Metriplectic geometry for gravitational subsystems

TL;DR

This work tackles the problem of defining quasi-local gravitational charges in a bounded region by examining two complementary frameworks. First, it analyzes the extended covariant phase space with dressing fields, where Komar charges become Hamiltonians for dressed diffeomorphisms and the associated charges close under a modified boundary-bracket, albeit with ambiguous physical interpretation for interior dynamics. Second, it introduces metriplectic geometry to incorporate dissipation directly, replacing the usual symplectic structure with a Leibniz bracket (Omega_ext - G) so that charges generate interior dynamics while accounting for environment exchange and entropy production. Overall, the paper demonstrates that both approaches give a rigorous meaning to quasi-local Komar charges at finite distance, but each faces challenges in linking these charges to unambiguous physical observables, suggesting rich avenues for black hole thermodynamics and possible implications for quantum gravity. The work highlights that dissipation in gravitational subsystems can be consistently modeled either by boundary extensions or by metriplectic dynamics, offering new tools to study energy, momentum, and memory effects in general relativity.

Abstract

In general relativity, it is difficult to localise observables such as energy, angular momentum, or centre of mass in a bounded region. The difficulty is that there is dissipation. A self-gravitating system, confined by its own gravity to a bounded region, radiates some of the charges away into the environment. At a formal level, dissipation implies that some diffeomorphisms are not Hamiltonian. In fact, there is no Hamiltonian on phase space that would move the region relative to the fields. Recently, an extension of the covariant phase space has been introduced to resolve the issue. On the extended phase space, the Komar charges are Hamiltonian. They are generators of dressed diffeomorphisms. While the construction is sound, the physical significance is unclear. We provide a critical review before developing a geometric approach that takes into account dissipation in a novel way. Our approach is based on metriplectic geometry, a framework used in the description of dissipative systems. Instead of the Poisson bracket, we introduce a Leibniz bracket - a sum of a skew-symmetric and a symmetric bracket. The symmetric term accounts for the loss of charge due to radiation. On the metriplectic space, the charges are Hamiltonian, yet they are not conserved under their own flow.