Birkhoff’s theorem in f(T) gravity up to the perturbative order

Abstract

f(T) gravity, a generally modified teleparallel gravity, has become very popular in recent times, as it is able to reproduce the unification of inflation and late-time acceleration without the need of a dark energy component or an inflation field. In this present work, we investigate specifically the range of validity of Birkhoff’s theorem with the general tetrad field via perturbative approach. At zero order, Birkhoff’s theorem is valid and the solution is the well known Schwarzschild–(A)dS metric. Then considering the special case of the diagonal tetrad field, we present a new spherically symmetric solution in the frame of f(T) gravity up to the perturbative order. The results with the diagonal tetrad field satisfy the physical equivalence between the Jordan and the so-called Einstein frames, which are realized via conformal transformation, at least up to the first perturbative order.

Appendix

Here, we deduce in detail the variation of the action for a simply scalar torsion T with respect to vierbein. Considering the basic relation e ^α _i e _β ⁱ=δ ^α _β in teleparallel gravity, we can define the algebraic complement C ^α _i of e _α ⁱ,

(58)

(59)

So we get C ^α _i =ee ^α _i and the variation of metric with respect to e _α ⁱ

(60)

and

(61)

Redefining the energy–momentum tensor formula of e _α ⁱ and $e = \sqrt{-g}$

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Maybe a more suitable form for this work is

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Differing from that in Einstein’s theory of general relativity, teleparallel gravity uses Weitzenböck connection, defined directly from the vierbein (3) and antisymmetric non-vanishing torsion (4).

Then we can deduce the variation of the torsion tensor with respect to e _α ⁱ and ∂ _μ e _α ⁱ, respectively,

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and the coupling with $\tilde{\partial}^{\mu}\varphi$

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The other two tensors are defined by (6) and (7). Then the torsion scalar as the teleparallel Lagrangian is defined by (8). One needs to pay attention to the S _ρ ^μν being a polynomial combination of the product of g ^μν and T ^ρ _μν , like

$$ S_\rho{}^{\mu\nu} = \sum g^{a b} \cdot T^{c}{}_{d e}. $$

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The indices a,b,c,d,e of the above definition are dummy indices. After summation of these five dummy indices, only ρ,μ,ν are left. Then we can use a step-by-step method for the binomial formula to deduce the variation of the torsion scalar with respect to e _α ⁱ and ∂ _μ e _α ⁱ, respectively,

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and

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Finally, we can get the variation equation (11) of the action (10) with respect to the vierbein.