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

This paper completes the analysis of degrees of freedom in New General Relativity by investigating Types 4, 7, and 9, revealing they possess five, zero, and three degrees of freedom respectively, while providing specific examples for handling irregular systems that lack general treatment methods.

The Gist

Imagine the universe as a giant, flexible trampoline. In standard physics (General Relativity), we describe gravity as the shape of this trampoline curving when you put a heavy bowling ball on it. But there's another way to look at it, called New General Relativity (NGR). Instead of just looking at the curves, NGR looks at how the trampoline fabric is twisted and stretched.

This paper is like a detective story where the author, Kyosuke Tomonari, is trying to figure out exactly how many "independent moves" or "degrees of freedom" this twisted trampoline can actually make.

Here is the breakdown of the paper using simple analogies:

1. The Big Picture: The Nine Types of Gravity

Think of NGR as a massive Lego set with three special knobs (parameters) you can turn. Depending on how you set these knobs, you get different versions of the theory. The author has already sorted these into nine distinct "Types" (like nine different Lego models).

• The Goal: In a previous paper, the author analyzed four of these models (Types 2, 3, 5, and 8). Those models were interesting because they could actually describe the gravity we see in the universe (like planets orbiting stars).
• This Paper: The author is now cleaning up the rest of the box. He is looking at the remaining three models: Type 4, Type 7, and Type 9.

2. The Problem: "Irregular" Systems

Most of the time, physics equations are well-behaved. You plug in numbers, and the rules stay consistent. But some of these NGR models are "irregular."

• The Analogy: Imagine a game of chess. In a normal game, every piece has a clear rule. In an "irregular" game, the rules for the King might change depending on whether the board is wet or dry, or if a specific piece is missing. The rules aren't consistent everywhere.
• The Challenge: Standard math tools (called Dirac-Bergmann analysis) usually break down when the rules are this messy. It's like trying to use a standard map to navigate a city where the streets keep changing names. The author had to invent a special way to "fix" the map (regularize the system) just to count the moves.

3. The Results: What Can These Models Do?

The author counted the "degrees of freedom" (DoF). Think of this as counting how many independent directions a dancer can move.

• Type 4: The Busy Dancer (5 Degrees of Freedom)

  • This model is a bit messy (irregular), but once the author fixed the rules, he found it has 5 independent moves.
  • The Catch: It doesn't have the "tensor" moves that describe real gravity (like ripples in spacetime). It's a valid mathematical system, but it's not the gravity we live in.

• Type 7: The Frozen Statue (0 Degrees of Freedom)

  • This is the most surprising result. In the middle of the universe (the "bulk" spacetime), this model has zero moves.
  • The Analogy: Imagine a sculpture that looks like a complex machine, but if you touch it, it doesn't move at all. It's a "purely topological" system. It exists, but it doesn't do anything dynamically in the middle of space.
  • The Twist: The author notes that while it's frozen in the middle, it might wiggle at the very edges (boundaries) of the universe. It's like a frozen lake that is solid in the middle but might have ripples at the shore.

• Type 9: The Minimalist (3 Degrees of Freedom)

  • This model is well-behaved (regular). It has 3 independent moves.
  • The Catch: Like Type 4, it lacks the specific "gravitational wave" moves needed to describe how gravity works in our universe.

4. Why Does This Matter?

You might ask, "If these models don't describe our gravity, why study them?"

1. Mathematical Gymnastics: Physics is full of "what if" scenarios. By studying these weird, irregular models, the author is teaching us how to solve math problems that usually break standard tools. It's like learning to juggle with flaming torches; even if you don't need to juggle torches every day, the skill helps you understand balance better.
2. Ruling Out Bad Ideas: To know what is gravity, we need to know what isn't. By proving these types don't have the right number of moves to describe real gravity, the author narrows down the search for a "Theory of Everything."
3. The "Ghost" Problem: In physics, some theories predict "ghosts"—particles that have negative energy and cause the universe to collapse instantly. The author checked these models and confirmed they are safe; they don't have these fatal ghosts.

Summary

The author took three weird, broken, or frozen versions of a gravity theory and used special math to figure out exactly how they move.

• Type 4 moves a lot but isn't our gravity.
• Type 7 is frozen solid in space (but might wiggle at the edges).
• Type 9 moves a little bit but isn't our gravity.

The real victory isn't just the numbers; it's showing the world how to handle "irregular" systems that don't play by the usual rules, paving the way for future discoveries in the messy, complex world of theoretical physics.

Technical Summary

1. Problem Statement

New General Relativity (NGR) is a three-parameter extension of the Teleparallel Equivalent of General Relativity (TEGR). It is classified into nine irreducible types based on the $SO(3)$-irreducible decomposition of canonical momenta on the spatial hypersurfaces of an ADM foliation.

While previous work (Tomonari & Blixt, 2025) analyzed the degrees of freedom (DoF) for Types 2, 3, 5, and 8 (which contain propagating gravitational tensor modes), the remaining types (4, 7, and 9) had not been fully analyzed. These specific types are generally considered unsuitable for describing standard gravity in cosmology because they lack tensor propagating modes. However, a complete classification of NGR is necessary for:

1. Understanding the full landscape of metric-affine gauge theories.
2. Investigating "irregular" constrained systems, where the functional independence of constraints is violated.
3. Exploring potential dynamical models in non-gravitational fields (e.g., topological systems or boundary dynamics).

The core challenge addressed in this paper is the rigorous counting of degrees of freedom for these specific types, particularly dealing with irregular systems where standard Dirac-Bergmann analysis fails due to the violation of constraint independence.

2. Methodology

The author employs the Dirac-Bergmann algorithm for constrained Hamiltonian systems, adapted to handle irregular constraints.

• Hamiltonian Formulation: The analysis is based on the Hamiltonian formulation of NGR in the Weitzenböck gauge. The configuration space includes the lapse function ($\alpha$), shift vector ($\beta^i$), and co-frame field components ($\theta^A_i$).
• Constraint Classification: The primary constraints are derived from the vanishing of momenta conjugate to $\alpha$ and $\beta^i$, and specific conditions on the parameters ($c_1, c_2, c_3$) that define the NGR types.
• Regularity Analysis: A critical step is determining if the system is regular (constraints are functionally independent) or irregular (functional independence is violated).
  • The paper identifies that the trace-free condition of the symmetric trace-free momentum part ($S\pi_{ij}$) is the source of irregularity.
  • If the trace of the constraint $SC_{ij}$ does not vanish as a strong equality, the system becomes irregular.
• Handling Irregular Systems: Since no general method exists for irregular systems, the author adopts a regularization strategy:
  • Reformulating the set of constraints to create a new, regular set.
  • Comparing the Poisson Bracket (PB) algebra of the original and regularized systems. If the algebra is preserved on the intersection of the constraint surfaces ($\Gamma_0 \cap \Gamma_1$), the dynamics are considered equivalent for the purpose of counting DoF.
• PB-Algebra Calculation: Detailed calculations of the Poisson brackets between primary constraints ($VC_i, AC_{ij}, SC_{ij}, T^C$) are performed to classify them as first-class (generating gauge symmetries) or second-class (reducing phase space dimensions).

3. Key Contributions

1. Classification of Regularity: The paper establishes a theorem classifying the regularity of all NGR types:
   • Regular: Types 2, 3, 5, 6, 8, and 9.
   • Irregular: Type 4 and Type 7.
   • This provides a rigorous framework for distinguishing between standard and irregular constrained systems in metric-affine gravity.
2. Treatment of Irregular Constraints: The paper provides specific examples of how to handle Type I irregular constraints (multi-linear constraints) in a gravitational context. It demonstrates how to regularize the system by introducing secondary constraints (e.g., treating the trace of $SC_{ij}$ as a constraint) to restore functional independence without altering the physical dynamics on the relevant subsurface.
3. Absence of Bifurcation: Unlike Type 8 (analyzed in previous work), the paper proves that Types 4, 7, and 9 do not exhibit bifurcation (branching of constraint emergence based on Lagrange multipliers). All constraints are strictly classified as either first-class or second-class.

4. Results: Degrees of Freedom (DoF)

The analysis yields the following non-linear degrees of freedom for the investigated types:

Type	Regularity	Constraint Structure	Degrees of Freedom	Physical Interpretation
Type 4	Irregular (Case C)	Purely Second-Class constraints. Requires regularization (introducing $T^C \approx 0$) to count.	5	Contains 5 DoF. No tensor modes. Likely contains scalar/vector modes.
Type 7	Irregular (Case B)	Purely First-Class constraints. The trace-free condition reduces independent constraints from 6 to 5.	0	Purely Topological in the bulk spacetime. No local propagating DoF.
Type 9	Regular	Purely Second-Class constraints.	3	Contains 3 DoF. No tensor modes.

• Type 4: The irregularity arises because the diagonal components of $SC_{ij}$ are not functionally independent. By treating the trace as a secondary second-class constraint, the system is regularized, yielding 5 DoF.
• Type 7: The system possesses only first-class constraints. However, due to irregularity, the diagonal components of $SC_{ij}$ are dependent, reducing the number of independent first-class constraints. The resulting DoF is 0, indicating a topological field theory in the bulk. The paper notes that gauge fixing could artificially introduce DoF (up to 3), but these are not intrinsic to the bulk theory.
• Type 9: Being a regular system, the standard counting applies. With 10 second-class constraints (3 from $VC_i$, 6 from $SC_{ij}$, 1 from $T^C$), the DoF is 3.

Ghost Analysis: The paper confirms that none of these types suffer from Ostrogradski ghosts (instabilities) because the Hamiltonian is bounded from below (quadratic theory). However, the absence of tensor modes implies that any propagating modes would be scalar or vector, potentially including ghost-like behavior in perturbative regimes (requiring further linear perturbation analysis).

5. Significance

• Completing the NGR Landscape: This work completes the DoF analysis for all nine types of NGR, providing a comprehensive reference for future theoretical developments in teleparallel and metric-affine gravity.
• Advancing Irregular System Theory: By providing concrete examples of irregular gravitational systems (Types 4 and 7) and demonstrating a regularization procedure that preserves the PB-algebra, the paper contributes to the broader mathematical physics community's understanding of how to handle non-standard constrained systems.
• Topological Gravity: The identification of Type 7 as a purely topological system (0 DoF in the bulk) suggests potential applications in topological field theories or as a model for boundary dynamics (holography), where non-vanishing DoF might emerge at the boundary.
• Implications for Cosmology: The results reinforce that Types 4, 7, and 9 are unsuitable for describing standard cosmological gravity (which requires 2 tensor DoF). However, they remain relevant for exploring alternative dynamical systems or specific limits of modified gravity theories.
• Foliation Considerations: The conclusion briefly touches upon recent debates regarding the ADM foliation in metric-affine theories, suggesting that while the current analysis holds, the global existence of foliation in broken Lorentz symmetry scenarios requires further investigation.

In summary, this paper rigorously resolves the degree of freedom counting for the remaining NGR types, highlighting the unique challenges of irregular constraints and establishing that Types 4, 7, and 9 possess 5, 0, and 3 DoF respectively, with no bifurcation phenomena.