Gravitational solitons and C 0 vacuum metrics in five-dimensional Lovelock gravity
aa r X i v : . [ g r- q c ] F e b Gravitational solitons and C vacuum metrics infive-dimensional Lovelock gravity C. Garraffo , G. Giribet , , E. Gravanis , S. Willison Instituto de Astronom´ıa y F´ısica del Espacio, IAFE, CONICET, Argentina.
Ciudad Universitaria, IAFE, C.C. 67, Suc. 28, 1428, Buenos Aires, Argentina . Center for Cosmology and Particle Physics, New York University, NYU, . Departamento de F´ısica, FCEN, Universidad de Buenos Aires, Argentina,
Ciudad Universitaria, Pabell´on 1, 1428, Buenos Aires, Argentina . Department of Physics, Kings College London, UK. Centro de Estudios Cient´ıficos CECS,
Casilla 1469, Valdivia, Chile.
Abstract
Junction conditions for vacuum solutions in five-dimensional Einstein-Gauss-Bonnet gravityare studied. We focus on those cases where two spherically symmetric regions of space-timeare joined in such a way that the induced stress tensor on the junction surface vanishes. So aspherical vacuum shell, containing no matter, arises as a boundary between two regions of thespace-time. A general analysis is given of solutions that can be constructed by this method ofgeometric surgery. Such solutions are a generalized kind of spherically symmetric empty spacesolutions, described by metric functions of the class C . New global structures arise with surpris-ing features. In particular, we show that vacuum spherically symmetric wormholes do exist inthis theory. These can be regarded as gravitational solitons, which connect two asymptotically(Anti) de-Sitter spaces with different masses and/or different effective cosmological constants.We prove the existence of both static and dynamical solutions and discuss their (in)stability un-der perturbations that preserve the symmetry. This leads us to discuss a new type of instabilitythat arises in five-dimensional Lovelock theory of gravity for certain values of the coupling ofthe Gauss-Bonnet term. The issues of existence and uniqueness of solutions and determinismin the dynamical evolution are also discussed. ontents = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Instantaneous shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Static spherical shells with Λ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 C metrics 29 Introduction
In this article we shall be concerned with the Einstein-Gauss-Bonnet theory of gravity, whose actionis given by the Einstein-Hilbert term plus the Einstein cosmological constant term Λ, and bothsupplemented with the Gauss-Bonnet term, quadratic in the curvature. The action of the theoryreads S = 12 κ Z d x √− g (cid:16) R −
2Λ + α (cid:0) R − R AB R AB + R ABCD R ABCD (cid:1) (cid:17) , (1)where κ = 8 πG and α represents the coupling constant of the quadratic term. The quadratic termis often called the Gauss-Bonnet term because it is the dimensional extension of the Gauss-Bonnettopological invariant in four dimensions.In five dimensions, the action (1) contains all of the non-zero terms of the Lovelock series. Itis thus the most general metric torsion-free theory of gravity which leads to conserved equationsof motion which are second order in derivatives [1]. The perturbation theory about the maximallysymmetric vacuum is free of ghosts [2, 3] which suggests that it could appear as a higher ordercorrection to Einstein’s theory in the effective action coming from some more fundamental quantumtheory. In fact, the Gauss-Bonnet term naturally arises as a higher order correction to gravity withinstring theory. Although the fourth-order derivative corrections are known to appear as the next-to-leading-order correction in the Type II strings[6], the quadratic corrections are present in both theheterotic and bosonic string theory [2, 4, 5]. In those cases, the coupling of the Gauss-Bonnet termis given by α ′ multiplied by a function of the dilaton, and so corresponding to powers of the stringcoupling. The five-dimensional Gauss-Bonnet term also arises in the Calabi-Yau compactification ofM-theory, where the coupling of the second-order corrections turns out to be given in terms of theK¨ahler moduli of the six-dimensional compact manifold [7].The presence of the Gauss-Bonnet term introduces some exotic features not found in GeneralRelativity. One such feature is related to the problem of causality; this was treated in Ref. [8] inthe Hamiltonian formalism (see also Ref. [9] for an alternative treatment of the Cauchy problem).Because of the non-linearity of the theory, the canonical momenta are not linear in the extrinsiccurvature; and there exist quite generically points in the phase space where the Hamiltonian turnsout to be multiple-valued. In such a situation, there is a breakdown in the deterministic evolution ofthe metric from the initial data. This can also be seen explicitly using the junction conditions [10, 11].In fact, it can be shown that there exist vacuum solutions where the extrinsic curvature can jumpspontaneously at some spacelike hypersurface in a way that is not predicted by the initial data . Thisbreakdown in predictability is induced by the presence of terms in the junction conditions which,unlike the Israel conditions valid for Einstein’s theory, contain non-linear contributions coming fromthe Gauss-Bonnet term.On the other hand, the timelike version of such a jump in the extrinsic curvature is also of greatinterest. This is realized by the existence of a kind of gravitational solitons in the theory, whichresemble a kink solution. These solitons correspond to spacetimes that contain timelike hypersurfaceswhere the metric is C continuous but where the extrinsic curvature jumps. Although the Riemanncurvature tensor contains delta-function singularities on the hypersurface, these spacetimes can stillbe vacuum solutions because of a nontrivial cancelation coming from additional terms in the junctionconditions. Some explicit examples have appeared in the literature [12], and a spherically symmetricrealization of such solutions were studied in detail in Ref. [13] for the case of pure Gauss-Bonnetgravitational theory. Here, the systematical analysis made in Ref. [13] will be extended to themore phenomenologically important case where Einstein-Hilbert term and cosmological constantare included in the gravitational action. We will show that vacuum shell solutions are indeed foundin Einstein-Gauss-Bonnet theory described by the action (1).So then we will consider the junction conditions for spherical thin shells in Einstein-Gauss-Bonnet theory in the case that the induced stress tensor on the shell vanishes. Then, we will showthat geometries associated with two different spherically symmetric spaces can be joined without Capital Roman letters A , B etc. have been employed for five-dimensional tensor indices. R ABCD is the five-dimensional Riemann tensor. The junction condition in vacuum gives precisely that the jump in the canonical momenta is zero. The existenceof solutions with non-zero jump in the extrinsic curvature at a spacelike shell is therefore equivalent to the problemof a multiple-valued Hamiltonian. C class metrics and the topology of the solutions. We also discuss there theuniqueness and staticity of the spherically symmetric solutions, concerning the global validity of theBirkhoff-type theorems in Lovelock gravity. Section 8 contains the conclusions.With respect to the style of presentation, we have chosen to organize our results in a series ofremarks, propositions and theorems in order to highlight key facts, but descriptions such as ‘theorem’should not be taken in the most strict mathematical sense. First, we will present some introductory material and notation and conventions. The sphericallysymmetric solutions of Einstein-Gauss-Bonnet gravitational theory will be reviewed. Then we willdiscuss the junction conditions in this theory.Then we will show how these junction conditions permit to join two spherically symmetric spaceswithout resorting to the introduction of matter source.
Let us consider the Einstein-Gauss-Bonnet theory. The field equations associated with the action(1) coupled to some matter action take the form G AB + Λ δ AB + αH AB = κ T AB , (2)where T AB is the stress tensor, G AB ≡ − δ ACDBEF R EFAB = R AB − δ AB R is the Einstein tensor and H AB ≡ − δ AC ...C BD ...D R D D C C R D D C C , and where the antisymmetrized Kronecker delta is defined as δ A ...A p B ...B p ≡ p ! δ A [ B · · · δ A p B p ] .We are mainly interested in the static spherically symmetric solution (without matter) to Einstein-Gauss-Bonnet theory in five dimensions. In this case, of space-times fibered over (constant radius)4-spheres, the solutions correspond to the analogues of the Schwarzschild geometry, and its form wasfound by D. Boulware and S. Deser in Ref. [14]. More generally, the solutions that correspond tofiber bundles over 3-surfaces of constant negative (or vanishing) curvature were subsequently studiedin Ref. [16] (and also Ref. [17] in a special class of Lovelock theories in arbitrary dimension [18]).Let us discuss these solutions here. First, let us write the ansatz for the metric as follows ds = − f ( r ) dt + dr f ( r ) + r d Ω k , (3)where d Ω k is the metric of the constant curvature three-manifold (of normalized curvature k = +1, − T = 0 (the other field equations are equivalent to it) one obtains f ′ (cid:8) r + 4 α ( k − f ) (cid:9) = − r Λ3 + 2 r ( k − f ) . (4)This is integrated for ( k − f ) to give f ( r ) = k + r α ξ r α M αr ! (5)where ξ = 1. The case ξ = +1 corresponds to the “exotic branch” of the Boulware-Deser metricswhich for Λ = 0 and M = 0 gives a “microscopic” anti-de Sitter or de Sitter metric, with f ( r ) =1 + r / α . It is usually argued that this exotic branch turns out to be an unstable vacuum of thetheory, containing ghost excitations [14, 2]. Unlike the case ξ = −
1, this branch does not have a welldefined α → M here is a constant of integration,and is also associated with the mass of the solution. In fact, when there is an asymptotic region atthe infinity of the coordinate r , i.e. 1 + α Λ3 ≥
0, the total energy w.r.t. each constant curvaturebackground is calculated to be mass = M π κ , (6)so that, in general, we will call M the mass parameter or simply the mass of the metric .The general features of the black holes (3)-(5), such as horizons structure, singularities, etc, werestudied systematically in Ref. [19]; for further details see the Appendix. Unlike General Relativity,the Einstein-Gauss-Bonnet theory admits massive solutions with no horizon but with a naked sin-gularity at the origin. From (5) we see that this always happens for the exotic branch ξ = +1, andmight also happen for the branch ξ = −
1, provided
M < α . A related feature occurs for electricallycharged solutions [20, 21]. Among other interesting properties, it can be seen that charged blackholes in Einstein-Gauss-Bonnet theory have a single horizon if the mass reaches a certain criticalvalue. Another substantial difference between the Schwarzschild solution and the Boulware-Desersolution concerns thermodynamics. Unlike black holes in General Relativity, the Einstein-Gauss-Bonnet black holes turn out to be eternal. The thermal evaporation process leads to eternal remnantsdue to a change of the sign in the specific heat for sufficiently small black holes. This and the otherunusual phenomena discussed above are ultimately due to the ultraviolet corrections introduced bythe Gauss-Bonnet term.The discussion about a spherically symmetric solution of a given theory of gravity immediatelyraises the obvious question about its uniqueness. Regarding this, there is a subtlety that deservesto be pointed out. The uniqueness of the Boulware-Deser solution, discussed previously in Refs.[22, 23] (see [24] for a uniqueness result in axi-dilaton gravity with Gauss-Bonnet term), is onlyvalid under certain assumptions. This was formalized in a theorem proven by R. Zegers [25], andwhich also holds for generic Lovelock theory in any dimension. Let us state the result as applies forEinstein-Gauss-Bonnet theory in five dimensions:
Theorem 1 (Ref. [25]) . Any solution with spherical (or planar or hyperbolic) symmetry in thesecond-order Einstein-Gauss-Bonnet theory of gravity has to be locally static and given by theBoulware-Deser solution provided two key conditions are satisfied: i) The coefficients of the Lovelockexpansion are generic enough, which means that the exceptional combination α Λ = − / is excluded;ii) the solution is C smooth. It should be kept in mind that the masses M in each branch ξ , by being the total energy w.r.t. the M = 0spacetime in that branch , can not be directly compared. i) is certainly a necessary assumption. Indeed, the non-uniqueness in the case of α Λ = − /
4, corresponding to the (A)dS-invariant Chern-Simons theory, is a well-known result andwas explicitly shown in Refs. [23, 26]. In this paper, we will see that condition ii) is also necessary.In fact, the vacuum shell solutions we will present are C spacetimes which are only piecewise of theBoulware-Deser form.In order to analyze C solutions, we will need to use the junction conditions in the theory, whichwill now be discussed. The next ingredient in our discussion is the junction conditions in Einstein-Gauss-Bonnet theory.These are the analogues of the Israel conditions [28] in General Relativity, and were worked out inRefs. [10, 11]. In particular, the junction conditions will be employed to join two different sphericallysymmetric spaces.We will organize the discussion as follows: First, we will discuss the timelike junction condition;namely, the case where the surgery is performed on a timelike hypersurface, which we shall call atimelike shell. After studying this we will briefly discuss its spacelike analogue.
Let Σ be a timelike hypersurface separating two bulk regions of spacetime, region V L and region V R (“left” and “right”). Conveniently, we introduce the coordinates ( t L , r L ) and ( t R , r R ) and themetrics ds L = − f L dt L + dr L f L + r L d Ω , (7) ds R = − f R dt R + dr R f R + r R d Ω , (8)in the respective regions. We shall be interested in the case where the bulk regions are empty ofmatter so f L ( r L ) and f R ( r R ) are the Boulware-Deser metric functions given by equation (5). Ingeneral, the mass parameter M R will be different from M L . Moreover, we will also consider thepossibility of having ξ R different from ξ L , so that the two different branches of the Boulware-Desersolution can be considered to the two spaces to be joined.It is convenient to parameterize the shell’s motion in the r − t plane using the proper time τ onΣ. In region V L we have r L = a ( τ ), t L = T L ( τ ) and in region V R we have r R = a ( τ ), t R = T R ( τ ).The induced metric on Σ induced from region V L is the same as that induced from region V R , andis given by d ˆ s = − dτ + a ( τ ) d Ω . (9)This guarantees the existence of a coordinate system where the metric is continuous ( C ).Here, d Ω will be chosen to be the line element of a 3-manifold with (intrinsic) curvature k = ± , k = 0and k = −
1. The hypersurface Σ is the shell’s world-volume, i.e. the four-dimensional history ofthe shell in spacetime. The intrinsic geometry is well defined on Σ and given by (9). However, sincethe metric is only C and not necessarily differentiable, the geometry of the embedding of Σ into V L is independent of the embedding of Σ into V R . The geometric information about the embedding isquantified by the extrinsic curvature as well as the orientation of Σ with respect to each bulk region.To be precise, let us consider the following conventions for a timelike shell outside of any eventhorizon: • The hypersurface Σ has a single unit normal vector n which points from left to right. • The orientation factor η of each bulk region is defined as follows: η = +1 if the radial coordinate r points from left to right, while η = − r points from right to left.This is depicted in Fig. 1. Notice that the wormhole depicted on the left of that figure is notthe only possibility for η L η R <
0. While this geometry roughly speaking corresponds to joining6igure 1: The figure on the left corresponds to a wormhole-like solution, defining the orientation η L η R <
0. The throat connects two different asymptotically (Anti) de-Sitter spaces. The figureon the right corresponds to a vacuum shell with standard orientation ( η L η R > η L η R < Definition 2.
The orientation defined by η L η R > will be called the standard orientation. A shellwith standard orientation will be called a standard shell. The orientation defined by η L η R < willbe called the wormhole orientation. [ This makes actual sense when η R = +1 . When η R = − thelatter case represents a closed universe, containing singularities. ]The components of the normal vector with respect to the basis e A := ( ∂ t L , ∂ r L , e θ , e χ , e ϕ ) of V L and the basis e A ′ := ( ∂ t R , ∂ r R , e θ , e χ , e ϕ ) of V R are respectively given by n A = η L (cid:18) ˙ af L , p f L + ˙ a , , , (cid:19) , n A ′ = η R (cid:18) ˙ af R , p f R + ˙ a , , , (cid:19) . where dot denotes differentiation with respect to τ . This formula for the normal vector extends thedefinition of the orientation factors to the situation where the shell is inside the horizon when r isa timelike coordinate.We can introduce the basis e a = ( ∂ τ , e θ , e χ , e ϕ ) intrinsic to Σ. The extrinsic curvature is thendefined as K ab := e a · ∇ e b n = − n · ∇ e b e a . In terms of a coordinate basis we have e Aa = ∂X A ∂ζ a andthe extrinsic curvature takes the explicit form K ab = − n A (cid:18) ∂ X A ∂ζ a ∂ζ b + Γ ABC ∂X B ∂ζ a ∂X C ∂ζ b (cid:19) , and in our case the components read K ττ = η ¨ a + f ′ p ˙ a + f , K θθ = K χχ = K ϕϕ = ηa p ˙ a + f . (10)We denote the extrinsic curvature with respect to the embedding into V L and V R by ( K L ) ab and( K R ) ab respectively. At a singular shell ( K L ) ab = ( K R ) ab , i.e. the extrinsic curvature jumps from7ne side to the other. This is a covariant way of expressing the fact that the metric is not C (i.e. there does not exist any coordinate system where the metric is C ). In General Relativitythis amounts to saying that (non-null) vacuum shells do not exist since Israel conditions cannotbe satisfied without the introduction of a induced stress tensor on the spherical shell. Things aredifferent in the case of the gravity theory defined by action (1). This is because the Gauss-Bonnetterm induces additional terms in the junction conditions, which supplements the Israel equation.In section 2.3 we will show how both contributions can be combined to yield vacuum sphericallysymmetric thin shells. First we briefly discuss spacelike shells. Solutions of a different sort are those constructed by joining two spaces through a spacelike juncture.Let us suppose now that Σ is now a spacelike hypersurface. The motion of the shell in the r − t plane is parameterized by ( t, r ) = ( T ( τ ) , a ( τ )), where it is necessary to remember that τ is now aspacelike coordinate on Σ. The induced metric on Σ is then given by d ˆ s = + dτ + a ( τ ) d Ω . (11)The components of the normal vector with respect to the basis e A := ( ∂ t L , ∂ r L , e θ , e χ , e ϕ ) of V L andthe basis e A ′ := ( ∂ t R , ∂ r R , e θ , e χ , e ϕ ) of V R are respectively: n A = η L (cid:18) ˙ af L , p ˙ a − f L , , , (cid:19) , n A ′ = η R (cid:18) ˙ af R , p ˙ a − f R , , , (cid:19) . This defines the orientation factors in the case of a spacelike shell. The components of the extrinsiccurvature are: K ττ = η ¨ a − f ′ p ˙ a − f , K θθ = K φφ = K χχ = ηa p ˙ a − f . (12) The Einstein-Gauss-Bonnet field equations are well-defined distributionally at Σ due to the propertyof quasi-linearity in second derivatives (see e.g. Refs [27, 13] ). Thus, one can define a distributionalstress tensor T AB = δ (Σ) e bA e bB S ab , where S ab is the intrinsic stress tensor induced on the shell and δ (Σ) denotes a Dirac delta function with support on the shell world-volume Σ.Integrating the field equations from left to right in an infinitesimally thin region across Σ oneobtains the junction condition. This relates the discontinuous change of spacetime geometry acrossΣ with the stress tensor S ba . For the Einstein-Gauss-Bonnet theory the general formulas can befound in the Refs. [10, 11, 13]. ( Q R ) ba − ( Q L ) ba = − κ S ba , (13)where the symmetric tensor Q ab is given by Q ab = ∓ δ acbd K dc + α δ acdebfgh (cid:16) ∓ K fc R ghde + 23 K fc K gd K he (cid:17) . (14)Above, the sign ∓ depends on the signature of the junction hypersurface: it is minus for the timelikecase and plus for the spacelike case. In this expression, lower case Roman letters from the beginningof the alphabet a , b etc. represent four-dimensional tensor indices on the tangent space of theworld-volume of the shell. The symbol K ab refers to the extrinsic curvature, while the symbol R abcd appearing here corresponds to the four-dimensional intrinsic curvature (see the appendix for details).Once applied to the spherically symmetric (or k = −
1, 0) case the tensor Q ba turns out to bediagonal with components Q ττ = − σ a − (cid:18) η a p ˙ a + f + 4 α η p ˙ a + f (cid:0) k + 23 σ ˙ a − f (cid:1)(cid:19) , (15) Q θθ = Q χχ = Q ϕϕ . (16)8he precise form of Q θθ will not be needed but is given in the appendix for completeness. The aboveformula was written in a way that is valid for both timelike and spacelike shells, where we havedefined σ = +1 (timelike shell) , σ = − . Also, let us be reminded of the fact that η L and η R (with η = 1) are the orientation factors in eachregion, which are independent one from each other. Above, the subscripts L , R signify the quantityevaluated on Σ induced by regions V L and V R respectively (e.g. Q L is a function of η L and f L ( a )) .One may verify that the following equation is satisfied ddτ (cid:0) a Q ττ (cid:1) = ˙ a a Q θθ , (17)which expresses the conservation of S ba . The reason why one obtains exact conservation, i.e. noenergy flow to the bulk, is that the normal-tangential components of the energy tensor in the bulkis the same in both sides of the junction hypersurface [10, 13].The main point here is that, unlike the Israel conditions in Einstein gravity, non-trivial solutionsto (C) are possible even when S ba = 0. That is, the extrinsic curvature can be discontinuous acrossΣ with no matter on the shell to serve as a source. The discontinuity is then self-supported gravi-tationally and this is due to non-trivial cancelations between the terms of the junction conditions.Similar configurations are impossible in Einstein gravity. From now on we consider the vacuum case S ba = 0 . (18)In the next section we will treat the static shell in detail. An exhaustive study of the space ofsolutions describing both static and dynamical shells is left until sections 5 and 6. Let us now firstbriefly introduce the basic features of the general solution for a dynamical shell.Equation (17) tells us that when ˙ a = 0, the components of the junction condition are notindependent; namely ( Q R ) ττ − ( Q L ) ττ = 0 ⇒ ( Q R ) θθ − ( Q L ) θθ = 0 . Therefore, for time-dependent solutions it suffices to impose only the first condition. This can befactorized as follows, (cid:16) η R p ˙ a + σf R − η L p ˙ a + σf L (cid:17) ×× (cid:26) a + 4 α ( k + σ ˙ a ) − σ α (cid:16) f R + f L + 2 σ ˙ a + η R η L p σf R + ˙ a p σf L + ˙ a (cid:17)(cid:27) = 0 . (19)Equation (19) contains all the information about the spherically symmetric junctions in emptyspace, which we generically call vacuum shells. Certainly, there exist several cases to be explored.First of all, there are the parameters k, M and ξ , which characterize each of the two Boulware-Deser metrics to be joined. On the other hand, there are two possible orientations for each one ofthe spaces, and this is given by the sign of the respective η . The solutions to (19) include bothwormhole-like and bubble-like geometries, depending on whether the orientation is η L η R < η L η R > σ , what tells us whether the signatureof the junction hypersurface is timelike ( σ = +1) or spacelike ( σ = − a = σ (cid:16) f R + f L − k + a / α ) (cid:17) − f R f L (cid:16) f R + f L − k + a / α ) (cid:17) =: − V ( a ) , (20) From now on we shall be concerned with f ( a ), i.e. the metric function evaluated at the shell. In an abuse ofnotation we shall just write f instead of f ( a ). V ( a ). Nevertheless, it isworth pointing out that, unlike the equation for a single particle, here we find that the energy h isunavoidably fixed to zero instead of arising as a constant of motion. An important difference arisesin the case where there is a minimum of V ( a ) precisely at V = 0. The constraint h = 0, providedthe fact that the minimum of V ( a ) is precisely at zero energy, would lead to the conclusion that theshell can not move but it would be stacked at the bottom of the potential. Actually, this is the caseif no external system acts as a perturbation. One such perturbation can be thought of as being anincoming particle which, after perturbing the shell, scatters back to infinity spending an energy δh through the process. This would provide energy for the vacuum shell to move. One can also thinkabout a slight change in the parameters of the solution yielding a shifting V ( a ) → V ( a ) − δh , see[31].Now, let us notice that since we have squared the junction condition, we must substitute (20)back into (19) to check the consistency. When doing so, the solutions of equation (20) are solutionsof the junction condition if and only if the following restrictions are obeyed − η R η L (2 f R + f L − k + a / α )) (2 f L + f R − k + a / α )) ≥ f R + f L − k + a / α )) > f R + f L − k + a / α )) < . (23)Furthermore, we also have an inequality which is not an extra condition but rather follows as aconsequence of equation (20). The fact that ˙ a is positive in (20) implies that (cid:16) f R + f L − k + a / α ) (cid:17) − f R f L ≥ . (24)for both timelike and spacelike. This inequality provides further information about the space ofsolutions of (20). Proposition 3.
For a dynamical vacuum shell with a timelike world-volume Σ , the scale factor ofthe metric (9) on Σ is governed by (20), under the inequalities (21) and (22).On the other hand, for a dynamical vacuum shell with a spacelike world-volume Σ , the scale factorof the metric (11) on Σ is governed by (20), under the inequalities (21) and (23). Now, let us begin by studying the inequalities to give idea of what kinds of solutions exist. Withthis in mind, let us translate the restrictive inequalities (21-23) into simpler terms. The metricfunction evaluated on the hypersurface is f L ( a ) = k + a α (cid:16) ξ L Y L ( a ) (cid:17) , Y L ( a ) ≡ r α Λ3 + 16 αM L a , (25)and similarly for f R . Recall that ξ L and ξ R are independent of each other, with ξ = +1 being theexotic branch of the Boulware-Deser solution. It is convenient to write the inequalities in terms ofthe square roots Y L ( a ) and Y R ( a ); namely − η R η L (2 ξ R Y R + ξ L Y L ) (2 ξ L Y L + ξ R Y R ) ≥ α ( ξ R Y R + ξ L Y L ) > α ( ξ R Y R + ξ L Y L ) < . (28)These inequalities contain relevant information about the global structure of the solutions. Let ussummarize this information in the following table Notice that the effective potential for the spacelike shell is simply minus the potential for the timelike shell. imelike Product of Product of Inequalities shells orientation branch signs imposed on( σ = +1) factors ( η L η R ) ( ξ L ξ R ) solutionsStandard +1 +1 No solutionorientation +1 -1 Y L ≤ Y R ( a ) ≤ Y L ( a ) ; ξ R ( M R − M L ) > αξ R > Y L , Y R > Y R ≥ Y L or Y R ≤ Y L ; ξ R ( M R − M L ) > Spacelike
Product of Product of Inequalities shells orientation branch signs imposed on( σ = −
1) factors ( η L η R ) ( ξ L ξ R ) solutionsStandard +1 +1 No solutionorientation +1 -1 Y L ≤ Y R ( a ) ≤ Y L ( a ) ; ξ R ( M R − M L ) > αξ R < Y L , Y R > Y R ≥ Y L or Y R ≤ Y L ; ξ R ( M R − M L ) < Remark 4.
Vacuum shells with the standard orientation always involve the gluing of a plus branch ( ξ = +1) metric with a minus branch ( ξ = − metric. Now the plus branch has a different effective cosmological constant to the minus branch. In thissense, standard shells are a kind of false vacuum bubble. This is discussed further in section 4.1.
Remark 5.
Vacuum shells which involve the gluing of two minus branch ( ξ = − metrics existonly when the Gauss-Bonnet coupling constant α satisfies α < . They always have the wormholeorientation. In the analysis above it has been explicitly assumed that ˙ a = 0. Nevertheless, the case ˙ a = 0 isalso of considerable interest. This describes static shells in the timelike case, and also an analogoussituation for the spacelike case which we call instantaneous shells. In the next section, the case ofconstant a shells is considered in detail. It can be checked that, as expected, all the informationabout the constant a solutions can be obtained from the dynamical case by imposing both V ( a ) = 0and V ′ ( a ) = 0. Thus, proposition 3 gives the general solution of all the vacuum shells, includingthe static ones.Closing the general discussion of the dynamical vacuum we note the following. The potential V ( a ) in (20) and the restrictive inequalities (21) and (22), (23) are symmetric in the exchange ξ L , M L ↔ ξ R , M R . (29)That is, the same kinds of motion are possible for the two situations obtained if we swap the valuesof the parameters ξ, M in V L and V R . In the constant a case, governed by V ( a ) = 0 = V ′ ( a ), thesymmetry means that the value of a is left unchanged under the swapping.11 Static vacuum shells
Now, let us discuss the solutions at constant a . That is, the static and instantaneous solutions,depending on whether the juncture corresponds to the timelike or spacelike case respectively.The bulk metric in each of the two region is assumed to be of the Boulware-Deser form (5)with ( k = ± ,
0) and considering a = a fixed. Although the main focus will be on the sphericallysymmetric case k = +1, the analysis can be straightforwardly extended to the cases k = − k = 0. Then, there are two possibilities to be distinguished; namely, • Static shell: For the timelike case the shell is located at fixed radius r L = r R = a . Theproper time on the shell’s world-volume is τ = t L p f L ( a ) = t R p f R ( a ) so that the inducedmetric on Σ turns out to be d ˆ s = − dτ + a d Ω . Then, the extrinsic curvature components are K ττ = η f ′ √ f , K θθ = K χχ = K ϕϕ = η √ fa and the intrinsic curvature components are R θϕθϕ = k/a ,etc. • Instantaneous shell: In the spacelike case there is an exotic kind of shell, which exists when f isnegative. The metric function is negative inside of an event horizon or outside of a cosmologicalhorizon, where r actually plays the role of a timelike coordinate. Matching two metrics at time r ± = a therefore describes an instantaneous transition from one smooth metric to another.We can introduce τ = t L p − f L ( a ) = t R p − f R ( a ) which is a spacelike intrinsic coordinate onthe shell, so that the induced metric on Σ is ds = + dτ + a d Ω . The extrinsic curvaturecomponents are K ττ = − η f ′ √− f , K θθ = K χχ = K ϕϕ = η √− fa .It is worth noticing that both the static and instantaneous shells can be analyzed together,provided the presence of σ in the equations. Recall that the sign of σ carries the information aboutthe signature of the junction hypersurface. Then, by considering the quantities introduced above,and by substituting this in the junction conditions with S ab = 0, we get S ττ = 0 ⇒ (cid:0) η R p f R − η L p f L (cid:1)(cid:16) a + 4 α (cid:8) k − f R − f L − ση L η R p f L f R (cid:9)(cid:17) = 0 . (30) S θθ = 0 ⇒ (cid:16) η R √ f R − η L √ f L (cid:17)(cid:16) k − Λ a − ση L η R p f L f R (cid:17) = 0 , (31)where σ = +1 is the static shell and σ = − θ − θ component of the junction condition and we have used it to eliminate thederivative of f from the formula. This is why Λ appears explicitly in equation (31).In both equations (30) and (31), the first factor vanishes if and only if the metric is smooth.Again, rejecting this as the trivial solution, we demand that the second factor vanishes in bothequations. So, we have Proposition 6.
A static vacuum shell is described by f L + f R = 2 k + 3 a α + Λ a , (32) η L η R p f L f R = k − Λ a , (33) under the condition f L , f R > . On the other hand, an instantaneous vacuum shell is described by f L + f R = 2 k + 3 a α + Λ a , (34) − η L η R p f L f R = k − Λ a , (35) under the condition f L , f R < . We have included for completeness the instantaneous shells. Now, let us consider some examplesof the static case with more attention. As mentioned, a more complete analysis of the space ofsolutions will be given in sections 5 and 6. 12 .1 The moduli space of solutions
Now, to continue the study of the different solutions we find it convenient to introduce some notation.For the rest of this section it is convenient to define the dimensionless parameters x ≡ α Λ3 , y ≡ Λ3 a , ¯ M ≡ Mα . (36)By x and y we measure the Gauss-Bonnet coupling and the vacuum shell radius respectively in unitsof Λ. The parameter y is useful for our purposes but it is meaningful only when Λ = 0. In terms ofthese parameters, the Boulware-Deser solution evaluated at r = a has the form f L,R ( a ) ≡ yx k + ξ L,R s x + x y ¯ M L,R ! . (37)The general solution will be derived in the following way: We will solve the junction conditionsfor ¯ M L and ¯ M R in terms of ( x, y ). The range of admissible values of ( x, y ) turns out to be restrictedby inequalities coming from demanding the metric to be real-valued. So there is a continuous spaceof solutions. Definition 7.
The range of values of ( x, y ) for which solutions exist will be called the moduli space. The parameters x and y are coordinates of this moduli space. The complete description of themoduli space will be given in more appropriate parameters introduced in section 6. For the moment,let us consider x , y and ¯ M .Since the moduli space is two dimensional, it can be plotted. So by obtaining a formula for themasses and by plotting the moduli space, we obtain all the solutions. Let us now do this explicitlyfor the case of non-vanishing cosmological constant. Λ = 0 Consider static spherically symmetric shells with Λ = 0. For definiteness, let us focus on the case oftimelike shells with k = 1. From Proposition 6 we have the following pair of equations f L + f R = yx (3 + x ) + 2 , (38) p f L f R = η L η R (1 − y ) , (39)where f L , f R >
0. We can see immediately from (39) that solutions with the wormhole orientation,i.e. η L η R = −
1, only exist for y ≡ Λ a / > Remark 8.
Static vacuum shell wormholes exist only when Λ > . Solving the equations above we see that f L and f R obey the same quadratic equation where one f has the + root of the solution and the other has the − root. So we define a solution f (+) whichcorresponds to the + root of the solution and an f ( − ) which corresponds to the − root. So there aretwo solutions to the problem: f L = f ( − ) , f R = f (+) or , f L = f (+) , f R = f ( − ) . (40)Substituting the explicit expression (37) for f L,R ( a ) we have: In the first case of (40), M L = M ( − ) , ξ L = ξ ( − ) and M R = M (+) , ξ R = ξ (+) , and in the second case + ↔ − , for constants ξ ( ± ) and M ( ± ) satisfying 1 + x − √ r x (1 + x ) (cid:16) y + 3 x − (cid:17) = 2 ξ ( − ) s x + x ¯ M ( − ) y , (41)1 + x + √ r x (1 + x ) (cid:16) y + 3 x − (cid:17) = 2 ξ (+) s x + x ¯ M (+) y . (42)13or a solution to exist, the square root in the l.h.s. of the above equations must be real, so thatwe demand x (1 + x ) (cid:16) y + 3 x − (cid:17) ≥ . (43)Since we have squared the equations we must substitute back to check the consistency. So we getthe following inequalities: yx (3 + x ) + 2 > y < , y > . (45)The above inequalities are plotted in figures 11 and 12. Also we find the regions of the moduli spacecorresponding to the allowed branch signs ( ξ ( − ) , ξ (+) ). ξ ( − ) ξ (+) Inequality+1 1 + x > T x (cid:16) y + x − (cid:17) < − x < S x (cid:16) y + x − (cid:17) > x > S x (cid:16) y + x − (cid:17) < − x < T x (cid:16) y + x − (cid:17) > − , +). The regions (+ , +), ( − , +) and ( − , − ) for the wormholes areshown in figure 13. Remark 9.
Provided Λ > and assuming the existence of two asymptotic regions we find α > .Consequently, at least one of the two spherically symmetric spaces connected through the throat turnsout to be asymptotically Anti-de Sitter. The next step is computing the masses. We can solve (41) and (42) to give the parameter M ineach region, namely¯ M ( − ) = y (1 + x )2 x (cid:26) x + 3 y − xy − y √ r x (1 + x ) (cid:16) y + 3 x − (cid:17) (cid:27) , (46)¯ M (+) = y (1 + x )2 x (cid:26) x + 3 y − xy + y √ r x (1 + x ) (cid:16) y + 3 x − (cid:17) (cid:27) . (47)As mentioned above, relations (40), the left-metric can be either a metric with parameters ( ξ ( − ) , M ( − ) )or a metric with ( ξ (+) , M (+) ), and the other way around for the right-metric. For wormholes thetwo solutions (40) correspond to the same spacetime looked at from the opposite way around. Inthe case of standard shells, they correspond to swapping the mass and branch sign of the interiorwith those of the exterior region.The metrics with parameters ( ξ ( − ) , M ( − ) ) and ( ξ (+) , M (+) ) as determined by the solutions wefound above have different properties. We will call these metrics minus- and plus-metrics respectively.Also we note the following useful expression: we can eliminate y to get an implicit equation forthe masses and x . The solution lies on sections of the curves1 + 94 x ( x + 1)3 − x M (+) + ¯ M ( − ) − p s − x ) x ( x + 1) ( ¯ M (+) + ¯ M ( − ) ) ! = ( − q s − x )(1 + x ) ( ¯ M (+) − ¯ M ( − ) ) ( ¯ M (+) + ¯ M ( − ) ) (48)where the signs ( − p and ( − q are to be determined by consistency. Note that the timelike condition (44), when combined with the reality condition (43) can be equivalently stated xy (1 + x ) > . This is useful for plotting the graphs. .3 Instantaneous shells Before concluding this section, let us briefly comment on spacelike junction conditions with ˙ a = 0.For instance, consider the case Λ = 0. From Proposition 6 we have the following pair of equations: f L + f R = yx (3 + x ) + 2 , (49) p f L f R = − η L η R (1 − y ) , (50)The solution is exactly the same as the above except that the inequalities (44) and (45) are reversed.That means yx (3 + x ) + 2 < y > , y < . (52)The inequality (43) and mass formulae are the same. The moduli space of these solutions is plottedin figure 14. They exist for α < Λ = 0
Now, we will consider the case of static spherically symmetric shells with Λ = 0. This is an interestingspecial case. The analysis simplifies considerably and, besides, there are some qualitative differencesbetween this and the case Λ = 0. In this case, the equations reduce to f L + f R = 2 + 3 a α , (53) η L η R p f L f R = 1 , (54)We see from the second equation that η L η R must be +1, i.e. static wormholes do not exist for Λ = 0.Then, the solution is either M L = M ( − ) , M R = M (+) or M L = M (+) , M R = M ( − ) where M ( ± ) α = 12 a α a α ± s a α + 9 (cid:18) a α (cid:19) . (55)The consistency of the solution requires α > , ( ξ ( − ) , ξ (+) ) = ( − , +1) , (56)so that M ( − ) and M (+) correspond to minus branch and exotic plus branch metrics respectively.There are solutions for all positive values of M ( − ) (the plus branch mass parameter is also positive butin that case the bulk spacetime asymptotically takes the form of a negative mass AdS-Schwarzschildsolution). When the throat radius is small compared to the scale set by the Gauss-Bonnet couplingconstant, a << α , the masses are also small compared to α , namely M ( − ) /α ∼ M (+) /α ∼ a / α .On the other hand, for large radius a >> α , the masses are large, M ( − ) /α ∼ a / α , M (+) /α ∼ a / α . Figure 10 shows a plot of the masses as a function of α and also an implicit plot of M (+) as a function of M ( − ) . In the previous section we have shown the existence of static vacuum shells in the spherically sym-metric case and found some basic qualitative features, as well as a formula for the mass parametersin each region. A more exhaustive treatment of the static shells will be left for section 6. Beforegoing any further let us summarize the catalogue of vacuum solutions that arise through the geomet-ric surgery we described above. The first cases of interest are those corresponding to the standardorientation η L η R >
0. 15igure 2: A spherically symmetric spacetime with metric of the class C . A vacuum shell with thestandard orientation always connects two regions with different branch signs ξ (and generically withdifferent mass parameters M ). Each region has a different effective cosmological constant. The vacuum shells with the standard orientation are always ( ξ ( − ) , ξ (+) ) = ( − , +1) branch. So region V L has a different effective cosmological constant to region V R , as can be seen from the expansionof the metric for large r . For example, when the bare cosmological constant Λ = 0 we have on oneside of the shell the effective cosmological constant Λ (+) d = − / α and on the other Λ ( − ) d = 0. In theregion with Λ (+) d the graviton is expected to have ghost instability. In this sense the shell is like thefalse vacuum bubbles studied in Refs. [33], but for a false vacuum which is of purely gravitationalorigin.These kind of solutions might lead to curious implications. For instance, let us consider thefollowing construction: Suppose we have a “well behaved” minus branch ( ξ L = −
1) solution withpositive mass M L ; where by “well behaved” we mean a solution in which the singularity is hiddenbehind an event horizon and for which we get a suitable GR limit for small α . Now, let us cut outthe black hole at some radius r = a ( τ ) > r H and then replace it with the interior of a plus branch( ξ R = +1) solution, i.e. a naked singularity. By doing this we would be constructing a vacuumsolution whose geometry, from the point of view of an external observer, would coincide with thatof a black hole but, instead, would not possess a horizon. A particle in free fall would not find ahorizon but rather a naked singularity as soon as it passes through the C junction hypersurfacelocated at r = a > r H . The solutions with Λ = 0 and α > M ( − ) ; so that we can indeed cut out the eventhorizon and replace it with a naked singularity!Also, for Λ = 0 “false vacuum bubble” solutions gluing a positive mass Boulware-Deser branch Strictly speaking, this label of false vacuum bubble would be correct if the minus branch metric were lower totalenergy with respect to the plus branch metric and if the classical transition were impossible.
So far, we have discussed different kinds of geometries constructed by a cut and paste procedure oftwo spaces that were initially provided with the Boulware-Deser metric on them. The strategy wasto make use of the junction conditions holding in Einstein-Gauss-Bonnet theory and, in particular,we have shown that solutions with non-trivial topology, which have no analogues in Einstein grav-ity, do arise through this method. A remarkable example is the existence of vacuum wormhole-likegeometries , corresponding to the case η L η R <
0. These “wormholes” can be thought of as be-longing to two different classes: The first class describes actual wormholes, presenting two differentasymptotic regions which are connected through a throat located at radius r L = r R = a ; the radiusof the throat being larger than the radius where the event horizons (or naked singularities) wouldbe. The two asymptotic regions are r L → ∞ and r R → ∞ as measured by the radial coordinate inthe respective sides of the junction. This type of geometry is an example of a vacuum sphericallysymmetric wormhole solution in Lovelock theory and its existence is a remarkable fact on its own.On the other hand, a second class of wormhole-like geometry with no asymptotic regions also exists.This second class is obtained also by considering the orientation η L η R <
0, this time cutting awaythe exterior region of both geometries and gluing the two interior regions together. We shall discussthis later; first let us discuss the static wormhole solutions with two asymptotic regions (actualwormholes).
Let us begin by emphasizing that such static wormhole solutions only exist if at least one of thetwo bulk regions corresponds to ξ = +1. That is, at least one of the two Boulware-Deser metricshas to correspond to what we have called the exotic branch. This could have deep implications inwhat regards semiclassical stability [14]. It is also remarkable that for these static wormholes toexist it is necessary that Λ >
0. Furthermore, the existence of two asymptotic regions demands α > α < ξ = +1, then at least one of the regions connected through the throat possesses a negative effectivecosmological constant.Another interesting feature concerns the stability under radial perturbations. This is seen in Fig15. In particular, it can be shown that stable static wormholes only exist for the case ξ L = ξ R = +1;namely, the case where both Boulware-Deser metrics correspond to the exotic branch. Nevertheless,no stable wormholes exist for the case M L = M R , and thus, concisely, the static symmetric wormholesare unstable under perturbations that preserve the spherical symmetry.An interesting possibility is that of having wormhole solutions whose Boulware-Deser metricswould correspond to negative mass parameters. For instance, one can construct a static wormholewith one side being of the “good branch” ξ L = − M L <
0. In that case,from the point of view of a naive external observer, the vacuum solution would seem to correspondto a naked singularity. However, now we know that the inclusion of non-trivial junctures makesit possible to replace such a singularity by an exterior region on the other side of a non-smoothwormhole throat. This has a deep implication in what concerns the “cosmic censorship principle”since for the appropriate values of the coupling constants, and unlike what usually happens inpure gravitational theories, the spherically symmetric vacuum solutions presenting naked singularitycannot be unambiguously classified (and consequently systematically excluded) in terms of the massparameter.Another particular case that deserves to be mentioned as a special one is that of having a masslesssolution in one of the sides of the wormhole geometry. For instance, such a construction is achieved ifthe massless side corresponds to the exotic branch ξ = +1 and the massive side to the branch ξ = − Smooth wormhole solutions in Lovelock theory have been found previously with matter source in refs. [34] andwithout matter for special choice of coupling constants in refs. [35, 26]. η L η R <
0. This solution presents two disconnectedasymptotically de-Sitter regions.In these cases, the wormhole throat turns out to be a kind of puncture of the (A)dS spacetime, letus call it a “hole in the vacuum”. Since (A)dS is homogeneously isotropic, a spherically symmetricmatching can be done anywhere: remarkably, several of these “holes” could be located at differentplaces in the spacetime and each “hole” would not influence the others. We shall discuss this kindof geometry in more detail in section 7.2. The massless side may then correspond to a microscopicde-Sitter geometry and, presumably, its cosmological horizon, yielding thermal radiation, could beseen from the massive sides. This is an intriguing possibility that deserves to be further explored.
Now, let us comment on the second class of wormholes; namely those with no asymptotic regions.As mentioned, these geometries are constructed by gluing the interior of the throat of both regions,instead of the exterior. One can perform the matching by keeping the region that is inside the throatbut still outside the horizons. Consequently, one gets a geometry that resembles a “static closeduniverse” with horizons. This exotic geometry has no asymptotic regions at all, and, because of this,this second type of geometry does not represent what one would usually call a wormhole. Never-theless, we shall abuse the notation and call “wormhole” any timelike junction with the orientation η L η R < α and x ≡ α Λ / < −
1. In this range of the coupling constants the Boulware-Deser metric develops a branch singularity at fixed radius r c = Mα | x |− , where the curvature diverges.This branch singularity represents the maximum three-sphere radius: the metric becomes non-realfor r > r c . In addition there is a curvature singularity at r = 0. In this region of the space ofparameters we would say that the Boulware-Deser geometry is somehow pathological. However,if junction conditions are appropriately applied, then a well-behaved C vacuum geometry can beconstructed by simply taking a pair of such pathological spaces, cutting out the naked singularitiesand joining them together. To see that this is possible, it is sufficient to consider the symmetricalcase. It can be checked from equations (46) and (47) that two bulk regions with equal masses M L /α = M R /α = x )( x − can be matched at a throat radius a = | α | − x . Consulting figure 13 (thesesolutions are located on the upper bounding curve of the left part of the moduli space) we see thatwormhole solutions exist when the bulk regions have branch signs ( ξ L , ξ R ) = ( − , − r H , which separates r = 0 (a timelike naked singularity) from r = r c , which is a18 c) (b) PSfrag replacements aa (a) PSfrag replacements r = 0 r H r = r c Figure 4: (a) The causal diagram of the smooth spherically symmetric solution for α <
0, 1+ α Λ3 < r = a cutting out the r = 0 singularity. (c) Causaldiagram of the resulting spacetime (a C , spherically symmetric vacuum solution).spacelike singularity. The static shell is located at a < r H . So by cutting out the regions r < a andjoining with the wormhole orientation the naked singularities can be removed. The causal diagramof the original pathological spacetimes and the extended causal diagram of the C closed universe,which results from the matching, with horizons are shown in figure 4.2.2. In general, vacuum shells will be dynamical objects. We discuss the dynamics here and also discussthe issue of radial stability of the static solutions.
Let us briefly recapitulate upon the equation (20), which governs the dynamics of the shells. Wecan treat both the timelike and spacelike together since, as we noticed in section 2, the analysis iscompletely analogous. A dynamical vacuum shell is governed by a differential equation of the form˙ a + V ( a ) = 0 ; (57)see (20) above. It is useful to express V ( a ) in terms of the non-negative quantity Y = q α Λ3 + Mαa ,and the effective potential then reads V ( a ) = σ (cid:18) k + a α (cid:19) − σa α (cid:18) ξ R Y R + ξ L Y L ) + ( ξ R Y R − ξ L Y L ) ξ R Y R + ξ L Y L ) (cid:19) . (58)In addition to the differential equation, the solution must obey the inequalities (26)-(28). It isconvenient to rewrite them as follows − η R η L (cid:16) ξ R Y R + ξ L Y L ) − ( ξ R Y R − ξ L Y L ) (cid:17) ≥ , (59) σα ( ξ R Y R + ξ L Y L ) > . (60)Note that the effective potential (58), V ( a ) = σa α + σk + ∆ V ( a ), consists of a quadratic piece,a constant determined by the three-dimensional curvature k of the shell, and another piece which,by inequality (60) obeys ∆ V < a = − σ a α h − α Λ / ξ R Y R + ξ L Y L i . (61)Considering the sign of this acceleration and making use of inequality (60) we can make some generalobservations: 19 emark 10. For a timelike shell ( σ = +1 ):When α Λ3 ≥ and α < a vacuum shell always experiences a repulsive force away from r = 0 ;When α Λ3 ≤ and α > a vacuum shell always experiences an attractive force towards r = 0 . In the situations not covered by Remark 10 the potential may have an extremum. From (61) wededuce that there is an extremum at r = a e iff ξ R Y R ( a e ) + ξ L Y L ( a e ) = 1 + 4 α Λ3 . (62)Recalling inequality (60), we conclude that an extremum can exist only if σα (cid:18) α Λ3 (cid:19) > . (63)The extremum will be a minimum or maximum depending on the sign of the second derivativeof the potential evaluated there, V ′′ ( a e ) = σα (cid:18) α Λ / ξ R ξ L Y R ( a e ) Y L ( a e ) − (cid:19) . (64)There is a general result for vacuum shells separating different branch metrics. From (63) we seethat V ′′ ( a e ) in (64) must be negative for 1 + α Λ3 ≥ Proposition 11.
In the range α Λ3 ≥ for the product of Gauss-Bonnet coupling and cos-mological constant: Let Σ be a vacuum shell such that ξ L ξ R = − . Then the potential never has aminimum. If Σ is a timelike shell it will either be in an (unstable) static state, or, if it is moving,will either expand or collapse, it can not be bound. We have already remarked in section 2 that any shell with standard orientation must match twobulk metrics of opposite branch sign ( ξ L ξ R = − Corollary 12.
Let α Λ3 ≥ . A timelike shell with standard orientation is either in an (unstable)static motion, or, if it is moving, will either expand or collapse, it can not be bound. We have already seen in section 3 that static shells with standard orientation are always in a stateof unstable equilibrium in the (physical) regime 1 + α Λ3 ≥
0. The proposition above strengthensthis result to include dynamical shells. A dynamical shell with standard orientation can not beoscillatory. It must either disappear into a singularity or fly out towards spatial infinity.There is not such a strong result for shells with the wormhole orientation. Indeed in section3 we found stable static wormholes for 1 + α Λ3 ≥ ξ L = ξ R = − α Λ3 ≥ Proposition 13.
Let α Λ3 ≥ and let Σ be a timelike vacuum shell with wormhole orientation,and V L and V R be minus branch bulk metrics ( ξ L , ξ R ) = ( − , − . Then the shell always experiencesa repulsive force away from r = 0 . So in summary, we have found some general results for the range of parameters 1 + α Λ3 ≥ | α Λ | << In the case of the wormhole orientation, by using the inequality (59), the result can be extended to apply to therange 1 + α Λ3 > − . .2 Comment on the stability of static shells Dynamical equation (57) resembles the equation for a particle moving under the influence of aneffective potential (58). Nevertheless, as pointed out in section 2, this is not strictly the case due tothe presence of the vanishing energy constraint. This is important for the case when the extremumof the potential is at a = a with V ( a ) = 0, i.e. when static solutions exist. When V ′′ ( a ) < a → a + δa will cause the shell to accelerate away from the (unstable) equilibrium radius.When V ′′ ( a ) > a when V >
0. Wecan consider spherically symmetric solutions which are close-by in the space of the solutions, i.e.with slightly different parameters M L,R and w such that the value of the potential at a e is slightlynegative: let us say V ( a e ) = 0 − . This means that such a solution oscillates between two radii around a e at which the potential vanishes. This is certainly a stable solution though not static, a ‘boundedexcursion’ [31]. Now if we let a e coincide with the a of the original static solution, this meansthat for slightly different parameters than those for which a is a static solution, there exists anoscillating solution around a . Therefore a static solution a which is a minimum of the potentialgives information about when infinitesimal bounded excursions can happen. More generally, thedynamics of the perturbed shell can be thought of as corresponding to a perturbation of the aboveequation V ( a ) → V ( a ) − δh , provided energy δh from an external excitation. The stable regions ofthe moduli space of static solutions are plotted in figures 15 and 16. The graph will take an elegantform in terms of the change of variables to be introduced in section 6 (see fig 8).In the rest of this section we present some illustrative examples of dynamical vacuum shells, firstin symmetrical wormhole solutions and then in the context of Chern-Simons gravity. Now let us consider the case where the masses in each bulk region are the same, being M L = M R = M . The inequalities (59) and (60) are equivalent to: Remark 14. If Σ is a vacuum shell joining two bulk regions with the same mass M R = M L then:i) The bulk solutions must have the same branch sign ξ L = ξ R = sign ( σα ) ;ii) The shell must have wormhole orientation. So the spacetime is completely left-right mirror-symmetric. The equation of motion reads σ ˙ a + a α h − ξ Y ( a )2 i + k = 0 . (65)The general solution is rather complicated. Next we proceed to consider a simple case where bothmasses vanish.The case where M L,R = 0 is an interesting special case of the symmetric wormholes, which existsfor 1 + α Λ3 >
0. The equation of motion reduces to σ ˙ a + a α − ξ q α Λ3 + k = 0 . (66) Remark 15.
Consider a timelike shell ( σ = +1) , that is sign ( α ) = ξ .Bounded motions: ξ = +1 , α Λ3 < , k = +1 ; ξ = +1 , α Λ3 = 3 , k = 0 .Unbounded motions: ξ = +1 , α Λ3 = 3 , k = − ; ξ = +1 , α Λ3 > , any k ; ξ = − , α Λ3 > − ,any k .The same bounded or unbounded configurations exist in the spacelike case provided one replaces k with − k , for the opposite sign of α . The hyperbolic shell, k = −
1, admits a stationary vacuum wormhole solution: for sign( α ) = ξ =+1 and 4 α Λ / a = 0 and ˙ a = 1.21hen Λ = 0 and ξ = − t − r = 8 α/ α > α < When 1 + α Λ3 = 0 some very special things happen. For this choice of coupling constants theEinstein-Gauss-Bonnet theory (in first order formalism) is equivalent to a Chern-Simons theory forthe deSitter ( α <
0) or Anti de Sitter ( α >
0) group . In this case the metric function takes thevery simple form f ( r ) = 1 + r α − µ , (67)where µ is a constant and the mass is proportional to µ − µ > µ < a ξY / α = − µ for eachbulk region is a constant and therefore the non-harmonic part of the potential ∆ V is a constant.The equation of motion takes the form˙ a + σ α a = E , E = − σ (cid:18) k + 3( µ R + µ L ) + ( µ R − µ L ) µ R + µ L ) (cid:19) . (68)The potential is like that of a harmonic oscillator potential (or an upside-down harmonic potentialif σα is negative), although it should be remembered that the origin r = 0 of the bulk spacetimesis singular so the shell can not really oscillate. The solution is constrained according to the twoinequalities (59) and (60), which now read − η L η R (9( µ R + µ L ) − ( µ R − µ L ) ) ≥ , (69) − σ ( µ R + µ L ) > . (70)The last inequality tells that E > − σk . These inequalities are generally consistent with E > µ R or µ L must be negative. So it is not possible to match two black hole spacetimes.From inequality (69) we see that shells with the standard orientation must obey µ R µ L < Remark 16.
For the Chern-Simons combination α Λ3 = 0 , timelike vacuum shells always rep-resent either:i) a matching between a bulk region of a black hole spacetime with bulk region of a naked singularityspacetime; orii) a matching, with wormhole orientation, between two bulk regions of naked singularity spacetimes. Now, let us analyze the de-Sitter invariant Chern-Simons gravity, which corresponds to α Λ = − / α <
0. In this case, the potential is like an inverted harmonic oscillator centered at theorigin. There are solutions for E positive, negative and zero.Let us just focus on the case E >
0. The trajectory of a timelike shell is then given by a ( τ ) = 2 p | α E| sinh (cid:16) ± τ p | α | + const. (cid:17) , (71)which is a shell either emerging from the past white hole or falling into the future black hole,depending on the sign ± in the argument.For E < E = 0 gives an increasing and a decreasingexponential.On the other hand, for E > k = 1, one could consider Euclideanization of the problem.Presumably, this could be relevant in describing the decay of the exotic negative µ spacetime. Definean angle χ by χ = τ E p | α | (72) The case of Poincar´e Chern-Simons theory was discussed in Ref. [13].
22p to a constant, where τ E is the Euclidean proper time of the shell. The metric on the Euclideanworld sheet of the shell reads ds = 4 | α | (cid:0) dχ + E sin χd Ω (cid:1) . (73)When E < E >
1, an excess. In both cases the space hasa curvature singularity at the poles χ = 0 and χ = π . Therefore, the smoothness of the Euclideanshell requires E = 1. This metric is spherically symmetric in the five dimensional sense in this case,whence it describes a 4-sphere. The 4-sphere separates a ball of Euclidean black-hole solution withmass parameter µ R from another solution with µ L , obeying the relation µ L + µ R + µ L µ R + 6( µ L + µ R ) = 0 . (74)It is interesting to note that the size of the Euclidean world sheet depends essentially only on α andnot on the µ L,R ; the latter change its shape, which is fixed to spherical by the above relation.The curve (74) is an ellipse. It is symmetrical around the line µ L = µ R and tangential with theline µ L + µ R = 0 at µ L = µ R = 0. It exists completely in the region µ L + µ R ≤
0. In view of theinequality (70) all points of the curve are included except µ L = µ R = 0. Therefore the 4-sphereEuclidean world sheet does exists for certain values of the parameters. Whether this interestingconfiguration is a mere curiosity or it is related to semiclassical transitions between the µ L and µ R spacetime is an open question.We can also consider the Anti-de Sitter invariant Chern-Simons theory, corresponding to α > V turns out to be a quadratic potential centered at the singularityat the origin. The analysis is similar to that of the dS case except that there are solutions only with E > α = − /
4. These shells represent a sudden classical transition from a spacetime withsome mass parameter µ in to another with a different mass parameter µ out . Such transitions occurfor quite general values of µ in , and this is a concise manifestation of the extreme degeneracy of thefield equations of the Chern-Simons theories. In this section we will perform an exhaustive analysis of the space of constant solutions, what wehave called the moduli space.
To begin, it will be convenient to introduce new dimensionless parameters, defined as follows u ≡ √ r x ( x + 1) (cid:16) y + 3 x − (cid:17) , w ≡ x + 1 . (75)On should think of u and w as functions of α , Λ and a , via the definitions (36). Here, u ≥ . The inverse transformation is given by y = 12 w ( w − u + 3( w − w ) , x = w − . (76)Each point on the ( w, u ) plane such that u + 3( w − w ) = 0 uniquely determines the values of thebasic dimensionless ratios x and y and therefore the solution . It is useful to remember that the radius a of the vacuum shell is given in terms of these variables by a = 4 α · wu + 3( w − w ) . (77) efinition 17. We will call the allowed domain on the ( u, w ) plane as the ( u, w ) parameter spacerepresenting the moduli space of the vacuum shell. Similarly the allowed domain on the plane of x and y is the ( x, y ) parameter space for the vacuum shell for non-zero Λ . The various possible pairsof parameters that uniquely represent all possible points of the moduli space can be thought of itscoordinates. In terms of these new variables we have that the vacuum shells are described by equations (seeProposition 6): f L + f R = 2( u + 9 w ) u + 3( w − w ) , p f L f R = ση R η L u − w u + 3( w − w ) . (78)Solving this (for details see appendix E) we find: Proposition 18.
The masses in the two bulk regions are M (+) and M ( − ) : ¯ M ( ± ) = 36 w (( w ± u ) − w )( u + 3( w − w )) . (79) The moduli space is divided into: timelike shell or spacelike shell solutions by the inequality u +3( w − w ) > or < respectively; standard orientation and wormhole orientation by u − w > or < respectively. (see Fig. 5). Furthermore the branch sign of the metric in each region is givenby ξ ( ± ) = sign ( w ± u ) . The points along the curves ± u = 3 w and u + 3( w − w ) are not regarded as part of the modulispace. Along the curve u − w = 0, which we will call the branch curve , one of the branch signs isundetermined. The line w = 0 is peculiar as it implies that either a = 0, or | α | = ∞ and Λ = 0.The latter is the case of pure Gauss-Bonnet gravity, whose solutions were found in Ref. [13]. For | α | < ∞ the line w = 0 is excluded from the moduli space as it corresponds to smooth geometries.We further note that the signs of α and Λ on the moduli space are given by sign( α ) = sign (cid:0) w ( u +3( w − w )) (cid:1) and sign(Λ) = sign( α ) sign( w − w ≈ M (+) and M ( − ) are related simply by u ↔ − u . Althoughthe true moduli space is the upper half plane u ≥
0, it is useful to formally extend to u < M ( − ) in the lower halfplane and for the region corresponding to M (+) in the upper half plane. Important physical properties of the solutions have to do with what values the masses M ( ± ) take,w.r.t. sign, magnitude and relative magnitude, over the moduli space. Let us comment on it below. A question with a very simple answer is where on the moduli space we could have M (+) = M ( − ) .We have seen that this happens at u = 0. Explicitly, from (79) we have¯ M (+) − ¯ M ( − ) = 144 uw (cid:0) u + 3( w − w ) (cid:1) . (80) Proposition 19. M (+) = M ( − ) only at the boundary u = 0 . Therefore such solutions exist only forwormholes. From Proposition 41 we have that at the points where M (+) = M ( − ) we have also that ξ (+) = ξ ( − ) . Lemma 20.
Symmetric configuration are such M L = M R and ξ L = ξ R . They exist only at theboundary u = 0 of the moduli space and they can be either time- or space-like shell wormholes. PSfrag replacements wwuu
Figure 5: The space of constant radius solutions, which we have called the moduli space, is depictedhere. The dimensionless variables w and u are defined at the beginning of section 6.1. The ellipsedivides solutions into spacelike (inside) and timelike (outside); The diagonal lines divide solutionsinto standard orientation (light grey) which have well-defined inner and outer region of the shell,and wormhole orientation (dark grey), where both regions can be thought of as exterior or interiordepending on whether a non-compact or compact region is maintained. Solutions exist for u ≥ w = 0, u > α does not actually belong to the moduli space as being trivial: thejunction condition require the metric across the shell must be continuous in this case. In terms ofthe couplings α and Λ, w is given simply by w ≡ α Λ3 . The combination of the couplings w = 0corresponds to the case where Einstein-Gauss-Bonnet gravity can be written as a Chern-Simonstheory with (A)dS gauge group. It for this combination that the smooth C metrics fail to beunique [23][25]. Note that the pure Gauss-Bonnet case, which formally corresponds to w = 0, Λ = 0in the limit that α → ∞ but M α is finite, does have nontrivial solutions, which were consideredseparately in Ref. [13].The equal mass ¯ M = ¯ M ( ± ) of the symmetric case reads¯ M = 4 ww − . (81)So, symmetric configurations exist for all w = 4 and ¯ M can take all real values except 4. The masses ¯ M ( ± ) change sign crossing the curves where they vanish, and of course these curves arewhere M ( ± ) vanish too. From the formula (79) and Proposition 41 we have that ¯ M ( ± ) = 0 along thecurves ( u ± w ) = 4 w , (82)respectively. They exist only for w >
0. The masses cannot vanish for w < u − w ) = 4 w , which alsoreads u = ± √ w + w , extended over the whole plane. The other mass is¯ M ( ± )0 = 9 w √ w ( w − √ w ± . (83)The curve ( u − w ) − w = 0 goes to negative values of u for 0 < w <
4. On the u > w = 0 and w = 4, fig. 6. Therefore25 Figure 6:
The moduli space showing the various curves listed in section 6.3. The basic division accordingto the time- or space-likeness of the world-volume of the shell and the orientation of the matching weredescribed in the figure 5 and are shown in black lines here: the points along them they do not belong to themoduli space.The diagonal (red) lines divide the space according to the branches of the bulk metrics which we classifyby the pair of signs ( ξ ( − ) , ξ (+) ) explained in section 3.2 and 6.1: in the region on the left the branch signsare (+ , +) i.e. both the bulk metrics on each side of the shell belong to the “exotic” Boulware-Deser branchwhich does not have a well-defined limit α →
0; the region in between the diagonal lines is ( − , +); in theregion on the left the branches are ( − , − ) i.e. both metrics belong to the branch with a well-defined α → w ≡ α Λ3 <
0, i.e. theyhave no asymptotics: certain curvature singularities appear at finite radius [26]).The hyperbola that exists on the outside of and touches the elliptic region of spacelike solutions only at theborder of the ellipse at the points w = 0 and w = 4 (blue line), is what we have called the stability curve:crossing this curve the second derivative V ′′ ( a ) of the potential (20) or (57) evaluated at the constant solu-tions a = a changes sign, which is a measure of (in)stability under perturbations. The constants solutionsfor which V ′′ ( a ) > u = 0. Each curve cor-responds to solutions such that one of the mass parameters vanishes i.e. one of the bulk regions is purevacuum. Note that they exist only for w = 1 + α Λ3 >
0. These configurations are discussed in sections 7.2and 7.3 as an interesting example of certain non-trivial features C metrics acquire when Einstein gravity issupplemented by the Gauss-Bonnet term is five dimensions. Figure 7:
In the upper half plain, the shaded region is where M (+) >
0. The inequality M ( − ) > M ( − ) ( − u ))). We note that M (+) and M ( − ) are both negative for all w < Proposition 21. ¯ M ( − ) = 0 for u = ± √ w + w > where ¯ M (+) = ¯ M ( ± )0 respectively. ¯ M (+) = 0 for u = 2 √ w − w > where ¯ M ( − ) = ¯ M ( − )0 . Independently of whether the mass that vanishes is an M (+) or an M ( − ) note also the following Remark 22.
When the zero mass is of branch ξ the massive side has mass ¯ M ( − ξ )0 and the matchinghappens according to u = | w − ξ √ w | > . The branch of the massive side depends, as always, onwhich side of the line u = w we are. Now, let us discuss the positivity of the mass parameters. The signs of ¯ M ( ± ) behave quite simply.From formula (79) we have: Proposition 23. ¯ M ( ± ) < at the convex region defined by the curves (82) i.e. where ( u ± w ) − w < respectively. They have an overlap for ≤ u < √ w − w , inside the spacelike shell wormhole region. ¯ M (+) < only in this overlap. Remark 24.
The entire curve u − w = 0 exists within the region where ¯ M ( − ) < . This is also seenby the fact that the r.h.s. of (E.3) vanishes there. ¯ M (+) > along u − w = 0 . The above mean that ¯ M ( − ) < w >
0. Thereforethe metrics f ( − ) will have inner branch singularities, discussed in appendix B. The signs of M ( ± ) = α ¯ M ( ± ) themselves are depicted in the figure 7 using also formula (E.6). We see from (79) that given a mass M ( ± ) , for given α and Λ, the radius a of the shell where thematching takes place is determined by a fourth order polynomial of u , namely( u + 3( w − w )) ¯ M ∗ − w ( u + 2 wu + w − w ) = 0 ; (84)27iven a u we can obtain a by (77). As discussed in Lemma 42, ¯ M ∗ is an ¯ M (+) when u is non-negativeand an ¯ M ( − ) when u is non-positive.The equation above does not seem to be very enlightening. However, we can combine it withsome pieces of information we have: First we know that u takes values on the entire real line.Secondly, there is at least one real solution u , since ¯ M ∗ is defined by (79) to correspond to some real u . Besides, ¯ M ∗ takes all real values itself as one may verify.So, one may ask the following: For a given ¯ M ∗ , and a given w , how many different real solutions u exist and what is their sign? Now, the l.h.s. of (84) is an even order polynomial. Then we knowthat there must be at least a second u producing the same M ∗ . What we a priori do not know iswhether the second u is of the same sign or of the opposite.There is one case where the second solution coincides with the first, and therefore has the samesign. This is when the root u is also an extremum of the polynomial. It is easy to verify when thishappens. We simply differentiate the polynomial w.r.t. u and use (79) to find the following answer( u + 3 w )( u − w + 4 w ) = 0 . (85)The points on the orientation curve u + 3 w = 0 are not included in the moduli space. Therefore wehave that there is a single u when ¯ M ∗ and w are such that u − w + 4 w = 0. We will see belowthat this is the stability curve i.e. the curve which separates the radially stable from the unstablesolutions on the moduli space as we will see below (see also figure 8).A related fact is given in the following Remark 25.
For fixed α and Λ we can think of the masses as functions of the radius of the shell a : M ( ± ) = M ( ± ) ( a ) . The function M (+) ( a ) has a global minimum and the function M ( − ) ( a ) hasa global maximum for radii a given by u − w + 4 w = 0 . Thus, there is simpler question one may ask: Given pair of masses M (+) and M ( − ) , when canthe matching happen at more than one shell radii a ?The answer is that this can never happen: Proposition 26.
For any w , any u such that ( w, u ) belongs to the moduli space gives a pair of massparameters ¯ M (+) and ¯ M ( − ) . Then, this is the unique u that gives these mass parameters. The proof is given in appendix E. Therefore, remembering that u is single valued in terms ofthe shell radius a , the junction conditions define a single-valued function a = a ( M ( − ) , M (+) ), infact one-to-one on the space of the allowed values of M ( ± ) . As we know from section 2.3 a is asymmetric function of M L and M R and it is given by a = a ( M ( − ) , M (+) ), via the correspondenceimplied in (40). Thus given the bulk metrics, the a =constant vacuum shell is unique . So we seethat a weakened version of uniqueness of solutions does survive. Note that for shells with standardorientation there are exactly two inequivalent configurations corresponding to the same shell radius,depending on whether M (+) is the mass of the inner or the outer region. We notice that, throughout the computations, the quantity W ≡ w − w = w ( w − , (86)appears often. Now we comment on how it turns out to be convenient to extract information onthe moduli space. First, notice that W clearly vanishes at w = 0 and w = 4. We also encounter thecurves u = ± W, u = ± W, u = ± w, u = ± w, u = 0 ; (87)28hich in detail correspond to u = 3( w − w ) : ¯ M (+) + ¯ M ( − ) = 2 ¯ M curve ,u = − w − w ) : causality curve ,u = ( w − w ) : stability curve ,u = − ( w − w ) : ¯ M (+) + ¯ M ( − ) = 0 curve , (88) u = ± w : orientation curve ,u = ± w : branch curve ,u = 0 : boundary curve (where ¯ M ( ± ) = ¯ M ) . We also found the curve where ¯ M ( ± ) = 0 to be( u − w ) − w = 0 , i.e. u = u ( ± ) = ± √ w + w : zero minus-mass curve , ( u + w ) − w = 0 , i.e. u = − u ( − ) = 2 √ w − w : zero plus-mass curve , respectively. And notice that in terms of W this simply reads u (+) u ( − ) = W . (89)The first four curves in our list, which involve W , are conic sections with symmetry axes thelines w = 2 and u = 0. The orientation and branch curves on the other hand have symmetry axesthe lines w = 0 and u = 0. The conic sections and especially the causality curve which is an ellipse, u + 3( w − = 12, break the symmetry between positive and negative values of w . The image ofthe causality curve around w = 0 would be centered at w = − x = − W ≡ w − w captures much importantinformation about the moduli space.Actually, the parameterization of the space of solutions in terms of variables ( u, w ) had shownto present advantages in order to classify the whole set of solutions. To emphasize this, and forcompleteness, let us also express the regions of radial stability over the moduli space in terms ofthese variables. Such regions are known to be characterized by the second derivative of the effectivepotential, which in terms of u and w is seen to be V ′′ ( a ) = − a w (cid:0) u − w + 4 w (cid:1) ( u − w ) (cid:0) u + 3( w − w ) (cid:1) . (90)The regions where V ′′ ( a ) > V ′′ ( a ) = 0 along the curve u − w + 4 w = 0which we have already called the stability curve, for reasons that become clear now. According toremark 25 this is where the mass ¯ M ( ± ) ( a ) have extrema. C metrics In Einstein gravity in four dimensions there is a variety of smooth, everywhere non-singular vacuumconfigurations in general characterized by some non-trivial topological property, e.g. Euler number.Any topological property they may have is an intrinsic feature of the smooth solution.In five dimensions and in Einstein-Gauss-Bonnet gravity similar configurations may exist aswell. The equations of motions though are such that one can manufacture, by cut and paste alongthe world-volume of vacuum shells, similar kind of solutions with the difference that they are notsmooth, i.e. not C . In this case there is no intrinsic property in the vacuum solution, we aresimply building objects which are much simpler locally. For that reason one may call these objectsnon-topological, though they certainly have non-trivial topological features. An analogy for thiswould be the difference existing between an object with an exactly given smooth metric which hasthe topology, e.g. of the sphere, and tetrahedra built out of flat pieces.29 Figure 8:
The shaded regions are where V ′′ ( a ) >
0. For the timelike shells (outside of the ellipse), theunshaded regions correspond to solutions unstable with respect to radial perturbations.
This digression leads us to recognize a great difference with respect to four dimensions. Unlike infour dimensions, in five-dimensional Lovelock theory, spacetimes which are defined by some simpleproperty locally, for example being vacuum and spherically symmetric, are by no means well definedglobally, if smoothness is given up. For each such metric, which may itself have non-trivial topologicalfeatures, one can construct infinitely many other spacetimes by cut and paste which locally are givenby the same simple property. That is, the theory allows for many different topologies where onewould expect it to allow only for different coordinates.A general analysis of the objects obtained by geometric surgery along vacuum shells is an inter-esting problem and contains much of the actual physics of five-dimensional Lovelock gravity (thatis, Einstein plus the Gauss-Bonnet term). In this work we mainly focus on the direct implicationsof their existence illustrated by appropriate examples. A systematic analysis is left for future work.Below we analyze how a constant curvature vacuum is modified by wormholes (and related config-urations). It turns out that, the smaller such constructions with wormholes are with respect to thescale set by α , the more complicated the topology can be. An interesting special case of a wormhole is when on one side we have pure vacuum, as mentionedalready in section 4. Starting from a constant curvature background, by introducing the vacuumwormhole we cut a hole in the constant curvature manifold, replacing it with an “outgoing” spacetimeregion of mass parameter M . Of course the topology of the vacuum is not the same anymore;there are now non-contractible 3-spheres. Nevertheless, it turns out that these configurations areeverywhere non-singular in the following sense: the only singularities that exist in spacetime areintegrable .These wormholes belong to a more general class of constructions: Depending on whether thevacuum shell is time- or space-like and the orientation of the matching (i.e. the different combinationsof the orientation signs η L and η R ), one obtains distinct types of configurations some of which containonly integrable singularities. Configurations we call “instantons” mentioned below belong to thismore general class. The curvature and Lovelock tensor are singular at the shell but only in the sense of delta functions. Local integralsof these quantities are finite and the physical laws defined by the field equations do not break down there. In thissense the solutions are not singular. f ( r ) = 1 − Kr . (91)This means that for the metric of branch ξ we have:4 αK = − (1 + ξ √ w ) . (92)These configurations exist for w > u = | w − ξ √ w | >
0. According to Proposition 38 the points on this curve such that u > w correspond to standard shell configurations. In detail, standard shells are the configurationscorresponding to: u = 2 √ w − w for w ∈ (0 , / u = 2 √ w + w for w ∈ (0 , u = | w − ξ √ w | is a continuous curve. The points with w = 0 and w = 4 do not belong in themoduli space. The same for the points with w = 1 / w = 1. So in all, from (92) we have that4 αK = − , − / , − , M = 9 α (4 αK + 1) (4 αK ) (4 αK + 3) . (93)Also Λ = 6 K + 12 αK . We have that this construction is possible when w >
0. Therefore, from Remark 44, we havethat the sign of α depends solely on the causal character of the vacuum shell. Namely, it is α > α < Remark 27.
All standard orientation shell configurations with zero mass in one of the bulk regionsare spacelike.
Remark 28.
The variable αK takes values on the entire real line with the exception of the points − , − / , − , . With these exceptions in mind we have:Spacelike shells i.e. α <
0: 4 αK ∈ ( − , . In the interval ( − / , exist all the standard shellconfigurations.Timelike shells i.e. α >
0: 4 αK ∈ ( −∞ , − ∪ (0 , ∞ ) . The mass M has poles at the boundary of the spacelike shell region. One may note that thoughtof as a function of α both poles are of first order.From the formula u = | w − ξ √ w | > a = K − (cid:18) αK (cid:19) − . (94)The vacuum of constant curvature K is a locally homogeneous spacetime and in particular islocally spatially homogeneous. Having placed one vacuum shell around some arbitrarily chosenorigin, we have seen that outside of the shell the homogeneity is everywhere maintained. As long asit does not cross the first, we may place a second vacuum shell and in fact an arbitrary number ofthem modifying the manifold in a way depicted in Fig. 9.Let K > α >
0. It is useful to rewrite this as αK = 38 (s αa + 1 − ) . (95)It is clear from both the last formula that in units of α the radius of the shell is an increasingfunction of the radius of universe 1 / √ K . When the shell is microscopic, i.e. small in units of α , wehave that Ka ≪
1. When the shell is macroscopic we have that Ka ≃
1. A microscopic universecould fit roughly ( Ka ) − = 14 s αa + 1 + 1 ! (96)vacuum shells of radius a . So the more microscopic the universe the more complicated its topologycan be. 31igure 9: K >
0. When α > α < < K < / | α | the solutions are multi-instantons. The radii of the instantons have a lower bound: a > | α | / Reversing in a sense our viewpoint from the previous section, we may think of the matching ofa massive metric with one of constant curvature along a vacuum shell, as a way to eliminate thesingularity at the origin, or better to replace it with an integrable singularity along the vacuum shell.Consider for example configurations along the curve u = 2 √ w + w and α > w > K >
0, that is
M > >
0, and the massive branch is anexotic branch ( ξ = +1). This metric alone has a naked singularity at the origin. By constructingthe vacuum wormhole we have managed to replace a region around the origin with a region of ade Sitter spacetime which contains the horizon. That is, the spacetime which asymptotically lookslike an exotic branch, massive, Boulware-Deser spacetime is actually everywhere non-singular andhas horizons. We might reasonably expect that thermal effects of the horizons will be felt in thiswould-be singular spacetime.The mass parameter M in the massive region is determined by the curvature K of the de Sitterregion. So then, the de Sitter space mimics a particle, or some fairly localized mass, as viewed fromsufficiently far away. The entropy S related to the existence of the de Sitter horizon depends on K and therefore on M . We expect ∂S/∂K >
0. We know that ∂M/∂K <
0. Therefore that entropywill decrease with M . This is not surprising since a positive mass in the exotic branch behaveseffectively like a negative gravitational mass.The previous example shows that the spectrum of black holes in Einstein gravity modified bythe Gauss-Bonnet term is not the same when C metrics are allowed, compared to the smoothBoulware-Deser metrics. A space which by an asymptotic observer who thinks in terms of smoothmetrics would not be recognized as a black hole might actually be one. Conversely, a spacetimewhich asymptotically would be a recognized as a Boulware-Deser black hole, could actually be aspacetime with a naked singularity, or a black hole different to the one expected.Consider, for example, the case Λ = 0 and α <
0. From the analysis in appendix B we seethat this spacetime is a black hole for
M > | α | (and ξ = − r is everywhere a timelike variable. Thus although outsidethe horizon spacetime looks like a specific Boulware-Deser black hole spacetime it can actually bea different one. The two different states have the same energy as measured at spatial infinity andhorizons with the same properties: as black holes they must be degenerate. Whether the usualentropy calculations take into account the effects of this degeneracy in the number of states is notclear to us. It is amusing to think that the modifications to the usual Bekenstein-Hawking formulain the presence of the Gauss-Bonnet term, see e.g. [16], are essentially due to such degeneracies. The analysis has focused on the spherically symmetric case ( k = 1). This can readily be extended tothe case of k = − k = 0 for toroidal black holes or naked singularity spacetimes,can be investigated.We have seen that spacelike shells exist, representing a sudden transition from one solution toanother. These present problems in terms of the predictability of the field equations. It would beuseful to know whether the shells are generic or if they only occur for a certain range of the couplingconstants and mass parameters.The Euclidean version of the C wormholes may be important for estimating the transition ratebetween the (unstable) plus branch and the (stable) minus branch solutions.These are left for future work. In this work we construct and analyze solutions of Einstein-Gauss-Bonnet gravity whose metric isclass C , piecewise analytic in the coordinates. The solutions are made by joining together twospherically symmetric pieces. Since the shell itself admits SO (4) isometry group, the resultingglobal spacetime is spherically symmetric. To put things into this context and discuss the specialimplications of low differentiability we start by reviewing the existing relevant theorems in Einsteinand Lovelock gravity.We start with a uniqueness and staticity theorem, applying to Lovelock gravity in general, whichimposes the stronger conditions on differentiability. Theorem 29 (Ref. [25]) . For generic values of the couplings (including the cosmological constant),class C solutions of the Lovelock gravity field equations with spherical, planar or hyperbolic symmetryare isometric to the corresponding static solutions.In particular, in Einstein-Gauss-Bonnet gravity in five dimensions C solutions with sphericalsymmetry are isometric to the Boulware-Deser solutions when Λ = − / α . When we let the metric become merely continuous at hypersurfaces, we have seen already thatone can construct many different time-independent solutions of the vacuum field equations: forexample, when Λ = 0 with α >
0, one can construct multiple concentric vacuum discontinuitiesseparating Boulware-Deser solutions. So uniqueness of black hole solutions does not hold for C metrics in Lovelock gravity. In fact neither does staticity. We return to discuss this below, after werevisit the corresponding theorems in Einstein gravity. Theorem 30 (Ref. [38][40]) . A differentiability class C and spherically symmetric vacuum solutionof Einstein gravity is: i ) static, ii ) equivalent to the Schwarzschild solution. That a spherically symmetric vacuum solution is static can be shown by finding a timelikeKilling vector, which also happens to be hypersurface orthogonal, even when the solution is givenin forms that don’t look very much like the usual Schwarzschild metric and which assume lowerdifferentiability [39], see [40]. 33 heorem 31 (Ref. [38]) . A C solution of the Einstein gravity field equations is well defined as thelimit of a sequence of (at least) C solutions. The metric is assumed to become C only at smoothhypersurfaces. Fields of low differentiability, e.g. with a discontinuous first derivative, can be understood assolutions of field equations in the weak sense, as limits of sequences of smoother fields. The factthat this limit is well defined makes the junction conditions of Israel well defined (the above workappeared earlier than Israel’s famous work). Now based on the junction conditions one may conclude:any hypersurfaces where the metric is not smooth must be a null hypersurfaces (we may call themshock waves). Then one may show that there are no spherically symmetric shock waves in Einsteingravity, see e.g. [40].The result regarding limits of smooth metrics holds in Einstein-Gauss-Bonnet and in fact inLovelock gravity in general (see the appendix of Ref. [13]).
Theorem 32. A C solution of Lovelock gravity field equations is well defined as the limit of asequence of (at least) C solutions. The metric is assumed to become C only at smooth hypersurfacesand their intersections. So considerations related to uniqueness or non-uniqueness similar to the above are meaningfulin Lovelock gravity as well. In this paper we have demonstrated:
Theorem 33.
There exist spherically symmetric C solutions of Einstein-Gauss-Bonnet gravity infive dimensions which are not given by the Boulware-Deser metric, but rather they are piecewise ofthe Boulware-Deser form. There exist solutions which are not static. In section 6 we found that for any value of the couplings α and Λ such that Λ > − / α , thereexist static (time-independent) vacuum shells: spherically symmetric C vacuum metrics are notunique for a wide range of couplings α and Λ in Einstein-Gauss-Bonnet gravity in five dimensions.One can in fact construct arbitrarily complicated spherically symmetric configurations by having aninfinity of concentric discontinuities. The exotic branch ( ξ = +1) is typically involved. Though theradius of a static vacuum shell is uniquely fixed by the metrics in the bulk, C static metrics are toa high degree non-unique as one does not a priori know how many vacuum shells there may be inspacetime.Now recall section 5.1. The time-dependent solutions, i.e. non-static ones, exist always : Forany non-zero value of α and any value of Λ there exists a time-dependent vacuum shell solution a ( τ ). The shell radius function a ( τ ) and the orientation signs η L and η R , completely define theworld-volume of the shell intrinsically as well as its embedding in spacetime (section 2.2). That is,they define a C metric in spacetime. Therefore a non-static C metric which respects everywherespherical symmetry can always be constructed in Einstein-Gauss-Bonnet gravity with cosmologicalconstant (which can be also zero). However in section 5.1 we have obtained a general result con-cerning shells with bulk metrics which have a well defined General Relativistic limit as α → ξ = − α Λ3 > C metrics. How is this to be interpreted?One could simply reject non-smooth metrics as unphysical. However, according to Theorem 32these C solutions are well defined as the limit of a family of smooth geometries. As such, theyapproximate arbitrarily closely to some smooth solution of the theory. Now suppose g ( n ) µν is a familyof smooth metrics which converge to a spherically symmetric vacuum shell solution as n → ∞ . Forfinite n , g ( n ) µν can not be a spherically symmetric vacuum solution, because the uniqueness theoremholds for smooth metrics. So it must either deviate slightly from spherical symmetry or have some According to that section, y ( w − w = yx ( x + 1) > yx ( x + 1) < x and y to the couplings these read for non-zero Λ: 3 / α + Λ > / α + Λ < In fact, it exists for a wide range of the bulk metric masses M L and M R , possibly for all values of the massesfor which the metrics are real. What is more important, for given values of the couplings α and Λ, for any givenBoulware-Deser metric one can construct a time-dependent vacuum shell for some other Boulware-Deser metric onthe other side. g ( n ) µν can be constructed which obeythe energy conditions, our results can be taken as evidence for the generic existence of such exoticfeatures as smooth wormholes in this theory. In this paper we have presented a method for generating new exact vacuum solutions of five-dimensional Lovelock theory of gravity. The solutions we obtained are spherically symmetric space-times whose metrics are C functions, and are composed by patches of different five-dimensionalBoulware-Deser spacetimes.The proof of the Birkhoff’s theorem for this theory (see e.g. Refs. [36], [22]) involves an as-sumption of differentiability [25]. We have seen that if this assumption is relaxed then there are C metrics (which satisfy the field equations in the distributional sense). The Birkhoff’s theorem stillholds, but merely in a (weaker) piece-wise form. Uniqueness and staticity of the metric turns out tobe valid only locally in regions where the metric is differentiable.We have used a geometric surgery procedure, employing the junction conditions of Einstein-Gauss-Bonnet theory to join spherically symmetric pieces of spacetime. This lead us to find differentgeometries with quite interesting global structure. In particular, we have shown that vacuum worm-holes do exist in this theory. The wormholes connect two different asymptotically (Anti)de-Sitterspaces and, in certain sense, they represent gravitational solitons in five-dimensions. Although theirmetrics are not C functions, they are globally static vacuum solutions of gravity equations of motionand have finite mass. These metrics, being non differentiable where the wormhole throat is located,still represent exact solutions defined everywhere, provided the junction conditions are obeyed. Thisis ultimately due to cancelations among different terms in the junction conditions.We have analyzed both static and dynamical solutions and, related to this, we have pointed out anew type of classical instability that arises in Einstein-Gauss-Bonnet gravity for certain range of theGauss-Bonnet coupling. This concerns fundamental aspects such as predictability and uniqueness. Acknowledgements:
C.G. and G.G. thank J. Oliva and R. Troncoso, for useful discussions.They are grateful to the Centro de Estudios Cient´ıficos CECS for the hospitality during their stays,where part of this work was done. Also, they specially thank M. Leston for helpful discussions.G.G. also thanks C. Bunster, A. Gurzinov, G. Gabadadze and M. Kleban for conversations; andthanks the financial support of Fulbright Commission, Universidad de Buenos Aires, CONICETand ANPCyT through grants UBACyT X816, PIP6160 and PICT34557. C.G. is doctoral fellow ofCONICET, Argentina. S.W. wishes to thank A. Giacomini, H. Maeda, J. Oliva and R. Troncosofor many useful discussions and IAFE and Universidad de Buenos Aires for warm hospitality. S.W.gratefully acknowledges funding through FONDECYT grant N o AppendixA Spherically symmetric solutions in Einstein-Gauss-Bonnetgravity
The spherically symmetric static solution of Einstein-Gauss-Bonnet theory of gravity was obtainedby Boulware and Deser in 1985, [Ref]. In five dimensions, and in terms of a suitable Schwarzschild-like ansatz (3), the metric is given by (5). On the other hand, if the higher dimensional theorywith the dimensionally extended quadratic Gauss-Bonnet term is considered, then the black holesolutions take a similar form, namely f ± ( r ) = k + r α ± r α r αM D r D − + α Λ D (A.1)35here M D is an integration constant, and Λ D is a numerical factor proportional to Λ and which onlydepends on the dimension D . The ambiguity in expressing the sign ± reflects the existence of twobranches, and corresponds to the parameter ξ introduced in section 2. The parameter k takes thevalue k = 1 in the case where the base manifold corresponds to the unitary ( D − d Ω D − ;besides, k may take the values − , if the base manifold is of negative or vanishing constantcurvature, respectively.Now, let us analyse the large distance limit of the solution (5). Asymptotically, this solutiontends to the five-dimensional Schwarzschild solution when α →
0, as it is naturally expected. Namely f − ( r ) ≃ − Mr − Λ6 r , (A.2)which represents a (A)dS-Schwarzschild black hole in five dimensions. Notice that the large r /α limit of the solution f + ( r ) acquires a large additional cosmological constant term ∼ r α . In particular,this implies that (A)dS space-time is a solution of the theory even for the case Λ = 0.It is also worth noting that in the case of non-vanishing cosmological constant, besides the leadingterm in the expansion (A.2), we find finite- α corrections to the black hole parameters [37]. Namely f ( r ) = 1 − m d πr − Λ d r + O ( αr − ) (A.3)where the dressed parameters m d and Λ d are given byΛ d = Λ ∞ X n =2 c n x n − ! = 1 − √ x, m d = m ∞ X n =2 n c n ( − x ) n − ! , with c n = (2 n − n − n ! , x := 43 Λ α. It is important to emphasize the difference existing between (A.2) and (A.3): While the first cor-responds to the actual limit α →
0, the second represents the large r /α regime which takes intoaccount finite- α contributions. For instance, the finite- α corrections to the mass are found by simplycollecting the coefficients of the Newtonian term ∼ r . The parameter x controls the dressing ofthe whole set of black hole parameters. The above power expansion converges for values such that x <
1. On the other hand, for x > dressed parameters in the large r regime m d = m p | x | ∞ X n =2 n c n ( − x ) − n ! Thus, we note that the Newtonian term ∼ m d r − vanishes in the limit | Λ α | → ∞ . The particularcase x = 1 is discussed below. Moreover, it is possible to see that, if one considers the case α Λ > x limit turns out to beΛ d = r α − α + O (1 / p | x | ) . One of the relevant differences existing between the black hole solutions in Einstein theory andin Einstein-Gauss-Bonnet theory is the fact that, in the latter, the metric does not diverge at theorigin of Schwarzschild coordinates, r = 0 , though its curvature is still singular. From (5), we easilyobserve f ± ( r = 0) = 1 ± r Mα .
In particular, this implies that the metric presents a angular deficit around the origin, and, also,that massive objects with no even horizon exist; thus, these correspond to naked singularities.Another interesting feature of the presence of the Gauss-Bonnet term is that, for the particularchoice of the parameters α Λ = − , the solution takes the form f ± ( r ) = r α − M (A.4)36here we have considered Λ < α >
0, and where M + 1 = q Mα . This solution resembles theBa˜nados-Teitelboim-Zanelli black hole [42, 41]. Actually, the solution (A.4) shares several propertieswith the three-dimensional black hole geometry, as it is the case of its thermodynamics properties.Parameter M in Eq. (A.4) plays the role of the mass M in the BTZ solution. For instance, justlike AdS space-time is obtained as a particular case of the BTZ geometry by setting the negativemass M = − (8 G ) − , the five-dimensional Anti-de Sitter space corresponds to setting M = − M − limit the solution becomes the metric to which AdS tends in the near boundary limit. Similarly, the massless BTZ corresponds to the boundary of AdS . Besides, as it was already mentioned, a conical singularity is found in the range 0 < M < α (corresponding to − < M < B Properties of the Boulware-Deser metric
The Boulware-Deser(-Cai) metric is given by (3) with metric function (5). The metric has twobranches for given cosmological constant Λ and energy M : ξ = ±
1. [These we call as the plus- andminus- branch respectively; also, more descriptively, as the “exotic” and the “good” branch.] There-fore solving the vacuum field equations for spherically symmetric metrics we obtain two solutions.Asymptotically they read f = 1 + ξ √ w α r + 1 + 2 ξM √ w r + O ( r − ) , (B.1)where we use our variable w ≡ α Λ3 > ξ = +1 branch depends on α asymptotically, while the asymptotically flat branch ξ = − M differently, i.e. the exotic metric of the Boulware-Deser solution doesnot reduce to Einstein solution in the “infrared” limit.The sign ξ is in some sense a charge which determines how a certain energy M enters a metricand thus if the field will be attractive or repulsive. As noted in [32], the graviton is a ghost on theasymptotic ξ = +1 branch, because the linear Einstein tensor appears to have the opposite overallsign (that is, this metric is classically unstable). This wrong sign is reflected in the inverted sign ofthe Schwarzschild term.An interesting issue about the Boulware-Deser solutions is that it contains a square root, whosereality imposes constraints. From (5) we see that: when w < maximum radius; when M/α < minimum radius in spacetime. At those finite radii there exists curvaturesingularities, known as branch singularities [26]. We call them outer and inner branch singularities,respectively to the cases above. These unusual spacetimes can also have horizons behind which thesingularities are hidden.We turn now to discuss the horizon structure of the Boulware-Deser spacetimes. The followingdoes not intend to be an exhaustive analysis, it is rather a list of general formulas in our notationuseful for our purposes. We will use the dimensionless parameters w and ¯ M . Recall the Boulware-Deser metric function f ( r ) given in (5) and define r H by f ( r H ) = 0. One finds that if w = 1 r H ± = 4 α ± p ¯ M ( w ) w − . (B.2)We have defined the useful quantity ¯ M ( w ) = w + (1 − w ) ¯ M , (B.3)which looks an interpolation between ¯ M and 1. Only the the spherically symmetric case k = 1 was discussed by Boulware and Deser. The cases k = 0 , − r H + we see that r H + > αw − > . (B.4)That is Λ >
0. Also r H − > M > α . Therefore we have:
Remark 34.
Elementary conditions for the existence of r H + is Λ > and for the existence of r H − the condition M > α . When 0 < | α | < ∞ , w = 1 ⇔ Λ = 0. So the previous formula holds for non-zero Λ. WhenΛ = 0, the correct result can be obtained as the limit w → r H − . Itreads r H − = 2 α ( ¯ M − . (B.5)We must substitute (B.2) back to f ( r H ) = 0 to solve for the signs. We have: − ξ = sign w ± p ¯ M ( w )1 ± p ¯ M ( w ) ! , (B.6)for r H ± respectively. Again the case w = 1 i.e. Λ = 0 can be correctly obtained from the limit w → r H − . Explicitly it reads − ξ = sign (cid:18) ¯ M + 1¯ M − (cid:19) . (B.7)We have used the sign function defined by sign( x ) = x/ | x | . When x = 0 it is ambiguous.Before continuing note the following. One implicit inequality that should be respected for hori-zons to exist is ¯ M ( w ) ≥ . (B.8)This is related to the reality of the square root of the Boulware-Deser metric function (5). ¯ M and w cannot be both negative. That is, if w ¯ M ≥ w + ¯ M ≥
0. This is precisely whatis guarantied by (B.8).From remark 34 we have
Remark 35. r H + > is equivalent to sign( α ) = sign( w − . r H − > is equivalent to sign( α ) =sign( ¯ M − . Note that, as we will solve the problem of existence for the real numbers r H ± / α the positivityconditions above essentially restrict the sign of α .Now r H + − r H − = 8 αw − q ¯ M ( w ) , (B.9)and (B.4) tell us that Remark 36. If r H + exists then r H + ≥ r H − . A solution r H corresponds to a horizon if r H > r < r H < r such that f ( r ) f ( r ) < Special cases : i ). w ± p ¯ M ( w ) = 0 ⇔ ¯ M = − w . (Note again that the correct result for w = 1 is obtained asthe limit). Then one of the two solutions r H ± coincides with a branch singularity . I.e. in this casethe branch singularity is null . [This is possible for α < r H = 4 α w +1 w − . It is a horizon of the branch ξ accordingto − ξ = sign( w ( w + 1)). [Of course the case w = 0 = ¯ M does not have two branches.] The case38 = − M = 1 does not have a horizon as r H vanishes (if ξ = − sign( α )), or the metric function f , which reads f = 1 + r α + ξ sign( α ) s − (cid:18) r α (cid:19) , (B.10)can vanish only at the branch singularity when α < r H = r H − when ¯ M = − w > r H = r H + when¯ M = − w < ii ). 1 − p ¯ M ( w ) = 0 ⇔ ¯ M ( w ) = ¯ M = 1. (Note again that the correct result for w = 1 is obtainedas the limit). We just learned that when we also we have w = − w = −
1. We observe that r H − = 0. This actually happens if ξ = − sign( α ) otherwisethis solution doesn’t exist.The single horizon solution is r H + = 4 α w − = . It is a horizon of the branch ξ according to − ξ = sign( w + 1). iii ). ¯ M ( w ) = 0. This is the saturated case where the two radii coincide: r H ± = 4 α/ ( w −
1) = 3 / Λ.Condition (B.6) works well in this case: ξ = − sign( w ). Also from r H > α ) =sign( w − r H is not a horizon radius. It is the (single) zero of f which has the same signeverywhere else. There are three non-trivial cases. w <
0: Then there is an outer branch singularityand f ( r ) ≥
0. 0 < w <
1: Then there is an inner branch singularity and f ( r ) ≤ w >
1: Then0 < r < ∞ and f ( r ) ≤ (cid:3) Recall (B.6).
Proposition 37.
With the exception of cases covered in i ) and ii ) we have: The radius r H + is ahorizon of the branch ξ if − ξ = sign (cid:18) w + q ¯ M ( w ) (cid:19) ; (B.11) the radius r H − is a horizon of the branch ξ if − ξ = sign(( ¯ M + w )( ¯ M − (cid:18) w + q ¯ M ( w ) (cid:19) . (B.12)The type of the horizon, i.e. whether it is black hole, inner or cosmological horizon, can bedetermined by the sign of the first derivative of f ( r ) (combined with Remark 36). We have r H ± f ′ ( r H ± ) = ∓ q ¯ M ( w ) · ± p ¯ M ( w ) w ± p ¯ M ( w ) . (B.13)Therefore for ¯ M ( w ) >
0, when r H − or r H + does correspond to a horizon, the type is determinedby sign( f ′ ( r H ± )) = ± ξ . (B.14)Remarks 35 and 36, Proposition 37, and formula (B.14) provide criteria for the existence andthe type of horizons for each branch ξ of the Boulware-Deser metric.For the exotic branch ( ξ = +1) a black hole horizon must be an r H + . This is not possible by(B.11). Thus there no black holes in the exotic branch. For the good branch ( ξ = −
1) a black holehorizon must be an r H − . From (34), this is possible only for M > α . C The Junction conditions
For our purposes, a singular shell Σ is a submanifold of codimension one at which the metric iscontinuous but the extrinsic curvature has a finite discontinuity. The field equations of Einstein-Gauss-Bonnet theory are given by (2). Integrating the field equations across Σ, one obtains thejunction conditions ( Q R ) ba − ( Q L ) ba = − κ S ba , Q ba given by (14). Lower case Roman letters from the beginning of the alphabet a , b etc.represent four-dimensional tensor indices on the tangent space of the world-volume of the shell.The R abcd appearing in the junction condition is the four-dimensional intrinsic curvature. Theantisymmetrized Kronecker delta is defined as δ a ...a p b ...b p ≡ p ! δ a [ b · · · δ a p b p ] .Now we calculate the intrinsic curvature of the world-volume of a spherical shell of radius a ( τ )and the extrinsic curvature (which takes a diagonal form). There are two cases: For the timelikecase the components are R τφτφ = .. aa , R φθφθ = R θχθχ = R χφχφ = ( k + . a ) a ,K ττ = η ¨ a + f ′ p ˙ a + f , K θθ = K φφ = K χχ = ηa p ˙ a + f ;while for the spacelike case these are R τφτφ = − .. aa , R φθφθ = ( k − . a ) a ,K ττ = η ¨ a − f ′ p ˙ a − f , K θθ = K φφ = K χχ = ηa p ˙ a − f . In this paper we are interested in pure vacuum shells, i.e. when S ab = 0. It is clear that in thiscase one can pull out a factor of ∆ K dc ≡ ( K + − K − ) dc , which is the jump in the extrinsic curvatureacross the shell. ∆ K dc ( · · · ) = S ab = 0 . (C.2)In the case of interest in this paper, the extrinsic curvature is diagonal. Thus, one expects eachcomponent of the junction conditions to factorize conveniently.Using the above formulae, we derive Q ττ given in (15). The angular components are, for thetimelike case: Q θθ = − a − ( η f ′ { a + 4 α ( k − f ) } p ˙ a + f + η a p ˙ a + f + η α ¨ a p ˙ a + f (cid:16) k + f + 2 ˙ a + a α (cid:17)) . (C.3) D The derivatives of the potential
As before, let us denote the derivative with respect to a by a prime. In analysing dynamical shellsand the stability of static shells it is useful to calculate the derivatives of V ( a ) with respect to a , V ′ , V ′′ etc. First we recall the definition of Y ( a ); namely f ( a ) ≡ k + a α (1 + ξY ( a )) , Y := r w + 16 M αa . (D.1) The notation of Ref. [13] has been used. However in that reference there was an unconventional sign conventionused (in equation A3) for the definition of extrinsic curvature. Although none of the results of that paper wereaffected by this, unfortunately the formulae B13-B17 for the Einstein-Gauss-Bonnet in the appendix were a mixtureof inconsistent sign conventions. Here we correct this sign error by choosing the standard sign convention as in Refs.[28] and [10]. The developed expression is: h σ ( K ab − δ ab K ) + 2 α “ J ab − δ ab J − ςP acbd K dc ” i + − = − κ S ab , (C.1)where σ is ± J ab := (2 KK ac K cb + K cd K d K ab − K ac K cd K db − K K ab ) / P abcd := R abcd + 2 R b [ c g d ] a − R a [ c g d ] b + Rg a [ c g d ] b is the trace-free part of the intrinsic curvature. In the case of atimelike shell ( σ = +1), this expression agrees with that given in Ref. [10, 11]. Y obeys the simple differential equation:( Y a ) ′ = 2 waY , (D.2)where we recall that w := 1 + α Λ3 .In terms of Y R and Y L , the effective potential defined in (20) takes the form: σV = (cid:18) k + a α (cid:19) − a α (cid:18) ξ R Y R + ξ L Y L − ξ R ξ L Y R Y L ξ R Y R + ξ L Y L (cid:19) . (D.3)This can be also written as V ( a ) = σ (cid:18) k + a α (cid:19) − σa α (cid:18) ξ R Y R + ξ L Y L ) + ( ξ R Y R − ξ L Y L ) ξ R Y R + ξ L Y L ) (cid:19) . (D.4)By repeated application of the differential equation (D.2) we obtain: σV ′ = a α (cid:18) − wξ R Y R + ξ L Y L (cid:19) , (D.5) σV ′′ = 12 α (cid:18) − wξ R Y R + ξ L Y L + 2 w ξ R ξ L Y R Y L ( ξ R Y R + ξ L Y L ) (cid:19) , (D.6)Note that the second derivative of V depends on a only implicitly through Y ( a ).Let a e be the radius at which V is an extremum, V ′ ( a e ) = 0. From (D.5) we have ξ R Y R ( a e ) + ξ L Y L ( a e ) = w . (D.7)It is of interest to know whether the extremum is minimum or maximum. The second derivativeevaluated at the extremum is: V ′′ ( a e ) = σα (cid:18) wξ R ξ L Y R ( a e ) Y L ( a e ) − (cid:19) , (D.8)If the right hand side of (D.8) is positive, the extremum is a minimum.Let us look for a solution where the minimum of the potential coincides with V = 0. Imposingat some radius a that V ( a ) = V ′ ( a ) = 0 implies: ξ R Y R + ξ L Y L = w , (D.9) ξ R ξ L Y R Y L = w − (cid:18) kαa (cid:19) w . (D.10)One can verify as a consistency check that the static and instantaneous shell solutions of section3 are recovered. In terms of the metric functions f the above two equations are: f R + f L = (cid:18) α + Λ3 (cid:19) a + 2 k , f R f L = (cid:18) Λ a − k (cid:19) , c.f. the junction conditions for static and instantaneous shells in proposition 6. Upon imposing theinequalities (21-23) we recover exactly the solutions of that section.It is important in analyzing the stability of the static ( σ = +1) vacuum shells to know the signof V ′′ evaluated at the static radius a . V ′′ ( a ) = 1 α ww − (cid:16) kαa (cid:17) − , (D.11)Note that this can also be written V ′′ ( a ) = − α ka α − α Λ3 ) ka α ! = − α xy − kx − yxy − kx − y in terms of the original variables and of the variables of section 3 respectively.41 Some details of the space of constant solutions
In terms of the variables w and u introduced in section 6, the junction conditions for static orinstantaneous shells are given by equation (78) with f L,R > σ = +1), and f L,R < σ = − f ( ± ) = (3 w ± u ) u + 3( w − w ) . (E.1)which turns out to be always real. The solution for f L,R is given by (40) as discussed there. Thenfrom (78) we first have
Proposition 38.
Let the total moduli space be described in the ( w, u ) parameter space. Then itnecessarily is a subset of the upper half plane u ≥ from which the points on the curves ± u = 3 w and u + 3( w − w ) = 0 are excluded. The four disconnected regions are divided according to thetype of the matching by combinations of the following. Timelike: u + 3( w − w ) > . Spacelike: u + 3( w − w ) < . Same orientation i.e. η L η R > u − w > . Opposite orientation, i.e. η L η R < u − w < . It is good to remember
Remark 39.
The points (0 , , (1 , and (0 , in the ( w, u ) plain do not belong to the moduli space.The point (1 , corresponds to the line x = 0 and y = 0 . We have already used the fact that f L,R ( r ) are the Boulware-Deser metric functions. In orderto completely solve our problem we must substitute for f L,R using the Boulware-Deser expressionevaluated at r = a given in equation (37), f L = f L ( a ) , f R = f R ( a ) . (E.2)Recall (40). Similarly to equations (41) and (42) we have that, within the space of Proposition 38,(E.2) amount to w ( w ± u ) = 2 w ξ ( ± ) s w + (cid:0) u + 3( w − w ) (cid:1) w ¯ M ( ± ) . (E.3)The solution w = 0 is possible only if ¯ M L,R = 0. Then for | α | < ∞ we have that M L,R = 0 andthe bulk metrics are simply f L,R ( r ) = 1 + r / (4 α ). We have Remark 40.
The line w = 0 , which lies in the “cone” u − w > and entirely within thetimelike standard shell region, is excluded from the moduli space as it merely corresponds to smoothgeometries. Therefore we work with non-zero w . Squaring the previous relation we find the mass parametersof f ( ± ) ( r ) which are consistent with the vacuum shell solution; they are given by equation (79).Substituting back into (E.3) we have the condition ξ ( ± ) | w ± u | = w ± u . (E.4)The sign of ξ (+) is completely determined over the moduli space if u + w = 0 by ξ (+) ( w + u ) > ξ ( − ) is determined if w − u = 0 by ξ ( − ) ( w − u ) >
0. Now for w + u = 0 we findthat ξ ( − ) | w | = w = − u <
0. Similarly for w − u = 0 we find that ξ (+) >
0, and this happens for w > ξ ± are specified for each point on the moduli space, i.e. a solution ofthe vacuum shell. We will say that this is a solution of the vacuum shell of type ( ξ ( − ) , ξ (+) ). Theexception is along the branch curve u − w = 0 where one of the signs is undetermined. We cansummarize Proposition 41.
The moduli space consists of the regions of the parameter space ( w, u ) given inProposition 38 such that: i ) the line w = 0 is excluded, ii ) according to the branch signs ( ξ ( − ) , ξ (+) ) of he bulk regions the parameter space is divided as follows: (+ , +) for u < w ; ( − , +) for − u < w < u , ( − , − ) for w < − u .The points along the branch curve u − w = 0 satisfy: if w > then ξ (+) > and ξ ( − ) arbitrary,if w < then ξ ( − ) < and ξ (+) arbitrary. The mass parameters M ( ± ) are well defined and givenover the moduli space by formula (79). Propositions 38 and 41 categorize the allowed spherically symmetric vacuum shell solution atconstant r in terms of spacelike/timelike and branch signs. This is plotted in figure 5.Note also the following: Formula (79) says that we can define a function¯ M ∗ ( w, u ) := 36 w (( w + u ) − w )( u + 3( w − w )) , (E.5)defined on the whole of the ( w, u ) plain (minus the curve u + 3( w − w ) = 0) and not only on theupper half. Then for u >
0, ¯ M (+) = ¯ M ∗ ( w, u ) and ¯ M ( − ) = ¯ M ∗ ( w, − u ). More generally, recallingalso equations (E.1), and (E.4), one may extend also f ( a ) and ξ , regarded as functions of w and u ,over the whole of the ( w, u ) plane. Lemma 42.
Let X denote any of the quantities ¯ M , f , ξ , or combinations of them. One may definea function X ∗ ( w, u ) such that X (+) = X ∗ ( w, u ) for u ≥ . Then, X ( − ) = X ∗ ( w, − u ) . At u = 0 wehave X (+) = X ( − ) i.e. X R = X L .The parameter space can be extended over the whole of the ( u, w ) plane. The mirror transforma-tion u → − u has the effect of sending (+) ↔ ( − ) . So one may specify one type of quantities ξ (+) , ¯ M (+) on the whole plane and mirror image the results to obtain the values of ξ ( − ) and ¯ M ( − ) . Now, we will also return to discuss in a more detailed manner the two most basic distinct typesof constructions here (recall Definition 2): matching with the same orientation, i.e. standard shellsolutions, and matching with opposite orientation, which we call collectively wormholes. Thoughthe following definition has been already in use in our work, it is useful to formalize the following
Definition 43.
A plus-metric, corresponding to the metric function f (+) ( r ) , is one whose massparameters is given by ¯ M ( w, u ) = M (+) and branch by ( w + u ) / | w + u | = ξ (+) over the moduli space.A minus-metric, corresponding to the metric function f ( − ) ( r ) ), is one whose mass parameters isgiven by ¯ M ( w, − u ) = ¯ M ( − ) and branch by ( w − u ) / | w − u | = ξ ( − ) over the moduli space. Now, let us make a remark on the sign of α . As a > y/x and is given by sign( α ) = sign (cid:0) w ( u + 3( w − w )) (cid:1) . (E.6)Therefore we have Remark 44. α > only for timelike vacuum shells and in the region w > (for standard orwormhole orientation). Inside the ellipse of the spacelike vacuum shells (see fig. 5), or for w < ,we have : α < . From the definition of w , the sign of the cosmological constant Λ is determined according tosign(Λ) = sign( α ) sign( w − . (E.7)When Λ = 0 i.e. w = 1 >
0, the sign of α depends on the whether the shell is time- or space-like aswe mentioned just above. Proof of Proposition 26:
For a given w , let u be such that the corresponding point ( u , w )belongs to the moduli space. We have¯ M (+) = 36 w ( u + w ) − w ( u + 3( w − w )) , ¯ M ( − ) = 36 w ( u − w ) − w ( u + 3( w − w )) . (E.8)Of course u ≥ M (+) = ¯ M ( − ) . Weknow that this is possible if and only if u = 0. So for non-unique solutions we may restrict ourselves43o u >
0. The second case is when ¯ M (+) + ¯ M ( − ) = 0. This happens in the moduli space along thecircle: u + w − w = 0. Clearly there is a unique positive u solving this equation.Therefore it is adequate to consider u > M (+) ± ¯ M ( − ) = 0. The proofis by contradiction. Let us suppose that u is not unique in the sense that there exists some u > u = u and which gives the same masses¯ M (+) = 36 w ( u + w ) − w ( u + 3( w − w )) , ¯ M ( − ) = 36 w ( u − w ) − w ( u + 3( w − w )) . (E.9)With a little rearranging subtracting the respective equations we have( u − u ) n ( u + u ) (cid:0) u + u + 6( w − w ) (cid:1) ¯ M (+) − w ( u + u + 2 w ) o = 0 , ( u − u ) n ( u + u ) (cid:0) u + u + 6( w − w ) (cid:1) ¯ M ( − ) − w ( u + u − w ) o = 0 .u = u so the quantities in the big brackets vanish. Adding and subtracting them we obtain theequations u + u + 6( w − w ) = 72 w ¯ M (+) + ¯ M ( − ) , u + u = 2 w ¯ M (+) + ¯ M ( − ) ¯ M (+) − ¯ M (+) . (E.10)Via (E.8) these equations express u in terms of u and w . The second of these tells us that u = − w u < . (E.11)So we conclude that u is negative, contradicting the assumption. (cid:3) Diagrams of the moduli space
Here we collect the diagrams referred to in section 3.
PSfrag replacements M ( − ) /αM (+) /α PSfrag replacements a / αM/α M ( − ) /αM (+) /α Figure 10: For Λ = 0, spherically symmetric shells exist only with standard orientation and for α >
0. Masses M ( − ) , M (+) and shell radius a are measured in units of the Gauss-Bonnet coupling, α . -15 -10 -5 5 10 -5 -15 -10 -5 5 10 -5 PSfrag replacements α Λ34 α Λ3 a Λ3 a Λ3 Figure 11: Static vacuum shells exist in the darkgrey region. Instantaneous vacuum shells with a = a exist in the light grey region. -15 -10 -5 5 10 -5 -15 -10 -5 5 10 -5 PSfrag replacements α Λ34 α Λ3 a Λ3 a Λ3 Figure 12: The static vacuum shells can have thestandard orientation η L η R > η L η R <
15 -10 -5 5 10 152468101214-15 -10 -5 5 10 152468101214
PSfrag replacements α Λ34 α Λ3 a Λ3 a Λ3 Figure 13: There are three types of static worm-holes according to the branch signs ( ξ L , ξ R ) in eachbulk region: ( − , − ) lightest grey; ( − , +) mediumgrey; (+ , +) dark grey. -15 -10 -5 5 10 15-551015-15 -10 -5 5 10 15-551015 PSfrag replacements α Λ34 α Λ3 a Λ3 a Λ3 Figure 14: The different types of constant a in-stantaneous shells are: ( − , +) branch standardorientation (light grey); ( − , +) branch wormholeorientation (medium grey); (+ , +) branchstan-dard orientation (dark grey). -15 -10 -5 5 10 15-5-2.52.557.51012.515-15 -10 -5 5 10 15-5-2.52.557.51012.515 PSfrag replacements α Λ34 α Λ3 a Λ3 a Λ3 Figure 15: The stable region V ′′ ( a ) > α this is a region of the (+,+) branch soluions. Fornegative α it includes all except a small region ofthe ( − , +) branch. -15 -10 -5 5 10 15-8-6-4-224-15 -10 -5 5 10 15-8-6-4-224 PSfrag replacements α Λ34 α Λ3 a Λ3 a Λ3 Figure 16: The stability of the standard shells.The stable regions are shown in light grey and theunstable regions in dark grey. All standard shellsare ( − , +).46 eferences [1] D. Lovelock, J. Math. Phys. , 498 (1971).[2] B. Zwiebach, Phys. Lett. B (1985) 315.[3] B. Zumino, Phys. Rept. (1986) 109.[4] D. Gross and J. Sloan, Nucl. Phys. B 291 , 41 (1987).[5] E. Fradkin and A. Tseytlin, Phys. Lett.
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