Electromagnetic and anisotropic extension of a plethora of well-known solutions describing relativistic compact objects
aa r X i v : . [ phy s i c s . g e n - ph ] N ov Electromagnetic and anisotropic extension of a plethoraof well-known solutions describing relativistic compactob jects
K. Komathiraj • Ranjan SharmaAbstract
We demonstrate a technique to generatenew class of exact solutions to the Einstein-Maxwellsystem describing a static spherically symmetric rela-tivistic star with anisotropic matter distribution. Aninteresting feature of the new class of solutions is thatone can easily switch off the electric and/or anisotropiceffects in this formulation. Consequently, we show thata plethora of well known stellar solutions can be iden-tified as sub-class of our class of solutions. We demon-strate that it is possible to express our class of solutionsin a simple closed form so as to examine its physical vi-ability for the studies of relativistic compact stars.
Keywords
Einstein-Maxwell system Exact solutionRelativistic star Anisotropy.
Exact solutions to Einstein-Maxwell system play a ma-jor role in the studies of relativistic compact objects.While the Reissner-Nordstr¨om solution uniquely de-scribes the exterior gravitational field of a static spher-ically isolated object in the presence of an electromag-netic field, a large class of interior solutions are availablein the literature which are regular, well behaved andphysically meaningful. In the uncharged case, a largeclass of such exact solutions and their physical viabilityhave been examined by Delgaty and Lake (1998). In
K. KomathirajDepartment of Mathematical Sciences, Faculty of Applied Sci-ences, South Eastern University of Sri Lanka, Sammanthurai3000, Sri Lanka.Email: [email protected] SharmaDepartment of Physics, Cooch Behar Panchanan Barma Univer-sity, Cooch Behar 736101, West Bengal, India.Email: [email protected] the charged case, Ivanov (2002) has compiled differentclass of exact solutions.In the recent past many new exact solutions havebeen developed some of which are, in fact, generaliza-tions of many of the well-known solutions. Most of theextensions have generally been done either by incorpo-rating an electromagnetic field or anisotropy or bothinto the system. The generalized models allow us tostudy the impacts of charge and/or anisotropy on thegross physical behaviour of a compact star. A primemotivating factor for such a generalization in most ofour previous works was to fine-tune the stellar observ-ables like mass and radius.Local anisotropy, as indicated by many investi-gators in the past, plays a significant role in thestudies of relativistic stellar objects (Ruderman 1972;Bowers and Liang 1974; Herrera and Santos 1997). Ina recent article, it has been argued that pressureanisotropy cannot be ignored in the studies of relativis-tic compact stars as it is usually expected to develop bythe physical processes inside such ultra-compact stars(Herrera 2020). Incorporation of an electromagneticfield in the studies of astrophysical objects is also well-motivated and many pioneering works have been donein this field in the past which includes the pioneer-ing works of Majumdar (1947), Papapetrou (1947),Cooperstock and Cruz (1977) and Bekenstein (1971),amongst others. Consequently, different stellar modelshave been developed by relaxing the pressure isotropycondition as well as by incorporating a net charge intothe system. Sharma et al (2001) have generalized thewidely used Vaidya and Tikekar (1982) stellar modelby assuming a particular form of the electric field.The investigation shows a wide range of causal be-haviour in the presence of the electric field. The Vaidyaand Tikekar model was generalized by Karmarkar et al (2007) to analyze the impact of anisotropy on the max-imum mass of a compact star. An anisotropic gener- alization of the Vaidya and Tikekar stellar model hasbeen made by Thirukkanesh et al (2019) recently. Ear-lier, Thirukkanesh et al (2018) developed an algorithmto generalize a plethora of well-known exact solutionsto Einstein field equations corresponding to a staticspherically symmetric star by relaxing the pressureisotropy condition. Komathiraj and Sharma (2018)have developed a formalism to generate a new classof interior solutions corresponding to the exterior R-Nmetric which contained many previously found solu-tions. Komathiraj et al (2019) also made an electro-magnetic generalization of the Durgapal and Fuloria(1985) stellar solution. By relaxing the pressureisotropy condition, Sharma et al (2017) generalized theFinch and Skea (1989) stellar model. For a specificcharge distribution, Ratanpal et al (2017) also made ageneralization of the Finch and Skea stellar model toanalyze the impact of the charge on the mass-radiusrelationship of a compact star, in particular. The rela-tivistic stellar model of Mak and Harko (2004) was ex-tended by Komathiraj and Maharaj (2007) to includecharge into the system. Maharaj et al (2014) made afurther generalization of the model by considering thesystem to be anisotropic as well. It is noteworthy thatmany of the static spherically symmetric anisotropicand/or charged stellar solutions available in the liter-ature do not possess isotropic and/or charge neutrallimits.In this paper, we intend to generate new class ofexact solutions corresponding to a static sphericallyanisotropic star possessing a net charge. The idea isthat once the anisotropy and/or charge are/is switchedoff we should be able to regain some of the well-behaved, physically interesting stellar solutions foundearlier. Such a generalization would allow us to in-vestigate the impacts of anisotropy and charge on thephysical features of a compact object in a neat man-ner. Moreover, physical acceptability of the general-ized solutions can be ensured by suitable choice of theanisotropic and/or charge parameters as their isotropicand uncharged counterparts have already been foundto be regular, well behaved and physically meaningful.The paper is organized as follows: In Section 2,we lay down the Einstein-Maxwell equations for ananisotropic fluid distribution. In Section 3, we pro-pose a technique to generate solutions to the systemof equations. We show how a large class of well knownsolutions can be regained from our general class of solu-tions. In Section 4, we express our solution in a closedform to analyze its features and physical viability. Weconclude by discussing the key results of our investiga-tion in Section 5.
We write the interior of a static spherically symmetricstar by the line element ds = − e ν ( r ) dt + e λ ( r ) dr + r ( dθ + sin θdφ ) , (1)in coordinates ( x a ) = ( t, r, θ, φ ), where ν ( r ) and λ ( r )are two unknown functions. For an anisotropic fluid inthe presence of an electromagnetic field, we assume theenergy momentum tensor in the form T ij = diag( − ρ − E , p r − E , p t + 12 E , p t + 12 E ) , (2)where ρ is the energy density, p r is the radial pressureand p t is the tangential pressure; measured relative tothe comoving fluid 4-velocity u i = e − ν δ i .For the line element (1) and matter distribution (2),the Einstein field equations are obtained as1 r (1 − e − λ ) + 2 λ ′ r e − λ = ρ + 12 E , (3) − r (1 − e − λ ) + 2 ν ′ r e − λ = p r − E , (4) e − λ (cid:18) ν ′′ + ν ′ + ν ′ r − ν ′ λ ′ − λ ′ r (cid:19) = p t + 12 E , (5)∆ = p t − p r , (6) σ = 1 r e − λ ( r E ) ′ , (7)where a prime ( ′ ) denotes derivative with respect tothe radial coordinate r . In the above, E is the electricfield, σ is the charge density and ∆ is the measure ofanisotropy or anisotropic factor. We shall use unitshaving 8 πG = 1 = c .The mass of the gravitating object within a stellarradius r is defined as m ( r ) = 12 Z r ˜ r ρ (˜ r ) d ˜ r. (8)Solutions of the above equations determine the phys-ical behaviour of the anisotropic fluid distribution. Thesystem (3)-(7) comprises five equations in eight un-knowns namely, ν, λ, ρ, p r , p t , E, ∆ and σ . Therefore,it is necessary to choose any three of these variables in-volved in the integration process to solve the system. Solutions to the system can be obtained by makingphysically reasonable choices for any three of the inde-pendent variables. Accordingly, we begin by assuming a particular form for one of the gravitational potentialsas e λ ( r ) = 1 − ar /R − br /R , (9)where a and b are nonzero arbitrary constants and R is the boundary of the star. A similar form ofthe metric potential was earlier used by Nasheeha et al (2020) for the modelling of a neutron star and alsoby Komathiraj and Sharma (2018) for a superdensecharged star. For particular choices of the parame-ters a and b , it is possible to identify the metric ansatzwith the following solutions: (i) charged stellar modelof Komathiraj and Maharaj (2007) for b = 1; (ii) stel-lar model of developed by Maharaj and Leach (1996)for b = 1; (iii) superdense stellar model developed byTikekar (1990) for a = − , b = 1; (iv) Vaidya andTikekar superdense stellar model (Vaidya and Tikekar1982) for a = − , b = 1 and (v) Durgapal and Ban-nerji neutron star model (Durgapal and Bannerji 1983)for a = 1 , b = 1 /
2. In other words, the general form (9)contains a large class of metric potentials which havebeen used to develop physically acceptable relativisticstellar models.Using (4) and (5), for the particular choice (9), weobtain (cid:18) − ar R (cid:19) (cid:18) − br R (cid:19) (cid:18) ν ′′ + ν ′ − ν ′ r (cid:19) − ( b − a ) (cid:16) rR (cid:17) (cid:18) ν ′ + 1 r (cid:19) + b − aR (cid:18) − ar R (cid:19) − (cid:18) − ar R (cid:19) E = (cid:18) − ar R (cid:19) ∆ , (10)which is a highly non linear differential equation withmore than one unknowns. To make the equationtractable, it is convenient at this stage to introduce thefollowing transformation e ν ( r ) = ψ ( x ) , x = 1 − br R . (11)The above transformation helps us to simplify the inte-gration procedure as demonstrated by Maharaj and Leach(1996) and Komathiraj and Maharaj (2007, 2010),amongst others.Under the transformation (11), equation (10) be-comes ( b − a + ax ) ¨ ψ − ax ˙ ψ + (cid:20) a ( a − b ) b + R ( b − a + ax ) b ( x −
1) ( E + ∆) (cid:21) ψ = 0 , (12)in terms of the new dependent and independent vari-ables ψ and x where a dot (.) denotes differentiation with respect to x . Under the transformation, the sys-tem (3) takes the following equivalent form ρ = b ( b − a ) R (3 b − a + ax )( b − a + ax ) − E , (13) p r = bR ( b − a + ax ) − bx ˙ ψψ + a − b ! + 12 E , (14) p t = p r + ∆ , (15) σ = b [2 xE − (1 − x ) ˙ E ] R (1 − x )( b − a + ax ) . (16)The mass function (8) becomes m ( x ) = − R b Z x x p − x ρ ( x ) dx, (17)in terms of the new variable x .Thus, we have essentially reduced the solution of thefield equations to integrating equation (12). The differ-ential equation (12) may be integrated once the elec-tric field E and the anisotropic factor ∆ are known.Since we have the freedom to choose two more vari-ables, we assume particular forms of the electric field E and anisotropic parameter ∆ at this stage. It isnoteworthy that though a variety of choices are possi-ble, the choices must ensure that they are regular, wellbehaved and can generate physically plausible stellarmodels. Keeping this in mind, we choose E ( x ) = αab ( x − R ( b − a + ax ) , (18)∆( x ) = βab ( a − b )( x − R ( b − a + ax ) , (19)where α and β are real constants. It should be pointedout that both the electric field and the anisotropic fac-tor given in (18) and (19) are regular at the centre ofthe star. Our plan is to make use of these assumptionstogether with the potential (9) to generate new class ofsolutions describing a stellar configuration with desir-able physical features. Using (18) and (19), we express(12) in the form( b − a + ax ) ¨ ψ − ax ˙ ψ + ab [(1 + β )( a − b ) + α ] ψ = 0 , (20)which is the master equation of the system and hasto be integrated to find an exact model for a chargedsphere with anisotropic pressure. Note that an un-charged and isotropic stellar solution can be regainedsimply by switching off the charge parameter α = 0 andthe anisotropic parameter β = 0 in (20). We intend tofind new class of solutions for α = 0 and β = 0. et al x = 0 is a regularsingular point of the differential equation (20). There-fore, the solution of the differential equation (20) can bewritten in the form of an infinite series by the methodof Frobenius: ψ ( x ) = ∞ X i =0 c i x i , (21)where c i are the coefficients of the series.To complete the solution we need to find the coeffi-cients c i explicitly. Substituting (21) in (20), we obtainthe recurrence relation a (cid:20) b [(1 + β )( a − b ) + α ] + ( i − i − (cid:21) c i − +( b − a ) i ( i − c i = 0 , i ≥ c and c respectively as: c i = (cid:18) aa − b (cid:19) i i )! i Y k =1 (cid:20) b [( a − b )(1 + β ) + α ]+(2 k − k − c (23) c i +1 = (cid:18) aa − b (cid:19) i i + 1)! i Y k =1 (cid:20) b [( a − b )(1 + β ) + α ]+(2 k − k − c . (24)It is possible to verify the results (23) and (24) by usingmathematical induction.Using (21), (23) and (24), we can now generate thegeneral solutions to (20), for the choice of the electricfield (18) and the anisotropic factor (19), as ψ ( x ) = c ψ ( x ) + c ψ ( x ) , (25)where we have set1 + ∞ X i =1 (cid:18) aa − b (cid:19) i i )! i Y k =1 (cid:20) b [( a − b )(1 + β ) + α ]+(2 k − k − x i = ψ ( x ) (26) x + ∞ X i =1 (cid:18) aa − b (cid:19) i i + 1)! i Y k =1 (cid:20) b [( a − b )(1 + β ) + α ]+(2 k − k − x i +1 = ψ ( x ) . (27)The general solution (25) can be expressed in terms ofelementary functions which is a more desirable form forthe physical description of a charged anisotropic rela-tivistic star. This is possible, in general, because theseries (26) and (27) terminate for restricted values ofthe parameters a, b, α and β so that elementary func-tions are possible.In our work, we develop two sets of general solutionsin terms of elementary functions by imposing the spe-cific restrictions on the quantity b [( a − b )(1+ β )+ α ] for aterminating series. The elementary functions, obtainedusing this method, can be written as polynomials andpolynomials with algebraic functions.We express the first category of solutions to (20) as ψ ( x ) = A n X i =0 γ i ( n + i − n − i )!(2 i )! x i + B ( b − a + ax ) / × n − X i =0 γ i ( n + i )!( n − i − i + 1)! x i +1 , (28)where γ = 4 a [4 n (1 − n ) + 1 + β ]4 an ( n −
1) + α , n (1 − n ) = 1 b [( a − b )(1 + β ) + α ] ,x = 1 − a (1 + β ) + α n (1 − n ) + 1 + β r R . The second category of solution is obtained as ψ ( x ) = A n X i =0 µ i ( n + i − n − i )!(2 i + 1)! x i +1 + B ( b − a + ax ) / n − X i =0 µ i ( n + i )!( n − i − i )! x i , (29)where µ = 4 a (2 − n + β ) a (4 n −
1) + α , − n = 1 b [( a − b )(1 + β ) + α ] ,x = 1 − a (1 + β ) + α − n + β r R . In the above, A and B are arbitrary constants.Thus, we generate two new class of solutions (28) and(29) in terms of elementary functions from the infiniteseries solution (25). It should be stressed that the newclass of solutions holds good for isotropic as well asanisotropic; charged as well as uncharged cases. In thefollowing, we demonstrate how our class of solutions canbe used to regain a wide variety of previously developedwell known stellar solutions: et al (2020) Class: I
We set a = − ˜ b, b = (˜ b − k ) / , α = 0 , and β = 2 k/ ( k − b ) ( n = 1) in (29) so that µ = 8˜ b/ ( k − b ). Equation(29) then takes the form ψ ( x ) = A q ˜ b (2 + ( k − ˜ b )˜ x )(5˜ b − k + 2˜ b (˜ b − k )˜ x )+ B (1 + ˜ b ˜ x ) / , (30)where, x = 1 − a (1+ β ) β − r R = 1 + ( k − ˜ b )2 ˜ x, ˜ x = Cr (= r R ) , and A , B are constants. The solution (30) wasthe first of its class solutions found by Nasheeha et al (2020). The exact solution (30) has been comprehen-sively studied (Nasheeha et al Class: II
We set a = − ˜ b, b = (˜ b − k ) / , α = 0 , and β =7 k/ ( k − b ) ( n = 2) in (28), so that γ = 28˜ b/ ( k − b ).Subsequently, equation (28) becomes ψ ( x ) = A [3( k − b ) + 12˜ b ( k − b )(7 + ( k − ˜ b )˜ x )+ 8˜ b (7 + ( k − ˜ b )˜ x ) ]+ B (1 + ˜ b ˜ x ) / q b − ˜ b (˜ b − k )˜ x, (31)where, x = 1 − a (1+ β ) β − r R = 1 + ( k − ˜ b )7 ˜ x, ˜ x = Cr (= r R ) , and A , B are constants. The so-lution (31) was the second class of solutions obtainedby Nasheeha et al (2020). Note that our solution (31)corrects a minor misprint in the result obtained by(Nasheeha et al If we set β = 0, equation (28) yields ψ ( x ) = A n X i =0 ( − γ ) i ( n + i − n − i )!(2 i )! x i + B ( b − a + ax ) / × n − X i =0 ( − γ ) i ( n + i )!( n − i − i + 1)! x i +1 , (32)where γ = 4 − b bn ( n − α , a + α = b [2 − (2 n − )] ,x = 1 − br R . If we set β = 0, equation (29) yields ψ ( x ) = A n X i =0 ( − µ ) i ( n + i − n − i )!(2 i + 1)! x i +1 + B ( b − a + ax ) / × n − X i =0 ( − µ ) i ( n + i )!( n − i − i )! x i , (33)where µ = 4 − bb (4 n − α , a + α = 2 b (1 − n ) ,x = 1 − br R .Solutions (32) and (33) correspond to the isotropiccharged stellar model of Komathiraj and Maharaj (2010).These solutions reduce to Komathiraj and Maharaj(2007) model if we set b = 1 . If we set b = 1 , α = 0 and β = 0, then equation (28)yields ψ ( x ) = A n X i =0 ( − γ ) i ( n + i − n − i )!(2 i )! x i + B (1 − a + ax ) / × n − X i =0 ( − γ ) i ( n + i )!( n − i − i + 1)! x i +1 , (34)where γ = 4 − n ( n − , a = [2 − (2 n − )] , x = 1 − r R . If we set b = 1 , α = 0 and β = 0, then (29) yields ψ ( x ) = A n X i =0 ( − µ ) i ( n + i − n − i )!(2 i + 1)! x i +1 + B (1 − a + ax ) / n − X i =0 ( − µ ) i ( n + i )!( n − i − i )! x i , (35)where µ = 4 − n − , a = 2(1 − n ) , x = 1 − r R . These two categories of solutions (34) and (35) corre-spond to Maharaj and Leach (1996) model describing arelativistic compact star. The Maharaj and Leach solu-tion has a simple form in terms of elementary functionsand provides a physically reasonable model for neutronstars.
If we set a = − , b = 1 , α = 0 , and β = 0 ( n = 2) in(28), then we get γ = − and subsequently equation(28) yields ψ ( x ) = A (cid:18) − x + 4924 x (cid:19) + B x (cid:18) − x (cid:19) / , (36)where, x = 1 − r R , A and B are constants. Thesolution (36) was found by Tikekar (1990) for the de-scription of compact stars like neutron stars. If we set a = − , b = 1 , α = 0 , and β = 0 ( n = 1) in(29), then we have µ = − and equation (29) becomes ψ ( x ) = A x (cid:18) − x (cid:19) + B (cid:18) − x (cid:19) / , (37)where, x = 1 − r R , A and B are constants. Theexact solution (37), developed by Vaidya and Tikekar(1982), has been widely used for the studies of rela-tivistic compact stars. If we set a = − , b = , α = 0 , and β = 0 ( n = 1) in(29), then we have µ = − and equation (29) yields ψ ( x ) = A (2 − ˜ x ) / (5 + 2˜ x ) + B (1 + ˜ x ) / , (38)where, x = 1 − r R = 1 − ˜ x, ˜ x = Cr (= r R ) , and A and B are constants which is the Durgapal and Bannerji(1983) stellar model. The model has been shown to sat-isfy all the physical requirements of a realistic star andhas got widespread attention for the modeling of rela-tivistic stellar configurations. In the previous section, it has been shown how a largeclass of previously reported solutions can be regainedfrom our general class of solutions. It is interesting tonote that the solutions can also be obtained in simpleanalytic forms which facilitates its physical analysis.This is demonstrated as follows.We set b = a (1 + β ) + αβ − n = 1)in (29) so that we have µ = 4 a ( β − a + α . Equation (29) then yields ψ ( x ) = Ax (cid:18) a ( β − a + α ) x (cid:19) + B (cid:18) a + αβ − ax (cid:19) / , (39)where, x = 1 − a (1+ β )+ αβ − r R . Using (9), (11), (13)-(16),(18) and (19), we obtain e λ = α + 3 a + a ( β − x ( α + a + aβ ) x , (40) e ν = ψ , (41) ρ = (3 a + α )( a + α + aβ )[3 α + a (5 + 2 β + ( β − x )] R ( β − a + α + a ( β − x ] − αa ( β − a + α + aβ )( x − R [3 a + α + a ( β − x ] , (42) p r = − ( a + α + aβ ) R ( β − a + α + a ( β − x ] × " a + α + 2( a + α + aβ ) x ˙ ψψ + αa ( β − a + α + aβ )( x − R [3 a + α + a ( β − x ] , (43) p t = p r + βa (3 a + α )( a + α + aβ )(1 − x ) R [3 a + α + a ( β − x ] , (44) E = αa ( β − a + α + aβ )( x − R [3 a + α + a ( β − x ] , (45)∆ = βa (3 a + α )( a + α + aβ )(1 − x ) R [3 a + α + a ( β − x ] , (46) σ = − αax ( a + α + aβ ) × [3 α + a (5 + 2 β + ( β − x )] R [3 a + α + a ( β − x ] , (47)where ψ is given in (39). The mass function(17) takes the form m ( x ) = − αR a ) tanh − "s a ( β − − x )( a + α + aβ ) − R p ( β − − x )8 a √ a + α + aβ [3 a + α + a ( β − x ] × [12 a ( x − − α − aα (11 + β + 2( β − x )] . (48)The simple closed-form nature of the above solutionfacilitates its physical analysis as discussed below.We note from (40) that e λ ( r = 0) = 1 , ( e λ ) ′ ( r = 0) = 0and from (41) we have e ν ( r = 0) = (cid:20) A (cid:16) a ( β − a + α ) (cid:17) + B (cid:16) a + αβ − + a (cid:17) / (cid:21) , ( e ν ) ′ ( r = 0) = 0.Obviously, the gravitational potentials are regular atthe origin.From (42), we obtain the central density as ρ = ρ ( r = 0) = a + α ) R ( β − , which implies that we must have(3 a + α )( β − > . (49)Using (43) and (44), we obtain the radial and tangentialpressures at r = 0 as p r ( r = 0) = p t ( r = 0)= − a + α + 2( a + α + aβ ) h ˙ ψψ i r =0 R ( β − , where ψ is given in (39). That density and pressureshould be positive puts the following bound on themodel parameters0 > a + α − β > a + α + aβ )( β − (cid:18) A a +3 α +2 aβ a + α ) + B (cid:16) a + α + aββ − (cid:17) (cid:19) × A α − a + 2 aβ a + α + 3 Ba s a + α + aββ − ! . (50)At the boundary of the star ( r = R ), the radial pres-sure must vanish, i.e. p r ( r = R ) = p r ( x = s − a (1 + β ) + αβ − . which yields BA = s a + α − β (1 − a )2 − β (cid:18) β − a + α + aβ (cid:19) × s s , (51) where s = 12(1 − a )( a + α + pβ )[ − α + a (1 + 2 α − β +2 a (1 + β ))] − [2 α + a (6 − a + α ( β − α + a (5 − α + 2 β − a (1 + β ))] and s = 3( a − a + α )[ − α +2 a (9 + 9 a + 8 α ) + aβ ( −
12 + 12 a − α )] . The exterior solution to the Einstein-Maxwell sys-tem for r > R is given by the Reissner-Nordstr¨om lineelement ds = − (cid:18) − Mr + Q r (cid:19) dt + (cid:18) − Mr + Q r (cid:19) − dr + r ( dθ + sin θdφ , (52)where, M and Q are the total mass and charge, respec-tively. Matching of the line element (1) and (52), acrossthe boundary r = R , we have (cid:18) − MR + Q R (cid:19) − = ( a − β − a + α + ( a − β − , (53) (cid:18) − MR + Q R (cid:19) = A s a + α + ( a − β − β × [3 α + a (5 − α + 2 β − a (1 + β ))]3(3 a + α )+ B (cid:18) (1 − a )( a + α + aβ ) β − (cid:19) . (54)These matching conditions and (51) help us to deter-mine the constants A and B explicitly in terms of themodel parameters a, α and β as follows: A = f ( a, α, β ) g ( a, α, β ) , (55) B = h ( a, α, β ) i ( a, α, β ) , (56)where, f ( a, α, β ) = − s − a + α (1 − a )( β − × [2 α − a (9 + 8 α )+ aβ (12 + α ) − a (3 + 2 β )] ,g ( a, α, β ) = 4(3 a + α ) s a + α − (1 − a ) β − β ,h ( a, α, β ) = s − a + α (1 − a )( β − × { [ − α + a ( − a − α ( β − × [ − α + a ( − α − β + 2 a (1 + β ))]+ 12( a − a + α + pβ ) × [ − α + a (1 + 2 α − β + 2 a (1 + β ))] } ,i ( a, α, β ) = 12(3 a + α ) (cid:20) (1 − a )( a + α + aβ ) β − (cid:21) . To analyze behaviour of the physical variables, fora star of radius R = 1 (with 8 πG = c = 1), we set a = − , α = 0 . β = − .
1, which are consistentwith the bounds (49)-(51). Using these values in (55)and (56), we determine the constants as A = 0 . B = 1 . . Making use of these values, we an-alyze the physical features of the model. Figures (1)and (2) show that the gravitational potentials e λ and e ν are continuous, regular and well-behaved at the in-terior of the star. Figure (3) shows that the energydensity ρ is positive, finite and monotonically decreasesradially outward from its maximum value at the centre.The behaviour of radial pressure p r and the tangentialpressure p t are plotted in Figures in (4) and (5) respec-tively which show that both the pressures are positiveand decreasing monotonically while the radial pressurevanishes at the boundary. In Figure (6), we show thatthe electric field intensity E is continuous throughoutthe interior and increases from the centre to the bound-ary, which is physically reasonable. The radial variationof the charge density is shown in figure (7). In Fig-ure (8), radial variation of the anisotropic factor ∆ isshown. We note that ∆ is positive and monotonicallyincreases from the centre until it attains a maximumvalue at the boundary of the star. This profile is similarto that obtained by Maurya and Maharaj (2017) andKomathiraj et al (2019). Figure (9) illustrates that theenergy conditions ρ + p r +2 p t > ρ − p r − p t > v r = dp r dρ and v t = dp t dρ are shown which con-firms that the causality condition is not violated in thismodel, a desirable feature for the modelling of a stellarstructure as pointed out by Delgaty and Lake (1998).For an anisotropic fluid sphere, a potentially stable con-figuration is ensured if we have − ≤ v t − v r ≤ et al ?? )shows the mass function profile within the stellar in-terior which is regular at the centre. Thus, we showthat there exists particular sets of model parametersfor which solution (40) satisfies all the requirements ofa realistic star. Through our investigation, we have provided a generalclass of charged anisotropic relativistic stellar solutionswhich is regular and well-behaved. The most inter-esting feature of the class of solutions is that many e λ Fig. 1
Behavior of metric potential e λ within the stellarinterior. e ν Fig. 2
Behavior of metric potential e ν within the stellarinterior. well known stellar solutions can be regained simply byswitching off the parameters specifying the anisotropyand/or charge distribution in this formulation.It is to be stressed that for physical analysis, we havegenerated one particular closed form solution by suit-ably fixing the model parameters. It will be interestingto probe what other combinations of the model parame-ters can provide new solutions in simple analytic forms.This will be taken up elsewhere.RS gratefully acknowledges support from the Inter-University Centre for Astronomy and Astrophysics (IU-CAA), Pune, India, under its Visiting Research Asso-ciateship Programme. ρ Fig. 3
Radial variation of energy density ρ . p r Fig. 4
Radial variation of radial pressure p r . p t Fig. 5
Radial variation of tangential pressure p t . E Fig. 6
Radial variation of electric field E . σ Fig. 7
Radial dependance of charge density σ . Δ Fig. 8
Radial variation of anisotropy ∆. ρ + p r + p t ρ - p r - p t Fig. 9
Fulfillment of energy conditions. v r = dp r d ρ v t = dp t d ρ Fig. 10
Sound speeds within the stellar interior. - - - - v t - v r Fig. 11
Difference of sound speeds. m Fig. 12
Radial variation of the mass function m ( r ). References