External Stability for Spherically Symmetric Solutions in Lorentz Breaking Massive Gravity

Abstract

We discuss spherically symmetric solutions for point-like sources in Lorentz-breaking massive gravity theories. This analysis is valid for Stückelberg’s effective field theory formulation, for Lorentz Breaking Massive Bigravity and general extensions of gravity leading to an extra term −S r ^γ added to the Newtonian potential. The approach consists in analyzing the stability of the geodesic equations, at the first order (deviation equation). The main result is a strong constrain in the space of parameters of the theories. This motivates higher order analysis of geodesic perturbations in order to understand if a class of spherically symmetric Lorentz-breaking massive gravity solutions, for self-gravitating systems, exists. Stable and phenomenologically acceptable solutions are discussed in the no-trivial case S ≠ 0.

Appendix: A: LBMG in Stückelberg’s Effective Field Theory Formulation and Bigravity

In this appendix, we sketch the main aspects of a class of LBMG realized as a Stückelberg’s effective field theory and Bigravity. All these equations can be found in [13] and [17]. This section is necessary to understand the model dependent assumptions of the results showed in Section 3.

Considering LBMG as an effective field theory, the graviton mass is generated by the interaction with a suitable set of Stückelberg fields. A set of four Stückelberg fields. Φ^A(A = 1, 2, 3, 4), is introduced by transformations under diffeomorphisms δ x ^μ = ζ ^μ(x). Such fields transforms as scalars and can realize a modified theory of gravity, at IR scales, which results manifestly invariant under diffeomorphism. Let us suppose to preserve the rotational invariance in the theory. The general action can be written as

$$ d^{4}x\sqrt{-g}{M_{P}^{2}}\left\{R+\mathcal{L}_{mat}+m^{2}\mathcal{V}(\mathcal{X},V^{i},S^{ij})\right\} S=\int $$

(27)

where $\mathcal {V}$ is a rotationally invariant potential and $\mathcal {X}=-g^{\mu \nu }\partial _{\mu }\phi ^{0}\partial _{\nu }\phi ^{0}$, V ⁱ = −g ^μν ∂ _μ Φⁱ ∂ _ν Φ⁰, S ^ij = −g ^μν ∂ _μ Φⁱ ∂ _ν Φ^j. m is the graviton mass scale. Lorentz breaking is manifest in the action. The Goldstone action can have additional symmetries and can single out a particular phase for LBMG; we can consider $\mathcal {V}=\mathcal {V}\left (\mathcal {X},W^{ij}\right )$ where $W^{ij}=S^{ij}-\mathcal {X}^{-1}V^{i}V^{j}$ [9]. The Goldstone action is invariant under $\Phi ^{i}\rightarrow \Phi ^{i}+\zeta ^{i}\left (\Phi ^{0}\right )$. The flat background is parametrizable as $\bar {g}_{\mu \nu }=\eta _{\mu \nu }$ and $\bar {\Phi }^{A}=\left (\alpha t,\beta x^{i}\right )$, preserving rotation and breaking Lorentz boosts if α ≠ β.

Assuming the spherical symmetry for the metric generated by a spherically symmetric source, it is possible to find out a set of coordinates in order to have metric and Goldstone fields in the simple form (m ₁ = 0 phase). That is

$$ ds^{2}=-dt^{2}J(r)+K(r)dr^{2}+r^{2}d\Omega^{2} $$

(28)

$$ \Phi^{0}=\alpha t+h(r),{\kern5pt} \Phi^{i}=\frac{x^{i}}{r}\phi(r). $$

(29)

As it was shown in [13], choosing the particular class of $\mathcal {V}$

$$ \mathcal{V}=\left[b_{0}+b_{1}\omega_{-1}+b_{2}\left(\omega_{-1}^{2}-\omega_{-2}\right)+b_{3}\left(\omega_{-1}^{3}-3\omega_{-2}\omega_{-1}+2\omega_{-3}\right)\right]\mathcal{X}^{-1}+ $$

(30)

$$a_{0}+a_{1}\omega_{1}+a_{2}\left({\omega_{1}^{2}}-\omega_{2}\right)+a_{3}\left({\omega_{1}^{3}}-3\omega_{2}\omega_{1}+2\omega_{3}\right)\,, $$

with ω _n = Tr(W ⁿ) one arrives to a Solution like (2), with $\Lambda ^{2}=\frac {1}{6}\left (12\bar {a}_{3}-6\bar {a}_{2}+\bar {a}_{0}-3\bar {b}_{1}+12\bar {b}_{2}-18\bar {b}_{3}\right )$ is the ”effective” cosmological constant; the exponent γ is $\gamma =-2\left (2\bar {a}_{2}-6\bar {a}_{3}+\bar {b}_{1}-2\bar {b}_{2}\right )/\left (\bar {a}_{1}-4\bar {a}_{2}+6\bar {a}_{3}\right )$. The last power law in (2) is the new term coming from the gravity modification, 1/r and r ², on the contrary, are also present in the Schwarzschild-de Sitter metric. Considering the case Λ=0, for γ < −2, the metric describes an asymptotically flat space. The masses of the fluctuations around the flat space are

$$m_{0}^{2}=\frac{3}{4}\left(\bar{a}_{1}-4\bar{a}_{2}+6\bar{a}_{3}-4\bar{b}_{2}-6\bar{b}_{3}\right)m^{2},\,\,\,\,\,{m_{1}^{2}}=0 $$

$$m_{2}^{2}=\frac{1}{2}\left(\bar{a}_{1}-2\bar{a}_{2}+\bar{b}_{1}-2\bar{b}_{2}\right)m^{2} $$

$$m_{3}^{2}=\frac{1}{4}\left(\bar{a}_{1}-6\bar{a}_{3}-\bar{b}_{1}+8\bar{b}_{2}-18\bar{b}_{3}\right)m^{2} $$

$$ {m_{4}^{2}}=\frac{1}{4}\left(\bar{a}_{1}-4\bar{a}_{2}+6\bar{a}_{3}-3\bar{b}_{1}+12\bar{b}_{2}-18\bar{b}_{3}\right)m^{2}. $$

(31)

In this case, the mass of the graviton is m ₂.

The same solution of spherically symmetric solution is found in bigravity where Lorentz invariance is spontaneously broken for a large class of potentials [17]. In these models, we consider a theory with two interacting metrics: g ₁ for our observable sector and g ₂ for another hidden sector. The two metrics are considered interacting through an effective potential. The resulting action is

$$ S=\int d^{4}x\left[\sqrt{-g_{1}}\left(M_{pl1}^{2}+\mathcal{L}_{1}\right)+\sqrt{-g_{2}}\left(M_{pl2}^{2}R_{2}+\mathcal{L}_{2}\right)-4(g_{1}g_{2})^{1/4}\mathcal{V}(\mathcal{X})\right]. $$

(32)

For symmetry, each rank-2 field is considered as coupled to its own matter field with the respective Lagrangians $\mathcal {L}_{1,2}$. For the sake of simplicity, we consider the interaction term without derivative couplings. Assuming two metrics, the only possible combination is $\mathcal {X}_{\nu }^{\mu }=g_{1}^{\mu \alpha }g_{2\nu \alpha }$. $\mathcal {V}$ is a function of the 4 independent scalar operators $\tau _{n}=\text {tr}(\mathcal {X}^{n}) n=1,2,3,4$. We can include the cosmological term in our observable sector through a term in the effective potential as $\mathcal {V}_{\Lambda _{1}}=\Lambda _{1}q^{-1/4}$ with $q=\mathrm { det} \mathcal {X}=g_{2}/g_{1}$. The modified Einstein equations result

$$ M_{pl1}^{2}E_{1\nu}^{\mu}+Q_{1\nu}^{\mu}=\frac{1}{2}T_{1\nu}^{\mu} $$

(33)

$$ M_{pl2}^{2}E_{2\nu}^{\mu}+Q_{2\nu}^{\mu}=\frac{1}{2}T_{2\nu}^{\mu} $$

(34)

where Q _1,2 are the effective energy-momentum tensors induced by the interaction

$$ Q_{1\nu}^{\mu}=q^{1/4}\left[\mathcal{V}\delta_{\nu}^{\mu}-4(\mathcal{V}^{\prime}\mathcal{X})_{\nu}^{\mu}\right] $$

(35)

$$ Q_{2\nu}^{\mu}=q^{-1/4}\left[\mathcal{V}\delta_{\nu}^{\mu}+4(\mathcal{V}^{\prime}\mathcal{X})_{\nu}^{\mu}\right] $$

(36)

with $(\mathcal {V}^{\prime })_{\nu }^{\mu }=\partial \mathcal {V}/\partial \mathcal {X}_{\mu }^{\nu }$.

The following Bianchi identities are satisfied:

$$ g_{1}^{\alpha \nu}\nabla_{1\alpha}E_{1\mu\nu}=\nabla_{1}^{\nu}E_{1\mu\nu}=0,{\kern7pt}g_{2}^{\alpha \nu}\nabla_{2\alpha}E_{2\mu\nu}=\nabla_{2}^{\nu}E_{2\mu\nu}=0\,. $$

(37)

They come from the invariance of the Einstein-Hilbert terms under diffeomorphisms

$$ \delta g_{1\mu\nu}=2g_{1\alpha\left(\mu\nabla_{\nu}\right)}\epsilon^{\alpha},{\kern7pt}\delta g_{2\mu\nu}=2g_{2\alpha\left(\mu\nabla_{\nu}\right)}\epsilon^{\alpha}\,. $$

(38)

Being the interaction term also invariant, it is $\nabla _{1,2}^{\nu }Q_{1,2\mu \nu }=0$ on shell for g _2,1.

Regarding the asymptotic solutions, we expect that at infinity from the sources g _1,2 are maximally symmetric. Inserting

$$ M_{Pl 1,2}^{2}=E_{\mu\nu1,2}=\mathcal{K}_{1,2}g_{\mu\nu\,2} $$

(39)

with $-\mathcal {K}_{1,2}/4$, the constant scalar curvature of g _{μ ν1,2}, into the equations (33) and (34), we obtain

$$ 2\mathcal{V}+\left(q^{-1/4}\mathcal{K}_{1}+q^{1/4}\mathcal{K}_{2}\right)=0 $$

(40)

$$ 8(\mathcal{V}^{\prime}\mathcal{X})_{\nu}^{\mu}+\delta_{\nu}^{\mu}\left(q^{1/4}\mathcal{K}_{2}-q^{-1/4}\mathcal{K}_{1}\right)=0\,. $$

(41)

Assuming asymptotically flat spaces $\mathcal {K}_{1,2}=0$, (40) and (41) are reduced to $\mathcal {V^{\prime }}_{\mu }^{\nu }=0$ and $\mathcal {V}=0$. So, assuming that rotational symmetry is preserved and the same signature for the two metrics (without any ’twist’ of coordinates), the bi-flat background can be written in the form

$$ \bar{g}_{1\mu\nu}=\eta_{\mu\nu}=\text{diag}(-1,1,1,1)\,, $$

(42)

$$ \bar{g}_{2\mu\nu}=\omega^{2}\text{diag}\left(-c^{2},1,1,1\right), $$

(43)

where ω is the relative conformal factor, c parametrizes the speed of light in sector 2. If c = 1, the two metrics are linearly dependent and we can locally diagonalize both of them by a coordinate transformation. This is the case of Lorentz Invariant Bigravity. On the other hand, in the case c ≠ 1, this simultaneous diagonalization is not possible and then the Lorentz symmetry is spontaneously broken.

Considering the family of potentials

$$ \mathcal{V}=\alpha_{0}\mathcal{V}_{1}+\alpha_{1}\mathcal{V}_{2}+\alpha_{3}\mathcal{V}_{3}+\beta_{1}\mathcal{V}_{-1}+\beta_{2}\mathcal{V}_{-2}+\beta_{3}\mathcal{V}_{-3}+\beta_{4}\mathcal{V}_{-4}+q^{-1/4}\Lambda_{1}+q^{1/4}\Lambda_{2} $$

(44)

where we introduced the combinations of the scalars $\tau _{n}=\text {tr}(\mathcal {X}^{n})$

$$ \mathcal{V}_{0}=\frac{1}{24|g_{2}|}(\epsilon\epsilon g_{2}g_{2}g_{2}g_{2})=1=\frac{1}{24q}\left({\tau_{1}^{4}}-6\tau_{2}{\tau_{1}^{2}}+8\tau_{1}\tau_{3}+3{\tau_{2}^{2}}-6\tau_{4}\right) $$

(45)

$$ \mathcal{V}_{1}=\frac{1}{6|g_{2}|}(\epsilon\epsilon g_{2}g_{2}g_{2}g_{1})=\tau_{-1}=\frac{1}{6q}\left({\tau_{1}^{3}}-3\tau_{2}\tau_{1}+2\tau_{3}\right) $$

(46)

$$ \mathcal{V}_{2}=\frac{1}{2|g_{2}|}(\epsilon\epsilon g_{2}g_{2}g_{1}g_{1})=\left(\tau_{-1}^{2}-\tau_{-2}\right)=q^{-1}\left({\tau_{1}^{2}}-\tau_{2}\right) $$

(47)

$$ \mathcal{V}_{3}=\frac{1}{|g_{2}|}(\epsilon\epsilon g_{2}g_{1}g_{1}g_{1})=\left(\tau_{-1}^{3}-3\tau_{-2}\tau_{-1}+2\tau_{-3}\right)=6q^{-1}\tau_{1} $$

(48)

$$ \mathcal{V}_{4}=\frac{1}{|g_{2}|}(\epsilon\epsilon g_{1}g_{1}g_{1}g_{1})=\left(\tau_{-1}^{4}-6\tau_{-2}\tau_{-1}^{2}+8\tau_{-1}\tau_{-3}+3\tau_{-2}^{2}-6\tau_{-4}\right)=24q^{-1} $$

(49)

where $\mathcal {V}_{-n}=\mathcal {V}\left (\mathcal {X}\rightarrow \mathcal {X}^{-1}\right )$; we have a solution as (2) plus the second metric:

$$ d{s_{2}^{2}}=-Cdt^{2}+Adr^{2}+2Ddtdr+Bd\Omega^{2} $$

(50)

with

$$ C=c^{2}\omega^{2}\left[1-\frac{2Gm_{2}}{\kappa r}+\mathcal{K}_{2}r^{2}\right]-\frac{2G}{c\omega^{2}\kappa}Sr^{\gamma},\,\,\,\,\,D^{2}+AC=c^{2}\omega^{4} $$

(51)

$$ B=\omega^{2}r^{2},\,\,\,\,A=\omega^{2}\frac{\tilde{J}-\tilde{C}-\tilde{J}\tilde{S}r^{\gamma-2}}{\tilde{J}^{2}} $$

(52)

The forms of M,S in (2) can be obtained by matching the exterior solution with an interior star solution as obtained in [13]. Assuming the mass density of the object as a constant, the following conditions

$$ M=M_{0}\left[1-\frac{8\mu^{2}R^{2}}{5(\gamma+1)(\gamma-2)}\right],\,\,\,S=\frac{24\mu^{2}M_{0}R^{1-\gamma}}{(\gamma-4)(\gamma+1)(2\gamma-1)(\gamma-2)} $$

(53)

$$ \mu^{2}={m_{2}^{2}}\frac{3{m_{4}^{4}}-{m_{0}^{2}}\left({m_{2}^{2}}-4{m_{3}^{2}}\right)}{{m_{4}^{4}}-{m_{0}^{2}}\left({m_{2}^{2}}-{m_{3}^{2}}\right)},\,\,\,M_{0}=\frac{4}{3}\pi R^{3}\rho_{0} $$

(54)

hold. As a consequence, the bare mass M ₀ is renormalized by the presence of the extra-term in S coming from modified gravity. Clearly, M is the mass and S depends on the size of the self-gravitating body. The mass shift ΔM = M−M ₀ and S will have the sign of μ ² in the case γ < −1. This means that the corrections are proportional to the physical size of the body. For μ ² > 0, we have positive corrections M > M ₀; for μ ² < 0 we have negative corrections. The parameter μ ² is proportional to the graviton mass scale m ²; the same holds for S and ΔM; in the limit of $m\rightarrow 0$, the deviations go to zero and LBMG theory converges to General Relativity.

Note that as a consequence of our analysis in Section 2, positive fluctuations are the only one permitted: the bound S > 0 excludes negative fluctuations.