Axially Symmetric Shear-free Fluids

Transcription

1 arxiv: v1 [gr-qc] 14 Jun 16 Axially Symmetric Shear-free Fluids in fr,t) Gravity Ifra Noureen a), M. Zubair b) a) Department of Mathematics, University of Management and Technology, Lahore, Pakistan. b) Department of Mathematics, OMSATS Institute of Information Technology, Lahore, Pakistan. Abstract In this work we have discussed the implications of shear-free condition on axially symmetric anisotropic gravitating objects in fr, T) theory. Restricted axial symmetry ignoring rotation and reflection enteries is taken into account for establishment of instability range. Implementation of linear perturbation on constitutive modified dynamical equations yield evolution equation. This equation associates adiabatic index Γ with material and dark source components defining stable and unstable regions in Newtonian N) and post-newtonian pn) approximations. Keywords: fr, T) gravity; Axial symmetry; Instability range; Shearfree ondition; Adiabatic index. 1 Introduction In a recent work [1], we have discussed the instability range of axial system in fr,t) gravity, R being Ricci scalar and T denoting trace of energy momentum. In continuation to [1], we plan to study the evolution of axially ifra.noureen@gmail.com; ifra.noureen@umt.edu.pk mzubairkk@gmail.com; drmzubair@ciitlahore.edu.pk 1

2 symmetric sources evolving under shear-free condition. Shear tensor has significant importance in the of structure formation, inhomogeneity factors and relativistic stellar phenomenons. Many authors []-[8] have discussed the impact of shear tensor in gravitational evolution and consequences of its vanishing value. It has been established recently [9] that the vanishing shear case has equivalence with the homology conditions and so has enormous significance in astrophysics [1, 11]. Possible aberrancies from most commonly studied spherical symmetry extended the concerns with non-spherical geometries of gravitating systems. Recent developments suggest that incidental variations in spherical symmetry prevails the more realistic scenarios. In this work, we have considered restricted axially symmetric system ignoring reflection and rotation about symmetry axis) with anisotropic matter configuration evolving under shearfree condition. This study is conducted in fr,t) gravity by taking into account a consistent fr,t) model satisfying df, d f ). dr dr Study of gravitational collapse and its end-state has gained great deal of attention in past years. Dominance of inward acting gravity force over outward acting pressure leads to the gravitational collapse due to which star contracts to a point. A gravitating system remains stable until pressure keeps on balancing the gravitational pull and collapses when pressure to gravity balance fluctuates. The range of stability/instability primarily depends on mass, supersessive stars lose stability more rapidly. Besides mass there are various factors that can modify instability range considerably such as shear, isotropy, anisotropy, radiation etc. Instability problem received enormous attention after the seminal contribution of handerashekar [1], he set stability criterion of gravitating sources in the form of adiabatic index Γ associating variation in pressure with varying energy density. The criterion for stability of gravitating systems with anisotropic matter distribution is established in [13]. Herrera and his collaborators [14]-[18] contributed greatly to study the dynamics of relativistic fluids to discuss the factors contributing in gravitational evolution. They also presented the general framework to deal with axially symmetric sources and their applications [19]-[1]. Gravitational interaction can be described in different manners, so far the scheme of general theory of relativity GR) being a self-consistent approach has been largely used. It explains gravitational phenomenons adequately at cosmological scales []. However, developments in observational situations

3 [3]-[7] suggests that GR cannot bethe only theory that can beappropriate for all scales. Many treatments have been made for suitable description of gravitational theories on large scales and deal with the issue of cosmic speed-up[8]-[37], by developing alternative gravitational theories [38]-[46] for e.g. fr), fr, T), fg), ft,t), Brans-Dicke theory and so on. Such alternative theories are dynamically equivalent to GR at cosmological levels. Herein, we workout the gravitational evolution in fr,t) theory introduced by Harko [47]. In this theory the matter content is considered to have interaction with the geometry. Many authors [48]-[5] have discussed the cosmological and thermodynamic implications in fr,t) gravity. The purpose of this work is to explore the implication of shear-free condition on dynamical instability of a restricted class of axially symmetric anisotropic systems in fr,t) gravity. The gravitational action in fr,t) includes an arbitrary function of R and T, written as [47] dx 4 g[ fr,t) 16πG +L m)], 1) where L m) is matter Lagrangian and g is the metric. A variety of choices of L m) can be dealt with, each of which represent a specific configuration of relativistic matter. The article is arranged as: Section includes the discussion of considered spacetime, matter configuration, kinematic variables along with shear-free condition and modified dynamical equations. In section 3, we introduced the fr, T) model and applied linear perturbation on conservation equations. Section 4 contains the discussion of weak field limit. Summary of the results is given in last section that is followed by an appendix. Modified Dynamical Equations With Vanishing Shear In this section, we develop the modified field field equations in framework of fr, T) gravity by considering axial system with anisotropic matter evolving under shear-free condition. 3

4 .1 Metric and Matter onfiguration The dynamical systems lacking spherical symmetry have received great attention in recent past, since such abberations describes the more realistic phases of gravitational evolution. The general line element for axially symmetric gravitating sources constitutes five metric functions independent), given by ds = A t,r,)dt +B t,r,)dr +r B t,r,)d + t,r,)dφ +Gt, r, )dtd + Ht, r, )dtdφ, ) Herein, our goal is to study the implications of vanishing shear scalar on collapsing phenomenon. We have restricted the character of spacetime by imposing the highest degree of symmetry to avoid the complexities generated from the terms of reflection and rotation about symmetry axis. Thus, we have restricted the general line element ) to somehow handle the problem under consideration and manipulate the results analytically. onsidering vorticityfree case i.e. ignoring dtd and dtdφ terms in ), the reduced form of line element containing three independent metric functions takes following form ds = A t,r,)dt +B t,r,)dr +r d )+ t,r,)dφ. 3) The matter distribution is taken to be anisotropic carrying unequal pressure stresses. The energy momentum tensor for usual matter is defined as [19] T m) uv = ρ+p )V u V v K u K v 1 3 h uv)p zz P xx ) L u L v 1 3 h uv)p zz P yy )+Pg uv +K u L v) P xy, 4) where P = 1 3 P xx +P yy +P zz ), h uv = g uv +V u V v, wherep xx,p yy,p zz andp xy denotecorrespondinganisotropicpressurestresses, satisfying P xy = P yx and P xx P yy P zz. The four vectors in radial and axial directions are described by K u and L u respectively, while V u stands for four-velocity and ρ is energy density. The above mentioned quantities are associated as V u = Aδ u, K u = Bδ 1 u, L u = rbδ u. 5) 4

5 . Kinematic Variables and Shear-free ondition Kinematic variables play significantly important role in the description of self-gravitating sources. Herein three kinematic variables are contributing in the systems evolution, i.e., acceleration a u, expansion scalar Θ and the shear tensor σ uv. These variables are obtained as follows The non-zero components of σ uv are a u = V v V u;v =, A A, A,), 6) A ) Θ = V;u u = 1 Ḃ A B + Ċ, 7) σ uv = V u;v) a u V v) 1 3 Θh uv). 8) σ 11 = B 3A σ = r B 3A σ 33 = 3A ) Ḃ B Ċ, 9) ) Ḃ B Ċ, 1) ) Ḃ B Ċ. 11) The expression for shear scalar is obtained as [] ) Ḃ σ = 1 A B Ċ. 1) Here dot and prime indicate derivatives w.r.t time and radial coordinates respectively. The shearing effects and inhomogeneity factors within the collapsing system play an essential role in the formation of the trapped surfaces and the apparent horizon. It is remarked in [6] that presence of shear causes distortion of trapped surface geometry which strongly implicates the development of naked singularities as end state of gravitational evolution. It is worth mentioning here that absence of shearing effects in collapsing cloud lead to the formation of a black hole. Thus, absence or presence of shear 5

6 have significant impression on kinematics of the gravitating source and its evolution. The gravitating source under consideration is assumed to have vanishing shear, i.e., σ =, that leads to Ḃ = Ċ. We shall employe shear-free condition B to study the dynamics of axially symmetric gravitating source..3 Modified Field Equations As mentioned earlier in introduction that different choices for matter Lagrangian can be taken to study the dynamics of a gravitating system, each choice represents aparticular set of fieldequations. Here, we takel m) = ρ, 8πG = 1 and vary the extended gravitational action 1) w.r.t metric g uv as follows G uv = 1 [ f T +1)T uv m) +ρg uv f T + f Rf R g uv f R + u v g uv )f R ], 13) where = u v, f R dfr,t)/dr, f T dfr,t)/dt, u stands for covariant derivative and T uv m) denote the usual matter stress energy tensor. orresponding components of effective Einstein tensor on account of shearfree condition are [ G f Rf R = 3 ) f R Ḃ A B f R 1 B r + B B G 1 = G = 1 ρ+ 1 A f R A f R f R B r B B )+ f R B ], 14) [ ] 1 A f A B f R A R + Ḃ B f R f R, 15) [ ] 1 A f r A B f R A R + Ḃ B f R f R, 16) 6

7 G 11 = G 1 = G = G 33 = [ 1 f P B xx f T +1)+ρf T R Ȧ f R A A f Rf R f R r B f R A ) )] + f R 1 B r A A + B B + f R 3B r B B A A, 17) [ ] 1 P r B 4 xy f T +1)+f R B f R B f R B B f R, 18) [ 1 P r B 4 yy f T +1)+ρf T + f R Ḃ f R A B A ) f + R A A f Rf R ) )] f R B f R A r B A B B + f R A B A B B +, 19) [ 1 f P zz f T +1)+ R f R A f R r B +ρf T f Rf R f R Ȧ A A B ) B f R B f R A B A B B 1 ) )] f R A r r B A B. ) B The equation for Ricci scalar is found as ) R = 3Ȧ Ḃ Ḃ A A B 3 B B B + [ A B A + A A + B B 1 A ) r A B B B B )] A r A + B B + B B ) + A. 1) A.4 onservation Equations In fr,t) gravity, the divergence of energy momentum tensor leads to the following expression [T uv +Θ uv ] u lnf T R,T)+ u Θ uv 1 ) g µν µ T. u T uv = f TR,T) κ f T R,T) ) It is clear from above equation that unlike GR the stress energy tensor remains non-conserved in fr,t) gravity. This is because of the direct matter 7

8 geometry coupling, non-zero divergence induces a force acting orthogonally to four velocity leading to the non-geodesic path of test particles in nonminimally coupled gravitational theories. Dynamics of a system can be explored by considering conservation laws, divergence free gravitational theories can be dealt by taking conservation of energy momentum tensor. However, in our case we can not do so since T uv is non-conserved that is why we take conservation of full form of field tensor i.e., effective Einstein tensor. Bianchi identities are used here to arrive at conservation equations that are vital in establishment of evolution equation analytically. Following equations are obtained from Bianchi identities G, +G 1,1 +G, +G A A + 3Ḃ B ) +G 1 3A A + B B ) r ) 3A +G A + B B + +G 11BḂ BḂ A +Gr +G 33Ċ =, 3) A A ) Ȧ G 1, +G11,1 +G1, +GAA B +G1 A + 5Ḃ A +G 11 B A + 3B B A )+G 1 ) G r r A + 4B B + + r B )+G 33 =,4) B B ) Ȧ G, +G1,1 +G, +GAA r B +G A + 5Ḃ B A B r B G11 + A + 4B B A )G 1 +G r A + 3B B + ) G 33 =. 5) r B The notion,1 and stands for t,r and. One can separate usual and dark source ingredients by substituting components of Einstein tensor from Eqs.14)-) in Eqs.3)-5). 3 Perturbation Scheme Theoretically and experimentally consistent fr, T) models can be constructed, such models shall be compatible with local gravity tests and cosmological constraints. These constraints are required to meet for consistent solar system tests, matter domination phase and stable high-curvature configuration. Generally, a consistent gravitational model corresponds to the choice of parameters that are in accordance with the observational scenarios [51]. A 8

9 viable fr,t) ghost-free model shall have positive first and second order derivatives for stability of cosmological perturbations [5]. In addition to this, any modified gravity model shall agree with weak field limit and must be stable at semiclassical and classical levels. The viable models in fr,t) gravity have classification based on fundamental form of f, as follows The model fr, T) = R + ft) describes the cosmological model accompanying time dependent and effective coupling, the term ft) characterizes the interaction of matter and curvature interaction in the extended gravitational action. fr,t) = f 1 + f, f 1 and f are arbitrary functions of R and T, respectively appearing explicitly. fr,t) = f 1 R)+f R)f 3 T) denotes themost general class of fr,t) models, carrying implicit form of R and T in extended gravitational action. Herein, we are dealing with axial system analytically, in order to arrive at some fruitful conclusion avoiding complexities we select fr,t) = f 1 R) + f T) type. More particularly, we take fr,t) = R + αr + λt, where α and λ denote positive constants. The origin of such constraints on model comes from the argument that non-linear terms of T in the model complicates the field equations that can not be tackled by analytic approach. ThefR,T)modelscontainingnon-linearenteries oft canbehandledbynumerical simulations but such analysis provide outcomes for a specific model. Whereas, more generic results can be found for a variety of models with arbitrary values of α and λ by using analytic approach.. Following scheme for perturbation of physical variables is employed to monitor the fluctuations in the gravitating object with the time transition. Metric coefficients and Ricci scalar are perturbed by using following pattern Lt,r,) = L r,)+ǫdt)lr,). 6) While energy density and pressures stresses are perturbed as follows ρt,r,) = ρ r,)+ǫ ρt,r,), 7) P ij t,r,) = P ij r,)+ǫ P ij t,r,). 8) 9

10 Variation in fr,t) model is taken in the following form fr,t) = [R r,)+αr r,)+λt r,)]+ǫdt)er,)[1 + αr r,)], 9) f R = 1+αR r,)+ǫαdt)er,), 3) f T = λ, 31) where < ǫ 1. Application of the first order perturbation on dynamical equations 3)-5) implies [ { a ρ+ ρ + 3λ ) 1b + λ } ] 1b P xx +P yy +P zz )+Z 1p Ḋ =, A B B [ λ 1 P b xx +λ ρ λ 1 P xx +λρ ) B ) ] D + eα + λ 1 P xx +λ ρ ) A + 3B B I,1 A B )+ 1r [ λ r 1Pxy b + eα ) ] P xy D + λ ) 1P xy A +4 B + + B I, r B A B λ1pyy +λ ρ ) ) 1 r + B + λ 1Pzz +λ ρ ) [ ) a +D λ 1 P xx +λρ ) + B A ) b b +4 + eα ) A + 3B )) ) b +λ 1 P yy +λρ ) B B I A B r B b + eα ) ) ) ) B 1 B I B r + B c b +λ 1 P zz +λρ ) + eα ) ) B B I ) ) a b b +λ 1 P xy +5 + eα ) )] +Z p =, 33) A B I 3) 1

11 [ λ 1Pyy b +λ ρ λ 1 P yy +λρ ) + eα B I B ) ] [ ] 1 D +, r BI λ 4 1P xy + ρ A A ) + λ 1Pyy +λ ρ ) A + λ 1 P xx +λ ρ ) B +λ 1 A Pxy + 4B B A B r A ) +3 B + + λ 1Pzz +λ ρ ) [ ) a b +D ρ + eα ) ) A B A B I A ) ) a b 4b +λ 1 P xy eα ) A + 4B + A B B I A B + 3 )) ) b b +λ 1 P xx +λρ ) 4 + eα ) ) B +λ 1 P yy +λρ ) r B B I B ) ) a b b +4 + eα ) ) ) A +3 B + A B B I A B ) b b +λ 1 P zz +λρ ) + eα ) )] +Z 3p =, 34) B I where expressions for Z 1p,Z p and Z 3p given in appendix. We take, I = 1 + αr and J = eαr for the sake of simplicity. The value of energy density ρ is derived from Eq.3) which turns out to be ρ = { ρ a A + 3λ 1b B ) + λ 1b B P xx +P yy +P zz )+Z 1p The expression for ρ and pressure stresses can be related as [1] } D. 35) P i = Γ p i ρ +p i ρ. 36) where Γ represents the variation of anisotropic pressure stresses with the varying energy density and index i = xx,yy,xy,zz. Making use of Eq.36) in Eq.35) and some algebraic manipulations provide linearly perturbed anisotropic stresses. An ordinary differential equation is obtained by using linearly perturbed form of Ricci scalar having solution as follows Dt) = e Z 4 t. 37) Theexpression forz 4 isgiven inappendix, Eq.37)isholdsforpositive values of Z 4. 11

12 4 Weak Field Limit In this section, we found the entries belonging to Newtonian N) and post Newtonian pn) eras. Equations 37) and 36) are inserted in Eq. 33) to arrive at evolution equation from which instability criterion is developed in terms of adiabatic index. 4.1 N Approximation In this regime, we take A = 1, B = 1, ρ p i and Schwarzschild coordinates = r in evolution equation, whose outcome reveals the following relation Γ < λn 3 N r λρ N ) P r xyn ) +λn N 3 N r +λp xy N 4 +Z N p λ 1 P xx N 1 ) + λ 1P r xy N 1 ) 1 N, r 1P xx +P yy +P zz ) 38) where Z N p represent those terms of Z p that belongs to the N-limit. Furthermore { N = ρ N 1 + λ } 1c r P xx+p yy +P zz )+Z 1 N p, N 1 = a+ 3λ 1c r, N = c r + αe I N 3 = a +4b + c r ), N 4 = a +4b + c r. Both usual matter and dark source entries are taking part in above inequality for Γ, gravitational sources maintain stability for those values of physical parameters for which the inequality 38) is satisfied. All entries appearing in 38) shall remain positive, to fulfil this requirement we need to constrain some of the physical parameters. The restrictions in N-limit are P xx +P yy +P zz < λ 1r N 1 P xx N 1 ) + 1 r, P xyn ) < λρ N ). Stability of the gravitating system distorts whenever above constraints are violated that leads to the instability range of collapsing stars. 1

13 4. pn Approximation In pn limit, we take A = 1 m r and B = 1+ m, use of these assumptions r in evolution equation leads to following inequality for Γ Γ < λn 1 +N 9 N 1 λρ N 6 ) P r xyn 6 ) +λρ N 7 3 N r 6 +λp xy N 8 +Z pn p λ 1 P xx N 5 ) + λ 1P r xy N 5 ) 1N, r 5P xx +P yy +P zz )+N 11 39) where ar N 5 = + λ 1br + λ ) 1c, N 6 = λ 1br + eα r m r +m r r +m I, ) ) ) ar br c N 7 = +4 + N 6 r m r+m r r + m ) ) 3r, r r +m ) ) ar br c ), 3 N 8 = + + N9 = r m r +m r r + m ) 3r r r +m { N 1 = ρ N 5 + λ 1br P xx +P yy )+ λ } 1c r +m r P zz+z 1 pn p, N 11 = P ) ) ) xyn 5 ar 4br +. r +m ) r m r +m Likewise N-limit the restrictions in metric functions and dark source entries are required to achieve stability of gravitating sources. System is exposed to gravitational collapse whenever the inequality 39) breaks down. Some of the constraints are [ λ1 br P xy + eα ] <, r +m I { ρ N 5 + λ 1br P xx +P yy )+ λ } 1c r +m r P zz +Z 1 pn p <. 5 Summary Study of non-spherical symmetries provide deep insight of more realistic settings such as weak lensing, large scale structures, Planck data, cosmic microwave background etc. In this work, we have studied the impact of restricted axial symmetry ignoring reflection and rotation) on anisotropic 13 ),

14 fr, T gravity model evolving under shear-free condition. Restricted character of spacetime incorporates three independent metric functions depending on t, r and, that leads to the vorticity-free case. We choose, fr,t) = R+αR +λt with positive values of α and λ as a viable fr,t) model to discuss the instability range. The variation of extended gravitational action 1) is considered to obtain the modified field equations in fr,t) gravity. We make use of shear-free condition to evaluate components of modified field equations. The energy momentum tensor remains non-conserved in fr,t) gravity that is why we take conservation of modified Einstein tensor. The conservation equations lead to highly non-linear complicated equations. To tackle with the complexities of dynamical equations, we implement linear perturbation on all physical parameters. Application of linear perturbation yield equations for perturbed energy density that is further used to evaluate perturbed anisotropic stresses. Insertion of perturbed stresses and energy density in second Bianchi identity leads to the collapse equation carrying adiabatic index that elaborates the variation of anisotropic pressures with energy density. It is found that imposed shear-free condition reduces the entries with negative sign in collapse equation that leads to less restricted and enhanced regions of stability. Thus it can be remarked that vanishing shear somehow delays the collapsing phenomenon. Moreover, weak field limit is checked by evaluating expression for adiabatic index in Newtonian and post Newtonian approximations. orrections to GR can be recovered by taking α,λ in evolution equation. While λ reduces results in fr) gravity for Starobinsky model. 14

15 Appendix { 1 Z 1p = e A A B I αer ) 1 b ) αer ) A B αer ) 1 b )} A ) αer B A A,1 r { A B I )} + α R [ e αer )) 4α A, I B r R R ) + R ) R ) a + b + αer { ) b )+I r A B I B b 3A + B 4 ) c A 1 ) + αer ) + B B A B r A r I B { 3 ) )}+e αer )) + I b b ) 3A + B r r B B A B + αer )} ) + B +αer ) B + 1 ) I B B r + αer ) ) B a + + b ) )I + I 3A + B + 1 r B A B r A B r ) + αer ) 1 b ) ) A αer ) A + αer ) αer ) 1 I B A I A I I b ) )] 3A ) + + B, 4) B A B 15

16 Z p = [[ 1 IB + J b I B 3B + B [ { 1 + r IB 4 { D 1 DA B ) A ) A + B + b B I +I { eb a +I A A ) )+ 1r J + B J b 1 B r ) ) ))}}] a b,1 ) A I B ) b B J + I +J + 1 ) ) }] b I IB 4 B B r B, e B + A [J + J B A r b ) I + I + J b ) I B B r B B { + 1 ) 1r ) b I J r B b ) )}] I B aa ) + b ) A αr B +I +I B B A B A B + 1 )) { )+ I B r r + I B αr B A +I + B r B A B 1 )} ) )+ I r r + I A ) ) + 3B a b +4 r A B A B A + + 3B + 1 ){ ) a A I + B A B r A A B + 1 ) J b I )+ 1r J + J b ) ) A I 3B + r B B A B + b ) ) ))} [ ) a b a I +I B A B A )) b ) ) I + B B I +I B ) A + 4B + B J B B r A B B ) b B J I J + 1 ))] )+I 1 B B B r bb r + 1 ) [ B B r A J + b ) a I I J b ) A I + B A B A 16 A D D J

17 B )+ 1r ) a I J b B A ))] b ) A I B B A B ) [ ) + LB + A +I + B ) I A B + B A B r A B [ I ]+ J b ) ) ) a b I +I + J b ) A I B B A B B A B { + 1 )+ 1r J + J b ) ) A I ) ) ) B a b +I r B A B A B b }] I c ) + b )[ A I B + 1 ) ) I A B B B A B r r A B ] + αr B +I + I DB J A J b ) I, 41) r DA I A B [ { ) Z 3p = Ir B 4 1 b B J + I +J + 1 ) ) }] b I r IB 4 B B r B,1 + DB A J + b ) [ { I J 1 DB + J J + b b I + I DA I A B Ir B 4 DA B B ) ) A J ) + B a I + 1 b I A B A r B J ) ) )}] A B + ) I A B aa, ) [J + I J + I e B + A B A r { + 1 ) 1r ) b I J r B b ) )}] I B aa ) + B B A r b B + J b B I ) b ) A αr B B A B B +I +I 17

18 B + 1 )) ) )+ I B r r + I { B b αr B A +I + r B B A B + 1 )} ) )+ I B r r + I A + 3B B {I r A B B aa A + B + 1 ) J b I )+ 1r b J I A B r B B J ) ) A 3B + + b ) ) ))} a b I +I A B B A B 1 [ ) ) ) a b + I + B B I +I + 1 )) A r A B B B r A ) ) + 4B + B b B J J I J + 1 ))] )+I B B B B r bb ) [ A + + 3B + B D A B A D J + b ) a I I B A J J b ) A I + B ) 1r J b ) A I B B A B B A B ) )] ) + a ) I + aa b +4 ) A ) ) [αr ] B A +I + B ) I A B + ) I B A B r A B [ J b ) ) ) a b I +I + J b ) A I B B A B B A B { + 1 J b ) ) A I ) ) ) B a b +I r B A A + 1 r +J b }] I B +I + I r I r r B c ) + b B A B A B )[ αr B B +I A A B B + 1 r )], 4) ) 18

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