On the Canonical Reduction of Spherically Symmetric Gravity

TL;DR

This paper interprets Kuchar's canonical reduction of spherically symmetric gravity using quasilocal energy concepts, extends it to 2D dilaton gravity, and applies it to Witten black holes, providing new insights and generalizations.

Contribution

It offers a geometric interpretation of Kuchar's canonical transformation and extends the reduction framework to 2D dilaton gravity and Witten black holes.

Findings

Kuchar's transformation is a sphere-dependent boost to the rest frame.

The formalism applies to vacuum 2D dilaton gravity.

New results on the canonical reduction of Witten-black-hole geometrodynamics.

Abstract

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. This reduction yields the true degrees of freedom for (vacuum) spherically symmetric general relativity. 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 $M_{S}$, expressed in terms of what are essentially Arnowitt-Deser-Misner variables, as a canonical coordinate. In this paper we discuss the geometric interpretation of Kuchar's canonical transformation in terms of 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," where the ``rest frame'' is defined by vanishing quasilocal momentum. Furthermore, our formalism is general enough to cover the case of (vacuum) two-dimensional dilaton gravity. Therefore, besides reviewing Kucha\v{r}'s original work for Schwarzschild black holes from the framework of hyperbolic geometry, we present new results concerning the canonical reduction of Witten-black-hole geometrodynamics.