Buchdahl model in f(R, T) gravity: A comparative study with standard Einstein's gravity
PPhysics of Dark Universe Vol. xx, No. x (2019) xxxxxx
Buchdahl model in f ( R, T ) gravity: A comparative study with standardEinstein’s gravity S. K. Maurya, Ayan Banerjee, Francisco Tello-Ortiz Department of Mathematical & Physical Sciences, College of Arts & Science, University of Nizwa, Nizwa, Sultanate of Oman Astrophysics and Cosmology Research Unit, University of KwaZulu Natal, Private Bag X54001, Durban 4000, South Africa Departamento de F´ısica, Facultad de ciencias b´asicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile
Abstract:
This paper is devoted in the study of the hydrostatic equilibrium of stellar structure in the frameworkof modified f ( R, T ) gravity theory that allows the non-conservation of energy-momentum, with possible implicationsfor several cosmological and astrophysical issues such as the late-time cosmic acceleration of the universe withoutappealing to exotic matter fields. For this purpose, we consider the gravitational Lagrangian by taking an arbitraryfunction of the Ricci scalar and the trace of the stress-energy tensor. We obtain a generic form for the gravitationalfield equations and derive the field equation for f ( R, T ) = R + 2 χT . Here we propose a particular metric potential Buchdahl ansatz [Phys. Rev. D 116, 1027 (1959)] in principle, of explaining almost all the known analytic solutionsto the spherically symmetric, static Einstein equations with a perfect fluid source. For the choice of f ( T ) = 2 χT one may observe that the pressure and energy density profiles are markedly different. Important cases, which havebeen analyzed in detail, are all possible Buchdahl solutions for spherical equilibrium configuration in f ( R, T ) gravityand compare them with standard gravity theory. We find that Buchdahl’s solution in Einstein gravity and f ( R, T )gravity behaves in a similar manner but in some situations Einstein gravity displays more pleasing behavior than its f ( R, T ) counterpart.
Key words:
Modified gravity; Dark Energy; gravitation, compact stars.
PACS:
Over the past decade, the late-time acceleration ofthe universe has led to new perspectives and scenariosin the field of modern cosmology and physics as a whole.Since this discovery was confirmed by several indepen-dent observations (see [1–7] for a detailed discussion ofthe recent astronomical observations). In order to ex-plain the accelerated expansion, there exist two differentapproaches to solve the debate. One is the existence ofmysterious dark energy (DE) and its possible extensionsı.e. modified theories of gravity (MTG). The idea ofDE which has negative pressure occupied approximately70% of the energy density of our universe exists in a non-matter form. The simplest example of the DE is the cos-mological constant Λ, as representing a constant energydensity of the vacuum which satisfy the cosmological ob-servations [8]. But it is plagued by a severe energy scaleproblem if it originates from the vacuum energy appear-ing in particle physics [9]. This problematic nature ofcosmological constant has motivated intense research for alternative theories of gravity extending the Einstein’stheory of gravity. This motivates the search for an alter-native gravity theory that can address the present accel-erating phase of the universe. It has been shown that al-ternative possibilities could give an adequate descriptionof cosmological observations [10]. One of the simplestpossible modification is the f ( R )-gravity [11, 12], has at-tracted serious attention possibly because of its (decep-tive) simplicity. A viable f ( R ) gravity, where f ( R ) is ageneric function of the Ricci scalar R . This theory comesinto the game as a straightforward extension of GR andto discuss a unified picture of both inflation and the ac-celerated expansion in more scientific ways [13–17]. InRef. [18], authors have studied a nonperturbative modelof strange spherical objects in f ( R ) = R + 2 α R gravitytheory, where α is a constant. They showed that themass of the spherical objects increases when the valueof the parameter α increases progressively. In addition,Capozziello et al [19, 20] have argued that the mass-radius profile undergoes modifications in f ( R ) domaindue to the presence of high order curvature terms such Submitted Oct. 20191) E-mail: [email protected]) E-mail: [email protected]) E-mail: [email protected] c (cid:13) a r X i v : . [ phy s i c s . g e n - ph ] D ec hysics of Dark Universe Vol. xx, No. x (2019) xxxxxx as R , R etc. For reviews of f ( R ) theory in more de-tails see Refs. [21–29]. In addition to this, the first f ( R )work with approximately viable f ( R ) to describe darkenergy and inflation was proposed by Nojiri and his col-laborators [30–32]. In this framework Hydrostatic equi-librium and stellar structure, stable neutron star modelsand extreme neutron stars have been discussed in details[33–35].In the same way, another alternative theory of grav-ity has been introduced, the so-called f ( R, T )-gravity[36]. This theory is based on the assumption of thegravitational field couples to the trace T of the energy-momentum tensor of the matter. An interesting aspectof f ( R, T ) theory is that it may provide an effective clas-sical description of the quantum properties of gravity.Apart from a better understanding at the fundamentallevel, some results have been obtained with this the-ory. In an argument Houndjo [37] discussed transitionof matter-dominated era to an accelerated phase by as-suming a special form of function f ( R, T ) = f ( R ) + f ( T ). The study of cosmological solutions of f ( R, T )gravity was performed through the phase space analy-sis [38]. The other motivation is related to reconstruct-ing f ( R, T ) gravity from holographic dark energy; see,e.g. [39]. Other issues as, for example, cosmologicaland solar System Consequences [40], anisotropic cosmol-ogy [41, 42], non-equilibrium picture of thermodynamics[43], a wormhole solution [44, 45], and some other rele-vant aspects [46–49].Therefore, it is not possible to confirm or to disprovesuch theories based on theresults of cosmology and com-pare them with the observational data. However, to es-tablish a satisfactory gravity theory, it is important tostudy on the astrophysical level, e.g. using the relativis-tic stars. Some arguments for these theories come fromthe assumption that relativistic stars in the strong grav-itational field could discriminate standard gravity fromits generalizations. Considering the case of f ( R, T ) grav-ity, a large number of works on the evolution of compactstars are available in different literature. In this frame-work, hydrostatic equilibrium configuration of neutronstars and strange stars have been studied [50]. The struc-ture of compact stars in f ( R, T ) gravity was investigatedrecently in refs. [51–56], whereas gravitational vacuumcondensate star (gravastars) solution has been obtainedin [57].Recently, Hansraj and Banerjee [58] have studiedstellar models within the context of f ( R, T ) gravity,and showed that in some situationthese theories displaysmore pleasing behavior than its Einstein counterpart.Motivated by these good antecedents, we extend Buch-dahl’s [59] spacetime from GR framework to the f ( R, T )gravity arena. The Buchdahl ansatz is well-known so-lutions in GR with a clear geometric characterization of the associated spacetime metric will be prescribed to de-termine the other. In particular, this ansatz contains awide range of models, each one is mathematically dif-ferent from the other. As a toy model, these solutionsaccurately describe the behavior of real astrophysicalobjects such as neutron stars, white dwarfs or strangestar families [60]. Thus, the information obtained fromthese celestial bodies has allowed a better understand-ing of the behavior of gravitational interaction in thestrong field regime and some intricate processes of cre-ation/annihilation of particles, among others.From the aforementioned discussion one natural ques-tion arises, does the Buchdahl model carry over to f ( R, T ) theory, which maintains the hydrostatic equi-librium and compatible with GR solution? To addressthis question, we would like to pass some remarks: I)We consider perfect fluid matter distribution as of GRcase. Because, our interest is to see the effects of f ( R, T )theory on compact stars model. In addition, we wouldlike to compare our results with there similes in GR; II)Despite its simplicity, the chosen f ( R, T ) model couldbe seen as Einstein gravity plus an effective cosmologi-cal constant; III) The Lagrangian matter density L m canbe taken as − p ı.e. the isotropic pressure. These are themain motivation for the extension of the Buchdahl modelto more complicated and general situations. An interest-ing aspect of our solution is that we use Gupta-Jasim twosteps methodology for solving the differential equations.Further, we explore the obtained solution by studyingall fundamental properties that any well behaved stellarstructure should satisfied. These constraints include: • (a) Positivity and finiteness of pressure and energydensity everywhere in the interior of the star in-cluding the origin and boundary:0 < p < ∞ , < ρ < ∞• (b) Inside the fluid sphere the pressure and den-sity should be monotonic decreasing functions withincreasing radius. The pressure vanishes at theboundary r = R : dpdr ≤ , dρdr ≤ , p ( R ) = 0 • (c) At the boundary of the star the interior solutionshould be matched with the Schwarzschild exteriorsolution, i.e. ds − = ds . If follows e ν ( R ) = e − λ ( R ) = 1 − MR . • (d) Inside the fluid sphere the velocity of soundshould everywhere be less than the speed of light0 ≤ v = dpdρ ≤ . • (e) The physical ways to characterize the energyconditions which are: – Null energy condition: ρ + p > – Weak energy condition: ρ > ρ + p > – Strong energy condition: ρ + 3 p > – Dominant energy condition: ρ ≥ | p | should be satisfied. • (f) The solution should be free from physical andgeometric singularities i.e. e ν and e λ in the range0 ≤ r ≤ R. • (h) It is the Buchdahl limit [59] for a perfect fluidsphere of radius R and mass M , if M/R ≤ , thenthere is no equilibrium solution whatsoever. Theimpact of this upper bound is that one cannot packmore matter into an object than the radius allows.As noticed in [61], this upper bound is larger thanthe Buchdahl-Bondi limit of GR in f ( R ) gravity,whenever f ( R ) (cid:54) = R . Moreover, Chakraborty andSengupta [62] established that an extra-massivestable star can exist in the context of Kalb-Ramondfield in a four dimensional spacetime.Finally, it is worth mentioning that, Buchdahl’s solu-tion has an enormous backrest as viable and tractablemodel describing compact configurations. In this re-spect, Vaidya and Tikekar [63] particularized the modelgiving a geometric meaning, prescribing specific 3-spheroidal geometries ( t = const.) to 4-dimensional hy-persurfaces. Note that this spheroidal condition hasbeen found very useful for finding exact analytic solu-tions of the Einstein field equations (EFEs) in GR andhave important applications ranging from singularity freeinterior solutions to the physical understanding of rela-tivistic phenomena [64]. Furthermore, using this situ-ation Kumar et al. [65, 66] have found an exact solu-tion to the EFEs with an anisotropic matter distributionand admitting conformal motion [67]. In the frameworkof anisotropic hypothesis “Buchdahl model” have beentested against astrophysical compact stellar objects (forreview, see [68]).This article is organized as follows: Starting with abrief introduction in Sect. , we make a review of theoriginal f ( R, T ) gravity in Sect. and present the for-malism that allows us to construct stellar structure forspherically symmetric solutions in Sect. . In Sect. , thecompact star models in frame of modified gravity with f ( R, T ) = R + 2 χT , are investigated in detail. For com-pact stars we use a well-known metric ansatz proposedby Buchdahl and find its solutions. In continuation withthis we derive the field equations by using Gupta-Jasim two steps method and studied solutions for positive and negative values of Buchdahl parameter K . We show ourresults and discussions in the same section and draw thefinal conclusions in Sect. . f ( R, T ) gravity theory
In this section, we concisely review the viable mod-ified theory of gravity, as in the case of f ( R, T ) gravitywith T being the trace of the stress-energy tensor, T µν .The full action is S = 116 π (cid:90) f ( R, T ) √− gd x + (cid:90) L m √− gd x, (1)where f ( R, T ) is the generic function of Ricci scalar R with g is the determinant of the metric tensor g µν .We define the matter Lagrangian density, related to theenergy-momentum tensor as T µν = − √− g δ ( √− g L m ) δg µν , (2)with the trace T = g µν T µν . Following the Ref [36], weconsider the case of Lagrangian density L m of matterdepends only on the metric tensor components g µν . Con-tracting Eq. (2) gives T µν = g µν L m − ∂ ( L m ) ∂g µν . (3)By varying the action (1) of the gravitational field withrespect to the metric tensor components g µν provides thefollowing relationship: δS = 116 π (cid:90) (cid:104) f R ( R, T ) R µν δg µν + f R ( R, T ) g µν (cid:3) δg µν − f R ( R, T ) ∇ µ ∇ ν δg µν + f T ( R, T ) δ ( g αβ T αβ ) δg µν δg µν − g µν f ( R, T ) δg µν + 16 π √− g δ ( √− g L m ) δg µν (cid:105) √− gd x, (4)where f R ( R, T ) = ∂f ( R, T ) /∂R and f T ( R, T ) = ∂f ( R, T ) /∂T . According to Ref [36], the variation of T with respect to the metric tensor as δ ( g αβ T αβ ) δg µν δg µν = T µν + Θ µν . (5)The ∇ µ denotes covariant derivative which is associatedwith the Levi-Civita connection of metric tensor g µν andbox operator (cid:3) is defined by (cid:3) ≡ ∂ µ ( √− gg µν ∂ ν ) / √− g, and Θ µν = g αβ δT αβ /δg µν . Now, partially integrating the second and third terms inEq. (4), one can obtain the field equations of the f ( R, T )gravity model as( R µν − ∇ µ ∇ ν ) f R ( R, T ) + (cid:3) f R ( R, T ) g µν − f ( R, T ) g µν = 8 π T µν − f T ( R, T ) ( T µν + Θ µν ) . (6) We see from this equation that when f ( R, T ) ≡ f ( R ), theabove equation reduces to f ( R ) gravity field equations.In particular, if f ( R, T ) ≡ R then standard Einstein’sfield equations are recovered in GR.To reach the expression of the covariant derivative ofthe energy-momentum tensor and extract the one of thealgebraic function, we perform the covariant derivativeof Eq.(6), as [69] ∇ µ T µν = f T ( R, T )8 π − f T ( R, T ) (cid:20) ( T µν + Θ µν ) ∇ µ ln f T ( R, T )+ ∇ µ Θ µν − g µν ∇ µ T (cid:21) . (7)It is straightforward to see that the stress-energy mo-mentum tensor T µν in f ( R, T ) gravity is not conservedas a view point of Einstein general relativity (GR) dueto presence of nonminimal matter-geometry coupling inthe formulation. By using Eq. (3), the tensor Θ µν isdefined asΘ µν = − T µν + g µν L m − g αβ ∂ L m ∂g µν ∂g αβ . (8)Henceforth, in order to facilitate a direct compari-son with the work of Buchdahal [59], we follow his con-ventions. For stellar configurations, one can assume aspherically symmetric metric with coordinates ( t , r, θ, φ )in the following form ds = e ν ( r ) dt − e λ ( r ) dr − r ( dθ + sin θ dφ ) , (9)where ν ( r ) and λ ( r ) are arbitrary functions of the ra-dial coordinate r only. We use the natural system ofunits with G = c =1. Further, we assume that the interiorof the star is filled with a perfect fluid source, and thestress tensor is given by T µν = ( ρ + p ) u µ u ν − pg µν , (10)where u ν is the four velocity, satisfying u µ u µ = 1 and u µ ∇ ν u µ = 0. Here, ρ is the matter density and p isthe isotropic pressure. Since, we choose a further as-sumption, namely, L m = − p , according to the definitionsuggested in [36], the tensor (8) yieldsΘ µν = − T µν − p g µν . (11)We then assume a more plausible and simple model,namely, f ( R, T ) = R + 2 χT , the usual Einstein-Hilbertterm plus a T dependent function f ( T ) [36]. This inter-esting model has some significant advantages, for exam-ple, this model can describe inflationary era as well asan accelerated expansion phase. In this framework weobtain singularity free spacetime. Thus, using the linearexpression and Eq. (6), the Einstein tensor reduce to G µν = 8 π T µν + χT g µν + 2 χ ( T µν + p g µν ) = 8 π ˜ T µν . (12) Note that field equation (6) reduce to Einstein fieldequations when f ( R, T ) ≡ R . Studying such particularlinear assumption ( f ( R, T ) = R + 2 χT ) has widely ac-cepted to address cosmological as well as astrophysicalsolutions (see introduction). By substituting the valueof f ( R, T ) = R + 2 χT in Eq. (7), we obtain(8 π + 2 χ ) ∇ µ T µν = − χ (cid:2) g µν ∇ µ T + 2 ∇ µ ( p g µν ) (cid:3) . (13)In that follows, if χ → Let us investigate the non-zero components of thefield equations for spherically symmetric line element (9),which are [58]˜ ρ = e − λ π (cid:18) − r + λ (cid:48) r + e λ r (cid:19) , (14)˜ p = e − λ π (cid:18) r + ν (cid:48) r − e λ r (cid:19) , (15)˜ p = e − λ π (cid:18) ν (cid:48)(cid:48) + ν (cid:48) − λ (cid:48) ν (cid:48) + 2 ν (cid:48) − λ (cid:48) r (cid:19) . (16)with, ˜ ρ = ρ + χ π (3 ρ − p ) , ˜ p = p − χ π ( ρ − p ) , where the prime denotes differentiation with respect tothe radial coordinate r . Using Eqs. (14-16), one canobtain another additional equation which is ν (cid:48) p + ρ ) + dpdr = χ (8 π + 2 χ ) ( p (cid:48) − ρ (cid:48) ) . (17)Note that Eq. (17) reduces to the hydrostatic equilib-rium condition of general relativity when χ = 0. Now,using Eqs. (14) and (15), we rewrite the modified equa-tions in terms of pressure ( p ) and energy density ( ρ ),which are ρ = π (8 π + 6 πχ + χ ) [(8 π + 3 χ ) ˜ ρ + χ ˜ p ] , (18) p = π (8 π + 6 πχ + χ ) [(8 π + 3 χ ) ˜ p + χ ˜ ρ ] . (19)where χ (cid:54) = − π & − π . While the equation of pressureisotropy G = G reduces to r (2 ν (cid:48)(cid:48) + ν (cid:48) − ν (cid:48) λ (cid:48) ) − r ( ν (cid:48) + λ (cid:48) ) + 4( e λ −
1) = 0 . (20) Interestingly the isotropy equation for f ( R, T ) gravityis the same for the ordinary Einstein’s equations with aperfect fluid source. The mass of the star is now due tothe total contribution of the energy density of the matterand that can be determined by the Eq. (14). The masstakes the new form as m = 4 π (cid:90) r ρ r dr = 4 π (8 π + 3 χ )(8 π + 6 πχ + χ ) (cid:90) r ˜ ρ r dr + 4 π χ (8 π + 6 πχ + χ ) (cid:90) r ˜ p r dr, (21)In this regard, constant parameter χ plays an importantrole for determining the stellar structure. Classically, the f ( R, T ) gravity recovers the same physics as the generalrelativity with χ = 0. Observing the Eq. (20), whichserves as the master differential equation in this analysiscontains two necessary constants of integration. How-ever, one can accommodate the constants in terms ofthe mass M and radius R of the distribution by solvingthe linear matching equations.As usual, all astrophysical objects are immersed invacuum spacetime and at the juncture interface wematch the interior spacetime to an appropriate exteriorvacuum region. Here, we are working with unchargedmatter distributions and then outer spacetime is de-scribed by Schwarzschild solution, ds = (cid:18) − Mr (cid:19) dt − dr − Mr − r ( dθ + sin θdφ ) , (22)where M is the Schwarzschild mass which concides withtotal mass within the fluid sphere at the surface of theobject defined by Σ = r = R . So, the first fundamen-tal form demands the continuity of the metric potentialsacross the boundary. Explicitly it reads e − λ ( R ) = e ν ( R ) = 1 − MR . (23)On the other hand the second fundamental form dictates p ( R ) = 0 , (24)this condition says that object can not expand indefi-nitely, in consequence this constraint determines the ob-ject size ı.e, its radius. In next, we will investigate stellarstructure with Buchdahl assumption as a metric poten-tial in the f ( R, T ) theory to solve the field equations.
The Buchdahl solution was generated through amathematical ansatz on the static, spherically symmet-ric fluid spheres of Einstein’s equations by Buchdahl [59],which cover almost all physically tenable known models.The widely studied metric ansatz given by [68] e λ = K (1 + Cr ) K + Cr , when K <
K > , (25) where K and C are two parameters that characterizethe geometry of the star. Here, we extend the follow-ing work [68], which was devoted in describing a classof relativistic stellar solutions for generalized Buchdahldimensionless parameter K .An interesting aspect of the Buchdahl solution is thatone can recover the interior Schwarzschild solution when K = 0 and for K = 1 the hypersurfaces { t = constant } areflat. Further, if we assume C = − K/R , one can recoverthe Vaidya and Tikekar [63] solution and for K = − e ν = Ψ and ξ = (cid:113) K + Cr K − [64, 66]. For more clear sighted, we consider the Gupta-Jasim [72] (see Refs. [68] for complete discussion) twostep method for solving the system of equations. Start-ing with the pressure isotropic Eq. (20), and using Eq.(25) the corresponding differential equation will then(1 − ξ ) d Ψ dξ + ξ d Ψ dξ + (1 − K )Ψ = 0 , (26)In this framework we consider mainly two cases for thespheroidal parameter K with C > K < i.e K is negative Now, differentiate the Eq. (26) with respect to ξ anduse another substitution ξ = sin x and d Ψ dξ = Y , we have d Ydx + (2 − K ) Y = 0 , for K < , ξ = sin x, (27)where dYdx = cos x d ψdξ and d Ydx = cos x d ψdξ − sin x d ψdξ , re-spectively. The above Eq. (27) is a homogeneous differ-ential equation of second order with constant coefficients.In this case Eq. (27) leads to Y ( x ) = a sin( n x ) + b cos( n x ) , if 2 − K = n , K < . (28)where a and b are arbitrary constants of integration.To obtain the complete solution we re-substitute d Ψ dξ = Y and d Ψ dξ = dYdx dxdξ into hypergeometric Eq. (26), we get ΨasΨ( x ) = sin x [ b + a tan( n x )] − n cos x [ b tan( n x ) − a ]sec( n x )(1 − n ) , (29)where n = √ − K and K < We calculate in detail the components of Eqs. (14-15)using the Eqs. (25) and (29), and we find ρ E = C [3 − K + ( K −
1) sin x ]8 π K ( K −
1) cos x , (30) p E = C π K cos x (cid:20) x [ a tan( n x ) + b ](1 − K ) Ψ( x ) sec( n x ) + 1 (cid:21) , (31) where ρ E and p E are the Einstein energy density andpressure, respectively. When the Einstein metric com-ponents are inserted into the f ( R, T ) counterparts, weobtain ρ f = C (8 π + 6 πχ + χ ) (cid:20) (8 π + 3 χ ) [3 − K + ( K −
1) sin x ]8 K ( K −
1) cos x + χ [2 sin x [ a sin( n x ) + b cos( n x )] + (1 − K ) Ψ( x )]8 K (1 − K ) Ψ( x ) cos x (cid:21) , (32) p f = C (8 π + 6 πχ + χ ) (cid:20) χ [3 − K + ( K −
1) sin x ]8 K ( K −
1) cos x + (8 π + 3 χ ) [2 sin x [ a sin( n x ) + b cos( n x )] + (1 − K ) Ψ( x )]8 K (1 − K ) Ψ( x ) cos x (cid:21) . (33) χ=0.8 χ=1.0 χ=0.6 χ=0.8χ=1.0 Fig. 1. The energy density and pressure against radial coordinate r have been plotted for modified gravity model f ( R, T ) and in GR for comparison for some different values χ and fixed negative value of K = − .
4. For plottingwe consider χ = 0 ( dashed black curve for GR case) reveals that for a mass of M = 1 . M (cid:12) , the radius goes upto R = 8 .
849 Km, whereas χ = 0 . , . , . . . .
141 and 9 . C = 1 . × − km − . χ=0.4 χ=0.6χ=0.8χ=1.0 χ=0.0χ=0.4 χ=0.6 χ=0.8χ=1.0 Fig. 2. Gravitational mass (left diagram) and energy condition (right diagram) versus radial coordinate r havebeen plotted. For NEC is determined by the condition ρ + p > χ=0.0χ=0.4 χ=0.6 χ=0.8χ=1.0 χ=0.0χ=0.4 χ=0.6 χ=0.8χ=1.0 Fig. 3. The SEC and DEC against the radial coordinate r have been plotted on the left and right panels for somevalues of χ . Of course, the physical form of the WEC is straightforward: total energy density of matter field ρ > ρ + p > Now, we study for relativistic compact stars depend-ing on two parameters K and χ , and keeping the otherparameters same for f ( R, T ) gravity and as of GR (when χ → ρ , the ra-dial pressure p and the mass m/M (cid:12) as a function of theradial coordinate are presented in Fig. (1-3) for differentvalues of χ . Finally, we move on to describe the resultsobtained from our calculations, which are illustrated inFigs. (1-2), for the GR case ( χ = 0) and for the f ( R, T )gravity ( different values of χ ). As one can see from Fig. 1that the pressure (energy density) inside the star is pos-itive and monotonically decreasing function towards theboundary, and reaches the value zero on the star surface.The mass-radius relation is represented in Fig. 2 (leftpanel). Note that if the radius is increasing with themass, the M/R ratio is also increasing, but much slowerfor the greater values of χ . But it is worth noticing thatthe maximum mass attended from standard GR for χ = 0as evident in Table 1. As a result GR has profound ef-fects on the critical mass of compact stars. For the caseof modified gravity the effect caused by the term 2 χT which affects the structure of the star. The stellar con-figurations have been analyzed for several values of theimportant physical parameters in the tabular format (seeTable 1). All properties of this class of stars have beenobtained by solving the system of Eqs. (30-33) for the χ =0, 0.4, 0.6, 0.8 and 1, respectively with K = − . M = 1 . M (cid:12) , the pressure value zero on the star sur-face goes as high as 8 . . .
141 and 9 . χ i.e. when the radius of starincreasing. One can see from Table 1 that the deviationof the central density and pressure from GR, in princi-ple, is higher than the f ( R, T ) model. In the plots thedashed black curve for GR case, while the other curvesrepresent the modified gravity throughout this work. Fi-nally, we report the results for the model obtained in[68]. It can be said that normal GR conditions ( χ = 0and K < etc , values of χ and K < f ( R, T )-gravity.In order to go further we discuss energy conditionsand physical quantities, respectively. In the case ofenergy conditions according to classical field theoriesof gravitation, we have analyzed the properties of thenull energy (NEC), weak energy (WEC), strong energy(SEC) and dominant energy conditions (DEC), respec-tively. It is interesting to point out that all conditionshold simultaneously in the framework of modified andclassical gravity, as evident in Fig. (2-3). We have usedthe same parameters values as indicating in Fig. 1.A more physical model should automatically accountfor sound speed for perfect fluid distribution. It is obvi-ous that the velocity of sound is less than the velocity oflight i.e. 0 < v = dp/dρ < For our stellar model the speed of sound is given by (for both cases) (cid:18) dpdρ (cid:19) E = (1 − K ) (cid:2) f ( x ) − a (cid:0) sin(2 n x ) v ( x ) + 2 Ω f ( x ) (cid:1)(cid:3) cot x cos x [ K − K −
1) cos(2 x )] (cid:2) sin( x )[ b cos( n x ) + a sin( n x )] + v ( x ) (cid:3) , (34) (cid:18) dpdρ (cid:19) f = (8 π + 3 χ ) [2 tan x v ( x ) − Ω f ( x )] + 2 χ tan x − χ Ω f ( x ) + 2 χ tan x Ω f ( x ) − χ Ω f ( x ) + (8 π + 3 χ ) [2 tan x − Ω f ( x ) + 2 tan x Ω f ( x )] + 2 tan x χ v ( x ) . (35) Fig. 4. The velocity of sound versus radial coordinate r have been plotted for χ = 0 (dashed black curve) χ = 0 . χ = 0 . χ = 0 . χ = 1 (yellowcurve). In our analytical approach, we use graphical repre-sentation to represent the velocity of sound due to com-plexity of the expression. In this case, we clearly observeform Fig. 4 that velocity of sound is decreasing away from the centre. Note that for GR case the sound speedis more closer to the velocity of light, but for this sit-uation is more realistic for f ( R, T ) gravity. The usedcoefficients in above equations are given in Appendix. < K < i.e. K is positive Here, we will report the solution for 0 < K <
1. As already mentioned that the energy density for Buchdahalmodel in GR is ˜ ρ = C ( K − Cr )32 π (1+ Cr ) . Thus, it seems interesting that within the limit of 0 < K <
1, the energy densityis negative due to the presence of ( K −
1) term. Therefore, given the value of K , obtained solution is not physicallyvalid in GR. Now, the density expression for f ( R, T ) gravity could be determined from Eq. (18), which involves thepressure term also. On the other hand, pressure term involves the metric potential e ν and e λ . However, in determin-ing the metric potential ν by solving of hypergeometric Eq. (26) we have to use the transformation ξ = (cid:113) K + Cr K − . Itis worth noticing that the transformation is not valid under the square root of 0 < K <
1, as there is no real solution.This implies that the pressure will not be physically valid. Therefore, such an analysis for GR and f ( R, T ) gravity,however, is not physically valid.
K > i.e. K is positive In order to conduct further investigations, we extend our analysis for positive values of K . With the purpose ofsolving the Eq. (26), we substitute d Ψ dξ = Y , and ξ = cosh y . Thus, we have d Ydy + ( K − Y = 0 , for K > , ξ = cosh y. (36)This is a homogeneous differential equation of second order with constant coefficients. Now, we will classify thesolution to the following cases Case IIIa: Y ( y ) = a cos( m y ) + b sin( m y ) , if K − m and K > . (37)Case IIIb: Y ( y ) = c cosh( m y ) + d sinh( m y ) , if 2 − K = m and 1 < K < . (38)Case IIIc: Y ( y ) = e + f y, if 2 − K = 0 . (39)where a , b , c , d , e and f are arbitrary constants of integration with y = cosh − ξ = cosh − (cid:113) K + Cr K − . Forsimplification we substitute d Ψ dξ = Y and d Ψ dξ = dYdy dydξ into hypergeometric Eq. (36), and we get Ψ as,Case IIIa : Ψ( y ) = [ a cos( m y ) + b sin( m y )] + m tanh y [ a sin( m y ) − b cos( m y )]( K − sechy for m = √ K − K > , (40)Case IIIb: Ψ( y ) = m tanh x [ c sinh( m y ) + d cosh( m y )] − [ d sinh( m y ) + c cosh( m y )](1 − K ) sech y for m = √ − K and 1 < K < , (41)Case IIIc: Ψ( y ) = f [ y cosh y − sinh y ] + e cosh y, for K = 2 . (42)where, Ψ ( x ) = cosh( m y ) + d sinh( m y ). Recalling the Eqs. (14) and (15) and plugged into the relevant equationwe obtain the expression of energy density and pressure corresponding to Einstein and f ( R, T )- gravity for threeseparate cases, as follows: 4.3.1
Case IIIa: K − m , and K > f ( R, T ) models with constant χ (cid:54) = 0. Thus,positive K -value provides a further set of expression for pressure and density, which are given by, ρ E = C [3 − K + ( K −
1) cosh y ]8 π K ( K −
1) sinh y , (43) p E = C π K sinh y (cid:20) a cos( m y ) + b sin( m y )]( K −
1) Ψ( y ) sechy − (cid:21) , (44) ρ f = C (8 π + 6 πχ + χ ) (cid:20) (8 π + 3 χ )[3 − K + ( K −
1) cosh y ]8 K ( K −
1) sinh y + χ K sinh y (cid:18) a cos( m y ) + b sin( m y )]( K −
1) Ψ( y ) sech y − (cid:19)(cid:21) , (45) p f = C (8 π + 6 πχ + χ ) (cid:20) χ [3 − K + ( K −
1) cosh y ]8 K ( K −
1) sinh y + (8 π + 3 χ )8 K sinh y (cid:18) a cos( m y ) + b sin( m y )]( K −
1) Ψ( y ) sech y − (cid:19)(cid:21) . (46)The sound speed index is given by (cid:18) dpdρ (cid:19) E = ( K −
1) tanh y [ a cos( m y ) + b m y )] [2 v ( y ) + Ω E ( y ) (cid:3) [9 − K + ( K −
1) cosh(2 y )] (cid:2) v ( y ) + coth y (cid:0) a cos( my ) + b sin( my ) (cid:1) ] , (47) (cid:18) dpdρ (cid:19) f = χ (cid:0) y csch y − f ( y ) (cid:1) − (8 π + 3 χ ) v ( y ) − coth( y ) v ( y )(8 π + 3 χ ) (cid:0) y csch y − f ( y ) (cid:1) − χ v ( y ) − coth( y ) v ( y ) . (48)In this framework, let us now discuss the stellar struc-ture with aim to study the physical validity and stabilityof the system under the f ( R, T ) gravity. The energy den-sity and pressure versus radial distance from the centerof the star have been plotted for each χ are depicted inFig. 5. As one can see that pressure and density for bothEinstein and f ( R, T ) model are maximum at the cen-ter and decreases monotonically towards the boundary.Fig. 5 confirms a well-behaved positive definite density.With the choice of the free parameters and dependingon matter content, one can increase and decrease the ra-dius of the stellar structure. For illustrating we assume that M = 1 . M (cid:12) , the constant C = 2 . × − km − .According to the results we observe that for χ = 0 theradius goes upto R = 8 .
849 Km, whereas χ = 0 . , . , . . . .
327 and9 . p ( r = R ) = 0. In Table 2, we showthe central density, central and surface pressure againstthe total radius for some different values of χ . Note thatthe energy density and pressure are in the same order ofmagnitude only near the centre of the star.In order to analyze the mass function (21), we inte-grate the system of equation considering many different χ=0.0 χ=0.4χ=0.6χ=0.8χ=1.0 χ=0.0χ=0.4 χ=0.6χ=0.8χ=1.0 Fig. 5. The energy density (left panel) and pressure (right panel) diagram in model f ( R, T ) = R + 2 χT and in GRfor compact stars with K = 3, M = 1 . M (cid:12) and C = 2 . × − km − . We find that for χ = 0 ( dashed black curvefor GR case) the radius goes upto R = 8 .
849 Km, whereas χ = 0 . , . , . . . .
327 and 9 . χ=0.4 χ=0.6χ=0.6χ=1.0 Fig. 6. Gravitational mass m ( r ) and Null energy condition (right diagram) versus radial coordinate r have beenplotted. The dashed black curves are the solutions of GR case, while the others for f ( R, T ) model. From a rapidinspection of these plots, the differences between GR and f ( R, T ) gravitational mass are clear and the tendency isthat at larger radius GR takes more masses.
Fig. 7. This diagram is for SEC and DEC against the radial coordinate r for different chosen values of χ .1-10hysics of Dark Universe Vol. xx, No. x (2019) xxxxxx χ=0.0 χ=0.4χ=0.6 χ=0.8 χ=1.0 Fig. 8. The sound speed obtained for parameters as on Fig. 6. values of χ . We omit the mass expressions as they areextremely lengthy. Fig. 6 shows the behavior of the totalmass, normalized in solar masses versus the radius of thestar. We find that variation in the parameters bring asignificant changes to central density and pressure, with-out bringing any huge effect to mass-radius relation, seeTable 2. From Fig. 6 and Table 2, we observe that themaximum gravitational mass M = 1 . M (cid:12) attended fromstandard GR when χ = 0 with radius R = 8 .
849 km.It is noteworthy from Table 2 that the modified effectsof pressure on mass density and self gravity of the starare extremely important when one seeks for high mass configurations of compact stars. At high mass, all thevariations of χ become gradually ineffective and finallyoverlap with the normal GR solutions. This situation isalso seen in [68].Analyzing the Figs. 6 and 7, our investigation showsthat the energy conditions in both cases are well behavedinside the star. In Fig. 8, one can see the sound speedin both cases are satisfied. Surprisingly the maximumvelocity, as seen from the models, is lowest for ordinaryGR (when χ =0).The value of parameters are enlisted inFigs. 5.4.3.2 Case IIIb: − K = m , and < K < < K <
2, and compare the obtained results with
K >
2. Under these assumptions,Eqs. (41) and (26) lead to the following set of equations ρ E = C [3 − K + ( K −
1) cosh y ]8 π K ( K −
1) sinh y , (49) p E = C πK sinh y (cid:20) c cosh( m y ) + d sinh( m y )]( K −
1) sech y Ψ( y ) − (cid:21) . (50) ρ f = C (8 π + 6 πχ + χ ) (cid:20) (8 π + 3 χ )[3 − K + ( K −
1) cosh y ]8 K ( K −
1) sinh y + χ K sinh y (cid:18) c cosh( m y ) + d sinh( m y )]( K −
1) sech y Ψ( y ) − (cid:19)(cid:21) , (51) p f = C (8 π + 6 πχ + χ ) (cid:20) χ [3 − K + ( K −
1) cosh y ]8 K ( K −
1) sinh y + (8 π + 3 χ )8 K sinh y (cid:18) c cosh( m y ) + d sinh( m y )]( K −
1) sech y Ψ( y ) − (cid:19)(cid:21) . (52)and the sound speed index inside the fluid is given by (cid:18) dpdρ (cid:19) E = 2 (1 − K ) (cid:2) Ω E ( y ) + c m cosh( m y ) csch y sinh( m y ) − d Ω E ( y ) (cid:3) csch y [9 − K + ( K −
1) cosh(2 y ) coth y ] [Ω ( y )] , (53) (cid:18) dpdρ (cid:19) f = χ v ( y ) + (8 π + 3 χ ) [Ω f ( y ) + Ω f ( y )] − coth y v ( y )(8 π + 3 χ ) v ( y ) + χ [Ω f ( y ) + Ω f ( y )] − coth y v ( y ) . (54)For the study of compact objects, Fig. 9 depictsthe energy density and isotropic pressure, respectively.We find that density and pressures are monotonicallydecreasing functions of the radial variable r which is expected from Buchdahal model of GR [68]. Comparing f ( R, T ) gravity with the previous model (
K > χ=0.0χ=0.4χ=0.6χ=0.8χ=1.0 χ=0.0χ=0.4χ=0.6χ=0.8χ=1.0
Fig. 9. The energy density and pressure diagram in model f ( R, T ) gravity and in GR for compact stars with K = 1 . M = 1 . M (cid:12) and C = 1 . km − . We find that for χ = 0 ( dashed black curve for GR case) the radiusgoes upto R = 8 .
849 Km, whereas χ = 0 . , . , . . . .
632 and 9 . χ=0.0χ=0.4χ=0.6χ=0.8χ=1.0 χ=0.0χ=0.4 χ=0.6 χ=0.8χ=1.0 Fig. 10. Variation of mass (top) and null energy condition (bottom) versus radial coordinate r have been plotted.We have used the same data set as of Fig. 9. χ=0.0χ=0.4 χ=0.6 χ=0.8χ=1.0 χ=0.0χ=0.4 χ=0.6 χ=0.8χ=1.0 Fig. 11. This diagram is for SEC and DEC against radial coordinate r for different chosen values of χ .1-12hysics of Dark Universe Vol. xx, No. x (2019) xxxxxx χ=0.0χ=0.4χ=0.6χ=0.8 χ=1.0 Fig. 12. The sound speed obtained for parameters as on Fig. 9. point of view, the difference between energy density atcentre and surface are of order (cid:118) magnitude onlynear the centre of the star (see Table 3). Comparingboth the Table 2 and 3, our conclusion is that for agiven radius with 1 < K < ∼ gm/cm . Theresults reported for a particular mass (in normalizedform) M = 1 . M (cid:12) with the constant C = 1 . km − .For larger value of χ = 1 the convergence is slower andthe radius goes upto 9 . K > χ . We furthermore recall the standard results of GR from our previous pa-per [68], and its solutions for K = 1 .
78, which can becompared to the analogous solutions coming from themodified gravity. It is possible to demonstrate that den-sity radial profiles coming from f ( R, T ) gravity analyticmodels and close to those coming from GR are compat-ible. The results reported in Fig. 10 and Fig. 11 areobtained for different values of χ , for the energy condi-tions. In figures it is observed that all energy conditionsare obeyed. The variation of square of sound speed isdisplayed in Fig. 12. It is also evident that the squareof sound speed is less than unity throughout the stellarinterior.4.3.3 Case IIIc: − K = 0We next analyze the result for K = 2. The density and pressure for the Einstein as well as f ( R, T ) gravity aregiven by ρ E = C [3 − K + ( K −
1) cosh y ]8 π K ( K −
1) sinh y , (55) p E = C π (cid:20) y ( f y + e ) − ( K −
1) Ψ( y ) K ( K −
1) sinh y Ψ( y ) (cid:21) , (56) ρ f = C (8 π + 6 πχ + χ ) (cid:20) (8 π + 3 χ )[3 − K + ( K −
1) cosh y ]8 K ( K −
1) sinh y + χ K (cid:18) y ( f y + e ) − ( K −
1) Ψ( y )( K −
1) sinh y Ψ( y ) (cid:19)(cid:21) , (57) p f = C (8 π + 6 πχ + χ ) (cid:20) χ [3 − K + ( K −
1) cosh y ]8 K ( K −
1) sinh y + (8 π + 3 χ )8 K (cid:18) y ( f y + e ) − ( K −
1) Ψ( y )( K −
1) sinh y Ψ( y ) (cid:19)(cid:21) . (58)The square of the acoustic speed dp/dρ becomes (cid:18) dpdρ (cid:19) E = 2 [2 (2 − K ) f ( e + f y ) coth y + Ω ( y )][9 − K + ( K −
1) cosh(2 y )] coth y csch y v ( y ) , (59) (cid:18) dpdρ (cid:19) f = 2 χ [( K −
1) coth y − ( y )] + (8 π + 3 χ )[ v ( y ) + 2 coth y Ω f ( y )]2 (8 π + 3 χ ) [( K −
1) coth y − ( y )] + χ v ( y ) + 2 χ coth y Ω f ( y )] . (60) χ=0.0χ=0.4 χ=0.6 χ=0.8χ=1.0 χ=0.0χ=0.4 χ=0.6 χ=0.8χ=1.0 Fig. 13. The energy density and pressure versus radial coordinate r have been plotted for different values of χ andfixed positive value of K = 2. For plotting we consider χ = 0 ( dashed black curve for GR case) reveals that for amass of M = 1 . M (cid:12) , the radius goes upto R = 8 .
849 Km, whereas χ = 0 . , . , . . . .
491 and 9 . C = 0 . km − . χ=0.0χ=0.4χ=0.6χ=0.8χ=1.0 χ=0.0 χ=0.4 χ=0.6 χ=0.8 χ=1.0 Fig. 14. Mass-radius relations in the GR and in the f ( R, T ) gravity are shown. The NEC (right diagram) isdetermined by the condition ρ + p >
0. We consider the same data set as of Fig. 13. χ=0.0 χ=0.4 χ=0.6 χ=0.8 χ=1.0 χ=0.0 χ=0.4 χ=0.6 χ=0.8 χ=1.0
Fig. 15. Variation of the SEC and DEC versus radial coordinate r have been plotted for χ = 0 . , . , . χ=0.0χ=0.4χ=0.6χ=0.8 χ=1.0 Fig. 16. Variation of square of the sound velocity v versus radial coordinate r for χ = 0 (dashed black curve) χ = 0 . χ = 0 . χ = 0 . χ = 1(yellow curve). Finally, we move on to describe the results obtainedfrom our calculations, which are illustrated in Fig. 13,for K = 2 and for different values of χ . We notice thatthe solution gives positive definite density and pressurewhich is also monotonic decreasing towards the bound-ary. In Fig. 14, we have depicted the variations of masswith respect to the distance from the center of a star. No-tice that the differences between GR and f ( R, T ) gravi-tational mass is clear from the figures and the tendencyis that at progressively decreasing values of χ , f ( R, T )model acquire more mass but always less than GR. Con-sidering Fig. 13 and 14, and the above Eqs. (55-58),our results are given in Table 4. According to Fig. 15,we observe that all energy conditions are satisfied. So,our assumption for star model introduced in this paperis suitable. Furthermore, the speed of sound is definedthrough the Eqs. (59-60), and presented at Fig. 16. Wesee from this figure that the speed of sound is less thanunity throughout the star. In conclusion we explore allthe mathematical and graphic analysis of our study. Insuch situations, a couple of comments are necessary. Re-garding the numerical values tabulated in Table (1-4),however, the results indicate that GR dominates all theresults obtained in the f ( R, T ) scenario. As consequence,in the framework of GR, the objects present a denser corethan in the f ( R, T ) theory, and therefore greater valueof gravitational redshift Z s do exists. In summary, fromTables (1-4), the tendency of the main physical param-eters that describe the micro-physical behavior (centralenergy-density, surface energy-density, central pressure)of the system tend to decreasing in magnitude when theparameter χ increase its value. To conclude our study, we highlight the most impor- tant results from the obtained solutions. First of all, itis worth mentioning that Buchdahl model [59] describ-ing a static spherically symmetric spacetime driven by anisotropic matter distribution is extended into the frame-work of modified gravity theories, specifically f ( R, T )gravity. To accomplish it, we have considered to mainingredients. The first one is f ( R, T ) = R + 2 χT ı.e. alinear function in both R and T , where χ is a dimension-less running parameter. The second one refers to theform of energy-momentum tensor T µν . In this respectwe have taken a perfect fluid matter distribution. Thefoundations of the mentioned elections lies on the fol-lowing statements: i) the linear f ( R, T ) function allowsa more tractable mathematical treatment, additionallyas pointed out earlier that extra piece 2 χT can be con-solidated as a running cosmological constant. ii) Ein-stein gravity theory and f ( R, T ) gravity share the sameisotropic condition (20). In other words, any perfect fluidsolution to Einstein field equations is also a solution inthe arena of f ( R, T ) gravity. Of course, it is only fromthe mathematical point of view, since the material con-tent is quite different. Precisely, this feature is the keystarting point to address cosmological issues from theperspective of compact structure within the astrophysi-cal framework. All things mentioned above allow us toexamine the viability of Buchdahl models in the frame-work of f ( R, T ) gravity and compare the results withpure GR solutions. In this respect, in our previous arti-cle, we examine the Buchdahl model for anisotropic fluidsphere [68] and showed that one can obtain an analyticsolution to the Einstein equations for positive and nega-tive values of Buchdahl parameter K .So, depending on the parameters of the model K and χ , we analyse the configuration from χ = 0 to χ reason-ably high. As applied to compact non-rotating stars, thefield equations are solved by applying Gupta-Jasim two f ( R, T ) gravity and GR for C = 1 . × − Km − , mass = 1 . M (cid:12) and K = − . χ . χ Radius Central pressure Central density Surface density M − R ratio Surface red R ( km ) p effc ( dyne/cm ) ρ effc ( gm/cm ) ρ effs ( gm/cm ) MR -shift ( Z s )0.0 8.849 6.86953 × × × × × × × × × × × × × × × Table 2. Comparative study of physical values of the compact star in f ( R, T ) gravity and GR for C = 2 . × − Km − , mass = 1 . M (cid:12) and K = 3 for different χ . χ Radius Central pressure Central density Surface density M − R Ratio Surface R ( km ) p effc ( dyne/cm ) ρ effc ( gm/cm ) ρ effs ( gm/cm ) MR redshift ( Z s )0.0 8.849 1.42038 × × × × × × × × × × × × × × × Table 3. Comparative study of physical values of the compact star in f ( R, T ) gravity and GR for C = 1 . Km − ,mass = 1 . M (cid:12) and K = 1 . Km for different χ . χ Radius Central pressure Central density Surface density M − R Ratio Surface red- R ( km ) p effc ( dyne/cm ) ρ effc ( gm/cm ) ρ effs ( gm/cm ) MR shift ( Z s )0.0 8.849 5.26129 × . × × × . × × × . × × × . × × × . × × Table 4. Comparative study of physical values of the compact star in f ( R, T ) gravity and GR for C = 0 . × − Km, mass = 1 . M (cid:12) and K = 2 for different χ . χ Radius Central pressure Central density Surface density M − R ratio Surface red- R ( km ) p effc ( dyne/cm ) ρ effc ( gm/cm ) ρ effs ( gm/cm ) MR shift ( Z s )0.0 8.849 3.46718 × × × × × × × × × × × × × × × step method (see review for details discussion [66, 68]).Here we report some progress in this direction: we derivemodified field equations and address issues regarding thechoice of χ . This has been done for two classes of models:the negative K and positive K . The obtained results aresummarized as follows: • In our analytical approach for
K <
0, we foundthat our stellar model is free from any geometricalsingularities. In figs. 1-4, we have shown all thephysical criteria for stellar structure as describedin the introduction. The results of calculations aregiven in Table 1. We found that central energydensity and surface density is much consistent forboth GR and modified gravity with same magni-tude. Also Table 1 exhibits the central pressure,the surface redshift and the mass-radius ratio forour predicted values. We find that the radius of thestar increase as the coupling parameter χ increases,but the maximal mass limit exists for χ = 0. Inconclusion, taking χ = 0 i.e. for GR the objectbecomes more compact, dense and massive. • In next, we have analysed the model for
K > χ = 0). Independently from Table2 one can see the decreasing value of the centraland surface density, central pressure, surface red-shift and the value of 2 M/R , with the increasingvalue of χ . We note here that variations in theparameters does not effect huge to mass-radius re-lation, however, it increases monotonically towardsthe boundary, as of Fig. 6. • For further precision we analyzed the model for1 < K <
2. The mass-radius relation and the varia-tion of density and pressure with radius of the starfor this f ( R, T ) model are depicted in Figs. 9-12.It is worth mentioning that the difference betweencentral density and surface density is very high (inTable 3), which is also higher than any other com-parative model. In the case χ = 0, we have thecentral pressure is about 5.7 × orders of mag-nitude which is larger than the χ = 1 for 4.7 × .It is clear that the masses and radii of the starschange with the increasing values of χ . • Finally, in Figs. 13-16 and Table 4 we show the ef-fects of f ( R, T ) theory in compact star propertiesobtained for K = 2. As evident form Fig. 14 that,for relatively small value of χ the model acquiremore masses but these value is less than GR. Ana-lyzing the maximum mass and its respective radiusfound in each curve we determine that these val- ues could change from 2% to 10%. Also, for smallvalue of χ , the velocity of sound is less than unity.For the sake of comparison with the results in theliterature, we arrive at conclusion that among the fourmodels, it is notable that, for 1 < K <
2, we obtainthe maximum density of order 10 gm/cm and pressure10 dyne/cm of stars. Our considerations show that fordescribing χ , the total gravitational mass increases andreach at maximum when χ = 0 i.e. GR case. The radii ofspheres are around around nine Km for realistic valuesof χ . It appears that self gravity has more pronouncedeffect on the gravitational mass because high mass con-figurations are obtained only when χ = 0. Furthermore,we see that the general trend predicts a larger radiusof a compact star corresponding to the higher valuesof χ . Increasing the values of χ > f ( R, T )-gravity is lowerthan the mass in GR. On the other hand, we concludethat the Buchdahal model gives a well behaved solutionfor all values of
K <
K > f ( R, T ) gravity theory. But unfortunately, both modelare not valid in the range of 0 < K <
Acknowledgments
S. K. Maurya acknowledge that this work is carriedout under TRC project-BFP/RGP/CBS/19/099 of theSultanate of Oman. F. Tello-Ortiz thanks the financialsupport by the CONICYT PFCHA/DOCTORADO-NACIONAL/2019-21190856, grant Fondecyt No.1161192, Chile and the projects ANT-1856 and SEM18-02 at the Universidad de Antofagasta, Chile.
A The expressions for used coefficientsin the above equations: Ω f ( x ) = (1 / b n sec x sin(2 n x )+cos ( n x ) [ − a b n +( a − b ) n tan x − b tan x ],Ω f ( x ) = 2 sin ( n x )[ − b n + a tan x + b n tan x tan( n x ) − n tan x (2 b + a tan( n x ))] + 2 n tan x [tan x (cid:0) b + a tan( n x ) (cid:1) + n (cid:0) a − b tan( n x ) (cid:1) ],Ω f ( x ) = n [2 a b cos(2 n x ) + ( a + b ) n sin(2 x ) + ( a − b ) sin(2 n x )] , Ω f ( x ) = sin x [ b cos( n x ) + a sin( n x )] + n cos x [ a cos( n x ) − b sin( n x )],Ω f ( x ) = 2 cos( n x ) sin x [ b + a tan( n x )] , Ω f ( x ) = Ω f ( x )[Ω f ( x )] , Ω f ( x ) = [ − K +( K −
1) cos(2 x )] tan( x )cos x ( K − , Ω f ( x ) = Ω f ( x )Ω f ( x ) , Ω f ( x ) = (3 − K +( K − x )( K −
1) cos x ) , Ω E ( y ) = [ (cid:0) − m +(1+ m ) cosh(2 x ) (cid:1) coth( x ) (cid:0) a cos( m x )+ b m x )] , Ω ( y ) = [ a cos( m y ) + b sin( m y )],Ω ( y ) = m [ a sin( my ) − b cos( my )] , Ω ( y ) = cosh y Ω ( y ) + Ω ( y ) sinh y, Ω ( y ) = [ b m cos( m y )+ a cos( m y ) coth y − a m sin( m y )+ b coth y sin( m y )] sinh y, Ω f ( y ) = (3 − K +( K − y )( K −
1) sinh y tanh y , Ω f ( y ) = (1+ m )Ω ( y )Ω ( y )[Ω ( y )] ,Ω f ( y ) = ( m +csch y )Ω ( y )+coth y Ω ( y )Ω ( y ) , Ω f ( y ) = Ω ( y )Ω ( y ) , Ω E ( y ) = cosh ( my ) [ − c d m csch y + coth y (cid:0) d m + c ( − m ) − c csch y (cid:1) ] , Ω E ( y ) = (1 /
2) csch y coth y sinh( m y ) [1 − m + (1 + m )cosh(2 y )] [2 c cosh( m y )+ d sinh( m y )]+ m [ − c sinh ( m y )+ d sinh(2 m y )] , Ω ( y ) = m sinh y [ d cosh( m y ) + c sinh( m y )] − cosh( y )[ c cosh( m y ) + d sinh( m y )] , Ω ( y ) = cosh( m y )[ d m coth y + c ( m − csch y )]+ ( c m coth y + d [ m − csch y )] sinh( m y ) , Ω ( y ) = ( m −
1) sinh y [cosh( m y )( d m + c coth( y )) +( c m + d coth y ) sinh( m y )] [ c cosh( m y ) + d sinh( m y )] , Ω ( y ) = cosh( m y )( d m + c coth y ) + ( c m + d coth y ) sinh( m y ) , Ω f ( y ) = Ω ( y ) / Ω ( y ) , Ω f ( y ) = Ω ( y ) / (Ω ( y )) ,Ω f ( y ) = Ω ( y ) / Ω ( y ),Ω ( y ) = f ( e + f y )csch y − coth y [ e + 2 e f y + f (2 − K + y ) + ( e + f y ) csch y ] + ( K −
2) ( e + f y ) coth y, Ω ( y ) = [3 − K + ( − a ) cosh( y ) ] coth( y ) csch y, Ω ( y ) = [( K − e + f y ) cosh y − ( K − f sinh y ] , Ω ( y ) = ( e + f y ) cosh y − f sinh y, Ω ( y ) = [2 f cosh y − ( K − e + f y ) sinh y ] , Ω f ( y ) = ( e + f y ) sinh y Ω ( y ) / [Ω ( y )] , Ω f ( y ) = Ω ( y ) / Ω ( y ) , Ω f ( y ) = Ω ( y ) / Ω ( y ) .v ( x ) = ( a n + 3 b n tan x + 2 b tan x ), v ( x ) = n cos x [ a cos( n x ) − b sin( n x )], v ( x ) = 2 tan x (cid:0) − Ω f ( x ) (cid:1) , v ( y ) = b m cos( m y ) + 2 a m sin( m y ), v ( y ) = [ χ Ω f ( y ) tanh y + (8 π + 3 χ ) Ω f ( y )], v ( y ) = [(8 π + 3 χ ) Ω f ( y ) tanh y + χ Ω f ( y )], v ( y ) = [2 coth y csch y − f ( y )], v ( y ) = [ χ Ω f ( y ) + (8 π + 3 χ ) Ω f ( y )], v ( y ) = [(8 π + 3 χ ) Ω f ( y ) + χ Ω f ( y )], v ( y ) = (cid:0) Ω f ( y ) + Ω f ( y ) (cid:1) , v ( y ) = [( e + f y ) cosh y − f sinh y ] ,v ( y ) = [Ω f ( y ) + Ω f ( y )] . References et al. [Supernova Search Team], Astron. J. (1998) 1009.2 S. Perlmutter, et al. [Supernova Cosmology Project Collabora-tion], Astrophys. J. (1999) 565.3 P. de Bernardis, et al. [Boomerang Collaboration], Nature (2000) 955.4 S. Hanany, et al.
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