Degrees of Freedom of New General Relativity I: Type 2, Type 3, Type 5, and Type 8

We investigate the degrees of freedom of new general relativity. This theory is a three-parameter theory and is classified into nine irreducible types according to the rotation symmetry of $SO(3)$ on each leaf of ADM-foliation. In this work, we focus on unveiling the degrees of freedom of the physically interesting types of NGR: Type 2, Type 3, Type 5, and Type 8, which contain the gravitational propagating degrees of freedom. First, we revisit the theory based on the gauge approach to gravity and reformulate the Lagrangian of the theory. Second, we review the irreducible decomposition of the theory while focusing on the Hamiltonian and the primary constraints in each type. Third, we perform the Dirac-Bergmann analysis to unveil the degrees of freedom of the theory in the case of Type 2, Type 3, Type 5, and Type 8. We find a novel new behavior of constraints in Type 8, which is classified as second-class but not to determine any Lagrange multipliers and to provide the gauge invariance of the theory under the satisfaction of a specific condition of the multipliers. The degrees of freedom of Type 2, Type 3, and Type 5 are unveiled as six, five, and seven, respectively. The degrees of freedom of Type 8 is either four under a specific condition to the Lagrange multipliers or six in the generic case. Finally, we conclude this work with five future perspectives.