Existence of Wormhole Solutions and Energy Conditions in F(R, T) Gravity

J. Astrophys. Astr. (2018) 39:69 © Indian Academy of Sciences https://doi.org/10.1007/s12036-018-9559-9

Existence of wormhole solutions and energy conditions in f (R, T) gravity

M. Z. BHATTI1, Z. YOUSAF1,∗ and M. ILYAS2

1 Department of Mathematics, University of the Punjab, Quaid-i-Azam Campus, Lahore 54590, Pakistan. 2Centre for High Energy Physics, University of the Punjab, Quaid-i-Azam Campus, Lahore 54590, Pakistan. ∗Corresponding author. E-mail: [email protected]

MS received 29 August 2018; accepted 3 October 2018; published online 23 November 2018

Abstract. This paper is devoted to investigate the spherically symmetric wormhole models in f (R, T ) gravity, where T and R are trace of stress energy tensor and the Ricci scalar, respectively. In this context, we discuss three distinct cases of fluid distributions viz, anisotropic, barotropic and isotropic matter contents. After considering the exponential f (R, T ) model, the behavior of energy conditions are analyzed that will help us to explore the general conditions for wormhole geometries in this gravity. It is inferred that the usual matter in the throat could obey the energy conditions but the gravitational field emerging from higher order terms of modified gravity favor the existence of the non-standard geometries of wormholes. The stability as well as the existence of wormholes are also analyzed in this theory.

Keywords. Gravitation—relativistic ﬂuids—self-gravitating systems.

1. Introduction describe the issue of the accelerating universe. In order to disclose the cosmic acceleration, along with distinct One of the greatest discoveries in modern physics is the interesting theories, f (R) modified gravity (where R is astonishing evidence of accelerating expansion of the the curvature scalar) has attracted many astrophysicists. universe established by different independent observa- Clifton et al. (2012) provided a comprehensive exposi- tional data-sets. The limits of general relativity (GR) tion of the distinct progress that have took place over have come into focus with the emergence of the dark the past few decades in the field of modified gravity. universe scenario. Due to which, GR requires modifi- There has been a lot of investigation over the last few cation on large distance scales and at late times in the decades in wormhole (WH) physics (Morris & Thorne universe. Many efforts have been carried out to investi- 1988; Morris et al. 1988). A WH is a topological fea- gate the mechanism behind this accelerated expansion. ture of spacetime connecting two distinct regions of In order to recognize the observed features of our uni- spacetime that would fundamentally be a shortcut. A verse, i.e, acceleration in the expanding cosmos as well WH may associate very short distances as few feet long as the structures of rotating curves of galaxies, the or very long distances such as billions of light years standard universal models (Friedmann-Lemaître uni- or more or different points in time. The exotic matter verse model, cold dark matter model, Bianchi models, is the fundamental element for wormhole geometries etc.) gained significance to describe both dark matter in the framework of GR for which energy conditions and energy. The idea of modifying GR on cosmologi- (ECs) are disobeyed while normal matter entertain these cal scales has really taken off over the last couple of conditions. Geometries, which involve inhomogeneous decades. Various candidates have been suggested in matter distributions, are supported by exotic matter literature to illustrate this process, setting from dark which violates null energy condition (NEC). It would energy models to modified gravity theories. Recently, be vital to minimize the usage of exotic form of mat- higher-order curvature gravity theories have experi- ter due to its unrealistic nature. Fukutaka et al. (1989) enced much attraction in relation with the probability found the WH solutions in a higher order theory which of giving rise to cosmological models in which one can includes the curvature squared term (R2) along with the 69 Page 2 of 11 J. Astrophys. Astr. (2018) 39:69 linear term, where R is curvature scalar. Various efforts (2017), Moraes & Sahoo (2017a,b)andMoraes et al. have been put forth to investigate the physically realistic (2017) studied few captivating features in the field of model describing the WHs in modified gravity theories f (R, T ) theory. like f (R) theory. Here, we are aiming to explore those WH solutions It is also observed that weak energy condition (WEC) which have no exotic matter by considering the behavior holds for certain dynamic WH solutions in the frame- of shape function. We examined the WECs and NECs work of non-linear electrodynamics (Arellano & Lobo to explore the suitable regions for WH solutions. In 2006). Lobo & Oliveira (2009) constructed WH geome- this regard, we consider three different fluid config- tries in the background of f (R) theory in such a way urations namely, anisotropic, barotropic and isotropic that it is the associate effective energy momentum fluids. The paper is outlined as follows. In Section 2,we tensor that fails to satisfy the NEC while the ordinary- review the basic concepts of f (R, T ) theory including matter energy momentum tensor satisfies it. They also the equations of motion. Also, we present the worm- explored distinct exact solutions by taking different hole geometries in the presence of three different fluid equations of state (EoS) and particular shape function. configurations with particular f (R, T ) model in this The violation of NEC via effective stress tensor may section. Section 3 is devoted to conclude our main allow the usual matter threading in the WH to sat- findings. isfy these constraints. Jamil et al. (2013) derived exact WH geometries in f (T ) theory and discussed particular cases with isotropic, barotropic and anisotropic 2. Modified field equations and WH geometry fluids and checked their viability through energy conditions. The f (R) theory can be modified by including In this section, we focus on the possibility that WHs can the contribution of matter contents as f (R, T ) theory be threaded by usual matter which satisfy the ECs in (Harko et al. 2011). In this theory, the cosmic accelera- the background of f (R, T ) theory. The central concept tion is due to the matter contents along with geometrical of this f (R, T ) theory is to use an algebraic general contribution of modified gravity whose cosmological function of Ricci as well trace of energy momentum and astrophysical applications have recently been stud- tensor in the standard Einstein-Hilbert action. It can be ied. Singh & Singh (2014) examined the influence of written as ( , ) √ f R T gravity on the evolution of universe and found = 4 − [ ( , ) + ], that this theory is capable to reproduce acceleration with S f (R,T ) d x g f R T L M the suitable form of modified model. here T, g are the traces of standard GR Harko et al. (2013) found that this is the extra energy-momentum tensor and the metric tensor, respec- curvature terms of f (R) gravity which support the tively while R is the Ricci scalar and L M is the matter WH geometries while the matter satisfies all the ECs. Langrangian. Choosing L M = ρ (where ρ is the sys- Capozziello et al. (2012) discussed the possibility for tem’s energy density) and making variation in the above the existence of WHs in hybrid metric-Palatini gravity equation with gαβ , the corresponding f (R, T ) field by exploring general conditions to violate the NEC at equations turns out to be the throat. They also studied some particular examples eff to support their investigation by using redshift function, Gμν = Tμν , potential as well as shape functions etc. Many authors where have worked on Bejarano et al. (2017); Mehdizadeh & eff (m) Tμν = (1 + fT (R, T ))Tμν − ρgμν fT (R, T ) Ziaie (2017); Mazharimousavi & Halilsoy (2016); Azizi (2013) the existence of WHs and energy conditions in f (R, T ) R − − f R(R, T ) various interesting scenarios. Alvarenga et al. (2013) R 2 tested particular f (R, T ) gravity models which satisfy 1 the ECs (which are worked out via the Raychaudhuri + ∇μ∇ν + gμν f (R, T ) R ( , ) equation for expansion) and found stable power-law and f R R T de-Sitter solutions for some values of the input parame- is the effective energy-momentum tensor showing ters. Yousaf et al. (2016); Bhatti (2016); Bhatti & Yousaf modified version of gravitational contribution emerging (2016, 2017); Yousaf (2017a,b) demonstrated the inho- due to f (R, T ) extra curvature ingredients while Gμν mogeneity factors of matter density for self-gravitating is the Einstein tensor. Also, ∇μ describes the covari- celestial stars evolving in the background of f (R, T ) ant derivative while fT (R, T ), , f R(R, T ) represent df(R,T ) , ∇ ∇ν df(R,T ) gravity and imperfects fluid configurations. Sahoo et al. dT μ and dR operators, respectively. J. Astrophys. Astr. (2018) 39:69 Page 3 of 11 69

We choose a static spherically symmetric geometry e−b(r) p = whose line element is of the form as t ( + ( , )) 1 fT R T a(r) b(r) 1 f (R, T ) ds2 = e dt2 − e dr2 − r 2 dθ 2 + sin2θdφ2 , r b(r) − a(r) + eb(r) − R 1 2 2 r ( ) here a r is an arbitrary function which calculate the 1 1 b(r) = + (a (r) − b (r)) + f R (R, T ) gravitational redshift of light particles while e 2 r −1 − β(r) β( ) ρ 1 r ,where r describes the geometry of 1 b(r) fT + e f R (R, T ) + f R (R, T ) − . (4) the WH called the shape function. The fundamental 2 (1 + fT ) properties of WHs impose the constraints on the red- The above field equations are quiet complicated to β−rβ > shift function as well as shape function that β2 0, explore the physical features of matter threading the called the flaring out condition while β (r0)<1where WH in this theory as there are three equations and six , ,ρ, , r0 is the throat radius of the WH solutions. Also, the unknowns namely, a b pr pt and f . Also, the − β > function f involves the contribution of matter via the shape function should satisfy the constraint 1 r 0 for the existence of WHs and β(r0) = r0 at the throat. trace of energy momentum tensor. One can choose a Further, the magnitude of the redshift function should particular form of f (R, T ) function to simplify these be constant in the entire region of the geometry. These equations as well as to separate the contribution of are the conditions in GR which make the usual matter to matter field from the geometric part, i.e., f (R, T ) = violate the WEC as well as NEC. We consider the mat- f (R) + f (T ). Further, we take f (T ) = λT ,whereλ is ter distribution to be anisotropic due to different stresses the coupling parameter, so that the field equations take having mathematical formulation as follows the form as 1 (2 + 5λ) = (ρ + ) − + , ρ = + λ + λ , Tμν pr VμVν pt gμν Xμ Xν (1) ( + λ) ( + λ) 1 2 2 3 (5) 2 1 2 1 − λ here ρ, pt , pr and are matter density, anisotropic 1 pr = 1 −(2 + 3λ) 2 + 2λ 3 , stress components and pr − pt , respectively. Further, 2(2λ + 1) (λ + 1) Xμ and Vμ are the fluid four vector and four velocity, (6) respectively.√ Mathematically, the quantity Xμ is defined −1 λ 1 = + λ − ( + λ) , as Xμ = g11δμ. Under comoving coordinate system, pt 1 2 2 1 3 μ μ 2(1 + 2λ) (1 + λ) these quantities obey V Vμ = 1andX Xμ =−1 (7) relations. Xμ is the unit four vector in radial direction. where −b(r) The modified field equations yield the following set 1 = e of equations for spherical metric as a (r) 1 1 a (r) − a(r)b(r) + a(r)2 + f ρ = −b(r) r R e 2 4 4 ( ) ( ) ( ) a r 1 1 2 a r b r 2 1 b(r) − a (r)b (r) + a (r) + + − × f R − e f R − f R , 2 4 4 r 2 r 2 b (r) 2 e−b(r) × f (R, T ) + − × f (R, T ) = R R 2 ( + λ) 2 r 1 ( ) ( ) 1 b(r) a r 1 1 2 b r − e f R (R, T ) − f R (R, T ) , (2) − − a (r)b (r) − a (r) + f R 2 2 4 4 r e−b(r) a (r) 2 p = + + r ( + ( , )) r 1 fT R T 2 ( ) ( ) a r 1 1 2 b r 1 b(r) − − a (r)b (r) − a (r) + × f R + e f R , 2 4 4 r 2 a (r) 2 e−b(r) f (R, T ) + + f (R, T ) = R R 3 ( + λ) 2 r 1 1 ρ f 1 f + eb(r) f (R, T ) − T , r b(r) − a(r) + eb(r) − R R (3) 1 2 2 (1 + fT ) 2 r 69 Page 4 of 11 J. Astrophys. Astr. (2018) 39:69 R 1 ( ) − ( ) + 1 − 6rαλR ) − 2e γ (−1 + α) γ 2 a r b r 2 r × 2λb + r (1 + 2λ) a + 2α(4γ (2 + 3λ) R 1 + + b(r) + . 2 f R e f R f R − 3r (2 + 3λ) R + γ b (2γλ 2 −r (2 + 3λ) R The ECs are utilized in variety of contexts to explore general results for distinct situations. These conditions + rγ (γ + 2γλ) a + 2 (2 + 3λ) R + R(2γ 2 can be evaluated via the Raychaudhuri equation for + R αλ + b γ 2 (− + α) γ ( + λ) expansion scalar which require the positivity of the 2 e r 1 1 α β α β expression Rαβ k k and leads to Tαβ k k ≥ 0. How- 2 − ebα 2λ + r 2γ (1 + λ) − rα(rγ 2 (1 + 2λ) a ever, in the background of modified theories, the usual − γ a (−4γ − 8γλ matter tensor turns out to be effective energy momentum tensor due to the contribution of extra curvature + rγ (1 + 3λ) b + 3r λR ingredients. The violation of EC imposes the condition + γ b (4γλ ef f α β T k k < 0, yielding ρef f + pef f < 0whichtakes αβ −r (2 + 3λ) R the following form + 2 (2γ (2 + 3λ) R − r (2 + 3λ) R 2 ef f 1 ρef f + = (ρ + )( + λ) pr pr 1 + rγ ((γ + 2γλ) a + (2 + 3λ) R , (10) f⎛R ⎞ − γ b+R β β − β γ 1 ⎝ r ⎠ e + 1 − f + f . (8) pr = R R β 4r 2γ 3 (1 + λ)(1 + 2λ) f R r 2r 2 1 − r R By making use of the modified field equations in the 4 −1 + eb (e γ (−1 + α) − α)γ 3λ above equation, we get R γ 2 ef f ef f 1 + rγ(e (−1 + α) γ ρ + pr = rβ − β . 3 r × ((r + 2rλ) a 2 − 4 (1 + λ) b Using the flaring out condition in the above equation, + r (1 + λ) a b + 2r (1 + 2λ) a ) we found that ρef f + pef f < 0 which indicates that the NEC is violated for the effective energy momentum ten- − α(rγ 2 (1 + 2λ) a 2 sor while the ordinary-matter energy momentum tensor + rγ a γ (1 + λ) b + 2 (2 + λ) R satisfies it, however, we have kept the usual matter to 2 satisfy the NEC. + 2(4γ (2 + 3λ) R + 3rλR Next, we take exponential form of f (R) model to + γ b (−2γ (1 + λ) continue our systematic analysis as follows (Astasheok et al. 2013) + rλR +rγ (γ + 2γλ) a − 2λR f (R) = R + αR (exp (−R/γ ) − 1) , (9) + R + R(2γ 2(−2αλ − eb γ r 2 (−1 + α) γ (1 + λ) here γ and α are arbitrary constants and this model ( ) correspond to quadratic f R model under the con- + ebα(2λ + r 2γ (1 + λ) straint γ>>R.Whenα = 0, this model coincides with the results of GR. The field equations, in the back- + rα(rγ 2 (1 + 2λ) a 2 ground of this exponential model takes the form as + γ b −4γ (1 + λ) +rλR + rγ a (γ − γ b+R e γ × (1 + λ) b + (2 + λ) R ) ρ = 4r 2γ 3 (1 + λ)(1 + 2λ) + 2 (2γ (2 + 3λ) R R 2 −4 −1 + eb (e γ (−1 + α) − α)γ 3λ + rγ + rλR + rγ (γ + 2γλ) a − λR , (11)

R × (−r e γ (−1 + α) − α γ 2 (1 + 2λ) a 2 and + γ a (αγ (4 + 8λ − (r + 3rλ) b ) γ b+R − γ R e b γ pt = −4 −1 + e + e (−1 + α) γ −4 − 8λ + (r + 3rλ) b 4r 2γ 3 (1 + λ)(1 + 2λ) J. Astrophys. Astr. (2018) 39:69 Page 5 of 11 69

R R × ( + λ) + α ( + λ) (e γ (−1 + α) − α)γ 3 (1 + λ) − e γ r (−1 + α) 1 2 12r 2 3

r0 m r0 m 3 − α ( + λ) − 3αγ − × γ 2b + a −2 − 4λ + rλb 15r0 2 3 mr 66rr0 r r 2m + rαγ(γa rγλb − 2(γ + 2γλ+ r (2 + λ) R ) 2 r0 × (2 + 3λ) + 69r0 (2 + 3λ) 2 r + − γ ( + λ) + ( + λ) m 2 4 1 R 3r 2 3 R 4 3 r0 +10r γ (2 + 3λ) − 4r r0 γ (7 + 10λ) , + γ b γ + r (2 + 3λ) R − 2rγ (2 + 3λ) R r (14) R b+ + ( γ 2(− γ 2 (− + α) γ r m R 2 e r 1 2mr 0 − 0 r + α + b − + 2γ ( + λ) −e r3γ r m 2 e 2 r 1 p = r 0 r 12γ 3 ( + λ)( + λ) 0 r 1 1 2 r + rα(4γ (1 + λ) R − 2r (2 + 3λ) R 2 ⎡ 2m m + γ a γ 2 + 4λ − rλb + r (2 + λ) R ⎣ 5 2 r0 r0 4m r0 −r + r0 αλ − γ b 2γ + r (2 + 3λ) R +2rγ (2 + 3λ) R ) . r r

(12) r m r m + m4r 0 α −24rr 0 For further investigation, we will keep the gravitational 0 r 0 r ( ) = redshift to be constant, i.e., a r c so that its deriva- r0 2m r0 m = + 24r 2 + 8r 4γ − 9r 3r γ λ tive vanishes (a 0). Also, we will discuss the energy 0 r 0 r conditions with three different cases of matter conﬁgu- ⎛ ⎞ r0 m rations as (i) Anisotropic Fluid; (ii) Isotropic Fluid; and 2mr0 r − 9γ 3 ( + λ) ⎝ r3γ (− + α) − α⎠ (iii) Barotropic Fluid. r 1 2 e 1 2.1 Anisotropic matter r 2m + m3α −36rr 2 0 λ 0 r In this case, we will continue our investigation of r0 3m r0 m exploring the viability regions of energy conditions with + 36r03 λ − 2r 7γ 2λ + 3r 6r0 γ 2λ r r the following choice of shape function. m + 4 r0 r0 m 1 + 2r r γ (4 + 31λ) β(r) = r , (13) 0 r r 2m 3 2 r0 so that the ﬁeld Eqs (10)Ð(12) take the form −2r r0 γ (4 + 33λ) r r m 2mr 0 2 3 4 − 0 r − m r αγ 2r γ (4 + 13λ) r3γ −e r0 m m ρ = mr 3 r0 12γ 3 ( + λ)( + λ) 0 − 2r r0 γ (3 + 14λ) r 1 1 2 r r r 2m r m m 2m 4 2 0 0 r0 2 r0 4m r0 −r + r0 α (2 + 3λ) − 6rr (4 + 19λ) + 3r (8 + 39λ) r r 0 0 r r m m 3 r0 r0 6 2 + m r α −24rr − mr αγ (12r (2 + 5λ) − r0 0 r 0 r ⎤ 2m m 2 r0 4 3 r0 r0 m ⎦ + 24r0 + 8r γ − 9r r0 γ (2+3λ) × (22 + 59λ)) , (15) r r r 2m 3m 2 2 r0 3 r0 r m + m α −36rr + 36r 2mr 0 0 r 0 r − 0 r e r3γ r m r m r 2m p = r 0 4 0 3 2 0 t 12 3 0 + 46r r0 γ − 50r r0 γ 2r γ (1 + λ)(1 + 2λ) r r r ⎡ ⎛ ⎞ r0 m r m 2mr0 r − 7γ 2 + 6 0 γ 2 ( + λ) 2r 3r r0 2 3 ⎣−r 9 ⎝e r3γ (−1 + α) −α⎠ γ 3 (1 + 2λ) ⎛ r ⎞ r0 m 2mr0 r 6 2 3 ⎝ 3γ ⎠ − r γ r e r (−1 + α) − α γ r0 2m r0 m + 8m5r 2 −r + r α (2 + 3λ) 0 r 0 r 69 Page 6 of 11 J. Astrophys. Astr. (2018) 39:69

Figure 1. Evolution of energy density with respect to r,α,γ.

m m 4 r0 r0 β( ) = / . + 2m r0 α (−24rr0 r r0 r0 r r r r 2m r m Now, let r = 1 and plot Eq. (14)forρ against (r,α,γ). + 2 0 + 4γ − 3 0 γ )( + λ) 0 24r0 8r 9r r0 2 3 We explore the plotted region where ρ>0asshownin ⎛r ⎛ r r0 m the left plot of Fig. 1. We see that for large values of r, 2mr0 r − mr6γ 2 ⎝r 3 ⎝e r3γ (−1 + α) one can have both positive as well as negative values of α so that the WEC holds. For very small values of r, i.e., 0 < r < 1, there are also some regions where WEC is −α) γ (1 + 2λ) + 24rα (3 + 5λ) valid as can be seen in the right plot of Fig. 1. We found r0 m that for 0 < r < 0.1, we can take any value of α and −2r0 α (41 + 67λ) r γ in f (R, T ) gravity to construct the WH regions with − 2m2r 3αγ 2r 4γ (12 + 19λ) usual matter distribution.

r m r m The validity of WEC imposes the condition on r − 6rr 0 (24+37λ) −r 3r 0 γ (33 + 50λ) 0 r 0 r having values in between 0 and 1. For the minimum 2m value of r = 1.14349, the WEC is valid for α = 2 r0 +3r0 (50 + 77λ) − . γ = . . r 20 291 and 9 12106 Similarly, for the minimum value of α =−100, the WEC is found to be r 2m + 2m3α −36rr 2 0 (2 + 3λ) valid by setting r = 1.65803 along with γ = 4.76133. 0 r Furthermore, the unit value of γ , the WEC is valid for 3m 3 r0 7 2 r = 2.50562 and α =−99.999. In a similar way, the + 36r0 (2 + 3λ) − 2r γ (2 + 3λ) r validity region for NEC are shown in Figs. 2 and 3 with m 6 r0 2 both some particular choices of λ. The minimum value + 3r r0 γ (2 + 3λ) r of r ﬁnds α =−20.291 and γ = 9.12106 for which m 4 r0 ρ + α =− + 2r r0 γ (48 + 73λ) pr is greater than zero, while for 100, the r ⎞⎤ values of r and γ are found to 1.65803 and 4.76133.

2m The condition ρ + pr > 0 provides γ = 1 along with 3 2 r0 ⎠⎦ − 2r r0 γ (52 + 79λ) . (16) r = . α =− . . r 5 06581 and 80 3193 We have plotted ρ, pr and pt for m = 0.5, r0 = 1,α=−0.1,γ= 5 and shown in Fig. 4. We observe that for small values It is worthy to note that one can take any shape of r, i.e., 0 < r < 1, the matter density and radial pres- function with particular value of m as studied in lit- sure have positive values (ρ>0andpr > 0), while erature. In order to satisfy the flaring out condition that the tangential pressure has negative value (pt < 0). It is the derivative of β is less than 1, one has to set m < 1. important to mention that one can have similar results Also, the constraint β(r0) = r0 and the condition of about the validity of ECs and the existence of WHs by asymptotically flat spacetime is trivially satisfied. Now, taking m = 1, − 1/2, 1/5. Figure 1 corresponds to we investigate the validity of NEC (ρ > 0) and WEC some specific values of ρ, say ρ0,andλ. All the conclu- (ρ + pr > 0,ρ+ pt > 0) for different shape functions sions drawn here are valid for ρ = ρ0. The same thing defined in Eq. (13). Initially, we take m = 1/2inEq. is done in Figs. 2 and 3, which are drawn some specific (13) so that the shape function turns out to be values of ρ + pr and ρ + pt . J. Astrophys. Astr. (2018) 39:69 Page 7 of 11 69

Figure 2. Evaluation of ρ + pr and ρ + pt with respect to r,α,γ with small λ .

Figure 3. Evolution of ρ + pr and ρ + pt with respect to r,α,γ with very small λ .

100 ρ

pr,

pt 50

0.0 0.2 0.4 0.6 0.8 1.0

Figure 4. Evolution of ρ, pr and pt for small regions.

2.2 Isotropic ﬂuid × (32 − 4eb(16 + 17r 2γ) Here,wetakeisotropicﬂuid(p = p = p or p − p = r t r t 2b 2 4 2 0), so that Eqs (11)and(12) yield + e 32 + 60r γ + 7r γ − 2r 2 −8 + eb 8 + 19r 2γ b ) −b + −b 1 − 2e −b 2 2e rb r2γ − r2γ 4α − + b e 4e r 4 e + 2γ + −b − γ ( + λ) 2(1 r b e rb ) 2+2e brb r 1 + rb −64e r2γ α − 64e r2γ α − 2+2e brb 4 5 4 + 9r 2γ b − 16e r2γ r 5αb − 2 −b 2+r2γ b+2e brb 2(1+r γ b+e rb ) 2γ 2γ 2 − + 128e r α − 32e r r αγ 2+2e brb r2γ 3 3 2b 2 2 − − − 4e r αb (−16 + 3e r γ(1 + r γ) 2+r2γ b+2e brb 2+3r2γ b+2e brb − + r2γ − r2γ 2αγ 2+2e brb 40e 8e r 2 + eb 16 + r 2γ − 8r 2b ) − 2e r2γ r 2αb 69 Page 8 of 11 J. Astrophys. Astr. (2018) 39:69

1.0 0.35

0.30 0.8

0.25

0.6 r r

r 0.20

β 0.4 0.15

0.2 0.10

0.0 0.05

0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r r Figure 5. Evolution of β(r) and β(r)/r for isotropic case. 2 −b − 2(1+r γ b+e rb ) 2+3r2γ b+2e brb (β(r)) to reduce the unknowns in the above equation 2γ 2γ 4 2 + 26e r − 10e r r αγ but still it is not be possible to solve it analytically. Therefore, we use numerical method to solve this equa- −b 2e +3b tion and plot these solutions. We can easily examine + e r2γ r 6γ 3 the three conditions on shape function for the valid- −b + 2γ + −b 2e +3b 2 3r b 2e rb ity of WH geometries. The evolution of shape function − e r2γ r 6αγ 3 + e r2γ r 6αγ 3 β(r) and β(r)/r verses r isshowninFig.5.From −b 2+2e rb left plot of Fig. 5, we can see that β(r) increases + 4e r2γ r 2α −16 + eb 16 − 3r 2γ with the increase in radial distance and also the con- − 2+2e brb β( )< 2 dition r r is satisfied. One can also observe + 7e2br 2γ 1 + r 2γ b − 16e r2γ r 4αb the asymptotic behavior in the right plot of Fig. 5 as − 2+r2γ b+2e brb β(r)/r → 0forr →∞. We found that the throat is 2γ 5 (3) b −8e r r αγb + 2 −1 + e located at r0 = 0.0993681999669 such that β(r0) = r0. The change with r in the shape function is plotted + −b −b 2 2e rb 2e +3b in the left of Fig. 6 which shows that the condition × − 32e r2γ α + e r2γ r 6 (−1 + α) γ 3 β (r0)<1 is obeyed while the right plot of Fig. − 2+3r2γ b+2e brb 6 represents the asymptotic behavior and show that − e r2γ r 4αγ 2 2 + r 2γ β(r) − r < 0 and consequently 1 − β(r)/r > 0. In − 2+r2γ b+2e brb the right plot of Fig. 7, the graph oscillates rapidly with + 16e r2γ α 4 + 5r 2γ growing amplitude as one approaches the throat located − 2(1+r2γ b+e brb ) at r0 = 0.0993681999669. The plot does not show the 2 2 4 2 − 2e r γ α(16 + 40r γ + 15r γ ) oscillations in the region r0 ≤ r ≤ 1.1 adjacent to the − throat (the plot is truncated there). These oscillations 2+2e brb + 8e r2γ may grow further there and become negative, therefore, we conclude that the WEC and NEC are violated. × r 2α 4 − eb 8 + 7r 2γ + 2b + 2γ + 4γ 2 e 4 7r r b 2.3 Barotropic fluid − 2+2e brb 2 − 4e r2γ r 4α(−2 + eb 2 + 3r 2γ )b It is well known that the density of the matter profile can − 2+r2γ b+2e brb be linked with radial pressure via realistic barotropic 2γ 5 b 2 (3) − 4e r r αγ −1+e 1+r γ b =0. equation of state (EoS), i.e., pr = kρ or pr − kρ = 0. Using the same method as deployed for the case of (17) isotropic fluid with this EoS, we get third order non linear differential equation. By making use of the same This is a non linear third-order differential equation. numerical technique, we plot β(r), β(r), β(r)/r and Now, we convert b(r) in terms of the shape function β(r) − r versus r as shown in Figs 8 and 9. We observe J. Astrophys. Astr. (2018) 39:69 Page 9 of 11 69

0.8 0

1 0.6

2 r

r 0.4 r β

β 3

0.2 4

0.0 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6 r r Figure 6. Evolution of β(r) and β(r) − r for isotropic case.

0.0085 0.0180

0.0080

0.0175

0.0075

0.0170 p 0.0070

0.0165 0.0065

0.0160 0.0060

1 2 3 4 5 6 1 2 3 4 5 6 r r Figure 7. Evolution of ρ and ρ + p .

0.4

0.3

0.2 r r

β 0.2 β

0.1

0.0

0.0 0 2 4 6 8 10 0 2 4 6 8 10 r r Figure 8. Evolution of β(r) and β(r) with barotropic EoS . that β(r)<1 while β(r) has an increasing behavior satisfied in this case which means that a realistic WH in this case. Also, β(r) − r remains less than zero which satisfies the ECs for usual matter in modified and the throat is located at r0 = 0.0245805 where gravity does not exist. β(0.0245805) = 0.0245805 while β(0.0245805)<1. Consequently, the flaring out condition along with other constraints, which support the WH geometries, hold 3. Discussion with barotropic EoS. We observe from the left plot of Fig. 9 that β(r)/r does not approaches to zero as Modified gravitational theories has received significant r →∞. We also found that the NEC and WEC are not attention as a possible alternative to GR during the last 69 Page 10 of 11 J. Astrophys. Astr. (2018) 39:69

0.04 0

2 0.03

r 4 r r

r 0.02

β β

0.01

0.00 0 2 4 6 8 0 2 4 6 8 10 r r Figure 9. Evolution of β(r)/r and β(r) − r with barotropic EoS.

0.02 0.005

0.00 0.000 t p

0.02 0.005

0.04 0.010

0 2 4 6 8 10 0 2 4 6 8 10 r r

Figure 10. Evaluation of ρ(r) and ρ + pt with barotropic EoS. few decades. A paramount feature in GR is the possible the behavior of that shape function. In the presence of existence of WH geometries which violate ECs sup- anisotropic fluid, we have done our analysis by using ported by exotic matter. Many researchers found this the first technique as the resulting equations are quiet quiet significant to examine whether different modified complicated with six unknowns. However, the com- gravity models via the effective energy momentum ten- plexity reduces somehow with isotropic and barotropic sor are responsible for the violation of ECs while the fluids and we were able to explore the behavior of usual matter satisfy these conditions. In this paper, we shape function via plots. We have considered a partic- have found that the WH geometries can exist even if the ular realistic form of f (R, T ) model, i.e., f (R, T ) = NEC is not violated by the usual matter (or the matter R + αR (exp (−R/γ ) − 1) + λT for the possible exis- threading the WH is not the exotic matter), but these tence of WH geometries. First of all, we have formulated are the extra curvature ingredients that sustained the the field equations with this choice of modified grav- WH in the context of modified gravity. For the sys- ity model filled with anisotropic matter contents. In all tematic investigation of this analysis, we have explored cases, we have chosen that the gravitational redshift, the behavior of WEC and NEC in the background of a(r), is constant so that its radial derivative vanishes. In f (R, T ) modified gravity with three different kind of the anisotropic matter case, the shape function is con- + matter configurations in the interior of a spherical star β( ) = r0 m 1 sidered as r r r ,wherem should be less namely, anisotropic, isotropic and barotropic (for which than 1 in order to satisfy the flaring out condition. the radial pressure satisfies the barotropic EoS) fluids. We have plotted the WEC and NEC and found that There exist various techniques in literature to there exist certain regions for the existence of WHs examine the WH geometries. One of these is to consider where these ECs are viable. We examined that for a particular shape function and then use it to analyze ECs 0 < r < 0.1, one can take any value of α and γ while the other one involves the exploration of shape in the modified gravity model for the possible appear- function via different assumptions and then examine ance of WH with normal matter. It is also observed J. Astrophys. Astr. (2018) 39:69 Page 11 of 11 69 that the evolution of tangential pressure does not Astasheok, A. V., Capozzielli, S., Odintsov, S. D. 2013, J. satisfy the condition of normal matter, i.e., pt > 0. Cosmol. Astropart. Phys. 12, 040 These results are shown in Figs.(1)-(4). For the isotropic Azizi, T. 2013, Int. J. Theor. Phys. 52, 3486 and barotropic fluid case, we have analyzed the behavior Bejarano, C., Lobo, F. S. N., Olmo, G. J., Garcia, D. R. 2017, of shape function by numerically solving the obtained Eur. Phys. J. C 77, 776 non-linear dynamical equation. We have examined the Bhatti, M. Z. 2016, Eur. Phys. J. Plus 131, 428 Bhatti, M. Z., Yousaf, Z. 2016, Eur. Phys. J. C 76, 219. viability of following necessary conditions for the exis- [arXiv:1604.01395 [gr-qc]] tence of WH geometries. Bhatti, M. Z., Yousaf, Z. 2017, Int. J. Mod. Phys. D 26, 1750029 • β( )< r0 1, where r0 is the throat radius for the Capozziello, S., Harko, T., Koivisto, T. S., Lobo, F. S. N., WH solution. Olmo, G. J. 2012, Phys. Rev. D 86, 127504 • β−rβ > The flaring out condition that β2 0 is satis- Clifton, T., Ferreira, P.G., Padilla, A., Skordis, C. 2012, Phys. fied. Rep. 513, 1 • − β(r) > Fukutaka, H., Ghoroku, K., Tanaka, K. 1989, Phys. Lett. B 1 r 0. 222, 191 There is one more condition of asymptotic flatness, Harko, T., Lobo, F. S. N., Nojiri, S., Odintsov, S. D. 2011, i.e., β(r)/r → 0asr →∞, which is satisfied in Phys. Rev. D 84, 024020 isotropic case but incompatible with barotropic EoS. Harko, T., Lobo, F. S. N., Mak, M. K., Sushkov, S. V. 2013, Phys. Rev. D 87, 067504 We also observed that ρ>0aswellasρ + p > 0in Jamil, M., Momeni, D., Myrzakulov, R. 2013, Eur. Phys. J. isotropic case and can be seen in Fig. 7 while ECs hold C 73, 2267 for very small values of r with barotropic fluid. We have Lobo F. S. N., Oliveira, M. A. 2009, Phys. Rev. D 80, also discussed the equilibrium picture of the WH solu- 104012 tion with anisotropic matter for different values of λ.We Mazharimousavi, S. H., Halilsoy, M. 2016, Mod. Phys. Lett. found that the anisotropic WH solutions are stable as A 31, 1650192 the anisotropic and hydrostatic forces counter balanced Mehdizadeh, M. R., Ziaie, A. H. 2017, Phys. Rev. D 95, each other. These results are represented via graphs and 064049 showninFig.10. Finally, we can conclude that the Moraes, P. H. R. S., Sahoo, P. K. 2017a, Eur. Phys. J. C 77, WH geometries can be constructed without exotic mat- 480 ter in the context of f (R, T ) gravity in certain regions Moraes, P. H. R. S., Sahoo, P. K. 2017b, Rev. D 96, of spacetime. Also, the WH geometries are stable and 044038 Moraes, P. H. R. S., Sahoo, P. K., Ribeiro, G., Correa, R. A. realistic in f (R, T ) theory with anisotropic fluid distri- C. arXiv:1712.07569 [gr-qc] bution. All our results are compatible and support the Morris, M. S., Thorne, K. S. 1988, Am. J. Phys. 56, 395 analysis of existing results in modified gravity theories. Morris, M. S., Thorne, K. S., Yurtsever, U. 1988, Phys. Rev. Lett. 61, 1446 Sahoo, P. K., Sahoo, P., Bishi, B. K. 2017, Int. J. Geom. Meth. References Mod. Phys. 14, 1750097 Singh, C. P., Singh, V. 2014, Gen. Relativ. Gravit. 46, 1696 Alvarenga, F. G., Houndjo M. J. S., Monwanou, A. V. Orou, Yousaf, Z. 2017a, Eur. Phys. J. Plus 132, 71 J. B. C. 2013, J. Mod. Phys. 4, 130 Yousaf, Z. 2017b, Eur. Phys. J. Plus 132, 276. Arellano, A. V. B., Lobo F. S. N. 2006, Class. Quantum Grav. Yousaf, Z., Bamba, K., Bhatti, M. Z. 2016, Phys. Rev. D 93, 23, 5811. 124048