Hamiltonian Approach to 2D Dilaton-Gravities and Invariant Adm Mass

TL;DR

This paper develops a Hamiltonian framework for 2D dilaton gravity theories, correcting the ADM mass formula to ensure it remains time-independent even in dynamic scenarios like infalling matter, by including appropriate boundary terms.

Contribution

It introduces a Hamiltonian approach that accurately computes the invariant ADM mass in 2D dilaton gravity, resolving previous inconsistencies in the literature.

Findings

Corrected the ADM mass formula for 2D dilaton gravity.

Demonstrated the importance of boundary terms for a well-defined Hamiltonian.

Ensured the mass remains time-independent in dynamic scenarios.

Abstract

The formula existing in the literature for the ADM mass of 2D dilaton gravity is incomplete. For example, in the case of an infalling matter shockwave this formula fails to give a time-independent mass, unless a very special coordinate system is chosen. We carefully carry out the canonical formulation of 2D dilaton gravity theories (classical, CGHS and RST). As in 4D general relativity one must add a boundary term to the bulk Hamiltonian to obtain a well-defined variational problem. This boundary term coincides with the numerical value of the Hamiltonian and gives the correct mass which obviously is time-independent.