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

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.

💡 Research Summary

This paper completes the systematic Hamiltonian analysis of New General Relativity (NGR), a three‑parameter extension of the Teleparallel Equivalent of General Relativity (TEGR). NGR can be classified into nine irreducible types according to the SO(3) decomposition of the canonical momentum on each ADM leaf. In a previous work (Phys. Rev. D 112 (2025) 084052) the authors studied the degrees of freedom (DoF) of Types 2, 3, 5, and 8, which contain propagating tensor modes. The present work turns to the remaining three types—Type 4, Type 7, and Type 9—thereby providing a full account of the DoF spectrum of NGR.

Hamiltonian formulation and primary constraints
The configuration variables are the lapse α, shift βⁱ, and co‑frame θᴬᵢ. Their conjugate momenta πᴬᵢ are decomposed into vector (Vπⁱ), antisymmetric (Aπⁱʲ), symmetric trace‑free (Sπⁱʲ), and trace (Tπ) parts. Vanishing of the corresponding coefficients V_A, A_A, S_A, T_A yields primary constraints V_Cⁱ, A_Cⁱʲ, S_Cⁱʲ, and T_C, respectively. The total Hamiltonian contains the diffeomorphism constraints (α, βⁱ) and the primary constraints multiplied by Lagrange multipliers.

Regularity and irregularity
A regular theory requires that the first‑order variation of every constraint be expressible as a linear combination of existing constraints; this guarantees functional independence of the constraint set. Type 9 satisfies this condition (regular, “Case A”), while Types 4 and 7 do not. In Type 4 the trace‑free condition of S_Cⁱʲ is only weakly satisfied; functional independence is lost unless the trace constraint T_C is added, thereby regularizing the system. Type 7 exhibits a different kind of irregularity (“Case B”) where the antisymmetric and symmetric trace‑free constraints coincide (A_Cⁱʲ = S_Cⁱʲ) and the constraint matrix loses rank. The authors resolve this by redefining an independent set of first‑class constraints without altering the Poisson‑bracket algebra.

Constraint algebra and classification
For Type 4 the Poisson brackets among the six independent components of S_Cⁱʲ form a non‑degenerate matrix; consequently all S_Cⁱʲ are second‑class constraints. Six second‑class constraints remove twelve phase‑space dimensions, leaving ten canonical variables, i.e. five physical DoF. Type 9 contains three sets of second‑class constraints: V_Cⁱ (3), S_Cⁱʲ (5), and T_C (1), totaling nine second‑class constraints. They eliminate eighteen phase‑space dimensions, leaving six canonical variables, i.e. three physical DoF. Type 7, by contrast, consists solely of first‑class constraints (the eight diffeomorphism constraints plus the six coincident A_Cⁱʲ = S_Cⁱʲ constraints). First‑class constraints each remove one canonical variable; thus all phase‑space degrees of freedom are eliminated, yielding zero physical DoF. This makes Type 7 a purely topological system in the bulk.

Absence of bifurcation
In none of the three types does the determinant of the constraint matrix vanish for any choice of the NGR parameters. Hence no bifurcation—i.e., no branching of the constraint structure—occurs, and the notion of “semi‑first‑class” constraints introduced for Type 8 does not appear here.

Physical implications
Type 4’s five DoF exceed the two tensor modes of GR, indicating the presence of additional scalar and vector excitations. Such extra modes are typically associated with strong‑coupling or ghost problems; further analysis would be required to assess their viability. Type 9’s three DoF correspond to GR plus a single extra scalar mode, reminiscent of scalar‑tensor theories that are often invoked in cosmology. Type 7, having no bulk propagating modes, is a candidate for a topological field theory; its dynamics may reside entirely on boundaries or defects, suggesting possible applications in holography or condensed‑matter analogues.

Conclusion
The authors have completed the DoF classification of all nine NGR types: Types 2, 3, 5, 6, 8 each have two tensor DoF; Type 4 has five, Type 9 has three, and Type 7 has none. The work also provides concrete examples of handling irregular constraint systems by regularization that preserves the Poisson algebra. These results furnish a solid foundation for future investigations into the quantum behavior of NGR, its cosmological phenomenology, and the broader problem of treating irregular gauge systems in gravitational theories.