Aspects of Geometrodynamics in the Jordan and Einstein Frames

TL;DR

This paper explores the Hamiltonian equivalence between Jordan and Einstein frames in Brans-Dicke theory, analyzing spherically symmetric solutions and boundary terms, and demonstrating their connection via canonical transformations.

Contribution

It provides a detailed Hamiltonian analysis and ADM formulation of spherically symmetric solutions, clarifying the relationship between the two frames through canonical transformations.

Findings

Hamiltonian equivalence established between frames

ADM analysis of spherically symmetric solutions

Explicit connection via canonical transformations

Abstract

We will summarize recent results on the Hamiltonian equivalence between the Jordan and Einstein frames based on the analysis of Brans-Dicke theory for both cases \omega\neq -\frac{3}{2} and \omega =-\frac{3}{2}. We will introduce and perform ADM analysis for spherically symmetric solutions of gravity. We will discuss with particular care the problem of the boundary terms to be introduced in the general case of spherical symmetry. These two frames are connected through a Hamiltonian canonical transformation on the reduced phase space obtained by gauge fixing the lapse and the radial shift functions. We introduce and discuss two static solutions (Fisher, Janis, Newman and Winicour solution in the Einstein frame and Bocharova-Bronnikov-Melnikov-Bekenstein black hole solution in the Jordan frame)