Scalar probes on wormholes in Lovelock theories with unique vacuum

In this paper we present a new family of wormhole geometries in Lovelock theories with a unique vacuum, and study their stability under scalar field perturbations. As the solutions already known in the literature for Chern-Simons gravity on AdS, the new wormholes are characterized by a single integration constant $\rho_{0}$, that parameterizes the contribution to the mass coming from each boundary, which cancel each other, leading to a new example of"mass without mass". The stability of these new configurations, as well as other previously constructed in the literature is explored under scalar field perturbations. The stability is ensured provided the mass of the scalar perturbation is above a deformed Breitenlohner-Freedman bound which is sensitive to the value of $\rho_{0}$.


I. INTRODUCTION
In the realm of four dimensional general relativity (GR), it is a difficult task to construct asymptotically flat, spherically symmetric wormhole geometries since they generally require violations of different energy conditions [1].In particular the averaged null energy condition must be circumvented for a wormhole to exist if the throat is to provide a shorter path connecting points of each asymptotic region through the bulk.The obstructions can be avoided, for instance, by going beyond GR (see, e.g., [2], [3] and references therein), by the inclusion of a NUT charge [4], or by constructing wormholes with long throats supported by Casimir energy [5].The inclusion of a negative cosmological constant allowed to construct asymptotically AdS wormholes which in the radial direction are foliated by warped AdS spacetimes and are devoid of closed timelike curves [6], which possess a noncontratible S 1 , and contain a family of BPS configurations [7].
The general features that a static metric must fulfill in order to describe a traversable, asymptotically flat wormhole were studied in the seminal papers [8], [9], and [10], and the metrics considered in such references have been used as toy models to study the propagation of probe fields on spacetimes with wormhole topology.For example, the propagation of scalar and electromagnetic waves may correctly lead to the interpretation of the wormhole geometry as an extended particle [11].Transmission and reflection coefficients for asymptotically flat, ultrastatic wormholes in 2+1 and 3+1 dimensions have been studied in Refs.[12], [13], and [14], featuring resonances for particular values of the wormhole parameters, and leading to almost reflectionless effective potentials 1 .In Ref. [14] it was also shown that a nonminimally coupled scalar field may lead to an instability since the effective Schrödinger potential for the perturbation turns out to be negative definite.Even more, when the wormhole is asymptotically flat in both asymptotic regions, outgoing boundary conditions at both infinities lead to a quasinormal spectrum which can mimic that of a black hole of mass M for a given wormhole mass M −1 [14] (see also the recent works [19,20] and references therein).
By imposing boundary conditions on the asymptotic AdS boundary as well as at the throat, 1 There is also an extensive literature on Euclidean wormholes (see, e.g., [15]) leading to instantons and on the use of scalar probes to test the stability of the configuration (see, e.g., [16], [17] and references therein).It has been recently established that geometries with such properties can be embedded in M-theory [18].In this paper we will be interested in traversable, Lorentzian wormholes.
a particular family of smooth wormholes in Einstein's theory coupled to a nonlinear electrodynamics has also been shown to be stable provided an effective Breitenlohner-Freedman bound [21][22][23] is fulfilled for the probe scalar field [24].
Asymptotically AdS wormholes in theories with quadratic terms in the curvature do exist in five dimensions [25], in vacuum.The action for the Einstein-Gauss-Bonnet theory in five dimensions reads where α is a coupling constant with mass dimension equal to −2.When formulated in first order, the local Lorentz invariance of the theory is enlarged to a local (A)dS group at the point Λα = −3/4 (see, e.g., [26] and references therein).In this case, the Lagrangian can be written as a Chern-Simons form where the spin connection as well as the vielbein transform as components of an (A)dS gauge connection.This structure can be extended to higher odd-dimensions as well as to dimension three, but in contrast to the latter, the higher dimensional case does describe a theory with local degrees of freedom in the bulk.
As shown in [27], all these theories contain generalizations of the static 2 + 1-dimensional BTZ black hole [28]- [29], characterized by an integration constant which can be identified with the mass.In [25] it was shown that this theory actually admits a larger family of static solutions, including wormhole geometries in vacuum with two asymptotically locally AdS 2n+1 regions for n ≥ 2. In five dimensions, the line element of these solutions is given by where The spacetime (2) represents a wormhole geometry with a throat located at ρ = 0, connecting two asymptotically locally AdS spacetimes with curvature radius l 2 = 4α = −3/Λ.
The constant ρ 0 is an arbitrary integration constant and determines the apparent mass of the wormhole as seen by an asymptotic observer at a given asymptotic region [25]- [30].The presence of ρ 0 ̸ = 0, creates a region which has interesting properties for particles with angular momentum, probing these geometries.As shown in [30], the effective potential for the geodesics has two contributions, one of which is due to the angular momentum of the particle around the S 1 factor parametrized by the coordinate ϕ in (2).This centrifugal contribution always points outward the throat located at ρ = 0, while the remaining contribution flips its direction at the surface ρ = ρ 0 and always points toward it.Therefore, the gravitational pull acting on a particle can be balanced by the centrifugal contribution only if the particle is in one of the regions −∞ < ρ < 0 or ρ 0 < ρ < +∞, for positive ρ 0 .Geodesic probes turn out to be expelled from the region 0 < ρ < ρ 0 .And for probe strings propagating in this background the surface ρ = ρ 0 /2 defines the turning point of strings with both ends attached to the same asymptotic region [31].
In arbitrary odd dimensions, d = 2n + 1, these wormholes can be embedded in Lovelock theories at the Chern-Simons point, at which the Lagrangian can be written as a Chern-Simons form for the SO(2n, 2) group.In this work, first we show that in even dimension, d = 2n, the wormhole geometry can also be embedded in Lovelock theory with a unique vacuum, when the maximum power in the curvature is present.Therefore we would have identified a sensible gravity theory in every dimension for which the wormhole geometry is a solution.Then we will study a minimally coupled massive scalar field on the wormhole geometry (2), revisiting the case with ρ 0 = 0 which can be solved analytically (see [32]).
Following a complementary approach to that in Ref. [32], by formulating the problem in a Schrödinger form, we show that the effective potential for the scalar probe corresponds to a Rosen-Morse potential, which explains the integrability of the spectrum [33].Remarkably, we find that also in the limit ρ 0 → ∞ the spectrum can be obtained in a closed form as well, which would be particularly useful for obtaining the normal frequencies of the scalar on wormholes with large ρ 0 .Then, we will numerically solve the spectrum for normal frequencies of the scalar field for finite, nonvanishing values of ρ 0 in dimension five, connecting the two exactly solvable models.We also show that there is an effective Breitenlohner-Freedman mass for the scalar probe, which depends on ρ 0 .In the last section, via a double Wick rotation, we construct a new family of wormholes, which maintain the properties of the asymptotic regions.We also show that the propagation of a scalar probe on the new family of wormholes, is equivalent to the propagation on the former geometries, by performing the double Wick rotation at the level of the quantum numbers, namely by setting ω → in and n → iω.Finally we obtain the mass of this new wormhole geometry.

MENSIONS
Before analysing the propagation of a scalar probe on the wormhole geometry in arbitrary dimensions, below we identify a Lovelock theory which admits the wormhole as a vacuum solution, in even dimension.As mentioned before, in odd dimensions it is enough to consider Lovelock theory with the couplings related in such a manner that the action can be written as a Chern-Simons form for the AdS group.Since Lovelock theories with generic couplings fulfill a Birkhoff's theorem (see [34,35]), we will have to consider some relation between the couplings in order to by-pass such uniqueness result.For concreteness, let us focus on the Lovelock theory that has a unique maximally symmetric solution and contains all the possible powers of the curvature allowed in dimension d, i.e., we consider all the Lovelock terms of the form R k with k ≤ [(d − 1)/2].The field equations of such theory can be written in a very compact manner where Here l is the curvature radius of the unique, maximally symmetric, AdS solution of the theory.Notice that all allowed powers of the curvature appear in the field equations.It is easy to see that for the wormhole geometry the components of the shifted curvature RAB CD are where Latin indices {i, j, k, l} are coordinate indices on the manifold Σ d−2 , which has an intrinsic Riemann tensor Rij kl .As explained in [25], in odd dimension d = 2k + 1 the field equations imply a single scalar constraint on the Euclidean manifold Σ d−2 which is solved for example by the manifold S 1 × H d−3 , with H d−3 with a suitable radius.
Here we are interested in the embedding of the wormhole geometry in a Lovelock theory with unique vacuum, in even dimension with d = 2k + 2. In this case the field equations ( 4) reduce to a tensor and a scalar constraint on the manifold Σ d−2 , which respectively read To fix ideas, if one considers these constraints in dimension six, they respectively reduce to that of an Einstein manifold Ri j = −3δ i j with a constant Kretchman scalar Rijkl Rijkl = 36.
In consequence, provided we fulfill these conditions, we would have found a new wormhole solution of Lovelock theory with a unique vacuum in even dimensions.A simple inspection of these equations show that one can solve them by considering suitable products of constant curvature spacetimes, or even products of homogeneous geometries (see [36,37]).

III. NORMAL MODES OF THE SCALAR PROBE
The scalar probe on the wormhole geometry can be separated as and hereafter we will drop the dependence of R ω,n (ρ) on ω and n.Notice that we have assumed the scalar probe to be independent of the coordinates parametrizing the manifold Σ d−3 .Since the equation is linear, one finds decoupled, second order, linear ODEs for each mode, leading to Had we considered dependence on the coordinates of Σ d−3 , by including in (13) an eigenfunction of the Laplace operator on Σ d−3 denoted by Y k (σ d−3 ), the Eq. ( 14) would have acquired a modification of the form n 2 → n 2 + k 2 , where −k 2 is the eigenvalue of the Laplace operator on Σ d−3 .
Even though scalar fields on asymptotically AdS spacetimes can have negative m 2 , we will focus on the case m 2 ≥ 0 (for a thorough analysis of the properties of a scalar field on the particular case of the wormhole with ρ 0 = 0 and the deformed Breitenlohner-Freedman bound, see [32]).Reflective boundary conditions require the field to vanish at infinity and therefore only the ∆ + branch in (15) and ( 16) is allowed.Under these boundary conditions the equation ( 14) defines a Sturm-Liouville problem which can be manifestly seen by transforming the radial problem ( 14) into a Schrödinger-like equation.
In order to simplify the presentation, here after, let us fix the dimension to d = 5.
Equation ( 14) can be written as where The radial coordinate ρ works as a "tortoise"-like coordinate in the sense that in terms of (t, ρ) the two-dimensional part of the metric (2) is manifestly conformally flat.The coordinate ρ connects both asymptotically AdS regions 0 < ρ < π.The effective Schrödinger potential takes the form This potential U (ρ) (depicted in Fig. 1), parametrically depends on the wormhole inte- gration constant ρ 0 , and for arbitrary values of the latter, the equation cannot be solved analytically.Even though we will solve the equation numerically to find the normal modes of the scalar probe on the wormhole, it is interesting to note that there are two particular values of ρ 0 for which the potential is shape invariant [33] and ( 18) can be solved analytically.
Those values are ρ 0 = 0 and ρ 0 → +∞, and the corresponding potentials U are given by The leading term of U (0) leads to an effective quantum mechanical problem with a Rosen-Morse potential, which can be solved analytically and with energies and bound states given by with p an arbitrary integration constant that can be fixed by normalization, and p = 0, 1, 2, 3, ... as the mode number.For the particular value ρ 0 = 0 the wormhole acquires a reflection symmetry with respect to the throat.P b a are the corresponding Jacobi polynomials.
Remarkably, for ρ 0 → ∞, the Schrödinger problem in (22) can also be integrated analytically since it corresponds to a Scarf potential, and leads to the following frequencies and eigenfunctions where the constant A > B are given by A (∞) p is an integration constant and p = 0, 1, 2, ... , is again the mode number.It is worth to mention that in this case the frequencies ω (∞) p lead to an equispaced, fully resonant spectrum.
This result may be particularly relevant for obtaining the spectrum of the scalar field on the wormhole, for large ρ 0 , by perturbative methods.The strict limit ρ 0 to infinity can be taken after suitable regularization of the geometry [30], leading to a wormhole that connects two different asymptotic regions.Since such wormhole is not asymptotically locally AdS we left its analysis for the Appendix.
Here we are interested in the effects of a finite value of ρ 0 on the propagation of a minimally coupled scalar field.Therefore, we are obligated to integrate equation ( 18) numerically, with reflective boundary conditions at both infinities which can be achieved only for a countably infinity number of frequencies ω p .This is done by using a variation of the so-called shooting method (see, for instance, [38]), which in our case consists in varying the value of ω in (18) and seeking those values which fulfill the boundary condition at ρ = π.More specifically, the Schrödinger equation is recast as a first order ODE system, which is then integrated "from left to right", i.e. from ρ = 0 to ρ = π for each value of ω.The integration is performed using the standard 4th order Runge-Kutta method [38], which thus determines the value of the solution at the right boundary, i.e. u(ρ = π, ω).The later can be considered as a function of ω, for which the sought after values ω p are zeros of, in virtue of the right boundary condition u(ρ = π, ω p ).After a solution for ω p is found, the value of p is determined by counting the number of nodes of the function u(ρ, ω p ).

IV. SPECTRA FOR THE MINIMALLY COUPLED SCALAR PROBE
Since the problem for the normal modes with ρ 0 ̸ = 0 is a Sturm-Liouville problem, with potential fixed by the angular momentum of the field n, as well as ρ 0 and m 2 , the eigenfunctions and eigenfrequencies will both be labeled by an integer p.In all the plots shown below, we have included small black bars near ρ 0 = 0 and ρ 0 large, marking the analytic values for the frequencies obtained in those cases.In what follows we plot some of the numerically obtained spectra, which present interesting features as ρ 0 varies.
The frequencies of the fundamental mode (p = 0) are shown on the left panel of Fig. 2, while the right panel shows a set of frequencies for the tenth overtone (p = 10).While the fundamental mode is a monotonic function of ρ 0 , we observe that the excited modes have a maximum frequency for a critical value of ρ 0 and then decay to a given value.As expected, the frequencies increase with the angular momentum of the field, and consistently with the asymptotic expression for U (∞) in Eq. ( 22), the frequencies merge regardless the value of n.

V. SCALARS PROBES WITH NONMINIMAL COUPLING
As shown in [32], for the case ρ 0 = 0 the equation for a scalar nonminimally coupled with the scalar curvature can also be solved in an analytic manner.This is remarkable since the FIG. 4: Fundamental and first two overtones for the scalar probe with n = 0, 1, 2, 3, 4, and m 2 = 6 (left-panel) and m 2 = 30 (right-panel).For a given value of ρ 0 the same frequency can be obtained for different modes since the effective potential depends explicitly on n.
Ricci scalar of the wormhole background is a nontrivial function of the radial coordinate ρ, therefore including a nonminimal coupling with the scalar curvature do not stand for a shift in the mass.Here, we explore the effect of a nonminimal coupling on the spectrum of the scalar for ρ 0 ̸ = 0, still within the context of dimension five.The equation for the scalar in this case is given by This scalar probe is invariant under local Weyl rescalings for ξ = 3/16.When ρ 0 vanishes, the equation for the radial dependence of the nonminimally coupled scalar can be obtained from that of the minimally coupled one by shifting the mass as well as the eigenvalues of the Laplace operator on the manifold Σ 3 = S 1 × H 2 /Γ.For nonvanishing ρ 0 , one cannot rely on this shift, and one is forced to numerically find the spectrum of normal modes.Using the separation (13) and the change of variables (19), one leads to a Schrödinger-like equation of the form (18) with an intricate potential given by Even with the nonminimal coupling with the scalar curvature, in the extremal values ρ 0 = 0 and ρ 0 → ∞ one also recovers shape invariant potentials, since corresponding to a Rosen-Morse potential and a Scarf potential, respectively.
It is interesting to note that for modes without angular momentum (n = 0) and ξ = 3/8 both potentials lead to a fully resonant, equispaced spectra, which might enhance the energy transfer between modes when nonlinearities are included [39], as it happens in AdS.
Note that in a truly quantum-mechanical problem nor the Rosen-Morse, neither the Scarf potential lead to equispaced energies since in that case the eigenvalue is quadratic in the principal quantum number, and actually the harmonic oscillator is the unique potential with such property.Nevertheless, in our relativistic theory the eigenvalue is quadratic in the frequencies, therefore all the potentials which are quadratic in the principal quantum number, may lead to a equispaced set of ω n (see [40]).
The asymptotic behaviors for the function R(z) with z = (1 − tanh(ρ))/2 that defined the radial dependence of the field (13), is the same as that given in equations ( 15) and ( 16), replacing the,, conformal weights by Regarding the asymptotic behavior one can define an effective mass m 2 eff := m 2 − 20ξ and we will be focused in the region m 2 eff ≥ 0, in which the reflective boundary conditions lead to a unique set of normal frequencies.Below we present the spectra for different values of the parameters characterizing the potential, i.e., (ξ, ρ 0 , m 2 , n).

VI. A NEW WORMHOLE SOLUTION OF EINSTEIN-GAUSS-BONNET AND LOVELOCK THEORIES
Before finishing, let us report on a new, wormhole solution of Einstein-Gauss-Bonnet or even Lovelock theories with a unique vacuum, provided (12), solve the corresponding field equations.The new wormhole geometry is given by where L and E, are the angular momentum and the energy of the particle and b = +1, 0, −1 for timelike, null or spacelike geodesics, respectively.From the geodesic radial equation (36), one can see that the existence of a value of the radial coordinate ρ = ρ c , for which the noncentrifugal (proportional to E 2 ) and the centrifugal (proportional to L 2 ) contributions balance, is nontrivial.In this case, the noncentrifugal contribution pushes toward the surface ρ = 0, while the centrifugal contribution pushes particles away from the surface ρ = ρ 0 , in consequence, in the region 0 < ρ < ρ 0 , it is impossible to balance both contributions.A similar mechanism precludes the existence of orbits within the same region for the wormhole (12) (see [25]).On the other hand, the equation for a scalar field probe (11), (13), on the new geometry (34), leads to the following equation for the radial dependence which exactly corresponds to (14) setting d = 5, ω → in and n → iω.
Before finishing this section, it is interesting to consider the energy content of the new wormhole geometry (34).For Einstein-Gauss-Bonnet, in five dimensions, at the Chern-Simons point, the regularized action principle of [41] leads, via Noether theorem, to a definition for the energy of an asymptotically locally AdS spacetime.The mass of the new wormhole (34) receives contributions from both boundaries ρ → ±∞, which as for the wormhole (12) (see [25]), vanishes, since the contributions from both boundaries cancel each other, and in this case are given by where σ 2 = 2πVol(Σ 2 ).In this normalization, the contribution to the mass coming from each boundary on the original wormhole geometry (2) in dimension five reads M old ±∞ = ± 3 σ 3 8πG sinh ρ 0 .As mentioned above, in both cases the total mass vanishes.
asymptotic region, which is independent of the value of the black hole mass.Nevertheless, when a negative cosmological constant is included, as for example in the four-dimensional Schwarzschild-AdS black hole, in Schwarzschild-like coordinates, a time dependence of the form e −iωt in a scalar probe implies that the frequencies correspond to those measured by an observer located at r = (2M l 2 ) 1/3 .Note that such observer is nongeodesic, in contraposition to our geodesic observer measuring frequencies in the wormhole.From the holographic viewpoint this coordinate time is actually the time in the dual CFT.
We have also included a nonminimal coupling between the scalar probe and the scalar curvature, which allows, for some particular values, to connect two fully resonant, equispaced spectra.It is worth to mention here also that fully resonant, equispaced spectra play a central role in the turbulent energy transfer that leads to nonperturbative AdS instability, see, e.g., [42][43][44][45][46][47].This interesting phenomenology has also been observed in other nonlinear models as gravitating scalars on a spherical cavity in 3+1 [48], on systems governed by the Gross-Pitaevskii equation [49], on conformal dynamics on the Einstein Universe [50] and on vortex precession in Bose-Einstein condensates [51].
Finally, we have also introduced a new family of wormhole geometries, that are obtained from the first set via a double Wick rotation.The propagation of a test particle on this new geometry, unveils a region where no geodesic, circular orbit exist, and the spectrum of a scalar field probe can also be obtained from the one on the seed geometries by a suitable Wick rotation in frequency/momentum domain.
It would be interesting to explore the sector with negative squared masses.Since there is an asymptotically locally AdS asymptotic behavior, it is natural to expect the existence of an effective Breitenlohner-Freedman bound that may depend on ρ 0 and that would have to be obtained numerically.We expect to report on this problem in the future.

100 FIG. 1 :
FIG.1: Effective potential for the radial dependence of the scalar probe, for different values of the parameters.As expected in the presence of a negative cosmological constant, the potential diverges at the boundaries ρ = 0, π.

Figure 3
Figure3shows the fundamental and ten excited modes for the "s-wave" of the scalar (n = 0) on the left panel, while the frequencies for a spinning scalar probe with n = 10 have been plotted on the right-panel, both as a function of ρ 0 .It is particularly interesting to notice that the "s-wave" fundamental and excited frequencies remain almost constant regardless the value of the integration constant ρ 0 .Finally, Fig.4shows the fundamental and first overtones for different values of the angular momentum and the mass of the scalar.Even though the effective Schrödinger problem is one-dimensional, there are degeneracies due to the fact the effective potential depends on n, therefore different values of n lead to different one-dimensional Sturm-Liouville problems.

20 FIG. 2 :
FIG.2: Spectra for the fundamental mode p = 0 (left-panel) and tenth overtone p = 10 (rightpanel) as a function of ρ 0 for different even values of the angular momentum of the scalar probe n = 0, 2, 4, ..., 10, for m 2 = 10.The fundamental mode has a monotonic behavior with ρ while excited frequencies present a maximum for a given critical value of ρ 0 which depends on the angular momentum of the field.

10 FIG. 3 :
FIG. 3: Spectra for vanishing angular momentum n = 0 (left-panel) and a spinning scalar probe with n = 10 (right-panel) as a function of ρ 0 the fundamental mode and the first 10 excited states (m 2 = 10).
identified, and dΣ 2 stands for the line-element of an Euclidean 2-dimensional manifold, whose local geometry is that of the hyperbolic space with radius 3 −1/2 , globally equivalent to the quotient of the hyperbolic space by a Fuchsian group, i.e., Σ 2 is homeomorphic to the quotient of H 2 by a freely acting,