A condition for a perfect-fluid space-time to be a generalized Robertson-Walker space-time
aa r X i v : . [ m a t h . DG ] M a r A CONDITION FOR A PERFECT-FLUID SPACE-TIMETO BE A GENERALIZED ROBERTSON-WALKER SPACE-TIME
CARLO ALBERTO MANTICA, LUCA GUIDO MOLINARI AND UDAY CHAND DE
Abstract.
A perfect-fluid space-time of dimension n ≥ Introduction
Standard cosmology is modelled on Robertson-Walker metrics for the high sym-metry imposed on space-time by the cosmological principle (spatial homogeneityand isotropy). A wide generalization are the ”generalized Robertson-Walker space-times”, introduced in 1995 by Al´ıas, Romero and S´anchez [1, 2]:
Definition 1.1. An n -dimensional Lorentzian manifold is a generalized Robertson-Walker space-time (GRW) if locally the metric may take the form: ds = − dt + q ( t ) g ∗ αβ ( x , . . . , x n ) dx α dx β , α, β = 2 . . . n (1) that is, it is the warped product ( − × q M ∗ , where M ∗ is a ( n − -dimensionalRiemannian manifold. If M ∗ has dimension 3 and has constant curvature, thespace-time is a Robertson-Walker space-time. Such spaces include the Einstein-de Sitter space-time, the Friedmann cosmologicalmodels, the static Einstein space-time and the de Sitter space-time. They are thestage for treatment of small perturbations of the Robertson - Walker metric. Werefer to the works by Romero et al. [29], S´anchez [30, 31], Gutierrez and Olea [19]for a comprehensive presentation of geometric properties and physical motivations.Recently Bang-Yen Chen proved the following deep result [8]: A Lorentzianmanifold of dimension n ≥ X j X j <
0, such that ∇ k X j = ρg kj . (2)According to Yano [37], a vector field v is torse-forming if ∇ k v j = ω k v j + f g jk ,where f is a scalar function and ω k is a 1-form. Its properties in pseudo-Riemannianmanifolds were studied by Mike˘s and Rach˚unek [26, 28]. The vector is named Date : 28 september 2015.2010
Mathematics Subject Classification.
Key words and phrases.
Generalized Robertson-Walker space-time, perfect fluid, concircularvector. concircular if ω k is a gradient (or locally a gradient); in this case v can be rescaledto a vector X with the property (2) [26].Mantica et al. [23] proved two sufficient conditions for a Lorentzian manifoldof dimension n ≥ u i u i = −
1. The other sufficient condition restricts theWeyl and Ricci tensors: ∇ m C jklm = 0 and R ij = Bu i u j where B is a scalar fieldand u is a time-like vector field.Lorentzian manifolds with a Ricci tensor of the form R ij = Ag ij + Bu i u j , (3)where A and B are scalar fields and u i u i = −
1, are often named perfect fluid space-times . It is well known that any Robertson-Walker space-time is a perfect fluidspace-time [27], and for n = 4 a GRW space-time is a perfect fluid if and only if itis a Robertson-Walker space-time.The form (3) of the Ricci tensor is implied by Einstein’s equation if the energy-matter content of space-time is a perfect fluid with velocity vector field u . Thescalars 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 becauseof the Bianchi identity ∇ m R im = ∇ i R , which translates into ∇ m ( Bu j u m ) = ∇ j [( n − A − B ] . (4)Geometers identify the special form (3) of the Ricci tensor as the defining propertyof quasi-Einstein manifolds (with any metric signature). The Riemannian ones wereintroduced by Defever and Deszcz in 1991 [13] (see also [15] and Chaki et al. [7]).In [16] Deszcz proved that a quasi-Einstein Riemannian manifold with null Weyltensor and few other conditions, is a warped product (+1) × q M ∗ , where M ∗ isa ( n − n = 4, withequation of state p = p ( µ ) and the additional condition that the Weyl tensor hasnull divergence, ∇ m C jklm = 0. They proved the following: the space-time isconformally 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-fluidspace-time in n = 4 with ∇ m C jklm = 0 admits a proper conformal Killing vector,i.e. ∇ i X j + ∇ j X i = 2 ρg ij , then it is conformally flat ( C ijkl = 0). In the frameworkof Yang’s gravitational theory, Guilfoyle and Nolan proved that a n = 4 perfectfluid space-time with p + µ = 0 is a Yang pure space (i.e. ∇ m C jklm = 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 abarotropic equation of state p = p ( µ ) and p + µ = 0, which admits a properconformal Killing vector parallel to the fluid 4-velocity, is locally a Friedmann-Robertson-Walker model [10].De et al. [12] showed that n = 4 conformally flat almost pseudo Ricci-symmetricspace-times, i.e. ∇ k R ij = ( a k + b k ) R ij + a j R ik + a j R jk , are Robertson-Walkerspace-times. RW PERFECT-FLUID SPACE-TIMES 3
Riemannian spaces equipped with a torse-forming vector field were studied byYano as early as 1944 [37]; his results were extended to pseudo-Riemannian spacesby Sinyukov [35]. They showed that the existence of such a vector implies thefollowing local shape of the metric: ds = ± ( dx ) + F ( x , . . . , x n ) d ˜ s , where d ˜ s is the metric of the submanifold parametrized by x , . . . , x n . If the vector field isconcircular (then it is rescalable to ∇ k X j = ρg kj ) then F is a function of x only.De and Ghosh [11] 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 Man-tica 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-fluidspace-times of dimension n ≥
4. The converse is also proven: a GRW space-time with ∇ m C jklm = 0 is a perfect-fluid space-time. In the conclusion, someconsequences for physics are presented.2. The theorem
Theorem 2.1.
Let M be perfect fluid-space-time, i.e. a Lorentzian manifold (ofdimension n > ) with Ricci tensor R kl = Ag kl + Bu k u l , where A and B are scalarfields, 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 jklm = 0 , then:i) u is a concircular vector and it is rescalable to a time-like conformal Killingvector X such that ∇ k X j = ρg kj and ∇ k ρ = A − B − n X k ;(5) 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 jklm = 0 implies: ∇ k R jl − ∇ l R jk = n − ( g jl ∇ k R − g jk ∇ l R ). With the explicit form of the Ricci tensor, it becomes ∇ k ( Bu j u l ) − ∇ l ( Bu j u k ) = − g jl ∇ k γ − g jk ∇ l γ n − γ = ( n − A + B . By transvecting with u j u l and using u l ∇ k u l = 0 we obtain( ∇ k + u k u l ∇ l ) B + Bu l ∇ l u k = 12( n −
1) ( ∇ k + u k u l ∇ l ) γ. (7)Contraction of the identity (4) with u j gives − B ∇ m u m = u m ∇ m γ , which rewritesidentity (4) as: ( ∇ k + u k u i ∇ i ) B + Bu m ∇ m u k = ( ∇ k + u k u i ∇ i ) γ. (8)Equations (7) and (8) imply: ( ∇ j + u j u k ∇ k ) γ = 0 , (9) ( ∇ j + u j u k ∇ k ) B + Bu m ∇ m u j = 0 . (10) 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 = − n −
1) ( u j ∇ k − g jk u l ∇ l ) γ By use of eq. (10) it simplifies to: B ( ∇ k + u k u m ∇ m ) u j = 12( n −
1) ( u j ∇ k − g jk u l ∇ l ) γ (11)If u is closed it is u m ∇ m u j = u m ∇ j u m = 0. Eq.(11) simplifies and shows that u isa torse-forming vector: ∇ k u j = ∇ k γ B ( n − u j − u m ∇ m γ B ( n − g kj ≡ ω k u j + f g kj (12)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 (10). 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 + f g kl ) = e − σ f g kl and consequently ∇ k X l = ρ g kl (13)being ρ = e − σ f and X j X j = − e − σ < ∇ k X j + ∇ j X k = 2 ρg kj shows that X j is a conformal Killing vector [34].According to Chen’s theorem, (13) is a sufficient condition for the space-timeto be a GRW. In appropriate coordinates M = ( − × q M ∗ . The additionalcondition ∇ m C jklm = 0 assures that the ( n − M ∗ is an Einstein space, by G¸ebarowski’s lemma [18].Another derivative and the Ricci identity give: ( ∇ j ∇ k −∇ k ∇ j ) X l = R jklm X m = g kl ∇ j ρ − g jl ∇ k ρ . Contraction with g kl : R jm X m = (1 − n ) ∇ j ρ . However, for theperfect fluid (3), R jm X m = ( A − B ) X j , then: ∇ j ρ = A − B − n X j (14)(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: R jklm X m = A − B − n ( X j g kl − X k g jl )(15)The Weyl tensor is: C jklm = R jklm + n − ( g jm R kl − g km R jl + R jm g kl − R km g jl ) − ( g jm g kl − g mk g jl ) R ( n − n − C jklm X m = 0, so that C jklm u m =0. It follows that the Weyl tensor is purely electric [21].In n = 4 the condition is equivalent to u i C jklm + u j C kilm + u k C ijlm = 0 (seeLovelock and Rund [22] page 128). Multiplication by u i gives C ijkl = 0. (cid:3) Remark 2.2.
Eq. (9) gives u j ∇ k γ = u k ∇ j γ . In the antisymmetric part of eq. (11) , 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 replacedwith the help of (10) to give: ∇ k ( Bu j ) = ∇ j ( Bu k )(16) RW 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 (16) imply that u is closed.If A = 0 the condition that u is closed is necessary for proving the theorem. How-ever, 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, 17] 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 R ij + ∇ i ∇ j θ − η ( ∇ i θ )( ∇ j θ ) = λg ij (17) where θ, η, λ are smooth functions. If λ = const and η = 0 it is named gradientRicci 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 thesame time, eq. (12) can be written ∇ i ∇ j θ = f ( ∇ i θ )( ∇ j θ ) + f g ij (since f = − u k ω k and ω i = − u i u k ω k by eq. (9) , it is ω i = f u i ). The sum of the equations yields aRicci tensor of the form (17) with λ = A + f and η = B + f , i.e. the manifoldis generalized quasi-Einstein in the sense of [5, 6, 17] . A gradient Ricci soliton isrecovered if A + f = const. and B + f = 0 .In [5] it was proven that locally conformally flat Lorentzian quasi-Einstein manifoldsare globally conformally equivalent to a space form, or locally isometric to a warpedproduct 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 manifoldwith harmonic Weyl tensor and zero radial curvature, is locally a warped productwith ( n − dimensional Einstein fibers. An inverse statement of the theorem is proven:
Theorem 2.5.
A generalized Robertson-Walker space-time with ∇ m C jklm = 0 isa quasi-Einstein space-time.Proof. A GRW is characterized by the metric (1). The explicit form of the Riccitensor R ij is reported for example in Arslan et al.[3]: R α = R α = 0, R = − ( n − q ′ q , R αβ = R ∗ αβ + g ∗ αβ (cid:2) q ′ ( n −
2) + qq ′′ (cid:3) , α, β = 2 . . . n. G¸ebarowski proved that ∇ m C jklm = 0 if and only if R ∗ αβ = g ∗ αβ R ∗ n − , then: R αβ = g ∗ αβ (cid:20) R ∗ n − q ′ ( n −
2) + qq ′′ (cid:21) . Following the trick in [9], in the local frame where (1) holds, define the vector u = 1 and u α = 0 ( u = − u j u j = − R ij = Ag ij + Bu i u j , where: A = 1 q (cid:20) R ∗ n − q ′ ( n −
2) + qq ′′ (cid:21) , B = − ( n − q ′ q + A (18)The expression is such in all coordinate frames, and characterizes a quasi-EinsteinLorentzian manifold. (cid:3) C. A. MANTICA, L. G. MOLINARI AND U. C. DE Some notes on physics
We transpose some of the results to physics (we use units c = 1). Consider aperfect fluid with energy momentum tensor T ij = pg ij + ( p + µ ) u i u j , where u j isthe velocity vector field, p is the isotropic pressure field and µ is the energy density.By Einstein’s equations R ij − Rg ij = κT ij ( κ = 8 πG is the gravitational constant)the Ricci tensor is: R ij = κ ( p + µ ) u i u j + κ p − µ − n g ij . Comparison with the form (3) identifies A = κ ( p − µ ) / (2 − n ), B = κ ( p + µ ). Then γ = ( n − A + B = 2 κµ .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 perfectfluid, the projection along u and its complementary part are: u k ∇ k µ + ( p + µ ) ∇ k u k = 0(19) ( ∇ j + u j u k ∇ k ) p + ( p + µ ) u k ∇ k u j = 0(20)By taking into account the results of the previous section we prove: Proposition 3.1.
A perfect fluid space-time in dimension n ≥ , with differentiableequation of state p = p ( µ ) , p + µ = 0 , and with null divergence of the Weyl tensor, ∇ m C jklm = 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 jklm u m = 0 ).Proof. We prove that u is irrotational and geodesic. Then, by the main theorem2.1 it follows that the manifold is a generalized Robertson-Walker space-time andthat 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 (16) 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 itfollows 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 isused 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. (19) again gives ( p + µ ) u k ∇ k u j = 0. (cid:3) The special case A = B in (14) characterizes a homothetic conformal Killingfield ( ∇ j X k = ρg jk with ∇ j ρ = 0). In terms of pressure and density this means p = 3 − nn − µ which, in n = 4, is p = − µ/ References [1] L. J. Al´ıas, A. Romero, M. S´anchez,
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On the torse-forming directions in Riemannian spaces , Proc. Imp. Acad. Tokyo, (1944) 340–345. C. A. Mantica: Physics Department, Universit`a degli Studi di Milano, Via Celoria16, 20133 Milano, Italy and I.I.S. Lagrange, Via L. Modignani 65, 20161, Milano, Italy– L. G. Molinari (corresponding author): Physics Department, Universit`a degli Studidi Milano and I.N.F.N. sez. Milano, Via Celoria 16, 20133 Milano, Italy, – U. C. De:Department of Pure Mathematics, University of Calcutta, 35 Ballygaunge CircularRoad, Kolkata 700019, West Bengala, India
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