Boundary Conditions and Quasilocal Energy in the Canonical Formulation of All 1+1 Models of Gravity

TL;DR

This paper analyzes boundary conditions and quasilocal energy in the canonical formulation of general two-dimensional gravity models, revealing conserved quantities related to mass and energy, including black-hole mass parameters, across various theories.

Contribution

It establishes a rigorous link between first-order 2D Einstein-Cartan models and generalized dilaton theories, identifying true degrees of freedom and conserved mass-like quantities.

Findings

Existence of a conserved quantity C related to mass in these models

C corresponds to black-hole mass in specific cases

Universal mass function incorporating matter contributions

Abstract

Within a first-order framework, we comprehensively examine the role played by boundary conditions in the canonical formulation of a completely general two-dimensional gravity model. Our analysis particularly elucidates the perennial themes of mass and energy. The gravity models for which our arguments are valid include theories with dynamical torsion and so-called generalized dilaton theories (GDTs). Our analysis of the canonical action principle (i) provides a rigorous correspondence between the most general first-order two-dimensional Einstein-Cartan model (ECM) and GDT and (ii) allows us to extract in a virtually simultaneous manner the ``true degrees of freedom'' for both ECMs and GDTs. For all such models, the existence of an absolutely conserved (in vacuo) quantity C is a generic feature, with (minus) C corresponding to the black-hole mass parameter in the important special cases of spherically symmetric four-dimensional general relativity and standard two-dimensional dilaton gravity. The mass C also includes (minimally coupled) matter into a ``universal mass function.'' We place particular emphasis on the (quite general) class of models within GDT possessing a Minkowski-like groundstate solution (allowing comparison between $C$ and the Arnowitt-Deser-Misner mass for such models).