We present the Hamiltonian formalism for f (T ) gravity, and prove that the theory has n(n−3) 2 + 1 degrees of freedom (d.o.f.) in n dimensions. We start from a scalar-tensor action for the theory, which represents a scalar field minimally coupled with the torsion scalar T that defines the teleparallel equivalent of general relativity (TEGR) Lagrangian. T is written as a quadratic form of the coefficients of anholonomy of the vierbein. We obtain the primary constraints through the analysis of the structure of the eigenvalues of the multi-index matrix involved in the definition of the canonical momenta. The auxiliary scalar field generates one extra primary constraint when compared with the TEGR case. The secondary constraints are the super-Hamiltonian and supermomenta constraints, that are preserved from the Arnowitt-Deser-Misner formulation of GR. There is a set of n(n−1) 2 primary constraints that represent the local Lorentz transformations of the theory, which can be combined to form a set of n(n−1) 2 − 1 first-class constraints, while one of them becomes second-class. This result is irrespective of the dimension, due to the structure of the matrix of the brackets between the constraints. The first-class canonical Hamiltonian is modified due to this local Lorentz violation, and the only one local Lorentz transformation that becomes second-class pairs up with the second-class constraint π ≈ 0 to remove one d.o.f. from the n 2 + 1 pairs of canonical variables. The remaining n(n−1) 2 + 2n − 1 primary constraints remove the same number of d.o.f., leaving the theory with n(n−3) 2 + 1 d.o.f. This means that f (T ) gravity has only one extra d.o.f., which could be interpreted as a scalar d.o.f.

The Hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR) is developed from an ordinary second-order Lagrangian, which is written as a quadratic form of the coefficients of anholonomy of the orthonormal frames (vielbeins). We analyze the structure of eigenvalues of the multi-index matrix entering the (linear) relation between canonical velocities and momenta to obtain the set of primary constraints. The canonical Hamiltonian is then built with the Moore-Penrose pseudo-inverse of that matrix. The set of constraints, including the subsequent secondary constraints, completes a first class algebra. This means that all of them generate gauge transformations. The gauge freedoms are basically the diffeomorphisms, and the (local) Lorentz transformations of the vielbein. In particular, the ADM algebra of general relativity is recovered as a sub-algebra.Comment: 13 pages, no figures, comments welcom

Null tetrads are shown to be a valuable tool in teleparallel theories of modified gravity. We use them to prove that Kerr geometry remains a solution for a wide family of f (T ) theories of gravity.

We show that McVittie geometry, which describes a black hole embedded in a FLRW universe, not only solves the Einstein equations but also remains as a non-deformable solution of f (T ) gravity. This search for GR solutions that survive in f (T ) gravity is facilitated by a null tetrad approach. We also show that flat FLRW geometry is a consistent solution of f (T ) dynamical equations not only for T = −6H 2 but also for T = 0, which could be a manifestation of the additional degrees of freedom involved in f (T ) theories.

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