Wormholes supported by phantom-like modified Chaplygin gas
WWormholes supported by phantom-like modifiedChaplygin gas
Mubasher Jamil ∗ , M. Umar Farooq and Muneer Ahmad Rashid † Center for Advanced Mathematics and Physics, National University of Sciences and TechnologyE&ME Campus, Peshawar Road, Rawalpindi, 46000, Pakistan
October 22, 2018
Abstract
We have examined the possible construction of a stationary, spherically symmetric andspatially inhomogeneous wormhole spacetime supported by the phantom energy. The later issupposed to be represented by the modified Chaplygin gas equation of state. The solutionsso obtained satisfy the flare out and the asymptotic flatness conditions. It is also shown thatthe averaged null energy condition has to be violated for the existence of the wormhole.
One of the most exotic geometries that arise as solutions of Einstein field equations isthe wormhole. A typical two mouth wormhole connects two arbitrary points of the samespacetime or two distinct spacetimes. One observes that any typical Schwarzschild space-time contains a singularity at r = 0 making it geodesically incomplete. Ellis [1] firstobserved that the coupling of the geometry of spacetime to a scalar field can produce astatic, spherically symmetric, geodesically complete and horizonless spacetime and thustermed it as a ‘drainhole’ that could serve as tunnel to traverse particles from one side tothe other. Later on Morris and Thorne [2, 3] proposed that wormholes could be thoughtof (imaginary) time machines that could render rapid interstellar travel for human be-ings. While a black hole possesses single horizon which forbids two way travel (in andout) of the black hole but this problem does not arise in the absence of horizon for a ∗ [email protected] † [email protected] a r X i v : . [ g r- q c ] F e b ormhole. Unfortunately the existence of wormholes require the violation of the mostcherished energy conditions of general relativity (null, weak, strong and dominant) whichare in fact satisfied by any normal matter or energy [4]. In particular, matter violatingnull energy condition is called ‘exotic matter’ [5]. Later it was proposed that wormholescould be constructed with arbitrary small quantities of exotic matter [6, 7]. A commonlyknown form of matter violating these energy conditions is dubbed as ‘phantom energy’characterized by the equation of state (EoS) p = ωρ , where p and ρ are respectively, thepressure and the energy density of the phantom energy, with ω < −
1. The existence ofthis matter remains hypothetical but the astrophysical observations of supernovae of typeIa and cosmic microwave background have suggested the presence of phantom energy inour observable universe [8, 9]. It can exhibit itself as a source that can induce an accel-eration in the expansion of the universe. The typical size of a wormhole can be of theorder of the Planck length but it can be stretched to a larger size if it is supported byexotic phantom like matter [10, 11]. The accretion of phantom energy can increase themass and size of the wormhole and hence guarantee the stability of the wormhole [12, 13].The astrophysical implications of wormholes are not exactly clear but it is suggested thatsome active galactic nuclei and other galactic objects may be current or former entrancesto wormholes [14]. It has been predicted that wormholes can also produce gravitationallensing events [15]. Since wormholes are horizonless, they can avoid undergoing any pro-cess of decay like Hawking evaporation and hence can survive over cosmological times.But a wormhole may form a black hole with a certain radial magnetic field (a form ofmagnetic monopole) if it accretes normal matter and consequently loses its structure.Earlier, Rahaman et al [16] investigated the evolution of wormhole using an averagednull energy condition (ANEC) violating phantom energy and a variable EoS parameter ω ( r ). We here investigate the same problem using a more general EoS for the pressuredensity namely the modified Chaplygin gas. It is well-known that the wormhole spacetimeis inhomogeneous and hence requires inhomogeneous distribution of matter. This can bemade by introducing two different pressures namely the radial and the transverse pressure.Our analysis shows that the parameters adopted in the equation of state for phantomenergy have to be tuned such that the radial pressure becomes negative in all directionsand for all radial distance. This result turns out to be consistent with Sushkov [17].The paper is organized as follows: In the second section, we have modeled the fieldequations for the wormhole spacetime and proposed the methodology that is adopted inthe later sections. Next, we have investigated the behavior of energy condition ANEC forall the wormhole solutions obtained. Finally, the last section is devoted for the conclusionand discussion of our results. Modeling of system
We start by assuming the static, stationary, spherically symmetric wormhole spacetimespecified by (in geometrized units G = 1 = c ): ds = − e f ( r ) dt + (cid:32) − b ( r ) r (cid:33) − dr + r ( dθ + sin θdφ ) . (1)Here f ( r ) is the ‘gravitational potential function’ while b ( r ) is called ‘shape function’ ofthe wormhole (see Ref. [18] for the consistent derivation of the above metric). The radialcoordinate r ranges over [ r o , ∞ ) where the minimum value r o corresponds to the radiusof the throat of the wormhole. If b ( r ) = 2 m ( r ), the later being the mass, then Eq. (1)represents a ‘dark energy star’ which may arise from a density fluctuation in the Chaplygingas cosmological background [19, 20]. Note that b ( r = r o ) = r o corresponds to the spatialposition of the wormhole throat. We shall, in this paper, assume f ( r ) = constant for theconvenience of our calculations. This choice, as a special case, is also physically motivatedand makes the time traveler to feel zero tidal force near the wormhole [16, 23]. A wormholewith small | f (cid:48) ( r ) | in the vicinity of the throat is likely to be traversable in the sense ofhaving low tidal forces. It also makes the wormhole to be horizon-free.We take the inhomogeneous phantom energy which is specified by the stress energytensor: T = ρ , T = p r , T = T = p t . Here p r and p t are, respectively, the radialand transverse component of the pressure while ρ is the energy density of the phantomenergy. It represents a perfect fluid (which is homogeneous and isotropic) if T = T = T = p r = p [21]. Note that in the stellar evolution, the difference p r − p t creates asurface tension inside star which makes it anisotropic. This feature is generically foundin more compact stars like neutron and quark stars [22], contrary to normal stars whichare majorally supported by radial pressure only against gravity.The Einstein field equations ( G αβ = 8 πT αβ ) for the metric (1) are b (cid:48) ( r ) r = 8 πρ ( r ) , (2) − br = 8 πp r ( r ) , (3) (cid:32) − br (cid:33) (cid:34) − b (cid:48) r + b r ( r − b ) (cid:35) = 8 πp t ( r ) . (4)The energy conservation equation is obtained from T αβ ; α = 0, which gives p (cid:48) r + 2 r p r − r p t = 0 . (5) his equation can be considered as the hydrodynamic equilibrium equation for the exoticphantom energy supporting the wormhole.Eq. (2) can be written in the form dbdr = 8 πr ρ. (6)Let us choose the modified Chaplygin gas (MCG) EoS for the radial pressure [25] p r ( r ) = Aρ ( r ) − Bρ ( r ) α . (7)Here A , B and α are constant parameters. The MCG best fits with the 3 − year WMAPand the SDSS data with the choice of parameters A = − .
085 and α = 1 .
724 [26] whichare improved constraints than the previous ones − . < A < .
025 [27]. Recently it isshown that the dynamical attractor for the MCG exists at ω = −
1, hence MCG crossesthis value from either side ω > − ω < −
1, independent to the choice of modelparameters [28]. Generally, α is constrained in the range [0 ,
1] but here we are assumingit to be a free parameter which can take values outside this narrow range, for instance α = − α < p t to be linearly proportional to the radial pressure p r as p t = np r , (8)where n is a non-zero constant. Thus p r is restricted to satisfy Eq. (7) for a given ρ while p t is arbitrary in nature due to free parameter n . Using Eq. (8) in (5), we obtain p (cid:48) r + 2 r p r − nr p r = 0 , (9)which gives p r = Cr n − . (10)Here C is a constant of integration. Since the wormhole is supported by a negativepressure inducing exotic phantom energy, it yields p r < C < < n < ∞ inorder to obtain finite negative radial pressure. Consequently p t < < n < ∞ . UsingEq. (10) in (7), we have Aρ − Bρ α = Cr n − , (11)which can be written as Aρ α +1 − Cr n − ρ α − B = 0 . (12)Note that Eq. (12) is a polynomial equation of degree α + 1 in variable ρ , which doesnot yield solutions for any arbitrary α . We shall, henceforth, solve Eq. (12) for specific hoices like α = −
1, 0 and 1. We shall further employ the following conditions on oursolutions given below [24]:1. The potential function f ( r ) must be finite for all values of r for the non-existence ofhorizon. In our model, this condition is trivially satisfied since f ( r ) is taken to be a finiteconstant throughout this paper.2. The shape function b ( r ) must satisfy b (cid:48) ( r = r o ) < r o , the so-called flare-out condition.3. Further b ( r ) < r outside the wormhole’s throat r > r o . This condition is a directconsequence of the flare-out condition.4. The spacetime must be asymptotically flat i.e. b ( r ) /r → | r | → ∞ .Now we shall consider the three cases for different choices of parameter α : Case-a: If α = 0 , then (12) gives ρ = BA + CA r n − . (13)Using Eq. (13) in (6), we get b ( r ) = 8 πA (cid:34) Br Cr n +1 n + 1 (cid:35) + C . (14)Here C is a constant of integration. Now b ( r ) r → | r | → ∞ if n = 1 and B = − C . Buthere b ( r ) = constant and hence gives ρ = 0 which is an acceptable solution and representsvacuum (empty space-time) outside the wormhole throat. This corresponds to vanishingpressures i.e. p r = p t = 0. This vacuum solution requires C = 0 which in turn leads to B = 0. Note that condition (4) can also be met if only B = 0 and n <
0. In figures 1 to4, we have plotted the ratio b ( r ) /r against the parameter r . The Fig. 1 shows that theratio declines as r → ∞ , although r is restricted to a certain range. The parameter A can assume the value in the range − . ≤ A ≤ .
025 [29], we choose A = 0 .
025 for ourwork.Further, flare-out condition (2) implies b (cid:48) ( r o ) = 8 πCA r no < , (15)which gives an upper limit on the size of throat’s radius as r o < (cid:18) A πC (cid:19) n . (16) his requires both A >
C >
0. The throat’s radius can be obtained by solving b ( r o ) = r o which gives r o = (cid:34) A (2 n + 1)8 πC (cid:35) n . (17)This quantity is positive if A/C > n + 1 > A/C < n + 1 <
0. Similarly,condition (3) translates into r < (cid:34) A (2 n + 1)8 πC (cid:35) n , r > r o . (18)Note that conditions (2) and (3) are satisfied if C = 0. Case-b: If α = 1 , then (12) gives Aρ − Cr n − ρ − B = 0 , (19)which is quadratic in ρ and gives two roots of the form ρ ± = Cr n − ± √ C r n − + 4 AB A . (20)These roots are real-valued if the quantity inside the square root is positive while theroots will be repeated if it is zero and complex valued otherwise. We next determine theshape function b ( r ) corresponding to these roots by substituting Eq. (20) in (6) to get b ± ( r ) = ± πr ( ± Cr n + r √ AB + C r n − ) A (1 + 2 n ) + C ± ± [2 Bπr (cid:115) ABr − n ) C (cid:113) AB + C r n − Γ (cid:32) n − n ) (cid:33) × F (cid:48) (cid:32) − n − n ) , , − n − n ) , − ABr − n ) C (cid:33) ] / [( n − ABr + C r n )] . (21)Here C ± are two constants of integration whereas F (cid:48) is the regularized hyper-geometricfunction. Figures 2 and 3 show the ratios b + ( r ) /r and b − ( r ) /r versus r , respectively. Bothratios decline for large values of r and approach zero, satisfying the asymptotic flatnesscondition for specific choice of the parameters. Case-c: If α = − , then (12) yields ρ = CA − B r n − , (22)Use of this in (6) enables us to write b ( r ) = 8 πC n + 1)( A − B ) r n +1) . (23) n figure 4, the ratio b ( r ) /r is plotted against r , showing its convergence to zero. Further,condition (2) implies b (cid:48) ( r o ) = 8 πCA − B r n +1 o < , (24)which gives the maximum size of the wormhole’s throat r o < (cid:18) A − B πC (cid:19) n +1 . (25)In other words, the throat’s radius is given by r o = (cid:34) n + 1)( A − B )8 πC (cid:35) n +1 . (26)It requires either A − B > C > n > − A − B <
C <
0. This laterchoice of parameters is consistent with the ones that are required for p r < (cid:90) γ T αβ k α k β dλ ≥ . (27)Here T αβ is the stress energy tensor, k α is the future directed null vector, γ is the nullgeodesic and λ is the arc-length parameter. In other words, the integrand must be positive.In an orthonormal frame of reference, we have k ˆ α = (1 , , , T ˆ α ˆ β k ˆ α k ˆ β = ρ + p r .We here adopt the ANEC integral from Visser et al [7] to analyze its violation in ourmodel: I = (cid:73) ( ρ + p r ) dV = 2 (cid:90) ∞ r o ( ρ + p r )4 πr dr. (28)The above integral is called the ‘volume integral quantifier’ [34]. It is obvious that theabove integral becomes negative if ρ + p r <
0, the violation of null energy condition see also [35]). The violation of ANEC is the requirement for any phantom matter andstability of a wormhole. Now we shall take different ρ and p r calculated in each of theabove cases to evaluate I . Our aim will be to find conditions under which I <
0. Weshall also consider the case of I → Case-a
Using Eqs. (7) and (13) in (28), we obtain I = 8 πC (1 + A ) A (1 + 2 n ) r n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ r o . (29)Note that the above integral gives a finite value if n < − /
2. Hence we obtain I = − (1 + A ) (cid:34) A (1 + 2 n )8 πC (cid:35) n . (30)Moreover, the above integral I <
A >
A > −
1. Further,
A/C <
C <
A >
C >
A <
0. Again the former case (1) isconsistent with p r <
0. Also I →
0, if either A → n → − / Case-b
Using Eqs. (7) and (21) in Eq. (28), we get I ± = 4 πr ((1 + 2 A ) Cr n ± r √ AB + C r n − ) A (1 + 2 n ) ∓ [32( n − Bπr (cid:115) ABr − n ) C (cid:113) AB + C r n − (31) × F (cid:48) (cid:32) − n − n ) , , − n − n ) , − ABr − n ) C (cid:33) ] / [(2 n + 1)(2 n − ABr + C r n )] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ r o . The ANEC is violated for particular choice of parameters like B = − C = − n = 3.Note that we have assumed C ± = 0 for the convenience of our calculations. Under thischoice of parameters, the two integrals I ± will be finite. Also figures (5) and (6) showthe behavior of the integrals I + < I − <
0, respectively. The plots suggest that thetwo integrals I ± tend to infinity for large r , so that an infinite amount of ANEC violatingmatter is necessary to sustain these geometries, which is a problematic issue. Howeverthis problem can be evaded by considering a matching to an exterior vacuum solutionwhich gives a thin-shell wormhole solution [32, 33, 35]. Further, the case of I ± → n → A → − / C + 4 AB → Case-c
Making use of Eqs. (7) and (22) in (28) yields I = 8 πC (1 + A − B ) r n ( A − B )(1 + 2 n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ r o . (32) he above integral is finite if n < − /
2. Therefore we obtain I = 1 + A − B n (cid:18) A − B πC (cid:19) n . (33)Further, the ANEC is violated I <
C < A − B <
0. The wormhole issupported by arbitrary small amount of phantom energy if A − B → In this paper, we have derived three solutions of wormhole by obtaining different formsof b ( r ). This is carried out by employing the modified Chaplygin gas for the pressure andusing three specific values of the parameter α . It needs to be mentioned that other valuesof α either don’t yield any b ( r ) or if it does exist than the stability conditions 1 to 4 are notverified. Hence we have restricted ourselves to these specific cases as shown in the figuresas well. The solutions so obtained also satisfy the stability conditions. The pressure andthe corresponding energy density obtained in each case, violate the null energy condition ρ + p r < p r = p t = p , for f ( r ) to be finite,one cannot construct asymptotically flat traversable wormhole [10]. In our work, werepresented the radial pressure by the modified Chaplygin gas and the transverse pressureto be linearly proportional to the radial one. The MCG has phantom nature with negativepressure. Earlier, Lobo [34] studied the Chaplygin traversable wormhole and concludedthat the Chaplygin gas needs to be confined around the wormhole throat neighborhood.That work was later extended in [35] using the modified Chaplygin gas and it was deducedthat modified Chaplygin wormholes may occur naturally and could be traversable. Ourresults are in conformity with their results since our solutions meet the criteria of wormholestability and traversability.In a recent paper, Gorini et al [36] have presented an interesting theorem which statesthat in a static spherically symmetric spacetime filled with the phantom Chaplygin gas,the scalar curvature becomes singular at some finite value of the radial coordinate r andhenceforth the spacetime is not asymptotically flat. This result apparently forbids theexistence of wormholes which are required to be non-singular. The theorem is based on theassumptions of homogeneity and isotropy of the spacetime. In case of anisotropy ( p t (cid:54) = 0),the above theorem is not applicable and wormhole spacetime appears naturally. For an sotropic and homogeneous spacetime filled with phantom Chaplygin gas, the asymptoticflatness can be achieved by cutting the spacetime at some spatial position r = R andglued with a vacuum spacetime, in particular, Schwarzschild exterior spacetime can beutilized [34]. Acknowledgment
One of us (MJ) would like to thank A. Qadir and M. Akbar for useful discussions duringthis work. We would also acknowledge anonymous referees for their useful criticism onthis work.
References [1] H.G. Ellis, J. Math. Phys. 14 (1973) 104.[2] M.S. Morris and K.S. Thorne, Am. J. Phys. 56 (1988) 395.[3] M.S. Morris et al, Phys. Rev. Lett. 61 (1988) 1446.[4] C.A. Pi´ c on, Phys. Rev. D 65 (2002) 104010.[5] C.G. B¨ o hmer et al, Phys. Rev. D 76 (2007) 084014.[6] P.K.F. Kuhfittig, Am. J. Phys. 67 (1999) 125[7] M. Visser et al, Phys. Rev. Lett. 90 (2003) 201102.[8] R.R. Caldwell, Phys. Lett. B 545 (2002) 23.[9] E. Babichev et al, Phys. Rev. Lett. 93 (2004) 021102.[10] F.S.N. Lobo, Phys. Rev. D 71 (2005) 084011; arXiv: gr-qc/0502099v2.[11] P.F. Gonz´ a lez-D´ i az, Phys. Rev. Lett. 93 (2004) 071301.[12] P.F. Gonz´ a lez-D´ i az, Phys. Lett. B 632 (2006) 159.[13] M. Jamil, Il Nuovo Cimento B 123 (2008) 599; astro-ph/0806.1319.[14] N.S. Kardashev et al, Int. J. Mod. Phys. D 16 (2007) 909.[15] T.K. Dey and S. Sen, gr-qc/0806.4059v1.[16] F. Rahaman, gr-qc/0701032v2.
17] S. Sushkov, Phys. Rev. D 71 (2005) 043520; gr-qc/0502084v1.[18] P.K.F. Kuhfittig, arXiv: 0802.3656v1 [gr-qc][19] F.S.N. Lobo, Class. Quant. Grav. 23 (2006) 1525; arXiv: 0508115 [gr-qc][20] A. DeBenedictis et al, gr-qc/0808.0839v1.[21] P.K.F. Kuhfittig, Class. Quant. Gravit. 23 (2006) 5853.[22] F. Rahaman, astro-ph/0808.2927.[23] M. Morris and K. Thorne, Am. J. Phys. 56 (1988) 39.[24] F. Rahaman et al, Phys. Scr. 76 (2007) 56.[25] M. Jamil, arXiv: gr-qc/0810.2896v2[26] J. Lu et al, Phys. Lett. B 662 (2008) 87.[27] L. Dao-Jun and L. Xin-Zhou, Chin. Phys. Lett. 22 (2005) 1600.[28] H. Jing et al, Chin. Phys. Lett. 25 (2008) 347.[29] Y.B. Wu et al Gen. Relativ. Gravit. 39 (2007) 653.[30] S.W. Hawking, Nature 248 (1974) 30[31] E. Poisson, A Relativist’s Toolkit: The mathematics of black hole mechanics, Cam-bridge University Press, 2004.[32] F.S.N. Lobo, Phys. Rev. D 71 (2005) 124022; arXiv: gr-qc/0506001v2.[33] E.F. Eiroa and C. Simeone, Phys. Rev. D 76 (2007) 024021; arXiv: 0704.1136v2[gr-qc][34] F.S.N. Lobo, Phys. Rev. D 73 (2006) 064028; arXiv: gr-qc/0511003v2.[35] S. Chakraborty and T. Bandyopadhay, arXiv: 0707.1183 [gr-qc][36] V. Gorini et al, Phys. Rev. D 78 (2008) 064064; arXiv: 0807.2740v2 [astro-ph] b ( r ) /r is plotted against r with C = 3 and for different values of n = − , − . , − , − . , − , − . b + ( r ) /r is plotted against r . The parameters are fixed at B = 6 , C = 7 and n = 3. 12igure 3: The ratio b − ( r ) /r is plotted against r . The parameters are fixed at B = − , C = 7and n = 3.Figure 4: The ratio b ( r ) /r is plotted against r for different values of n = − , − , − , − , − , − B = 1, A = − C = 2 which correspond to curves in right to leftorder. 13igure 5: The ANEC integral I + is plotted against r . The apparent negative values of I + showthe violation of the ANEC condition. The parameters are chosen as B = − C = − n = 3Figure 6: The ANEC integral I − is plotted against r . The apparent negative values of I −−