Hamiltonian equations of motion of quadratic gravity

Imagine the universe as a giant, stretchy trampoline. In the standard story of gravity (Einstein's General Relativity), this trampoline bends and warps based on how heavy the objects sitting on it are. The rules for how it bends are simple and well-understood.

But what if the trampoline wasn't just stretchy, but also had a complex internal structure? What if, instead of just bending, it could also "twist" or "ripple" in more complicated ways? This is the idea behind Quadratic Gravity. It's a more advanced, "super-charged" version of Einstein's theory that tries to fix some problems we have when we try to mix gravity with quantum mechanics (the physics of the very small).

Here is a breakdown of what Jorge Bellorin's paper does, using everyday analogies:

1. The Problem: The "Too Complicated" Recipe

Einstein's theory is like a recipe with one main ingredient: curvature. Quadratic Gravity adds two extra, very spicy ingredients (terms involving the square of the curvature).

The Catch: When you add these extra ingredients, the math gets messy. In the standard way of doing physics (the "Lagrangian" method), the equations become incredibly high-order, meaning they depend on how fast the trampoline is changing, how fast that is changing, and so on. It's like trying to predict the weather by looking at the wind speed, the acceleration of the wind, the jerk of the wind, and the snap of the wind. It's a nightmare to solve.

2. The Solution: The "Hamiltonian" Switch

The author uses a different mathematical toolkit called the Hamiltonian formulation.

The Analogy: Think of the standard method as trying to drive a car by only looking at the road ahead (position). The Hamiltonian method is like looking at both the road and the speedometer (momentum) at the same time.
Why it helps: By splitting the problem into "position" and "speed," the messy, high-order equations get broken down into a set of simpler, first-order equations. It turns a terrifying, complex monster into a manageable set of instructions.

3. The Tool: The "Magic Calculator" (Cadabra)

Doing this math by hand is like trying to solve a Rubik's Cube while juggling chainsaws. The author used a computer program called Cadabra.

The Analogy: Cadabra is like a super-smart, magical calculator that understands the language of shapes and indices (the little numbers and letters physicists use). It can handle thousands of terms without getting tired or making a typo, allowing the author to derive the exact rules of motion for this complex gravity theory.

4. The Big Discovery: The "Traceless" Rule

The most important finding in the paper is about consistency.

The Scenario: The author compared the new "Hamiltonian" rules (the speedometer method) with the old "Covariant" rules (the road-ahead method).
The Conflict: When the standard gravity terms are active (like having a heavy rock on the trampoline), the two methods didn't quite match up. They were like two maps of the same city that disagreed on where the river was.
The Fix: The author discovered that for the two maps to match, you have to impose a specific rule: The ripples on the trampoline must be "traceless."
- Simple Translation: Imagine the trampoline fabric. If you stretch it in one direction, it usually shrinks in another. The "traceless" condition means that for the math to work in this specific theory, the stretching and shrinking must perfectly cancel each other out in a specific way. It's a constraint you have to manually apply to make the theory work.

5. The Application: A Perfectly Smooth Universe

Finally, the author tested these new rules on a simple scenario: a universe that is perfectly smooth and expanding (like the Big Bang model).

The Result: They found explicit solutions. They showed how this "super-charged" gravity theory would behave in a universe filled with a perfect fluid (like a gas of particles). They found that the universe could expand in specific, predictable ways, proving that the math actually works and produces physical results.

Summary: Why Does This Matter?

Think of this paper as the instruction manual for a new, complex engine.

Before this paper, we knew the engine existed (Quadratic Gravity), but we didn't have the step-by-step instructions on how to run it (the equations of motion).
The author wrote down the manual, showed us how to fix a glitch in the instructions (the "traceless" rule), and even drove the car to show it works (the expanding universe solution).

This is a crucial step because if we ever want to understand the very beginning of the universe or the center of a black hole using quantum gravity, we need to know exactly how this "super-gravity" behaves. This paper gives us the first clear, step-by-step guide.

1. Problem Statement

Quadratic gravity is a theory extending General Relativity (GR) by including terms quadratic in the curvature tensor ( $R_{\mu\nu}R^{\mu\nu}$ and $R^2$ ) in the Lagrangian. While this theory is renormalizable, it suffers from Ostrogradsky instabilities due to higher-order time derivatives.

The Gap: Although the Hamiltonian formulation of quadratic gravity was previously established by Buchbinder and Lyahovich (providing constraints and counting degrees of freedom), the explicit equations of motion derived from this Hamiltonian formulation were missing in the literature.
The Challenge: The presence of higher-order time derivatives requires an extended phase space (Ostrogradsky method or its generalization). Furthermore, establishing the equivalence between the Hamiltonian equations of motion and the standard covariant (Lagrangian) field equations is non-trivial, particularly regarding the handling of arbitrary functions (gauge freedoms) and the consistency of the linearized limit.

2. Methodology

The author employs the Buchbinder-Lyakhovich generalization of the Ostrogradsky method, adapted to the Arnowitt-Deser-Misner (ADM) formalism.

Variables: The phase space is extended to include the spatial metric $g_{ij}$ , its conjugate momentum $\pi_{ij}$ , the extrinsic curvature $K_{ij}$ (treated as an independent variable), and its conjugate momentum $P^{ij}$ . The lapse $N$ and shift $N^i$ act as Lagrange multipliers.
Computational Tool: Due to the extreme algebraic complexity of the calculations (involving high-order derivatives, tensor contractions, and index manipulations), the author utilizes Cadabra, a symbolic computational tool specialized for field theory. This tool handles index notation, metric contractions, and variational calculus automatically.
Analysis Steps:
1. Derive the canonical momenta and identify primary constraints ( $P_A = 0$ ).
2. Construct the Hamiltonian density and identify secondary constraints ( $T_A = 0$ ).
3. Compute the Poisson brackets between canonical variables and constraints to establish the constraint algebra.
4. Derive the explicit non-perturbative equations of motion.
5. Perform a linearized analysis using longitudinal-transverse decomposition to compare with covariant field equations.
6. Apply the formalism to homogeneous and isotropic (FLRW) configurations.

3. Key Contributions

Explicit Equations of Motion: The paper provides the first explicit form of the Hamiltonian equations of motion for quadratic gravity in the generic case (where the coupling matrix $G_{ijkl}$ is invertible).
Constraint Algebra: The author explicitly calculates the Poisson brackets between the canonical fields and the constraints $T_A$ . It is confirmed that the constraints form a first-class algebra identical to that of General Relativity, generating spatial diffeomorphisms and time evolution (on-shell).
Linearized Equivalence & Traceless Condition: A critical finding is the condition required for the linearized Hamiltonian formulation to be equivalent to the covariant field equations.
- When the GR term ( $\kappa^{-2} \neq 0$ ) is active, the equivalence holds only if the trace of the perturbative spatial metric is set to zero ( $h_L = -h_T$ , or $h^i_i = 0$ ).
- This condition fixes one of the arbitrary functions associated with the gauge freedom, a requirement that was not previously explicit.
Homogeneous Solutions: The paper derives explicit analytical solutions for a flat, homogeneous, and isotropic universe coupled to a perfect fluid, demonstrating the consistency of the Hamiltonian approach with covariant results.

4. Key Results

A. Hamiltonian Structure

Degrees of Freedom: The theory possesses 8 physical modes (16 canonical variables minus $2 \times 4$ first-class constraints).
Constraints: The system has four first-class constraints ( $T_0$ and $T_i$ ). $T_i$ generates spatial diffeomorphisms, while $T_0$ generates time evolution (orthogonal diffeomorphisms on-shell).
Equations of Motion: The paper presents the full set of four coupled differential equations for $\dot{g}_{ij}$ , $\dot{\pi}_{ij}$ , $\dot{K}_{ij}$ , and $\dot{P}^{ij}$ (Eqs. 5.1–5.4). These are highly complex, involving spatial derivatives up to fourth order and non-linear couplings.

B. Linearized Analysis

Decomposition: The equations are decomposed into longitudinal ( $L$ ) and transverse ( $T$ ) modes.
Equivalence Check:
- In the pure quadratic limit ( $\kappa^{-2} = 0$ ), the Hamiltonian and covariant equations are equivalent without fixing arbitrary functions.
- In the presence of GR terms ( $\kappa^{-2} \neq 0$ ), the Hamiltonian equations reproduce the covariant equations only if the condition $h_L = -h_T$ (traceless spatial perturbation) is imposed. Without this, a discrepancy term proportional to $\kappa^{-2}\Delta(h_L + h_T)$ appears.
Dispersion Relations: The linearized equations yield dispersion relations corresponding to massive and massless modes. Stability (real frequency solutions) requires specific bounds on coupling constants: $\kappa^{-2} \geq 0$ , $\alpha > 0$ , and $\beta < -\alpha/3$ .

C. Cosmological Solutions

For a flat FLRW metric with a perfect fluid ( $P=k\rho$ ), the Hamiltonian equations reduce to a single evolution equation for the scale factor $a(t)$ .
Power-law solutions: Explicit solutions of the form $a(t) \propto t^p$ $a (t) \propto t^{p}$ were found.
- For radiation ( $k=1/3$ ), $a(t) \propto t$ .
- For dust ( $k=0$ ), $a(t) \propto t^{4/3}$ .
These solutions match the results obtained from the covariant field equations, validating the Hamiltonian formalism in a non-perturbative, cosmological context.

5. Significance

Foundational Advance: This work fills a significant gap in the literature by providing the explicit dynamical equations for quadratic gravity in the Hamiltonian framework. This is essential for numerical relativity applications, where first-order time evolution systems are preferred.
Quantum Implications: The analysis of the linearized equivalence highlights that the "arbitrary functions" (gauge choices) in the Hamiltonian formalism are not entirely free when matching to the covariant theory in the presence of GR terms. This has direct implications for canonical quantization, as the classical Hamiltonian must reproduce the correct dynamics for the quantum theory to be consistent.
Tool Validation: The paper serves as a practical demonstration of the utility of Cadabra for handling the complex tensor algebra inherent in higher-order gravity theories.
Limitations: The analysis assumes the coupling matrix $G_{ijkl}$ is invertible (generic case). It notes that the limit $\alpha \to 0$ (recovering pure GR) is singular in this formulation, meaning the GR Hamiltonian cannot be obtained as a smooth limit of this specific quadratic gravity Hamiltonian.

In summary, Bellorin successfully constructs the complete Hamiltonian dynamics of quadratic gravity, validates it against covariant results, and identifies the specific gauge conditions necessary for consistency, providing a robust foundation for future classical and quantum investigations of higher-curvature gravity.