Gabriele Gionti (Vatican Observatory)

Title : ON THE CANONICAL EQUIVALENCE OF JORDAN AND EINSTEIN FRAMES

Abstract :

Following a seminal paper by Dicke, physics should be invariant under redefinition of units of measurement, which implies invariance under conformal transformation of the metric coefficients. This implies the longstanding issue of the equivalence between the Jordan and the Einstein frames. It is believed, but not completely proved, that the cosmological physical observables are the same in the two frames. Our aim is to tackle this problem from the perspective of the Hamiltonian formalism. For this reason, we will perform the Hamiltonian analysis of the Brans-Dicke theory with Gibbons-Hawking-York boundary term both in the Jordan and the Einstein frames. The Hamiltonian Dirac’s constraint analysis will be carefully carried out in both cases and, contrary to several claims made in the literature in the past, it will be shown that the transformations from the Jordan to the Einstein frames do not strictly appear Hamiltonian canonical transformations on the extended phase space. We will show that if we perform a gauge fixing on the lapse and shift functions and implement them as secondary Dirac’s constraints in the ADM formalism, the primary first-class constraints become second class. Replacing Poisson Brackets with Dirac’s Brackets and solve “strongly” the second-class constraints, the transformation from the Jordan to the Einstein frames is Hamiltonian canonical transformation. We will briefly mention the study of the ADM formalism in the case of spherical symmetry both in the Jordan and the Einstein frames with appropriate boundary terms. We derive the equations of motion in both frames. Weintroduce the Fisher–Janis–Newman–Winicour naked singularity solution in the Einstein frame and Bocharova–Bronnikov-Melnikov-Bekenstein black hole solution in the Jordan frame. They are, respectively, solution of the equations of motion in the two frames and are connected by the Hamiltonian canonical transformation from the Jordan to the Einstein frames on the reduced phase space. This very fact highlights that the transformation from the Jordan to the Einstein frames is useful to generate solutions of the equations of motion in gravity.

The seminar will take place in Room S08 at the Faculty of Sciences.