CARLO ALBERTO MANTICA, LUCA GUIDO MOLINARI AND UDAY CHAND DE

Transcription

1 A CONDITION FOR A PERFECT-FLUID SPACE-TIME TO BE A GENERALIZED ROBERTSON-WALKER SPACE-TIME arxiv: v3 [math.dg] 4 Mar 206 CARLO ALBERTO MANTICA, LUCA GUIDO MOLINARI AND UDAY CHAND DE Abstract. A perfect-fluid space-time of dimension n 4 with ) irrotational velocity vector field, 2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with Einstein fiber. Condition ) is verified whenever pressure and energy density are related by an equation of state. The contraction of the Weyl tensor with the velocity vector field is zero. Conversely, a generalized Robertson-Walker space-time with null divergence of the Weyl tensor is a perfect-fluid space-time.. Introduction Standard cosmology is modelled on Robertson-Walker metrics for the high symmetry imposed on space-time by the cosmological principle (spatial homogeneity and isotropy). A wide generalization are the generalized Robertson-Walker spacetimes, introduced in 995 by Alías, Romero and Sánchez [, 2]: Definition.. An n-dimensional Lorentzian manifold is a generalized Robertson- Walker space-time (GRW) if locally the metric may take the form: () ds 2 = dt 2 +q(t) 2 g αβ(x 2,...,x n )dx α dx β, α,β = 2...n that is, it is the warped product ( ) q 2 M, where M is a (n )-dimensional Riemannian manifold. If M has dimension 3 and has constant curvature, the space-time is a Robertson-Walker space-time. Such spaces include the Einstein-de Sitter space-time, the Friedmann cosmological models, the static Einstein space-time and the de Sitter space-time. They are the stage for treatment of small perturbations of the Robertson - Walker metric. We refer to the works by Romero et al. [29], Sánchez [30, 3], Gutierrez and Olea [9] for a comprehensive presentation of geometric properties and physical motivations. Recently Bang-Yen Chen proved the following deep result [8]: A Lorentzian manifold of dimension n 4 is a GRW space-time if and only if it admits a timelike vector, X j X j < 0, such that (2) k X j = ρg kj. According to Yano [37], a vector field v is torse-forming if k v j = ω k v j + fg jk, wheref is ascalarfunction andω k is a-form. Itspropertiesinpseudo-Riemannian manifolds were studied by Mike s and Rachůnek [26, 28]. The vector is named Date: 28 september Mathematics Subject Classification. 83F20, 53C30, 53C50. Key words and phrases. Generalized Robertson-Walker space-time, perfect fluid, concircular vector.

2 2 C. A. MANTICA, L. G. MOLINARI AND U. C. DE concircular if ω k is a gradient (or locally a gradient); in this case v can be rescaled to a vector X with the property (2) [26]. Mantica et al. [23] proved two sufficient conditions for a Lorentzian manifold of dimension n 4 to be a GRW space-time: the first one is the existence of a concircular vector such that u i u i =. The other sufficient condition restricts the Weyl and Ricci tensors: m C jkl m = 0 and R ij = Bu i u j where B is a scalar field and u is a time-like vector field. (3) Lorentzian manifolds with a Ricci tensor of the form R ij = Ag ij +Bu i u j, where A and B are scalar fields and u i u i =, are often named perfect fluid spacetimes. It is well known that any Robertson-Walker space-time is a perfect fluid space-time [27], and for n = 4 a GRW space-time is a perfect fluid if and only if it is a Robertson-Walker space-time. The form (3) of the Ricci tensor is implied by Einstein s equation if the energymatter content of space-time is a perfect fluid with velocity vector field u. The scalars A and B are linearly related to the pressure p and the energy density µ measured in the locally comoving inertial frame. They are not independent because of the Bianchi identity m R im = 2 ir, which translates into (4) m (Bu j u m ) = 2 j[(n 2)A B]. Geometers identify the special form (3) of the Ricci tensor as the defining property of quasi-einstein manifolds(with any metric signature). The Riemannian ones were introduced by Defever and Deszcz in 99 [3] (see also [5] and Chaki et al. [7]). In [6] Deszcz proved that a quasi-einstein Riemannian manifold with null Weyl tensor and few other conditions, is a warped product (+) q 2 M, where M is a (n )-dimensional Riemannian manifold of constant curvature. Pseudo-Riemannian quasi-einstein spaces arose in the study of exact solutions of Einstein s equations. Robertson-Walker space-times are quasi-einstein (see [4], [34] and references therein). Shepley and Taub studied a perfect-fluid space-time in dimension n = 4, with equation of state p = p(µ) and the additional condition that the Weyl tensor has null divergence, m C jkl m = 0. They proved the following: the space-time is conformally flat C jklm = 0, the metric is Robertson-Walker, the flow is irrotational, shear-free and geodesic [33]. A related result was obtained by Sharma [32] (corollary p.3584): if a perfect-fluid space-time in n = 4 with m C jkl m = 0 admits a proper conformal Killing vector, i.e. i X j + j X i = 2ρg ij, then it is conformally flat (C ijkl = 0). In the framework of Yang s gravitational theory, Guilfoyle and Nolan proved that a n = 4 perfect fluid space-time with p + µ 0 is a Yang pure space (i.e. m C jkl m = 0 and k R = 0) if and only if it is a Robertson-Walker space-time [20]. Coley proved that any perfect fluid solution of Einstein s equations satisfying a barotropic equation of state p = p(µ) and p + µ 0, which admits a proper conformal Killing vector parallel to the fluid 4-velocity, is locally a Friedmann- Robertson-Walker model [0]. De et al. [2] showed that n = 4 conformally flat almost pseudo Ricci-symmetric space-times, i.e. k R ij = (a k + b k )R ij + a j R ik + a j R jk, are Robertson-Walker space-times.

3 GRW PERFECT-FLUID SPACE-TIMES 3 Riemannian spaces equipped with a torse-forming vector field were studied by Yano as early as 944 [37]; his results were extended to pseudo-riemannian spaces by Sinyukov [35]. They showed that the existence of such a vector implies the following local shape of the metric: ds 2 = ±(dx ) 2 +F(x,...,x n )d s 2, where d s 2 is the metric of the submanifold parametrized by x 2,...,x n. If the vector field is concircular (then it is rescalable to k X j = ρg kj ) then F is a function of x only. De and Ghosh [] showed that if R ij = Ag ij + Bu i u j with u i closed and C ijkl = 0, then u is a concircular vector. The results were extended by Mantica et al. to pseudo Z-symmetric spaces [24] and to weakly Z-symmetric spaces [25]. In this paper the theorem by Shepley and Taub is generalised to perfect-fluid space-times of dimension n 4. The converse is also proven: a GRW spacetime with m C jkl m = 0 is a perfect-fluid space-time. In the conclusion, some consequences for physics are presented. 2. The theorem Theorem 2.. Let M be perfect fluid-space-time, i.e. a Lorentzian manifold (of dimension n > 3) with Ricci tensor R kl = Ag kl +Bu k u l, where A and B are scalar fields, u is a time-like unit vector field u j u j =. If k u j j u k = 0 (u is closed) and if m C jkl m = 0, then: i) u is a concircular vector and it is rescalable to a time-like conformal Killing vector X such that (5) k X j = ρg kj and k ρ = A B n X k; ii) M is a generalised Robertson-Walker space-time whose sub-manifold (M,g ) is a Riemannian Einstein space. iii) C jklm u m = 0. Proof. The condition m C jkl m = 0 implies: k R jl l R jk = 2(n ) (g jl k R g jk l R). With the explicit form of the Ricci tensor, it becomes (6) k (Bu j u l ) l (Bu j u k ) = g jl k γ g jk l γ 2(n ) being γ = (n 2)A+B. By transvecting with u j u l and using u l k u l = 0 we obtain (7) ( k +u k u l l )B +Bu l l u k = 2(n ) ( k +u k u l l )γ. Contractionof the identity (4) with u j gives B m u m = 2 um m γ, which rewrites identity (4) as: (8) ( k +u k u i i )B +Bu m m u k = 2 ( k +u k u i i )γ. Equations (7) and (8) imply: (9) (0) ( j +u j u k k )γ = 0, ( j +u j u k k )B +Bu m m u j = 0.

4 4 C. A. MANTICA, L. G. MOLINARI AND U. C. DE Contraction of (6) with u l gives: u j ( k +u k u l l )B B k u j u j Bu l l u k u k Bu l l u j = 2(n ) (u j k g jk u l l )γ By use of eq. (0) it simplifies to: () B( k +u k u m m )u j = 2(n ) (u j k g jk u l l )γ If u is closed it is u m m u j = u m j u m = 0. Eq.() simplifies and shows that u is a torse-forming vector: k γ k u j = 2B(n ) u j um m γ (2) 2B(n ) g kj ω k u j +fg kj Let us show that u is a concircular vector, i.e. that ω k is closed: j ω k k ω j = B (ω k j ω j k )B = (ω k u j ω j u k )u m m B by (0). Eq. (9) gives the relation ω k = u k u m ω m, then ω k u j ω j u k = 0. Being closed, ω k is locally the gradient of a scalar function: ω k = k σ. Let X l = u l e σ ; we have k X l = e σ ( u l k σ +ω k u l +fg kl ) = e σ fg kl and consequently (3) k X l = ρg kl being ρ = e σ f and X j X j = e 2σ < 0 (time-like vector). The symmetrized equation k X j + j X k = 2ρg kj shows that X j is a conformal Killing vector [34]. According to Chen s theorem, (3) is a sufficient condition for the space-time to be a GRW. In appropriate coordinates M = ( ) q 2 M. The additional condition m C jkl m = 0 assures that the (n )-dimensional Riemannian space M is an Einstein space, by Gȩbarowski s lemma [8]. AnotherderivativeandtheRicciidentity give: ( j k k j )X l = R jkl m X m = g kl j ρ g jl k ρ. Contraction with g kl : R jm X m = ( n) j ρ. However, for the perfect fluid (3), R jm X m = (A B)X j, then: (4) j ρ = A B n X j (this is an explicit expression for a relation obtained by Chen). Therefore, if A B the conformal killing vector X is proper; if A = B it is homothetic. Moreover: (5) R jklm X m = A B n (X jg kl X k g jl ) The Weyl tensor is: C jklm = R jklm + n 2 (g jmr kl g km R jl +R jm g kl R km g jl ) (g jmg kl g mk g jl )R (n )(n 2) ThepreviousequationsandlittlealgebraimplythatC jklm X m = 0,sothatC m jkl u m = 0. It follows that the Weyl tensor is purely electric [2]. In n = 4 the condition is equivalent to u i C jklm + u j C kilm + u k C ijlm = 0 (see Lovelock and Rund [22] page 28). Multiplication by u i gives C ijkl = 0. Remark 2.2. Eq.(9) gives u j k γ = u k j γ. In the antisymmetric part of eq.(), B( k u j j u k ) + B(u k u m m u j u j u m m u k ) = 0, the last terms are replaced with the help of (0) to give: (6) k (Bu j ) = j (Bu k )

5 GRW PERFECT-FLUID SPACE-TIMES 5 Remark 2.3. The case A = 0, i.e. R ij = Bu i u j, was studied in [23]. Since γ = B, the property u j k γ = u k j γ and (6) imply that u is closed. If A 0 the condition that u is closed is necessary for proving the theorem. However, if a one-to-one differentiable relation A(x) = F(B(x)) exists, one proves that u is closed. Remark 2.4. In [5, 6, 7] a notion of quasi-einstein manifold different from (3) was introduced. It emerges from generalizations of Ricci solitons. More generally, they defined a generalized quasi - Einstein manifold by the condition (7) R ij + i j θ η( i θ)( j θ) = λg ij where θ,η,λ are smooth functions. If λ = const and η = 0 it is named gradient Ricci soliton, if λ =const. and η =const. it is named quasi-einstein. In the present case, the condition that u is closed means that locally u k = k θ, for some function θ. Then (3) takes the form R ij = Ag ij +B( i θ)( j θ). At the same time, eq.(2) can be written i j θ = f( i θ)( j θ)+fg ij (since f = u k ω k and ω i = u i u k ω k by eq.(9), it is ω i = fu i ). The sum of the equations yields a Ricci tensor of the form (7) with λ = A + f and η = B + f, i.e. the manifold is generalized quasi-einstein in the sense of [5, 6, 7]. A gradient Ricci soliton is recovered if A+f =const. and B +f = 0. In [5] it was proven that locally conformally flat Lorentzian quasi-einstein manifolds are globally conformally equivalent to a space form, or locally isometric to a warped product of Robertson-Walker type, or a pp-wave. Catino [6] proved that a complete (i.e. A + f is a smooth function) generalized quasi-einstein Riemannian manifold with harmonic Weyl tensor and zero radial curvature, is locally a warped product with (n ) dimensional Einstein fibers. An inverse statement of the theorem is proven: Theorem 2.5. A generalized Robertson-Walker space-time with m C jkl m = 0 is a quasi-einstein space-time. Proof. A GRW is characterized by the metric (). The explicit form of the Ricci tensor R ij is reported for example in Arslan et al.[3]: R α = R α = 0, R = (n ) q q, R αβ = Rαβ +gαβ [ q 2 (n 2)+qq ], α,β = 2...n. Gȩbarowski proved that m C m jkl = 0 if and only if Rαβ = g αβ R n, then: [ ] R R αβ = gαβ n + q 2 (n 2)+qq. Following the trick in [9], in the local frame where () holds, define the vector u = and u α = 0 (u = ). It is u j u j = in any frame. The components of the Ricci tensor gain the covariant expression R ij = Ag ij +Bu i u j, where: (8) A = q 2 [ R n + q 2 (n 2)+qq ], B = (n ) q q +A The expression is such in all coordinate frames, and characterizes a quasi-einstein Lorentzian manifold.

6 6 C. A. MANTICA, L. G. MOLINARI AND U. C. DE 3. Some notes on physics We transpose some of the results to physics (we use units c = ). Consider a perfect fluid with energy momentum tensor T ij = pg ij + (p+µ)u i u j, where u j is the velocity vector field, p is the isotropic pressure field and µ is the energy density. By Einstein s equations R ij 2 Rg ij = κt ij (κ = 8πG is the gravitational constant) the Ricci tensor is: R ij = κ(p+µ)u i u j +κ p µ 2 n g ij. Comparison with the form (3) identifies A = κ(p µ)/(2 n), B = κ(p+µ). Then γ = (n 2)A+B = 2κµ. As is well known (see Wald [36]) in General Relativity the equations of motion k T kj = 0 result from the Bianchi identity in Einstein s equations. For a perfect fluid, the projection along u and its complementary part are: (9) (20) u k k µ+(p+µ) k u k = 0 ( j +u j u k k )p+(p+µ)u k k u j = 0 By taking into account the results of the previous section we prove: Proposition 3.. A perfect fluid space-time in dimension n 4, with differentiable equation of state p = p(µ), p+µ 0, and with null divergence of the Weyl tensor, m C jkl m = 0, is a generalized Robertson-Walker space-time. The velocity vector field is irrotational ( k u l l u k = 0), geodesic (u k k u j = 0) and it annihilates the Weyl tensor (C jkl m u m = 0). Proof. We prove that u is irrotational and geodesic. Then, by the main theorem 2. it follows that the manifold is a generalized Robertson-Walker space-time and that u annihilates the Weyl tensor. If p (µ) 0 then k p = p (µ) k µ. The eqs. u j k γ = u k j γ and (6) become: u j k µ = u k j µ and j [(p + µ)u k ] = k [(p + µ)u j ]. Being k p = p (µ) k µ it follows that k u j = j u k. Eq.(9) is j µ + u j u m m µ = 0, and translates to j p + u j u m m p = 0. This is used in (20) to annihilate the first term. The equation of a geodesic is obtained: (p+µ)u k k u j = 0. If k p = 0, eq. (9) again gives (p+µ)u k k u j = 0. The special case A = B in (4) characterizes a homothetic conformal Killing field ( j X k = ρg jk with j ρ = 0). In terms of pressure and density this means which, in n = 4, is p = µ/3. p = 3 n n µ References [] L. J. Alías, A. Romero, M. Sánchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson-Walker space-times, Gen. Relativ. Gravit. 27 n. (995) [2] L. Alías, A. Romero, M. Sánchez, Compact spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes. In: Dillen F. editor. Geometry and Topology of Submanifolds VII. River Edge N.J., USA, World Scientific, 995, pp [3] K. Arslan, R. Deszcz, R. Ezentaş, M. Hotloś, C. Murathan, On generalized Robertson-Walker spacetimes satisfying some curvature condition, Turk. J. Math. 38 (204)

7 GRW PERFECT-FLUID SPACE-TIMES 7 [4] J. K. Beem, P. E. Ehrlich, K. L. Easley, Global Lorentzian Geometry, 2nd ed. Pure and Applied Mathematics, vol. 202, 996 Marcel Dekker, New York. [5] M. Brozos-Vázquez, E. García-Río, S. Gavino Fernández, Locally conformally flat Lorentzian quasi-einstein manifolds, Monatshefte für Mathematik 73 (204) 2. [6] G. Catino, Generalized quasi-einstein manifolds with harmonic Weyl tensor, Mathematische Zeitschrift 27 n.3-4 (202) [7] M. C. Chaki and R. K. Maity, On quasi Einstein manifolds, Publ. Math. Debrecen 57 (2000), [8] B-Y. Chen, A simple characterization of generalized Robertson-Walker manifolds, Gen. Relativ. Gravit. 46 (204) 833. [9] J. Chojnacka-Dulas, R. Deszcz, M. G logowska and M. Prvanovic, On warped product manifolds satisfying some curvature conditions, J. Geom. Phys. 74 (203) [0] A. A. Coley, Fluid spacetimes admitting a conformal Killing vector parallel to the velocity vector, Class. Quantum Grav. 8 (99) [] U. C. De and S. K. Ghosh, On conformally flat pseudo-symmetric spaces, Balk. J. of Geom. and its Appl. 5 n.2 (2000) [2] A. De, C. Özgür and U. C. De, On conformally flat almost pseudo-ricci symmetric spacetimes, Int. J. Theor. Phys. 5 n.9 (202) [3] F. Defever and R. Deszcz, On semi-riemannian manifolds satisfying a condition R R = Q(S, R), in Geometry and Topology of submanifolds III, World Scientific Publish. Singapore (99) [4] R. Deszcz, F. Dillen, L. Verstraelen and L. Vrancken, Quasi-Einstein totally real submanifolds of the nearly Kähler 6-sphere, Tohoku Math. J. 5 (4) (999) [5] R. Deszcz, M. G logowska, M. Hotloś and Z. Sentürk, On certain quasi-einstein semisymmetric hypersurfaces, Ann. Univ. Sci. Budapest Eötvös Sect. Math. 4 (998) [6] R. Deszcz, On conformally flat Riemannian manifolds satisfying certain curvature conditions, Tensor (N.S.) 49 (990) [7] S. Gavino-Fernández, The geometry of Lorentzian Ricci solitons, Ph. D. Thesis, Publicaciones del Departamento de Geometria y Topologia, Universidade de Santiago de Compostela, (202) 05pp. [8] A. Gȩbarowski, On nearly conformally symmetric warped product spacetimes, Soochow J. Math. 20 n. (994) [9] M. Gutiérrez and B. Olea, Global decomposition of a Lorentzian manifold as a Generalized Robertson-Walker space, Differential Geom. Appl. 27 (2009) [20] B. S. Guilfoyle and B. C. Nolan, Yang s gravitational theory, Gen. Relativ. Gravit. 30 n.3 (998) [2] S. Hervik, M. Ortaggio, and L. Wylleman, Minimal tensors and purely electric and magnetic spacetimes of arbitrary dimensions, Class. Quantum Grav. 30 (203) 6504 (50pp). [22] D. Lovelock and H. Rund, Tensors, Differential Forms and Variational Principles, reprint Dover Ed. (988). [23] C. A. Mantica, Y. J. Suh and U. C. De, A note on generalized Robertson - Walker spacetimes, (private communication). [24] C. A. Mantica and Y. J. Suh, Pseudo Z symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Meth. Mod. Phys. 9 n. (202) (2pp). [25] C. A. Mantica and L. G. Molinari, Weakly Z symmetric manifolds, Acta Math. Hungar. 35 n.-2 (202) [26] J. Mike s and L. Rachůnek, Torse-forming vector fields in T-symmetric Riemannian spaces, Steps in Differential Geometry, Proc. Colloq. Diff. Geom., Debrecen, July 2000, [27] B. O Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 983. [28] L. Rachůnek and J. Mike s, On tensor fields semi-conjugated with torse-forming vector fields Acta Univ. Palacki. Olomuc., Fac. rer. nat. Mathematica 44 (2005) [29] A. Romero, R. N. Rubio and J. J. Salamanca, Uniqueness of complete maximal hypersurfaces in spatially parabolic generalised Robertson-Walker space times, Class. Quantum Grav. 30 n. (203) 5007 (3pp). [30] M. Sánchez, On the geometry of generalized Robertson-Walker spacetimes: geodesics, Gen. Relativ. Gravit. 30 (998)

8 8 C. A. MANTICA, L. G. MOLINARI AND U. C. DE [3] M. Sánchez, On the geometry of generalized Robertson-Walker spacetimes: curvature and Killing fields, Gen. Relativ. Gravit. 3 (999) 5. [32] R. Sharma, Proper conformal symmetries of space-times with divergence-free Weyl tensor, J. Math. Phys. 34 (993) [33] L. C. Shepley and A. H. Taub, Space-times containing perfect fluids and having a vanishing conformal divergence, Commun. Math. Phys. 5 (967) [34] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Hertl, Exact solutions of Einstein s Field Equations, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2nd ed. (2003). [35] N. S. Sinyukov, Geodesic mappings of Riemannian spaces, Nauka, Moscow 979. [36] R. M. Wald, General Relativity, The University of Chicago Press (984). [37] K. Yano, On the torse-forming directions in Riemannian spaces, Proc. Imp. Acad. Tokyo, 20 (944) C. A. Mantica: Physics Department, Università degli Studi di Milano, Via Celoria 6, 2033 Milano, Italy and I.I.S. Lagrange, Via L. Modignani 65, 206, Milano, Italy L. G. Molinari (corresponding author): Physics Department, Università degli Studi di Milano and I.N.F.N. sez. Milano, Via Celoria 6, 2033 Milano, Italy, U. C. De: Department of Pure Mathematics, University of Calcutta, 35 Ballygaunge Circular Road, Kolkata 70009, West Bengala, India address: carloalberto.mantica@libero.it, luca.molinari@mi.infn.it, uc de@yahoo.com