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The Seven Classes of the Einstein Equations Sergey E. Stepanov Professor, Chair of Mathematics, Finance Academy under the Government of the Russian Federation, 49, Leningradsky Prospect, Moscow, 125993 Russia. e-mail: [email protected] Irina I. Tsyganok Professor, Chair of General Scientific Disciplines, Vladimir branch of Russian Uni- 0 versity of Cooperation, 14, Vorovskogo str., Vladimir, 6000001 Russia. 1 0 e-mail: [email protected] 2 Abstract. In currentpaperwe referto thegeometrical classification oftheEinstein n equationswhich has beendeveloped byoneof theauthorsof thispaper. Thisclassi- a fication was based on the classical theory for decomposition of the tensor product of J representations into irreducible components, which is studied in the elementary rep- 6 resentation theory for orthogonal groups. We return to this result for more detailed 2 investigation of classes of theEinstein equations. ] Mathematics Subject Classification (2000): 53Z05; 83C22 G Keywords: Einsteinequations,classification, invarianttensorandmetriccharacter- D istics. . h t a m Introduction [ 1 Method of classification of Riemannian manifolds, endowed with tensor struc- v ture, on the basis of decomposition of irreducible tensor product of represen- 3 tations (irreducible with respect to the action of orthogonal group) into irre- 7 ducible components has become traditional in differential geometry (see, e.g., 6 [1] - [5]). 4 . Thismethodisusedintheoreticalphysics(see,forexample,[6];[7];[8]). For 1 example,inthepaper[7]Einstein-Cartanmanifoldswereclassifiedonthebasis 0 ofirreducibledecompositionofthetorsiontensorofanaffine-metricconnection 0 1 (irreducible with respect to the action of the Lorentz group). Current results : in this modern today researchdirection in the General Relativity Theory were v i systemized in paper [7]. X At current paper we return back to the work [8] of one of the authors and r study in details proposed in this work classification of The Einstein equations a on the basis of method mentioned above. 1 1 The Einstein equations The space-time (M,g) in the General Relativity Theory is a smooth four- dimensional manifold M, endowed with metrics g of the Lorentz signature (- + + +). Inanarbitrarymap(U,ϕ)withthe localcoordinatesystem{x0,x1,x2,x3} the metric g of (M,g) is defined by its components g = g(∂ ,∂ ), where ij i j ∂ =∂/∂xi for i,j,k,l,...=0,1,2,3. These components define the Christoffel i symbolsΓk =2−1gkl(∂ g +∂ g −∂ g )oftheLevi-Civitaconnection∇and ij i lj j li l ij the operator of the covariantdifferention defined by ∇ Xk =∂ Xk+ΓjXl for i i il X =Xk∂ . k In the General Relativity Theory a metric tensor g is interpreted as gravi- tational potential and is related by the Einstein equations 1 1 Ric− sg =T : R − sg =T (1.1) ij ij ij 2 2 to mass-energy distribution, which generates the gravitational field. Here Ric is the Ricci tensorof metric g ofa space-time (M,g), which canbe found from identity Ric(∂ ,∂ ) = R = Rk for components Rl of torsion tensor R, i j ij ikj ikj which,intheirturn,canbefoundfromequalityRk Xl =∇ ∇ Xk−∇ ∇ Xk, lij i j j i s := gijR – is scalar curvature of metrics g for (gij) = (g )−1 and, finally, ij ij T = T(∂ ,∂ ) – are components of known energy-momentum tensor T of ij i j matter. The curvature tensor R satisfies well-known the Bianchi identities dR=0: ∇ Rk +∇ Rk +∇ Rk =0. (1.2) m lij i ljm j lmi From these identities, in particular, following equalities can be found d∗R=−dRic: −∇ Rk =∇ R −∇ R =0, (1.3) k lij j li i lj which lead to other the Bianchi identities 2d∗Ric=−ds: 2gkj∇ R =∇ s. (1.4) j ki i Equations (1.1) are complemented by ”conservation laws” d∗T =0: −gik∇ T =0, (1.5) i kj which are derived from The Einstein equations on the basis of the Bianchi identities (1.4). 2 2 Invariantly defined seven classes of the Einstein equations In the paper [8] was proved that on a pseudo-Riemannian manifold (M,g) the bundle Ω(M) ⊂ T∗M ⊗ S2M, which fiber in each point x ∈ M con- sists of linear mappings Ω : T M → R such that Ω(X,Y,Z) = Ω(X,Z,Y) x n and Ω(e ,e ,X) = 0 for arbitrary vectors X,Y and Z orthonormal basis i i iP=1 {e1,e2,...,en} of space TxM, has point-wise irreducible decomposition (ir- reducible with respect to the action of pseudo-orthogonal group) Ω(M) = Ω1(M)⊕Ω2(M)⊕Ω3(M). If as(M,g) we take space-time, then ∇T ∈Ω(M). As the result six classes of the Einstein equations can be determined via invariant approach Ω and α Ω ⊕Ω (α,β,γ = 1,2,3 β < γ), for each of them covariant derivatives ∇T β γ of energy-momentum tensors T of matter are cross-section of relevant invari- antsubbundles Ω1(M),Ω2(M),Ω3(M)andtheir directsums Ω1(M)⊕Ω2(M), Ω1(M)⊕Ω3(M) Ω2(M)⊕Ω3(M). Seventh classes determined in [8] satisfy the condition ∇T =0. 3 The class Ω and integrals of 1 geodesic equations Class Ω1 of the Einstein equations is selected via condition (see, [8]) ∗ δ T =0: ∇ T +∇ T +∇ T =0. (3.1) k ij i jk j ki ∗ As trace T = −s, then from (3.1) follows that d(trace T) = 2d T = 0, i.e. g g s=const. In this case equations (3.1) can be re-written in the following form ∗ δ Ric=0:∇ R +∇ R +∇ R =0. Reverseisevident,i.e. fromcondition k ij i jk j ki ∗ δ Ric=0 can be deduced that s=const, and then equations (3.1). If each solution xk = xk(s) of geodesic equations on pseudo-Riemannian manifold (M,g) satisfies condition a dxidxj = const for symmetric tensor ij ds ds field a with local components a , then it’s said that such geodesic equations ij admit first quadratic integral (see, for example, [9]. In fact, following identity ∗ needs to be true δ a = 0. At the same time tensor field a, determined by identityabove,iscalledtheKillingtensor(see[10]). Inourcase,firstquadratic integral of geodesic equations will be R dxidxj = const and Ricci tensor will ij ds ds be consequently the Killing tensor. Following theorem was proved: 3 Theorem 1. The Einstein equations belong to the class Ω2 if and only if in the space-time (M,g) geodesic lines equations admit following first quadratic integral R dxidxj =const for Ricci tensor R =Ric(∂ ,∂ ) of metrics g. ij ds ds ij i j Remark. Letusunderlinethattheoryoffirstintegralsequationsofgeodesic andsymmetrictensorKillingfieldshasmanyapplicationsinmechanics,general relativity theory and other physics research(see, for example, [10] and [11]). 4 Class Ω and the Young-Mills equations 2 Class Ω2 of The Einstein equations is selected via condition (see, [8]) dT =0: ∇ T =∇ T . (4.1) k ij i kj Considering equations (3.1) together with The Einstein equations we get that s=const, and consequently we get following equations dRic=0: ∇ R =∇ R . (4.2) k ij i kj Equations (4.2) are known as Codazzi equations (see, for details, [12]). Easy to prove that equations (4.1) follow from Codazzi equations (4.2). In fact, ∗ from equations (4.2) follows that d Ric=−ds. Comparing this equation with Bianchi identities (1.4) we conclude that s = const. As the result equations (4.1) will be true. So, equations (4.1) and (4.2) are equivalent. For constructing an example of space-time (M,g) with ∇T ∈ Ω2(M), we will take n-dimensional (n ≥ 4) conformally flat pseudo-Riemannian manifold (M,g), for which as known (see, for example, [9]) following equations are ful- filled 1 d Ric− s·g =0. (cid:20) 2(n−1) (cid:21) If we suppose here that s = const we will get equations (4.2). Therefore, on conformally flat space-time (M,g) with the constant scalar curvature s the Einstein equations belong to the class Ω2. On arbitrary n-dimensional pseudo-Riemannian manifold (M,g) the Weyl projective curvature tensor P is defined by (see, for example, [9]) compo- nents P =R − 1 (g R −g R ) in arbitrarymap(U,ϕ) with a local lijk lijk n−1 lj ik lk ij coordinate system {x0,x1,x2,x3}. For n > 2 conversion into zero of tensor P characterises manifold of constant curvature (see [9]). Easy to show that due to (1.3) covariant differential is d∗P = −n−2d Ric. We will say that n−1 Weyl projective curvature tensor is a harmonic tensor if d∗P = 0. Definition ∗ is explained by the fact that from condition d P = 0 automatically follows Bianchi identities dP = 0. Therefore if tensor P is considered as second form 4 P :Λ2(TM)→Λ2(TM), then this form will be simultaneously closed and co- closedandso a harmonicform(see, for example,[13]). Conditionfor the Weyl projective curvature tensor P to be harmonic leads us to Codazzi equations (4.2), which are equivalent as was proved to (4.1). Following theorem is true: Theorem 2. The Einstein equations belong to the class Ω2 if and only if the Weyl projective curvature tensor P of space-time (M,g) is harmonic. It is well known (see, for example, [13]) that connection ∇ˆ in main bundle π : E → M over pseudo-Riemannian manifold (M,g) with fiber metrics g is E called the Young-Mills field, if its curvature Rˆ together with Bianchiidentities dRˆ =0 satisfies the Young-Mills equations d∗Rˆ =0. IfweconsiderthatE =TM,g =g andwetakeLevi-Civitaconnection∇ E of pseudo-Riemannian metrics g as connection ∇ˆ, then Young-Mills equations ∗ d R=0 due to equalities (1.3)willbecome Codazziequations(4.2), whichare equivalent to equations (4.1). Following theorem is true: Theorem 3. The Einstein equations belong to the class Ω2 if and only if the Levi-Civita connection ∇ of metrics g of space-time (M,g) considered as connection in bundle TM is the Young-Mills field. Remark. Yang-Mills theory and the other variational theory as Seiberg- Witten theory have been developed greatly and influenced to topology and physics(see[14];[15];[16];[17]andetc),especiallyinthecaseof4-dimensional manifolds. Yang-Millstheoryappearedindifferentialgeometry(see,for exam- ple, [18], p. 443) as pseudo-Riemannian manifolds with harmonic curvature. This means that pseudo-Riemannianmanifolds (M,g) of which curvature ten- ∗ sor R of the Levi-Civita connection ∇ satisfies d R=0, i.e. ∇ is a Yang-Mills connection, taking E = TM, the tangent bundle of M, and g = g as in our E pieces 4. 5 The class Ω and geodesic mappings 3 The class Ω2 of the Einstein equations is selected via condition (see, [8]) 1 ∇ R = (4(∂ s)g +(∂ s)g +(∂ s)g ). (5.1) k ij k ij i kj j ik 18 Let us recall that diffeomorphism f : (M,g) → (M¯,g¯) of pseudo-Riemannian manifolds of dimension n ≥ 2 is called geodesic mapping (see [19], p. 70), if each geodesic curve of manifold (M,g) is transferred via this mapping onto geodesic curve of manifold (M¯,g¯). An arbitrary (n + 1)-dimensional (n ≥ 1) pseudo-Riemannian manifold (M,g) which scalar curvature s 6= const and the Ricci tensor Ric satisfies the 5 following equations: n−2 2n ∇ R = (∂ s)g +(∂ s)g +(∂ s)g (5.2) k ij k ij i kj j ik 2(n−1)(n+2)(cid:18)n−2 (cid:19) foralocalcoordinatesystem{x0,x1,...,xn}iscalledamanifolds Ln+1. These manifolds were introduced by N.S. Sinyukov (see [19], pp. 131-132). Manifold Ln+1 are an example of pseudo-Riemannian manifolds with non-constant cur- vaturewhichadmitnon-trivialgeodesicmappings. Easytocheckthatforn=3 equations(5.1)and(5.2)areequivalentandthatiswhyspaces-times(M,g)for whichT ∈Ω3 areSinyukovspacesLn+1andduetothisreasonadmitnontrivial geodesic mappings. Following theorem is true: Theorem 4. The Einstein equations belong to the class Ω2 if and only if the space-time (M,g) is a Sinyukov space L4. In addition, we call that if (M,g) is an (n+1)-dimensional Sinyukov man- ifold then the metric form ds2 of the manifold (M,g) has the following form (see [19], pp. 111-117and [20]) n ds2 =g00(x0)(dx0)2+f(x0) gab(x1,...,xn)dxadxb (5.3) aX,b=1 for special local coordinate system {x0,x1,...,xn}. The function f(x0), com- ponents g00(x0) and gab(x1,...,xn) of the metric form (5.3) satisfy some de- fined properties which can be found in the paper [20]. From this we conclude thataspace-time(M,g)withtheEinsteinequationsoftheclassΩ2 isawarped product space-time and the Robertson-Wolker space-time, in particular (see, for example, [21]). 6 Three other classes In brief we will describe remaining three classes of the Einstein equations. At the beginning lets consider class of the Einstein equations Ω1 ⊕ Ω2 defined by the following condition gij∇ T = 0 or equivalent condition s = const. k ij Following theorem is true: Theorem 5. The Einstein equations belong to the class Ω1 ⊕Ω2 if and only if the metric g of the space-time (M,g) has constant scalar curvature. Next class of the Einstein equations Ω2 ⊕Ω3 is defined by the following condition d T − 1(trace T)g = 0. To obtain its analytical characterisation 3 g lets use the(cid:2)Weyl tensor W o(cid:3)f conformal curvature (see, for example, [9]). It 6 is known that on arbitrary pseudo-Riemannian manifold (M,g) of dimension n≥3 this tensor follows equation (see, for example, [9] and [18]) n−3 1 ∗ d W =− d Ric− s·g . (6.1) n−2 (cid:20) 2(n−1) (cid:21) We’ll say that the Weyl tensor of conformal curvature is a harmonic tensor if ∗ d W =0(seealso[18]). Definitionisexplainedbythefactthatfromcondition ∗ d W = 0 (see [9]) automatically follows Bianchi identities dW = 0. Therefore if the Weyl tensor W of conformal curvature is considered as second form W : Λ2(TM) → Λ2(TM), then this form will be simultaneously closed and coclosed and so harmonic (see [13]). If the equation T − 1(trace T)g =Ric− 1s·g is true, then for space-time 3 g 6 (M,g)dueto(6.1)condition∇T ∈Ω2(M)⊕Ω3(M)isequivalenttoharmonicity ∗ requirement d W =0 for the Weyl tensor W of conformal curvature. Following was proved: Theorem 6. The Einstein equations belongtotheclass Ω2⊕Ω3 if andonly if the tensor W of conformal curvature of the space-time (M,g) is harmonic. The example of a space-time (M,g) with ∇T ∈ Ω2(M)⊕Ω3(M) is a con- formally flat space-time (M,g), this means that W ≡ 0 (see [9], p 116) and, ∗ consequently, condition d W =0 is fulfilled automatically. The last, third, class Ω1⊕Ω3 is defined by the following condition δ∗ T − 1(trace T)g = 0. On the basis of the identity T − 1(trace T)g = 6 g 6 g Ri(cid:2)c− 1s·g, our con(cid:3)dition turns into following differential equations 3 1 ∗ δ (Ric− s·g)=0: 3 1 ∇ R +∇ R +∇ R = ((∂ s)g +(∂ s)g +(∂ s)g ). (6.2) k ij i jk j ki k ij i jk j ki 3 In this case R dxidxj = const will be first quadratic integral of light-like ij ds ds geodesic equations xk = xk(s) of the space-time (M,g). It is obvious that on a Sinyukov manifold L4 equations (6.2) become identity. Remark. The seventh class of the Einstein equations is defined by the property ∇T = 0. Chaki and Ray have studied a space-time with covariant- constant energy-momentum tensor T (see [22]). In addition, two particular types of such space-times were considered and the nature of each was deter- mined (see also [23]). References [1] Gray A. Einstein-like manifolds which are not Einstein. Geometriae dedicate, 1978, Vol.7, pp.259-280. 7 [2] Gray A., Hervella L. The sixteen class of almost Hermitean manifolds. Ann. Math. Pura e Appl.,1980, No. 123, pp.35-58. [3] Naveira A.M. A classification of Riemannian almost product manifolds. Rend. Mat. Appl.,1983, Vol.3, No. 3, pp.577-592. [4] TricerriF.,VanheckeL.Homogeneousstructures.Progressinmathematics(Dif- ferential geometry), 1983, Vol. 32, pp. 234-246. [5] Stepanov S.E., Shandra I.G. Seven classes of harmonic diffeomorphisms. Math- ematical Notes, 2003, Vol. 74, 5, pp. 708-716. [6] Hehl F.W., McCrea J.D., Mielke E.W., Ne’eman Y. Metric-affine gauge theory of gravity: field equations, Noether identities, word Spinors, and breaking of dilation invariance. Phys.Reports, 1995, Vol. 258, pp.1-171. [7] CapozzielloS.,LambiaseG.,StornaioloC.Geometricclassificationofthetorsion tensor in space-time. Annalen Phys., 2001, Vol.10, pp.713-727. [8] StepanovS.E.OnagroupapproachtostudyingtheEinsteinandMaxwellequa- tions. Theoretical and Mathematical Physics, 1997, Vol. 111, 1, pp.419-427. [9] Eisenhart L.P. Riemannian geometry. Princeton Univ.Press: Princeton, 1926. [10] Stephani H., Kramer D., MacCallum M., Hoenselaers C., Herlt E.. Exact Solu- tionsofEinstein’sFieldEquations: SecondEdition.CambridgeUniversityPress: Cambridge, 2003. [11] IvancevicV.G., Tvancevic T.T. Applied differential geometry: A modern intro- duction. Word ScientificPublishing Co., 2007. [12] Derdzinski A., Shen C.-L.Codazzi tensor fields, curvature and Pontrygin forms. Proc. London Math. Soc., 1983, Vol.47, No. 3, pp. 15-26. [13] Alexeevskii D.V., Vinogradov A.M., Lychagin V.V. Basis ideas and concepts of differentialgeometry.GeometryI.Encycl.Math.Sci., 1991, Vol.28 (translation fromItogiNaukiTekh.,Ser.Sovrem.Probl.Mat.,Fundam.Napravleniya,1988, Vol. 28, pp.5-189). [14] Dubois-Violette M. The Einstein equations, Yang-Mills equations and classical field theory as compatibility conditions of partial differential operators. Physics Letters B, 1982, Vol. 119, Issues 1-3, pp.157-161. [15] ItohM.Self-dualYang-MillsequationsandTaubes’theorem.TsukubaJ.Math., 1984, Vol.8, No. 1, pp.1-29. [16] Tafel J. Null solutions of the Yang-Mills equations. Letters in Math. Physics, 1986, Vol.12, No. 2, pp. 167. [17] Sibner L.M., Sibner R.J., Uhlenbeck K. Solutions to Yang-Mills equations are not self-dual. Proc. Natl. Acad. Sci. USA,1989, Vol. 86, pp.8610-8613. [18] Besse A.L. Einstein manifolds. Springer-Verlag: Berlin, Heidelberg, 1987. [19] Sinyukov N.S. Geodesic mappings of Riemannien manifolds. Nauka: Moscow, 1979. 8 [20] Formella S. Generalized Einstein manifolds. Proc. 9th Winter Sch. ”Geometry and physics”, Suppl.Rend.Circ. Mat. Palermo, II, Ser.22, 1990, pp.49-58. [21] Beem J., Erlich P. Global Lorentzian geometry. Marcel Dekker Inc.: New York and Basel, 1981. [22] Chaki M.C., Ray S.Space-time with covariant constant energy-momentum ten- sor. International Journal of Theoretical Physics, 1996, Vol. 35, Issue 5, pp. 1027-1032.A [23] Bhattacharyya A., De T., Debnath D. Space time with generalized covariant recurrent energy-momentum tensor. Tamsui Oxford Journal of Mathematical Sciences, 2009, Vol. 25, No.3, 269-276. 9

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