Spherically Symmetric Geometrodynamics in Jordan and Einstein frames

TL;DR

This paper explores spherically symmetric geometrodynamics within scalar-tensor theories and General Relativity, analyzing the Hamiltonian formalism in both Jordan and Einstein frames and examining the impact of singular transformations on static solutions.

Contribution

It derives the Hamiltonian equations of motion in both frames and investigates the effects of singular transformations on static solutions in scalar-tensor theories.

Findings

Hamiltonian formalism established in both frames

Transformation singularities affect static solutions

Connection between frames via canonical transformation

Abstract

Spherically symmetric geometrodynamics is studied for scalar-tensor theory and Einstein General Relativity minimally coupled to a scalar field. We discussed the importance of boundary terms and derived the equations of motion in the Hamiltonian canonical formalism both in the Jordan and Einstein frames. These two frames are connected through an Hamiltonian canonical transformation on the reduced phase space obtained gauge-fixing the lapse and the radial shift functions. We discussed the effects of the singularity of the Hamiltonian canonical transformation connecting Jordan and Einstein frames for 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).