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

Abstract

It has recently been shown that f(T) gravity has 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.