On the canonical reduction of spherically symmetric gravity

In a thorough paper Kuchar has examined the canonical reduction of the most general action functional describing the geometrodynamics of the maximally extended Schwarzschild geometry. The essential technical ingredient in Kuchar's analysis is a canonical transformation to a certain chart on the gravitational phase space which features the Schwarzschild mass parameter MS (expressed off-shell in terms of ADM-like variables) as a canonical coordinate. In this paper we reveal the geometric interpretation of Kuchar's canonical transformation. We do this by appealing to the theory of quasilocal energy-momentum in general relativity given by Brown and York. We find Kuchar's transformation to be a "sphere-dependent boost to the rest frame" (defined by vanishing quasilocal momentum). Furthermore, our formalism is robust enough to include the pure-dilaton model of Callan, Giddings, Harvey, Strominger, and Witten. Therefore, besides reviewing Kuchar's original work for the Schwarzschild case from the framework of hyperbolic geometry, we present new results concerning the canonical reduction of Witten-black-hole geometrodynamics. Finally, addressing a recent work of Louko and Whiting, we discuss some delicate points concerning the canonical reduction of the "thermodynamical action," which is of central importance in the path-integral formulation of gravitational thermodynamics.