Post-Newtonian Approximation of Teleparallel Gravity Coupled with a Scalar Field

We use the parameterized post-Newtonian (PPN) formalism to explore the weak field approximation of teleparallel gravity non-minimally coupling to a scalar field $\phi$, with arbitrary coupling function $\omega(\phi)$ and potential $V(\phi)$. We find that all the PPN parameters are identical to general relativity (GR), which makes this class of theories compatible with the Solar System experiments. This feature also makes the theories quite different from the scalar-tensor theories, which might be subject to stringent constraints on the parameter space, or need some screening mechanisms to pass the Solar System experimental constraints.


I. INTRODUCTION
more general scalar-tensor theories and f (R) theories, the well-known Chameleon mechanism is invoked to screen the fifth force [72][73][74][75], and hence they have no significant deviation from GR on small scale, while they can still drive the acceleration of the universe on cosmological scale. Similarly, the Vainshtein mechanism [89][90][91] and the Symmetron mechanism [92,93] are also extensively invoked in other types of modified gravity theories to pass the local tests in Solar System. Motivated by the above discussions, it is necessary and worth to explore the weak field behaviours of modified gravities. Recently, the PPN parameters for the teleparallel dark energy model have been explicitly calculated in [63], and it is claimed that the potential of the scalar field has no effect on PPN parameters and hence this model can be compatible with the local tests in Solar System. Note that in [63] the coupling is chosen to be a particular form. In the present work, we try to generalize the work of [63] and explore the weak field approximation of teleparallel gravity non-minimally coupling to a scalar field φ with arbitrary coupling function ω(φ) and potential V (φ), by explicitly calculating the corresponding PPN parameters. This paper is organized as follows. We give a brief review of teleparallel gravity in Sec. II. Next, we present the action functional for the teleparallel gravity coupled with a scalar field and derive the corresponding field equations in Sec. III. We then expand the field equations to sufficient orders and solve the perturbations to obtain the post-Newtonian approximation in Sec. IV. Finally, some concluding remarks are given in Sec. V.

II. TELEPARALLEL GRAVITY
Here we give a brief review of teleparallel gravity. Teleparallel gravity uses a vierbein field e a = e µ a ∂ µ as dynamical quantity, with Latin indices a, b, · · · = 0, · · · , 3, and i, j, · · · = 1, · · · , 3, Greek indices µ, ν, · · · = 0, · · · , 3, and ∂ µ coordinate bases. The vierbein is an orthonormal basis for the tangent space at each point x µ of the manifold, namely e a · e b = η ab , with η ab = diag (−1, 1, 1, 1). Then the metric tensor can be expressed in the dual vierbein e a µ as Rather than using the torsionless Levi-Civita connection in GR, teleparallel gravity uses the Weitzenböck connection Γ λ µν [46], which is defined by Note that the lower indices µ and ν are not symmetric in general, thus the torsion tensor (will be defined below) is non-vanishing in the teleparallel spacetime. The Weitzenböck torsion tensor is defined by In teleparallel gravity, the gravitational action is given by the torsion scalar instead of the the Ricci scalar in GR. The torsion scalar is basically the square of the Weitzenböck torsion tensor, and reads with the super-potential tensor S ρ µν defined by The gravitational field is driven by the torsion scalar T , and the action reads where e = det (e a µ ) = √ −g and κ 2 = 8πG N , with g the determinant of the metric g µν and G N the Newtonian constant. Note that we have used the units in which the speed of light c = 1, and the reduced Planck constant = 1. S m [e µ a , χ m ] is the matter part of the action, and χ m denotes all matter fields collectively.

III. TELEPARALLEL GRAVITY WITH A SCALAR
We will study the theories of teleparallel gravity coupled with a scalar in which gravity is described by a dynamical scalar φ in addition to the vierbein e µ a . Without loss of generality, we consider the Brans-Dicke-like theories, whose actions are given by where the coupling function ω(φ) and the potential V (φ) are two arbitrary functions of φ. At first glance, one might consider that this action is not so general. In fact, we can make it more familiar. Introducing a new scalarφ according to (∂φ) 2 = −ω(φ)(∂φ) 2 /(κ 2 φ), Eq. (3.1) can be recast as Obviously, ifω(φ) = 1 + ξκ 2φ2 , Eq. (3.2) reduces to the action considered in [63]. So, the action (3.1) is general enough in fact (see Sec. V for further discussion). The variation of the action (3.1) with respect to the scalar field φ yields where a prime denotes a derivative with respect to φ, and = g µν ∇ µ ∇ ν is the d'Alembert operator, with ∇ µ the covariant derivative associated with the Weitzenböck connection. The variation of the action (3.1) with respect to the dual vierbein e a ν yields where we have used the vierbein (or dual vierbein) to switch from Latin to Greek indices and back, for example T ν µ = e a µ T ν a . Taking the trace of Eq. (3.5) yields with T = T µ µ . Multiplying Eq. (3.6) by −δ ν µ /2 , then adding Eq. (3.5), we get The gravitational fields are truly governed by the field equations (3.3) and (3.7). We will expand these two equations in the post-Newtonian approximation in the following section.

IV. POST-NEWTONIAN APPROXIMATION
The post-Newtonian approximation of GR on the behavior of hydrodynamic systems has been systematically investigated in e.g. [79]. In analogy to [79], we assume that the gravitating source matter is contributed by a perfect fluid which obeys the post-Newtonian hydrodynamics. We will use the PPN formalism to expand the field equations (3.3) and (3.7) perturbatively by assigning appropriate orders of magnitude to all dynamical variables appearing in the field equations. The resulting perturbation equations can then be subsequently solved order by order.

A. General framework
Conventionally, the velocity of the source matter | v| characterize the smallness of the system. So, we will perturbatively expand all dynamical quantities in orders of O(n) ∼ | v| n . We will firstly find out the perturbations for the vierbein following [63], and then expand the energy-momentum tensor to sufficient orders. Finally, the perturbations of all functions of φ are obtained by using Taylor expansion.
For the gravitational sector, we expand the dual vierbein fields around the flat background as where each term B (n) a µ is of order O(n). By using Eq. (2.1), this decomposition gives the usual metric as an expansion around the flat Minkowski background, where η µν is the Minkowski metric and each symmetric term (n) h µν is of order O(n). For our purpose, it is sufficient to expand the metric up to the order of O(4). A detailed analysis (see e.g. [80]) shows that (1) h µν = 0, which corresponds to B (1) a µ = 0 (nb. Eq. (2.1)), and the only non-vanishing components of the metric perturbations are (2) h 00 , (2) We now can raise and lower the spacetime indices of the perturbations of vierbein (or dual vierbein) by the Minkowski metric η µν , As a result, B µν is symmetric, and the non-vanishing components are In addition, (2) B ij is diagonal [63]. For convenience, we introduce a time-independent function A, such that (2) B ij = Aδ ij . We also give the relations between the metric perturbations and the vierbein perturbations [63], (2) h 00 = 2 (2) B 00 , (4.6a) From the definitions, we see that T ρ µν and S ρ µν are at least O(2) quantities, and the torsion scalar T is an at least O(4) quantity.
The energy-momentum tensor of a perfect fluid takes the form where ρ, Π, p and u µ are the energy density, the specific internal energy, the pressure, and the fourvelocity of the fluid, respectively. Note that the velocity of the source matter is given by v i = u i /u 0 . We assign the velocity orders ρ ∼ Π ∼ O(2), and p ∼ O(4) by considering their orders of magnitude in the Solar System [80]. Then we get the perturbations of energy-momentum tenor in Eq. (4.7) as We also note that T = g µν T µν = −ρ − ρΠ + 3p. In addition, we assume the gravitational field is quasistatic, so the time derivative ∂ 0 = ∂/∂t of the vierbein or other fields are weighted with an additional velocity order O(1).
For the scalar field φ, we expand it around its cosmological background value φ 0 , where we assume φ 0 to be of order O(0) and the perturbations ψ (n) are of order O(n) as usual. We also need to expand the functions ω(φ) and V (φ) around φ 0 . To this end, we expand them using Taylor expansion to sufficient orders, . We assume all these expansion coefficients to be of order O(0). We also give the expansion of ω ′ and V ′ for further convenience,

B. Solving the perturbed equations
Here we will solve the perturbed equations order by order. We refer to the Appendix A for a detailed computation of the corresponding quantities up to the appropriate orders. Below we just give the results.
Expanding Eqs. for the scalar field perturbation ψ (2) , where ∇ 2 = δ ij ∂ i ∂ j and m ψ = 2κ V2φ0 2ω0 . Eq. (4.11) is a screened Poisson equation. Since we demand that φ to take its cosmological value at large scale, which is equivalent to saying that the perturbation should vanish at cosmological distance due to the absence of the gravitational field and the matter source, i.e., ψ (2) → 0 as r → ∞ (r is the distance from the Sun), we get the solution of Eq. (4.11) as ψ (2) = 0. (4.12) In order to get the corresponding vierbein perturbations, we use the ansatz where γ(r) is a PPN parameter measuring the amount of space curvature produced by unit rest mass [80]. We also adopt the gauge conditions for the vierbein perturbation B µ ν as [81] We should verify the consistency of these gauge conditions after obtaining the solutions. Actually, as we will see later, our results are identical to GR, so these conditions are just the Newtonian continuity equations [82], and are satisfied automatically. Expanding (0, 0) component of Eq. (3.7) to O(2), we get in which the gravitational potential U is defined by The solution to this equation is Taking the trace of Eq. (4.19) yields Noting that T = 2∂ i A∂ i A (see Eq. (A26)), the above equation can be simplified to where we have used the identity and Φ 2 is defined by Eq. (4.23) is a screened Poisson equation and can be solved by The solution to this equation is with V i and W i defined as in [79], and Expanding (0, 0) component of Eq. (3.7) to O(4), we obtain The solution to this equation is where Φ 1 , Φ 3 , and Φ 4 are defined as in [80], In summary, we get the corresponding metric perturbations as From above equations, it is easy to see that the effective Newtonian constant G eff = G N /φ 0 , and the PPN parameter β(r) is given by We note that the PPN parameter β(r) measures the amount of "non-linearity" in the superposition law for gravity [80]. Notice that Eqs. (4.21) and (4.35) are the main results of this work.

V. CONCLUSIONS AND DISCUSSIONS
We have studied the post-Newtonian approximation of teleparallel gravity coupling to a scalar field φ with arbitrary coupling function ω(φ) and arbitrary potential V (φ). We have chosen a frame in which the Sun is at rest in both the coordinate system and the orthonormal bases, such that the vierbein (dual vierbein) can be perturbatively expanded around the flat spacetime, which leads to the usual expanding of the metric around the Minkowski spacetime. The functions ω(φ) and V (φ) are characterized by the coefficients of Taylor expansion. Interestingly, the only non-vanishing PPN parameters β and γ are all equal to 1, indicating that these models are indistinguishable from GR in the Solar System distance up to the post-Newtonian order. In addition, we can rescale the cosmological background value φ 0 of the scalar to φ 0 = 1, and then G eff = G N . Since the rescaling can be done globally, we conclude that the effective Newtonian constant has also no contribution to the Solar System experiments.
This feature makes the theories we studied quite different from the scalar-tensor theories, which might be subject to stringent constraints on the parameter space, or need some screening mechanisms to pass the Solar System experimental constraints. We might conclude that the coupling between the scalar field and the torsion scalar in teleparallel gravity is less strong as that between the scalar and the Ricci scalar in GR. This can be seen from the relationship between the torsion scalar constructed from the Weitzenböck connection and the Ricci scalar constructed from the Levi-Civita connection [84], in which D µ is associated with the Levi-Civita connection. Although the second term on the right hand side of Eq. (5.1) is a boundary term in the TEGR case, it will be nontrivial when a scalar field φ is coupled to the torsion, which makes the theories quite different from the scalar-tensor theories. In addition, T is at least O(4), while R is at least O(2) when perturbated around the flat spacetime. This fact makes the gravitational sector have no effect on the ψ (2) when Eq. (3.3) is expanded up to O(2), thus leading to the PPN parameter γ(r) equals to 1. This indirect coupling between the scalar field and the gravitational sector is the meaning of less strong coupling we proposed. One might note that the action (3.1) considered in this work could be further generalized to However, it is an illusion. Introducing a new scalarφ = ξ(φ), Eq. (5.3) can be recast aŝ which reduces to the action (3.1) actually. So, the conclusions do not change for the action (5.3). This indicates that the action (3.1) considered in this work is general enough. Finally, from the viewpoint of symmetry, black holes have similar environments like the Solar System. So, we might speculate our theories will have the same solutions as GR when applying to black holes. Thus, it would be interesting to study the black hole solutions in the future works.
order. Note that the ansatz (4.13) is equivalent to (2) where each term C (n) µ a is of order O(n). Noting that g µν (x) = η ab e µ a (x)e ν b (x) and using Eq. (2.1), we can easily get C We then expand the torsion tensor T λ µν up to O(4), For convenience, we also present the definition of the super-potential tensor S ρ µν here, In addition, we use the anti-symmetric properties of the torsion tensor T λ µν and the super-potential tensor S ρ µν to simplify our calculations. Since the space-space component of metric g ij is expanded around the usual Euclidean metric δ ij , we do not distinguish the upper indices and the lower indices of the perturbation quantities up to appropriate order. Instead, we use the upper indices and the lower indices interchangeably, e.g. B (2) i j = B (2) ij = B (2) j i = B (2) ij , up to O(2).

Up to O(2)
The expansion of torsion tensor to O(2) can be read from Eq. (A3) as Some of its components can be obtained directly, And the expansion for some components of the super-potential tensor S ρ µν is also obtained, and The expansion of torsion tensor to O(3) can be read from Eq. (A3) as Some of its components read and When we derive above equations, the gauge conditions (4.14) have been used. Some components of the super-potential read and

Up to O(4)
The expansion of torsion tensor to O(4) can be read from Eq. (A3) as which can directly lead to and (2) α i + B (2) j α ∂ i B (2) α j = ∂ j B (2) j i − ∂ i B (2) j j − B (2) j k ∂ j B (2) k i + B (2) j The components of the super-potential for our interest are also given, and Finally, we expand the torsion scalar T up to O(4) as (A26)