Stability of Schwarzschild- f(R) gravity thin-shell wormholes
aa r X i v : . [ g r- q c ] O c t Stability of Schwarzschild- f ( R ) gravity thin-shell wormholes Alina Khaybullina ∗ and Gulira Tuleganova † Zel’dovich International Center for Astrophysics, Bashkir State Pedagogical University,3A, October Revolution Street, Ufa 450008, RB, Russia (Dated: October 23, 2018)Mazharimousavi and Halilsoy [1] recently proposed wormhole solutions in f ( R )-gravity that satisfyenergy conditions but are unstable. We show here that stability could still be achieved for thin-shell wormholes obtained by gluing the wormholes in f ( R )-gravity with the exterior Schwarzschildvacuum. Using the new geometrical constraints from thin-shell ”mass” and from external ”force”developed by Garcia, Lobo and Visser, we demarcate and analyze the stability regions. I. INTRODUCTION
Wormholes are geometrical handles in spacetime that connect two universes or two distant regions of spacetime andare solutions of Einstein’s general relativity including other theories of gravity. The subject has received considerableattention after the influential theoretical work by Morris and Thorne [2]. While there exist enormous work on blackholes, especially directed to observing the signatures of a supermassive black hole at our galactic center, relatively lesswork is available on wormholes. This scenario has now drastically changed due to the recent discovery that thin-shellwormholes have the ability to mimic recently observed gravitational ring-down post-merger waves that have heretoforebeen believed to be characteristic exclusively of black hole horizon [3-6]. Therefore, the study of stability of thin-shellwormholes is of paramount importance.Morris-Thorne wormholes within Einstein’s theory of general relativity (GR) require exotic ghost matter (i.e.,matter that violates the Null Energy Condition ) for their construction, as shown by Hochberg and Visser [7]. Whileall experiments to date have failed to directly detect astrophysical ghost matter, modified gravity theories such asthe f ( R )-gravity (where R is the Ricci scalar) seems to provide a promising avenue to look at gravitational from amore general perspective beyond Einstein’s theory (for which f ( R ) = R ). There exist extensive works on black holesand wormholes in f ( R )-gravity. Some key references, though by no means exhaustive, might be mentioned, see e.g.,[8-12]. Excellent reviews on f ( R )-gravity are also available [13-16].On the state-of-the-art of f ( R )-gravity, a thorough update on the existence, stability and thermodynamics ofdifferent configurations is in order. The existence problem concerns the conditions necessary for the existence ofconfigurations in f ( R )-gravity that are analogous to the ones in GR. The strategy to obtain those conditions is toemploy what is called a near-horizon test in [17,18] exploiting the regularity of GR black hole horizon. The regularityallows the metric functions to be analytically expanded around the horizon and when they are put back in the fieldequations of f ( R )-gravity, they yield the necessary conditions on the form of f ( R ) admitting analog black holes or f ( R )-black holes. The remarkable result obtained in [17,18] is that just any arbitrary polynomial form of f ( R ) donot admit analog black holes. A classic example is that the existence of analog Schwarzschild black hole requires that f ( R )-gravity must be of the specific form f ( R ) = α √ R + β , where α , β are constants. Analog Reissner-Nordstr¨omblack hole requires an infinite series form of f ( R ) with the coefficents determined by the horizon radius [19].As to the question of stability, a viable f ( R )-gravity solution should be stable, ghost-free and should admit Newto-nian and Post-Newtonian limits including correct cosmological dynamics. This means that for the stability of analogblack holes, the near-horizon test yielding the form of f ( R ) should be further constrained by the conditions that dfdR > d fdR > f ( R )-gravity intoa Brans-Dicke theory with massive scalaron and showed that the f ( R ) analog of Schwarzschild black hole is stableagainst the external perturbations if the scalaron mass squared is not negative (tachyonic mass) that is ensured bythe condition d fdR (cid:12)(cid:12)(cid:12) R =0 >
0. The opposite of this stability inequality is the Dolgov-Kawasaki instability condition d fdR < f ( R )-gravity has also been studied [26]. The status of Jebsen-Birkoff theorem and its stability in f ( R )-gravity hasbeen studied in [27]. ∗ Electronic address: [email protected] † Electronic address: [email protected] That is, matter having ρ + p r < ρ + p t <
0, where ρ is the energy density, p r and p t are radial and tranverse components of pressure. Thermodynamics of analog black holes have been studied [19,28-31]. In [28], using the Hawking temperatureand entropy, the exact expression for heat capacity and the first law of thermodynamics have been found by usingthe Misner-Sharp formalism [19]. Similar thermodynamics dealing with analog Reissner-Nordstr¨om black hole alsoexists [29]. Another novel method of studying thermodynamics and phase transitions of analog black holes is to usegeometrothermodynamical methods [30]. Yokokura [31] generalized the space-time thermodynamics and then from theequation for entropy balance for nonequilibrium processes found the new entropy production terms in f ( R )-gravity.Wormhole solutions including the thin-shell variants have also been obtained in f ( R )-gravity [1,32]. Evolvingwormholes have been recently found by Bhattacharya and Chakraborty [33]. Wormholes in the Palatini formulationof f ( R ) gravity have been studied in [34]. Necessary conditions for having wormholes in f ( R )-gravity have beenobtained in [35]. Some thin-shell wormholes and their stabilities have been studied in [36-38]. The focus in thispaper is on the two traversable analytical wormhole solutions recently found by Mazharimousavi and Halilsoy (MH)[1]. These wormholes are however unstable since, as generically argued in [39], there cannot be ghost-free, stablewormholes in f ( R )-gravity. This being the case, it would be of interest to study the possibility of stability of thin-shell wormholes obtained by gluing the exterior Schwarzschild vacuum with MH wormholes of f ( R )-gravity .In this Letter, our purpose is to use the new constraints developed by Garcia, Lobo and Visser (GLV) [40] for iden-tifying the stability regions of the linearly perturbed spherical motion of the thin-shell moving in the bulk spacetime.For this purpose, we obtain the thin-shell by gluing the relevant solutions at some suitable “standard” coordinateradius. In other words, the asymptotic masses on one side will be the mass of the wormhole in f ( R )-gravity and onthe other side the Schwarzschild mass. Henceforth, we take G = 1 , c = 1 unless specifically restored. II. GLV FORMALISM FOR STABILITY
This formalism being relatively new, for the benefit of readers, we explain below in slightly more detail the GLVformalism [40] for stability of the thin-shell and the attendent new concepts. The formalism is quite generic, flexibleand robust that can be applied to general spherically symmetric spacetimes in 4-dimensions. The idea is to surgicallygraft together two bulk spacetimes in such a way that no event horizon is formed. This surgery generates a thin shell(the wormhole throat, where all the exotic matter is concentrated) between the two bulk spacetimes on either sideof the throat. The thin shell will be free to move in the bulk spacetimes allowing a fully dynamic stability analysisagainst spherically symmetric perturbations. The stability will then be dictated by the properties of the exotic matterresiding on the wormhole throat. The novelty of the formalism is that, apart from a ”mass” constraint from the massof the shell, it introduces an entirely new hitherto unknown geometrical constraint of external ”force”. The motion ofthe shell is driven by a ”potential” V ( a ) appearing in the equation of motion ˙ a + V ( a ) = 0 and the stability undersmall oscillation about a static solution a = a (initial gluing radius) will be given by the condition V ′′ ( a ) ≥
0. Thisis, in short, the strategy of GLV formalism.We note that GLV method is developed within Einstein’s GR having second order equations, wheras f ( R )-gravityequations are of fourth order. To justify the use of GLV method gluing the two solutions from two different theories,we recall the conformal equivalence of f ( R )-gravity with GR minimally coupled to a scalar field with a potential.Mathematically, the conformal transformation converts the fourth-order f ( R ) equations into two second-order equa-tions, one for the Einstein frame metric and the other for a scalar field ϕ . The needed junction condition on extrinsiccurvature in the Einstein frame is the same as in GR [41].Omitting lengthy details, we shall try to reproduce the salient features of GLV formalism as cogently as possible,especially the steps leading to the usable stability constraints. The method starts with gluing together two spheri-cally symmetric spacetimes by ”cut and paste” surgery developed by Visser [42]. GLV take two generic sphericallysymmetric spacetimes M ± possessing the metrics in each as ds ± = − e ± ( r ± ) (cid:20) − b ± ( r ± ) r ± (cid:21) c dt ± + (cid:20) − b ± ( r ± ) r ± (cid:21) − dr ± + r ± d Ω ± , (1)where Ω = dθ + sin θdφ . A single manifold M is obtained by gluing the two manifolds M + and M − at Σ, i.e., at f ( r, τ ) = r − a ( τ ) = 0. That is, the points r < a ( τ ) are excised out of M and the intrinsic metric on Σ is assumed tohave a form ds = − dτ + a ( τ ) ( dθ + sin θ dφ ) . (2) As appropriately commented by an anonymous referee, the present Letter provides a ”graceful exit” to the usual wormhole instabilities.
The position of the junction surface is given by x µ ( τ, θ, φ ) = [ t ( τ ) , a ( τ ) , θ, φ ], and the non-trivial extrinsic curvaturecomponents on both sides of the shell are given by K θ ± θ = ± a r − b ± ( a ) a + ˙ a , (3) K τ ± τ = ± ¨ a + b ± ( a ) − b ′± ( a ) a a q − b ± ( a ) a + ˙ a + Φ ′± ( a ) r − b ± ( a ) a + ˙ a , (4)where the prime now denotes a derivative with respect to the coordinate a and overdot denotes derivative with respectto τ . Note that K ij is not continuous across Σ, so one defines κ ij = K + ij − K − ij that satisfy Lanczos equations on theshell as S ij = − π ( κ ij − δ ij κ kk ) , (5)where S ij is the surface stress-energy tensor on Σ defined by S ij = diag( − σ, P , P ), σ is the surface energy density and P is the surface pressure σ = − πa "r − b + ( a ) a + ˙ a + r − b − ( a ) a + ˙ a , (6) P = 18 πa a + a ¨ a − b + ( a )+ ab ′ + ( a )2 a q − b + ( a ) a + ˙ a + r − b + ( a ) a + ˙ a a Φ ′ + ( a )+ 1 + ˙ a + a ¨ a − b − ( a )+ ab ′− ( a )2 a q − b − ( a ) a + ˙ a + r − b − ( a ) a + ˙ a a Φ ′− ( a ) . (7) T he surface stress S ij satisfies a conservation identity via Lanczos and Gauss-Codazzi equation as S ij | i = h T µν e µ ( j ) n ν i + − , (8)where n ν is the unite normal to Σ and e µ ( j ) are the orthonormal basis vectors such that g ij = g µν e µ ( i ) e ν ( j ) | ± . Defininga new term Ξ = 14 πa " Φ ′ + ( a ) r − b + ( a ) a + ˙ a + Φ ′− ( a ) r − b − ( a ) a + ˙ a , (9)the conservation identity (8) can be written as d ( σA ) dτ + P dAdτ = Ξ A ˙ a , (10)where ˙ a = dadτ , A = 4 πa is the surface area of the shell. The first term represents change in the internal energy of theshell, while the second term represents the work done by the shell’s internal force, and the third term represents thework done by the external forces (genesis of the ”force” constraint). Assuming the existence of a suitable function σ ( a ), the conservation identity can be rewritten as σ ′ = − a ( σ + P ) + Ξ , (11)where σ ′ = dσ/da . The right hand side is the net discontinuity of the bulk momentum flux and is physically interpretedas the work done by external ”forces” on the thin shell occurring due to Φ ′± = 0. We can rearrange Eq.(6) into theform r − b + ( a ) a + ˙ a = − r − b − ( a ) a + ˙ a − πa σ ( a ) , (12)which yields the thin-shell equation of motion given by12 ˙ a + V ( a ) = 0 , (13)where the potential V ( a ) is given by V ( a ) = 12 ( − ¯ b ( a ) a − (cid:20) m s ( a )2 a (cid:21) − (cid:20) ∆( a ) m s ( a ) (cid:21) ) , (14)¯ b ( a ) = b + ( a ) + b − ( a )2 , ∆( a ) = b + ( a ) − b − ( a )2 . The potential V ( a ) is seen as a function of the thin-shell mass m s ( a ) = 4 πa σ ( a ) and is the key for stability analysis.It follows from Eq.(14) that m s ( a ) = − a "r − b + ( a ) a − V ( a ) + r − b − ( a ) a − V ( a ) , (15)the negative sign is required for compatibility with the Lanczos equations (5). Assume a static solution (at a ) to theequation of motion ˙ a + V ( a ) = 0, then a Taylor expansion of V ( a ) around a to second order yields V ( a ) = V ( a ) + V ′ ( a )( a − a ) + 12 V ′′ ( a )( a − a ) + O [( a − a ) ] . (16)But since we are expanding around a static solution, ˙ a = ¨ a = 0, we automatically have V ( a ) = V ′ ( a ) = 0, so it issufficient to consider V ( a ) = 12 V ′′ ( a )( a − a ) + O [( a − a ) ] . (17)The static solution at a is stable if and only if V ′′ ( a ) ≥
0. Since V ( a ) = 0, it follows from Eq.(15) that m s ( a ) = − a s − b + ( a ) a + s − b − ( a ) a . (18)Linearized stability using the condition V ′′ ( a ) ≥ m ′′ s ( a ) | a , that is called the ”mass”constraint given by m ′′ s ( a ) | a ≥ a (cid:20) [ b + ( a ) − a b ′ + ( a )] [1 − b + ( a ) /a ] / + [ b − ( a ) − a b ′− ( a )] [1 − b − ( a ) /a ] / (cid:21) + 12 " b ′′ + ( a ) p − b + ( a ) /a + b ′′− ( a ) p − b − ( a ) /a . (19)Assuming integrability of Eq.(11), lengthy calculations yield the parametric solution: σ ≡ σ ( a ), P ≡ P ( a ) andΞ ≡ Ξ( a ). The same linearized stability analysis with V ′′ ( a ) ≥ σ ′ , σ ′′ , P ′ , P ′′ , Ξ ′ , Ξ ′′ butthe last one is the most relevant. When Φ ± = 0, there will appear an additional constraint analogous to the one on m ′′ s ( a ) | a , called the ”force” constraint given by[4 π Ξ( a ) a ] ′′ | a ≤ n Φ ′′′ + ( a ) p − b + ( a ) /a + Φ ′′′− ( a ) p − b − ( a ) /a o(cid:12)(cid:12)(cid:12) a − ( Φ ′′ + ( a ) ( b + ( a ) /a ) ′ p − b + ( a ) /a + Φ ′′− ( a ) ( b − ( a ) /a ) ′ p − b − ( a ) /a )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a − (cid:26) Φ ′ + ( a ) [( b + ( a ) /a ) ′ [1 − b + ( a ) /a ] / + Φ ′− ( a ) [( b − ( a ) /a ) ′ [1 − b − ( a ) /a ] / (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) a − ( Φ ′ + ( a ) ( b + ( a ) /a ) ′′ p − b + ( a ) /a + Φ ′− ( a ) ( b − ( a ) /a ) ′′ p − b − ( a ) /a )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a . (20)The same force constraint appears for Φ ′± ( a ) ≤ reversed . The threeinequalities, (19),(20) and the reversed one, are the GLV master inequalities for deciding stability zones of thin-shellmotion. III. STABILITY OF THIN-SHELL FROM SCHWARZSCHILD BLACK HOLE- f ( R ) WORMHOLEGLUING (a) Schwarzschild black hole-MH1 wormhole
We glue at a common radius a above the Schwarzschild horizon r hor = 2 M and wormhole throat r , i.e., at acertain radius r = a > M, r . The interior regions r ≤ M , r are surgically excised out of the respective spacetimesbecause we don’t want the presence of any horizon in the resultant wormhole. Thus linear spherical perturbationswill be assumed to take place about the radius a . Casting the Schwarzschild metric into the GLV form of Eq.(1),one obtains b + = 2 M, Φ + = 0 , (21)and similarly casting the first metric of MH1 wormhole [1] of f ( R )-gravity, viz., dτ = − (cid:16) r r (cid:17) − dt + dr − r r + r ( dθ + sin θdφ ) , (22)in the form of GLV metric, we get b − = r r , Φ − = −
12 ln (cid:20)(cid:16) − r r (cid:17) (cid:16) r r (cid:17) (cid:21) . (23)The redshift function Φ − ( r ), now denoted without subscript simply as Φ( r ), happens to be a logarithmic functionshowing divergence at r = r but from the physical standpoint this does not pose any problem since we already madethe choice that the accessible region has r > r - the throat radius of the wormhole.Stability zone of thin shell depends on the sign of Φ ′ ( a ), which for the MH1 solution is given byΦ ′ ( a ) = ( a − r ) r ( a − r ) a . (24)There occurs two cases. Case 1: r = a ∈ ( r , r ], for which we get Φ ′ ( a ) ≤ r = a ∈ [2 r , ∞ ),for which we get Φ ′ ( a ) ≥
0. In either case, there will occur the effect of external ”force” influencing the thin shellmotion. Introducing the above functions in the inequalities (19,20), and defining the dimensionless variables x = Ma , y = r M , (25)we find, respectively m ′′ s ( a ) | a ≥ f ( x, y ) = x (cid:18) √ − x + x (1 − x ) / − xy (2 x y − − x y ) / (cid:19) , (26)[4 π Ξ( a ) a ] ′′ | a ≥ g ( x, y ) = xy (6 − xy − x y + 30 x y + 2 x y − x y )(1 − x y ) / , (27) (cid:0) Φ ′± ( a ) ≤ , Case 1 (cid:1) (28)[4 π Ξ( a ) a ] ′′ | a ≤ g ( x, y ) . (cid:0) Φ ′± ( a ) ≥ , Case 2 (cid:1) (29)It can be noticed from Eq.(26) that for f ( x, y ) to be real, two conditions must be fulfilled: one is x < , whichimplies a > M as required, and the other is y < x or a > r . The latter condition guarantees the reality of also g ( x, y ). Note further that y = r M has no upper bound but has a lower bound 2 such that the wormhole throat satisfies r > M as required. The above Cases then split the xy plane into two areas: In Case 1, x ∈ ( , ), y ∈ (2 ,
4) and inCase 2, x ∈ (0 , ), y ∈ (4 , ∞ ). In the contour plots, the two areas are combined such that x ∈ (0 , ), y ∈ (2 , ∞ ). (b) Schwarzschild black hole-MH2 wormhole FIG. 1: 2D contour plot of f ( x, y ) for x ∈ (0 , ) and y ∈ (2 , ∞ ). The shaded region represents the zone of stability of thethin-shell exclusively under ”mass constraint”.FIG. 2: 2D contour plot of g ( x, y ). The relevant area is defined by x ∈ ( , ), y ∈ (2 ,
4) in Case 1 [Sec.3 (a) ]. For Case 2[Sec.3 (a) ], the relevant area is x ∈ (0 , ), y ∈ (4 , ∞ ). In either case, the exclusive ”force” constraint indicates that the shadedregion is the stable zone. The second wormhole MH2, cast in the GLV form, is ds = − (cid:18) rr (cid:19) dt + dr − r r + r ( dθ + sin θdφ ) , (30) b − = r r , Φ − = 12 ln (cid:20) r r ( r − r ) (cid:21) . (31)Like in (a) , stability zone of the thin-shell depends on the sign of Φ ′ ( a ), which for the MH2 solution is given byΦ ′ ( a ) = 2 a − r ( a − r ) a . (32) FIG. 3: Thin-shell stability under simultaneous constraints is restricted to parameter values ( x, y ) on the left of the curveobtained by the intersections f ( x, y ) ∩ g ( x, y ). There again occurs two cases. Case 1: r = a ∈ ( r , q r ], for which we get Φ ′ ( a ) ≤ r = a ∈ [ q r , ∞ ), for which we get Φ ′ ( a ) ≥
0. In either case, there will occur the effect of external ”force” influencing thethin shell motion. These Cases, like in (a) , will split the xy plane into two areas: In Case 1, x ∈ ( √ , ), y ∈ (2 , √ x ∈ (0 , √ ), y ∈ ( √ , ∞ ). In the contour plots to be given below, the two cases are combined suchthat x ∈ (0 , ), y ∈ (2 , ∞ ). Introducing the above metric functions in the inequalities (19,20), we find m ′′ s ( a ) ≥ f ( x, y ) = x (cid:18) √ − x + x (1 − x ) / + xy (2 x y − − x y ) / (cid:19) , (33)[4 π Ξ( a ) a ] ′′ | a ≥ g ( x, y ) = 4 − x y + 45 x y − x y (1 − x y ) / , (34) (cid:0) Φ ′± ( a ) ≤ , Case 1 (cid:1) (35)[4 π Ξ( a ) a ] ′′ | a ≤ g ( x, y ) , (cid:0) Φ ′± ( a ) ≥ , Case 2 (cid:1) (36)It can be seen that f ( x, y ) in both (a) and (b) is the same, so exactly the same mass constraint applies to both.However, the functions g ( x, y ) and g ( x, y ) differ so that the force constraints different. As a result, the combinedeffect of two constraints would also differ leading to different zones of stability. The results are discussed below. IV. RESULTS AND DISCUSSION
The goal of this work was to demarcate the stability regions of the thin-shell formed by gluing the Schwarzschildblack hole with f ( R )-gravity wormholes derived by MH. The remarkable feature of analytical MH wormholes is thatthey do not require elusive exotic matter but unfortunately are unstable. A limited stability (”graceful exit”) canhowever be explored within the GLV formalism, which is robust providing an excellent way to tackle wider classesof thin-shell wormholes surgically born out of two static spherically symmetric spacetimes. The present study yieldspossible parameter ranges of the participating solutions (Schwarzschild -MH) for which stability in some sense canbe achieved. Earlier, stability of thin-shell from Schwarzschild-Schwarzschild gluing using the approach of a potentialwas studied before by Poisson and Visser [43]. The stability constraints on the classes of thin-shells investigated herehave relevance for their very existence, since stable analytic wormholes are rather scarce. The study of the properties FIG. 4: 2D contour plot of g ( x, y ). Qualitatively, the stability picture is the same as in Fig.2. For Case 1 [Sec.3 (b) ], the areaof g ( x, y ) is demarcated by x ∈ ( √ , ), y ∈ (2 , √
6) and for Case 2 [Sec.3 (b) ], the relevant area is x ∈ (0 , √ ), y ∈ ( √ , ∞ ).The ”force” constraint implies that the shaded area is the stable zone.FIG. 5: Thin-shell stability under simultaneous constraints is restricted to parameter values ( x, y ) to the left of the curveobtained by the intersections f ( x, y ) ∩ g ( x, y ). of light rings on either side of the shell would be interesting as a future task in connection with their characteristicring-down modes [3-6].The thin-shell examples considered above are interesting in their own right since by construction the exterior hasa Schwarzschild vaccum ( M = 0). Physically, this thin-shell configuration resembles a gravastar having an interior f ( R )-MH spacetime but the exterior vacuum is indistinguishable from the Schwarzschild black hole. The stability ofthe thin-shell is dictated by two constraints, one is ”mass” constraint symbolized by an explicit inequality involvingthe second derivative of the mass of the throat, m ′′ s ( a ) and the other is a external ”force” constraint through theinequalities on [4 π Ξ( a ) a ] ′′ depending on the signs of Φ ′± ( a ). Note that the ”mass” constraint (19) does not involveΦ ± and its derivatives, while the ”force” constraint (20 and its reverse) exists due to Φ ± = 0. The latter constraintis a new discovery of GLV. Using these two constraints, we had demarcated the zones of stability in the parameterspace.The analyses in this paper bring out some important characteristics of Schwarzschild-MH thin-shell wormholes.Even though the two independent MH wormholes have different metrics, the stability picture of thin-shells underdiscussion is remarkably similar. Especially, the contour for f ( x, y ) is the same for both thin-shells since b ± are thesame for both MH solutions, only the Φ ± are different. The variables for the contour plots are chosen as x = Ma ,y = r M , where x ∈ (0 , ), y ∈ (2 , ∞ ), the ranges being dictated by the requirement that the gluing radius a > M and that the wormhole throat r > M . The reality requirements of the functions f ( x, y ), g ( x, y ) and g ( x, y ) yieldparameter ranges x < and y < x . It is clear that as x increases from 0 to , the gluing radius a shrinks from ∞ to 2 M . Similarly, when y increases indefinitely from 2, the wormhole throat radius r increases from 2 M to ∞ .The following are our new results. The shaded regions in the 2D contour plots represent the ”topography” of thesurfaces f ( x, y ), g ( x, y ) and g ( x, y ). The higher values on the surfaces are marked by layers of lighter regions. Thezones of stability depend on the Cases discussed in Sec.3. The ”mass” constraint however does not involve suchCases as it does not involve Φ. Now consider (a) of Sec.3 analyzing Schwarzschild-MH1 thin-shell. The contourplot of the function f ( x, y ) is given in Fig.1 for the parameter values x ∈ (0 , ) and y ∈ (2 , ∞ ). The condition (26)then implies that the shaded region in Fig.1 is the zone of stability. The same conclusion holds for (b) too, i.e., forSchwarzschild-MH2 thin-shell.The contour plot of the surface g ( x, y ) is given in Fig.2, where the ”force” constraint is divided also into two casesas pointed out in Sec.3 (a) . For Case 1, the area is demarcated by x ∈ ( , ), y ∈ (2 ,
4) and for Case 2, area isdefined by x ∈ (0 , ), y ∈ (4 , ∞ ). (Note that the lines inside plots make boundaries of points of same heights andnot the boundaries of the areas mentioned here). In either case, the shaded region in Fig.2 indicates the stabilty zonepurely due to ”force” constraint. The contour plot of the surface g ( x, y ) is given in Fig.4 and again there occurs twoCases of inequalities as pointed out in Sec.3 (b) . Qualitatively, the stability picture is similar. For Case 1, the area isdemarcated by x ∈ ( √ , ), y ∈ (2 , √
6) and for Case 2, area is x ∈ (0 , √ ), y ∈ ( √ , ∞ ) implying that the shadedarea in Fig.4 is the stable zone. However, the individual constraints do not completely describe the zones of stability.The complete stability zone should be determined by the combined application of ”mass” and ”force” constraints.In this case, the stability zone is restricted to parameter values ( x, y ) on the left of the curves obtained by theintersections f ( x, y ) ∩ g ( x, y ) (Fig.3) and f ( x, y ) ∩ g ( x, y ) (Fig.5) respectively. The interesting result that followsfrom it is that Schwarzschild-MH2 thin-shell has a larger stability zone (Fig.5) than that of Schwarzschild-MH1 (Fig.3).The intriguing result that emerges is that the thin-shell motion is stable for x < . x < . a is rather far away from the horizon a = 2 M or x = . It can be verified that all theabove stability zones can be read off from the 3D plots as well. Such a complete stability picture is possible essentiallydue to the effect of the ”external force” constraint, a new discovery by GLV. Acknowledgments
The reported study was funded by RFBR according to the research project No. 18-32-00377. [1] S.H. Mazharimousavi and M. Halilsoy, Mod. Phys. Lett. A , 1650192 (2016).[2] M.S. Morris and K.S. Thorne, Am. J. Phys. , 395 (1988).[3] V. Cardoso, E. Franzin and P. Pani, Phys. Rev. Lett. , 171101 (2016); , 089902E (2016).[4] V. Cardoso and P. Pani, Nature Astronomy , 586 (2017).[5] R.A. Konoplya and A. Zhidenko, JCAP 12 (2016) 043.[6] K.K. Nandi, R. N. Izmailov, A.A. Yanbekov and A.A. Shayakhmetov, Phys. Rev. D , 104011 (2017).[7] D. Hochberg and M. Visser, Phys. Rev. Lett. , 746 (1998).[8] S. Nojiri and S.D. Odintsov, Phys. Rev. D , 086005 (2006).[9] S.H. Mazharimousavi and M. Halilsoy, Phys. Rev. D , 0640328 (2011);A. Sheykhi, Phys. Rev. D , 024013 (2012);M. Cvetic, S. Nojiri and S.D. Odintsov, Nucl. Phys. B , 295 (2002);R.G. Cai, Phys. Rev. D , 084014 (2002).[10] A. de la Cruz-Dombriz and A. Dobado, Phys. Rev. D , 087501 (2006);A. de la Cruz-Dombriz, A. Dobado and A.L. Maroto, Phys. Rev. D , 124011 (2009); Phys. Rev. D , 029903E (2011);A. de la Cruz-Dombriz, A. Dobado and A.L. Maroto, Phys. Rev. Lett. , 179001 (2009);P.K.S. Dunsby, V.C. Busti, S. Kandhai, Phys. Rev. D , 064029 (2014);J.A.R. Cembranos, A. de la Cruz-Dombriz and P. Jimeno Romero, Int. J. Geom. Meth. Mod. Phys. , 1450001 (2014);A. de la Cruz-Dombriz, P.K.S. Dunsby, S. Kandhai, D. S´aez-G´omez, Phys. Rev. D , 084016 (2016).[11] G.J. Olmo and D. Rubiera-Garcia, Phys. Rev. D , 124059 (2011);D. Bazeia, L. Losano, G.J. Olmo and D. Rubiera-Garcia, Phys. Rev. D , 044011 (2014); E. Barrientos, F.S.N. Lobo, S. Mendoza, G.J. Olmo and D. Rubiera-Garcia, Phys. Rev. D , 104041 (2018);D. Bazeia, L. Losano, R. Menezes , G.J. Olmo and D. Rubiera-Garcia, Eur. Phys. J. C , 569 (2015).[12] R. Goswami, A.M. Nzioki, S.D. Maharaj and S.G. Ghosh, Phys. Rev. D , 084011 (2014);A.M. Nzioki, R. Goswami and P.K.S. Dunsby, Int. J. Mod. Phys. D , 1750048 (2016);A.M. Nzioki, S. Carloni, R. Goswami and P.K.S. Dunsby, Phys.Rev. D , 084028 (2010);A.M. Nzioki, P.K.S. Dunsby, R. Goswami and S. Carloni, Phys. Rev. D , 024030 (2011);T. Clifton, P. Dunsby, R. Goswami and A.M. Nzioki, Phys. Rev. D , 063517 (2013).[13] T.P. Sotiriou and V. Faraoni, Rev. Mod. Phys. , 451 (2010).[14] S. Nojiri and S.D. Odintsov, Phys. Rept. , 59 (2011).[15] S. Capozziello and V. Faraoni, Beyond Einstein gravity , Fundamental Theories of Physics 170 (Springer, 2011).[16] L. Sebastiani and S. Zerbini, Eur. Phys. J. C , 1591 (2011).[17] S.E.P. Bergliaffa and Y.E.C. de O. Nunes, Phys. Rev. D , 084006 (2011).[18] S.H. Mazharimousavi and M. Halilsoy, Phys. Rev. D , 088501 (2012).[19] S.H. Mazharimousavi, M. Kerachian and M. Halilsoy, Int. J. Mod. Phys. D , 1350057 (2013).[20] Y.S. Myung, T. Moon and E.J. Son, Phys. Rev. D , 124009 (2011).[21] A.D. Dolgov and M. Kawasaki, Phys. Lett. B , 1 (2003).[22] G.J. Olmo, Phys. Rev. Lett. , 261102 (2005).[23] V. Faraoni and S. Nadeau Phys. Rev. D , 124005 (2005).[24] G.J. Olmo, Phys. Rev. D , 083505 (2005); D , 023511 (2007).[25] Y.S. Myung, Eur. Phys. J. C , 1550 (2011).[26] Y.S. Myung, Phys. Rev. D , 104017 (2013).[27] A.M. Nzioki, R. Goswami and P.K.S. Dunsby, Phys. Rev. D , 064050 (2014).[28] S.H. Mazharimousavi and M. Halilsoy, Phys. Rev. D , 064032 (2011).[29] S.H. Mazharimousavi, M. Halilsoy and T. Tahamtan, Eur. Phys. J. C. , 1851 (2012).[30] S. Soroushfar, R. Saffari and N. Kamvar, Eur. Phys. J. C , 476 (2016).[31] Y. Yokokura, Int. J. Mod. Phys. A , 1250160 (2012).[32] F.S.N. Lobo and M.A. Oliveira, Phys. Rev. D , 104012 (2009).[33] S. Bhattacharya and S. Chakraborty, Eur. Phys. J. C , 558 (2017).[34] C. Bambi , A. Cardenas-Avendano, G.J. Olmo and D. Rubiera-Garcia, Phys. Rev. D , 064016 (2016).[35] S.H. Mazharimousavi and M. Halilsoy, Mod. Phys. Lett. A , 1650203 (2016).[36] F. S. N. Lobo and P. Crawford, Class. Quant. Grav. , 4869 (2005);E.F. Eiroa, M.G. Richarte and C. Simeone, Phys. Lett. A , 1 (2008);E.F. Eiroa and G.F. Aguirre, Eur. Phys. J. C , 132 (2016); Phys. Rev. D , 044016 (2016).[37] E. F. Eiroa and C. Simeone, Phys. Rev. D , 044008 (2004); Phys. Rev. D , 127501 (2005);F. Rahaman, M. Kalam and S. Chakraborti, Int. J. Mod. Phys. D , 1669 (2007);Z. Amirabi, M. Halilsoy, S.H. Mazharimousavi, Mod. Phys. Lett. A , 1850049 (2018);S.D. Forghani, S.H. Mazharimousavi and M. Halilsoy, Eur. Phys. J. C , 469 (2018);S.H. Mazharimousavi, M. Halilsoy and S.N.H. Amen. Int. J. Mod. Phys. D , 1750158 (2017);S.H. Mazharimousavi and M. Halilsoy, Int. J. Mod. Phys. D , 1850028 (2017);A. Eid, Indian J. Phys. , 1065 (2018).[38] A.R. Khaybullina, G.F. Akhtaryanova, R.F. Mingazova, D. Saha and R.N. Izmailov, Int. J. Theor. Phys. , 1590 (2014);J.P.S. Lemos and F.S.N. Lobo, Phys. Rev. D , 044030 (2008).[39] K.A. Bronnikov, M.V. Skvortsova and A.A. Starobinsky, Grav. Cosmol. , 216 (2010).[40] N.M. Garcia, F.S.N. Lobo and M. Visser, Phys. Rev. D , 044026 (2012).[41] N. Deruelle, M. Sasaki and Y. Sendouda, Prog. Theor. Phys. , 237 (2008).[42] M. Visser, Lorentzian Wormholes-From Einstein To Hawking (AIP, New York, 1995).[43] E. Poisson and M. Visser, Phys. Rev. D52