Self-dual cosmological Bianchi IX and VIII metrics
aa r X i v : . [ g r- q c ] F e b Self-dual cosmological Bianchi IX and VIIImetrics
A. Miković ∗ and N. Manojlović † ∗† Grupo de Física Matemática da Universidade de LisboaDepartamento de Matemática, Faculdade de CiênciasCampo Grande, Edifício C6, PT-1749-016 Lisboa, Portugal ∗ Departamento de Matemática e COPELABSUniversidade LusófonaAv. do Campo Grande, 376, 1749-024 Lisboa, Portugal † Departamento de Matemática, Faculdade de Ciências e TecnologiaUniversidade do Algarve, Campus de GambelasPT-8005-139 Faro, Portugal
Abstract
We show that self-dual Bianchi IX and VIII cosmological models are de-scribed by the Nahm dynamical system for an appropriate type of matrices.We construct the general solutions in the case of the diagonal reductionsof the corresponding Nahm equations and give the explicit expressions forthe corresponding self-dual metrics in the Euclidean and in the Minkowskisignature cases. ∗ E-mail address: [email protected] † E-mail address: [email protected]
Introduction
Imposing self-duality on the solutions of the Einstein equations is a way to gener-ate new solutions, since one often ends up with an integrable system of differentialequations [1, 2]. However, one can use this method only for the eucledean Gen-eral Relativity (GR) or for the complex GR. Still, the eucledean or complex GRsolutions are usefull, since they play a role in quantum gravity [3, 4].Self-dual Bianchi metrics have been mainly studied in the context of self-dualityof the Weyl tensor and spherical symmetry, see [5–12]. Self-dual spherically sym-metric Bianchi metrics were studied in the context of self-duality of the Riemanntensor in [13]. In this paper we will analyze the case of self-dual cosmologicalBianchi metrics and a good framework for such a study is the Ashtekar formal-ism for self-dual metrics [16], as well as the Ashtekar formalism for the Bianchicosmological spacetimes [17].Bianchi metric reductions of the Einstein equations give dynamical systemsfrom classical mechanics [18, 19], and imposing the Riemann tensor self-dualityis expected to give integrable dynamical systems. This is reasonable to espect,because the spherically symmetric reduction of a self-dual metric leads to theLagrange or the Halphen system of ordinary differential equations (ODE), see[13–15]. We will show that in the case of Bianchi IX and Bianchi VIII cosmologicalspacetimes one obtains the Nahm systems of ODE for the matrices form the Liealgebras so (3) and so (2 , , respectively. Also, in the case of the complex GR, theself-dual Bianchi IX and VIII reductions give the Nahm system for matrices fromcomplex so (3) and so (2 , Lie algebras. In the case of other Bianchi cosmologicalspacetimes, the self-dualty restriction gives linear systems of ODE, so that self-dual Bianchi IX and VIII cosmological models are the interesting cases as far asthe integrability is concerned.In section 2 we review the self-dual metrics in the Ashtekar formulation of GR.In section 3 we review the cosmological Bianchi models in the Ashtekar formulationand show that in the Bianchi IX case the dynamical equations are given by the 3-dimensional real (or 2-dimensional complex in the Minkowski case) Nahm systemof differential equations. In section 4 we analyze the integrability of the real andthe complex Nahm system, and solve the diagonally reduced system, which is theLagrange dynamical system. In section 5 we study the case of self-dual BianchiVIII model, and solve the corresponding diagonally reduced Nahm system. Insection 6 we construct the Bianchi IX and VIII self-dual metrics in the diagonallyreduced cases. In section 7 we present our conclussions.2
Self-dual metrics in the Ashtekar formulation
Let M = Σ × R be a 4-manifold where Σ is a 3-manifold. Let g be a metric on M and let h be an induced metric on Σ . These two metrics are related by ds = g µν dx µ dx ν = (cid:0) ξ N + h ij n i n j (cid:1) dt + 2 dtdx i h ij n j + h ij dx i dx j , (2.1)where N is the laps, n i are the components of the shift vector and ξ = 1 in theEuclidean case while ξ = − in the Minkowski case.By using the metric (2.1) one can obtain the canonical formulation of theEinstein-Hilbert action, i.e. the Arnowitt-Deser-Misner formulation [20], so that p | det g | R ( g ) ∼ = π ij ˙ h ij − n i C i ( π, h ) − N C ( π, h ) , (2.2)where R ( g ) is the scalar curvature, ∼ = is up to a surface term (when Σ is non-compact), π ij are the canonically conjugate momenta for the 3-metric components h ij and ˙ X = dX/dt . The constraints C i and C are given by C i = ∇ j π ji (2.3) C = 1 √ det h (cid:18) π − π ij π ij (cid:19) + √ det h R ( h ) , (2.4)where π = h ij π ij .One can change the canonical variables ( π ij , h ij ) to ( p iα , e αi ) canonical variables,where e αi are the triads, so that h ij = e αi e jα . Furthermore, one can pass to (˜ p αi , ˜ e iα ) canonical variables, where ˜ e iα are densitized inverse triads given by ˜ e iα = √ det h e iα . (2.5)Consequently p | det g | R ( g ) ∼ = ˜ p αi d ˜ e iα dt − N α ˜ C α (˜ p, ˜ e ) − n i ˜ C i (˜ p, ˜ e ) − N ˜ C (˜ p, ˜ e ) , (2.6)where ˜ C µ (˜ p, ˜ e ) = C µ ( p, e ) = C µ ( π, h ) , µ = 0 or µ = i , and ˜ C α = ǫ αβγ ˜ e iβ ˜ p γi . (2.7)Here ǫ αβγ is the totally antisymmetric 3-dimensional symbol.The Ashtekar variables are given by the canonical transformation (˜ p αi , ˜ e iα ) → ( A αi , E iα ) such that A αi = Γ αi (˜ e ) + z ˜ p αi , E iα = ˜ e iα , (2.8)3here − ξz = 0 , Γ αi (˜ e ) = ω αi ( e ) and ω αi ( e ) is a spin connection on Σ , whosedependence on the triads is given by the vanishing torsion equations T α = de α + ǫ αβγ ω β ∧ e γ = 0 . The one-forms A α are real in the eucledean gravity case ( z = ± ), while they arecomplex in the Minkowski case ( z = ± i), and they are known as the Ashtekarconnections [21].By using (2.8), one can show that (2.6) becomes p | det g | R ( g ) ∼ = − ξzE iα ˙ A αi − N α G α − n i G i − ˜ N G , (2.9)where ˜ N = N/ √ det h , G α = D i E iα , G i = F αij E jα , G = ǫ αβγ F αij E iβ E jγ ,D i X = ∂ i X + [ A i , X ] and F = dA + [ A, A ] is the curvature 2-form for a real SO (3) or SU (2) connection A in the Euclidean case, while in the Minkowski case we havea complex SO (3) or SU (2) connection A .In the Euclidean gravity case the self-dual (SD) metrics are defined as R ∗ ab = R ab , (2.10)where R ab is the curvature 2-form for the torsion-free spin connection ω ab on M and R ∗ ab = ǫ cdab R cd , where ǫ abcd is the 4-dimensional totally antisymmetric symbol. The self-duality ofthe curvature is equivalent to the self-duality of the connection ω ∗ ab = ω ab , (2.11)see [22].In the Minkowski gravity case the definition of self-duality as X ∗ = X has tobe modified, because ( X ∗ ) ∗ = − X . One can then define self-duality conditions as X ∗ = ± i X , so that R ∗ ab = ± i R ab ⇔ ω ∗ ab = ± i ω ab . (2.12)The self-duality conditions (2.12) can be realized if we use complex metrics, whichis also reflected by the fact that the Ashtekar connection is complex in the Minkowskisignature case.In the Euclidean case we can choose the gauge N α = 0 , n i = 0 , ˜ N = 1 , (2.13)4o that the 4-metric is given by ds = N dt + h ij dx i dx j . (2.14)The Einstein equations are then given by ˙ A αi = ǫ αβγ F βij E jγ , (2.15) ˙ E iα = ǫ αβγ E jβ D j E iγ , (2.16)plus the Gauss and the 3-diffeomorphism constraints D i E iα = 0 , F αij E jα = 0 . (2.17)If we impose A αi = 0 , then F αij = 0 , which corresponds to the vanishing of the(anti) self-dual piece of the Riemann tensor, see [23] for the Minkowski case. Inthe Euclidean case, ( ω ∗ ab ) ∗ = ω ab , so that ω ab = 12 ( ω ab + ω ∗ ab ) + 12 ( ω ab − ω ∗ ab ) = ω + ab + ω − ab , where ( ω ± ab ) ∗ = ± ω ± ab are the self-dual and the anti self-dual piece of the spinconnection. Then ω ∗ ab = ω ab corresponds to R ∗ ab = R ab , which corresponds to R − ab = 0 or ω − ab = 0 . Hence the vanishing of the anti self-dual piece of the spinconnection is equivalent to vanishing of the Ashtekar connection, since ω − abµ = 0 ⇔ ω − αβi = 12 (cid:16) ω αβi − ( ω ∗ ) αβi (cid:17) = 0 ⇔ A αi = Γ αi + z ˜ p αi = 0 . Hence the SD gravity equations are given by ˙ E α = ǫ αβγ [ E β , E γ ] , ∂ i E iα = 0 , (2.18)where E α = E iα ∂ i , which was the main result of [16]. The SD metric is given by(2.14), where N = √ det h , h ij = e αi e jα , e iα = E iα √ det h , det h = det( E iα ) . (2.19)Note that the anti self-dual (ASD) metric equations are given by ˙ E α = − ǫ αβγ [ E β , E γ ] , while the other equations are the same as in the SD case.In the Minkowski case the SD/ASD equations are given by ˙ E α = ± i ǫ αβγ [ E β , E γ ] , (2.20)while the other equations are the same as in the Eucledean case except the expres-sion for the 4-metric, which is given by ds = − N dt + h ij dx i dx j . (2.21)5 Self-dual Bianchi cosmological models
Bianchi cosmological spacetimes have topology Σ × R and globally defined 1-forms χ I on Σ such that dχ I + C IJK χ J ∧ χ K = 0 , where C IJK are the structure constants of a 3-dimensional Lie algebra [24]. Bianchishowed that C IJK = ǫ JKL S LI + δ I [ J v K ] , where S is a diagonal matrix whose values can be or ± and v K = ( v, , [24].The inverse χ Ii vector fields L I = L iI ∂ i satisfy the Lie algebra [24] [ L I , L J ] = C KIJ L K . (3.1)In the Ashtekar formulation for GR, one can write A αi ( x, t ) = A αI ( t ) χ Ii ( x ) , E iα ( x, t ) = E Iα ( t ) L iI ( x ) , see [17], so that the self-dual equations of motion (EOM) (2.18) become ˙ E Iα = ǫ βγα E Jβ E Kγ C IJK , E Iα v I = 0 , (3.2)and A = 0 .For the class A Bianchi models v = 0 , so that there is only one equation, whilefor the class B, v = 0 , so that a non-trivial solution ( E 6 = 0 ) requires det E = 0 .The dynamical equation from (3.2) can be rewritten in the case of the BianchiIX model ( S = diag (1 , , , v = 0 ) as the Nahm equation ˙ V α = 12 ǫ βγα [ V β , V γ ] , (3.3)where V α = 2 E α = 2 E Iα T I and T I are n × n real matricies which generate the Liealgebra (3.1).For C IJK = ǫ IJK we can take ( T I ) JK = − ǫ IJK , so that n = 3 .We can also take T I = − i σ I , where σ I are Pauli matrices, so that C IJK = ǫ IJK and n = 2 . This choice requires complex Nahm matrices, but if we restrict the Liealgebra coefficients to real numbers, we can still get real metric components. Ifwe consider complex metrics, we can take the same Lie algebra generators for the n = 2 and n = 3 case and allow the coefficients to take complex values. In thiscase ˙ E α = i ǫ βγα [ E β , E γ ] , (3.4)and V α = 2 i E α , then obeys the Nahm equation (3.3) for complex matricies V α .As far as the integrability is concerned, the only non-trivial SD Bianchi modelsare Bianchi IX and VIII, since the other Bianchi models give linear systems ofODE. 6 Self-dual Bianchi IX model
The Nahm equations (3.3) can be written as ˙ V = [ V , V ] , ˙ V = [ V , V ] , ˙ V = [ V , V ] . (4.1)The corresponding Lax pair is given by L = A + + λA + λ A − , M = 12 A + λA − , (4.2)where A ± = V ± i V , A = 2 i V , so that the EOM (4.1) are equivalent to ˙ L = [ M, L ] . The integrals of motion I mn ( V ) can be determined from tr ( L n ) = X m I mn ( V ) λ m , (4.3)where n = 1 , , , ... . We need to find 8 independent integrals of motion from theequation (4.3) in order to construct the general solution. However, due to theidentity L = 12 tr ( L ) L , the only independent integrals of motion which are generated by tr ( L n ) are the 5integrals coming from tr ( L ) , and these are I = a − b , I = a + b − c , I = ~a · ~b , I = ~b · ~c , I = ~a · ~c , where ~a = ( u , v , w ) , ~b = ( u , v , w ) , ~c = ( u , v , w ) , a = ( ~a ) and b = ( ~b ) .The components of these vectors are related to the elements of the 3-dimensionalmatrices V α as V α = − w α v α w α − u α − v α u α , where V α = u α T + v α T + w α T . By choosing u, v, w ∈ R we get the Euclideangravity case, while u, v, w ∈ C gives the Minkowski complex gravity case.In the 2-dimensional case, we have V α = 12 (cid:18) − i w α v α − i u α v α + i u α i w α (cid:19) ,
7o that even when u, v, w ∈ R one obtains complex matrices. In order to avoidconfusion, we will work with the 3-dimensional representation.In terms of the vectors ~a , ~b and ~c the Nahm system takes the form ˙ ~a = ~b × ~c , ˙ ~b = ~c × ~a , ˙ ~c = ~a × ~b . (4.4)Although we cannot find a general solution of the Nahm system, we can find ageneral solution for a reduced Nahm system given by V = xT , V = yT , V = zT . (4.5)In this case we obtain the Lagrange system ˙ x = y z , ˙ y = x z , ˙ z = x y . (4.6)The Lagrange system has quadratic integrals of motion I = αx + βy + γz , α + β + γ = 0 , so that there are 2 independent quadratic integrals of motion. By taking C = x − z and C = y − z we obtain x i = ±√ C i + z , so that ˙ z = ± p ( C + z )( C + z ) . Hence t + C = Z dz p ( C + z )( C + z ) , (4.7)and a real solution is obtained in the following cases C > , C > , (4.8) C > , C < , z > − C , (4.9) C < , C > , z > − C , (4.10) C < , C < , z > max {− C , − C } . (4.11)In all these cases the solutions can be written as Jacobi elliptic functions [25–27]. Let pq ( u, k ) = pn ( u, k ) qn ( u, k ) , where pn ( u, k ) = qn ( u, k ) ∈ { sn ( u, k ) , cn ( u, k ) , dn ( u, k ) } , are the 3 basic Jacobi elliptic functions ( u ∈ R , < k < , see [27]).8et C = α , C = β , α < β and u = β ( t + C ) . Then (4.7) becomes u = Z z/α (1 + t ) − / [1 + ( k ′ ) t ] − / dt = arcsc ( z/α, k ) , where k ′ = √ − k = α/β . Consequently z = α sc ( u, k ) , x = β dc ( u, k ) , y = α nc ( u, k ) . (4.12)If α ≥ β then u = α ( t + t ) , k ′ = β/α , and x, y and z are given by (4.12) with ( α, β ) → ( β, α ) .In the case C = − α , C = β , we have u = γ ( t + t ) and u = Z + ∞ z/γ ( t − k ′ ) − / ( t + k ) − / dt = arcds ( z/γ, k ) , where γ = p α + β and k = αγ . Then z = − γ ds ( u, k ) , x = − γ cs ( u, k ) , y = − γ ns ( u, k ) . (4.13)In the last case we have C = − α , C = − β and for β < α we obtain u = α ( t + t ) and u = Z + ∞ z/α ( t − − / ( t − k ) − / dt = arcdc ( z/α, k ) , where k = β/α . Consequently z = α dc ( u, k ) , x = γ sc ( u, k ) , y = γ nc ( u, k ) , (4.14)where γ = p α − β .When α < β , the solution is given by (4.14) with ( α, β ) → ( β, α ) , so that u = β ( t + t ) and k = α/β .In the complex case one can use the solution of (4.6) given in [28], which canbe written as x = p I sn ( u, k ) , y = p − I cn ( u, k ) , z = − p − I dn ( u, k ) , (4.15)where k = √ I − √ I − I √ I , u = p I ( t + t ) , and I = x − y , I = x − z . α and β by ±√ C and ±√ C .For example, the real solution (4.12) gives a complex solution x = p C dc ( u, k ) , y = p C nc ( u, k ) , z = p C sc ( u, k ) , (4.16)where k = r C − C C , u = p C ( t + t ) , and there is no sign restriction on C and C . In the Bianchi VIII case S = diag (1 , , − and v = 0 , so that we have the system ˙ ~a = ~b × ~c , ˙ ~b = ~c × ~a , ˙ ~c = − ~a × ~b . (5.1)The Lax pair is given by (4.2), where now the matrices V α belong to the so (2 , Lie algebra, so that V α = − w α v α w α u α v α u α . The quadratic integrals of motion are then given by I = a − b , I = a + b + 2 c , I = ~a · ~b , I = ~b · ~c , I = ~a · ~c . (5.2)Note that I and I are two independent integrals of motion coming from a setof quadratic integrals of motion I = α a + β b + γ c , α + β − γ = 0 . Hence we can also chose the pair I ′ = a + c and I ′ = b + c instead of the ( I , I ) pair.As in the Bianchi IX case, the integrals (5.2) are not sufficient to solve thesystem (5.1), and we can make the reduction ~a = ( x, , , ~b = (0 , y, , ~c = (0 , , z ) . In this case we get a modified Lagrange system ˙ x = y z , ˙ y = x z , ˙ z = − x y . (5.3)10y using the integrals of motion a + b and b + c , we obtain C = x + z , C = y + z , (5.4)so that ˙ z = − p ( C − z )( C − z ) . (5.5)Hence t + t = − Z dz p ( C − z )( C − z ) , (5.6)and real solutions are obtained for C > , C > , z < min { C , C } . (5.7)Let C = α , C = β , α < β and u = β ( t + t ) then (5.6) becomes − u = Z z/α (1 − t ) − / (1 − k t ) − / dt = arcsn ( z/α, k ) , where k = α/β . Hence z = α sn ( − u, k ) , x = α cn ( − u, k ) , y = β dn ( − u, k ) . (5.8)There is an alternative form of the reduced Bianchi VIII solution: if we takethe integrals x − y = C ′ , y + z = C , then we obtain ˙ y = p ( C ′ + y )( C − y ) . The real solutions are obtained for C ′ > and C > or for C ′ < and C > . In the first case let C ′ = α and C = β , then u = Z y/γ (1 − t ) − / ( k ′ + k t ) − / dt = arccn ( y/β, k ) , where γ = p α + β , u = γ ( t + t ) , k = β/γ . Consequently y = β cn ( u, k ) , x = γ dn ( u, k ) , z = − β sn ( u, k ) . (5.9)In the second case let C ′ = − α , C = β and β > α , then u = Z y/α ( t − − / (1 − k ′ t ) − / dt = arcnd ( y/α, k ) , u = β ( t + t ) , k ′ = α/β . Consequently y = α nd ( u, k ) , x = α ′ sd ( u, k ) , z = β ′ cd ( u, k ) , (5.10)where α ′ = α p − ( α/β ) and β ′ = β p − ( α/β ) .In the complex metric case one can take any of the real solutions (5.8), (5.9)or (5.10), and put α = √ C or α = ± p C ′ , β = √ C whith no restriction onthe signs of the constants. For example, the real solution (5.8) gives a complexsolution x = p C cn ( − u, k ) , y = p C dn ( − u, k ) , z = p C sn ( − u, k ) , (5.11)where k = r C C , u = p C ( t + t ) , and there is no sign restriction on C and C . By using the definitions (2.14), (2.19) and (4.5) one obtains that the metric in thereduced (Lagrange) case is ds = Ω Ω Ω (det L ) (cid:18) dt + χ Ω + χ Ω + χ Ω (cid:19) , (6.1)where Ω i = x i ( t ) / and det L = det( L iα ) .In the Bianchi IX case we have L = sin ψ ∂ θ − cos ψ sin θ ∂ φ + cot θ cos ψ ∂ ψ ,L = cos ψ ∂ θ + sin ψ sin θ ∂ φ − cot θ sin ψ ∂ ψ , (6.2) L = ∂ ψ , where θ, φ, ψ are the Euler angles , while x i ( t ) are given by (4.12),(4.13) and (4.14).Consequently det( L iα ) = 1sin θ , We use the parametrization R ( φ, θ, ψ ) = R ( ψ ) R ( θ ) R ( φ ) for an SO (3) group element,where R and R are rotations around the x and the z axis. χ = sin ψ dθ − cos ψ sin θ dφ ,χ = cos ψ dθ + sin ψ sin θ dφ , (6.3) χ = cos θ dφ + dψ , so that ds = Ω Ω Ω sin θ (cid:18) dt + χ Ω + χ Ω + χ Ω (cid:19) . (6.4)In the Bianchi VIII case we have L = sin ψ ∂ θ ′ − cos ψ sinh θ ′ ∂ φ + coth θ ′ cos ψ ∂ ψ ,L = cos ψ ∂ θ ′ + sin ψ sinh θ ′ ∂ φ − coth θ ′ sin ψ ∂ ψ , (6.5) L = ∂ ψ , where the angles are defined by an SO (2 , group element parametrization R ( φ, θ, ψ ) = R ( ψ ) ˜ R ( θ ′ ) R ( φ ) such that ˜ R ( θ ′ ) = θ ′ sinh θ ′ θ ′ cosh θ ′ , and θ ′ ∈ R .Consequently det( L iα ) = 1sinh θ ′ , and χ = sin ψ dθ ′ − cos ψ sinh θ ′ dφ ,χ = cos ψ dθ ′ + sin ψ sinh θ ′ dφ , (6.6) χ = cosh θ ′ dφ + dψ , while the metric is given by (6.4). The Ω i ( t ) functions in the Bianchi VIII caseare given by (5.8).In the case of a Minkowski signature self-dual Bianchi IX complex metric, wehave the same formulas for the L ’s and the χ ’s as in the eucledean case, while Ω i ( t ) are given by (4.15) or by (4.16) and ds = Ω Ω Ω sinh θ ′ (cid:18) − dt + χ Ω + χ Ω + χ Ω (cid:19) . (6.7)In the case of a complex Bianchi VIII self-dual metric we have (6.7) with Ω i ( t ) given by (5.11). 13 Conclusions
The main results are the self-dual Bianchi IX and VII cosmological metrics givenby the expression (6.4) in the Euclidean case, and by the expression (6.7) in theMinkowski case. The forms χ I are given by (6.3) in the Bianchi IX case and by(6.6) in the Bianchi VIII case. The functions i ( t ) are given by (4.12),(4.13) and(4.14) in the Bianchi IX case, while in the Bianchi VIII case these functions aregiven by (5.8). In the Minkowski case, the functions i ( t ) are given by (4.16) forthe Bianchi IX case, while in the Bianchi VII case these functions are given by(5.11).Note that we solved the reduced Nahm’s equations (4.6) and (5.3), so that anatural next step would be to solve the complete set (4.4) or (5.1). This wouldrequire the knowledge of 8 independent integrals of motion, and the Lax methodgives only 5. This means that one could solve a 6-variables reduction of the Nahm’sequations. However, it is not clear how to implement a 6-variables reduction suchthat it is preserved by the time evolution. As far as solving the complete set ofNahm’s equations is concerned, one would need to find additional 3 integrals ofmotion, and there are indications that these conserved quantities cannot be localfunctions.Note that in [13] it was considered a selfdual spherically-symmetric metric ofthe form ds = Ω Ω Ω (cid:18) dr + σ Ω + σ Ω + σ Ω (cid:19) , (7.1)where Ω i = f i ( r ) , r = x + y + z + t , σ i = 1 r η iµν x µ dx ν , and η iµν are the t’Hooft coefficients.The metric (7.1) has a similar structure as the SD Bianchi IX metric (6.1), butthe variables are different. In [13] it was also showed that the self-duality of theconnection associated to the metric (7.1) gives the Lagrange system Ω ′ i = Ω j Ω k , i = j = k . Hence our Lagrange system solutions (4.12),(4.13) and (4.14) can be used to con-struct self-dual metrics of the type (7.1), simply by replacing the variable t in x i ( t ) with the variable r . References [1] M. Dunajski, L.J. Mason and N.M.J. Woodhouse, J. Phys. A: Math. Gen. (1998)6019
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