Degrees of Freedom of New General Relativity 2: Type 4, Type 7, and Type 9

Abstract: 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 the previous work~[{\it 2410.15056[gr-qc]}], we investigated the degrees of freedom in types of NGR that are interests for describing gravity: Type 2, Type 3, Type 5, and Type 8. In this work, we focus on unveiling those numbers in all other types to complete the analysis of NGR. After providing the Hamiltonian formulation of NGR, we perform the analysis on Type 4, Type 7, and Type 9 according to the method that is provided in the previous work~[{\it 2410.15056[gr-qc]}]. Then we unveil that the degrees of freedom of Type 4, Type 7, and Type 9 are five, null, and three, respectively. Type 4 and Type 9 have second-class constraint densities only. Type 7 has first-class constraint densities only, but which is over constraint. In every type, no bifurcation occurs unlikely to Type 8 in the previous work~[2410.15056[gr-qc]]. Finally, we summarize this work and give a concluding remark for this series of works.