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

TL;DR

This paper completes the analysis of degrees of freedom in New General Relativity types, revealing specific counts for Types 4, 7, and 9, and discussing irregular systems and their constraints.

Contribution

It provides the Hamiltonian analysis of NGR Types 4, 7, and 9, including irregular systems, and clarifies their degrees of freedom and constraint structures.

Findings

Type 4 has five degrees of freedom.

Type 7 is a purely topological system with zero degrees of freedom.

Type 9 has three degrees of freedom.

Abstract

We investigate degrees of freedom in New General Relativity. This theory is the three-parameter extension of Teleparallel Equivalent to GR and classified into nine irreducible types according to the rotation symmetry $SO(3)$ on each leaf of ADM-foliation. In the previous work~[{\it Phys. Rev. D 112 (2025) 8, 084052}], we investigated the degrees of freedom in NGR types that are of interest in 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 and considering in detail the regularity of NGR, we perform the analysis of Type 4, Type 7, and Type 9. We reveal that the degrees of freedom of Type 4, Type 7, and Type 9 are five, zero (purely topological system in bulk spacetime), and three, respectively. Type 4 and Type 9 have second-class constraint densities only, whereas Type 7 has first-class constraint densities only. In every type, no bifurcation occurs. In particular, Type 4 and Type 7 are irregular and provide specific examples of handling irregular systems. Since no general method is known for treating an irregular system, this work contributes to furthering the understanding of irregular systems.