We present the Hamiltonian formalism for $f(T)$ gravity, and prove that the theory has $\frac{n(n-3)}{2}+1$ 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 $\frac{n(n-1)}{2}$ primary constraints that represent the local Lorentz transformations of the theory, which can be combined to form a set of $\frac{n(n-1)}{2}-1$ 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 $\pi \approx 0$ to remove one d.o.f. from the $n^2+1$ pairs of canonical variables. The remaining $\frac{n(n-1)}{2}+2n-1$ primary constraints remove the same number of d.o.f., leaving the theory with $\frac{n(n-3)}{2}+1$ 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.

Spherically symmetric solutions of theories of gravity built one fundamental class of solutions to describe compact objects like black holes and stars. Moreover, they serve as starting point for the search of more realistic axially symmetric solutions which are capable to describe rotating compact objects. Theories of gravity that do not possess spherically symmetric solutions which meet all observational constraints are easily falsified. In this article, we discuss classes of exact and perturbative spherically symmetric solutions in f(T,B)-gravity. The perturbative solutions add to the ones which have already been found in the literature, while the exact solutions are presented here for the first time. Moreover, we present general methods and strategies, like generalized Bianchi identities, to find spherically solutions in modified teleparallel theories of gravity.

The Hamiltonian formulation of the teleparallel equivalent of general relativity 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 pseudoinverse 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 Arnowitt, Deser, and Misner algebra of general relativity is recovered as a subalgebra.

It has recently been shown that $f(T)$ gravity has $\frac{n(n-3)}{2}+1$ physical degrees of freedom (d.o.f.) in $n$ dimensions, contrary to previous claims. The simplest physical interpretation of this fact is that the theory possesses a scalar d.o.f. This is the case of $f(R)$ gravity, a theory that can be understood in the Einstein frame as general relativity plus a scalaron. The scalar field that represents the extra d.o.f. in $f(T)$ gravity encodes information about the parallelization of the spacetime, which is detected through a reinterpretation of the equations of motion in both the teleparallel Jordan and Einstein frames. The trace of the equations of motion in $f(T)$ gravity shows the propagation of the scalar d.o.f., giving an accurate proof of its existence. We also provide a simple toy model of a physical system with rotational pseudoinvariance, like $f(T)$ gravity, which gives insights into the physical interpretation of the extra d.o.f. We discuss some implications and unusual features of the previously worked out Hamiltonian formalism for $f(T)$ gravity. Finally we show some mathematical tools to implement the Hamiltonian formulation in the Einstein frame of $f(T)$ gravity, which exhibits some problems that should be addressed in future works.

Boosted and rotated tetrad backgrounds for the Minkowski space are studied in $f(\mathbb{T})$ gravity. We perform Lorentzian perturbations at first order around nontrivial backgrounds and show that some Lorentz modes can exhibit nontrivial dynamics and can propagate in time. This remarkable feature gives evidence of additional mode(s) in the Lorentzian sector which have no precedent in Lorentz-violating modified gravities; however, they can cast doubts onto even theoretical viability of these models.

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.