More about stable wormholes in beyond Horndeski theory

TL;DR

This paper investigates whether beyond Horndeski theories can support stable, traversable wormholes, finding that these theories may indeed allow such solutions, unlike traditional Horndeski theories.

Contribution

It derives stability conditions for non-spherical perturbations in beyond Horndeski theories and provides an example of a wormhole solution satisfying some of these conditions.

Findings

Beyond Horndeski theories can evade previous no-go theorems against wormholes.

Conditions for absence of ghost and gradient instabilities are established for certain perturbations.

An example wormhole solution satisfying some stability criteria is presented.

Abstract

It is known that Horndeski theories, like many other scalar-tensor gravities, do not support static, spherically symmetric wormholes: they always have either ghosts or gradient instabilities among parity-even linearized perturbations. Here we address the issue of whether or not this no-go theorem is valid in "beyond Horndeski" theories. We derive, in the latter class of theories, the conditions for the absence of ghost and gradient instabilities for non-spherical parity even perturbations propagating in the radial direction. We find, in agreement with existing arguments, that the proof of the above no-go theorem does not go through beyond Horndeski. We also obtain conditions ensuring the absence of ghosts and gradient instabilities for all parity odd modes. We give an example of beyond Horndeski Lagrangian which admits a wormhole solution obeying our (incomplete set of) stability conditions. Even though our stability analysis is incomplete, as we do not consider spherically symmetric parity even modes and parity even perturbations propagating in angular directions, as well as "slow" tachyonic instabilities, our findings indicate that beyond Horndeski theories may be viable candidates to support traversable wormholes.

Equations

$$S=\int {\mathrm{d}}^{4}x\sqrt{-g}\left({\mathcal{L}}_2+{\mathcal{L}}_3+{\mathcal{L}}_4+{\mathcal{L}}_5+{\mathcal{L}}_{\mathcal{B}\mathcal{H}}\right),$$

$${\mathcal{L}}_2=F(\pi ,X),$$

$${\mathcal{L}}_3=-K(\pi ,X)\square \pi ,$$

$${\mathcal{L}}_4=G_4(\pi ,X)R+G_{4X}(\pi ,X)\left[{\left(\square \pi \right)}^2-{\pi }_{;\mu \nu }{\pi }^{;\mu \nu }\right],$$

$${\mathcal{L}}_5=G_5(\pi ,X)G^{\mu \nu }{\pi }_{;\mu \nu }-\frac{1}{6}{G}_{5X}\left[{\left(\square \pi \right)}^3-3\square \pi {\pi }_{;\mu \nu }{\pi }^{;\mu \nu }+2{\pi }_{;\mu \nu }{\pi }^{;\mu \rho }{\pi }_{;\rho }^{\nu }\right],$$

$${\mathcal{L}}_{\mathcal{B}\mathcal{H}}=F_4(\pi ,X){ϵ}_{\sigma }^{\mu \nu \rho }{ϵ}^{{\mu }^{\prime }{\nu }^{\prime }{\rho }^{\prime }\sigma }{\pi }_{,\mu }{\pi }_{,{\mu }^{\prime }}{\pi }_{;\nu {\nu }^{\prime }}{\pi }_{;\rho {\rho }^{\prime }}+$$

$$+F_5(\pi ,X){ϵ}^{\mu \nu \rho \sigma }{ϵ}^{{\mu }^{\prime }{\nu }^{\prime }{\rho }^{\prime }{\sigma }^{\prime }}{\pi }_{,\mu }{\pi }_{,{\mu }^{\prime }}{\pi }_{;\nu {\nu }^{\prime }}{\pi }_{;\rho {\rho }^{\prime }}{\pi }_{;\sigma {\sigma }^{\prime }},$$

$$F_5(\pi ,X)=0,$$

$$8\pi M_{Pl}^2=1.$$

$$d{s}^2=-A(r)d{t}^2+\frac{d{r}^2}{B(r)}+J^2(r)\left(d{\theta }^2+{\sin }^2\theta d{\phi }^2\right),$$

$$A(r)\ge A_{min}>0,B(r)\ge B_{min}>0,J(r)\ge R_{min}>0,$$

$$A(r)\to 1,B(r)\to 1,J(r)=|r|+O(1)\text{mbox}asr\to \pm \infty .$$

$${\mathcal{E}}_A=0,{\mathcal{E}}_B=0,{\mathcal{E}}_J=0,{\mathcal{E}}_{\pi }=0,$$

$$g_{\mu \nu }={\bar{g}}_{\mu \nu }+h_{\mu \nu },$$

$$\text{mbox}Parityodd\{\begin{array}{l}\begin{array}{rl}& h_{tt}=0,h_{tr}=0,h_{rr}=0,\\ & h_{ta}=\sum _{\ell ,m}{h}_{0,\ell m}(t,r)E_{ab}{\partial }^b{Y}_{\ell m}(\theta ,\phi ),\\ & h_{ra}=\sum _{\ell ,m}{h}_{1,\ell m}(t,r)E_{ab}{\partial }^b{Y}_{\ell m}(\theta ,\phi ),\\ & h_{ab}=\frac{1}{2}\sum _{\ell ,m}{h}_{2,\ell m}(t,r)\left[E_a^c{\nabla }_c{\nabla }_b{Y}_{\ell m}(\theta ,\phi )+E_b^c{\nabla }_c{\nabla }_a{Y}_{\ell m}(\theta ,\phi )\right],\end{array}\end{array}$$

$$\text{mbox}Parityeven\{\begin{array}{l}\begin{array}{rl}{h}_{tt}=& A(r)\sum _{\ell ,m}{H}_{0,\ell m}(t,r)Y_{\ell m}(\theta ,\phi ),\\ h_{tr}=& \sum _{\ell ,m}{H}_{1,\ell m}(t,r)Y_{\ell m}(\theta ,\phi ),\\ h_{rr}=& \frac{1}{B(r)}\sum _{\ell ,m}{H}_{2,\ell m}(t,r)Y_{\ell m}(\theta ,\phi ),\\ h_{ta}=& \sum _{\ell ,m}{\beta }_{\ell m}(t,r){\partial }_a{Y}_{\ell m}(\theta ,\phi ),\\ h_{ra}=& \sum _{\ell ,m}{\alpha }_{\ell m}(t,r){\partial }_a{Y}_{\ell m}(\theta ,\phi ),\\ h_{ab}=& \sum _{\ell ,m}{K}_{\ell m}(t,r)g_{ab}{Y}_{\ell m}(\theta ,\phi )+\sum _{\ell ,m}{G}_{\ell m}(t,r){\nabla }_a{\nabla }_b{Y}_{\ell m}(\theta ,\phi ).\end{array}\end{array}$$

$$\pi (t,r,\theta ,\phi )=\pi (r)+\sum _{\ell ,m}{\chi }_{\ell m}(t,r)Y_{\ell m}(\theta ,\phi ),$$

$$h_{2,\ell m}(t,r)=0,$$

$${\beta }_{\ell m}(t,r)=0,K_{\ell m}(t,r)=0,G_{\ell m}(t,r)=0.$$

$$\ell >1.$$

$$S_{odd}^{(2)}=\int \text{mbox}dt\text{mbox}dr\left[A_1{h}_0^2+A_2{h}_1^2+A_3\left({{\dot{h}}_1}^2-2{\dot{h}}_1{h}_0^{\prime }+h_0^{\prime 2}+\frac{4J^{\prime }}{J}{\dot{h}}_1{h}_0\right)\right],$$

$$A_1$$

$$A_2$$

$$A_3$$

$$\mathcal{F}$$

$$\mathcal{G}$$

$$\mathcal{H}$$

$$S_{odd}^{(2)}=\int \text{mbox}dt\text{mbox}dr\left[A_1{h}_0^2+A_2{h}_1^2+A_3\left(2Q\left({\dot{h}}_1-h_0^{\prime }+\frac{2J^{\prime }}{J}{h}_0\right)-Q^2\right)-\frac{2}{J^2}\frac{d}{dr}\left(A_3{J}^{\prime }J\right)h_0^2\right].$$

$$h_0=\frac{d}{dr}\left(A_{3}Q{J}^2\right)\cdot {\left[2\frac{d}{dr}\left(A_3{J}^{\prime }J\right)-A_1{J}^2\right]}^{-1},h_1=\frac{A_3}{A_2}\dot{Q}.$$

$$S_{odd}^{(2)}=\int \text{mbox}dt\text{mbox}dr\sqrt{\frac{A}{B}}{J}^2\frac{\ell (\ell +1)}{2(\ell -1)(\ell +2)}\cdot \frac{B}{A}\left[\frac{{\mathcal{H}}^2}{A\mathcal{G}}{\dot{Q}}^2-\frac{B{\mathcal{H}}^2}{\mathcal{F}}(Q^{\prime }{)}^2-\frac{l(l+1)}{J^2}\cdot \mathcal{H}{Q}^2-V(r)Q^2\right],$$

$$V(r)$$

$$-\left(\frac{A^{\prime 2}}{A^2}-\frac{A^{\prime }}{A}\frac{B^{\prime }}{B}-\frac{A^{\prime \prime }}{A}+\frac{B^{\prime \prime }}{B}+4\frac{B^{\prime }}{B}\frac{J^{\prime }}{J}-4\frac{J^{\prime 2}}{J^2}+4\frac{J^{\prime \prime }}{J}\right)-\frac{4}{J^{2}B}\cdot \frac{\mathcal{F}}{\mathcal{H}}].$$

$$\mathcal{G}>0,$$

$$\mathcal{F}>0,$$

$$\mathcal{H}>0,$$

$$c_r^2=\frac{\mathcal{G}}{\mathcal{F}},c_{\theta }^2=\frac{\mathcal{G}}{\mathcal{H}}.$$

$$\mathcal{F}\ge \mathcal{G}>0$$

Full text

1 Introduction

Traversable wormholes are intriguing objects whose hypothetical existence has fascinated the scientific community for many years [1, 2, 3, 4, 5]. Within General Relativity, one of the non-trivial features of the traversable wormholes is the necessity to fill the space near the throat with exotic matter, which violates the Null Energy Condition (NEC). Since the NEC is quite robust and the majority of forms of matter comply with it, obtaining stable traversable wormhole is challenging. As an example, wormhole solutions exist in theories involving scalar field with wrong sign kinetic term [1, 2, 3, 4, 5, 6, 7], but these ghost theories are catastrophically unstable in quantum theory [8] (see Ref. [9] for a review). Once gravity is modified, the NEC is no longer relevant; one replaces it with the Null Convergence Condition (NCC) [10, 11], which says that $R_{\mu\nu}n^{\mu}n^{\nu}>0$ for all null vectors $n^{\mu}$. The NCC is also hard to violate 444 There is a claim [12, 13] that there exists a stable static wormhole in a dilaton–Gauss–Bonnet gravity. This solution, however, appears unstable even against spherically symmetric perturbations [14]. Furthermore, the construction of Refs.  [12, 13] involves a negative tension 2-brane at the wormhole throat. It is known in other contexts that a brane bending mode (not considered in Refs.  [12, 13, 14]) in the perturbation spectrum of a negative tension brane tends to be a ghost (see, e.g., Ref. [15]). Also, the stability analysis of the non-spherical perturbations about the wormhole of Refs. [12, 13] is still lacking..

Recently, it has been suggested that it might be possible to construct a wormhole by employing Horndeski theories [16], which are remarkable due to their ability to violate the NEC and NCC without introducing ghost or gradient instabilities. Horndeski theory is a scalar-tensor gravity whose Lagrangian involves terms with second derivatives but the equations of motion remain second order. There is an extension of Horndeski theories usually referred to as “beyond Horndeski” theories [17]. The major difference between the general Horndeski theory and its extension is that the latter has equations of motion of the third order in derivatives. Nevertheless, the number of dynamical degrees of freedom is the same in Horndeski and beyond Horndeski theory  [18].

Since (beyond) Horndeski theories are capable of violating the NEC and NCC without introducing ghost or gradient instabilities, they have become quite popular in studies of scenarios requiring the NEC/NCC violation, e.g., the Universe with a bounce, see Ref. [19] for review. Attempts to construct traversable wormholes within various subclasses of Horndeski theories were made, for instance, in Refs. [20, 21, 22, 23, 24], but stability of the solutions has always been a troublesome issue. Finally, a no-go theorem has been proven in Ref. [25], which forbids the existence of a stable, static, spherically symmetric wormhole solution in the general Horndeski theory. Namely, static, spherically symmetric wormholes in Horndeski theory inevitably have a region (typically near the throat) where non-spherical parity even (symmetric under reflection of 2-sphere) modes of linear perturbations have wrong sign time-derivative term in their quadratic action, meaning either ghost or gradient instability. Note that this instability is local in space, as it is associated with high spatial momenta, and that it is catastrophic, since frequencies of unstable modes are arbitrarily high.

Interestingly, this no-go theorem has its close analogue in the cosmological setting, namely, similar no-go argument holds for non-singular homogeneous, isotropic, spatially flat solutions in Horndeski theory, for instance, the cosmological bounce [26, 27]. The analogy between the cosmological bounce and wormhole is not utterly surprising: the radial profile of a wormhole resembles the evolution of the scale factor in the bouncing Universe. Although imperfect, this wormhole/bounce analogy becomes even more tempting because of the recent resolution of the stability issue in non-singular cosmological scenarios: in that case, the no-go theorem has been evaded [28, 29] by going beyond Horndeski, and complete stable bouncing solutions have been constructed [30, 31, 32]. One may conjecture that beyond Horndeski theories may admit stable wormholes as well. A necessary (possibly, crucial) step in checking this conjecture is to construct a wormhole in a beyond Horndeski theory that explicitly evades the no-go theorem proven in the Horndeski case. Such a construction has been attempted in Ref. [33] where we presented static, spherically symmetric wormhole in beyond Horndeski theory which, we argued, did not have ghost instabilities among parity even modes, and also among parity odd modes555Ref. [33] had a computational error which lead to a wrong conclusion concerning fine-tuning. We correct the error in this paper.. Independently, by making use of the effective field theory (EFT) approach based on ADM formalism, it was shown in Ref. [34] that the proof of the no-go theorem does not go through in beyond Horndeski theory, and that the parity odd modes may be completely stable.

In this paper, we obtain part of the stability conditions for a static, spherically symmetric solution in beyond Horndeski theory in a covariant form (as opposed to ADM EFT form of Ref. [34]). Namely, we derive the conditions ensuring the absence of ghosts and absence of gradient instabilities for non-spherical parity even modes propagating in radial direction. We also find the conditions for the absence of ghosts and gradient instabilities for all parity odd modes. These results extend the stability analysis carried out for Horndeski theory in Refs. [35, 36] to the beyond Horndeski case; note that neither we nor Refs. [34, 35, 36] study the full set of stability issues which includes the absence of gradient instability in angular directions in parity even sector as well as the absence of “slow” tachyonic modes. Yet our analysis is sufficient to show explicitly that beyond Horndeski terms modify the stability conditions in such a way that the no-go argument of Ref. [25] no longer applies. Not surprizingly, the form of the modified stability conditions in spherically symmetric background is similar to that in the homogeneous cosmological setting, which deepens the analogy between the two cases.

We then suggest a model with a specific Lagrangian of beyond Horndeski type, which admits a wormhole solution free of all those instabilities that we consider in this paper. This explicit example (which, in fact, is inspired by our original model given in Ref. [33]) shows that the no-go theorem can indeed be circumvented in the beyond Horndeski theories. However, the example is not completely satisfactory: even though space-time is asymptotically Minkowskian, weak gravity regime is grossly different from General Relativity even at large distances from the wormhole. Yet another possibly unphysical feature is that our wormhole has vanishing mass.

Thus, we give both general arguments and explicit example showing that the obstacle for constructing stable, static, spherically symmetric wormholes in Horndeski theories is absent in beyond Horndeski case. We emphasize, however, that we do not pretend to construct a fully stable wormhole. Indeed, we do not consider spherically symmetric parity even modes and parity even perturbations propagating in angular directions, as well as “slow” tachyonic instabilities. Note that the latter instabilities are potentially quite dangerous: as an example, they ruin classical linear stability of wormholes in theories with ghost fields [37] (over and beyond quantum catastrophic instability of these theories), whose geometries are very similar to that of our wormhole. Yet we think that our findings indicate that beyond Horndeski theories may be viable candidates to support traversable wormholes.

This paper is organized as follows. We give generalities of the beyond Horndeski theory and spherically symmetric Ansatz in section 2, derive our subset of stability conditions and show that the wormhole no-go theorem can be circumvented in section 3. We construct the wormhole solution obeying our subset of stability conditions in section 4. We conclude in section 5. Explicit formulas, which are often cumbersome, are collected in Appendices.

2 Beyond Horndeski Lagrangian

In what follows we generalize the stability analysis of Refs. [35, 36] by introducing the beyond Horndeski terms in the Lagrangian. The most general form of the beyond Horndeski Lagrangian is (with mostly positive signature of the metric)

[TABLE]

where $\pi$ is a scalar field (sometimes dubbed generalized Galileon), $X=-\frac{1}{2}\>g^{\mu\nu}\pi_{,\mu}\pi_{,\nu}$, $\pi_{,\mu}=\partial_{\mu}\pi$, $\pi_{;\mu\nu}=\triangledown_{\nu}\triangledown_{\mu}\pi$, $\Box\pi=g^{\mu\nu}\triangledown_{\nu}\triangledown_{\mu}\pi$, $G_{iX}=\partial G_{i}/\partial X$. Functions $F_{4}(\pi,X)$ and $F_{5}(\pi,X)$ in (1f) are characteristic of beyond Horndeski case, while the general Horndeski theory involves $\mathcal{L}_{2}-\mathcal{L}_{5}$. In this paper we set

[TABLE]

since the function $F_{4}(\pi,X)$ is sufficient to reveal the difference between Horndeski and beyond Horndeski cases. When generalizing the stability conditions found in Refs. [35, 36] to beyond Horndeski theory, we intensely use their notations. In what follows we set

[TABLE]

We consider static, spherically symmetric background geometry with metric of the following form:

[TABLE]

where the radial coordinate $r$ runs from $-\infty$ to $+\infty$, and the functions $A(r)$, $B(r)$ and $J(r)$ are positive and bounded from below,

[TABLE]

with $R_{min}$ standing for the radius of the wormhole throat. Note that $r$ is a globally defined coordinate, so that the horizon absence and “flaring-out” conditions are satisfied automatically. Wormhole is asymptotically flat at both sides provided that

[TABLE]

Even though $B(r)$ in eq. (4) can be set equal to 1 by coordinate transformation, we keep it arbitrary. The scalar field $\pi$ is static and depends on the radial coordinate only, $\pi=\pi(r)$ and hence $X=-B(r)\pi^{\prime 2}/2$, where prime denotes the derivative with respect to $r$.

To carry out the stability analysis, we adopt a bottom up approach, namely, we derive the background equations for the action (1) first, linearize them and then reconstruct the quadratic action for perturbations. We keep the notations of Ref. [36] for the equations of motion:

[TABLE]

where ${\cal E}_{A}$ is obtained by varying the action with respect to $A$, etc. The explicit forms of ${\cal E}_{A},{\cal E}_{B},{\cal E}_{J}$ and ${\cal E}_{\pi}$ are given in Appendix A, eqs. (Appendix A)–(81). In the next section we introduce the parametrization of perturbations and develop the corresponding linearized theory.

3 Linearized theory

3.1 Parametrizing the perturbations

In the same manner as in Refs. [35, 36], we make use of the Regge–Wheeler classification of perturbations and decompose them into parity odd and parity even sectors [38], according to their transformation laws under the two-dimensional reflection. Perturbations are further expanded in spherical harmonics $Y_{\ell m}(\theta,\varphi)$. This approach to parametrization simplifies calculations, since not only the odd and even modes evolve independently, but also perturbations with different $\ell$ and $m$ do not mix at the linearized level.

The perturbed metric reads

[TABLE]

where $\bar{g}_{\mu\nu}$ stands for the background metric (4) and $h_{\mu\nu}$ are linear perturbations. The parity odd sector of metric perturbations has the following parametrization:

[TABLE]

where $a,b=\theta,\varphi$, $E_{ab}=\sqrt{\det\gamma}\>\epsilon_{ab}$, with $\gamma_{ab}=\mbox{diag}(1,\;\sin^{2}\theta)$; $\epsilon_{ab}$ is totally antisymmetric symbol ($\epsilon_{\theta\varphi}=1$) and $\nabla_{a}$ is covariant derivative on 2-sphere. Decomposition in the parity even sector reads

[TABLE]

The perturbation of the scalar field $\pi$ is non-vanishing only in the parity even sector:

[TABLE]

where $\pi(r)$ is the spherically symmetric background field.

To proceed further, one either makes use of gauge-invariant variables [39] or imposes gauge conditions. We follow the latter route and adopt the gauge choice made in Refs. [35, 36], namely, in the parity odd sector we set

[TABLE]

while in the parity even sector we impose the gauge conditions

[TABLE]

Note that our gauge choice differs from that of Ref. [34], but despite this fact we expect to have the same linearized spectrum in the end.

We consider the linearized theory for parity odd and parity even perturbations separately in the next two subsections.

3.2 Parity odd sector

To warm up, we consider parity odd sector first. As outlined above, to obtain the quadratic action for perturbations we linearize equations of motion (eqs. (Appendix A)–(81) in Appendix A) and reconstruct the corresponding quadratic action. Note that the perturbations with $\ell=0$ do not exist, while perturbations with $\ell=1$ are pure gauges [7, 35]. So, we are left with

[TABLE]

Making use of the notations (9) for parity odd perturbations and gauge condition (12) and performing integration by parts whenever necessary, we obtain the quadratic action for the parity odd sector:

[TABLE]

where dot and prime stand for the temporal and radial derivatives, respectively. Since modes with different angular momenta $\ell$ and $m$ decouple, hereafter we drop subscripts $\ell$ and $m$ in $h_{0}$, $h_{1}$ and other perturbations; summation over $\ell$ and $m$ is implicit in (15) and in similar expressions below. We integrated over the angular coordinates in eq. (15). The coefficients $A_{1}$, $A_{2}$ and $A_{3}$ include $\sqrt{-g}$ and read

[TABLE]

We retain the left hand sides of equations of motion ${\cal E}_{A}$ and ${\cal E}_{B}$ in the expressions for $A_{1}$ and $A_{2}$ for book keeping purposes. The coefficients in eqs. (16)–(18) get modified as compared to their counterparts in Ref. [35]:

[TABLE]

Both ${\cal H}$ and ${\cal G}$ involve the beyond Horndeski function $F_{4}(\pi,X)$.

The expression (15) for the action shows that $h_{0}$ is a non-dynamical degree of freedom, but the corresponding constraint is a second-order differential equation. To avoid solving the differential equation as it is, it was suggested in Ref. [40] to make use of the following trick for rewriting the action (15) in terms of a single variable. First, we gather the terms in the action (15) containing derivatives of $h_{0}$ and $h_{1}$, and introduce the Lagrange multiplier $Q$:

[TABLE]

The equations of motion for $h_{0}$ and $h_{1}$ following from eq. (22) read, respectively:

[TABLE]

Upon substituting eqs. (23) into the action (22) and making use of expressions for coefficients $A_{i}$ given by eqs. (16)– (18), we obtain the quadratic action in terms of the dynamical degree of freedom $Q$:

[TABLE]

which is the desired result. The original variables $h_{0}$ and $h_{1}$ are found from (23) as soon as $Q$ is known. Note that the third term in eq. (24) corresponds to the angular part of the Laplace operator and governs the stability in the angular direction, while $V(r)$ stands for the ”potential” and reads:

[TABLE]

The conditions for the absence of ghosts and gradient instabilities in the parity odd sector immediately follow from the quadratic action (24):

[TABLE]

The sound speeds squared for the modes propagating in the radial and angular directions are, respectively:

[TABLE]

To ensure that all modes propagate at subluminal speed, one requires that

[TABLE]

and

[TABLE]

These inequalities are similar to their counterparts in the general Horndeski case. However, the expressions for ${\cal G}$ and ${\cal H}$ now involve the beyond Horndeski contributions.

We note that the conditions (26) – (28) do not guarantee that the parity odd sector is completely stable. Indeed, there may occur “slow” tachyonic instabilities due to possible negative eigenvalues of the pertinent eigenvalue problem involving the “potential” $V$. It has been argued, though, that requirement of the absence of these instabilities may not be particularly restrictive [34].

3.3 Parity even sector: circumventing the no-go

theorem

Let us now consider the parity even sector of perturbations. For technical reasons, we consider non-spherical perturbations, $\ell\neq 0$; we do not expect anything particularly pathological in $\ell=0$ sector666The case $\ell=1$ is special too [36]. In that case, there is one dynamical mode, rather than two at $\ell>1$. Upon imposing appropriate gauge conditions, we find that the coefficients in its quadratic action similar to (37) are given by (39) and (47). Therefore, our stability analysis is valid for $\ell=1$ as well.. In full analogy with the parity odd case, we linearize the equations of motion (Appendix A)–(81), adopting the parametrization of the perturbations (10) and (11) and gauge conditions (13), and arrive at the following quadratic action:

[TABLE]

where the subscripts $\ell$, $m$ are again omitted, $j^{2}=\ell(\ell+1)$, and we have integrated over $\theta$ and $\phi$. The explicit expressions for coefficients $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$ and $e_{i}$ with $\sqrt{-g}$ included are given in Appendix B. Unlike in the homogeneous case, there are no new structures in the quadratic action arising due to adding the beyond Horndeski term. But the expressions for the coefficients $a_{2}$, $a_{6}$, $c_{1}$, $c_{4}$, $e_{1}$ and $e_{4}$ in (32) change significantly (see Appendix B for details). This substantially modifies the stability conditions, as we show below.

According to the form of the action (32), $H_{0}$ is a Lagrange multiplier that gives the following constraint:

[TABLE]

Following Ref. [36] we introduce a new variable $\psi$ such that

[TABLE]

then both $\chi^{\prime\prime}$ and $\alpha^{\prime}$ terms get cancelled out in eq. (33) upon substitution of $H_{2}$ from eq. (34). The resulting equation can be solved to give777It appears that there is a misprint in eq.(29) of Ref. [36].

[TABLE]

where we made use of the relation $a_{3}^{\prime}=a_{7}$ (see Appendix B). $H_{1}$ is a also a non-dynamical degree of freedom in the action (32). The corresponding constraint reads

[TABLE]

Equation (35) enables one to express $H_{2}$ and $H_{1}$, using eqs. (34) and (36), in terms of $\psi$ and $\chi$. Hence, upon substituting eqs. (34)-(36) into the action (32) and integrating by parts, the quadratic action is written in terms of two dynamical degrees of freedom:

[TABLE]

where $i=1,2$ and $v^{1}=\psi$, $v^{2}=\chi$. We note that terms which are higher order in derivatives, like $\dot{\psi}^{\prime}\dot{\chi}$, safely disappear upon integrating by parts.

To ensure the absence of ghosts in the parity even sector, we require that the quadratic form ${\cal K}_{ij}$ is positive definite, i.e.,

[TABLE]

The explicit expressions for $\mathcal{K}_{11}$ and $\det(\mathcal{K})$ read

[TABLE]

[TABLE]

where ${\cal F}$ and $\mathcal{H}$ are given by (19) and (21),

[TABLE]

and

[TABLE]

The expressions for $\mathcal{K}_{12}$ and $\mathcal{K}_{22}$ are given in Appendix C for completeness. According to eqs. (39) and (40), both no-ghost conditions (38) reduce to the following requirement:

[TABLE]

where ${\cal F}$ is positive by the stability conditions in the parity odd sector (see eq. (27)). Importantly, ${\cal P}_{1}$ in eq. (42a) significantly differs from its analogue in Horndeski theory due to the explicit presence of the function $F_{4}$ in the numerator. It is this $F_{4}$-term that enables one to circumvent the no-go theorem.

The general structure of the no-ghost condition for parity even perturbations is very similar to the stability condition in non-singular cosmological scenarios (see Ref. [33] for discussion). Indeed, the condition (43) has the same form as the central relation in the cosmological case, with ${\cal P}_{1}$ being proportional to the derivative of a certain function $\xi$ ($\dot{\xi}$ in the cosmological case, see, e.g., Ref. [32]). In complete analogy to the cosmological setting, the condition (43) requires that ${\cal P}_{1}$ is bounded from below by a positive function:

[TABLE]

where

[TABLE]

Thus, $\xi$ must be a monotonously growing function of radial coordinate and has to cross zero somewhere888A possible loophole is that $\mathcal{F}$, and hence $\mathcal{G}$, vanish as $r\to-\infty$ and/or $r\to+\infty$, see Refs. [26, 27, 41, 42] for the discussion of similar possibility in the cosmological context. This is potentially dangerous because of possible strong coupling far away from the wormhole, and certainly requires strong deviation from General Relativity there.. In Horndeski theory, the no-go theorem [25] is obtained by noting that $\xi$ cannot cross zero in a healthy way (the numerator in (45) is manifestly positive for $F_{4}=0$), which means that the condition (43) is inevitably violated at some point and the solution is plagued by ghosts. The situation in beyond Horndeski theory is entirely different: the extra $F_{4}$-term in the numerator of $\xi$ makes it possible that $({\cal H}-2F_{4}B^{2}\pi^{\prime 4})$ crosses zero, while ${\cal H}$ remains safely positive, as required by the stability condition (28) in the parity odd sector. Hence, in beyond Horndeski theory, the no-ghost condition (43) may be satisfied throughout the whole space.

Let us now discuss radial gradient instabilities. They are absent, provided that the matrix $\mathcal{G}_{ij}$ is positive-definite:

[TABLE]

Here

[TABLE]

[TABLE]

with

[TABLE]

and

[TABLE]

The expressions for $\mathcal{G}_{12}$ and $\mathcal{G}_{22}$ are given in Appendix C. Both conditions (46) are satisfied provided that

[TABLE]

The sound speeds squared of the even-parity modes along the radial direction are equal to the eigenvalues of the matrix $(AB)^{-1}(\mathcal{K})^{-1}\mathcal{G}$:

[TABLE]

Note that $c_{s1}^{2}$ coincides with the radial speed squared of the odd parity mode $c_{r}^{2}$ in eq. (29), which enables one to interpret it as the radial propagation speed of two tensor degrees of freedom.

Thus, the parity even modes have neither ghosts nor radial gradient instabilities provided that the conditions (43), (51) are satisfied. These modes are subluminal along the radial direction when $c_{s1},c_{s2}\leq 1$. We will see in the next section that all these constraints can indeed be satisfied.

As we pointed out in section 1, our stability analysis (like the ones in Refs. [34, 36]) is incomplete, as we do not study angular gradient instabilities as well as “slow” tachyonic instabilities in the parity even sector. In other words, the stability conditions associated with matrices $\mathcal{M}_{ij}$ and $\mathcal{Q}_{ij}$ in the action (32) are yet to be addressed. This issue, technically quite challenging, is left for the future.

4 Wormhole beyond Horndeski: an example

Let us give a specific example of beyond Horndeski theory admitting a wormhole solution which is free of all ghost and gradient instabilities in parity odd sector and free of ghosts and radial gradient instabilities in parity even sector. Similarly to the cosmological setting, we adopt a “reconstruction” approach. Namely, we arbitrarily choose the background metric of wormhole form, and cook up the Lagrangian functions in such a way that the equations for background and stability conditions (27)-(28),  (43) and (51) are satisfied. At the same time we ensure that the sound speeds in (29) and (52) are at most luminal. The whole procedure is very similar to that carried out for the bouncing solution in Ref. [32], modulo the form of asymptotics. Here we require only that the space-time is asymptotically flat and do not impose any restrictions on the asymptotic behavior of the Lagrangian.

We begin with choosing the same specific form of the metric functions in (4) as in Refs. [1, 2]:

[TABLE]

where the parameter $\tau\equiv R_{min}$ regulates the size of the wormhole throat at $r=0$. Equation (53) immediately implies that gravity is modified, as compared to General Relativity, even far away from the wormhole: the wormhole mass vanishes, and yet the metric is not exactly flat at large $|r|$.

The scalar field, supporting the wormhole, is also static and spherically symmetric:

[TABLE]

For monotonously growing scalar field background, the solution $\pi(r)$ can be always transformed into the form (54) by field redefinition. In what follows we choose

[TABLE]

although this choice is not obligatory.

To find the Lagrangian, we choose the following Ansatz:

[TABLE]

while $K(\pi,X)=0$, $G_{5}(\pi,X)=0$ and $F_{5}(\pi,X)=0$ in full analogy with the bouncing setup of Ref. [32].

Our main requirement is that there are no ghost and gradient instabilities of the types considered in this paper in both parity even and parity odd perturbation sectors about the wormhole set-up (53). To this end, we make use of constraints (26)–(28),  (30)–(31),  (38), (46) and also $c_{s2}\leq 1$, where $c_{s2}^{2}$ is given by (52). Let us recall the no-ghost condition (38) for the parity even modes:

[TABLE]

where we use the definition (42a) of ${\cal P}_{1}$. Making use of (56), we express ${\cal H}$, ${\cal F}$, ${\cal G}$, $\Xi$, $\Sigma$ and $\Gamma$ in terms of $g_{4i}$, $f_{4i}$ and $f_{j}$:

[TABLE]

Here and in what follows we keep $\pi$ in the argument of functions to make the expressions easy to read. However, we have chosen the coordinate dependence of $\pi$ in eq. (54), so, in fact we work with functions of $r$. Note that ${\cal H}={\cal G}$ due to our choice $G_{5}(\pi,X)=0$ (see eqs. (21) and (20)). To avoid superluminal propagation, we set for simplicity

[TABLE]

which immediately gives $c_{r}^{2}=c_{\theta}^{2}=c_{s1}^{2}=1$. The choice in eq. (59) together with relations (58a) and (58b) enables one to express the functions $g_{41}(\pi)$ and $f_{40}(\pi)$ through $g_{40}$ anf $f_{41}$:

[TABLE]

As we discussed in section 3.3, in order to circumvent the no-go theorem one has to make sure that $\left(\mathcal{H}-2F_{4}B^{2}\pi^{\prime 4}\right)$ in eq.(57) crosses zero at some point. Since we have fixed ${\cal H}$ in eq.(59), let us choose $F_{4}(\pi,X)$ as follows:

[TABLE]

where $w$ and $u$ are parameters and $\tau$ still defines the size of the wormhole throat. Again $f_{40}$ and $f_{41}$ are in fact functions of $r$, see eq. (54). For sufficiently large $w$, the above choice leads to the required zero-crossing of the numerator in eq. (57). In our numerical examples below we set

[TABLE]

Note that for $u\neq 0$, our set-up is not invariant under reflection $r\to-r$. We introduced the parameter $u$ into our Ansatz to emphasize that there is no fine-tuning (while with $u=0$ one has $F_{4}^{\prime}=0$ at the wormhole throat $r=0$).

Making use of eqs. (60a) and (60b), we obtain both $g_{40}(\pi)$ and $g_{41}(\pi)$ explicitly from eq. (61):

[TABLE]

Hence, we have completely defined $G_{4}(\pi,X)$.

Now we turn to the denominator of the no-ghost condition (57), where $\Xi$ involves the yet undetermined function $f_{41}(\pi)$, see eq. (58c). We choose the function $f_{41}(\pi)$ is such a way that the denominator of the inequality (57) behaves as follows (except for the vicinity of $r=0$):

[TABLE]

which is sufficient for satisfying the no-ghost condition (57) everywhere. The approximate equality (64) holds for the function $f_{41}(\pi)$ given by

[TABLE]

Note that $f_{41}(\pi)$ is non-singular at $r=0$, where $\pi=0$. Thus, we have chosen $g_{40}(\pi)$, $g_{41}(\pi)$, $f_{40}(\pi)$ and $f_{41}(\pi)$ in such a way that the constraints (27)–(28) and (38) are satisfied.

The functions yet undefined are $f_{0}(\pi)$, $f_{1}(\pi)$ and $f_{2}(\pi)$. We find $f_{0}(\pi)$ in terms of $f_{1}(\pi)$ and $f_{2}(\pi)$ by solving the background equation ${\cal E}_{A}=0$:

[TABLE]

In the same manner, $f_{1}(\pi)$ is found from the background equation ${\cal E}_{B}=0$ with $f_{0}(\pi)$ substituted from eq. (LABEL:D:f0). Thus, the solution for $f_{1}(\pi)$ involves only $f_{2}(\pi)$ and reads:

[TABLE]

Then the background equation ${\cal E}_{J}=0$ is satisfied due to our specific choice of metric functions in eq. (53) and ${\cal E}_{\pi}=0$ is valid automatically.

Now, we still have to ensure that there are no radial gradient instabilities, i.e. inequality (46) holds. An additional constraint is imposed by the requirement that $c_{s2}^{2}\leq 1$ in eq. (52). These requirements are satisfied by a judicial choice of the remaining function $f_{2}(\pi)$ entering $\Sigma$ (see eq. (58d)). Indeed, the conditions (46) and (52) boil down to

[TABLE]

We have already ensured that the right hand side here is positive, so the latter inequalities can be satisfied by choosing $\Sigma$, and hence $f_{2}$ an an appropriate way. As an example, we choose $\Sigma(r)$ in such a way that

[TABLE]

One can check that the inequality (68) is indeed satisfied by this choice. The corresponding function $f_{2}$ is given in Appendix D.

The reconstructed functions entering eqs. (56a)–(56c) are shown in Fig.1 They have the following asymptotic behaviour as $r\rightarrow\pm\infty$:

[TABLE]

which results in the following asymptotics of $F(\pi,X)$, $G_{4}(\pi,X)$ and $F_{4}(\pi,X)$ far away from the throat, where $|\pi(r)|=c_{1}\log|r|$:

[TABLE]

where $q_{i}$, $i=1,\dots,5$ are positive coefficients, whose values are not important for us. Even though the asymptotic value of $G_{4}$ is that of General Relativity, weak gravity regime is grossly different from General Relativity even asymptotically. Indeed, non-vanishing $F_{4}(\pi,X)$ gives a non-trivial contribution to the sound speeds of tensor-like modes in eqs. (29).

Let us demonstrate explicitly that our solution satisfies our subset of stability conditions. We have arranged the Lagrangian functions so that ${\cal H}={\cal G}={\cal F}=1$, which automatically ensures that the parity odd modes obey (26)-(28) and (30), (31). As for the parity even modes, it follows from eqs. (39), (40), (47) and (48) that the absence of ghosts and gradient instabilities in radial direction amounts to satisfying the following inequalities:

[TABLE]

Here we simplified expressions for ${\cal K}_{11}$, etc., and introduced the factor $J^{-2}$ to match the asymptotics: $\tilde{\cal K}_{11}={\cal K}_{11}$ as $r\rightarrow\pm\infty$, etc. The functions (72)–(75) are shown in Fig.2.

The corresponding sound speed squared $c_{s2}^{2}$ for parity even modes (52) is shown in Fig.3 (recall that another sound speed $c_{s1}^{2}$ in parity even sector is equal to 1).

Thus, parity odd perturbations around the constructed wormhole solution are healthy against ghosts and gradient instabilities, while parity even sector is free of ghosts and radial gradient instabilities. As we already mentioned in section 1, our solution is not completely satisfactory due to vanishing mass of the wormhole (see eq. (53)) and non-trivial asymptotics,with gravity deviating strongly from General Relativity as $|r|\to\infty$ even in the weak field limit. Nevertheless, our findings suggest that beyond Horndeski theories may admit stable, and possibly phenomenologically viable wormholes.

5 Conclusions

To summarize, generalizations of Horndeski theories known as “beyond Horndeski” are interesting candidates for theories admitting stable Lorentzian wormholes. Indeed, this work, together with the independent analysis of Ref. [34] and our earlier study [33] has shown that beyond Horndeski theories circumvent the no-go theorem forbidding the existence of stable, static, spherically symmetric wormholes in Horndeski theories, the latter being a fairly general class of scalar-tensor gravities. Moreover, we have seen that ghost and gradient instabilities may be absent both in the parity odd sector of perturbations and among parity even perturbations travelling along radial direction. A particularly new point of this paper is the construction of an explicit example of such a wormhole. It remains to be seen whether one can construct completely stable wormholes in beyond Horndeski theories, and if so, whether the wormholes of this sort are viable in the sense that physics in the large distance asymptotics is consistent with experimental and observational tests.

Acknowledgements

The authors are grateful to Evgeny Babichev and Sergei Sibiryakov for helpful discussions and Enrico Trincherini for useful correspondence. We are indebted to the anonymous referees for instructive criticism. The work of S.M. and V.V. was supported in part by the RFBR grant 18-32-00812 and the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” grant.

Appendix A

In this Appendix we collect the Einstein and Galileon equations of motion derived for beyond Horndeski theory (1) and evaluated on the background (4):

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

with

[TABLE]

[TABLE]

Appendix B

In this Appendix we give the explicit expressions for coefficients entering the quadratic action (10) for the parity even modes. The coefficients below involve the structures ${\cal F}$, ${\cal G}$, ${\cal H}$, $\Xi$, $\Gamma$ and $\Sigma$, which are the combinations introduced in the main body of the text (see eqs. (19)–(21),  (41), (49) and (50)). Most importantly, upon introducing beyond Horndeski term to the Lagrangian, not only ${\cal G}$, ${\cal H}$, $\Xi$, $\Gamma$ and $\Sigma$ get modified as it might have been anticipated, but also additional contributions appear in the coefficients of the action (32). To emphasize the way beyond Horndeski terms alter the structure of coefficients, we introduce new terms $a_{i}^{BH}$ ($i=2,6,8$), $c_{j}^{BH}$ and $e_{j}^{BH}$ ($j=1,4$). Thus, we have

[TABLE]

[TABLE]

[TABLE]

Appendix C

In this Appendix we give the elements of matrices ${\cal K}_{ij}$ and ${\cal G}_{ij}$ entering the quadratic action (37) for parity even modes, which are used for deriving the analytic expressions for sound speeds squared in eq. (52):

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

Appendix D

This Appendix contains the explicit expression for function $f_{2}(\pi)$, which is found by solving eq. (69):

[TABLE]