PHYSICAL REVIEW D 100, 044011 (2019)
Traversable wormholes in five-dimensional Lovelock theory
Gaston Giribet, Emilio Rubín de Celis, and Claudio Simeone Physics Department, University of Buenos Aires and IFIBA-CONICET Ciudad Universitaria, pabellón 1, 1428 Buenos Aires, Argentina
(Received 11 June 2019; published 8 August 2019)
In general relativity, traversable wormholes are possible provided they do not represent shortcuts in the spacetime. Einstein equations, together with the achronal averaged null energy condition, demand to take longer for an observer to go through the wormhole than through the ambient space. This forbids wormholes from connecting two distant regions in the space. The situation is different when higher-curvature corrections are considered. Here, we construct a traversable wormhole solution connecting two asymptotically flat regions, otherwise disconnected. This geometry is an electrovacuum solution to the Lovelock theory of gravity coupled to an Abelian gauge field. The electric flux suffices to support the wormhole throat and to stabilize the solution. In fact, we show that, in contrast to other wormhole solutions previously found in this theory, the one constructed here turns out to be stable under scalar perturbations. We also consider wormholes in anti–de Sitter (AdS). We present a protection argument showing that, while stable traversable wormholes connecting two asymptotically locally AdS5 spaces do exist in the higher-curvature theory, the region of the parameter space where such solutions are admitted lies outside the causality bounds coming from AdS=CFT.
DOI: 10.1103/PhysRevD.100.044011
I. INTRODUCTION describes a pair of extremal magnetically charged black holes connected by a long throat, in such a way that it takes Wormholes are one of the most fascinating solutions of longer for an observer to go through the wormhole than gravitational field equations. Originally conceived as a through the ambient space, as demanded by the inalienable hypothesis on the structure of matter in classical physics achronal averaged null energy condition [18]. Einstein [1], wormholes have served as illustrative examples on how equations, once such an energy condition on the matter abstruse the topology of spacetime can be [2], allowing us fields was imposed, do not allow for wormholes with a to investigate to what extent causality, locality, and energy short throat. In particular, wormholes connecting two conditions are interrelated [3]. These geometries have also distinct asymptotically flat regions are excluded. The provided a particularly rich source of inspiration for science situation is different when higher-curvature corrections fiction literature. The science fiction wormholes, however, to the gravitational action are taken into account. In that are radically different from those seriously considered in case, the higher-degree2 terms can effectively act as the theoretical physics; the main difference being their invi- exotic matter contribution needed for such wormholes ability at macroscopic scales. The existence of wormholes to exist. demands the violation of certain energy conditions, which Here, we will explicitly show that higher-curvature is only possible in quantum physics. models in five dimensions do allow for stable wormholes In the past few years, the interest on wormhole geom- that connect two asymptotically flat—or asymptotically etries have been renewed. Both traversable and nontravers- anti–de Sitter (AdS)—regions without introducing extra able wormhole geometries were recently considered in exotic matter. As a working example, we will consider the relation to quantum gravity and notably to holography Einstein-Maxwell theory supplemented with the quadratic [4–13].In[13], an explicit example of a metastable, Gauss-Bonnet higher-curvature terms; namely traversable wormhole was constructed.1 The solution Z 1 5 pffiffiffiffiffiffi μν 1 S d x −g R − 2Λ − FμνF There have been other interesting papers recently on worm- ¼ 16π ð – hole geometries; see for instance [14 17] and references therein μνρσ μν 2 and thereof. þ αðRμνρσR − 4RμνR þ R ÞÞ þ B; ð1Þ
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. 2In dimensions greater than four, higher-curvature terms do not Further distribution of this work must maintain attribution to necessarily yield higher-order terms in the field equations. They the author(s) and the published article’s title, journal citation, can well lead to second-order field equations that are nonlinear in and DOI. Funded by SCOAP3. the second derivative of the fields.
2470-0010=2019=100(4)=044011(11) 044011-1 Published by the American Physical Society GIRIBET, DE CELIS, and SIMEONE PHYS. REV. D 100, 044011 (2019) where B stands for the boundary term that renders the the curvature would contribute, and so they cannot be variational problem well posed. This theory is the favorite neglected. If we think of (1) as the truncation of an model to study the effects of higher-order terms in the ultraviolet complete theory, the effective theory would context of AdS5=CFT4 correspondence [19,20], the reason break down at that scale. In addition, we will see that being that it is analytically solvable in many physically the geometric construction of the higher-curvature worm- attractive scenarios such as AdS black holes. Besides, holes requires one to consider certain internal patches of the action (1) can be motivated from string theory: it resembles spacetime solutions to (1) that are nonperturbative, in the α0 ∼ 1 α— the -corrected low energy effective action of heterotic sense of having Son-shell = these solutions are some- string theory, and it also appears in Calabi-Yau compacti- how analogous to the self-accelerated solutions of higher- 3 fication of M theory to five dimensions. The specific derivative models. However, one can in principle answer combination of quadratic terms in (1) is the only one that these objections in the following way: Despite that higher- pffiffiffi yields second-order field equations [24,25]. As a conse- curvature terms cannot be neglected at length scales α, quence, the theory is free of Ostrogradsky instabilities. It one has the expectation that the qualitative features intro- also results as free of ghosts around its perturbative duced by the terms quadratic in the curvature will not be maximally symmetric vacuum [26]. The theory, however, drastically affected by the introduction of, say, quartic or exhibits some causality issues [27] and other notorious higher terms. More precisely, one can imagine a window in features [28].In[27], the causality constraints of higher- the parameter space such that the wormhole throat, curvature models were studied, and it was shown in although microscopic, is still larger than the length at particular that a theory such as (1) has to be supplemented which quartic terms become relevant. We will adopt here a with massive higher-spin fields in order to be free of pragmatic point of view: As in other works in which higher- causality problems. Causal structure of Einstein-Gauss- curvature actions such as (1) are considered—notably in the Bonnet (EGB) theory has also been studied in [29,30], context of AdS5=CFT4—and even when such actions seem where different notions closely connected to causality are to violate the logic of effective field theory and require fine- studied in detail, such as the relation between Killing tuning, we will take into account that holography does not horizons and characteristic hypersurfaces, hyperbolicity in exclude such models, and so they might in principle be the near horizon regions. considered [35]. Theory (1) is a particular case of the so-called Lovelock We should probably add that wormhole solutions in five- theory of gravity [31], which is a natural generalization of dimensional EGB theory (1) have already been discussed in Einstein theory to higher dimensions [25]. It is defined as the literature [36–40]. Our solutions, however, differ from the dimensional extension of a topological invariant that, in those considered before in many aspects. For example, the virtue of the Chern-Weil generalization of the Gauss- main differences between the wormhole solutions consid- Bonnet theorem, computes the Euler characteristic of a ered in [36–38] and ours are the following: First, the 4-manifold. In such a theory, we will construct traversable solutions considered therein only exist at the so-called wormhole geometries supported by a Maxwell field with- Chern-Simons point Λα ¼ −3=4 [41], while our solutions out introducing exotic matter and, as a consequence, exist for a continuous range of the parameter α after the satisfying the energy conditions. Our wormhole solution cosmological constant has been fixed. Second, the base connects two different asymptotically flat or AdS regions. manifold in the solution of [36–38] is different from those In other words, it represents a “short” wormhole. The of our examples: while we consider three-dimensional impediments that one finds when trying to construct such a constant curvature manifolds for the constant-time, solution in Einstein theory are circumvented here due to the constant-radius foliation of the geometry, the base manifold presence of higher-curvature terms, which suffice to sup- of the solution considered in [36–38] is the product of port the throat. This implies, in particular, that the worm-ffiffiffi p a two-dimensional locally hyperbolic space H2=Γ and a hole will be microscopic, i.e., with a throat of the size α. 1 The geometry still represents an exact solution to (1) which, circle S . A third difference is that our solutions are – constructed by gluing together different patches, all of in contrast to other solutions found previously [32 34],is ξ stable under S-perturbations. It is the presence of the Uð1Þ them corresponding to different values of in the spheri- charge that stabilizes the wormhole and makes the asymp- cally symmetric solution (3) below. In contrast, the solution of [36–38] is given by a smooth C∞ single manifold—at totically flat solution possible. pffiffiffi The fact that the wormhole throat is of order α raises an the Chern-Simons point, the different effective cosmological immediate objection: At that scale, terms that are higher in constants of the higher-curvature Lovelock theory coincide. Our solutions also differ from the class of solutions studied in
3 [39]. There, the authors find some conditions and no-go The first higher-curvature correction of M theory in 11 propositions for the existence of wormhole-type solutions in dimensions is a quartic term, R4 [21,22]. When compactifying the theory on a six-dimensional Calabi-Yau, the effective five- EGB theory coupled to matter. Those results are certainly dimensional theory exhibits R2 terms as those in (1); see for consistent with our finding: For instance, in proposition 2 of instance Eqs. (2.6)–(2.9) in [21]; see also Eq. (1) in [23]. [39], the point is made that the null energy condition has to be
044011-2 TRAVERSABLE WORMHOLES IN FIVE-DIMENSIONAL … PHYS. REV. D 100, 044011 (2019) violated if thewormhole’s throat is embedded in the so-called α > 0 and, mainly, asymptotically flat spacetimes which general relativity (GR) branch of EGB theory [which correspond to ξ ¼ −1 solutions and vanishing cosmologi- corresponds to the choice ξ ¼ 1 in (3) below]. Our solution cal constants. These are k ¼ 1 spherical geometries with a does obey the null energy condition on the throat precisely branch singularity at r r , shielded by two horizons for p¼ffiffiffi S because its throat is embedded in the opposite branch [the charges jQj Qc there is a naked singularity at the radius those considered in Ref. [32], where non-GR branches were where the radicant in (3) vanishes; i.e., at the surface also considered: In contrast to those in [32], the solutions we r ¼ rS, where rS is the largest real solution to the equation construct here have attached an asymptotic region with an 6 1 4αΛ 3 216α − 8α 2 3 0 rSð þ = ÞþrS M Q = ¼ . The outer hori- arbitrary small effective cosmological constant. In addition, zon for jQj
044011-3 GIRIBET, DE CELIS, and SIMEONE PHYS. REV. D 100, 044011 (2019) asymptotically flat configurations, but we keep track of a we set Λ ¼ 0. The hypersurface at the throat determines nonvanishing cosmological constant to later extend it to the that the inner regions must both be of the ξi ¼þ1 branch to 5 case of wormholes in AdS5 and make some comments in construct the symmetric wormhole. The configuration relation to AdS=CFT. implies a throat whose radial coordinate is greater than
Our solution is constructed with four bulk pieces, four the branch singularity of the interior metrics, a>rSi , while distinct patches joined by three 4-surfaces. Each piece is a bubbles’ radii must be greater than the would-be horizons region obtained from metric (2) by removing part of the of the external black hole metrics or their singularity geometry on one side of some codimension-one hypersur- surfaces; b>MaxfrSe ;rHg. face defined at a fixed radial coordinate. The four bulk We are looking for solutions without matter sources, so regions are pasted to construct the geodesically complete that the construction requires zero energy density and manifold M describing our traversable wormhole. We pressure at the bubbles and throat, implying vanishing Me denote the four regions as left exterior L, left interior induced stress tensor at the shells. A Gauss law integration Mi Mi Me L, right interior R, and right exterior R, with their of MaxwellH equations across the noncharged surfaces of the α μν μν ½μ ν respective coordinates xL;R for the left (L) and right (R) shells, FμνdX ¼ 0 with dX ¼ r t dΩ, gives the bulks. Explicitly, they are continuity of the electromagnetic field in the temporal tμ and radial rν directions, as Me α ≥ Mi α ≥ ≥ L ¼fxL=rL bg; L ¼fxL=b rL ag; μ ν þ ðηFμνt r Þj− ¼ 0; ð8Þ Mi α ≤ ≤ Me α ≤ R ¼fxR=a rR bg; R ¼fxR=b rRg; − M Me ∪Mi ∪Mi ∪Me where þ and indicate the bulks at each side of the shell, and the complete manifold is ¼ L L R R. η η 1 Each external region Me is joined to its corresponding with the orientation factor of each bulk region as ¼þ L;R for radial coordinates pointing from left to right in the inner region Mi at the hypersurface of a bubble Σb .In L;R L;R wormhole manifold, or η ¼ −1 if the radial coordinates Σb Mi ∩ Me other words, L;R ¼ L;R L;R. The left bubble is point in the opposite direction. The latter condition can be placed at r ¼ b and the right bubble is placed symmet- η η L given in an orthonormal frame as ð Ftˆ rˆÞþ ¼ð Ftˆ rˆÞ− or, rically at r ¼ b. Inner regions are also glued to each other −1 R using the metric function f ¼ −gtt ¼ grr , directly as at the throat located at r ¼ a ¼ r , with a where the dots stand for derivatives with respect to τ. For a static timelike shell at r ¼ ρ, such that ρ˜_ ¼ 0 and ρ̈˜ ¼ 0,we obtain 5A static throat with null or positive energy density (and α > 0) is incompatible with bulk regions of the ξ ¼ −1 branch at both FIG. 1. Scheme of the wormhole geometry. sides of the hypersurface [32]. 044011-4 TRAVERSABLE WORMHOLES IN FIVE-DIMENSIONAL … PHYS. REV. D 100, 044011 (2019) α pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi τ The latter four equations determine the possible configu- Sτ η f ρ −η− f− ρ ¼ 2πρ3 þ þð Þ ð Þ rations for the wormhole spacetime. A priori, from the ρ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi simultaneous requirements in (15) and (17), we see that 3 − ρ − ρ −η η ρ ρ × kþ4α fþð Þ f−ð Þ þ − fþð Þf−ð Þ ; there are no solutions with a vanishing charge and a vanishing cosmological constant. Considering a nonvan- ð11Þ ishing charge only, k ¼ −1 and k ¼ 0 curvatures are not admissible either. The remaining possibility with Λ ¼ 0 θ ϕ χ ≠ 0 1 Sθ ¼Sϕ ¼Sχ and Q is the spherical (k ¼ ) wormholes which are η η h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i shown to exist and are studied below. The inclusion of þ − Λ ≠ 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffi−pffiffiffiffiffiffiffiffiffiffiffiffi k−ζðρÞ− f ðρÞf−ðρÞη η− ; generates a variety of possibilities commented in the ρ ρ þ þ fþð Þ f−ð Þ last section. ð12Þ Now, let us study the space of solutions. To analyze the wormhole construction we use the following definitions for where we used the definition the configuration parameters: 2 2 2α ρ2 ≡ a ≡ b ζ ρ ≡ − − ρ 0 ρ − ρ x ;y ; ð18Þ ð Þ ρ k þ 4α fð Þ f ð Þþk fð Þ 4α 4α 2 2 Q ρ Λ ≡ Mi ≡ M ; mi ;m ; ð19Þ ¼ 3ρ4 þ 3 ð13Þ α α 2 2 1 Q 4αΛ and in the last equality of (12) it is assumed that Q and Λ 2 ≡ λ ≡ q 3 4α ; 3 : ð20Þ are the same at both sides of the hypersurface. We recall that, despite that we are mainly interested in Λ ¼ 0 and k ¼ 1, we keep track of cosmological constant solutions Squaring conveniently Eq. (14), and combining it with and curvature parameter k in some general expressions to (15), the static equations for a general symmetric vacuum be able to compare cases, as well as for comments throat are on Sec. V. 4 2 4 2 − 3 2 0 We first consider vanishing stress tensor components in kx þð k miÞx þ q ¼ ; ð21Þ the static case to apply them to the construction of the 3 3 − λ 2 16 4 2 − 0 throat. The inner regions of our wormhole have metric ð Þx þ kx þð k miÞ¼ ; ð22Þ functions fiðaÞ at the position of the throat, at rL ¼ rR ¼ a, 0 and the orientation factor for each bulk region at the throat with the condition x þ k> , not to lose the information of the sign in the original unsquared equation. Additionally, is given by ηLi ¼ −1 and ηRi ¼ 1. Putting all together in the requirement x>x, where x ¼ðr Þ2=ð4αÞ is the (11) and (12) the following two equations are obtained: s s Si branch singularity corresponding to the interior radial 2 coordinate r , must be considered. The latter equations a Si f ðaÞ¼3 k þ ; ð14Þ λ i 4α establish the relation among x, mi, q, and as compatible with a generic symmetric vacuum throat. It is worth pointing out that the existence of the worm- f ðaÞ¼ζðaÞ − k: ð15Þ i hole solutions demands the presence of the higher- curvature terms. This is why the GR limit α → 0 in the On the other hand, at the left and right side bubbles placed equations above does not yield any configuration of this at r b and r b, the exterior regions have metric L ¼ R ¼ class. Both finite values of the higher-curvature coupling α functions f b , while the left and right interior regions eð Þ and the electric charge Q are necessary for having solutions have f b . The orientation factor for the bulk regions at ið Þ like the ones we presented here. Nevertheless, the role played each bubble is given by η −1 and η −1 for the left Li ¼ Le ¼ by the higher-curvature terms and by the charge are different: side bubble, and η 1 and η 1 for the right side Re ¼ Ri ¼ while the former are needed and are sufficient to have bubble. By demanding a vanishing stress tensor, the wormhole solutions in five-dimensional EGB theory [46], following two equations are obtained: the latter is what, as we will see below, suffices to render the solution stable under radial perturbations; cf. [32]. pffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffi b2 feðbÞ fiðbÞ ¼ 3 k þ − feðbÞ − fiðbÞ; ð16Þ On the other hand, the vacuum bubbles at radial coor- 4α dinates b are constructed from the junction of an interior pffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffi metric with fiðbÞ¼k þ yð1 þ IyÞ and an exterior feðbÞ¼ 2 2 feðbÞ fiðbÞ ¼ k − ζðbÞ: ð17Þ 1 − ≡ 1 mi − q λ 1=2 k þ yð EyÞ, where Iy ð þ y2 y3 þ Þ and 044011-5 GIRIBET, DE CELIS, and SIMEONE PHYS. REV. D 100, 044011 (2019) 2 m 2q 1=2 and (26), the interior mass and charge in this approximation E ≡ð1þ 2 − 3 þλÞ , respectively. From Eqs. (16) and y y y ≃ 4 16 2 ≃ 4 2 0 are mi þ x and q x . Under this consideration (17) of the shell we read the general conditions kþy> 2 2 2 ζy ¼ðq=yÞ ≃ ð2x=yÞ . Besides, considering the inequality and k>ζy, with ζy ≡ ζðbÞ¼ðq=yÞ þ yλ. Additionally, Ey >Iy=2,wehave after some manipulations, we have the inequalities Iy > 2 Ey >Iy= . Considering these conditions, the combined 3 1 4 6 2 3 2 2 − 2 2 3 3 vacuum bubble equations can be expressed as m> þ x þ x =y þ 2 ð x = y = þ x =yÞ; ð30Þ 3 − − − yIyEy ¼½ ðk þ yÞ yðIy EyÞ ðIy EyÞ; ð23Þ for the exterior mass. Using the latter and Eqs. (28) and (29) to express the sum m þ m to first order in the small ζ − i y þ y ¼ yðIy EyÞ: ð24Þ parameters, we have 2 4α 2 We stress that y>yh ¼ rH=ð Þ if the charge is lower than −ζy þ 6ζy þ yθ ≳ 5; ð31Þ the critical value for the exterior metric, or if not, then 2 4α θ 13 ζ − 10ζ 1=2 6 ∼ O 1 y>ys ¼ðrSe Þ =ð Þ to avoid branch singularity. The latter where ¼ 2 y y þ ð Þ, and positive, for equations and conditions give the relations among y, mi, m, 0 < ζy < 1. Defining ϵ as a positive quantity of the same q,andλ compatible with the static vacuum bubble. Note that order as x, y, and q in the small wormhole, the inequality the positivity of each side in (24) is fixed by the aforemen- 1=2 (31) establishes that 1 > ζy > 1 − ϵ, which is consistent tioned conditions and, therefore, M 044011-6 TRAVERSABLE WORMHOLES IN FIVE-DIMENSIONAL … PHYS. REV. D 100, 044011 (2019) (a) (b) FIG. 2. Solid curves represent stable configurations (only possible with supercritical charge values). Dashed curves are unstable solutions. (a) Bubble radius against throat radius. (b) Interior mass (upper curve) and exterior mass (lower curve). The dynamics of the throat at radius a˜ ≡ aðτÞ is potential evaluated at the static radius a is V0ðaÞ¼0, 2 described by a˜_ þ Vða˜Þ¼0, with the effective potential while for the second derivative we obtain ˜ 2 2 Ix 3q − 4kx Vða˜Þ¼k þ x˜ − x˜ ; ð34Þ 00 2 V ðaÞ¼ 2 ; ð35Þ 4αx ðk þ xÞ τ α2ðτÞ which follows from Sτ ¼ 0 in (9), where we used x˜ ≡ , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4α where the prime stands for the derivative with respect to the 2 2 ˜ ≡ 1 mi − q λ and Ix þ x˜2 x˜3 þ . The first derivative of the radius of the shell. Evaluating the latter for the wormhole (a) (b) FIG. 3. Solid curves represent stable configurations (only possible with supercritical charge values). Dashed curves are unstable solutions. (a) Charge against external mass. (b) Bubble radii (upper curve) and throat radius (lower curve). 044011-7 GIRIBET, DE CELIS, and SIMEONE PHYS. REV. D 100, 044011 (2019) configurations, with parameters given as in (25) and (26) in dimensionless mass parameters against dimensionless pffiffiffi the throat, the second derivative of the potential at the throat radius a= α. Figure 3 shows the dimensionless equilibrium position is charge and shells radii against dimensionless external mass M=α. The curves in the different sets of parameter axes 8 þ 9x represent the wormhole solutions. Solid line curves re- V00ðaÞ¼ > 0;q≠ 0; λ ¼ 0;k¼ 1; ð36Þ 4αð1 þ xÞ present stable configurations, and dashed lines are unstable. Stable configurations occur only with charges greater than yielding stable symmetric vacuum throats in every case; the critical value given by the external asymptotically flat cf. [40]. The complete stability of shells in the wormhole black hole metric, the cases with jQj >Qc are painted in ≤ we are interested in will exclusively depend on stability of blue, while jQj Qc are in red. 00 0 the bubbles.6 The neutral equilibrium condition V ðbÞ¼ for the Dynamics of the bubbles with radius b˜ ≡ bðτÞ is configuration space of the bubble can be obtained by 2 equating to zero the left-hand side of (39) with k ¼ 1 and ˜_ ˜ 0 described by b þ VðbÞ¼ , with the effective potential λ ¼ 0. This gives the relation between y and the dimen- sionless charge q in the configuration of neutral equilib- ˜ I˜ E˜ ˜ ˜ − y ˜ − ˜ y y rium of the bubble, namely VðbÞ¼k þ y 3 Iy Ey þ ˜ ˜ ; ð37Þ Iy − Ey 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 yne 3 2 − 9 6 − 2 2 2 q ¼ þ yne þ yne yne : ð40Þ b ðτÞ ne 2 where the dynamical parameter is y˜ ≡ 4α . Evaluating at ˜ the static position of radius b we have y˜ ¼ y, Iy ¼ Iy, and ˜ By introducing this charge in the formulas for the static Ey ¼ Ey. The first derivative of the potential at the configurations (28) and (29) we would obtain the mass 0 equilibrium position is V ðbÞ¼0, while for the second parameters for the neutral equilibrium curves of the bubble; derivative we obtain these are m ¼ mðyneÞ and mi ¼ miðyneÞ. On the other hand, considering Eqs. (25) and (26) for the static throat of 1 3q2=y3 I − E our wormhole we can write the charge in terms of interior V00ðbÞ¼ − y y − 1 : ð38Þ α Iy − Ey IyEy mass as 1 The stability requirement V00ðbÞ > 0 can be expressed 2 3=2 q ¼ ½ð28 þ 9miÞ − 80 − 108mi ð41Þ generically as 243 2 2 2 for mi > 4. Requiring these two square charges to be 9q ðk þ yÞ − ðζy þ yÞ½3q þ y ð3ðk þ yÞ − ζyÞ > 0; ð39Þ the same, by evaluating the latter at mi ¼ miðyneÞ,we achieve the neutral equilibrium point in the wormhole where we used Eqs. (23) and (24), together with the space of solutions. Solving accordingly we obtain that ζ 0 general condition y þ y> for vacuum static bubbles. ≃ 0 70 yne . and, for the original metric parametersffiffiffi at From the latter, a general feature can be mentioned; ≃ 0 94pα the neutralffiffiffi equilibrium point, we have ane . , bubbles are unstable without electromagnetic field for ≃ 1 68pα ≃ 3 28α ≃ 2 82α ≃ bne . , jQjne . , Mne . , and Mine any possible value of the cosmological constant. Then, 7.94α. Each of these neutral equilibrium parameters can only charged configurations would admit stable vacuum be read in the corresponding plotted figure. Stable worm- bubbles. holes occur with bubble radius bQc. This might be relevant for some questions on λ 3 4 − 2 3 4 2 4 3 2 ¼ þ k=x q =x , mi ¼ k þ kx þ q =x and is stable quantum gravity: In the path integral approach, one 3 2 4 2 if q > kx ,asitisgivenby(35) by using the general condition formally defines the theory as the fluctuations about saddle k þ x>0. We note that if there were a vanishing charge, the stability would depend exclusively on the value of the curvature points that obey certain asymptotic conditions that define k. Particularly, if q ¼ 0 and k ¼ 1 solutions exist only for λ > 0 charges at infinity. One picks such saddles from a set of and are unstable under radial perturbations. physically sensible classical solutions, typically excluding 044011-8 TRAVERSABLE WORMHOLES IN FIVE-DIMENSIONAL … PHYS. REV. D 100, 044011 (2019) naked singularities and other pathological configurations. the stable AdS5 wormholes are possible with the causality This means that certain regions of the space of charges will segment8 in principle be excluded. For example, solutions (2) and (3) 9 7 with jQj >Qc are expected to develop a branch singularity, − < λ < ð43Þ and therefore one excludes such values of the charge. 25 9 — Here, we are finding that completely regular solutions that comes from AdS5=CFT4. It turns out that stable worm- with two asymptotic regions—actually exist for supercriti- hole solutions can be seen to exist within the range cal values of the charge, and thus such a bound should be −1 ≤ λ ≲ −0.83, which means that the causality bound reconsidered. Conversely, the analysis above also tells us excludes such solutions. An interesting particular case, which that wormhole solutions with jQj 044011-9 GIRIBET, DE CELIS, and SIMEONE PHYS. REV. D 100, 044011 (2019) [1] C. W. Misner and J. A. 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