Quest for the extra degree of freedom in f(T) gravity

Abstract: 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.