General Relativity and Weyl Frames

TL;DR

This paper develops a gravitational theory within Weyl geometry, showing it can reproduce general relativity in a specific frame and offers a new perspective on gravitational phenomena through Weyl frames.

Contribution

It introduces a Weyl-invariant scalar-tensor gravitational theory that generalizes general relativity using non-Riemannian geometry and explores its relation to Brans-Dicke theory.

Findings

The theory is invariant under Weyl transformations.

It reduces to general relativity in the Riemann frame.

A known Brans-Dicke solution can be reinterpreted in a Weyl frame.

Abstract

We present the general theory of relativity in the language of a non-Riemannian geometry, namely, Weyl geometry. We show that the new mathematical formalism may lead to different pictures of the same gravitational phenomena, by making use of the concept of Weyl frames. We show that, in this formalism, it is possible to construct a scalar-tensor gravitational theory that is invariant with respect to the so-called Weyl tranformations and reduces to general relativity in a particular frame, the Riemann frame. In this approach the Weyl geometry plays a fundamental role since it appears as the natural geometrical setting of the theory when viewed in an arbitrary frame. Our starting point is to build an action that is manifestly invariant with respect to Weyl transformations. When this action is expressed in more familiar terms of Riemannian geometry we find that the theory has some similarities with Brans-Dicke theory of gravity. We illustrate this point with an example in which a known Brans-Dicke vacuum solution may appear when reinterpreted in a particular Weyl frame.