Kaluza-Klein theory for teleparallel gravity

We study teleparallel gravity in the \emph{original} Kaluza-Klein (KK) scenario. Our calculation of the KK reduction of teleparallel gravity indicates that the 5-dimensional torsion scalar $^{(5)}T$ generates the non-Brans-Dicke type effective Lagrangian in 4-dimension due to an additional coupling between the derivative of the scalar field and torsion, but the result is equivalent to that in general relativity. We also discuss the cosmological behavior in the FLRW universe based on the effective teleparallel gravity.

It is convenient to calculate torsion by the Cartan structure equation T I = dθ I . The nonvanishing components of the torsion 2-form in the orthonormal frame areT k ij ,T k 5j and T 5 i5 . The 5-dimensional torsion scalar (5) T can be given in a similar way as Eq. (2.1b) in 5-dimension, decomposed as [12] whereT is the induced 4-dimensional torsion scalar.

B. Kaluza-Klein Theory
In the original Kaluza-Klein theory, the 5-dimensional spacetime bulk V = M × S 1 is a product space of a hypersurface M and a circle S 1 with small radius r representing the compactified extra dimension. Because of the embedding, we can always write the 5dimensional metricḡ in the coordinate system as where y = x 5 and ε = −1. In the orthonormal frame, the vielbein fields are e i µ and e5 5 = φ(x µ , y). Consequently, the 5-dimensional torsion scalar in the coordinate frame is whereT = T is a pure 4-dimensional object due toT ρ µν = T ρ µν . Note that as there only exists the KK zero mode in the effective low-energy theory based on the cylindrical condition, all fields have no dependence on the 5th-dimensional component, i.e., g µν,5 = 0, which is referred to as KK ansatz [3]. Within the ansatz, the metric is reduced tō The residual components of the non-vanishing torsion tensor are T ρ µν andT 5 which can be used to construct the 5-dimensional action. The extra dimension y = r θ is the ground state ofḡ 55 with the radius r. The invariant volume element is (5) e d 5 x = e φ r dθ d 4 x by the dimensional reduction with (5) e = det(e I M ). As a result, we obtain the 4-dimensional effective action where the effective coupling κ 4 := κ 5 /2πr = 8π G 5 /2πr and T µ := T σ σ µ is the torsion trace vector. Hence, we have the KK reduction procedure in teleparallel gravity which replaces the 5-dimensional torsion scalar (5) For F (T ) gravity in the KK theory, where F (T ) is an arbitrary function of T , the La- (2.9) By taking F ( (5) T ) to be linear in the torsion scalar with a cosmological constant, i.e., respectively. By choosing φ = Ω 2 , the action reads where ψ = (6/κ 4 ) 1/2 ln Ω is a dilaton field. As seen from Eq. It is interesting to note that in the 5-dimensional general relativity, the effective action is (−1/2κ 4 ) d 4 x √ −g φR which is a specific case of the Brans-Dicke theory [3,14,15]. It is clear that the form of the effective action in Eq. (2.8) is not a scalar-tensor-like due to the additional non-minimal coupling 2 T µ ∂ µ φ, which is different from the reduction action in general relativity although it is still equivalent to the Lagrangian φR up to the total derivative term as shown in Eq. (2.13). We also note that our action in Eq. (2.9) is clearly different from Eq. (5) in Ref. [11], in which the function variable is T + φ −2 ∂ µ φ ∂ µ φ without coupling between T and φ instead of T + 2 φ −1 T µ ∂ µ φ.
The equation of motion of the scalar field φ is where Γ ν νµ = e ν i ∂ µ e i ν with the non-vanishing component being e α a ∂ 0 e a α = 3 (ȧ/a) in the coordinate. Subsequently, we obtain aä +ȧ 2 = 0. The equation of motion is independent of the scalar field and the scale factor can be solved directly. By taking the solution to be proportional to t m , we find that m = 0 and 1/2, leading to For (i), we obtain a = a s , which corresponds to a static universe, where a s is the scale factor of the valid energy scale for the effective teleparallel gravity in Eq. (2.8). For (ii), we get a = b √ t. In this case, t has to be larger than t cut which represents the cut-off energy scale for the low-energy effective teleparallel gravity. Consequently,ä = −b/(4 t 2/3 ) and H = 1/(2 t). For b < 0 (> 0), the universe is accelerated (decelerated) expanding.
Comparing to the effective Lagrangian in general relativity, the equation of motion of φ isR(e) = 0, which also leads to the same solution for the scale factor. We conclude that teleparallel gravity and general relativity are equivalent in the KK scenario.
Finally, we remark that the KK reduction generates an effective low-energy theory that is improper to be applied to the inflationary stage. At the high energy scale, it has to consider the KK modes of gravitational and scalar fields, i.e., there exist massive gravitational and scalar fields. Clearly, the situation is much more complicated than the one discussed in Ref. [11].

IV. CONCLUSIONS
We have examined the KK reduction of telaparallel gravity. Our result in Eq. (2.8) has shown that there is a coupled term between the derivative of the scalar field and torsion trace vector, which implies that the KK reduction procedure can not be applied in telaparallelism to obtain the Brans-Dicke type theory. The effective Lagrangian is different from general relativity although it is equivalent to φR up to the total derivative term. The property of the additional coupling leads to an Einstein frame by the conformal transformation for the nonminimal coupled teleparallel gravity, which is different from the result in the literature [11].
In cosmology, we have obtained the equation of motion in the FLRW universe and found that the accelerated expansion of the universe can be achieved by the effective teleparallel gravity, which is the same as the effective KK scenario in general relativity.