } has singular points on S X , and (b) the set of groupparameters corresponding to the identity group element is a proper subset ofthe set (cid:0) S q ∈ U p T q ( M ) (cid:1) T S X ≡ K .Under the previous assumption, the set { X A } can now serve as the frame vectorson S q ∈K T q ( M ) , while the set comprised of their duals, say, { σ A } : X A y σ B = δ BA (3.0.24)which is also characterised by the property: £ ξ A σ B = 0 , ∀ A, B ∈ { , . . . , n } (3.0.25)following immediately by a Lie differentiation with respect to ξ A , of (3.0.24) andimplementation of (3.0.23a), can serve as the coframe vectors on S q ∈K T ∗ q ( M ) .Moreover, given the fact that all the inner products amongst the frame vectors { X A } are, by virtue of (3.0.23a), constant (or dependent on outer variable(s)),it is deduced that the frame metric is also constant (or dependent on outervariable(s)).Thus, a first fundamental form like: g red. = h AB σ A ⊗ σ B (3.0.26)defines a smooth metric tensor field throughout S q ∈K T ∗ q ( M ) , which triviallysatisfies, the equivalent to, the isometry condition: £ ξ A g red. = h MN (cid:0) ( £ ξ A σ M ) ⊗ σ N + σ M ⊗ ( £ ξ A σ N ) (cid:1) (3.0.25) = 0 (3.0.27a) dh AB = 0 (3.0.27b)rendering the homogeneity manifest.Both problems, i.e., that of an irreducible form for the compatible metrictensor field g , and that of the knowledge of the full symmetry group of thelatter, have been reduced to the search for an irreducible form for the framemetric h and its symmetries. See the first of references [9]. Here, outer stands for variables irrelevant to M and without any prejudice regarding theircharacter. .1 Irreducible forms for the frame metric tensor field in n -dimensional LHS In the calculus on differentiable manifolds, there are two distinct categories ofgauge freedom. Both categories can be given the structure of a (local) continuousgroup –along, with their disconnected components, corresponding to discretesymmetries : GF The category related to changes in local coordinate systems in a givenatlas on the manifold, constituting the diffeomorphisms group . GF The category related to changes in basis in the tangent space, i.e., thegroup associated with the (co)tangent bundle. This could also be thoughtof as being the symmetry group of the CSEs . Further, there is alwaysa residual, yet trivial, freedom: that reflected in the action of the group GL ( n, R ) .Although a rare phenomenon, nothing prevents these two categories from havingcommon members; frame basis changes inducing diffeomorphisms might exist.If O ( h ) and o ( h ) are a Lie group and its Lie algebra respectively , describedin GF category: O ( h ) = { Q AB : h AB Q AM Q BN = h MN } , in general: dQ AB = 0 (3.1.1a) o ( h ) = { q AB : h AM q MB + h MB q MA = 0 } , in general: dq AB = 0 (3.1.1b)then it is clear that an irreducible frame metric is presupposed, for spuriousdegrees of freedom in h might lead to a different group.Apparently, category GF —by its definition, as a symmetry of the metrictensor field h — should not contribute to the search for an irreducible form forthe latter. On the other hand, the residual freedom provided by GL ( n, R ) , asit stands, does not suffice; not only it does not cover the case where h mightdepend on irrelevant parameters, but it might also lead to wrong results. Forinstance, since SO ( n , n − n , R ) ⊳ GL ( n, R ) (i.e., a subgroup), rotations can beused to diagonalise any matrix h AB with signature ( n , n − n ) , but this is ageneral result only when the homogeneity corresponds to an abelian group –asit is well known. The solution towards the search for an irreducible form for theframe metric in an n -dimensional LHS is provided by the following: Proposition 3.1.1.
Let a LHS be characterised by a Lie algebra (3.0.19) , alongwith a fundamental form: g red. = h AB σ A ⊗ σ B (3.1.2) defined throughout S q ∈K T ∗ q ( M ) . Then, a class of irreducible frame metrics isgiven, in infinitesimal form, by the family: { h AB + h AM C MNB ε N + h MB C MNA ε N } (3.1.3) where the set { ε N } is a collection of, at most n , functions depending on theouter variables of h —if any— multiplied by parameters close to zero. Here, the word categories stands for a class or division of things regarded as havingparticular shared characteristics; a class which might have a structure like that of a group. See: [5], the last two of references [9] and, of course, [10]. Irrespectively of a torsion and/or a metric tensor field on the tangent space. Of order n ( n − / . The disconnected, to the identity, components of the initial continuousgroup have been neglected. roof. Let F be the set: F = ( S AB : U AB → Y AB , U AB , Y AB ⊆ R , analytic ∀ A, B ) ∧ ( | S AB | 6 = 0) ∧ ( S AB C BMN = C AKL S KM S LN ) (3.1.4)i.e., the set of all, structure constants form preserving, invertible matrices, witheach component of which defining a different, analytic mapping from a subsetof R to another subset. In total, n analytic mappings and two families { U AB } and { Y AB } , with n members each, of subsets of R are needed.It should be obvious that F defines multi parametric subsets of GL ( n, R ) : eachfor every n -tuple of points, say { w α } α ∈{ ,...,n } , on the family of sets { U AB } .This set, endowed with the usual operations, can be given the structure of alocal Lie group; its corresponding Lie algebra F will be spanned by a subspaceof gl ( n, R ) .A trivial calculation proves that the set: { f X B } = { X A ( S − ) AB , S AB ∈ F } (3.1.5)and only that, preserves the form of the system of PDEs (3.0.23a) along withthe (consequences of its) initial conditions, i.e., (3.0.23b), thus constituting aLie-point symmetry of this system. The duality expressed in (3.0.24) inducesexactly the same symmetry on the corresponding cotangent space; the set: { e σ A } = { S AB σ B , S AB ∈ F } (3.1.6)preserves the manifest homogeneity of the first fundamental form: g red. = h AB σ A ⊗ σ B g red. = e h MN e σ M ⊗ e σ N ) (3.1.6) ⇒ e h MN = h AB ( S − ) AM ( S − ) BN (3.1.7)It is trivial to prove that this relation defines an equivalence class for framemetrics; i.e., h AB and e h AB determine the same local Riemannian geometry –yeta redundancy, hidden in the matrices S AB , might (dis)appear at will.Next, it would be most useful to find the Lie algebra F , and for this scopeit is necessary to consider those matrices which are connected to the identity: S AB (0) = δ AB (3.1.8)Then, substitution to: S AB ( w α ) C BMN = C AKL S KM ( w α ) S LN ( w α ) (3.1.9) As within the context of differential equations –see the 7th from references [9]. The proof of that statement is trivial; one has only to consider the CSEs corresponding totwo first fundamental forms, g and e g in terms of the same coframe vectors, with frame metricsrelated as in (3.1.7), and the defining property of the matrices S AB , i.e., that of the structureconstants form preservation. w α i at zero, resultsin: λ A ( α i ) B C BMN = λ Q ( α i ) M C AQN + λ Q ( α i ) N C AMQ (3.1.10a)where: λ A ( α i ) B ≡ dS AB ( w α ) dw α i (cid:12)(cid:12)(cid:12) w α =0 , α i ∈ { , . . . , n } (3.1.10b)are the requested generators. The number of independent solutions to thislinear system determines the number of the independent parameters (i.e., theupper bound for α i –say q ≤ n ), and thus the number of the generators of F .Essentially, F defines the automorphism group corresponding to G n . Due to theJacobi identities, the space of solutions to (3.1.10) can be written as: { λ A ( α i ) B } = { C A ( a I ) B , E A ( α J ) B } , ( a I ∈ { , . . . , n } α J ∈ { n + 1 , . . . , q } (3.1.11)The subset of { C A ( a I ) B } , as generators, defines a Lie algebra corresponding tothe inner automorphism group; a subgroup of the automorphism group.Since the interest is focused on the infinitesimal form for the transformationof the frame metrics (3.1.7), it suffices to consider automorphic group elementsnear the identity: S AB ( w α ) ≃ δ AB + w α i λ A ( α i ) B , w α i → (3.1.12)Then, condition (3.1.7) becomes (up to the first order): e h AB ≃ h AB − h AM λ M ( α i ) B w α i − h MB λ M ( α i ) A w α i , w α i → (3.1.13)At this point, it is necessary to remember that these changes upon the form ofthe frame metric can be divided into two categories: those which are inducedby diffeomorphisms, and those which are symmetries of the frame metric (i.e.,they leave it form invariant as a Lorentz transformation leaves form invariantthe Minkowski metric on the tangent space). Since an irreducible form for h AB is not known (as this is the desired result), it is not possible —at thisstage— to identify which part(s) of (3.1.13) belong to the first and which to thesecond category. But, the first category must be induced by equally infinitesimaltransformations.It is, therefore, deemed appropriate to find which generators, contributingto (3.1.12), are induced by infinitesimal GCTs.By construction, the set S q ∈K T q ( M ) not only contains the set { X A } , withoutany singular points, but also constitutes a real vector space. Thus, the generatorof any desired infinitesimal GCT will be a linear combination of those vectorfields. There are two possibilities for this linear combination: having eitherconstant or non constant coefficients. The first possibility not only ensures, inprinciple, the existence of well defined vector fields, throughout S q ∈K T q ( M ) ,but also preserves the form invariance of the initially adopted KVFs; indeed, theLie derivative of the KVFs with respect to any generator of the desired GCTsvanishes if and only if the coefficients in the linear combination are constant. Thesecond possibility, which corresponds to a kind of self similarity of the initiallyadopted KVFs, leads to a more rich structure. Yet, this second possibility ismore restrictive in the following sense: a linear combination with non constant12oefficients might not be well defined everywhere. Also, the resulted freedom,being of local character (with respect to S q ∈K T q ( M ) ), can not be extended inmany cases –like that of embedding to higher dimensions. For all these reasons,the first possibility —which obviously constitutes a minimum requirement— isadopted.Let Π α i be a member of a class of vector fields operating as generators for afamily of infinitesimal (i.e., multi parametric) GCTs: Π α i = Σ Aα i X A , d Σ Aα i = 0 ( d Σ Aα i = 0 (3.0.23a) ⇐⇒ £ Π αi ξ A = 0) (3.1.14a) { x a } → { e x a } : e x a + w α i Π aα i ( x b ) , w α i → (3.1.14b)The quantities Σ Aα i may (and will) depend on those outer parameters uponwhich h is dependent. Before continuing, a simple observation: by (3.1.12), theinfinitesimal version of (3.1.6) can easily be read off: e σ A ≃ σ A + w α i λ A ( α i ) B σ B , w α i → (3.1.15)Thus, if the change in form of the coframe vectors is supposed to be induced by(3.1.14b), then the condition: £ Π αi σ A = λ A ( α i ) B σ B (3.1.16)must hold on S q ∈K T ∗ q ( M ) . By virtue of the Jacobi identities, the integrabilitycondition of (3.1.16) is empty, thus this system always admits a well definedsolution of the form (3.1.14b). Therefore, the setting of the system (3.1.14b)and (3.1.16) is well posed and always admits a solution, because of both thedefinition for the set S q ∈K T ∗ q ( M ) and its (empty) integrability conditions.Using, (3.1.14a), (3.0.24), and a few standard properties of the Lie derivative ,it is inferred that: C AMB Σ Mα i = λ Aα i B , and thus: α i = a I (3.1.17)In other words, only the inner automorphisms can be induced by infinitesimaldiffeomorphisms –cf. [6]. Substitution of the last result to (3.1.13) and theallocation ε N ≡ − Σ Na I w a I complete the proof. Q.E.D.At this point, it should be stressed, once again, that a non constant linearcombination for the generators Π α i (i.e., d Σ Aα i = 0 ) would have led to bothinner and outer automorphisms (the rich structure mentioned earlier in the lastpart of the proof), but then one would be in position of implementing thatstructure only in the case where the study would exclusively be restricted onthe homogenous spaces per se, without any ambition to consider them as a partof a more complex system –like when a homogeneous space is embedded intoanother higher dimensional space.Therefore, from this point of view, the Proposition offers a class rather thana unique family of irreducible frame metrics. Although it seems that not theentire “symmetry (i.e., inner and outer automorphisms) of the symmetry (i.e.,Lie group of transformations acting simply transitively)” is exploited, a use ofwhich would be done at the expense of a restricted study, the results are basedon the least minimum requirement and are valid, without any modification, evenwithin other more complex frameworks –see, e.g., the third reference of [6]. See, Section 2. { ε A } has been implemented. This result stands on its own right: indeed, ithas been proven that a (sub)group of the automorphism group —that of theinner automorphisms— not only is induced by the action of another local Liegroup —that of diffeomorphisms— but also gives a geometric definition of gaugesymmetries. Of course, this is not quite a new result; the symmetry of thesymmetry is of central importance in many research works. An indicative ideaof the general interest might be found in [6] and the references therein. On theother hand though, in many instances, there is a kind of disagreement on whatis (or should be) considered as gauge freedom; see the results found in [8] asthey are opposed to those in the first of references [6].In any case, the first goal towards the discovery of the full symmetry group,i.e., the irreducible form for the frame metric, has been achieved. Let, again, a LHS be characterised by a Lie algebra (3.0.19), along with afundamental form: g irred. = h irred. AB σ A ⊗ σ B (3.2.1)defined throughout S q ∈K T ∗ q ( M ) . It is meant that the freedom provided by theinner automorphisms (sub)group of G n has been implemented in order to castthe frame metric in an irreducible (though not unique) form.If a metric tensor field g irred. admits a KVF(s) ζ , irrelevant to the initialKVFs which describe homogeneity, then the system of the Killing equations: £ ζ g irred. = 0 (3.2.2)is completely equivalent to: £ ζ g irred. = 0 (3.2.3)for the corresponding first fundamental form g . Following [11], this system is,in turn, equivalent to: £ ζ g irred. = 0 plusIntegrabilityConditions ⇔ g irred. = h irred. AB σ A ⊗ σ B dh irred. AB = 0 £ ζ σ A = Ψ AB σ B , in general: d Ψ AB = 0 h irred. AM Ψ MB + h irred. MB Ψ MA = 0 £ ζ Γ AB = Ψ AM Γ MB − Γ AM Ψ MB − d Ψ AB £ ζ Ω AB = Ψ AM Ω MB − Ω AM Ψ MB plusCSEs (3.2.4)A further simplification is possible by two observations: Ob Because of: (3.0.23b), (3.0.24), the first set of the CSEs becomes: d σ A = 12 C AMN σ M ∧ σ N (3.2.5)also, cf. (2.0.10). 14 b The condition h irred. AM Ψ MB + h irred. MB Ψ MA = 0 denotes the antisymmetry, inits indices, of the matrix Ψ AB ≡ h irred. AM Ψ MB . Therefore, the latter can beparametrized by an antisymmetric matrix: h irred. AM Ψ MB = F AB ⇒ Ψ AB = h AM irred. F MB (3.2.6a) F ( AB ) = 0 (3.2.6b)Using these observations, (3.2.4) assumes its simplest form: g irred. = h irred. AB σ A ⊗ σ B dh irred. AB = 0 £ ζ σ A = h AN irred. F NB σ B , in general: dF AB = 0 F ( AB ) = 0 d σ A = 12 C AMN σ M ∧ σ N Γ AB = γ ABM σ M , γ A [ BM ] = 12 C ABM h irred. AS γ SBM + h irred. SB γ SAM = 0 £ ζ Γ AB = ( h AN irred. F NM ) Γ MB − Γ AM ( h MN irred. F NB ) − h AN irred. dF NB Ω AB = 12 R ABMN σ M ∧ σ N Ω AB = d Γ AB + Γ AM ∧ Γ MB £ ζ Ω AB = ( h AN irred. F NM ) Ω MB − Ω AM ( h MN irred. F NB ) (3.2.7)The merits of this approach are both simplicity and clarity. Simplicity, becausethe last integrability condition imposes algebraic constraints upon the matrix F AB –something which renders the solving process, regarding the other steps,easier. Clarity, for the nature of the matrix F AB encodes much information:since it is antisymmetric, only up to n ( n − / independent components mayexist, each for every extra KVF ζ . Thus, if F AB = 0 then the only KVFsadmitted by the space are those given initially. On the other hand, when thenumber of components is the maximum, then the total number of the admittedKVFs is: n (those expressing homogeneity)+ n ( n − / (those corresponding tothe matrix F AB )= n ( n + 1) / , i.e., the case of maximally symmetric spaces [9]. Conventions adopted.
Lower case Greek indices are used for any coordinatespace in dimensions. It is assumed that all 3 dimensional LHS have metrictensor fields of Euclidean signature: (+ , + , +) .This section is divided into two parts: in the first part, the analysis developedin the previous section, i.e., the Proposition regarding the irreducible form forthe frame metric h along with the symmetry system (3.2.7), is applied when n = 3 . In this case (which is of physical interest), there are nine distinct15ontinuous Lie groups of transformations associated with nine real Lie algebras.These groups and their algebras have been classified by L. Bianchi –although S.Lie found them first (see the corresponding reference in [9]). Thus, the BianchiTypes I-IX emerged.In the second part, the results found in the first part are used in the searchfor special solutions to the EFEs. No new —to the literature— solutions areobtained, yet this pedantic application carries the ambition to exhibit a simpleapplication of the entire analysis, something which might be of interest in otherareas.
For every Bianchi Type, the information given includes the structure constants C EAB of the corresponding Lie algebra, the Killing vector fields { ξ A } , the set ofboth left invariant fields { X A } , and right invariants { σ A } —as in [13]— and anirreducible form for the frame metric h irred. AB along with the further KVF(s). Ofcourse, a local coordinate system { x A } = { x, y, z } has been adopted to expressnot only all these vector fields with the coordinate basis { ∂ x A ≡ ∂/∂x A } , butalso the dual basis { dx A } . Also, Bianchi Types VI and VII are each a family ofgroups parametrized by q within the limits given.Finally, it should be stressed that only the most generic case will be of interesteach time. To this end, the initial reducible frame metric will be assumed to bethe most general: h red. AB = e h e h e h e h e h e h e h e h e h i.e., neither further relations amongst the e h ij nor discriminating cases will beconsidered. The only restriction is that all the frame metrics h (i.e., bothreducible and irreducible) are supposed to be positive definite. It is obvious, thatspecializations —like that corresponding to biaxial symmetry (i.e., e h = e h ),etc.— on the frame metric components may lead to further symmetries. Bianchi Type I
This type is characterised by: C EAB = 0 (4.1.1a) { ξ A } = { ∂ x , ∂ y , ∂ z } (4.1.1b) { X A } = { ∂ x , ∂ y , ∂ z } (4.1.1c) { σ A } = { dx, dy, dz } (4.1.1d) h irred. AB = h h
00 0 h (4.1.1e) L. Bianchi:1. Mem. della Soc. Italiana delle Scienze Ser. 3a, , (1897), 267;2. Lezioni Sulla Teoria Dei Gruppi Continui Finiti Di Transformazioni , Spoerri, Pisa, 1918 GL (3 , R ) can still be implemented. In fact, the maximum number (i.e., 3) of GCTs canbe used to bring the frame metric to a diagonal form; the degrees of freedomleft are just the three eigenvalues of the matrix. This is the only case withsuch a singular behaviour –a characteristic feature of all the abelian prototypes,irrespectively of the dimensions.There are three more KVFs: { ζ A } = {− zh ∂ y + yh ∂ z , − zh ∂ x + xh ∂ z , − yh ∂ x + xh ∂ y } (4.1.2)Therefore, Bianchi Type I admits a G local Lie group of transformations, withthe corresponding algebra (only non vanishing commutators are given): [ ζ , ζ ] = h ζ [ ζ , ζ ] = h ζ [ ζ , ζ ] = h ζ [ ξ , ζ ] = h ξ [ ξ , ζ ] = − h ξ [ ξ , ζ ] = h ξ [ ξ , ζ ] = − h ξ [ ξ , ζ ] = h ξ [ ξ , ζ ] = − h ξ (4.1.3)Of course, the space is maximally symmetric –as expected. Bianchi Type II
This type is characterised by: C = 1 (4.1.4a) { ξ A } = { ∂ y , ∂ z , ∂ x + z∂ y } (4.1.4b) { X A } = { ∂ y , x∂ y + ∂ z , ∂ x } (4.1.4c) { σ A } = { dy − xdz, dz, dx } (4.1.4d) h irred. AB = h h h h h (4.1.4e)Only two, out of the available three, GCTs can be used. There is one moreKVF: ζ = zh + xh h h − h ∂ x + z h − x h h h − h ) ∂ y − zh + xh h h − h ∂z (4.1.5)Therefore, Bianchi Type II admits a G local Lie group of transformations, withthe corresponding algebra (only non vanishing commutators are given): [ ξ , ξ ] = ξ [ ξ , ζ ] = − h h h − h ξ + h h h − h ξ [ ξ , ζ ] = − h h h − h ξ + h h h − h ξ (4.1.6)17 ianchi Type III This type is characterised by: C = 1 (4.1.7a) { ξ A } = { ∂ y , ∂ z , ∂ x + y∂ y } (4.1.7b) { X A } = { e x ∂ y , ∂ z , ∂ x } (4.1.7c) { σ A } = { e − x dy, dz, dx } (4.1.7d) h irred. AB = h h h h
00 0 h (4.1.7e)Only two, out of the available three, GCTs can be used. There is one moreKVF: ζ = yh ∂ x + (cid:16) y h − e x h h h − h ) (cid:17) ∂ y + e x h h h − h ∂ z (4.1.8)Therefore, Bianchi Type III admits a G local Lie group of transformations,with the corresponding algebra (only non vanishing commutators are given): [ ξ , ξ ] = ξ [ ξ , ζ ] = h ξ [ ξ , ζ ] = ζ (4.1.9) Bianchi Type IV
This type is characterised by: C = C = C = 1 (4.1.10a) { ξ A } = { ∂ y , ∂ z , ∂ x + ( y + z ) ∂ y + z∂ z } (4.1.10b) { X A } = { e x ∂ y , xe x ∂ y + e x ∂ z , ∂ x } (4.1.10c) { σ A } = { e − x dy − xe − x dz, e − x dz, dx } (4.1.10d) h irred. AB = h h
00 0 h (4.1.10e)All the three available GCTs can be used to bring the frame metric to a diagonalform; the degrees of freedom left are just the three eigenvalues of the matrix.There is no further KVF, since the integrability conditions imply that F AB = 0 . Bianchi Type V
This type is characterised by: C = C = 1 (4.1.11a) { ξ A } = { ∂ y , ∂ z , ∂ x + y∂ y + z∂ z } (4.1.11b) { X A } = { e x ∂ y , e x ∂ z , ∂ x } (4.1.11c) { σ A } = { e − x dy, e − x dz, dx } (4.1.11d) h irred. AB = h h h h
00 0 p h h − h (4.1.11e)18ll the three available GCTs can be used to bring the frame metric to a blockdiagonal form. There are three more KVFs: ζ = zh / ∂ x + (cid:16) y h h + 2 yzh h + z h h h + e x h h / (cid:17) ∂ y + (cid:16) − y h − yzh h + z ( h h − h )2 h − e x h h / (cid:17) ∂ z (4.1.12a) ζ = yh / ∂ x + (cid:16) − y h + y h h − yzh h − z h h − e x h h / (cid:17) ∂ y + (cid:16) y h h + 2 yzh h + z h h h + e x h h / (cid:17) ∂ z (4.1.12b) ζ = − (cid:16) yh + zh h / (cid:17) ∂ y + (cid:16) yh + zh h / (cid:17) ∂ z (4.1.12c) h = ( h h − h ) / (4.1.12d)Therefore, Bianchi Type V admits a G local Lie group of transformations, withthe corresponding algebra (only non vanishing commutators are given): [ ξ , ξ ] = ξ [ ξ , ξ ] = ξ [ ζ , ζ ] = h h / ζ + h h / ζ [ ζ , ζ ] = h h / ζ + h h / ζ [ ξ , ζ ] = − h h / ζ [ ξ , ζ ] = h / ξ − h h / ζ [ ξ , ζ ] = ζ [ ξ , ζ ] = h h / ζ + h / ξ [ ξ , ζ ] = h h / ζ [ ξ , ζ ] = ζ [ ξ , ζ ] = h h / ξ − h h / ξ [ ξ , ζ ] = h h / ξ − h h / ξ (4.1.13)This result is in full agreement with the fact that the space is not only of constantcurvature but also conformally flat. Bianchi Type VI
This type is characterised by: C = 1 , C = q ( = 0 , (4.1.14a) { ξ A } = { ∂ y , ∂ z , ∂ x + y∂ y + qz∂ z } (4.1.14b) { X A } = { e x ∂ y , e qx ∂ z , ∂ x } (4.1.14c) { σ A } = { e − x dy, e − qx dz, dx } (4.1.14d) h irred. AB | q = − = h h
00 0 h (4.1.14e) h irred. AB | q = − = h h h h
00 0 h (4.1.14f)All the three available GCTs can be used to bring the frame metric to a diagonalform if q = − , and to a block diagonal form if q = − . There is no further19VF, since the integrability conditions imply that F AB = 0 , for every value ofthe group parameter within the range given (and for both sectors: q = − , and q = − ). Bianchi Type VII
This type is characterised by: C = 1 , C = − , C = q ( q < (4.1.15a) { ξ A } = { ∂ y , ∂ z , ∂ x − z∂ y + ( y + qz ) ∂ z } (4.1.15b) { X A } = { A ∂ y − B∂ z , B∂ y + A ∂ z , ∂ x } (4.1.15c) { σ A } = { C dy − Ddz, Ddy + C dz, dx } (4.1.15d)where: (4.1.15e) A , = e kx cos ( ax ) ± kB, B = − a e kx sin ( ax ) , (4.1.15f) C , = e − kx cos ( ax ) ∓ kD, D = − a e − kx sin ( ax ) , (4.1.15g) k = q , a = 12 (4 − q ) / (4.1.15h) h irred. AB = h h
00 0 h (4.1.15i)All the three available GCTs can be used to bring the frame metric to a diagonalform. There is no further KVF, since the integrability conditions imply that F AB = 0 , for every value of the group parameter within the range given (andfor both sectors: q = 0 , and q = 0 ). Bianchi Type VIII
This type is characterised by: C = − C = − C = − (4.1.16a) { ξ A } = { e − z ∂ x + 12 ( e z − y e − z ) ∂ y − ye − z ∂ z , ∂ z ,e − z ∂ x −
12 ( e z + y e − z ) ∂ y − ye − z ∂ z } (4.1.16b) { X A } = {
12 (1 + x ) ∂ x + 12 (1 − xy ) ∂ y − x∂ z , − x∂ x + y∂ y + ∂ z ,
12 (1 − x ) ∂ x + 12 ( − xy ) ∂ y + x∂ z } (4.1.16c) { σ A } = { dx + (1 + x ) dy + ( x − y − x y ) dz, xdy + (1 − xy ) dz,dx + ( − x ) dy + ( x + y − x y ) dz } (4.1.16d) h irred. AB = h h
00 0 h (4.1.16e)All the three available GCTs can be used to bring the frame metric to a diagonalform; the degrees of freedom left are just the three eigenvalues of the matrix.20he generators of those infinitesimal GCTs define the so (2 , Lie algebra. Thereis no further KVF, since the integrability conditions imply that F AB = 0 . Bianchi Type IX
This type is characterised by: C = C = C = 1 (4.1.17a) { ξ A } = { ∂ y , cos ( y ) ∂ x − cot ( x ) sin ( y ) ∂ y + sin ( y ) sin ( x ) ∂ z , − sin ( y ) ∂ x − cot ( x ) cos ( y ) ∂ y + cos ( y ) sin ( x ) ∂ z } (4.1.17b) { X A } = { − sin ( z ) ∂ x + cos ( z ) sin ( x ) ∂ y − cot ( x ) cos ( z ) ∂ z ,cos ( z ) ∂ x + sin ( z ) sin ( x ) ∂ y − cot ( x ) sin ( z ) ∂ z , ∂ z } (4.1.17c) { σ A } = { − sin ( z ) dx + sin ( x ) cos ( z ) dy, cos ( z ) dx + sin ( x ) sin ( z ) dy,cos ( x ) dy + dz } (4.1.17d) h irred. AB = h h
00 0 h (4.1.17e)All the three available GCTs can be used to bring the frame metric to a diagonalform; the degrees of freedom left are just the three eigenvalues of the matrix.The generators of those infinitesimal GCTs define the so (3) Lie algebra. Thereis no further KVF, since the integrability conditions imply that F AB = 0 .At this point, it should be noted that all the results found thus far are in fullagreement with those concerning the corresponding generic cases, in reference[3]. This coincidence on the results obtained via different approaches exhibitsthe correctness of the method. As it is mentioned in the Introduction, in many instances symmetry can beconsidered as an exact assumption towards a simplification of the problem underconsideration. Of course, the EFEs —as a system of entangled PDEs of the 2ndorder— constitute a prominent example where this practice is applied. Thus,various prototypes characterised by some symmetry (of either higher order, likelocally homogeneous space(times), or lower order, like space times with axialsymmetry, etc.) emerge, offering a useful insight into some basic features of boththe mathematical structure and the physical system described by the theory.In mathematical cosmology, spatial homogeneity is —in most studies— asine qua non approximation to reality, since the adopted symmetry is not ofhigher order (thus the corresponding theory is not that much artificial), yet thisorder is enough for a simplification of the initial system.21or the development, the following would be useful:
Definition 4.2.1.
Let a structure ( M × R , (4) g , G ) be such that: SH M is a 3 dimensional, (usually) simply connected, Hausdorff and C ∞ manifold, SH g is a C m metric tensor field, defined on the entire product space thatis a non degenerate, covariant tensor field of order 2, with the propertythat at each point of M × R one can choose a frame of 4 real vectors { e , . . . , e } , such that g ( e α , e β ) = η αβ where η αβ is a (possibly constant)symmetric matrix of Lorentzian signature, SH G is a local Lie group of transformations (associated with a Lie algebra g ) acting simply transitively on M .Then this structure defines a spatially (and locally) homogeneous space time.A standard choice for the first fundamental form, associated with (4) g , is (ina local coordinate system { x α } = { t, x, y, z } ): (4) g = η αβ θ α ⊗ θ β η αβ = (cid:18) − h irred. AB ( t ) (cid:19) { θ α } = { N ( t ) dt, N A ( t ) dt + σ A } (4.2.1)where each pair ( h irred. AB , { σ A } ) is to be attributed according to the Bianchi Typechosen. It should be noted that, now, there is one outer parameter entering theframe metric h irred. , which is of a time like character.Regarding the KVFs, representing the action of G at an infinitesimal level, acommon technique is to “promote” them —by construction (or definition) of thespatially homogeneous prototype– into KVFs of the product space; this is done,at a local level, by adding a zero (temporal) component: { ξ A = ξ aA ( x b ) ∂ a } → { (4) ξ A = 0 ∂ + ξ aA ( x b ) ∂ a } (4.2.2)The result of this technique is that the prolonged vector fields are KVFs of themetric tensor field (4) g , as well: £ (4) ξ A ( (4) g ) = 0 (4.2.3)Of course, this is the right point for one to ask whether such a techniquecould (or should) be implemented for the further KVFs { ζ A } –if any.The fact that there is a subclass of the Szekeres family in which the inducedmetric tensor field g , defined on 3 dimensional space like hypersurfaces (whichare conformally flat), is invariant under a G while the metric tensor field (4) g is not invariant under any isometry group G n (see [14]) clearly shows that —ingeneral— such a technique should not be implemented; at least, not when thegoal is to find general solutions to EFEs with the initial symmetry setting.On the other hand, prolonging structures is a very subtle and delicate matter.If one observes those cases which do admit further KVF(s) (i.e., Bianchi TypesI, II, III, and V), one will see that both the KVF(s) { ζ A } , and the enlarged Liealgebra g A + B (i.e., the algebra spanned by the set { ξ A } S { ζ B } ) depend on the22omponents of the irreducible frame metric h irred. ; this is tantamount to the factthat they all be time dependent. Time dependence is an admissible feature whenKVFs are concerned, but not when it refers to the (enlarged) Lie algebra. Thus,the only reasonable and minimum requirement is that the structure constants ofthe enlarged Lie algebra must be valid in the 4 dimensional product space. Thecorresponding commutators can be thought of as being systems of PDEs withinitial conditions described by the eigenvalues of the irreducible frame metric h irred. at a given instant of time. This constraint will have much impact on theform of the metric tensor field (4) g , through the extended symmetry condition: £ (4) ζ A ( (4) g ) = 0 (4.2.4)for the aforementioned initial conditions must be preserved on time. Since theirreducible frame metric h irred. AB (at a point) is a positive definite matrix, it willhave three strictly positive eigenvalues (or eigenfunctions at a given instantof time), say { λ , λ , λ } . Yet, this is not enough information because in manytypes the automorphism group is not adequate to diagonalise the matrices underconsideration. Thus, in most cases the irreducible frame metric h irred. AB (at apoint) will be fully equivalent to the set { λ , λ , λ } S { off diagonal components } (the 2nd set, estimated at a given instant of time), where λ i > .Naturally, two (or more) branches emerge: B The first is characterised by the fact that the set { λ , λ , λ } is sufficient;then, without any loss of generality, one can set: λ = λ = λ = 1 (i.e., h irred. AB = δ AB ) B The other(s) is(are) characterised by the fact that the set { λ , λ , λ } isnot sufficient, but rather arbitrary constant values must be assigned tothe off diagonal terms –along with some convenient and positive valuesfor the set { λ , λ , λ } .So, in order to prolong the extra KVF(s) a series of steps is needed: S The matrix h irred. AB , at a given instant of time, is diagonalised. Thus, thethree strictly positive eigenvalues { λ i } are associated with the diagonalcomponents h irred. AA , while the off diagonal components are parametrizedby a constant –say S . S The substitutions { h irred. AA → λ A , h irred.off diagonal → S } are applied to the extraKVFs, { ζ A } . S A proper prolongation for the KVF(s) { ζ A } is considered: { ζ A = ζ aA ( h irred. AB ( t ) , x b ) ∂ a } → { (4) ζ A = ζ A ( t ) ∂ t + ζ aA ( λ i , S, x b ) ∂ a } and various branches must be discriminated along the lines above.Finally, the demand on constancy upon the structure coefficients determined byall the commutators of the enlarged Lie algebra, together with (4.2.4) will resultin constraints not only upon the quantities ζ A ( t ) but also on the metric tensorfield (4) g as well. Of course, even in the case where the extra symmetry can In those Bianchi Types where the frame metric has not been brought to a diagonal form,there is only one off diagonal term.
23e promoted to the product space, this by no means implies that the resultedmetric tensor fields (4) g will be consistent with the EFEs.With all these in mind, one can explore the possibility of adopting the fullsymmetry group admitted by M , as a symmetry for the product space endowedwith a metric tensor field satisfying the EFEs for the vacuum (i.e., Ricci flat)case. Indeed, since —as stated earlier— out of the nine Bianchi Types only I,II, III and V admit further KVFs, only these four cases have to be considered.Leaving the tedious but straightforward calculations aside, it is found: Bianchi Type I : Adoption of g corresponding to (4.1.3) as a symmetryon the product space leads to one branch only, that of λ = λ = λ = 1 .Then, according to the steps described above, the extra KVFs result intrivial subclasses like conformally flat or Minkowski space time –as specialmembers of the Kasner family of solutions [5]. Bianchi Type II : Adoption of g corresponding to (4.1.6) as a symmetryon the product space leads to two branches: one for S = 0 and one for S = 0 . In any case, the result is trivial leading to either a special memberof the Taub family of solutions [5] or space times with symmetry higherthan the Taub family –but incompatible with the EFEs. Bianchi Type III : Adoption of g corresponding to (4.1.9) as a symmetryon the product space leads to two branches: one for S = 0 and one for S = 0 . Demanding the ensuing space times to be solutions to the EFEs,the first branch results in special members of the Ellis - MacCallum family,while the second leads to special members of the Kinnersley family. It isvery interesting that both special families constitute 2 disjoint classes ofsolutions with Type III symmetry –see [1]. Bianchi Type V : Adoption of g corresponding to (4.1.13) as a symmetryon the product space leads to two branches: one for S = 0 and one for S = 0 . In any case, the result is trivial leading to either a special memberof the Joseph family of solutions [5] or space times with symmetry higherthan the Joseph family –but incompatible with the EFEs. A quite old discovery in the area of special solutions to the EFEs was thespark for inspiration towards a simple question: that of whether the impositionof a symmetry, as a working hypothesis, can induce further symmetry. Theframework chosen is not abstract; it is that of differential geometry of locallyhomogeneous spaces, since these are of interest in many areas of physical theorieswith a geometrical flavour. Indeed, almost any such physical theory implementssymmetry, in many instances, as an exact assumption; analytical cosmologicalprototypes in gravitation constitute a prominent example.Both the initial question, adapted to this framework, and the analysis towardits answer, turn out to be quite fruitful. In the present work, the initial problemis divided into two parts: the first part concerns the need for an irreducible formfor the system upon which the assumption of symmetry is implemented, whilethe second part concerns the search for induced symmetries admitted by that(irreducible) system. In the literature (see e.g., [3]) these induced symmetries,24hen they exist, are called internal and have played an important rôle in thevarious classification schemes for space times in General Relativity per se. Thefirst part of the problem, i.e., the need for an irreducible form, has led to aProposition which attributes a geometric nature to the gauge degrees of freedom.From this point of view, this work is closely related to the spirit of [6]. On theother hand, and in contrast to previous work presented in [3, 4], the analysis isquite general and applicable to any number of dimensions. Moreover, the gaugedegrees of freedom have been detached, in a way, from the context of classicaldynamics and have been given a purely geometrical character.The second part of the problem exhibits the merits and the power of a frameapproach to symmetries. Indeed, prompted by [11], the use of É. Cartan’smoving frames even when dealing with symmetries rendered the search for theseclearer: the initial and induced symmetry are separated in both qualitative andquantitative terms (cf. comments on the matrix F AB in the symmetry system,Section 3). Thus, combining the two parts, the initial problem has been solved.At the level of applications, only a sample regarding spatially homogeneousspace times has been given, since the simplest cases (i.e., when the number ofdimensions is 3) either have already been attacked in the literature or have ledto trivial cases. Yet, this is done in order not only to exhibit in a manifestway the consistency and the truth of the analysis by fully recovering Szafron’sresults, but also to provide a pedantic example –as the 3-dimensional Bianchitypes constitute, due to the simplicity of the calculations. Indeed, for the n = 4 case there are 30 real Lie algebras or “Bianchi Types” and such a presentationnot only would be very tedious and complicated to follow, but also it wouldnot offer more, compared to the n = 3 case, regarding the comprehension.Besides, the paradigm of the Bianchi Type III justifies the cause: in a conciseand uniform way, two very special and completely disjoint classes of resultsnaturally emerged. So, even if the particular example is old enough, this specialfeature revitalises the interest in the method.Worth mentioning is the fact that in those Bianchi Types where extraKVF(s) exist, two very special features manifest themselves: • The totality of the KVFs, leading to non trivial cases upon embedding intoa 4-dimensional Lorentzian space, forms a g Lie algebra (correspondingto a G group), implying locally rotationally symmetric space times (LRS)(see, e.g., [15]). By no means, this fact should be interpreted as if LRSwere exhausting the extra symmetry, when existing, for this phenomenonis an accidental feature taking place only because the number of (spatial)dimensions is 3. Therefore, when n > richer groups may be found, whichobviously will include (generalised) LRS as special sub cases. • The extra KVFs present a smooth behaviour –something which makesfeasible the hope of extending the present analysis in order to includeglobal/topological considerations.An extended analysis based in this very last feature, as well as applications tohigher dimensions or in cases where the degree of symmetry is lower, might bethe objective of a future work. 25 cknowledgement.
The authors are much indebted to Associate Professor T.Christodoulakis not only for introducing them to the key observation (i.e., theexistence of an extra symmetry admitted by the Bianchi Type III prototype–see [1]) and thus initiating the entire problematic, but also for enlighteningdiscussions during the preparation of this work.
Note added. ∼ sbonano/. References [1] For example, see:T. Christodoulakis & Petros A. Terzis, Class. Quantum Grav. (2007)875-887and the references therein for further details.[2] W. Kinnersley, J. Math. Phys. (7) (1969) 1195-1203[3] D.A. Szafron, J. Math. Phys. (3) (1981) 543-548[4] J. Milnor, Adv. Math. (1976) 293-329[5] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Hertl, Exact Solutions of Einstein’s Field Equations ,2nd ed. Cambridge University Press, Cambridge, 2003[6] For extended discussions on gauge freedom in the case of (all) the 3-dimensional homogeneous space(time)s, see, e.g.,:O. Coussaert & M. Henneaux, Class. Quantum Grav. (1993) 1607-1617;J. M. Pons & L. C. Shepley, Phys. Rev. D, (1998) 024001-17;T. Christodoulakis, G. Kofinas, E. Korfiatis, G. O. Papadopoulos, & A.Paschos, J. Math. Phys. (8) (2001) 3580-3608;T. Christodoulakis, E. Korfiatis & G. O. Papadopoulos, Commun. Math.Phys. (2002) 377-391and the references therein.[7] For accounts on various global/topological considerations in the case of(all) the 3-dimensional homogeneous space(time)s, see, e.g.,:G. F. R. Ellis, Gen. Rel. Grav. (1) (1971) 7-21;W. P. Thurston, Bull. Amer. Math. Soc. (3) (1982) 357-381and the references therein.[8] For extended discussions on both global/topological considerations andgauge freedom in the case of (all) the 3-dim homogeneous space(time)s,see, e.g.,:A. Ashtekar & J. Samuel, Class. Quantum Grav. (1991) 2191-2215;H. Kodama, Prog. Theor. Phys. (2) (2002) 305-362and the references therein. 269] An extended and clear treatise, without global/topological considerations,is:L.P. Eisenhart, Continuous Groups of Transformations ,Dover Publications, New York, 2003, unaltered reprint of the 1961 Doverunabridged and corrected republication of the work first published byPrinceton Univesity Press in 1931While, classical references with global/topological considerations are:R. Gilmore,
Lie Groups, Lie Algebras, and Some of Their Applications ,Dover Publications, New York, 2006;C. Chevalley,
Theory of Lie Groups , Vol. I, Princeton, N. J.: PrincetonUniv. Press, 1946;C. Chevalley,
Théorie des Groupes de Lie , Vol. II.
Groupes Algébriques ,Paris: Hermann, 1951;C. Chevalley,
Théorie des Groupes de Lie , Vol. III.
Théorèmes Générauxsur les Algèbres de Lie , Paris: Hermann, 1955;L. S. Pontryagin,
Topological Groups , Translated from the second Russianedition by Arlen Brown, Gordon and Breach, Inc., New York, 1966;S. Lie (unter Mitwirkung von F. Engel),
Theorie der Transformationsgrup-pen , Bände I(1888), II(1890), III(1893), Leipzig: Teubner;F.W. Warner,
Foundations of Differentiable Manifolds and Lie Groups ,Corrected reprint of the 1971 edition, Springer, New York, 1983;S. Helgason,
Differential Geometry, Lie groups, and Symmetric Spaces ,Corrected reprint of the 1978 original, Amer. Math. Soc., Providence, RI,2001[10] É. Cartan,
Leçons sur la Géométrie des Espaces de Riemann , 2nd ed.Gauthier-Villars, Paris, 1951;or:
Geometry of Riemannian Spaces , (english translation by J. Glazebrook,note and appendices by R. Hermann), Math. Sci. Press, Brookline MA, 1983[11] F.J. Chinea, Class. Quantum Grav. (1988) 135-145[12] W. Ambrose & I.M. Singer, Duke Math. J. (4) (1958) 647-669;I.M. Singer, Comm. Pure Appl. Math. (1960) 685-697[13] M.P. Ryan & L.C. Shepley, Homogeneous Relativistic Cosmologies ,Princeton University Press, Princeton NJ, 1975[14] P. Szekeres, Commun. Math. Phys. (1975) 55-64;W.B. Bonnor, A.H. Sulaiman, & N. Tomimura, Gen. Rel. Grav. (8) (1977)549-559;D.A. Szafron, C.B. Collins, J. Math. Phys. (11) (1979) 2354-2361[15] M.A.H. MacCallum, Anisotropic and inhomogeneous relativistic cosmolo-gies (pp. 533–580)in:
General Relativity, An Einstein centenary survey , ed. S.W. Hawking &W. Israel, Cambridge University Press, Cambridge, 1979;M.A.H. MacCallum,
The mathematics of anisotropic spatially-homogeneouscosmologies (pp. 1–59)in: