Variational Problem and Bigravity Nature of Modified Teleparallel Theories

Variational Problem and Bigravity Nature of Modified Teleparallel Theories Martin Krˇsˇsák∗ Institute of Physics, University of Tartu, W. Ostwaldi 1, Tartu 50411, Estonia August 15, 2017 Abstract We consider the variational principle in the covariant formulation of modified teleparallel theories with second order field equations. We vary the action with respect to the spin connection and obtain a consistency condition relating the spin connection with the tetrad. We argue that since the spin connection can be calculated using an additional reference tetrad, modified teleparallel theories can be interpreted as effectively bigravity theories. We conclude with discussion about the relation of our results and those obtained in the usual, non-covariant, formulation of teleparallel theories and present the solution to the problem of choosing the tetrad in f(T ) gravity theories. 1 Introduction Teleparallel gravity is an alternative formulation of general relativity that can be traced back to Einstein’s attempt to formulate the unified field theory [1–9]. Over the last decade, various modifications of teleparallel gravity became a popular tool to address the problem of the accelerated expansion of the Universe without invoking the dark sector [10–38]. The attractiveness of modifying teleparallel gravity–rather than the usual general relativity–lies in the fact that we obtain an entirely new class of modified gravity theories, which have second arXiv:1705.01072v3 [gr-qc] 13 Aug 2017 order field equations. See [39] for the extensive review. A well-known shortcoming of the original formulation of teleparallel theories is the problem of local Lorentz symmetry violation [40, 41]. This is particularly serious in the modified case, where we often encounter situations that out of two tetrads related by a local Lorentz transformation, only one solves the field equations. Since both tetrads correspond to the same metric, it is not the metric tensor but a specific tetrad that solves a problem in modified teleparallel theories. This introduces a new complication of how to choose the tetrad. As it turns out, the tetrad must be chosen in a specific way or otherwise the field equations lead us to the condition ∗ Electronic address: [email protected] 1 fT T = 0, which restricts the theory to its general relativity limit. Tetrads that lead to this restriction were nicknamed bad tetrads, while those that avoid it and lead to some interesting new dynamics as good tetrads [42, 43]. An intriguing sub-class of good tetrads are those that lead to T = const, which can be used to show that the solutions of the ordinary GR are the universal solutions of an arbitrary f(T ) gravity [44, 45]. Recently [46], it was shown that local Lorentz invariance can be restored in modified teleparallel theories, if the starting point is a more general formulation of teleparallel gravity [47–54], in which teleparallelism is defined by a condition of vanishing curvature. This introduces the purely inertial spin connection to the theory that can be calculated consistently in the ordinary teleparallel gravity [52, 53]. However, in the modified case it was argued to be possible only to “guess” the connection by making some reasonable assumptions about the asymptotic properties of the ansatz tetrad [46]. While it was shown to work, and even to be rather easily achievable, in many physically interesting cases [46, 55–57], it is obviously not a satisfactory situation and better understanding of covariant modified teleparallel theories is needed. In this paper, we discuss various approaches to the the variational principle and derive the condition (27) relating the spin connection with tetrad by varying the action with respect to the spin connection. We argue that in many situations the solutions of this condition correspond to the “guesses” discussed in Ref. [46]. We further obtain understanding of why T = const can be used to show the universality of general relativity solutions in f(T ) gravity. We show that all these results can be always interpreted in the original, non-covariant, formulation and explain the physical origin of why some tetrads are “good”, and some other “bad”. Moreover, we argue that we can view modified teleparallel gravity theories as effectively bigravity theories with two tetrads, where the first tetrad determines the spacetime metric, while the second tetrad generates the spin connection and corresponds to the non-dynamical, “reference”, metric. Notation: We follow notation, where the Latin indices a, b, ... run over the tangent space, while the Greek indices µ, ν, ... run over spacetime coordinates. The spacetime indices are raised/lowered using the spacetime metric gµν , while the tangent indices using the Minkowski metric ηab of the tangent space. 2 Covariant formulation of teleparallel theories Teleparallel theories are formulated in the framework of tetrad formalism, where the funda- a mental variable is the tetrad, h µ, related to the spacetime metric through the relation a b gµν = ηabh µh ν . (1) The parallel transport is determined by the spin connection, which is fully characterized by its curvature and torsion in the metric compatible case. In general relativity, we use the connection with vanishing torsion known as the Levi-Civita connection. In teleparallel a gravity, we follow a complimentary approach and define the teleparallel connection, ω bµ, by the condition of zero curvature a a a a a c a c R bµν (ω bµ)= ∂µω bν − ∂ν ω bµ + ω cµω bν − ω cνω bµ ≡ 0. (2) 2 The most general connection satisfying this constraint is the pure gauge-like connection [47] a a c ω bµ = Λ c∂µΛb , (3) c −1 c where Λb = (Λ ) b. The torsion tensor of this connection a a a a a a b a b T µν(h µ,ω bµ)= ∂µh ν − ∂ν h µ + ω bµh ν − ω bνh µ, (4) is in general non-vanishing, and under the local Lorentz transformations ′a a b ′a a c d a c h µ = Λ bh µ, and ω bµ = Λ cω dµΛb + Λ c∂µΛb , (5) transforms covariantly. Therefore, this approach to teleparallel theories is called the covariant approach [52, 46]; in contrast with the more common non-covariant approach that will be discussed in section 6. The Lagrangian of teleparallel gravity is given by [8] h L (ha ,ωa )= T, (6) TG µ bµ 4κ a where h = det h µ, κ = 8πG is the gravitational constant (in c = 1 units), and we have defined the torsion scalar a a a µν T = T (h µ,ω bµ)= T µνSa , (7) ρσ where Sa is the superpotential 1 S µν = (T νµ + T µν − T µν ) − h νT σµ + h µT σν . (8) a 2 a a a a σ a σ A straightforward calculation shows that the teleparallel Lagrangian (6) and the Einstein- Hilbert one are equivalent up to a total derivative term [8] ◦ h µ LTG = LEH − ∂µ T , (9) κ from where follows that the field equations of general relativity and teleparallel gravity are equivalent. Since the Lagrangian (6) contains only the first derivatives of the tetrad, we can straight- forwardly construct modified gravity theories with second order field equations. The most popular of these theories is the so-called f(T ) gravity [10–13] defined by the Lagrangian h L (ha ,ωa )= f(T ), (10) f µ bµ 4κ where f(T ) is an arbitrary function of the torsion scalar (7). The variation with respect to the tetrad yields the field equations [46] µ a a µ a Ea (h µ,ω bµ)=Θa (h µ, Φ), (11) 3 where the term on the right-hand side is the energy-momentum tensor defined by a µ 1 δLM(h µ, Φ) Θa = a . (12) h δh µ a and LM(h µ, Φ) is the matter Lagrangian not depending on derivatives of the tetrad. On the left-hand side we have introduced a shortened notation for the Euler-Lagrange expression a a µ a a κ δLf (h µ,ω bµ) Ea (h µ,ω bµ) ≡ a , (13) h δh µ where − 1 E µ(ha ,ωa ) ≡ f S µν ∂ T +h 1f ∂ (hS µν)−f T b S νµ+f ωb S νµ+ f(T )h µ, (14) a µ bµ T T a ν T ν a T νa b T aν b 4 a and fT and fT T denote first and second order derivatives of f(T ) with respect to the torsion scalar T . The field equations of the ordinary teleparallel gravity are recovered in the case f(T )= T . 3 Variational problem in teleparallel gravity The difficulty of the variational problem in general relativity is related to the presence of the second derivatives of the metric tensor (or tetrads in tetrad formalism) in the Einstein- Hilbert action. The straightforward variation of the action leads to the field equations that are satisfied only if we require vanishing variations of both the metric and their normal derivatives on the boundary. To satisfy both these requirements simultaneously is too strong condition that leads to the variation problem that is not well-posed, what is usually solved by adding the boundary term to the action [58–60]. This problem is absent in teleparallel gravity as the teleparallel action contains just the first derivatives of the tetrad. However, we face a new difficulty since the action depends on the tetrad and the spin connection, and needs to be varied with respect to both. The straightforward variation of the teleparallel action with respect to the spin connection as independent variable does not take into account the teleparallel constraint (2), and can be shown to be inconsistent [61, 53, 54]. A viable solution is to enforce the teleparallel constraint (2) through the Lagrange multiplier [62, 61, 50, 54], but this introduces additional gauge symmetries associated with the multiplier and the theory becomes significantly more complicated and hard to handle [63, 64].

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