Hamiltonian formalism for f(T) gravity

Rafael Ferraro, María José Guzmán

Published 2018-02-06Version 1

We present the Hamiltonian formalism for $f(T)$ gravity, and prove that the theory has $\frac{n(n-3)}{2}+1$ degrees of freedom in $n$ dimensions. We start from a scalar-tensor action for the theory, that represents a scalar field minimally coupled with the torsion scalar $T$ that defines the 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 auxiliar scalar field generates one extra primary constraint when compared with the TEGR case. The secondary constraints are the super-Hamiltonian and super-momenta constraints, that are preserved from the ADM 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 become 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 degree of freedom from the $n^2+1$ pairs of canonical variables. The remaining $2n-1+\frac{n(n-1)}{2} - 1 $ primary constraint remove the same number of degrees of freedom, leaving the theory with $\frac{n(n-3)}{2}+1$ degrees of freedom. This means that $f(T)$ gravity has only one extra degree of freedom, which could be interpreted as a scalar degree of freedom.