Conformally flat spacetimes and Weyl frames

TL;DR

This paper explores the equivalence of Weyl and Riemann frames in metric gravity, focusing on conformally flat spacetimes, and shows how gravity can be described by a scalar field in the Weyl frame, linking it to Newtonian potential and scalar theories.

Contribution

It introduces a new perspective on conformally flat spacetimes using Weyl frames, relating gravity to scalar fields and revisiting classical scalar gravity theories.

Findings

Weyl and Riemann frames are equivalent for geodesic motion.

In Weyl frames, gravity is described by a scalar field.

The Weyl scalar field can be identified with the Newtonian potential in weak fields.

Abstract

We discuss the concepts of Weyl and Riemann frames in the context of metric theories of gravity and state the fact that they are completely equivalent as far as geodesic motion is concerned. We apply this result to conformally flat spacetimes and show that a new picture arises when a Riemannian spacetime is taken by means of geometrical gauge transformations into a Minkowskian flat spacetime. We find out that in the Weyl frame gravity is described by a scalar field. We give some examples of how conformally flat spacetime configurations look when viewed from the standpoint of a Weyl frame. We show that in the non-relativistic and weak field regime the Weyl scalar field may be identified with the Newtonian gravitational potential. We suggest an equation for the scalar field by varying the Einstein-Hilbert action restricted to the class of conformally-flat spacetimes. We revisit Einstein and Fokker's interpretation of Nordstr\"om scalar gravity theory and draw an analogy between this approach and the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as viewed in the Weyl frame and address the question of quantizing a conformally flat spacetime by going to the Weyl frame.