Metriplectic geometry for gravitational subsystems

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