A Perturbative Approach to Neutron Stars in f(T, \mathcal{T})-Gravity
AA Perturbative Approach to Neutron Stars in f ( T, T ) − Gravity
Mark Pace ∗
1, 2 and Jackson Levi Said †
1, 2 Department of Physics, University of Malta, Msida, MSD 2080, Malta Institute of Space Sciences and Astronomy,University of Malta, Msida, MSD 2080, Malta (Dated: 15 November 2018)We derive a Tolman-Oppenheimer-Volkoff equation in neutron star systems within the modified f ( T, T )-gravity class of models using a perturbative approach. In our approach f ( T, T )-gravity isconsidered to be a static spherically symmetric space-time. In this instance the metric is built froma more fundamental tetrad vierbein which can be used to relate inertial and global coordinates.A linear function f = T ( r ) + T ( r ) + χh ( T, T ) + O ( χ ) is taken as the Lagrangian density forthe gravitational action. Finally we impose the polytropic equation of state of neutron star uponthe derived equations in order to derive the mass profile and mass-central density relations of theneutron star in f ( T, T )-gravity. PACS numbers: 04.40.Dg, 04.50.Kd
I. INTRODUCTION
Recently it has been shown that the Universe isaccelerating in its expansion [1, 2]. The conceptof the cosmological constant together with theinclusion of dark matter yield the ΛCDM modelwhich explains a whole host of phenomenawithin the universe. [3–5]. We may also explainthis acceleration by instead modifying the gravi-tational theory itself with alternative theories ofgravity an example of which is f ( R )-gravity [6–9].Our focus of this paper is on one alterna-tive theory of gravity called f ( T )-gravity whichmakes use of a “teleparallel” equivalent of GR(TEGR) [10] approach, in which instead ofthe torsion-less Levi-Civita connection, theWeitzenb¨ock connection is used, with the dy-namical objects being four linearly independentvierbeins [11, 12]. The Weitzenb¨ock connectionis curvature-free and describes the torsion of amanifold.The differences between f ( T ) class of grav-ity and other gravity forms such as f ( R ) andTEGR is in the choice of the function f ( T )which is taken [8]. Comparing f ( T )-gravitywith f ( R )-gravity it is noted that f ( T )-gravitycannot be reforumlated as a teleparallel actionplus a scalar field through the conformal trans-formation due to the appearance of additionalscalar-torsion coupling terms [13, 14]. Theobvious difference is that f ( T )-gravity has a ∗ [email protected] † [email protected] class of equations which is easier to work withbecause the field equations are second orderrather than fourth-order like in f ( R )-gravityclass scenarios [8]. In f ( T )-gravity more degreesof freedom are obtained which thus correspondsto one massive vector field [15, 16].We make use of a pure tetrad [17], whichmeans that the torsion tensor is formed by amultiple of the tetrad and its first derivativeonly. Under the assumption of invarianceunder general coordinate transformations, globalLorentz transformations, and the parity op-eration we construct the Lagrangian densityfrom this torsion tensor [9, 10, 12, 17]. Alsothe Lagrangian density is second order in thetorsion tensor [10, 12]. Thus f ( T )-gravitygeneralises the above TEGR formalism, makingthe gravitational lagrangian a function of T [8–10].Our goal for this paper is to derive a work-ing model for the TOV equations within anew modification of f ( T ) class gravity, called f ( T, T )-gravity in a perturbative manner. Wemake use of a perturbative approach due to thefact that a non physical assumption had to betaken whilst deriving the TOV equations in ananalytical manner. f ( T, T )-gravity couples the gravitationalsector and the standard matter one [8]. Insteadof having the Ricci scalar coupled with the traceof the energy momentum tensor T as is donein f ( R, T )-gravity, f ( T, T )-gravity couples thetorsion scalar T with the trace of the matterenergy-momentum tensor T [8, 9, 18]. Recentlya modification to this theory has been propose, a r X i v : . [ g r- q c ] A p r that of allowing for a general functional depen-dence on the energy momentum trace scalar, T µµ = T [8, 9].Our interest is in studying the behaviourof spherically symmetric compact objects inthis theory. We propose the use of a linearfunction, namely f ( T, T ) = αT ( r ) + β T ( r ) + ϕ ,where α and β are arbitrary constants whichmay be varied to align our star’s behaviour withcurrent observations. ϕ is then considered tobe the cosmological constant. We consider thelinear modification since it is the natural firstfunctional form to consider, and the right placeto start to understand how the trace of thestress-energy tensor might effect f ( T, T ) gravity.In particular, our focus is on neutron stars in f ( T, T ) gravity.Besides the possibility of the existence ofthese exotic stars, this is also a good placeto study the behavior of modified theories ofgravity in terms of constraints. Moreover, thisalso opens the door to considerations of stiffmatter in early phase transitions [19].The plan of this paper is as follows; in sec-tion 2 we go over the mathematical tools andgive an overview of f ( T, T ) − gravity. In section3 we discuss the rotated tetrad taken and discusshow the equations of motion in f ( T, T ) − gravityare derived perturbatively. In section 4 the twoTOV equations are derived and discussed alongwith the schwarzschild solution, while the resultsare then used in section 5 where we output thenumerical results given by the yielded TOVequations. Finally we discuss the results insection 6. II. f ( T, T ) -GRAVITY OVERVIEW f ( T, T )-gravity generalises f ( T )-gravity andthus is based on the Weitzenbock’s geometry.We will follow a similar notation style as thatgiven in Ref. [7, 8, 10, 20–22]. Using: Greekindices µ, ν, . . . and capital Latin indices i, κ, . . . over all general coordinate and inertial coordi-nate labels respectively [7, 8, 20, 21].Torsion tensor [20, 21, 23] is given by T λµν (cid:0) e λµ , ω λiµ (cid:1) = ∂ µ e λν − ∂ ν e λµ + ω λiµ e i ν − ω i λν e iµ , (1) where ω λiµ is the spin connection [23]. The tor-sion tensor has vanishing curvature. Therefore bydoing so all the information of the gravitationalfield is embedded in the torsion tensor [22], whilethe gravitational Lagrangian is the torsion scalar[23]. The contorsion tensor is then defined as K µνρ = − (cid:0) T µνρ − T νµρ − T µνρ (cid:1) , (2)while the superpotential of teleparallel gravity isdefined by [20, 21] S µνρ = 12 (cid:0) K µνρ + δ µρ T ανα − δ νρ T αµα (cid:1) . (3)Unlike the contorsion tensor, the superpotentialtensor does not have any apparent physicalmeaning, instead is it purely introduced toreduce the size of the Lagrangian.The torsion scalar [20–22] is then given as T = S µνρ T ρµν . (4)As in the analogous f ( R, T ) theories [24], wefurther generalised upon the gravitational la-grangian by taking an arbitrary function f andthus giving [25, 26] S = − πG (cid:90) d xe [ f ( T, T ) + L m ] . (5)The function f ( T, T ) is taken to be equal to T ( r ) + T ( r ) + χh ( T, T ) + O ( χ ) where χ is avery small paramter which will aid in differenti-ating between zeroth and first order term [27],and h ( T, T ) is an arbitatary function of the tor-sion scalar T and the trace T of the energy mo-mentum tensor e-m T given by T = δ νµ T µν . L m isthe matter Lagrangian density [22, 25]. In thisinstance f is an arbitrary function of the torsionscalar T and the trace of the energy-momentumtensor T [25]. The variation of the action definedin Eq.(5) with respect to the tetrad leads to thefield equations [27] e ρi S µνρ ∂ µ T χh
T T + e ρi S µνρ χh T T T + e − ∂ µ (cid:0) ee ρi S µνρ (cid:1) (1 + χh T ) + e µi T λµκ S νκλ (1 + χh T ) − e νi T ( r ) + T ( r ) + χh ( T, T )4 + (1 + χh T ) ω i λν S νµi − (1 + χh T )2 (cid:0) e λi T νλ + p ( r ) e νi (cid:1) = − πe λi e-m T νλ , (6)where h T = ∂h∂T , h T = ∂h∂ T , and h T T = ∂ h∂T ∂ T . III. PERTURBATIVE EQUATIONS OFMOTION IN f ( T, T ) -GRAVITY In perturbative theory the field equationsmay be expanded perturbatively in χ [27] andtherefore the metric components take on theexpansions A ( r ) χ = A ( r ) + χA ( r ) + . . . and B ( r ) χ = B ( r ) + χB ( r ) + . . . [27]. The energy-momentum tensor in the field equations, is stillthe energy-momentum tensor of the perfect fluid.The hydrodynamic quantities are also definedperturbatively by ρ ( r ) χ = ρ ( r ) + χρ ( r ) + . . . and p ( r ) χ = p ( r ) + χp ( r ) + . . . [27].A spherically symmetric metric which hasa diagonal structure is considered for our system[28], ds = − e A ( r ) χ dt + e B ( r ) χ dr + r dθ + r sin dφ , (7) and we consider the fluid inside the star to bethat of a perfect fluid which yields a diagonalenergy-momentum tensor e-m T νλ = diag ( − ρ ( r ) χ , p ( r ) χ , p ( r ) χ , p ( r ) χ ) , (8)where ρ ( r ) χ and p ( r ) χ are the energy densityand pressure of the fluid respectivel [28]. Thesealso make up the matter functions which, alongwith the metric functions, A ( r ) and B ( r ), arealso taken to be independent of time [22]. Thusthe system is taken to be in equilibrium [5, 28].The equation of conservation of energy isgiven by dp ( r ) dr = − ( ρ ( r ) + p ( r )) dA ( r ) dr . (9)Following Ref. [17] the following rotated tetradis used e aµ = e A ( r ) χ e B ( r ) χ θ cos φ e B ( r ) χ θ sin φ e B ( r ) χ θ − r cos θ cos φ − r cos θ sin φ r sin θ r sin θ sin φ − r sin θ cos φ This form of vierbein is considered because itallows us more degrees of freedom [29] and itallows us to acquire a static and sphericallysymmetric wormhole solution in our standardformulation of f ( T, T )-gravity [29, 30].Also because this is a pure form of tetrad[23], the spin connection elements of the tetradvanish and thus ensure that the spin connection terms need not be included [23].Inserting this vierbein into the field equa-tions, from Eq.(4) we get the resulting torsionscalar T ( r ) =2 e − B ( r ) r − e B ( r )2 − e B ( r )2 + rA (cid:48) ( r ) , (10) where the prime denotes derivative with respectto r . The resulting field equation componentsturn out to be.The t − t component, given by i = ν = 0 results in4 πρ ( r ) χ = T ( r ) χ + T ( r ) χ + χh e − B ( r ) r (1 + χh T ) − e B ( r )2 + rA (cid:48) ( r ) e B ( r )2 − + rB (cid:48) ( r ) + (1 + χh T )2 ( ρ ( r ) − p ( r )) + e − B ( r ) χr e B ( r )2 − ( h T T T (cid:48) ( r ) + h T T T (cid:48) ( r )) . (11)While the r − r component, given by i = ν = 1 results in4 πp ( r ) χ = T ( r ) χ + T ( r ) χ + χh e − B ( r ) r (1 + χh T ) e B ( r )2 − + rA (cid:48) ( r ) e B ( r )2 − − p ( r ) (1 + χh T ) . (12)Note that the zeroth order quantities are givenwithout a subscript. IV. PERTURBATIVE DERIVATION IN f ( T, T ) − GRAVITY
We will now make use of the equations of motiongiven by Eq. (11) and Eq. (12) by first consider-ing a solution for ρ ( r ) χ and p ( r ) χ up to order χ .The zeroth order quantities are considered fromthese two equations and given by4 πρ ( r ) = T ( r ) + T ( r )4 + e − B ( r ) r (cid:20) − e B ( r )2+ rA (cid:48) ( r ) e B ( r )2 − + rB (cid:48) ( r ) (cid:21) + ( ρ ( r ) − p ( r ))2 , (13)and 4 πp ( r ) = T ( r ) + T ( r )4 + e − B ( r ) r (cid:20) e B ( r )2 − + rA (cid:48) ( r ) e B ( r )2 − (cid:21) − p ( r ) . (14)At this point the torsion scalar given by Eq.(10) and T ( r ) = ρ ( r ) − p ( r ) are inserted into thetwo equations which after manipulation result in4 πρ ( r ) = e − B ( r ) r (cid:16) − e B ( r ) + rB (cid:48) ( r ) (cid:17) + 14 (3 ρ ( r ) − p ( r )) , (15)and4 πp ( r ) = e − B ( r ) r (cid:16) − e B ( r ) − rA (cid:48) ( r ) (cid:17) + 14 ( ρ ( r ) − p ( r )) , (16)respectively.At this point it is convenient to take a massparameter ansatz. The solution is assumed tohave the same form of the exterior solution forthe metric function B χ . In order to render ametric ansatz in line with the Schwarzschildmetric we take the following [27]. e − B ( r ) χ = 1 − Ω M ( r ) χ r + (cid:15) ( r ) χ , (17)where Ω is an arbitrary constant and (cid:15) ( r ) istaken to be a function of r . Similar to ρ χ , M ( r ) χ is expanded in χ as M χ = M + χM + . . . , [27]where M is the zeroth order solution. Taking a derivative of M χ with respect to r the following is obtained dM χ dr = 1Ω (cid:18) − e − B ( r ) χ + (cid:15) ( r ) χ + e − B ( r ) χ rB (cid:48) ( r ) χ + r(cid:15) (cid:48) ( r ) χ (cid:19) . (18)Now we focus on the first equation of motiongiven by Eq. (11) where we insert the torsionscalar equation and the energy momentum com-ponents and thus obtain the following equation4 πρ ( r ) χ = Ω2 r dM χ dr − ( (cid:15) χ ( r )) (cid:48) r + 14 (3 ρ ( r ) χ − p ( r ) χ )+ χ (cid:40) h e − B ( r ) h T r − e B ( r )2 + rA (cid:48) ( r ) e B ( r )2 − + rB (cid:48) ( r ) + h T ρ ( r ) − p ( r )) + e − B ( r ) r e B ( r )2 − ( h T T T (cid:48) ( r ) + h T T T (cid:48) ( r )) (cid:41) . (19)Here we invoke a linear parameter for h , givenby αT ( r ) + β T ( r ) + ϕ which after being inserted into this equation and further reduced yields dM χ dr = 8 πr ρ ( r ) χ Ω + ( (cid:15) χ ( r )) (cid:48) r − r
2Ω (3 ρ ( r ) χ − p ( r ) χ ) − r χ (cid:40) αe − B ( r ) r e B ( r )2 − e B ( r ) + rA (cid:48) ( r ) e B ( r )2 − + 2 rB (cid:48) ( r ) + ϕ + β (3 ρ ( r ) − p ( r )) (cid:41) . (20)The main task at this point is to reduce the valuesof A (cid:48) ( r ) and B (cid:48) ( r ) where the definitions givenby Eq. (15), Eq. (16), and Eq. (17) will be substituted and thus resulting in our first TOVequation dM χ dr = 8 πr ρ ( r ) χ Ω + ( (cid:15) χ ( r )) (cid:48) r − r
2Ω (3 ρ ( r ) χ − p ( r ) χ ) − r χϕ χα (cid:18) − Ω M ( r ) r + (cid:15) ( r ) (cid:19) (cid:40) − r (cid:20) r + r(cid:15) ( r ) − M ( r )Ω (cid:21) − (cid:20) r ( p ( r )(7 + 16 π ) − ρ ( r )) − (cid:21) + (cid:18) (cid:15) ( r ) − Ω M ( r ) r (cid:19) − (cid:20) − r ( p ( r )(17 + 16 π ) + ρ ( r )(32 π − (cid:21)(cid:41) . (21)Now we shift our focus into deriving thepressure-radius relation of the TOV equations.For this purpose Eq. (12) is considered where a similar treatment will be given i.e. we substi-tute the torsion scalar equation and the energymomentum definition to give A (cid:48) ( r ) χ = e B ( r ) χ (cid:40) r (cid:16) − e − B ( r ) χ (cid:17) + r ρ ( r ) χ − p ( r ) χ ) − πp ( r ) χ r + 2 rχ (cid:26) h h T e − B ( r ) χ r (cid:20) e B ( r )2 − + rA (cid:48) ( r ) e B ( r )2 − − h T p ( r ) (cid:27)(cid:41) . (22)Inserting the definition of h and Eq. (17) andreducing further yields A (cid:48) ( r ) χ = (cid:40) p ( r ) πr ( αχ − M ( r ) r − (cid:15) ( r ) r + rχϕ − r p ( r ) − ρ ( r )) (1 + β − αχ ) (cid:41)(cid:20) − Ω M ( r ) χ r + (cid:15) ( r ) χ (cid:21) − . (23)This result is then inserted into the continuityequation given by Eq. (9) and thus results in thesecond TOV equation required dp ( r ) χ dr = − ( ρ ( r ) χ + p ( r ) χ ) (cid:40) p ( r ) πr ( αχ −
1) + Ω M ( r ) r − (cid:15) ( r ) r + rχϕ − r p ( r ) − ρ ( r )) (1 + β − αχ ) (cid:41)(cid:20) − Ω M ( r ) χ r + (cid:15) ( r ) χ (cid:21) − . (24) V. NUMERICAL MODELING OFNEUTRON STARS
Through Eq. (21) and Eq. (24) any spheri-cally symmetric mass in f ( T, T )-gravity can be investigated in terms of its physical properties.In order to obtain a mass profile relation forthe TOV equations, we numerically integrate ourTOV equations of stellar structure to build mod-els of neutron stars in f ( T, T )-gravity. Here wetake the relativistic energy density ρ ( r ) as equalto ρ ( r ) = ρ ( r ) + p ( r )Γ − , (25)where ρ ( r ) is the rest matter density [31]. Wetake the initial conditions as equal to m (0) = 0and p (0) = Kρ Γ0 ,c where ρ Γ0 ,c ( r ) = 10 gm/cm is the central density [31]. We take Γ = 4 / G = 1 and c = 1 [31].The value of Ω is taken to be 2 as the lit-erature in Ref. [27] suggests and the value of ϕ is taken to be the cosmological constant, as2 . × − [32]. Here we also consider ourvalue of χ as being a very small but non-zerovalue ∼ − cm − [27]. The value of α in Eq.(21) and Eq. (24) is taken to be − β is varied. We vary the value of β soas to manipulate the dominance of the function T ( r ). A. Mass Profile Curve
In Fig.(1) we show the mass profile curve of aneutron star. We take β = − β .As the function T ( r ) = ρ ( r ) − p ( r ) [26] isincluded i.e. when β = − km mark it surpasses the neutronstar generated by the GR case to yield a largerstellar structure.Current observations shown in the litera-ture by [33] state that such massive neutronstars may exist. In fact the figures show thatthere is ≈ .
67% increase in the maximum massvalue of the neutron star gained.We further magnify this effect by taking β = −
10 and as may be seen from the Fig.(1),the neutron star at first behaves exactly likethe previous case and again surpasses the othercases at the 17 km mark to yield a more massiveneutron star. In fact by considering β = −
10 theallowable maximum mass of the neutron star isincreased by 5 .
43% over the β = − .
35% over the GR case. In order to better understand the physicalbehaviour of neutron stars we investigate howmass varies over radius for different settings of β in Fig.(1). Whichever values of β are takenthe graphical output is similar in structure tothat of GR. Another aspect to note is that thegeneral behavior of the stellar system remainsthe same however a new degree of freedom isallowed depending on the maximum mass andradius of these stars.In contrast the results gained in this paperare similar to those gained by various otherauthors such as found in Ref. [34–37]. Theresulting mass profile curves behave in a similarmanner where mass steadily increases withradius to plateau at an instance. FIG. 1: Mass profile graph of a neutron star where α = − β . The valueof Γ = 4 /
3, Ω = 2 [27] and ϕ = 2 . × − [32] B. Radius-Central Density Curve
Fig.(1) is heavily dependent on the centraldensity, not in behaviour but in terms of theparticular values being produced. To contrastthis we plot the radius-central density curvewhich was generated from the TOV equations.This plot is given by Fig.(2).Again we contrast with the GR case whentaking β = −
1. When we decrease the value of β by taking β = − km .To magnify this we again take a lower valueof β by taking β = −
10, this shows that thecentral density declines somewhat rapidly at firstinitially with radius and then slowly declinessteadily similar to the previous cases however itreaches the lowest value the fastest out of thethree cases considered. Thus this shows that aslightly larger neutron star is allowed in such agravitational framework.
FIG. 2: Radius-central density graph of a neutronstar where α = − β . The value of Γ = 4 /
3, Ω = 2 [27] and ϕ = 2 . × − [32] Much like the mass profile curves the resultsgained from Ref. [34–37] exhibit similar resultsas gained in this study. The central density valueincreases significantly with radius at first to gaina maximum value as we get closer to the centreof the star.
VI. CONCLUSION
In this study the TOV equations are derived ina perturbative way for f ( T, T ) − gravity. Laterthe two equations are applied to a polytropicequation of state which yielded the characteris-tics of the neutron star in such a gravitationalframework.Our main goal throughout this research isto derive a working model which involved littleto no assumptions in the derivation. We alsowanted to retain and include as many generalterms as possible. We did this also because we would like to further fine tune our results tocurrent observations.A reasonable boundary condition was takenin order to solve the TOV equations by nu-merical techniques. We apply the polytropicequation of state in order to reduce our TOVequations from a four variable equation to athree variable equation by making one of thevariables dependent on the others.Our approach considered a value of χ which isnon-zero however very small ∼ − cm − . Theliterature shown in Ref. [27] shows that thetypical value of the Ricci curvature is calculatedto be roughly on a similar order. Thus assumingour value of χ to be so small is reasonable.Our graphical representations are inspiredby the work carried out by the authors of Ref.[8, 38]. The graphs show that a larger neutronstar is allowed in such a gravitational framework.We vary the values of β accordingly to outputthe variations occurring when we include the T ( r ) term. By taking a lower value of this termwe note that it allows for a larger neutron star.This value will require future fine tuning in orderto align with current observations.More values of β were considered in test-ing. The yielded results showed that whenpositive values of β were considered no tangibleneutron star would be yielded in such a grav-itational framework. When lower values of β were considered the yielded stellar structuresdid not behave in accordance to the theory asexplained in Ref. [39]. Thus the range of valuesfor β considered to yield a tangible and properneutron star would be − (cid:28) β (cid:28) − f ( T, T ) − gravity in ananalytical manner. VII. ACKNOWLEDGMENTS
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