Phantom Wormholes in (2+1)-dimensions
PPhantom Wormholes in (2+1)-dimensions
Mubasher Jamil ∗ and M. Umar Farooq † Center for Advanced Mathematics and Physics,National University of Sciences and Technology, Rawalpindi, 46000, Pakistan
Abstract:
In this paper, we have constructed a (2+1)-dimensional wormholeusing inhomogeneous and anisotropic distribution of phantom energy. We have de-termined the exact form of the equation of state of phantom energy that supportsthe wormhole structure. Interestingly, this equation of state is linear but variableone and is dependent only on the radial parameter of the model.
Keywords:
Lower dimensional gravity; Wormhole; Phantom energy ∗ Electronic address: [email protected] † Electronic address: m˙[email protected] a r X i v : . [ g r- q c ] J a n A typical wormhole is characterized by a tunnel in spacetime connecting two arbitraryspacetime sections. These sections could either belong to the same spacetime or to twodifferent spacetimes. The wormhole geometry arises naturally as a solution of the Einsteinfield equations [1–3]. Interest in wormhole physics was initiated when Morris and Thorneinvestigated the wormhole structure and proposed that the material required to constructit has to be exotic, i.e. its (negative) radial pressure and energy density must satisfy theinequality | p | > ρ [4]. They also concluded that this structure could also serve as a timetravel machine if it is horizon-free.From the cosmological perspective, a candidate for exotic matter exists namely the phan-tom energy. Presently it is well-motivated from the observational data that the observableuniverse is pervaded with the phantom energy, which is characterized by ω = p/ρ < − θ is fixed so that dθ = 0. Consequently, this reduces the complexity of the field equations.The metric of a (2+1)-dimensional Morris-Thorne (MT) wormhole is given by [13–15] ds = − e f ( r ) dt + 11 − b ( r ) r dr + r dφ , (1)where f ( r ) is called potential function while b ( r ) is the shape function. These functions arearbitrary functions of radial coordinate r and will be determined below for a specific choiceof matter distribution. The radial coordinate has a range that increases from a minimumvalue at r , corresponding to the wormhole throat, and extends to infinity. For the wormholeto be traversable, conditions commonly termed stability and traversability, are imposed onthese two functions, namely: f ( r ) must be bounded for all values of r ; b (cid:48) ( r ) < r = r ; b ( r ) < r for all r > r and b/r → | r | → ∞ . The stress energy components in anorthonormal frame of reference are T = ρ , T = p and T = p t . Here ρ is the energydensity, p is the radial pressure while p t is the transverse pressure.The Einstein field equations become (units are c = 1 = G ) ρ ( r ) = b (cid:48) r − b πr , (2) p ( r ) = (1 − b/r ) f (cid:48) πr , (3) p t ( r ) = (1 − b/r )8 π (cid:104) f (cid:48)(cid:48) − ( b (cid:48) r − b )2 r ( r − b ) f (cid:48) + f (cid:48) (cid:105) . (4)Here prime ( (cid:48) ) denotes differentiation with respect to r . The energy conservation equationis obtained by evaluating T AB ; A = 0, with A, B = 0 , ,
2. It gives p (cid:48) + f (cid:48) ρ + (cid:16) f (cid:48) + 1 r (cid:17) p − p t r = 0 . (5)Note that in the above equations (3) to (5), the function f must not be a constant (i.e. f (cid:48) (cid:54) = 0) otherwise the field equations become identically zero. Below we shall determineexplicit form of f for a specific choice of two parameters. To solve the field equations, wechoose the following ansatz for the shape function and pressures b ( r ) = b r m , m = 0 , , , ... (6) p t = αp, (7)where α and b are constants. Notice that α is dimensionless while b possesses dimensionsof ( length ) m +1 . It is easy to check that Eq. (6) satisfies the stability conditions for thewormhole. The second ansatz (7) says that the ratio of transverse to radial pressure willremain constant although both can vary differently. Our task is to find ρ , f , p and p t usingEqs. (2) to (7).Inserting (6) in (2), we have ρ ( r ) = − b ( m + 1)16 πr m +3 . (8)Since ρ is always positive so we require b <
0. Making use of Eqs. (2) to (7) and aftersimplification, we arrive at f (cid:48)(cid:48) + (cid:104) b ( m + 1)2 r ( r m +1 − b ) − αr (cid:105) f (cid:48) + f (cid:48) = 0 . (9)To solve this equation, we rewrite it as f (cid:48)(cid:48) f (cid:48) − m + 12 (cid:16) r − r m r m +1 − b (cid:17) − αr = − f (cid:48) . (10)Integrating it we get f (cid:48) e f = Cr α + + m √ r m +1 − b , (11)where C is a constant of integration and for the sake of convenience we fix C = 1. Integrationonce more leads to f ( r ) = ln r (3+ m +2 α ) (cid:113) − r m b F (cid:16) m +2 α m , , m +2 α m , r m b (cid:17) √ r m − b (3 + m + 2 α ) . (12)Here F is a hypergeometric function representing a series expression. It should be notedthat logarithmic form for f ( r ) have been obtained for a MT wormhole in (3+1)-dimensionsas well [10]. Also notice that f ( r ) does not give a finite value as r → ∞ , so the solution isnot asymptotically flat. Hence we may match this interior solution to an exterior vacuumspacetime at a junction radius R [17]. Notice that in (2+1) dimensions, the only exteriorvacuum solution is the stationary BTZ spacetime [18, 19], given by ds = − (cid:16) − M + r l (cid:17) dt + (cid:16) − M + r l (cid:17) − dr + r dφ . (13)Here M corresponds to mass of the spherically symmetrical object while l = 1 / √− Λ > < g AB ,across a surface S , i.e. g int AB | S = g ext AB | S . (14)The wormhole metric is continuous from the throat radius r = r to a finite distance r = R .Explicitly Eq. (14) can be written as g int00 | S = g ext00 | S , (15) g int11 | S = g ext11 | S . (16)Notice that g is already continuous, so we don’t need any matching equation for it. Thelast two equations yield respectively2 R (3+ m +2 α ) (cid:113) − R m b F (cid:16) m +2 α m , , m +2 α m , R m b (cid:17) √ R m − b (3 + m + 2 α ) = − M + R l , (17)1 − b R m = − M + R l . (18)Here M now refers to the mass of wormhole. Using (6) and (12) in (3), we get p = − b r − (3+ m ) (cid:113) − r m b (3 + m + 2 α )16 π F (cid:16) m +2 α m , , m +2 α m , r m b (cid:17) . (19)Putting (19) in (7), we obtain p t = − b r − (3+ m ) (cid:113) − r m b α (3 + m + 2 α )16 π F (cid:16) m +2 α m , , m +2 α m , r m b (cid:17) . (20)Alternatively, Eq. (20) can be obtained by inserting (6) and (12) in (4). In Fig. 1 and 2,we have plotted the magnitudes of the radial and the transverse pressures against the radialcoordinate. These show that both the pressures have arbitrary large values near the throatwhile these vanish in the asymptotic limit of r . This shows that the matter distributionalso satisfies the condition of asymptotic flatness, consistently with the wormhole geometry.The difference of radial and transverse pressures represents surface tension which plays verycrucial role in compact stars [20]. Fig. 3 shows that the behavior of surface tension isanalogous to the two pressures.Comparison of Eqs. (8) and (19) yields a relationship between pressure and energydensity, given by p = (cid:113) − r m b (3 + m + 2 α )( m + 1) F (cid:16) m +2 α m , , m +2 α m , r m b (cid:17) ρ. (21)On comparing Eq. (21) with p = ωρ , we get ω ( r ) = (cid:113) − r m b (3 + m + 2 α )( m + 1) F (cid:16) m +2 α m , , m +2 α m , r m b (cid:17) . (22)It shows that the wormhole under consideration satisfies a variable equation of state. Thevariable EoS arises naturally while solving the field equations for the wormhole. Interestinglya variable EoS unifies various forms of dark energy including phantom energy and Chaplygingas, both of which support the wormhole spacetime [21]. The behavior of ω ( r ) is given inFig. 4 and it shows that the EoS parameter ω has to be negative to model a wormhole. Itnaturally yields negative radial pressure and positive energy density.The case of isotropic pressure p = p t is obtained by fixing α = 1. We have p = p t = (cid:113) − r m b (5 + m )( m + 1) F (cid:16) m m , , m m , r m b (cid:17) ρ. (23)Similarly, the EoS parameter ω becomes ω ( r ) = (cid:113) − r m b (5 + m )( m + 1) F (cid:16) m m , , m m , r m b (cid:17) . (24)We would also comment that the case of vanishing pressure p = 0 (dust) is not allowed inthe present formalism since it will make f ( r ) unbounded. The dust cases in the frameworkof braneworld wormholes are investigated in [22, 23].The specific dimensionless parameter ξ , defined by ξ = ( p − ρ ) / | ρ | , characterizes how theexotic or normal matters are distributed around the wormhole’s throat [4, 15]. The exoticity at or near the throat of the wormhole is required to be non-negative, ξ >
0. The positivityof the exoticity ensures that wormhole will satisfy the flare-out condition as well. From Eq.(15), the exoticity becomes | ρ | ξ = − b ( m + 1)16 πr m +3 (cid:113) − r m b (3 + m + 2 α )( m + 1) F (cid:16) m +2 α m , , m +2 α m , r m b (cid:17) − . (25)In Fig. 5, we have plotted the exoticity against the radial coordinate. It shows thatthe exoticity remains positive while it converges to zero for large r . Hence the wormholeis surrounded by the exotic phantom energy, right from its throat to a sufficiently largedistance. This also suggests that we can construct a wormhole with a sufficiently largeradius that could be traversable for two dimensional beings.In summary, our objective in this article has been to present a mathematical prescriptionfor obtaining a wormhole in low dimensional spacetime. The wormhole is supported byan external source of phantom energy which is anistropically distributed. The wormholespacetime satisfies a variable equation of state, which is in good agreement with earliermodels available in the literature. It is also shown that the otherwise asymptotically non-flat wormhole could be converted to an asymptotically flat one by matching the variouscomponents of its metric with the exterior BTZ spacetime. [1] M. Visser, Lorentzian Wormholes: From Einstein to Hawking, (Springer-Verlag New York,Inc, 1996)[2] M. Jamil et al, arXiv:0906.2142v1 [gr-qc] [3] M. Jamil et al, Eur.Phys.J.C59:907-912,2009[4] M.S. Morris and K.S. Thorne, Am. J. Phys. 56 (1988) 395[5] R.R. Caldwell, Phys. Lett. B 545 (2002) 23[6] R.R. Caldwell et al, Phys. Rev. Lett. 91 (2003) 071301[7] A. DeBenedicitis et al, Phys. Rev. D 78 (2008) 104003[8] S. Sushkov, Phys. Rev. D 71 (2005) 043520[9] O.B. Zaslavskii, Phys. Rev. D 72 (2005) 061303[10] P.K.F. Kuhfittig, Class. Quantum. Grav. 23 (2006) 5853[11] M. Cataldo et al, Phys.Rev.D78:104006,2008[12] S. Carlip, Quantum Gravity in 2+1 dimensions, (Cambridge University Press, 1998)[13] F. Rahaman et al., Phys. Scr. 76 (2007) 56[14] E.A.L. Rubio, Rev. Col. Fis. 40,2 (2008) 222-224 [arXiv:0707.0900][15] W.T. Kim et al, Phys. Rev. D 70 (2004) 044006[16] M.S.R. Delgaty and R.B. Mann, Int.J.Mod.Phys. D4 (1995) 231-246[17] S. Chakraborty and T. Bandyopadhyay, Phys.Rev.D75:064027,2007[18] M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849[19] M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D 48 (1993) 1506[20] F. Rahaman et al, arXiv:0808.2927 [astro-ph][21] P.K.F. Kuhfittig, Gen. Relativ. Gravit. 41 (2009) 1485[22] S. Chakraborty and T. Bandyopadhyay, Astrophys. Space Sci. 317:209-212,2008[23] F.S.N. Lobo, Phys.Rev.D75:064027,2007 FIG. 1: The radial pressure is plotted against radial coordinate while other parameters are fixedat b = − m = 2 and α = 1 , , b = − m = 2 and α = 1 , , FIG. 3: The surface tension is plotted against radial coordinate while other parameters are fixedat b = − m = 2 and α = − , , ω is plotted against radial coordinate while other param-eters are fixed at b = − m = 2 and α = 1 , , FIG. 5: The exoticity | ρ | ξ is plotted against radial coordinate while other parameters are fixed at b = − m = 2 and α = 1 ,,
0. The positivityof the exoticity ensures that wormhole will satisfy the flare-out condition as well. From Eq.(15), the exoticity becomes | ρ | ξ = − b ( m + 1)16 πr m +3 (cid:113) − r m b (3 + m + 2 α )( m + 1) F (cid:16) m +2 α m , , m +2 α m , r m b (cid:17) − . (25)In Fig. 5, we have plotted the exoticity against the radial coordinate. It shows thatthe exoticity remains positive while it converges to zero for large r . Hence the wormholeis surrounded by the exotic phantom energy, right from its throat to a sufficiently largedistance. This also suggests that we can construct a wormhole with a sufficiently largeradius that could be traversable for two dimensional beings.In summary, our objective in this article has been to present a mathematical prescriptionfor obtaining a wormhole in low dimensional spacetime. The wormhole is supported byan external source of phantom energy which is anistropically distributed. The wormholespacetime satisfies a variable equation of state, which is in good agreement with earliermodels available in the literature. It is also shown that the otherwise asymptotically non-flat wormhole could be converted to an asymptotically flat one by matching the variouscomponents of its metric with the exterior BTZ spacetime. [1] M. Visser, Lorentzian Wormholes: From Einstein to Hawking, (Springer-Verlag New York,Inc, 1996)[2] M. Jamil et al, arXiv:0906.2142v1 [gr-qc] [3] M. Jamil et al, Eur.Phys.J.C59:907-912,2009[4] M.S. Morris and K.S. Thorne, Am. J. Phys. 56 (1988) 395[5] R.R. Caldwell, Phys. Lett. B 545 (2002) 23[6] R.R. Caldwell et al, Phys. Rev. Lett. 91 (2003) 071301[7] A. DeBenedicitis et al, Phys. Rev. D 78 (2008) 104003[8] S. Sushkov, Phys. Rev. D 71 (2005) 043520[9] O.B. Zaslavskii, Phys. Rev. D 72 (2005) 061303[10] P.K.F. Kuhfittig, Class. Quantum. Grav. 23 (2006) 5853[11] M. Cataldo et al, Phys.Rev.D78:104006,2008[12] S. Carlip, Quantum Gravity in 2+1 dimensions, (Cambridge University Press, 1998)[13] F. Rahaman et al., Phys. Scr. 76 (2007) 56[14] E.A.L. Rubio, Rev. Col. Fis. 40,2 (2008) 222-224 [arXiv:0707.0900][15] W.T. Kim et al, Phys. Rev. D 70 (2004) 044006[16] M.S.R. Delgaty and R.B. Mann, Int.J.Mod.Phys. D4 (1995) 231-246[17] S. Chakraborty and T. Bandyopadhyay, Phys.Rev.D75:064027,2007[18] M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849[19] M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D 48 (1993) 1506[20] F. Rahaman et al, arXiv:0808.2927 [astro-ph][21] P.K.F. Kuhfittig, Gen. Relativ. Gravit. 41 (2009) 1485[22] S. Chakraborty and T. Bandyopadhyay, Astrophys. Space Sci. 317:209-212,2008[23] F.S.N. Lobo, Phys.Rev.D75:064027,2007 FIG. 1: The radial pressure is plotted against radial coordinate while other parameters are fixedat b = − m = 2 and α = 1 , , b = − m = 2 and α = 1 , , FIG. 3: The surface tension is plotted against radial coordinate while other parameters are fixedat b = − m = 2 and α = − , , ω is plotted against radial coordinate while other param-eters are fixed at b = − m = 2 and α = 1 , , FIG. 5: The exoticity | ρ | ξ is plotted against radial coordinate while other parameters are fixed at b = − m = 2 and α = 1 ,, ,,