Noncommutative-geometry wormholes with isotropic pressure
aa r X i v : . [ g r- q c ] F e b Noncommutative-geometry wormholes with isotropicpressure
Peter K. F. Kuhfittig* ∗ Department of Mathematics, Milwaukee School of Engineering,Milwaukee, Wisconsin 53202-3109, USA
Abstract
The strategy adopted in the original Morris-Thorne wormhole was to retain completecontrol over the geometry at the expense of certain engineering considerations. Thepurpose of this paper is to obtain several complete wormhole solutions by assuming anoncommutative-geometry background with a concomitant isotropic-pressure condi-tion. This condition allows us to consider a cosmological setting with a perfect-fluidequation of state. An extended form of the equation generalizes the first solutionand subsequently leads to the generalized Chaplygin-gas model and hence to a thirdsolution. The solutions obtained extend several previous results. This paper alsoreiterates the need for a noncommutative-geometry background to account for theenormous radial tension that is characteristic of Morris-Thorne wormholes.
Keywords: traversable wormholes; noncommutative geometry; isotropic pressurePACS (2020): 04.20-q; 04.20.Jb
Wormholes are tunnel-like structures in spacetime that connect widely separated regionsof our Universe or different universes altogether. Although not entirely new, a detailedanalysis of traversable wormholes was first performed by Morris and Thorne [1] in 1988.They had proposed the following static and spherically symmetric line element for awormhole spacetime: ds = − e r ) dt + dr − b ( r ) /r + r ( dθ + sin θ dφ ) , (1)using units in which c = G = 1. Here b = b ( r ) is called the shape function and Φ = Φ( r )is called the redshift function ; Φ( r ) must be everywhere finite to avoid an event horizon.The shape function must also have certain properties, including the fixed-point property b ( r ) = r , where r = r is the radius of the throat of the wormhole. An important ∗ kuhfi[email protected] flare-out condition at the throat: b ′ ( r ) <
1, while b ( r ) < r near thethroat. For the wormhole spacetime as a whole, the most important physical property isasymptotic flatness, which demands that lim r →∞ Φ( r ) = 0 and lim r →∞ b ( r ) /r = 0.The flare-out condition refers to the tunnel-like shape of b ( r ) when viewed, for example,in an embedding diagram [1]. This condition can only be met by violating the null energycondition (NEC) T αβ k α k β ≥ k α , where T αβ is the energy-momentum tensor. Matter that violates theNEC is called “exotic” in Ref. [1]. In particular, for the outgoing null vector (1 , , , T αβ k α k β = ρ + p r < . (3)Here T t t = − ρ is the energy density, T r r = p r is the radial pressure, and T θ θ = T φ φ = p t is the lateral pressure.Regarding the theoretical construction of a wormhole, Morris and Thorne adopted thefollowing stategy: specify the functions b = b ( r ) and Φ = Φ( r ) to produce the desiredgeometric properties. This strategy retains complete control over the geometry but leadsto enormous practical problems: the members of the engineering team must manufactureor search the Universe for matter or fields that yield the required energy-momentumtensor. There are theoretical problems as well, as can be seen from the Einstein fieldequations, listed next. ρ ( r ) = b ′ πr , (4) p r ( r ) = 18 π (cid:20) − br + 2 (cid:18) − br (cid:19) Φ ′ r (cid:21) , (5) p t ( r ) = 18 π (cid:18) − br (cid:19) (cid:20) Φ ′′ − b ′ r − b r ( r − b ) Φ ′ + (Φ ′ ) + Φ ′ r − b ′ r − b r ( r − b ) (cid:21) . (6)Since Eq. (6) can be obtained from the conservation of the stress-energy tensor T µν ; ν = 0,only Eqs. (4) and (5) are actually needed.. These can be written in the following forms: b ′ = 8 πρr , (7)and Φ ′ = 8 πp r r + b r ( r − b ) . (8)It now becomes apparent that due to the condition b ( r ) = r , Φ ′ ( r ) and hence Φ( r ) arenot likely to exist. (There are exceptions, however, for certain special forms of b ( r ), asshown by Lobo [2].)The purpose of this paper is to obtain several complete wormhole solutions by assum-ing a noncommutative-geometry background in conjunction with an isotropic-pressurecondition, discussed in the next section. 2 Noncommutative geometry
An important outcome of string theory is the realization that coordinates may becomenoncommutative operators on a D -brane [3, 4]. Noncommutativity replaces point-likeobjects by smeared objects [5, 6, 7] with the aim of eliminating the divergences thatinvariably appear in general relativity. As a consequence, spacetime can be encoded inthe commutator [ x µ , x ν ] = iθ µν , where θ µν is an antisymmetric matrix that determinesthe fundamental cell discretization of spacetime in the same way that Planck’s constantdiscretizes phase space [6]. The smearing can be modeled using a Gaussian distributionof minimal length √ β instead of the Dirac delta function [6, 8, 9, 10]. An equally effectiveway is to assume that the energy density of the static and spherically symmetric andparticle-like gravitational source is given by [11, 12] ρ ( r ) = µ √ βπ ( r + β ) , (9)which can be interpreted to mean that the gravitational source causes the mass µ of a par-ticle to be diffused throughout the region of linear dimension √ β due to the uncertainty; so √ β has units of length. Here it is important to note that the noncommutative effects canbe implemented in the Einstein field equations by modifying only the energy-momentumtensor, while leaving the Einstein tensor unchanged. The reason is that, according toRef. [6], a metric field is a geometric structure defined over an underlying manifold. Itsstrength is measured by curvature, which is a response to the presence of a mass-energydistribution. Here the key observation is that noncommutativity is an intrinsic prop-erty of spacetime (rather than a superimposed geometric structure), thereby affecting themass-energy and momentum distributions. But the energy-momentum density, in turn,determines the spacetime curvature, which explains why the Einstein tensor can be leftunchanged. An important consequence is that the length scales can be macroscopic. Wecan therefore use Eq. (9) to determine the mass distribution Z r π ( r ′ ) ρ ( r ′ ) dr ′ = 2 Mπ (cid:18) tan − r √ β − r √ βr + β (cid:19) , (10)where M is now the total mass of the source.Suppose we return the the conservation law T α γ ; α = 0. If γ = r , then we obtain ∂∂r T rr = − g tt ∂g tt ∂r ( T rr − T t t ) − g θθ ∂g θθ ∂r ( T rr − T θθ ) . (11)According to Ref. [6], to preserve the property g tt = − g − rr , we require that T rr = T t t = − ρ ( r ), while T θθ = T φφ = − ρ ( r ) − r ∂ρ ( r ) ∂r . (12)Furthermore, a massive structureless point is replaced by a self-gravitating droplet ofanisotropic fluid of density ρ , which yields the radial pressure p r ( r ) = − ρ ( r ) , (13)3hereby preventing the collapse to a matter point. The tangential pressure is given by p t ( r ) = − ρ ( r ) − r ∂ρ ( r ) ∂r . (14)Since the length scales can be macroscopic, we can retain Eq. (13) and then use Eq. (14)to determine p t ( r ) = − ρ ( r ) − r ∂ρ ( r ) ∂r = p r ( r ) + 2 µr √ βπ ( r + β ) (15)by Eq. (9). So for larger r , we have p t ( r ) ≈ p r ( r ). Since the pressure becomes isotropic,we can write the equation of state in the form p ( r ) = − ρ ( r ) . (16)In the present context, this is an important observation since Morris-Thorne wormholesare normally characterized by an anisotropic pressure. The conservation law T µν ; ν = 0 yields the following equation: p ′ r = − Φ ′ ρ − Φ ′ p r − r p r + 2 r p t . (17)From Eq. (16) we obtain p ′ = − ( p + ρ )Φ ′ . (18)Given the isotropic pressure, we can safely assume the perfect-fluid equation of state p = ωρ, (19)which, in turn, allows us to consider a cosmological setting. Moreover, in Sec. 4 we aregoing to use the more general equation of state ρ = 1 / | ω | p α ; (20)if α = −
1, we obtain p = | ω | ρ , which is a special case of Eq. (19). Eq. (20) notonly generalizes the equation of state, it can be adapted to a cosmological model usuallyreferred to as generalized Chaplygin gas, discussed in Sec. 5.Returning now to Eq. (19), if we substitute p = ωρ in Eq. (18), we find that ωρ ′ = − ρ (1 + ω )Φ ′ and hence Φ ′ = − ω ω ρ ′ ρ . (21)Integrating, we obtain the redshift functionΦ = − ω ω (ln ρ + ln c ) = − ω ω ln cρ, c is an arbitrary positive constant. Substituting ρ ( r ) from Eq. (9) yieldsΦ = − ω ω ln cµ √ βπ ( r + β ) ;so e r ) = (cid:20) cµ √ βπ ( r + β ) (cid:21) − ω ω . (22)The next step is to determine the shape function from Eqs. (7) and (9): b ( r ) = Z rr π ( r ′ ) ρ ( r ′ ) dr ′ = 4 m √ βπ (cid:18) √ β tan − r √ β − rr + β − √ β tan − r √ β + r r + β (cid:19) + r ; (23)observe that b ( r ) = r , as required. Here m is the total mass of the source; so b ( r ) is themass distribution covering both sides of the throat.One of the requirements of a valid wormhole solution is asymptotic flatness. For theshape function, we evidently have lim r →∞ b ( r ) /r = 0. In Eq. (22), we are faced with amore complicated stuation: if − < ω <
0, then lim r →∞ e = 0; if ω < − ω ≥ r →∞ e = + ∞ . Either way, our wormhole spacetime is not asymptotically flat.Accordingly, the wormhole material must be cut off at some r = a and joined to theexterior Schwarzschild solution ds = − (cid:18) − Mr (cid:19) dt + dr − M/r + r ( dθ + sin θ dφ ) . (24)From the shape function b = b ( r ), we have11 − b ( a ) a = 11 − Ma . (25)So the effective mass of the wormhole is M = b ( a ). For the redshift function, we thenobtain e a ) = 1 − Ma = 1 − b ( a ) a . (26)This junction condition now yields the constant c by letting r = a in Eq. (22): e a ) = (cid:20) cµ √ βπ ( a + β ) (cid:21) − ω ω = 1 − b ( a ) a . (27)Solving for c , we have c = π ( a + β ) µ √ β (cid:18) − b ( a ) a (cid:19) − ω ω . (28)Substituting in Eq. (22) and simplifying yields the final form e r ) = (cid:18) a + βr + β (cid:19) − ω ω (cid:18) − b ( a ) a (cid:19) . (29)5bserve that at r = a , we have indeed e a ) = 1 − b ( a ) a = 1 − Ma . (30)The complete solution can now be written as ds = − e r ) dt + dr − b ( r ) /r + r ( dθ + sin θ dφ ) , (31)for r ≤ a ; here e r ) is given by Eq. (29) and b ( r ) by Eq. (23). For r > a , ds = − (cid:18) − b ( a ) r (cid:19) dt + dr − b ( a ) /r + r ( dθ + sin θ dφ ) . (32)Since the interior solution, Eq. (31), extends only from r = r to r = a , the solutionis valid for all ω , except for ω = −
1, as can be seen from Eq. (29).Returning to the junction surface r = a , while our metric is continuous at r = a , thederivatives may not be. This behavior needs to be taken into account when discussingthe surface stresses. The following forms, proposed by Lobo [2, 13], are suitable for thispurpose: σ = − πa r − Ma − r − b ( a ) a ! (33)and P = 18 πa − Ma q − Ma − [1 + a Φ ′ ( a )] r − b ( a ) a . (34)Since b ( a ) = 2 M , the surface stress-energy σ is zero. From Eq. (29), we obtainΦ ′ ( a ) = ω ω aa + β . (35)Substituting in Eq. (34) and simplifying, we get P = 18 πa q − Ma (cid:20) Ma + ω ω a a + β (cid:18) − b ( a ) a (cid:19)(cid:21) . (36)Due to the condition b ( r ) < r near the throat, P is positive whenever − < ω ≤ . (37) In this section, we return to Eq. (20), restated here for convenience: ρ = 1 / | ω | p α , α > − . (38)6rom Eq. (18), Φ ′ = − p ′ p + ρ = − p α p ′ p α +1 + | ω | . (39)Integrating, we obtain 2Φ = − α + 1 ln (cid:20) c (cid:18) p α +1 + 1 | ω | (cid:19)(cid:21) (40)and hence e = (cid:20) c (cid:18) p α +1 + 1 | ω | (cid:19)(cid:21) − α +1 , (41)where p α +1 = (cid:18) π ( r + β ) | ω | µ √ β (cid:19) α +1 α (42)from Eq. (9). As before, the resulting wormhole spacetime is not asymptotically flat andneeds to be joined to an external Schwarzschild spacetime, i.e., (cid:20) c (cid:18) p ( a ) α +1 + 1 | ω | (cid:19)(cid:21) − α +1 = 1 − b ( a ) a . (43)Solving for c , we get c = (cid:16) − b ( a ) a (cid:17) − α +12 p ( a ) α +1 + | ω | (44)to be substituted into Eq. (41). The result is e r ) = " p ( r ) α +1 + | ω | p ( a ) α +1 + | ω | − α +1 (cid:18) − b ( a ) a (cid:19) . (45)Using Eq. (42) and reducing, we obtain the final form e r ) = (cid:16) π ( r + β ) µ √ β (cid:17) α +1 α | ω | − α + 1 (cid:16) π ( a + β ) µ √ β (cid:17) α +1 α | ω | − α + 1 − α +1 (cid:18) − b ( a ) a (cid:19) . (46)So e a ) = 1 − b ( a ) a . The shape function is still given by Eq. (23).We saw earlier that if α = − p = | ω | ρ , a special form of theperfect-fluid equation, but e r ) in Eq. (46) is not defined for α = −
1. However, the firstfactor on the right takes on the indeterminate form 1 ∞ at α = −
1. So lim r →− e r ) canbe evaluated by means of L’Hospital’s rule. The result is e r ) = (cid:18) a + βr + β (cid:19) − || ω | | ω | (cid:18) − b ( a ) a (cid:19) , (47)which is consistent with our previous form, Eq. (29).7 The generalized Chaplygin-gas model
Another equation of state of interest to us is p = − Kρ α , < α ≤ . (48)The model with this equation of state is referred to as generalized Chaplygin gas [14, 15,16, 17]. Cosmologists became interested in this form of matter when it turned out to be acandidate for combining dark matter and dark energy. To support a wormhole, we musthave K < πr ) α +1 . (49)Suppose we rewrite Eq. (38) in the form p = (1 / | ω | ) /α ρ /α . (50)Now assume that α ≥ α = 1 /α . Then p = (1 / | ω | ) /α ρ α , < α ≤ . (51)If we now choose K = − (cid:18) | ω | (cid:19) /α = −| ω | − α , (52)we have a valid equation of state for the Chaplygin model since K can be determinedfrom the free parameter ω .It now follows directly from Eq. (46) that for the generalized Chaplygin model, e r ) = (cid:16) π ( r + β ) µ √ β (cid:17) α +1 K − (cid:16) π ( a + β ) µ √ β (cid:17) α +1 K − − α α (cid:18) − b ( a ) a (cid:19) . (53)As before, e a ) = 1 − b ( a ) a . The shape function is again given by Eq. (23), therebyproducing a complete wormhole solution.
While noncommutative geometry was instrumental in obtaining a wormhole solution,its role may extend well beyond the present study. For example, there is an aspect ofwormhole physics that goes back to Ref. [1] but has not been discussed so far: recalling8hat the radial tension τ ( r ) is the negative of p r ( r ) and reintroducing c and G , the radialtension is given by [1] τ ( r ) = b ( r ) /r − r − b ( r )]Φ ′ ( r )8 πGc − r . (54)From this condition it follows that the radial tension at the throat is τ = 18 πGc − r ≈ × dyncm (cid:18)
10 m r (cid:19) . (55)In particular, for r = 3 km, τ has the same magnitude as the pressure at the center ofa massive neutron star. Attributing this outcome to exotic matter is problematical atbest since exotic matter was introduced for a completely different reason: matter is calledexotic if it violates the NEC.It was shown in a recent paper (Kuhfittig [20]) that noncommutative geometry canaccount for the large radial tension. As an offshoot of string theory, it can therefore beviewed as a foray into quantum gravity, thereby going beyond classical general relativity. The strategy for the theoretical construction of wormholes adopted by Morris and Thorne[1] was to specify the desired geometric properties of the wormhole and then manufac-ture or search the Universe for matter or fields that would produce the correspondingenergy-momentum tensor. The purpose of this paper is to determine complete wormholesolutions by starting with a noncommutative-geometry background. Given that the non-commutative effects can be implemented in the Einstein field equations without changingthe Einstein tensor, it follows that the length scales can be macroscopic. It was alsoconcluded that for larger r , the radial and transverse pressures become equal, therebyyielding the equation of state p = ωρ , allowing us to consider a cosmological setting. Thiscase is discussed in Sec. 3. The solution obtained is not asymptotically flat and needs tobe cut off and joined to an exterior Schwarzschild solution at the junction surface r = a .It was found that the solution is valid for all ω = −
1. (An examination of the surfacestresses has shown, however, that a positive surface pressure at the junction surface isobtained only if − < ω ≤ ρ = (1 / | ω | ) p − α , α > −
1. This equation of state not only generalizes the previous solution, it can alsobe adapted to another cosmological model, generalized Chaplygin gas. The result is twoadditional complete wormhole solutions.Previous studies [2, 18] have shown that wormhole solutions exist for ω < − b = b ( r ), asshown in Ref. [19], but it does not yield the redshift function Φ = Φ( r ).As a final remark, Sec. 6 reiterates another aspect of noncommutative geometry,the ability to account for the enormous radial tension in a typical traversable wormhole,discussed in Ref. [20]. 9 eferences [1] M. S. Morris and K. S. Thorne, Am. J. Phys. , 395-412 (1988).[2] F. S. N. Lobo, Phys. Rev. D , 0844011 (2005).[3] E. Witten, Nucl. Phys. B , 335-350 (1996).[4] N. Seiberg and E. Witten,
J. High Energy Phys. , 032 (1999).[5] A. Smailagic and E. Spallucci,
J. Phys. A , L-467-L-471 (2003).[6] P. Nicolini, A. Smailagic and E. Spallucci, Phys. Lett. B , 547-551 (2006).[7] P. Nicolini and E. Spallucci,
Class. Quant. Grav. , 015019 (2010).[8] M. Rinaldi, Class. Quant. Grav. , 105022 (2011).[9] F. Rahaman, P. K. F. Kuhfittig, S. Ray and S. Islam, Phys. Rev. D , 106101(2012).[10] P. K. F. Kuhfittig, Int. J. Pure Appl. Math. , 401-408 (2013).[11] K. Nozari and S. H. Mehdipour, Class. Quant. Grav. , 175015 (2008).[12] J. Liang and B. Liu, EPL , 30001 (2012).[13] F. S. N. Lobo,
Class. Quant. Grav. , 4811 (2004).[14] F. S. N. Lobo, Phys. Rev. D , 064028 (2006).[15] A. Y. Kamenshchik, U. Moschella and V. Pasquier Phys. Lett. B , 265-268 (2001).[16] M. C. Bento, O. Bertolami and A. A. Sen,
Phys. Rev. D , 043507 (2002).[17] P. K. F. Kuhfittig, arXiv: 0802.365.[18] S. V. Sushkov, Phys. Rev. D , 043520 (2005).[19] P. K. F. Kuhfittig, Int. J. Mod. Phys. D , 1550023 (2015).[20] P. K. F. Kuhfittig, Eur. Phys. J. Plus135