Hamiltonian approach to two-dimensional dilaton gravities and invariant ADM mass

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Abstract
The formula existing in the literature for the Arnowitt-Deser-Misner (ADM) mass of two-dimensional (2D) dilaton gravity is incomplete. For example, in the case of an infalling matter shock wave 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, Callan-Giddings-Harvey-Strominger, and Russo-Susskind-Thorlacius). 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.