Shear and expansion evolution for dissipative fluids 1

Modern Physics Letters A Vol. 33, No. 20 (2018) 1850111 (18 pages) c World Scientific Publishing Company DOI: 10.1142/S0217732318501110

Shear and expansion evolution for dissipative fluids

S. Ahmad∗,‡ , A. Rehman Jami† and Z. Aas∗,§ ∗Department

of Mathematics, University of Management and Technology, Sialkot 51310, Pakistan †Department of Mathematics, University of Sharjah, United Arab Emirates ‡[email protected] †[email protected]; [email protected] §[email protected]

Received 30 March 2018 Revised 7 May 2018 Accepted 22 May 2018 Published 21 June 2018 The aim of this work is to analyze the role of shear evolution equation in the modeling of relativistic spheres in f (R) gravity. We assume that non-static diagonally symmetric geometry is coupled with dissipative anisotropic viscous fluid distributions in the presence of f (R) dark source terms. A specific distribution of f (R) cosmic model has been assumed and the spherical mass function through generic formula introduced by Misner– Sharp has been formulated. Some very important relations regarding Weyl scalar, matter variables and mass functions are being computed. After decomposing orthogonally the Riemann tensor, some scalar variables in the presence of f (R) extra degrees of freedom are calculated. The effects of the three parametric modified structure scalars in the modeling of Weyl, shear, expansion scalar differential equations are investigated. The energy density irregularity factor has been calculated for both anisotropic radiating viscous with varying Ricci scalar and dust cloud with present Ricci scalar corrections. Keywords: Relativistic fluids; anisotropic fluids; dissipative fluids. PACS Nos.: 04.40.Dg, 04.40.Nr

1. Introduction In 1915, Professor Albert Einstein proposed a totally new model for the description of gravitation dynamics, widely known as general relativity (GR). The purpose of this model was to understand the dynamics induced by the accelerated frame of references, for which he calculated field equations to introduce a peculiar association between relativistic matter and its geometry. This gravitational model has, ‡ Corresponding

author 1850111-1

S. Ahmad, A. R. Jami & Z. Aas

undoubtedly, gained a lot of successes in mathematical physics and relativistic astrophysics. Recent observations about the evolution of our cosmos states that our universe is in the epoch of accelerating expansion. Qadir et al.1 studied the effects of dark source terms on the discussion of black hole dynamics, evolution of the universe, etc. and proposed that GR may need to be modified to resolve quantum gravity and dark matter problems. The modified gravity theories (MGT) can be introduced by modifying the geometric portion of the Einstein–Hilbert (EH) action (for further reviews on DE and modified gravity, see, for instance, Refs. 2–14). Very interesting and captivating MGT models have been introduced for the exploration of accelerating expanding cosmos.15 The study of structure formation of the universe has been performed in f (R),16 f (R, T )17 (T is the trace of energy– momentum tensor) and f (R, T, Rμν T μν ) gravity.18 Harko et al.19 proposed a new MGT known as f (R, T ) theory. They proposed that creation of large-scale structures in the universe, field of scalars and anisotropic models play an important role. Felice and Suyama20 considered a vacuum case in which they studied the cosmological deviation for f (R, G) function. They demonstrated their results under the stable modes of scalar field. Fabris et al.21 presented some cosmic outcomes from the Starobinsky model of expansion and found some outcomes that match with that of experimental data. Glazer22 studied the distribution frequencies of electrohydrodynamics by generating the theory of relativistic oscillating equation. He also applied their results to the case of charged dust. Zhang et al.23 solved the set of equations for the non-ideal state of matter. They also described that the concept of critical mass does not change by the presence of total charge. Zhao24 explained that supernova are formed by the implosion of the dust clouds. He described the four cases for the formation of black holes (BH), in which three cases are discussed by invoking the effects of DE. Abreu et al.25 performed the cracking method to identify potentially changeable anisotropic material arrangements in the evolutionary phases of self-gravitating system. They also discussed its relation with the speed of sound. Burko26 demonstrated the dependents of multipole expansion on the coordinate system. So, he adapted a coordinate system for a charge distribution. He also elaborated that in such coordinate system, potential contains only a monopole expression which is used to eliminate all the complex multipoles. He also suggested that monopole term can be used to solve the potential problems by choosing the surface to coincide. Cruz-Dombriz et al.27 investigated the existence of BH solutions with and without electromagnetic field by using perturbations scheme in f (R) theory. They connected this solution to the state for the effective Newton’s constant. They also discussed the consequences and physical properties of the f (R) model in local and global stability of the BH. Ivanov28 presented the effective anisotropic fluid models with heat transfer flow, ideal fluid, thickness, charge and beam of light. Jamil et al.29 derived the equation of motion of f-essence and solved for different types of metrics numerically. Then, they discussed both classical and quantum models in this framework. Ganguly et al.30 described the exact solution for a BH in the 1850111-2


Starobinsky model with the Schwarzschild solution. They said that their solution is unique with asymptotically flat background. They introduced two additional junction functions, that have made the problem more precise for studying the quark star. Nojiri and Odintsov31 worked out various features of the exterior RN dense object in MGT. They confronted the phenomenon of anti-vaporization but only in the region with very strong gravitational connection. In addition, they constructed the general unvarying result in the Einstein frame of f (R) gravity. Cembranos et al.32 examined the structure formation of compact objects with an inverse electrodynamics background. They concluded that presence of U(1) invariant terms in the action function are likely to host modified form of the charged BH. Takisa et al.33 analyzed a regular exact model of the Einstein–Maxwell system. They used this model for electropositive compact stellar objects. They compared the result of uncharged case and mass of the compact objects, which is derived from current error-free examinations. Tripathy34 described that when models totally act dissimilar to their Hubble constant, then asymmetric imbalance shows interesting resemblance to the behavior of the evolution equation. Das et al.35 considered the stellar solution obeying conformal motions under f (T ) theory of gravity. They studied the behavior of some physical features involved with the validity of their calculated model. Bhar and Rahaman36 discovered a category of accurate interior solutions. They claimed that this solution described a model of hypothetical compact object consisting of five regions. Islam et al.37 found the solution to the Einstein equation for the static spherical hypersurface by studying the barotropic equation of state. They asserted that imagination and empirical data which are used to describe the exotic dense star could behave as a reliable guide for forming a relationship between mass and radius. Pimentel et al.38 presented some viability conditions for the relativistic configuration of energy–momentum tensor. They consider ideal matter content and provided the detailed description of magnetic fields in the presence of charge. Maurya et al.39 elaborated a spherically anisotropic charge distribution and found a new solution of their model by using a particular source function. Silva et al.40 considered a specific GR formulation widely known as Horndeski gravity. They reviewed some stellar models with a single scalar field in Horndeski gravity and claimed that their solutions could be interpreted as new non-spinning BH models. Malaver41 presented a new model of compact exotic star with a charged anisotropic self-gravitating system. He confronted that this model may be beneficial for the description of charged anisotropic system. Zhang et al.42 investigated various dynamical properties of the charged stellar bodies in Einstein gravity. They found some properties that are universal for any kind of stellar collapse. Maurya and Govender43 considered 4D Einstein spacetime and inserted this geometry into a 5D flat spacetime by using some viable necessary and sufficient conditions. They also discussed the gravitational behavior of stellar structure by using Einstein field equation. Ulhoa44 described that Einstein–Rosen suggested a theory to narrate the elementary particles in respect of geometry of 1850111-3


spacetime. He explained the application of quantization procedure to the expression of gravitational energy which is defined in terms of teleparallelism. Mishra et al.45 constructed different types of metric in 4D space and described the anisotropic behavior of accelerated universe. They also obtained the cosmological motion from early retardation to late-time increasing speed in the context of multiplier factor. Quevedo46 described the exterior and interior regions of the celestial bodies by using Einstein equation regarding multipole structure. Sunzu et al.47 introduced accurate result to the Einstein equation with state function for anisotropic charged strange matter. They build two new models by describing a particular form of the potential energy and difference of anisotropy. Then they informed that there exists some relativistic matter variable that could be well ordered all over the interior. Hendi and Armanfard48 explained the thermodynamic state of matter for the charged supernova in Brans–Dick theory. They performed their analysis through phase space analysis and concluded that the critical values can be obtained from two different type of diagrams which represent phase transition. Murad and Fatema49 described a new analytical anisotropic charged compact stellar structure with generalized Tolman IV model background. They obtained some constraints that play an important role in the description of the internal structure of the exotic star. Oda50 found a mathematical formulation of a stellar solution that can be differentiable as many times as in the unimodular gravity. Bhatti51 presented the shear free model of the self-gravitating body with ideal fluid and explained the quantitative solution. They constructed the matching process to explain the two regions by applying Darmois conditions. Finally, they analyzed various physical features of their results through mathematical analysis. Aoki et al.52 presented a rotating star solution in a bimetric gravity. They described the existence of neutron star with the help of a gauge coupling parameter. They categorized their result into two classes. For class one, they considered that neutron star does not exist. But for class two, neutron star always exist in a regular solution and also Vainshtein screening components hold. Maurya et al.53 recently established a model for directionally-dependent charged solution. They found wellbehaved physical parameters in the modeling of compact stellar objects. Maurya et al.54 constructed the universal anisotropic solution of the relativistic compact star. They elaborated the component of anisotropy on the basis of metric potential. They showed that mass–radius relationship of the celestial objects is important because they compared it with the observational data. Abreu et al.55 explained the effects of density variations on the stability of matter composition. They considered two types of equation of state which may lead to the occurrence of cracking within the configuration. They compared the result of perturbation on these types of equation of state. They described that influence of thickness and anisotropic deviation are qualitatively different. Pappas and Kanti56 studied the excretion of Hawking radiation by an SdS black hole, in the form of a field of scalar, either negligibly or non-negligibly combined to gravity. They applied the six non-identical temperatures for the SdS background and the effective 1850111-4


temperature does not exist in the literature. Herrera et al.57 studied the effects of anisotropic pressure on the collapse of spherical geometry. The examination about the emergence of irregularities in the initially regular energy density was performed by Herrera et al.59 Herrera et al.60 explored the role of cosmological constant in this problem by relating the Weyl scalar with the matter functions for the case of dissipative compact stars. Yousaf and his colleagues62 modified their results within the background of various modified gravitational models. Bhatti and his team63 discussed the influences of present choice of the Ricci scalar on the rate of stellar collapse and inferred that such mathematical figures could behave as the repulsive forces induced by cosmological constant. Di Prisco et al.64 found some expansion-free collapsing spherical models by considering some specific combinations of stellar mass function. The impact of titled congruences in some dynamical properties of radiating as well as non-radiating systems are observed by Herrera et al.65 and Herrera.66 Yousaf et al.67 found some irregularity factors for the spherical relativistic geometries as observed by titled observers. Herrera68 discussed the emergence of gravitational radiations observed by the non-tilted observers during the evolutionary stages of isentropic self-gravitating bodies. Yousaf 69 explored the analytical model with the expansion-free conditions. They considered four solutions; two of these solutions are used to fulfill the Darmois conditions on the surface. They also explained the connection between Weyl tensor and density. The same author70 found exact analytical model under the free expansion condition. They described the shear free problems for exploring the error-free model. They considered two types of equations by imposing condition on the mass function and physical force. They derived the five solutions which are helpful to explain the junction problems. Bhatti et al.71 considered the role of charge and the stability of WH solution by small disturbance with f (R) expression. In f (R) model, they constructed the WH using the copying process. Finally, they explored the stability of WH reducing the equation from f (R) to GR. Yousaf et al.72 considered various elements of matter distribution creating the density inhomogeneity of the self-gravitating astronomical object in modified gravity background. Furthermore, they constructed the generalized Ellis equations. Bhatti considered the planar73 and cylindrical74 geometry in order to explore instability epochs with line symmetry filled with directionally dependent source. They found the adiabatic terms which effected the instability ranges under mathematical approximation. They concluded that the unstable regions increases by the reflection term. This work studies some dynamical features of spherical relativistic structures in MGT. Firstly, the non-static diagonally symmetric spherical line element is taken into account. Then, it is assumed that it is filled with locally anisotropic viscous radiating matter distribution in f (R) gravity. After calculating the corresponding equations of motion and mass function, the Weyl scalar is related to fluid content 1850111-5


parameters. Further, the MSS are being calculated and the role of shear evolution equation is investigated through MSS. The paper is outlined as follows. In Sec. 3, the role of shear evolution equation (SEE) and expansion evolution equations (EEE) in the modeling of radiating stars in MGT are analyzed after calculating MSS. Section 4 studies the whole discussion for the dust ball with constant R values. The main findings are discussed in Sec. 5. 2. Anisotropic and Radiating Geometry The Einstein–Hilbert (EH) in modified gravity, i.e. f (R) gravity is stated as √ Sf (R) = (2κ)−1 d4 x −gf (R) + SM ,

(1)

where κ, f (R) and SM stand for coupling constant, the Ricci scalar function and action of the matter, respectively. To calculate the metric f (R) field equations, we now vary above action with gαβ to obtain 1 (M) Rαβ fR − gαβ f + gαβ fR − ∇α ∇β fR = κTαβ , 2

(2)

d f , = ∇α ∇α with ∇α as a covariant derivative. A description of where fR ≡ dR f (R) field equations can be calculated by rearranging Eq. (2) with Einstein tensor Gαβ as

Gαβ =

κ (D) (M) (Tαβ + Tαβ ) , fR

(3)

(D)

where Tαβ describes the contribution of metric f (R) dark sources quantities that could be dubbed as effective energy–momentum tensor. This dark source term vanishes on applying the GR limits. Its formulations are found to be (D) 1 gαβ − fR gαβ . Tαβ = ∇α ∇β fR + (f − RfR ) κ 2 The four-dimensional irrotational non-static spherical geometry can be given as follows: ds2 = A2 dt2 − B 2 dr2 − C 2 (dθ2 + sin2 θ dφ2 ) ,

(4)

where A, B and C are the functions of r and t. We assume locally anisotropic and radiating self-gravitating fluid in the above relativistic geometry. The mathematical form of such energy–momentum tensor is given by (M)

Tαβ = (μ + P⊥ )Vα Vβ + (Pr − P⊥ )χα χβ − P⊥ gαβ + qα Vβ + εlα lβ − 2ησαβ + Vα qβ .

(5)

We suppose that above matter content is in the state of dissipation that is radiating energy by means of heat and free streaming. In the above expression, the quantities 1850111-6


Pr and P⊥ indicate radial and perpendicular pressure components, while μ describes the fluid’s energy density and η, σαβ , qβ stand for coefficient of shear viscosity, shear tensor and heat conducting vector, respectively, while ε represents radiation density. Further, V β , lβ and χβ are four vectors, which under comoving reference frame are 1 β 1 β δ0 , χβ = B1 δ1β , lβ = A δ0 + B1 δ1β , q β = q(t, r)χβ . These vectors defined as V β = A are satisfying V α Vα = −1 ,

χα χα = 1 ,

χα Vα = 0 ,

V α qα = 0 ,

lα Vα = −1

l α lα = 0 .

The shear tensor with the help of fluid’s four velocity and four acceleration (aγ = Vγ;δ V δ ) can be written as V ρ;ρ (gγδ + Vγ Vδ ) + a(γ Vδ) . 3 The scalar associated with the above tensor for the case of spherically symmetric radiating geometry can be given as R˙ 1 B˙ − , σ= A B R σγδ = V(γ;δ) −

while the nonvanishing σγδ components are found as 2σ 2 σ σ33 σ22 = − C 2 = B , , 3 3 sin2 θ The corresponding value of the expansion scalar is B˙ 1 2C˙ Θ= + . A C B

σ2 =

σ11 =

3 αβ σ σαβ . 2

∂ ∂ In this paper, we use “overdot” and “prime” notations for ∂t and ∂r operations, respectively. The f (R) equations of motion (3) in the background of metric approach, after using Eqs. (4) and (5) provide

B˙ B 2C˙ f˙R 2C fR A2 f − RfR fR κ 2 2 − + − − + 2 μA + εA + fR κ 2 B C A2 B C B2 B

=

2B˙ C˙ + C B

C˙ + C

B C

2

C − C

2B C − C B

2R − R

˙ A f˙R κ Bf 1 ˙ R f − − qBA + εBA − fR κ R B A C˙ A C˙ C B˙ =2 − − , C CA BC 1850111-7

A B

2 ,

(6)

(7)


A˙ 2C˙ f˙R κ 4 B 2 f − RfR 2 2 2 + − Pr B − ησB + εB − fR 3 κ 2 A C A2 +

A 2C + A C

B2 C − 2 + C C

fR f¨R − B2 A2

=

C˙ 2A˙ − A C

C˙ 2C¨ − C C

B2 A2

C 2A + , C A

(8)

B˙ C˙ f˙R A˙ f¨R κ 2 C 2 f − RfR f 2 2 − 2+ − + + R2 P⊥ C + ησC − 2 fR 3 κ 2 A A B C A B + +

C B A − + C B A A A

C B − C B

fR B2

=

C˙ C

A˙ B˙ − A B

C BC A − + C BC A

¨ C¨ B˙ A˙ B − + − C BA B

C2 . B2

C2 A2 (9)

The mass function of the relativistic spheres via Misner–Sharp formula gives75 ⎧ 2 2 ⎫ ⎨ C C˙ C ⎬ C . (10) m(t, r) = (1 − g αβ C,α C,β ) = 1 + − ⎩ 2 A B ⎭ 2 Now, we take the derivatives of the mass function with the help of radial and time ∂ 1 ∂ coordinates derivative operators, DC = C1 ∂r , DT = A ∂t . With the help of this, the ˙ fluid velocity can be mentioned through another operator as U = DT C = C A . This velocity would turn out to be negative for the systems undergoing the collapsing process. It follows from the expression of mass function as 1/2 2m(t, r) C = 1 + U2 − . (11) E≡ B C Then, the operator DC on mass function gives ⎧ ⎞ ⎫ ⎛ (D) (D) ⎪ ⎪ ⎨ T00 T01 ⎟ U ⎬ 2 κ ⎜ μ ˆ+ C , + ⎝qˆ − DC m = ⎠ 2fR ⎪ A2 AB E⎪ ⎩ ⎭

(12)

In the above expression, the “over hat” notation on the corresponding terms describes gˆ = g + ε. The ratio 3m C 3 can be evaluated from Eq. (12) for the spherical radiating geometry in metric f (R) gravity as ⎧ ⎞ ⎫ ⎡ ⎤ ⎛ (D) (D) ⎪ ⎪ r ⎨ ⎬ 3m T00 T01 ⎟ U 3κ ⎢ 1 ⎥ ⎜ μ ˆ+ C 2 C ⎦ dr . (13) = + ⎝qˆ − ⎠ ⎣ 3 3 2 C 2C 0 fR ⎪ A AB E⎪ ⎩ ⎭ 1850111-8


This equation has related spherical mass function with the relativistic matter parameters in the presence of f (R) extra dark source terms. It is worthy to note that in literature, there exist two components of the Weyl tensor which are named as the electric and magnetic parts. The former component can be given through its scalar function, E, as ! 1 Eαβ = E χα χβ − (gαβ + Vα Vβ ) , 3 where

E=

C¨ + C

B˙ C˙ − B C

C − − C

C˙ A˙ + C A

C B + C B

¨ 1 B 1 − − 2 B 2A 2C 2

A C − A C

1 A . − A 2B 2

(14)

The Weyl scalar E can be written with the help of Eqs. (10) and (13) as ⎛ ⎞ (D)

(D)

(D)

κ ⎜ ˆ + 2ησ + T00 − T11 + T22 ⎟ E = ˆ−Π ⎝μ ⎠ 2fR A2 B2 C2

−

3κ 2C 3

0

⎧ ⎤ ⎞ ⎫ ⎛ (D) (D) ⎪ ⎪ ⎨ ⎬ T00 T01 ⎟ U ⎢ 1 ⎥ ⎜ μ ˆ+ C 2 C ⎦ dr , + ⎝qˆ − ⎣ ⎠ 2 ⎪ fR ⎩ A AB E⎪ ⎭ ⎡

r

(15)

where “over hat” Π notation is the difference of Pˆr and P⊥ . Equation (15) has related the Weyl scalar with anisotropic pressure, IED, radiation densities (both heat and free streaming), cosmic dark source quantities and shear viscosity. 3. Spherical Stars and Its Evolution This section is devoted to split orthogonally the Riemann curvature tensor and the formulation of modified structure scalars with extra curvature corrections. We, then use these quantities in the modeling of EEE and SEE. In this framework, we use formulations of two well-known tensorial forms, i.e. Xγδ and Yγδ that were obtained from the splitting of the Riemann curvature tensor as ∗ Xγδ = ∗ Rγμδν V μV ν =

1 ρ η R∗ V μ V ν , 2 γμ ρδν

Yγδ = Rγμδν V μ V ν .

(16)

The detailed description as well as the derivations of the tensors, Xγδ and Yγδ , are found in Refs. 58–60. Here, the superscript ∗ describes the operations of dual derivations. The right, double and left dual of the Rμνγδ can be written as follows: ∗ ≡ Rγδμν

1 η ρμν R ργδ , 2

∗

Rγδμν ≡

1 ηγδ ρ R ρμν , 2

1850111-9

∗

∗ Rγδμν ≡

1 ρ ∗ η R . 2 γδ ρμν


In order to introduce corrections of f (R) gravity, we assume three parametric f (R) model61 ⎡ −n ⎤ 2 R ⎦, (17) f (R) = R + λRc ⎣1 − 1 + 2 Rc where λ ∈ R+ and n ∈ R+ with R denotes the set of real numbers. Further, the quantity Rc has values in the order of the present Ricci invariant. This model provides the dynamics of null Λ in the context of flat metric at R = 0. However, ˜ = R0 x1 with x1 > 0 de Sitter models can be achieved by taking R = constant = R along with λ to be λ=

2[(x21

(x21 + 1)n+1 x1 . + 1)n+1 − (n + 1)x21 − 1]

This value can be attained by performing first derivation of the particular value of x1 accompanied by substitution and manipulation for λ. It can be seen that in the above expression x1 < 2λ which further gives Λ(R1 ) = R41 < Λ(∞) at de Sitter point. Further, by assuming x1 with n 1 and by fixing n with x1 1, the constraint x1 → 2λ can be easily calculated. This values favor the outcomes of the ΛCDM cosmic evolution model. Equation (16) can be rewritten by means of four vectors and metric tensor as

(gγδ + Vγ Vδ ) gγδ + Vγ Vδ X T + χα χβ − (18) Xγδ = XTF , 3 3 Yγδ

(gγδ + Vγ Vδ ) gγδ + Vγ Vδ YT + χα χβ − = YTF . 3 3

(19)

In the above expression, the quantities XTF and XT are the trace-free and trace components of Xγδ , while YTF and YT are the trace-free and trace-free components of the tensor Yγδ , respectively. The quantities, XTF , XT , YTF and YT are known widely as structure scalars. Since, we wish to evaluate these in the realm of a f (R) gravity, therefore we label these terms with modified structure scalars (MSS). These MSS, after using Eqs. (6)–(9), (18) and (19) can be found as

κRc ψμ μ ˆ+ 2 , (20) XT = Rc − 2nλRc R(1 − R2 Rc2 )−n−1 A

ψPr κRc ψP⊥ ˆ Π − 2ση + 2 − 2 , XTF = −E − (21) 2[Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ] B C

κRc ψPr 2ψP⊥ ψμ ˆr − 2Π ˆ , YT = + + + 3 P μ ˆ + 2[Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ] A2 B2 C2 (22) YTF = E −

Rc κ ˆ − 2ησ + ψPr − ψP⊥ . Π 2[Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ] B2 C2 1850111-10

(23)


Equations (15) and (23) provide another helpful formulations of the MSS, YTF as

κRc 2ψPr 2ψP⊥ ψμ ˆ + YTF = μ ˆ − 2Π + 4ησ + 2 − 2[Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ] A B2 C2 r 3κ Rc − 2C 3 0 [Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ]

ψμ ψq U 2 × μ ˆ + 2 + qˆ − C C dr . A AB E

(24)

In order to see clearly the effects of fluid variables, we shall introduce some effective configurations of these variables. These variables are being introduced by putting some dark source variables as well as some sort of dissipations in them. The use of such variables could be helpful to understand the role of dark source three parametric f (R) corrections in the mathematical definitions of SEE, EEE and Weyl scalar differential equation. We shall also use these variables in f (R) MSS. These are defined as ψP 2ψP⊥ ψμ 4 2 ˆ+ 2 , Preff = Pˆr − ησ + 2r , P⊥eff = P⊥ + ησ + , μeff = μ A 3 B 3 C2 Πeff = Preff − P⊥eff = Π − 2ησ +

ψPr ψP − 2⊥ . 2 B C

Equations (20)–(23) provide

r 3κ ψq U Rc XTF = μeff + qˆ − C 2 C dr 2C 3 0 [Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ] AB E −

κRc μeff , 2[Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ] YTF

(25)

κ 4ψPr eff = μeff − 2Π + B2 2[1 − ρ sech2 ( RRc )] r 3κ Rc − 2C 3 0 [Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ]

ψq U 2 × μeff + qˆ − C C dr , AB E

(26)

YT =

Rc κ (μeff + 3Preff − 2Πeff ) , 2[Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ]

(27)

XT =

Rc κμeff . [Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ]

(28)

1850111-11


It is noteworthy that these MSS are having very important role in the discussion of various properties of celestial structures, like, stability of compact bodies,9,58,59,63 inhomogeneous energy density (IED),10,77–79 gravitational collapse,72,80–82 etc. For example, the implosion rate of spherical cosmic bodies, quantity of matter content as suggested by Tolman, the emergence of IED, all of these dynamical phenomena could be dealt via three parametric Ricci scalar MSS. We shall use these MSS in the modeling of the notable Raychaudhuri equation (EEE) (the equation which is also calculated independently by Landau).76 Thus, we have 2 Θ2 −YT = V α Θ;α + σ αβ σαβ + (29) − aα;α . 3 3 This indicates that the Raychaudhuri expression has been expressed through one of the MSS, YT . The very helpful phenomenon of the gravitational theories like the calculation of some exact analytical stellar models and for the Penrose–Hawking singularity theorems can be discussed through YT . This YT contains three parametric Ricci scalar corrections. This further suggests that three parametric f (R) along with some other fluid parameters are fully involved in controlling the effects related to EEE. The SEE through MSS is found as follows: 2 aC 1 − Θσ − V α σ;α − σ 2 . (30) BC 3 3 This indicates that the three parametric dark source curvature quantities affect the shear evolution of the radiating spherical self-gravitating fluids. Herrera et al.77 investigated the stability of shear-free condition on the dynamical evolution of spherical system with the help of structure scalar in Einstein gravity. It is worthy to stress that the above SEE has been found to be identical with that found in GR by Herrera et al.77 Moreover, the tidal forces produced by our system and its fluid variables can be connected through the following differential equation as κRc μeff XTF + 2[Rc − 2nλRc R(1 − R2 Rc2 )−n−1 ] YTF = a2 + χα a;α −

3C ψq 2 2 −n−1 −1 + [κRc (Θ − σ)][2{Rc − 2nλRc R(1 − R Rc ) = −XTF }] qˆB − . C A (31) One can see XTF to be a factor controlling IED of the regular distribution of spherical spacetime under the following limits. (i) If anisotropic and viscous spherical structure dynamically behave in such a way that the role of the shear scalar is equal to that of expansion scalar, then XTF will be fully involved in inducing pits/irregularities in the initially homogeneous stellar body. (ii) If the effects of dissipation as well as three parametric Ricci scalar corrections are not present in the homogeneous spheres, then XTF will be a factor controlling IED of our system. 1850111-12


Equation (31) and the subsequent discussion, leading to the identification of XTF as the inhomogeneity factor were explicitly discussed in Einstein gravity by Herrera et al.58 Furthermore, a much more deep discussion about this issue is given in Ref. 78. Here, Herrera calculated inhomogeneity factors for four different types of matter distributions, i.e. dust, isotropic, anisotropic and dust radiating spherical systems. He found XTF as an inhomogeneity factor for anisotropic spherical system. 4. Relativistic Dust Ball This section is devoted to study some dynamical properties of the initially homogenous dust cloud in the background of present Ricci scalar corrections. We shall also formulate some physically realistic features of the relativistic dust system through SEE and EEE. We shall use the notation “tilde” for those equations and quantities that are evaluated via constant Ricci scalar limit. The trace of Eq. (2) gives (32) fR × R − 2f + 3fR = κT , √ (M) where fR = −g−1/2 ∂γ ( −gg γδ ∂δ fR ) while T = g γδ Tγδ . The curvature term fR will be zero, if one takes f (R) = R. The corrections introduced by this term will indicate the propagation as well as the evolution of the scalar variable named as scalaron. In this scenario, Eq. (13) turns out to be r Rc κ μC 2 C dr m= ˜ 2 Rc2 )−n−1 ] 0 2[Rc − 2nλRc R(1 − R ⎧ −n ⎨ λR ˜2 R Rc λRc c − − 1− 2 ˜ 2 Rc2 )−n−1 ] ⎩ 2 2 Rc 4[Rc − 2nλRc R(1 − R

˜2 2nR × 1− 2 Rc

−1 ⎫ ⎬ 2 2 ˜ ˜ R nλR + 1− 2 , ⎭ Rc Rc

while Eqs. (13) and (15) boil down to

(33)

r 1 3 μ C dr μ− 3 C 0 ⎧ −n ⎨ λR ˜2 R λRc c 2 2 −n−1 −1 ˜ − 1− 2 − Rc [4Rc − 2nλRc R(1 − R Rc ) ] ⎩ 2 2 Rc

3m Rc κ = 3 ˜ 2 R2 )−n−1 ] C 2[Rc − 2nλRc R(1 − R c

×

˜2 2nR 1− Rc2

E=

˜2 nλR + Rc

˜2 R 1− 2 Rc

−1 ⎫ ⎬ ⎭

,

Rc κ 3 ˜ 2 R2 )−n−1 ] 2C [Rc − 2nλRc R(1 − R c 1850111-13

(34) 0

r

μ C 3 dr .

(35)


The constant Ricci scalar three parametric MSS for the dust self-gravitating system are found as ⎧ ⎡ −n ⎨ λR ˜2 R R λR κ R c c c c ⎣μ + + 1− 2 Y˜T = − ˜ 2 Rc2 )−n−1 ] 2 ⎩ 2 2 Rc 2[Rc − 2nλRc R(1 − R

˜2 2nR × 1− Rc2

−1 ⎫⎤ ⎬ ˜2 ˜2 R nλR ⎦, − 1− 2 ⎭ Rc Rc

(36)

˜ TF = Y˜TF = E , (37) −X ⎧ ⎡ −n ⎨ λR ˜2 R λR κ R R c c c c ˜T = ⎣μ − X − 1− 2 ˜ 2 Rc2 )−n−1 ] 2 ⎩ 2 2 Rc [Rc − 2nλRc R(1 − R

˜2 2nR × 1− 2 Rc

−1 ⎫⎤ ⎬ 2 2 ˜ ˜ R nλR ⎦. + 1− 2 ⎭ Rc Rc

(38)

The corresponding EEE, SEE and the differential equation for the tidal forces are 2 Θ2 −Rc κ − aα V α Θ;α + σ 2 + ;α = ˜ 2 Rc2 )−n−1 ] 3 3 2[Rc − 2nλRc R(1 − R ⎧ ⎡ −n ⎨ λR ˜2 R λR R c c c × ⎣μ + + − 1− 2 2 ⎩ 2 2 Rc

˜2 2nR × 1− Rc2

˜2 nλR − Rc

˜2 R 1− 2 Rc

−1 ⎫⎤ ⎬ ⎦ = −Y˜T , ⎭ (39)

2 σ2 + σΘ = −E = −Y˜TF , V α σ;α + 3 3

⎡ ˜ TF + ⎣X

×

(40)

κRc μ ρRc − 2 2 −n−1 ˜ ˜ 2 Rc2 )−n−1 ] 2[Rc − 2nλRc R(1 − R Rc ) ] 4[Rc − 2nλRc R(1 − R

⎧ ⎨ λR ⎩ 2

c

−

λRc 2

˜2 R 1− 2 Rc

−n

˜2 2nR 1− Rc2

˜2 nλR + Rc

˜2 R 1− 2 Rc

−1 ⎫⎤ ⎬ ⎦ ⎭

3C . (41) C One can observe from the last of the above equations that μ = 0, if and only if ˜ TF = 0, thereby pointing X ˜ TF as a parameter for producing as well as controlling X IED of the dust ball. ˜ TF = −X

1850111-14


5. Discussion This work is devoted to analyze the influences of the three parametric forms of f (R) model on the expressions of SEE, EEE and the differential equation of tidal forces. The well-known modified gravitational theory, f (R) gravity has occupied foremost role in the study of various foremost cosmic issues like, late-time cosmic acceleratory phase and inflationary era. In this direction, we have considered a particular f (R) combination that was first formulated by Ref. 61. This model could provide gravitational dynamics consistent with ΛCDM model. The aim of this work is to explore the role of fourth-order metric f (R) theory on the dynamical aspects of the initially regular distributions of celestial body. It is worthy to note that Raychaudhuri equation (EEE) has a direct relevance in the understanding of exact analytical stellar models of GR. The well-known Penrose–Hawking singularity theorems can be well discussed through EEE. We have provided an effective platform to derive such equations through set of MSS. In this framework, we have considered a diagonally symmetric non-static irrotational body of spherical in shape in cosmos. We have assumed that it is filled with a shearing viscous locally anisotropic radiating fluid. We have calculated the corresponding metric f (R) equations of motion. After calculating the Misner–Sharp mass function, we have developed few very important relations between matter variables, mass function and the Weyl scalar. With the help of the orthogonal decomposition of the Riemann tensor, the MSS are defined with the three parametric Ricci scalar corrections. By following this technique, we have evaluated four different three f (R) parametric scalar variables. These are labeled as XTF , XT , YTF and YT . Equations (20)–(23) describe the mathematical formulations of such variables and all of these variables are mediating extra degrees of freedom coming from the three parametric f (R) model. The influences of such MSS are also analyzed via formulating very important EEE and SEE. Since f (R) gravity is a nonlinear fourth-order gravity, therefore in order to have deep understanding on the role of scalar functions, we have boiled down our results by assuming constant values of the Ricci scalar. We have seen that XTF occupies foremost importance in the analysis of the emergence of IED on the surface of the smooth environment of the relativistic dust balls. Moreover, the MSS XT has been explored to be the control center for estimating the fluctuations emerging in the congruences of the collapsing dust sphere in this gravity. The Raychaudhuri equation, i.e. EEE has been found to be expressed in terms of one of the structure scalars, YT . Thus, in order to study the evolutionary phases of irregularities in the radiating spherical object, one needs to study the behavior of YT since such a quantity is actually controlling all the influences of Raychaudhuri equation. The SEE can be fully expressed through the scalar function YTF . All of our results reduce to GR under f (R)→R limit.60 Our results are based on the stability of IED and the formulations of EEE and SEE for the spherically symmetric systems. For this purpose, we assume that our geometry is coupled with a viscous and radiating anisotropic fluid distributions. 1850111-15


We have discussed our results that are based on the specific choice of f (R) model. Our considered f (R) model could be considered as a viable tool to discuss the accelerating expansion of the universe. We noticed that under some constraints, our considered three parametric f (R) model could favor the outcomes of the ΛCDM model. Thus, our results of inhomogeneity factors are quite broad and describe the evolution of energy density inhomogeneities over the spherical systems by keeping the validity of ΛCDM observational values. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

17.

18.

19. 20. 21. 22. 23. 24.

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