Primary Constraints of Newer General Relativity

This paper analyzes the primary constraint structure of Newer General Relativity by performing a fully nonlinear decomposition of its Lagrangian, recovering known tensor and vector constraints while identifying a previously unreported degeneracy in the scalar sector that yields one or two additional constraints depending on the theory's parameters.

The Gist

Imagine the universe as a giant, flexible trampoline. For decades, physicists have used Albert Einstein's General Relativity to describe how heavy objects (like stars) warp this trampoline, creating the force we call gravity. Einstein's model has passed every test thrown at it, from tracking planets to listening to gravitational waves.

However, just like a car that runs perfectly but has a mysterious engine noise, our current understanding of gravity has some "noise." We see things in the universe (like dark matter and dark energy) that our equations can't quite explain without adding invisible ingredients. We also know that Einstein's math breaks down at the very center of black holes. So, scientists are building new, experimental models of gravity to see if they can fix these issues.

This paper is a technical "safety inspection" of one such new model called Newer General Relativity.

The New Model: A Different Kind of Stretching

In Einstein's original theory, gravity is caused by the curvature of spacetime (the trampoline bending). In "Newer General Relativity," the authors explore a different idea: what if gravity comes from the stretching or distortion of the grid lines on the trampoline, rather than the bending itself?

They call this Nonmetricity. Imagine the trampoline's grid lines getting stretched or squashed unevenly as you move across it, even if the surface itself doesn't curve. This theory allows for five different "knobs" (mathematical coefficients called $c_1$ through $c_5$) that control how much stretching happens in different ways.

The Problem: Too Many Moving Parts?

When you build a new theory of physics, you have to make sure it doesn't have "ghosts" (mathematical errors that predict impossible things) or "extra wheels" (unnecessary variables that make the math unstable).

To check this, the authors perform a Hamiltonian analysis. Think of this as taking the engine apart to see how the pistons move. They look at the relationship between:

1. Velocity: How fast the "grid lines" are changing.
2. Momentum: The "push" or energy associated with that change.

In a healthy theory, if you know the push, you can figure out the speed, and vice versa. But in this new theory, depending on how you turn those five "knobs," the map between push and speed might break. If the map breaks, it means the system has Primary Constraints.

The Discovery: The "Constraint" Filter

The authors found that this new theory acts like a complex filter with three different sections: Tensor, Vector, and Scalar.

1. The Tensor Section (The 5-Filter):
   Imagine a sieve with five holes. If the authors turn the knobs just right (specifically, if the first knob $c_1$ is zero), five of the "push" variables get stuck. They can't move freely. This creates 5 constraints. This part was already known to other scientists.

2. The Vector Section (The 3-Filter):
   This is like a sieve with three holes. If the knobs are set to a specific combination ($2c_1 + c_2 + c_4 = 0$), three more variables get stuck. This creates 3 constraints. This was also known.

3. The Scalar Section (The New Discovery):
   This is the most interesting part. The authors found a previously hidden "trap" in the math.

   • Scenario A: For most settings, this section acts like a sieve with 1 hole, creating 1 constraint.
   • Scenario B: But, if the knobs are turned to a very specific, rare combination, the entire section collapses. It acts like a sieve with 2 holes, creating 2 constraints.

   The Big Reveal: Previous studies missed this second possibility. They thought there was only ever one constraint in this section. The authors proved that depending on the settings, you can have either one or two "stuck" variables here.

The Final Count: How Many Rules?

By mixing and matching these three sections, the authors created a complete "menu" of possible versions of this theory. They found that the total number of rules (constraints) the theory must follow can be:

• 1, 2, 3, 4, 5, 6, 8, 9, or 10.

Wait, where is 7?
The authors point out a funny algebraic quirk: You can never have exactly 7 constraints. If you try to set the knobs to get 7, the math forces you to accidentally get 8, 9, or 10 instead. It's like trying to build a tower with 7 blocks, but the laws of physics force you to either add an 8th block or remove one to make it 6.

Why Does This Matter?

This paper doesn't say "this theory is the winner" or "this explains dark energy." Instead, it acts as a diagnostic tool.

• It tells us exactly which versions of "Newer General Relativity" are mathematically stable and which ones might be broken.
• It corrects errors in previous papers that missed the "two-constraint" possibility in the scalar section.
• It provides the foundation for future work. Before we can ask if this theory explains the universe, we first need to know exactly how many "moving parts" (degrees of freedom) it has. This paper counts those parts for us.

In short, the authors took a complex, experimental theory of gravity, took it apart, and mapped out exactly where the "brakes" (constraints) are located, revealing a hidden complexity that no one had seen before.

Technical Summary

Technical Summary: Primary Constraints of Newer General Relativity

Problem Statement
Newer General Relativity (Newer GR) is a gravity theory based on a torsionless teleparallel geometry where the gravitational action is constructed from quadratic combinations of the nonmetricity tensor with arbitrary coefficients $c_i$. While previous studies have explored specific aspects of this theory, such as gravitational wave propagation and the post-Newtonian limit, the number of propagating degrees of freedom for the most general models remains unclear. Determining this number is essential for characterizing potential pathologies, such as ghosts, strong coupling issues, and instabilities. A critical first step in this characterization is the Hamiltonian analysis, specifically the identification of primary constraints arising from the singular nature of the Legendre transform. Previous attempts to classify these constraints, notably in Ref. [41], contained inconsistencies regarding the Hessian determinant and the scalar sector degeneracy conditions.

Methodology
The authors perform the initial stage of the Dirac-Bergmann algorithm to identify primary constraints. The methodology proceeds as follows:

1. Canonical Variables: The theory is treated within a metric-affine framework where the metric $g_{\mu\nu}$ and the connection are independent, though the connection is constrained to be torsionless and flat (symmetric teleparallel). The authors focus on the momenta conjugate to the metric.
2. Hessian Analysis: The canonical momenta are derived by differentiating the Lagrangian with respect to the metric velocities. The relationship between velocities and momenta is governed by the Hessian matrix. The primary constraints correspond to the kernel (null space) of this Hessian.
3. Spectral and Irreducible Decomposition: To analyze the kernel, the authors employ two complementary approaches:
   • Spectral Decomposition: They decompose the nonmetricity tensor and the Hessian operator into invariant subspaces corresponding to vector, tensor, and scalar sectors. This allows for the diagonalization of the operator and the identification of eigenvalues that vanish under specific parameter conditions.
   • Irreducible Decomposition (ADM): They utilize the ADM decomposition of the metric to express the Hessian in terms of lapse, shift, and spatial metric variables. They explicitly construct the kernel elements in terms of scalar, vector, and tensor modes to verify the spectral results.
4. Covariant Formulation: The paper also explores a covariant Hamiltonian treatment by introducing a tetrad-like variable $e^a_\mu = \partial_\mu \xi^a$ to reduce the order of derivatives in the connection, avoiding Ostrogradsky instabilities associated with treating the connection as a fundamental second-derivative variable.

Key Contributions and Results
The paper provides a complete classification of the primary constraints for Newer GR based on the degeneracy of the Hessian. The main findings are:

• Sector Decomposition: The Hessian naturally decomposes into three irreducible sectors with distinct degeneracy conditions:

  • Tensor Sector: Degenerate when $c_1 = 0$. This yields five primary constraints corresponding to the five independent components of a spatial symmetric trace-free tensor.
  • Vector Sector: Degenerate when $2c_1 + c_2 + c_4 = 0$. This yields three primary constraints associated with spatial vectors.
  • Scalar Sector: Governed by a $2 \times 2$ matrix block. The degeneracy condition is given by the vanishing of the determinant:
    $$c_1^2 + c_1(c_2 + c_4 + 4c_3 + c_5) + 3c_3(c_2 + c_4) - \frac{3}{4}c_5^2 = 0$$
    This sector is more complex than previously understood. It generically yields one scalar primary constraint. However, on a specific sub-locus of parameters (where the entire $2 \times 2$ block vanishes), it yields two independent scalar primary constraints.

• Classification of Models: By combining the degeneracies of these sectors, the authors identify nine nontrivial patterns of primary constraints (excluding the non-degenerate case). The total number of primary constraints can be 1, 2, 3, 4, 5, 6, 8, 9, or 10. Notably, a seven-constraint sector is algebraically impossible; if the scalar sector is fully degenerate (2 constraints), the tensor and vector sectors are forced to be degenerate as well, leading to a maximally degenerate case with 10 constraints.

• Correction of Literature: The authors identify errors in Ref. [41], specifically regarding the calculation of the Hessian determinant and the classification of scalar constraints. They demonstrate that the conditions found in Ref. [41] are special cases of their more general quadratic equation (54). They also clarify that the scalar sector can support two constraints, a feature not isolated in previous works.

• Covariant Constraints: In the covariant formulation using the $e^a_\mu$ variables, the authors derive new primary constraints (Eq. 131) that relate the momenta conjugate to the connection variables to the metric momenta. These constraints are covariant analogues to those found in metric teleparallel gravity.

Significance
The paper establishes the foundational step for the Hamiltonian analysis of Newer General Relativity. By rigorously decomposing the Hessian and identifying the precise conditions for degeneracy, the authors provide a necessary framework for determining the number of propagating degrees of freedom in future work. The discovery of the "two-scalar-constraint" regime and the algebraic exclusion of the seven-constraint case refine the theoretical landscape of symmetric teleparallel gravity. The authors note that while this work does not yet determine the final count of propagating degrees of freedom (which requires the full Dirac-Bergmann algorithm including secondary constraints), it resolves the primary constraint structure and corrects previous inconsistencies in the literature, serving as a prerequisite for analyzing the well-posedness and stability of viable Newer GR models.