General Principles of Hamiltonian Formulations of the Metric Gravity

TL;DR

This paper develops a Hamiltonian framework for metric gravity, introducing momenta, deriving Hamiltonians, and simplifying the equations to facilitate analytical and numerical solutions, including the Jacobi equation for free gravitational fields.

Contribution

It formulates principles for Hamiltonian approaches in metric gravity, introduces methods to determine Poisson brackets, and reduces Hamiltonians to natural forms for easier analysis.

Findings

Derived canonical and total Hamiltonians for metric GR.

Developed a method to compute Poisson brackets between dynamical variables.

Reduced Hamiltonian to natural form enabling simplified equations like the Jacobi equation.

Abstract

Principles of successful Hamiltonian approaches, which were developed to describe free gravitational field(s) in the metric gravity, are formulated and discussed. By using the standard $\Gamma-\Gamma$ Lagrangian ${\cal L}_{\Gamma-\Gamma}$ of the metric GR we properly introduce all momenta of the metric gravitational field and derive the both canonical $H_C$ and total $H_t$ Hamiltonians of the metric GR. We also developed an effective method which is used to determine various Poisson brackets between analytical functions of the basic dynamical variables, i.e., generalized coordinates $g_{\alpha\beta}$ and momenta $\pi^{\mu\nu}$. In general, such variables can be chosen either from the straight $\{ g_{\alpha\beta}, \pi^{\mu\nu} \}$, or dual $\{ g^{\alpha\beta}, \pi_{\mu\nu} \}$ sets of symplectic dynamical variables which always arise (and complete each other) in any Hamiltonian formulation developed for the coupled system of tensor fields. By applying canonical transformation(s) of dynamical variables we reduce the canonical Hamiltonian $H_C$ to its natural form. The natural form of canonical Hamiltonian provides numerous advantages in actual applications to the metric GR, since the general theory of dynamical systems with such Hamiltonians is well developed. Furthermore, many analytical and numerically exact solutions have been found and described in detail for dynamical systems with the Hamiltonians already reduced to their natural forms. In particular, reduction of the canonical Hamiltonian $H_C$ to its natural form allows one to derive the Jacobi equation for the free gravitational field(s), which takes a particularly simple form.