Static and stationary dark fluid universes: A gravitoelectromagnetic perspective
aa r X i v : . [ g r- q c ] J un Static and stationary dark fluid universes: Agravitoelectromagnetic perspective M . Nouri-Zonoz ( a ) ∗ and A . Nouri-Zonoz ( b ) † (a): Department of Physics, University of Tehran,North Karegar Ave., Tehran 14395-547, Iran.(b): Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran. Abstract
We introduce a physical characterization of the static and stationary perfect fluid solutions ofthe Einstein field equations with a single or 2-component perfect fluid sources, according to theirgravitoelectric and gravitomagnetic fields. The absence or presence of either or both of these fieldscould restrict the equations of state of the underlying perfect fluid sources. As an example andrepresentative of each class we consider solutions that include the cosmological term as a fluidsource with the equation of state p = − ρ = constant . All these solutions share the feature that goover smoothly into the Minkowski spacetime as Λ → ∗ Electronic address: [email protected] (Corresponding author) † Electronic address: [email protected] . INTRODUCTION AND MOTIVATION There are detailed discussions of exact solutions of Einstein field equations (EFE) andtheir characterization based on different symmetry groups of either geometric objects (suchas Weyl and Ricci tensors) or the energy-momentum tensor in the literature [1]. Static andstationary perfect fluid solutions, on the other hand have played a pivotal role in the evo-lution of the cosmological models and have been discussed extensively in the exact solutionliterature [1, 2]. There could be more than one perfect fluid source with different barotropicequations of state (EOS), for static and stationary perfect fluid solutions. Employing thequasi-Maxwell form of the Einstein field equations for multi-component perfect fluid sources,here we show how a combination of different choices for the gravitoelectromagnetic (GEM)fields, along with different EOS for different perfect fluid sources, could naturally lead towell-known static and stationary perfect fluid spacetimes as the representative of each class,hence furnishing a physical characterization of these spacetimes. The presence or absenceof either or both of the gravitoelectric (GE) and gravitomagnetic (GM) fields could in somecases, not only restrict the minimum number of the perfect fluid sources, but also fix theirEOS. We will treat the cosmological term, Λ g ab , as a perfect (dark) fluid source with EOS p Λ = − ρ Λ , in which ρ Λ = Λ8 π . Interestingly enough we will find out that in some cases thesign of the cosmological constant, or equivalently ρ Λ , is fixed by our choice of the GEMfields. Indeed, as an interesting example of the above characterization, it has already beenshown that the de Sitter space, and the so called de Sitter-type spacetimes are the only staticsingle-component perfect fluid solutions of EFE in the non-comoving frames [3]. Character-izing them in this way, the apparent paradox raised by some authors [4, 5] on why there aredifferent static spacetimes with Λ as their only parameter was resolved. De Sitter-type space-times are axially and cylindrically symmetric static Einstein spaces (solutions of R ab = Λ g ab )with Λ as their only parameter, so that they were first expected to be the good old de Sitterspacetime just in different coordinate systems. But they were found to be genuinely differentfrom de Sitter space, when their curvature invariants as well as their dynamical forms inthe comoving synchronous coordinate systems were calculated. These findings motivatedthe idea that one should consider a perfect fluid nature for the cosmological term and assigna 4-velocity to the fluid particles, in order to be able to interpret the preferred directionalexpansion of the de Sitter-type spacetimes in their dynamical forms [3].2ere in light of the recent introduction of the stationary analogs of de Sitter and anti-deSitter spacetimes [6] we will show how the static and stationary dark fluid universes couldbe characterized in terms of their gravitoelectromagnetic fields.The outline of the paper is as follows. In the next section we introduce the threadingformulation of spacetime decomposition, and the quasi-Maxwell form of the Einstein fieldequations. In four subsections of section III, using the characterization based on the quasi-Maxwell form of EFE, and the gravitoelectromagnetic fields, we show how the homogeneousstatic and stationary perfect fluid solutions, with linear barotropic equations of state (EOS)could be categorized. The Einstein static universe, the (anti-)de Sitter spacetime, the G¨odeluniverse and the above-mentioned stationary analogs of de Sitter and anti-de Sitter space-times will act as the representatives of these four classes of solutions.Throughout, the Latin indices run from 0 to 3 while the Greek ones run from 1 to 3, andwe will use the units in which c = G = 1. II. GRAVITOELECTROMAGNETISM AND THE QUASI-MAXWELL FORM OFTHE EINSTEIN FIELD EQUATIONS
The 1 + 3 or threading formulation of spacetime decomposition is the decomposition ofspacetime by the worldlines of fundamental observers who are at fixed spatial points in agravitational field. In other words, these worldlines, sweeping the history of the spatialpositions of the fundamental observers, decompose the underlying spacetime into timelikethreads [7]. In stationary asymptotically flat spacetimes, these observers are at rest withrespect to the distant observers in the asymptotically flat region. Employing propagation ofradar signals between two nearby fundamental observers (i.e ignoring spacetime curvature),the spacetime metric could be expressed in the following general form, ds = dτ sy − dl = g ( dx − g α dx α ) − γ αβ dx α dx β , (1)where g α = − g α g and γ αβ = − g αβ + g α g β g ; γ αβ = − g αβ , (2)is the spatial metric of a 3-space Σ , on which dl gives the element of spatial distancebetween any two nearby events. Also, dτ sy = √ g ( dx − g α dx α ) gives the infinitesimalinterval of the so-called synchronized proper time between any two events. In other words3 B Test particle trajectoryLight signals
FIG. 1: A congruence of nearby worldlines of fundamental observers and a test particle crossingthem. The observers A and B exchange radar signals to define spatial distances and the 3-velocityof the test particle in terms of the synchronized proper time. any two simultaneous events have a world-time difference of dx = g α dx α . The origin ofthis definition of a time interval could be explained through the following procedure. If theparticle departs from point B (with spatial coordinates x α ) at the moment of world time x and arrives at the infinitesimally distant point A (with spatial coordinates x α + dx α ) atthe moment x + dx , then to determine the velocity we must now take, difference between x + dx and the moment x − g α g dx α which is simultaneous at the point B with the moment x at the point A (Fig. 1). Now upon dividing the infinitesimal spatial coordinate interval dx α by this time difference the 3-velocity of a particle in the underlying spacetime is givenby [7, 8] v α = dx α dτ sy = dx α √ g ( dx − g α dx α ) . (3)Obviously, in the case of static spacetimes (i.e., g α = 0) the above definition reduces tothe proper velocity defined by v α = √ g dx α dx [25].Substituting the above definition of 3-velocity in Eq. (1), one can show the following relationbetween the proper and synchronized proper times dτ = g ( dx − g α dx α ) [1 − v ] = dτ sy. (1 − v ) . (4)4lso the components of the 4-velocity u i = dx i /dτ of a test particle, in terms of the compo-nents of its 3-velocity, are given by u α = v α √ − v , u = 1 √ − v (cid:18) √ g + g α v α (cid:19) . (5)Obviously the comoving frame is defined by v α = 0 leading to u i = ( √ g , , ,
0) as expected.Applying the above formalism we define the 3-force acting on a test particle in a stationarygravitational field as the 3-dimensional covariant derivative of the particle’s 3-momentumwith respect to the synchronized proper time [7, 8], i.e, f µ ≡ Dp µ dτ sy = √ − v Dp µ dτ , (6)in which we used equation (4) to write it in terms of the proper time. Since by definition p µ = mu µ , we use the spatial components of the geodesic equation for a test particle, namely du µ dτ = − Γ µab u a u b = − Γ µ ( u ) − µ β u u β − Γ µαβ u α u β . (7)and substitute expressions for the connection coefficients in terms of the 3-dimensional ob-jects and the 4-velocity components from (5), to arrive at the following expression for theGEM Lorentz-type 3-force, f µ = √ − v ddτ mv µ √ − v + λ µαβ mv α v β √ − v , (8)in which λ µαβ is the 3-dimensional Christoffel symbol constructed from γ αβ . Intuitively, thisshows that test particles moving on the geodesics of a stationary spacetime depart from thegeodesics of the 3-space Σ as if acted on by the above-defined 3-force. Lowering the index,in its vectorial form the above expression could be written in the following form, f g = m √ − v ( E g + v × √ g B g ) , (9)in which the gravitoelectric (GE) and gravitomagnetic (GM) 3-fields (with lower and upperindices respectively), are defined as follows [26] B g = curl ( A g ) ; ( A gα ≡ g α ) (10) E g = −∇ ln √ h ; ( h ≡ g ) , (11)in which ln √ h and A g are the so-called gravitoelectric and gravitomagnetic potentials re-spectively [10]. We notice that gravitoelectric part of the GEM Lorentz force is the general5elativistic version of the gravitational force in Newtonian gravity [11], while its gravito-magnetic part has no counterpart in Newtonian gravity. Obviously by their definition, theysatisfy the following constraints ∇ × E g = 0 , ∇ · B g = 0 . (12)Now in terms of the GEM fields measured by the fundamental observers, the Einstein fieldequations for a multi-component fluid sources, each having an energy-momentum tensor T ab = ( p + ρ ) u a u b − pg ab with u a u a = 1, could be written in the following quasi-Maxwellform [8], ∇ · E g = 12 hB g + E g − π Σ i (cid:18) p i + ρ i − v i − ρ i − p i (cid:19) (13) ∇ × ( √ h B g ) = 2 E g × ( √ h B g ) − π Σ i (cid:18) p i + ρ i − v i (cid:19) v i (14) (3) P µν = − E µ ; νg + 12 h ( B µg B νg − B g γ µν ) + E µg E νg + 8 π Σ i (cid:18) p i + ρ i − v i v iµ v iν + ρ i − p i γ µν (cid:19) , (15)in which v i is the 3-velocity of the i -th component of the source fluid as defined in (3). Also (3) P µν is the three-dimensional Ricci tensor made out of the 3-d metric γ µν . Here we focuson 2-component fluid sources so that i = 1 , III. STATIC AND STATIONARY PERFECT FLUID SOLUTIONS
Using the quasi-Maxwell form of the Einstein field equations (13)-(15), in what followswe will employ the following three criteria to characterize well-known static and stationaryperfect fluid solutions:I-Vanishing of either or both of the gravitoelectric ( E g ) and gravitomagnetic ( B g ) fields.II-Number of perfect fluid components and their corresponding EOS.III-Fluid components and their frames: either a comoving frame or a non-comoving one.Indeed in what follows we will find out that applying the first criterion to Eqs. (13) and(14), will automatically restrict both the minimum number of the fluid components as wellas their EOS in a given frame. 6V-In the case of static spacetimes we only consider spherically symmetric solutions whereasin the case of stationary spacetimes we restrict our study to cylindrically symmetric cases. A. Spacetimes without gravitoelectromagnetic fields E g and B g : Einstein StaticUniverse Substituting E g = 0 and B g = 0 in Eqs. (13)-(14) we end up with the following equations,Σ i (cid:18) p i + ρ i − v i − ρ i − p i (cid:19) = 0 (16)Σ i (cid:18) p i + ρ i − v i (cid:19) v i = 0 (17) (3) P µν = 8 π Σ i (cid:18) p i + ρ i − v i v iµ v iν + ρ i − p i γ µν (cid:19) . (18)As pointed out above, in what follows, we will only consider spherically symmetric solutionsas the prototype solutions. We notice that the first two equations only include the sourcespecifications and any solution has the following characteristics:1-It is a static spacetime because it has a vanishing gravitomagnetic field [3].2-With E g = B g = 0 in the GEM Lorentz force (9), there will be no gravitational forceacting on test particles in this spacetime, i.e. particles stay where they are.Now Eq. (17) seems to be satisfied for a single component perfect fluid, either withA-any EOS in a comoving frame ( v = 0) or B- a dark fluid with EOS p = − ρ .If we take the first case and substitute v = 0 in Eq. (16), that will fix the fluid EOS to p = ρ/ incoherent radiation . Of course photons as particles of radiationare not timelike and do not satisfy u a u a = 1. Now if we choose the second single componentfluid with EOS p = − ρ , that will not satisfy Eq. (16). Also it is noticed that we have foundthese results without recourse to the last equation and in fact none of these choices satisfyEq. (18) which takes the forms (3) P µν = ± πpγ µν (with the minus sign for the dark fluid)for a constant pressure.From a physical point of view, that a single-component fluid does not lead to a solution isexpected, since any kind of normal matter will produce attractive gravity, and hence leadsto a collapsing system with F g = 0, hence contradicting the second point above. Indeed thiswas the problem Einstein faced in his 1917 effort to find an static Universe.So to have a solution we need at least a 2-component fluid which, when plugged into Eqs.716)-(18), leads to the following equations; (cid:18) p + ρ − v − ρ − p (cid:19) + (cid:18) p + ρ − v − ρ − p (cid:19) = 0 (19) (cid:18) p + ρ − v (cid:19) v + (cid:18) p + ρ − v (cid:19) v = 0 (20) (3) P µν = 8 π (cid:18) p + ρ − v v µ v ν + ρ − p γ µν (cid:19) + 8 π (cid:18) p + ρ − v v µ v ν + ρ − p γ µν (cid:19) . (21)Looking at Eq. (20), we notice that one can always satisfy it by choosing one of the fluidcomponents (with any well-known EOS) to be in the comoving frame (say v = 0), and thesecond component to have an EOS p = − ρ , that of a dark fluid. Obviously the next step isto put these values in Eq. (19) to find the relation between the two component densities (orpressures). The last equation, Eq. (21) serves for the application of the required symmetry.Now we could have for the fluid in the comoving frame either 1- dust ( p = 0), 2-radiation( p = ρ/
3) or 3- stiff matter ( p = ρ ) leading respectively to:1-Einstein static universe in which the relation between the two fluid densities is given by ρ Λ = ρ dust or equivalently Λ = 4 πρ dust .2-Static universe filled with incoherent radiation in which the relation between the two fluiddensities is given by ρ Λ = ρ radiation or equivalently Λ = 8 πρ radiation [27].3-Static universe filled with stiff matter (SM) in which the relation between the two fluiddensities is given by ρ Λ = 2 ρ SM or equivalently Λ = 16 πρ SM .In terms of the cosmological constant, the metric of the above three static spherically sym-metric spacetimes are given by, ds = dt − dr − Λ β r − r ( dθ + sin θdφ ) , (22)in which β = 1 , / , →
0, and the obvious fact that Λ β gives the spacetime curvature for different values of β . In summary, vanishing of both E g and B g is consistent with the static nature of this solution where the repulsion of the darkfluid counterbalances the attraction of the non-dark element which could be dust, incoherentradiation or stiff matter. 8 . Spacetimes without a gravitomagnetic field B g : de Sitter spacetime Starting from Eqs. (13)-(15) and setting B g = 0, we end up with the following equations; ∇ · E g = E g − π Σ i (cid:18) p i + ρ i − v i − ρ i − p i (cid:19) (23)Σ i (cid:18) p i + ρ i − v i (cid:19) v i = 0 (24) (3) P µν = − E µ ; νg + E µg E νg + 8 π Σ i (cid:18) p i + ρ i − v i v iµ v iν + ρ i − p i γ µν (cid:19) . (25)Again looking at Eq. (24), it seems that we could have a one-component fluid solution eitherwith any EOS in a comoving frame, or if we are looking for a solution in a non-comovingframe, then the only choice would be a dark fluid, namely p = − ρ , but now, unlike theprevious case in the last section, such a choice is not forbidden by the other two equations.Indeed this case has been thoroughly discussed in [3], where it is shown that it leads to aunique characterization of de Sitter and de Sitter-type spacetimes as the only one-component static perfect fluid solutions of Einstein field equations in a non-comoving frame . The wellknown de sitter spacetime ds = (1 − Λ r c dt − (1 − Λ r − dr − r ( dθ + sin θdφ ) , (26)is the spherically symmetric member of this family, and indeed their representative, whichcould be easily shown to satisfy Eqs. (23) and (25). The axially and cylindrically symmetricmembers of the same family are given by [4, 15, 16] ds = (1 − Λ z ) c dt − (1 − Λ z ) − dz − Λ4 ρ ) ( dρ + ρ dφ ) , (27)and ds = cos / (cid:18) √ ρ (cid:19) ( dt − dz ) − dρ −
43Λ sin (cid:18) √ ρ (cid:19) cos − / (cid:18) √ ρ (cid:19) dφ , (28)respectively. Finally it should be noted that the same approach could also be applied to darkfluids with ρ Λ <
0, leading to the anti-de Sitter spacetime and its axially and cylindricallysymmetric counterparts [17, 18].
C. Spacetimes without a gravitoelectric field E g : The G¨odel Universe Spacetimes with a gravitomagnetic field are stationary spacetimes and the absence of thegravitoelectric field requires a constant time-time component of the metric, i.e h ≡ a =9 onstant . Looking for cylindrically symmetric solutions [28], these observations reduce thegeneral form of the metric (in a cylindrically symmetric coordinate system) into [1], ds = a [ dt + A ( r ) dφ ] − dρ − e K ( r ) dz − G ( r ) dφ , , (29)which has a gravitomagnetic field along the z-axis . Starting from Eqs. (13)-(15) and setting E g = 0, we end up with the following equations;12 a B g = 8 π Σ i (cid:18) p i + ρ i − v i − ρ i − p i (cid:19) (30) a ∇ × ( B g ) = − π Σ i (cid:18) p i + ρ i − v i (cid:19) v i (31) (3) P µν = 12 a ( B µg B νg − B g γ µν ) + 8 π Σ i (cid:18) p i + ρ i − v i v iµ v iν + ρ i − p i γ µν (cid:19) . (32)Lets try a single perfect fluid source where, with any linear barotropic EOS with constantpressure (density), and not of a dark-type ( p = − ρ = constant ), then Eqs. (30) and (31)are simultaneously satisfied, only in a comoving frame ( v = 0), leading to a uniform andcurl-free gravitomagnetic field. This includes for example stiff matter ( p = ρ = constant ),which when plugged into (31) leads to the following equations B g = 32 πa ρ SM (33) ∇ × ( B g ) = 0 (34) (3) P µν = 12 a ( B µg B νg − B g γ µν ) , (35)in which, as mentioned, the first two equations refer to a uniform gravitomagnetic field.Indeed this form of a source matter, satisfying the last equation (35), will result in a solutionwhich is the famous G¨odel universe [20] in which the source of the spacetime is stiff matterin a comoving frame.The one-component perfect fluid of the dark-type with EOS p = − ρ = constant , althoughsatisfying Eqs. (30) and (31) for ρ Λ < < (3) P µν = a B µg B νg + 16 πρ Λ γ µν [29].If on the other hand we insist on having a dark fluid component, as we have done so far, then10e should look for a solution of the above equations with two perfect fluid sources namely,12 a B g = 8 π (cid:18) ( p + ρ − v − ρ − p p + ρ − v − ρ − p (cid:19) (36) a ∇ × ( B g ) = − π (cid:18) p + ρ − v v + p + ρ − v v (cid:19) (37) (3) P µν = 12 a ( B µg B νg − B g γ µν )+8 π (cid:18) ( p + ρ − v v µ v ν + ρ − p γ µν ) + ( p + ρ − v v µ v ν + ρ − p γ µν ) (cid:19) . (38)To have a curl-free gravitomagnetic field, Eq. (37) invite us to choose, as in the case of thestatic universes discussed in section ( ), a dust component [30] in the comoving frame, plusa dark component ( p = − ρ ). These two sources substituted in the above equations lead to, B g = 16 πa ( ρ dust − ρ Λ ) (39) ∇ × ( B g ) = 0 (40) (3) P µν = 12 a ( B µg B νg − B g γ µν ) + 8 πγ µν ( ρ dust ρ Λ ) . (41)Now if we choose the relation, ρ Λ = − ρ dust − ρ SM < , (42)the above set of equations will be equivalent to the Eqs. (33)-(35), and consequently leadsto the same solution which is the G¨odel universe, given in the Cartesian coordinates as, ds = a ( dt − e x dy ) − a dx − a e x dy − a dz , (43)where a = − . This is the form of the metric which was originally introduced by G¨odelhimself. The above form written already in the 1 + 3 form, clearly indicates a uniformgravitomagnetic field B g = √ a ˆ z . In terms of the cosmological constant it could be writtenas follows ds = ( dT − e √ | Λ | X dY ) − dX − e √ | Λ | X dY − dZ , (44)showing clearly the flat space limit | Λ | →
0. The metric (43) could also be written in thecylindrical coordinates of the form (29), as follows, ds = [ dt − √ a sinh ( r a ) dφ ] − dr − dz − a sinh ( ra ) dφ . (45)11his form clearly indicates the regular flat space behavior near the axis ( r → negative density dark fluid component ( negative cosmologicalconstant), where the corresponding densities satisfy the first equation in (42), or equiva-lently Λ = − πρ dust . In other words in this second choice for the source of the spacetime,the requirement of having a dark fluid component, automatically results in a negative cos-mological constant. We also note that in this sense it is just the opposite of what we had inthe case of Einstein static universe. D. Stationary spacetimes with non-vanishing E g and B g : Stationary analogs of deSitter and anti-de Sitter spacetimes Obviously keeping both fields E g and B g will leave us with more degrees of freedom, andspecially one could look for stationary cylindrically symmetric Einstein spaces . These areequivalent to the stationary, cylindrically symmetric solutions of the Eqs. (13)-(15) with asingle dark fluid source ( p Λ = − ρ Λ = constant ), which take the following forms, ∇ · E g = 12 hB g + E g + 8 πρ (46) ∇ × ( √ h B g ) = 2 E g × ( √ h B g ) (47) (3) P µν = − E µ ; νg + 12 h ( B µg B νg − B g γ µν ) + E µg E νg + 8 πργ µν . (48)The solutions of the above equations for both positive and negative densities (cosmologicalconstant) have already been discussed extensively in the literature [21–23]. As expected,exact solutions of the above equations having the general form ds = F ( ρ )[ dt + A ( r ) dφ ] − dr − e K ( r ) dz − G ( r ) dφ , (49)contain a large family, so here, as in the previous sections, we only consider a special set ofsolutions as the family’s representative. These solutions, which are the stationary analogsof de Sitter and anti-de Sitter spacetimes, introduced very recently [6], and are given by, ds = cos − / ( ar ) cos(2 ar ) (cid:18) dt − sin ( ar ) a cos(2 ar ) dφ (cid:19) − dr − cos / ( ar ) dz − a sin ( ar ) cos / ( ar )cos(2 ar ) dφ ; Λ > < r < π √ , and ds = cosh − / ( ar ) (cid:18) dt − sinh ( ar ) a dφ (cid:19) − dr − cosh / ( ar ) dz − a sinh ( ar ) (cid:18) cosh − / ( ar ) + sinh ( ar ) (cid:19) dφ ; Λ < < r < ∞ , and in both solutions a = √ | Λ | . These solutions were obtained bychoosing special values for all the constant parameters in the general form of such metricsdiscussed in [23]. The only criterion employed in [6] to fix all the constant parameters, wasthat the solution reduces smoothly to the flat spacetime in the limit Λ →
0. It is interestingthat both spacetimes are regularly flat on the axis r = 0. The positive Λ solution, (50), hasthe following non-zero components of gravitoelectromagnetic fields, E rg = √ ( √ r ) (cid:0) ( √ r ) (cid:1) (52) B zg = 2 √
3Λ cos − / ( √ r ) cos − / ( √ r ) (53)So a test particle released near the axis will start rotating around the axis with ever increasingradius [24]. In other words these metrics, through Λ, as their only parameter, produce anisotropic fields which are not consistent with the idea of Λ as a dynamic-free geometricconstant. This indicates that, as in the case of their static counterparts, one should treat thecosmological term as a dark fluid and assign the source of the spacetime anisotropy to thefluid’s velocity [3]. It should be noticed that although setting v = 0 in the above equations(i.e going to the comoving frame), gives formally the same result as setting p = ρ , the factthat these solutions are in non-comoving frame is evident from their non-zero Christoffelsymbol Γ ρtt .One also notices the similarity of the gravitomagnetic potential of the above negative Λsolution to that of the G¨odel universe which leads to similar motion for test particles neartheir symmetry axis. But unlike the case of the G¨odel universe which apart from Λ hasa dust component, to which its anisotropic behavior could be assigned, here it is only bytreating Λ as a dark fluid with a preferred direction, that one could interpret its anisotropicfeature. 13 V. SUMMARY
We have shown that the well known static and stationary single and two componentperfect fluid solutions of Einstein field equations, which all include at least a dark componentwith EOS p = − ρ ( acting as a cosmological constant), could be categorized in terms of theirgravitoelectromagnetic fields. All these solution share the same flat space limit as | Λ | → E g = 0 E g = 0 B g = 0 Einstein Static universeΛ = 4 πρ dust > < > B g = 0 G¨odel universeΛ = − πρ dust < < > TABLE I: All the above solutions reduce to the Minkowski space as | Λ | → cknowledgments M. N-Z thanks University of Tehran for supporting this project under the grants providedby the research council. [1] H. Stephani et. al.
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223 (1975).[22] N. O. Santos, Class. Quantum Grav. , 1627 (1998).[24] M. Nouri-Zonoz, B. Nazari and A. Nouri-Zonoz, Geodesics of the stationary analogs of deSitter and anti-de Sitter spacetimes, in preparation.[25] For a detailed discussion on the definition of 3-velocity refer to [9].[26] We note that the differential operations in these relations are defined in the 3-space Σ with metric γ µν . Specifically, divergence and curl of a vector are defined as div V = √ γ ∂∂x i ( √ γ V i ) and (curl V ) i = √ γ ǫ ijk ( ∂V k ∂x j − ∂V j ∂x k ) , respectively with γ = det γ ij .[27] We notice that this case could be treated in the present formalism, if we consider massiverelativistic particles as incoherent radiation.[28] For a recent review on cylindrical gravitational fields refer to [19].[29] Indeed one could show that equations for (3) P ρρ and (3) P zz lead to ρ Λ = 0.[30] we note that unlike the case of static universes, here we are not allowed to choose incoherentradiation, as it will not be consistent with the cylindrical symmetry.= 0.[30] we note that unlike the case of static universes, here we are not allowed to choose incoherentradiation, as it will not be consistent with the cylindrical symmetry.