Gravitational Energy for GR and Poincare Gauge Theories: a Covariant Hamiltonian Approach

TL;DR

This paper develops a covariant Hamiltonian approach to define quasi-local gravitational energy, momentum, and angular momentum in Poincaré gauge theories, extending previous work in general relativity and addressing longstanding issues.

Contribution

It introduces a covariant Hamiltonian formalism for Poincaré gauge theories, providing new insights into quasi-local gravitational energy and boundary conditions.

Findings

Hamiltonian boundary term determines quasi-local quantities

Energy-momentum linked to Poincaré symmetries in Riemann-Cartan spacetime

New results on boundary conditions for gravitational variables

Abstract

Our topic concerns a long standing puzzle: the energy of gravitating systems. More precisely we want to consider, for gravitating systems, how to best describe energy-momentum and angular momentum/center-of-mass momentum (CoMM). It is known that these quantities cannot be given by a local density. The modern understanding is that (i) they are quasi-local (associated with a closed 2-surface), (ii) they have no unique formula, (iii) they have no reference frame independent description. In the first part of this work we review some early history, much of it not so well known, on the subject of gravitational energy in Einstein's general relativity (GR), noting especially Noether's contribution. In the second part we review (including some new results) much of our covariant Hamiltonian formalism and apply it to Poincar\'e gauge theories (GR is a special case). The key point is that the Hamiltonian boundary term has two roles, it determines the quasi-local quantities, and, furthermore it determines the boundary conditions for the dynamical variables. Energy-momentum and angular momentum/CoMM are associated with the geometric symmetries under Poincar\'e transformations. They are best described in a local Poincar\'e gauge theory. The type of spacetime that naturally has this symmetry is Riemann-Cartan spacetime, with a metric compatible connection having, in general, both curvature and torsion. Thus our expression for the energy-momentum of physical systems is obtained via our covariant Hamiltonian formulation applied to Poincar\'e gauge theories.