Manifest Covariant Hamiltonian Theory of General Relativity

TL;DR

This paper develops a covariant Hamiltonian formulation of General Relativity using an extended DeDonder-Weyl formalism, preserving gauge invariance and scalar properties in curved spacetime.

Contribution

It introduces a non-perturbative, covariant Hamiltonian framework for Einstein's equations based on a synchronous variational principle with superabundant variables.

Findings

Derived the covariant Hamiltonian structure for Einstein's equations.

Established a continuum Poisson bracket representation.

Maintained gauge invariance and scalar properties in the formulation.

Abstract

The problem of formulating a manifest covariant Hamiltonian theory of General Relativity in the presence of source fields is addressed, by extending the so-called "DeDonder-Weyl" formalism to the treatment of classical fields in curved space-time. The theory is based on a synchronous variational principle for the Einstein equation, formulated in terms of superabundant variables. The technique permits one to determine the continuum covariant Hamiltonian structure associated with the Einstein equation. The corresponding continuum Poisson bracket representation is also determined. The theory relies on first-principles, in the sense that the conclusions are reached in the framework of a non-perturbative covariant approach, which allows one to preserve both the 4-scalar nature of Lagrangian and Hamiltonian densities as well as the gauge invariance property of the theory.