Tolman IV Fluid Sphere in Bigravity

Home , Bimetric gravity

arXiv:2006.07135v1 [gr-qc] 10 Jun 2020 dia. dia. n opc udshr nbgaiy eew aeex- parameter have scale we of Here effect bigravity. the plored in sphere fluid compact ing Abstract Rahaman Farook s.Nwo Singh Newton bigravity Ksh. in sphere fluid IV Tolman al,Pn,Mhrsta412,India Maharashtra-411023, Pune, wasla, httesudsedadteaibtcidxaemore observed are of index be values adiabatic can the lower It and for speed sound analysis. the graphical that the lower by for ported that shows it of three values and for stars model have compact We well-known stars. compact of distribution hsaetn h o.Oecnas hn httepa- the that think also can rameter One EoS. background, the curvature affecting constant thus the from term metric extra background by a of presence γ the the satisfying in still equations are field equations field Einstein’s of tions h o.As EoS. the and Stability Keywords curvature spacetime. scalar background the the reducing stellar of by local enhanced the is of stability structure The gravity forces. under hydrostatic equilibrium condition, and and causality conditions energy the the satisfy all solution The coupling the vanishes. and spacetime Minkowski’s to reduces time email:[email protected] eateto ahmtc,Jdvu nvriy Kolkata, University, Jadavpur Mathematics, of Department Rahaman Farook email:[email protected] eateto ahmtc,Jdvu nvriy Kolkata, University, Jadavpur Mathematics, of Department Sarkar Susmita Khadak- [email protected] Academy, email: Defence National Physics, of Department Singh Newton Ksh. µν oee,tedniyadpesr osmodified does pressure and density the However, . γ µν α n osqetymr h opigsie is stiffer coupling the more consequently and k ≡ epeetTla Vsaeierepresent- spacetime IV Tolman present We ed osie o.Ti li sas sup- also is claim This EoS. stiffer to leads irvt omnI opc tr; star Compact ; IV Tolman ; Bigravity 1 k /k ∞ → 2 sculn osatbtenthe between constant coupling as k ti lose htaltesolu- the all that seen also is It . h akrudd-itrspace- de-Sitter background the , • umt Sarkar Susmita k ntelclmatter local the in • g In- In- µν ogyasbc,N oe ( Rosen N. back, years long A Introduction 1 2012 1978 er1981, year and ( sub- gravity a has massive gravity with bimetric relationship the tle that investigated is It investigated were 1989 aspects various be- and years ( BGTR few proposal within The the researcher the of among the results. famous model a different comes Ein- star yield from collapsed BGTR theories a obtained two for for from result However, that the obtained GTR. from proved stein’s results difference also the no They has star, a ordinary star. obtain an to compact curvature, of constant model us- of by universe of obtained de-Sitter are procedure which the 1985, equations, ing year field the the bimet- In solved additional form an term. the with in ric equations equations field field Einstein’s static of the written initially eea hoyo eaiiy(T)ivligaback- a involving (GTR) metric relativity ground of theory general oei h rmwr fBT in BGTR of were charge framework electric the an in of done field symmetric spherically metric γ rudmetric ground relativity of theory ihrsett h akrudmti.Both metric. derivatives background covariant the the to by phys- respect Einstein’s replaced with of are of derivatives metrics ordinary similar ical the are with equations however, GTR field the BGTR, eat ihmte.I h er1978, year the In matter. with teracts hdkr&Krd 1989 Karade & Khadekar µν r rsn ntefil qain u only but equations field the in present are rpsdanwmdfidter oEinstein’s to theory modified new a proposed ) .Suiso nt iecag n h static the and charge size finite a of Studies ). ; oe Rotbart & Rosen oda oe 1977 Rosen & Goldman g µν hc skona h “ the as known is which , ai Rosen & Falik γ γ µν µν BT rbgaiy” h back- The bigravity)”. or (BGTR sgvre yaparameter a by governed is nadto oteuulphysical usual the to addition in Rosen ( 1979 ( ; 1981 ed Venkateswarlu & Reddy ,rsetvl.I the In respectively. ), apz&Rosen & Harpaz oe 1973 Rosen ( 1980 ; Akrami aeivsiae a investigated have ) orsodn to corresponding ) iercgeneral bimetric Rotbart Baccetti Rosen tal. et , 1974 g g µν , ( ( ( tal. et 2015 µν k 1979 1985 1978 1975 In . and in- ). ) ) ) , 2 charged point particle and then Rosen, himself dis- been renewed interest in bigravity due to their accel- cussed a classical model for elementary particles in erating cosmological solutions (Damour et al. 2002). BGTR (Rosen 1989). Hassan & Rosen (2012) showed that the introduction In the context of the Hassan-Rosen theory, spheri- of a kinetic term for the background metric in the ghost- cally symmetric systems were first studied by Comelli et al. free massive gravity leads to a bigravity theory which (2012), where in particular, the perturbative solutions is also ghost free. A details study of cosmological solu- to the equations of motion was published. In pres- tions in bigravity can be found in refs. Volkov (2012); ence of bimetric gravity, Volkov (2012) performed Strauss et al. (2012); Comelli et al. (2012). an extensive numerical study, and gave conditions for In present paper, we have developed a model of com- the existence of asymptotically flat black hole solu- pact star in bimetric gravity where the metric potentials tions. Star solutions and the so-called Vainshtein are chosen as Tolman IV metric potential. In one of our mechanism was studied by Babichev & Crisostomi previous paper, we have investigated a new model of (2013). Solutions for charged black holes and for ro- anisotropic compact star in (3+1)-dimensional space- tating black holes can be found in Brito et al. (2013), time. The model was obtained in the background of Babichev & Fabbri (2014), respectively. In massive bi- Tolman IV grr metric potential as input and the field gravity, a general review of black holes was proposed by equatios were solved by assuming a suitable expression Babichev & Brito (2222). Enander & Mörtsella pro- for radial pressure pr (Bhar et al. 2016). The present posed the phenomenology of stars and galaxies in mas- paper is organized as follows: In sect. 2, field equations sive bigravity by applying a parameter conditions for in bigravity is developed. In sect. 3 and 4, the model the existence of viable star solutions when the radius of parameters are obtained by solving the field equations the star is much smaller than the Compton wavelength and physical properties are discussed, respectively. In of the graviton. sect. 5, we match our interior spacetime to the exte- Avakian et al. (1991) have studied on superdense rior Schwarzschild line element. Final two sections are celestial objects in generalized bimetric gravity with devoted on a brief discussion. a variable gravitational constant. In the framework of bimetric gravity, the solution for spherically sym- 2 Field equations in bimetric gravity metric self-gravitating anisotropic matter distribution was investigated by Khadekar & Kandalkar (2004) The bigravity modifies the Einstein’s theory by ac- and that solution agrees with the Einstein’s GRT counting a background space-time. Therefore, one can for a physical system compared to the solar system defined the two symmetric tensor g and γ associ- size of universe. A charged fluid was also studied µν µν ated with the two spacetime and are defined as in BGTR (Kandalkar & Gawande 2010). Recently, et al. 2 µ ν 2 µ ν Grigorian (2018) have shown that the masses ds− = gµν dx dx , dsb = γµν dx dx . (1) for superdense compact objects in bimetric theory can be essentially larger compared to the objects in GTR These two metric tensor are having non-vanishing de- depending on the value of the bimetric parameter. terminants. Adopting the notation used by Rosen (1940), λ is the Chistoffel symbol in g different- It is well known that, in the presence of an extra µ ν − iation and Γλ in γ differentiation. The relationship spin-2 field, bimetric theory describes gravitational in- µν − teractions. Recently, Hassan et al. (Hassan & Rosen between these two symbols are given as 2012,a) have proposed a particular bimetric theory (or λ λ λ bigravity) to avoid the ghost instability. This theory µ ν =Γµν + ∆µν , (2) describes a nonlinear interactions of the gravitational n o λ metric with an additional spin-2 field. By assuming where, ∆µν is found to be related to gµν as the Planck mass Mf of the second metric, the ghost 1 ∆λ = gλσ g + g g . (3) instability can be avoid by pushing back to early un- µν 2 µσ,ν νσ,µ − µν,σ observably times. This limit the uses an effective cosmological constant in general relativity (Akrami et al. Now the Riemann tensor related to the two metric func- 2015). tions can be written as It is quite familiar that non-linear bimetric theories λ = λ ∆λ + ∆λ + ∆λ ∆α ∆λ ∆α of gravity suffer from the same Boulware-Deser ghost Rµνσ Pµνσ − µν,σ µσ,ν αν µσ − ασ µν = ∆λ + ∆λ + ∆λ ∆α ∆λ ∆α . (4) instability (Boulware & Deser 1972). To describe the − µν,σ µσ,ν αν µσ − ασ µν interaction of gravity with a massive spin-2 meson, One can easily see that the tensor ∆λ is related to such theories were introduced. Recently, there has µν the normal Riemann curvature tensor. The curvature 3

1.00 450

0.95

0.90 -λ 400 e 3

0.85 fm /

0.80 MeV 350 k = 100 Blue

0.75 ρ = Metric Potentials k 150 Red

0.70 k = 200 Black e ν 300 0.65

2 4 6 8 0 2 4 6 8 r km r km

Fig. 1 Varaition of metric functions with respect to the Fig. 2 Variation of density with respect to the radial co- radial coordinate r for LMC X-4 (M = 1.04M⊙, R = 8.3km ordinate r for LMC X-4 (M = 1.04M⊙, R = 8.3km and and b = 0.001896, B = 0.788065) with (a = 0.00266, k = b = 0.001896, B = 0.788065) with (a = 0.00266, k = 100), 100), (a = 0.0029, k = 150) and (a = 0.003, k = 200). (a = 0.0029, k = 150) and (a = 0.003, k = 200).

tensor of the constant curvature spacetime can Following Falik & Rosen (1980) we can write the Pλµνσ be written as field equations as 1 ′ −ν −λ λ 1 1 3e λµνσ = γµν γλσ γµσγλν . (5) 8πρ = e + + (12) P k2 − r − r2 r2 2k2 ν′ 1 1 3e−ν By using (4), one can also find the difference in the 8πp = e−λ + + (13) Ricci tensor i.e. = which satisfy the r r2 − r2 2k2 Kµν Rµν −Pµν same form of Einstein’s field equations. e−λ ν′2 ν′ λ′ ν′λ′ 3e−ν 8πp = ν′′ + + − + (14). Finally, the field equations in bigravity can be writ- 2 2 r − r 2k2 ten as Rosen (1978) Due to the constant curvature of the background met- Gµν = Sµν 8πTµν , (6) ric, the density and pressure are modified as − where, 3e−ν ρ (r) = ρ(r) (15) e − 16πk2 1 −ν Gµν = Rµν R gµν (7) 3e − 2 pe(r) = p(r) 2 (16) 3 1 − 16πk S = γ g gαβγ (8) µν k2 µν − 2 µν αβ which also satisfies the TOV-equation i.e. Tµν = (ρ + p)uµuν p gµν . (9) ′ − ν dpe (ρe + pe) =0. (17) Now the interior and background de-Sitter space- − 2 − dr time are taken as To analyze in deeper aspects, we will ansatz the Tol- 2 ν 2 λ 2 2 2 2 2 man IV spacetime and discuss its behavior w.r.t. the ds− = e dt e dr r (dθ + sin θ dφ ) (10) − − − scale factor k. r2 r2 1 ds2 = 1 dt2 1 dr2 r2(dθ2 b − k2 − − k2 − + sin2 θ dφ2). (11) 3 Tolman IV solution in bigravity

Here k represents the scale parameter in the de-Sitter The Tolman IV (Tolman 1939) spacetime is given by universe. 2ar2 +1 ds2 = B2(1 + ar2)dt2 dr2 − − (ar2 + 1)(1 br2) − r2(dθ2 + sin2 θ dφ2). (18) − The nature of the metric functions are shown in Fig. 1. 4

30 k = 100 Blue k = 150 Red 25 § ¨© 3 k = 200 Black

fm £ ¤¥ ¦ / 20 2

v 0 ¡ ¢ MeV 15 k = 100 Blue p 0.22 10 k = 150 Red 0.21 5 k = 200 Black 0.20 0 0 2 4 6 8 0 2 4 6 8 r km r km Fig. 4 Varaition of sound speed with respect to the radial Fig. 3 Variation of pressure with respect to the radial coordinate r for LMC X-4 (M = 1.04M⊙, R = 8.3km and coordinate r for for LMC X-4 (M = 1.04M⊙, R = 8.3km b = 0.001896, B = 0.788065) with (a = 0.00266, k = 100), and b = 0.001896, B = 0.788065) with (a = 0.00266, k = (a = 0.0029, k = 150) and (a = 0.003, k = 200). 100), (a = 0.0029, k = 150) and (a = 0.003, k = 200).

For the Tolman IV space-time the density and pres- The causality condition can be verify by computing sure of the stellar conﬁguration can be written as the speed of sound as

2 dp 2 3 2 2 4 2 2 2 2 1 3 2 2 4 v = = (2ar + 1) 4a B k r +2a B k r ρ(r) = 2 4a B k r dρ 16B2k2 (2ar2 + 1) (πar2 + π) h (br2 +4)+6r4 +4a B2k2(br2 +1)+3 r2 + 3br2 +1 +2a2 B2k2r2 13br2 +5 +6r4 2bB2k2 +3 8a 4B2k2r6 +4a3 B2k2r4(br2 + 9) +2 a B2k2 10br2 +3 +6r2 +6bB2k2 +3 6 ih2 2 2 2 2 4 +6r +6a B k r (3br +8)+6 r +2a (19) −1 2B2 k2(6br2+5)+9r2 + 10bB2k2 +3 (23). i 1 3 3ar2 +1 br2 1 To satisfy the causality condition the speed of sound v p(r) = − 8π 2B2k2 (ar2 + 1) − 2ar4 + r2 should be less than unity (in c =1= G unit). The 1 trend of the sound speed is shown in Fig. 4. . (20) −r2 To analyze the energy conditions we can calculate the following parameters: The graphical representations of our obtained den- 6a2br4 +6abr2 + a +2b sity and pressure are shown in Figs. 2 and 3 with re- ρ(r) p(r) = (24) spect to the radial coordinate r for the compact star − 4π(2ar2 + 1)2 LMC X-4. Fig. 1 ρ(r) 3p(r) = 2bB2 The density and pressure gradients are calculated as − 8B2k2(2ar2 + 1)2(πar2 + π) k2(12a3r6 + 23a2r4 + 14ar2 + 3) dρ ar 4 2 2 6 = 8a B k r 3 2 2 4 2 2 2 2 4 dr − 2 2 2 2 2 3 4a B k r 4a (B k r +3r ) 8πB k (ar + 1) (2ar + 1) − − +4a3 B2k2r4(br2 +9)+6r6 +6a2 B2k2r2 12ar2 3 . (25) 2 4 2 2 2 − − (3br +8)+6r +2a 2B k (6br +5)+ For any physical matters, the strong, weak, domi- 2 2 2 9r + 10bB k +3 (21) nant and null energy conditions need to be satisfy i.e. dp ar = 4a3B2k2r4 ρ 0; ρ p 0; ρ 3p 0; ρ p . dr − 2 2 2 2 2 2 ≥ − ≥ − ≥ ≥ | | 8πB k (ar + 1) (2ar + 1) +2a2 B2k2r2(br2 +4)+6r4 +4a B2k2 The results obtained in Eqs. (24) and (25) are shown graphically in Fig. 5. (br2 +1)+3 r2 +2bB2k2 +3 . (22)

400 k = 100 Blue 14 k = 100 Blue k = 150 Red 380 12 k = 150 Red k = 200 Black 3 ρ - pr k = 200 Black

fm 360 10 / ) r (

340 Γ 8

ECs320 MeV 6

ρ - 3p 300 r 4

280 2 0 2 4 6 8 0 2 4 6 8 r km r km

Fig. 5 Varaition of energy conditions with respect to the Fig. 6 Varaition of adiabatic index with respect to the radial coordinate r for LMC X-4 (M = 1.04M⊙, R = 8.3km radial coordinate r for LMC X-4 (M = 1.04M⊙, R = 8.3km and b = 0.001896, B = 0.788065) with (a = 0.00266, k = and b = 0.001896, B = 0.788065) with (a = 0.00266, k = 100), (a = 0.0029, k = 150) and (a = 0.003, k = 200). 100), (a = 0.0029, k = 150) and (a = 0.003, k = 200).

1.0 Now we can deﬁned the adiabatic index, mass func- k = 100 Blue tion and compactness parameter as 0.8 k = 150 Red ) r

( k = 200 Black r m ( )

ρ + p dp u Γ = (26) 0.6

p dρ &

r ) 1 r 0.4 m(r) = 4π ρ(r)r2dr = (

3/2 2 2 2 m 0 4a B k (2ar + 1) Z 0.2 √ar 2a2B2k2r2(br2 +1)+2ar2(bB2k2 + 3) u ( ) 0.0 0 2 4 6 8 +3 3(2ar2 + 1) tan−1(√ar) (27) − r km Fig. 7 Varaition of mass function and compactness pa- r 2m(r) 1 rameter with respect to the radial coordinate for LMC X-4 (M = 1.04M⊙, R = 8.3km and b = 0.001896, B = u(r) = = 3/2 2 2 3 r 2a B k (2ar + r) 0.788065) with (a = 0.00266, k = 100), (a = 0.0029, k = 150) and (a = 0.003, k = 200). √ar 2a2B2k2r2(br2 +1)+2ar2(bB2k2 +3)+ 3 3(2ar2 + 1) tan−1(√ar) . (28) For any physical matter, the Zeldovich’s condi- − tion (Zeldovich & Novikov 1971) has to be satisfy i.e. p /ρ 1, which implies The variation of the above physical quantities can be c c ≤ seen in Figs. 6 and 7. 0 a +2b. (31) ≤ The TOV-equation of hydrostatic equilibrium can be 4 Physical properties of the solution written as ν′ dp The non-singular nature of the solution can be seen (ρ + p) = 0 from the central values of density and pressure. The − 2 − dr central values of these parameter can be written as or Fg + Fh = 0. (32)

6aB2k2 +6bB2k2 +3 ρ = > 0 (29) c 16πB2k2 2a 2b +3/B2k2 p = − > 0. (30) c 16π 6 0.00001 mass as a function of its central can be given as

Fh × -6 5. 10 R 2 18 k = 100 Blue m(ρc) = 1+ br 2 2 3 4 " − 2B k (3b 8πρ)+3 − k = 150 Red − km 0.00000 / 2 i k = 200 Black 3 br 1 F − 2 2 2 2 [2r (3b 8πρc) 3]+3r /B k # -5.×10-6 − − Fg 9B2k2 24πρ 9b 9/2B2k2 c − − − 2 2 2 -0.00001 [2Bpk (3b 8πρc) + 3] 0 2 4 6 8 − 8πρ 1 r km −1 c tan r b 2 2 . (35) r 3 − − 2B k ! Fig. 8 Varaition of various forces in TOV-equation with respect to the radial coordinate r for LMC X-4 (M = 1.04M⊙, The variation of mass with central density is shown in R = 8.3km and b = 0.001896, B = 0.788065) with Fig. 9. It signiﬁes that the solution gain its stability (a = 0.00266, k = 100), (a = 0.0029, k = 150) and when k increases i.e. whenever the scalar curvature of (a = 0.003, k = 200). the background de-Sitter spacetime (R = 12/k2) de- ceases, the stability of the local stellar system is en- Here the gravitational and hydrostatic forces can be hanced. This is because, as k increases the stable range deﬁned as of density increases thereby the change in density dur- ing radial oscillations does not trigger gravitational col- ν′ ar Fg = (ρ + p)= − lapse. − 2 8πB2k2 (ar2 + 1)2 (2ar2 + 1)2

3+4a3B2k2r4 +2a2 B2k2r2(br2 +4)+6r4 5 Matching of interior and exterior boundary 2 2 2 2 2 2 +4a B k (br +1)+3r +2bB k (33) Assume the exterior metric as the Schwarzschild vac- uum for the region r > R i.e. dp ar 3 2 Fh = = 4a B − dr 8πB2k2 (ar2 + 1)2 (2ar2 + 1)2 −1 2 2M 2 2M 2 2 4 2 2 2 2 ds = 1 dt 1 dr k r +2a B k r + − r − − r 2 4 2 2 2 2 2 2 2 2 (br +4)+6 r +4a B k (br +1)+3r r (dθ + sin θ dφ ). (36) − 2 2 +2bB k +3 . (34) Matching at the boundary r = R we get ν(R) −λ(R) 2M The exact profiles of the gravitational force Fg and hy- e = e =1 (37) − R drostatics force Fh for our model are displayed in Fig. p(R) = 0. (38) 8, which indicates that solution represents the equilibrium matter configuration. On using the boundary conditions (37) and (38) we To check the stability of the solution and how the get parameter k affect it, one can adopt the static stability criterion. As per this criterion, the mass of the system bR3 2M a = − (39) must be increasing function of its central density i.e. R2(4M bR3 R) − − ∂m/∂ρc > 0 or otherwise unstable. This insures the 2a2B2k2R2 +2aB2k2 +6aR2 +3 stability of the system under radial perturbations. The b = (40) 2B2k2(aR2 + 1)(3aR2 + 1) R 2M B = − (41) sR (aR2 + 1)

Here we have used M, R and k as free parameters and rest of the constants are found using the above three equations. 7

0.08 1.4

1.2 0.06 1.0

) 0.8 c 0.8 ρ ρ / ( 0.04

k = 100 Blue m 0.6 0.6 k = 100 Blue k = 150 Red k = 150 Red 0.4 k = 200 Black 0.02 0.4 k = 200 Black 0.2

0.2 0.0 0.00 0.002 0.004 0.006 0.008 0.010 0 2 4 6 8

2 ρ c / km r km

Fig. 9 Varaition of mass with respect to the radial central Fig. 11 Varaition of equation of state parameter with re- density ρc for LMC X-4 (M = 1.04M⊙, R = 8.3km and spect to the radial coordinate r for LMC X-4 (M = 1.04M⊙, b = 0.001896, B = 0.788065) with (a = 0.00266, k = 100), R = 8.3km and b = 0.001896, B = 0.788065) with (a = 0.0029, k = 150) and (a = 0.003, k = 200). (a = 0.00266, k = 100), (a = 0.0029, k = 150) and (a = 0.003, k = 200). 0.28

0.26 framework of bigravity as well, the Tolman IV space- 0.24 time behaves in good agreement as in GR. The obtained physical parameters, density and pres- )

r 0.22 k = 100 Blue ( sure are as expected positive, maximum at the centre z k = 150 Red 0.20 and then monotonically decreasing in nature towards k = 200 Black the surface , Figs. 1 (Right) and 2 (Left). The mass and 0.18 the compactness parameter are monotonically increas- 0.16 ing in nature toward the surface, shown in Fig. 4 (Left).

0 2 4 6 8 Therefore, all the physical parameters are well-behaved r km and physically acceptable. satisfies the TOV-equation even in the presence of the background spacetime, (see Fig. 10 Varaition of redshift with respect to the radial Fig. 4, Right). Hence, the solution can represent static coordinate r for LMC X-4 (M = 1.04M⊙, R = 8.3km and and equilibrium relativistic fluid spheres. The nature of b = 0.001896, B = 0.788065) with (a = 0.00266, k = 100), the equation of state parameter (EoS) i.e. p/ρ < 1 indi- (a = 0.0029, k = 150) and (a = 0.003, k = 200). cates the solution represents the physically acceptable matter distribution (Rahaman et al. 2010). Now the red-shift and equation of state parameter of The scale parameter k can affect the solution and compact stars can be determined as thus affect the corresponding equation of state (EoS) or the internal compositions of the compact fluid ob- 1 z(r) = e−ν/2 1= 1 (42) ject. We have provided the central values of some phys- 2 − B√1+ ar − ical quantities in Table 2, whereas the Table-1 display p ω = 1. (43) the compatible constant parameter to model SMC X-4, ρ ≤ LMC X-1 and PSR J1614-2230. As we can see, smaller Fig. 10 and 11 clarifies the behavior of redshift function value of k corresponds to stiffer EoS i.e. k = 33 along and the equation of state parameter with respect to the with the given parameters, corresponds to mass and ra- radial coordinate function r respectively. dius of PSR J1614-2230 (1.97M⊙, 9.69km) with compactness parameter of 0.407 and for k = 200 corresponds to LMC X-4 (1.04M⊙, 8.3km) with u =0.251. 6 Results and discussions This claim is further supported by the graphical representations. The velocity of sound and central values In this article, we have explored the behavior of the Tol- adiabatic index are more for k = 100 than k = 200 (see man IV solution in bimetric gravity describing relativis- Figs. 2 Right and 3 Right). This suggest that as k tic fluid sphere. To demonstrate the effect of the scale increases the stiffness of the solution decreases. parameter k on the solution we used graphical meth- Furhter, the solution also satisfy the causality and ods for a specific compact star i.e. LMC X-4. In the all the energy conditions (see Figs. 2 Right, 3 Left). 8

Table 1 All the parameters corresponds to well-behaved solution representing few well-known compact stars. −2 −2 Objects a km b km B k M/M⊙ R km Mobs/M⊙ Robs km SMC X-4 0.003 0.0020484 0.757390 80 1.29 8.831 1.29 0.05 8.831 0.09 ± ± LMC X-4 0.003 0.0018955 0.788065 200 1.04 8.3 1.04 0.09 8.301 0.2 ± ± PSR J1614-2230 0.002 0.00332169 0.706809 33 1.97 9.69 1.97 0.04 9.69 0.2 ± ±

Table 2 All the parameters corresponds to well-behaved solution representing few well-known compact stars.

Objects u zc zs ρc ρs pc Γc g/cm3 g/cm3 dyne/cm2 SMC X-4 0.292 0.32 0.19 8.34 1014 5.30 1014 6.54 1034 2.592 × × × LMC X-4 0.251 0.269 0.155 7.89 1014 5.16 1014 5.59 1034 2.726 × × × PSR J1614-2230 0.407 0.415 0.298 10.03 1014 7.40 1014 6.99 1034 3.601 × × ×

Consequently, the solution representing matter distri- distances (scale of clusters of galaxies) the scale param- bution is physical. The stability of compact star is eter affect the field equations and thus the dynamics. one of the most vital requirement, for that, we have As mentioned by Harpaz & Rosen (1985), all the so- focused to discuss the stability with respect to the vari- lutions of field equations in general relativity does satis- ation of adiabatic index Γ inside the compact star. For fies the field equations with a background and the same a Newtonian fluid, stable configuration can be achieved is also supported by the current work with more details if Γ 4/3 (Bondi 1964). In our model, the stable analysis. The internal structure of compact stars can ≥ nature of the configuration can be convinced from the also be influence by the background metric. We have graphical representation of the Γ, provided in Fig. 3 shown that the local structures can also be influence by (Right). Further, the solution also satisfies the static the background spacetime. Hence, the bimetric GTR stability criterion showing that the stability is enhance can inspire many researchers and also can contribute to when the curvature of the background metric decreases new physics in future. (Fig. 5). Therefore, the stability will be maximum when assumed a flat or Minkowski’s spacetime. Acknowledgement

7 Conclusion Farook Rahaman would like to thank the authorities of the Inter-University Centre for Astronomy and As- Since the local spacetime interior to the compact star trophysics, Pune, India for providing research facili- is part of the global structure in the universal scale, the ties. Susmita Sarkar is grateful to UGC (Grant No.: physics of such object can be influence by the scale pa- 1162/(sc)(CSIR-UGC NET , DEC 2016)), Govt. of In- rameter of the universe. Even though the effect of the dia, for financial support. scale parameter may be small, however, the physics of such small perturbations can be interesting and worth complete analysis. Harpaz & Rosen (1985) have shown Conflict of interest that there is possible to exist a configuration in hydrostatic equilibrium for a collapsing star filling its The authors declare no conflict of interest. Schwarzschild sphere when no such configuration exist in Einstein’s gravity. It is also mentioned that in bigravity even leads a cosmology without a “Big Bang” (Rosen 1980). Rosen (1980) also solved the field equations with background metric near the Schwarzschild sphere and found that the field differs from that of the Einstein’s GTR. Instead of a black hole they have obtained an impenetrable sphere. It is also found that the bigravity is identical with the ordinary general relativity at large distances (scale of solar system), however, at very large 9 References

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