Hamiltonian equations of motion of quadratic gravity

TL;DR

This work provides the missing explicit Hamiltonian equations of motion for quadratic gravity, i.e., the most general theory with terms quadratic in the curvature, derived in the ADM framework using the Buchbinder–Lyahovich/Ostrogradsky approach and verified with Cadabra. It establishes the full constraint algebra, showing all constraints are first-class and that the Hamiltonian is a sum of these constraints, while also presenting both nonperturbative and linearized evolution equations for the canonical variables. A key result is that, when general-relativity terms are present (i.e., $\kappa^{-2}\neq 0$), the linearized Hamiltonian dynamics are equivalent to covariant field equations only if the perturbative spatial metric is traceless, i.e., $h^L = - h^T$; in the special case $\kappa^{-2}=0$, equivalence holds without fixing arbitrary functions. The paper also demonstrates concrete homogeneous and isotropic solutions, clarifying the physical content (eight degrees of freedom) and highlighting the nontrivial parameter constraints required for stability and consistency, thereby providing a practical canonical-toolset for quadratic gravity analyses.

Abstract

We compute explicitly the equations of motion of the Hamiltonian formulation of quadratic gravity. This is the theory with the most general Lagrangian with terms of quadratic order in the curvature tensor. We employ the symbolic computational tool Cadabra. We present the linearized version of the equations of motion, performing the longitudinal-transverse decomposition. We compare the linear equations with the covariant field equations, finding that, if general-relativity terms are active, the linear Hamiltonian formulation is valid only if the perturbative spatial metric is traceless, a condition that can be freely imposed by recurring to an arbitrary function. We apply the equations of motion on homogeneous and isotropic configurations, finding explicit solutions.