On the possibility of traversable wormhole formation in the Galactic halo in the presence of scalar field
aa r X i v : . [ phy s i c s . g e n - ph ] A ug Noname manuscript No. (will be inserted by the editor)
On the possibility of traversable wormhole formation inthe Galactic halo in the presence of scalar field
B. C. Paul
Received: date / Accepted: date
Abstract
In the paper we obtain traversable wormhole (TW) solutions in theEinstein gravity with a functional form of the dark matter in the galactic halo. Thedark matter model is pseudo isothermal which is derived from modified gravity.For a given central density the possibility of TW is explored determining the shapefunctions. The null energy condition (NEC) and the weak energy conditions in themodel is probed. We also study the existence of TWs considering homogeneousscalar field in addition to the dark matter halo in the galaxies. An interestingobservation is that TW solutions exist even if NEC is not violated in the presenceof scalar field.
General Theory of Relativity (GTR) permits wormhole solutions of the Einsteinfield equations for a special composition of matter different from perfect fluid.Wormholes are tunnel like objects which may connect different spacetime re-gions within the universe, or different universes [1, 2]. In recent times the studyof traversable wormholes (TW) has attracted a lot of interest with the pioneeringwork of Ellis [3, 4], Bronnikov [5] including the seminal work of Morris and Thorne[6]. The geometry of TW requires exotic matter concentrated at the wormholethroat implying violation of the energy conditions, namely null energy condition(NEC) [7]. It is speculated that the exotic matter can exist in the context of quan-tum field theory. The concept of dark matter, a hypothetical form of matter thatmakes up about 25 % of the matter composition of the Universe is predicted fromobservations. A number of possible candidates from particle physics and super-symmetric string theory such as axions and weakly interacting massive particles(WIMP) are considered to describe as the candidate of the dark matter though
B. C. PaulE-mail: [email protected] of Physics, University of North Bengal, Siliguri, Dist. : Darjeeling 734 013, WestBengal, India B. C. Paul there is no direct experimental detection of dark matter. Nevertheless, its exis-tence are hinted in the galactic rotation curves [8], the galaxy clusters dynamics[9], and also at cosmological scales of anisotropies encoded in the cosmic microwavebackground measured by PLANCK [10].In the literature [11, 12, 13] considering characteristics of dark matter halos andgalaxies formation of traversable wormholes are considered based on the Navarro-Frenk-White (NFW) profile [14] of matter distribution. TWs may form in theouter halo of galaxies which are interesting to describe time machines. Rahamanet al. [15] first proposed the possible existence of wormholes in the outer regions ofthe galactic halo based on the NFW density profile, then they used the UniversalRotation Curve (URC) dark matter model to obtain analogous results for the cen-tral parts of the halo [16], DM has been adapted as a non-relativistic phenomenonsuch as NFW profile and King profile to study WH construction [20]. Recently,Jusufi et. al. [21] pointed out the possibility of traversable wormhole formationwith a Bose-Einstein condensation dark matter halo, however, in this case approx-imate solutions are found. The existence of TW in the spherical stellar systemsbased on the fact that the dark matter halos produced in computer simulationsare best described by such a profile is also investigated. Assuming a number ofparameters that exists in the Einasto profile [22, 23] it is shown that TWs in theouter regions of spiral galaxies are possible while the inner part regions prohibitssuch formations. The Bose-Einstein Condensation dark matter (BEC-DM) modelshows that the model is more realistic on the small scales of galaxies comparedto the CDM model. For instance, the interactions between dark matter particlesare very strong in the inner regions of galaxies, and thus the dark matter will nolonger be cold. For the BEC-DM model, the dark matter density profile can bedescribed by the Thomas-Fermi (TF) profile [24]. The possibility of formation ofTWs in the dark matter halo is studied with isotropic pressure with TF profile[25]. The BEC-DM model predicts much less dark matter density in the centralregions of galaxies compared to that found with the NFW profile. There existsanother class of DM described by pseudo isothermal (PI) profile in addition to theCDM model and the BEC-DM model which is associated with the modified grav-ity, such as Modified Newtonian Dynamics (MOND) [26]. In the MOND model,the dark matter density profile is given by ρ DM = ρ (cid:16) rR c (cid:17) (1)where ρ is the central dark matter density and R c is the scale radius. Usingthe above formalism, we adapted the DM density distribution and used to studywormhole construction for MOND. Subsequently we introduce homogeneous scalarfield in addition to MOND dark matter to investigate the TW models and thematter required for their existence.The present paper is organized as follows. In Section II, the Einstein gravityand field equations are obtained. The traversable wormhole solutions are obtainedwith MOND dark energy profile. In Section III, the existence of wormhole wthMOND and scalar field is discussed and finally a brief discussion in Section IV. itle Suppressed Due to Excessive Length 3 The Einstein field equation is given by R µν − g µν R = T µν (2)where µ, ν = 0 , , , T µν = diag ( − ρ, P r , P t , P t ) is the energy-momentum ten-sor, in gravitational unit c = 8 πG = 1. The spacetime metric of a sphericallysymmetric traversable wormhole is given by ds = − e Λ ( r ) dt + dr − α ( r ) + r ( dθ + sin θdφ ) (3)where Λ is the redshift function. For wormhole we consider α ( r ) = b ( r ) r , where b ( r )is the shape function. In order to ensure a wormhole to be traversable, there shouldbe no event horizon. In this case Λ ( r ) should be finite which tends to zero when r → ∞ . The geometry of the wormhole is determined by the shape function b ( r ),which at the throat of the wormhole r = r should satisfy the condition b ( r ) = r .To keep the wormhole’s throat open, the shape function b ( r ) should satisfy theflare out condition b ( r ) − rb ′ ( r ) b ( r ) > b ′ ( r ) < ρ ( r ) = b ′ ( r )8 πr (4)8 πP r ( r ) = − b ( r ) r + 2 (cid:18) − b ( r ) r (cid:19) Λ ′ ( r ) r (5)8 πP t ( r ) = (cid:18) − br (cid:19) (cid:20) Λ ′′ ( r ) + Λ ′ − UΛ ′ − Ur + Λ ′ r (cid:21) (6)setting U = rb ′ ( r ) − b ( r )2 r ( r − b ( r )) where ρ ( r ) is the energy density for dark matter, P r and P t are radial and transverse pressures. For a flat rotation curve for the circular stablegeodesic motion in the equatorial plane of a galaxy one finds e Λ ( r ) = E r l . (7)In the above l = 2( v φ ) where v φ represents the rotational velocity and E is anintegration constant for a large r [16, 17, 18]. For a typical galaxy it is found that v φ ∼ − (300 km/s) within 300 kpc [19]. We assume that the density of thewormhole matter is described by the density profile of the MOND. We considercritical value E = R lc where R c is the scale radius, the spacetime metric can bewritten as ds = − (cid:16) rR c (cid:17) l dt + dr − b ( r ) r + r ( dθ + sin θdφ ) (8)Solving Einstein field eq. (4) using the DM density given by eq. (1), we obtain b ( r ) = ρ R c h rR c − tan − rR c i + C (9)where C is an integration constant. B. C. Paul H r L Fig. 1
Variation b ( r ) with r in the unit of R c for ρ R c = 1 (red), 2 (green), 3 ( blue ) (cid:144) r Fig. 2
Radial variation of b ( r ) r for ρ R c == 0 . thickline ), 1 ( blue ), 1 . orange ),1 . green ) - Ρ+R _r Fig. 3
Variation NEC with r in the unit of R c with ρ R c = 1 for l = 0 . green ) 1 ( blue ),2 ( red ) Using the boundary condition b ( r ) = r , the integration constant C can befixed for the shape function which is given by b ( r ) = ρ R c h rR c − r R c − (cid:16) tan − rR c − tan − r R c (cid:17)i + r , (10)where r represents the throat of the wormhole. The flaring out condition is re-quired to be valid to keep the mouth of the wormhole open which is satisfied hereas b ′ ( r ) = ρ R c r R c (cid:16) r R c (cid:17) < itle Suppressed Due to Excessive Length 5 - - Ρ+R _r Fig. 4
Variation NEC with r for R c with l = 2 for ρ R c = 1 ( red ), 1 . green ), 1 . cyan ),1 . blue ) DM < Fig. 5
Variation mass profile with r in unit of R c with 4 πρ R c = 1 ( green ), 5 ( red ), 10 ( Blue ) for ρ R c ∼
1. In Fig. (1) we plot radial variation of b ( r ) with different ρ R c . It isfound that as the central density is increased b ( r ) is found to increases for a givenradial distance. The radial variation of the flaring out condition is checked in Fig.(2) it is found that b ( r ) r < ρ R c ≤ T µν U µ U ν ≥ i.e., ρ ≥ , ρ ( r ) + P r ≥ U µ denotes the time like vector. This means that local energy density ispositive and it gives rise to the continuity of NEC, which is defined by T µν k µ k ν ≥ i.e., ρ ( r ) + P r ≥ k µ represents a null vector. The NEC is checked in Fig. (3), it is found thatnear the throat NEC is not obeyed but away from the throat it always satisfiedwhich is shown for different power law index ( l ) in eq. (7). We note that for l = 2,NEC is always obeyed for the central density ρ < . R c . It is evident from Fig. (4)that traversable wormholes exists with exotic matter near the throat but awayfrom the throat exotic matter is not required in a low central density galaxy butfor higher central density galaxies initially there is exotic matter then again exoticmatter followed by normal matter at r → ∞ .The mass profile of the galactic halo is M DM ( r ) = 4 π Z r ρ DM ( r ′ ) r ′ dr ′ (14) B. C. Paul which yields M DM ( r ) = 4 πρ R c h rR c − tan − (cid:16) rR c (cid:17)i . (15)In Fig. (5), it is observed that dark matter increases with the increase in radialsize in unit of R c for increasing values of 4 πρ R c = m . It is evident that as themass parameter ( m ) is increased From the eq. (15), one can find the tangentialvelocity [ ? ] v tg ( r ) = GM DM ( r ) r for a test particle moving in the dark halo given by v tg ( r ) = 4 πρ R c (cid:20) − R c r tan − (cid:16) rR c (cid:17)(cid:21) , (16)which represents a constant rotational curve away from the centre of the galaxy.The rotational velocity of a test particle within the equatorial plane is determinedby v tg ( r ) = rΛ ′ ( r ) (17)Combining eq.(17) with the expression for the test particle moving in the darkhalo given by eq. (16) we get rΛ ′ ( r ) = 4 πρ R c (cid:20) − R c r tan − (cid:16) rR c (cid:17)(cid:21) . (18)On integrating Λ ( r ) = 4 πρ R c ln rR c r (cid:16) rR c (cid:17) − R c tan − (cid:16) rR c (cid:17) r + C (19)where C is an integration constant. Thus the g component of the metric tensorbecome e Λ ( r ) = De πρ R c ln rRc r ( rRc ) − R c tan − ( rRc ) r +2 C , (20)the new integration constant D can be absorbed by rescaling t → Dt . For finiteredshift we get a constraint equation which determines C as follows lim r → R c e Λ ( r ) = 1 . The limiting value lim r → R c e πρ R c ln rRc r ( rRc ) − R c tan − ( rRc ) r +2 C = 1determines the constant as C = 2 πR c (cid:16) π ln (cid:17) . (21)The wormhole metric is not asymptotically flat, therefore, we employ matchingconditions by truncating the wormhole metric at radius R and connecting withthe exterior Schwarzschild black hole metric as the later corresponds to vacuum itle Suppressed Due to Excessive Length 7 solution and asymptotically flat. Now imposing the matching condition at r = R ,we get e πρ R c ln RRc r ( RRc ) − R c tan − ( RRc ) R +4 πR c ( π + ln ) = 1 − MR (22)1 − b ( R ) R = 1 − MR (23)where the active mass of the galaxy ( M ) is determined as M = ρ R c (cid:20) RR c − r R c − tan − RR c + tan − r R c (cid:21) + r ρ R c ∼
1. The eqs. (22) and (24) implicitly provide thevalue of truncated radius ( R ) where the matching occurs. Note that ρ representsgalactic dark matter in the unit of M DM halo/kpc , r has units of M DM halo and R and r are in the units of kpc . From eq. (22), R c is determined for density atthe throat of the wormhole knowing the central dark matter density ρ . In this section we consider wormhole solutions in the presence of scalar field andMOND dark matter density profile. Therefore the energy momentum tensor con-sists of two terms T µν = T µνDM + T fµν . To determine the second part of stressenergy tensor we consider a minimally coupled massless scalar field described bythe Lagrangian: L = 12 √− gg µν φ ; µ φ ; ν . (25)The equation of motion for φ is (cid:3) φ = 0 . (26)The stress energy tensor for the scalar field is obtained from eq. (25) which is T fµν = φ ; µ φ ; ν − g µν g ση φ ; σ φ ; η . (27)The conservation eq. (26) of φ is given by φ ” φ ′ + 12 (1 − br ) ′ (1 − br ) + 2 r = 0 . (28)On integrating we get φ ′ r (cid:18) − br (cid:19) = φ π (29)where φ o >
0. Now the Einstein field equations are given by b ′ ( r )8 πr = ρ ( r ) + 12 φ ′ (cid:18) − b ( r ) r (cid:19) (30) − b ( r )8 πr + (cid:18) − b ( r ) r (cid:19) Λ ′ ( r )4 πr = P r ( r ) − φ ′ (cid:18) − b ( r ) r (cid:19) (31) B. C. Paul φ r NEC ( r ≥ r )0.05 0.376 √ √ √ √ Table 1
NEC for ρ R c = 10 and throat radius r = 0 .
5, ”X” means violation and ” √ ” meanssatisfied Φ o - Fig. 6
Variation of NEC with r in unit of R c with 4 πR c = 1 and different φ π (cid:18) − br (cid:19) (cid:20) Λ ′′ ( r ) + Λ ′ − UΛ ′ − Ur + Λ ′ r (cid:21) = P t ( r ) − φ ′ (cid:18) − b ( r ) r (cid:19) (32)setting U = rb ′ ( r ) − b ( r )2 r ( r − b ( r )) where ρ ( r ) is the energy density for dark matter, P r and P t are radial and transverse pressures. Using the solution given by eq. (29) in thefield eq. (30) we get b ′ ( r )8 πr = ρ ( r ) + φ πr . (33)Integrating we determine the shape function as b ( r ) = ρ R c h rR c − r R c − (cid:16) tan − rR c − tan − r R c (cid:17)i + φ (cid:18) r − rrr (cid:19) + r (34)For Λ →
0, no horizon occurs and the radial variation of NEC is drawn in Fig. (6)for a range of values of φ . It is found that near the throat NEC is violated butaway from the throat it is obeyed which depends on φ . The NEC is checked for agiven φ R c and throat radius numerically which are tabulated in the Tables I-II.It is evident that as φ R c is increased NEC is satisfied at lower φ of the scalarfield. In Table-III, we display the values of φ for a given throat radius r withdensity parameter of the galaxy. It is found that if the throat radius is big then itle Suppressed Due to Excessive Length 9 φ r NEC ( r ≥ r )0.10 0.339 √ √ √ √ Table 2
NEC for ρ R c = 8 and throat radius r = 0 .
5, ”X” means violation and ” √ ” meanssatisfied ρ R c r ≥ φ
10 0.5 0.2510 0.4 0.0658 0.5 0.56 0.5 0.0515 0.5 ∼ Table 3
Tabulation of φ with ρ R c and radial distance r from which NEC is satisfied. one needs a scalar field with high φ scalar field value for accommodating TWwith normal matter for a given density of the galactic halo. It is also noted thatfor same throat radius if the density is high then one requires large value of thescalar field. In the present paper we study the possibility of traversable wormhole in the Ein-stein gravity with MOND dark matter profile in the presence and absence of scalarfield. It is found that in the absence of scalar field one requires exotic matter at thethroat of the wormhole. It is also noted that there exists two scenario (i) exoticmatter at the throat and then normal matter away from the throat as NEC isobeyed and (ii) exotic matter at the throat, subsequently away from the throatNEC violates followed by the region where NEC is obeyed depending on the valueof the central density. It is also found that if the central density is large the scaleradius will be low for a given galactic halo mass. In the presence of homogeneousscalar field in addition to dark matter represented by MOND, we obtain a classof TW with exotic or without exotic matter depending on the values of the scalarfield φ and density of the galactic halo. The later result represents a new class ofTW not shown earlier. The TW in higher dimensional gravity [27] is important toprobe in this context which will be taken up elsewhere. The author would like to thank IUCAA , Pune and IUCAA Centre for AstronomyResearch and Development (ICARD), NBU for extending research facilities andDST-SERB Govt. of India (File No.:EMR/2016/005734).
References
1. J. A. Wheeler,
Phys. Rev. , 511 (1995)2. C. W. Misner, J. A. Wheeler, Ann. Phys. , 525 (1957)3. H. G. Ellis, J. Math. Phys. , 104 (1973).4. H. G. Ellis, J. Math. Phys. , 520 (1974)(Erratum).5. K.A. Bronnikov, Acta Phys. Polon.
B 4 , 251 (1973)6. M. S. Morris, K. S. Thorne,
Am. J. Phys. , 395 (1988)7. M. Visser, Lorentzian Wormholes (AIP Press, New York, 1996).8. V. C. Rubin, Jr. W. K. Ford and N. Thonnard, Astro. Phys. J. , 471 (1980).9. F. Zwicky,
Helvetica Physica Acta , 110 (1933).10. P. A. R. Ade, N. Aghanim, et al., A& A , A13 (2016).11. F. Rahaman, P.K.F. Kuhfittig, S. Ray, N. Islam, Eur. Phys. J. C , 2750 (2014).12. F. Rahaman, P. Salucci, P. K. F. Kuhfittig, S. Ray, M. Rahaman, Ann. Phys. , 561(2014).13. N. Sarkar, S. Sarkar, F. Rahaman, P. K. F. Kuhfittig, G. Khadekar
Mod. Phys. Lett. A , 1950188 (2019).14. J. F. Navarro, C. S. Frenk, S. D. M. White, Astrophys. J. , 563 (1996).15. F. Rahaman, G. C. Shit, B. Sen, S. Ray,
Astrophys. Space Sci.
Ann. Phys. , 561(2014).17. K. K. Nandi et. al. , Mon. Not. R. Astron. Soc. , 2079 (2009).18. S. Chandrasekhar,
Mathematical Theory of Black Holes (Oxford Classic Texts) (1983).19. U. Nucamendi, M. Salgado, D. Sudarsky,
Phys. Rev.
D 63 , 125016 (2001).20. S. Islam, F. Rahaman, A. Ovgun, M. Halilsoy,
Canadian Journal of Physics , , 241(2019).21. K. Jusufi, M. Jamil, M. Rizwan, Gen Relativ Gravit. , 102 (2019).22. J. Einasto, Tr. Inst. Astrofiz. Alma-Ata , 87 (1965).23. J. Einasto, U. Haud, Galaxy Astron. Astrophys. , 89 (1989).24. C. G. Bhmer, T. Harko,
J. Cosmol. Astropart. Phys. , 025 (2007)25. Z. Xu, M. Tang, G. Cao, S-N, Zhang,
Eur. Phys. J. C , 70 (2020).26. K.G. Begeman, A.H. Broeils, R.H. Sanders, Mon. Not. R. Astron. Soc. , 523 (1991).27. P. K. Chattopadhyay and B. C. Paul,
Pramana74