Geometrodynamical Formulation of Two-Dimensional Dilaton Gravity

TL;DR

This work develops a geometrodynamical reformulation of general matterless two-dimensional dilaton gravity with an arbitrary dilatonic potential. By a Lagrangian-level transformation inspired by the topological character of 2D gravity, the theory is recast in terms of a conserved mass $M$ and the dilaton $oldsymbolphi$, yielding a Lagrangian and a canonical structure that directly reflect spacetime geometry. The canonical transformation to geometrodynamical variables generalizes the Kuchař–Varadarajan construction to arbitrary $V(oldsymbolphi)$ and reduces the quantum theory to a 0+1 dimensional system governed by the boundary mass $m$, thereby establishing a quantum Birkhoff theorem. The framework clarifies the physical observables, facilitates quantization, and provides a robust foundation for exploring quantum aspects of black holes and gravitational collapse in two dimensions, including the role of boundary terms and the potential extension to thermodynamics with matter.

Abstract

Two-dimensional matterless dilaton gravity with arbitrary dilatonic potential can be discussed in a unitary way, both in the Lagrangian and canonical frameworks, by introducing suitable field redefinitions. The new fields are directly related to the original spacetime geometry and in the canonical picture they generalize the well-known geometrodynamical variables used in the discussion of the Schwarzschild black hole. So the model can be quantized using the techniques developed for the latter case. The resulting quantum theory exhibits the Birkhoff theorem at the quantum level.