# Weyl relativity: A novel approach to Weyl's ideas

## TL;DR

This paper introduces Weyl relativity, a new Weyl-invariant theory unifying gravity and electromagnetism with second-order equations, compatible with known tests, and offering potential cosmological insights.

## Contribution

It develops a Weyl-invariant framework that unifies Einstein and Maxwell equations with second-order field equations and stable cosmological constant.

## Key findings

- Equivalent to trace-free Einstein equations
- Radiatively stable cosmological constant
- Potential cosmological effects from non-integrability of spacetime

## Abstract

In this paper we revisit the motivation and construction of a unified theory of gravity and electromagnetism, following Weyl's insights regarding the appealing potential connection between the gauge invariance of electromagnetism and the conformal invariance of the gravitational field. We highlight that changing the local symmetry group of spacetime permits to construct a theory in which these two symmetries are combined into a putative gauge symmetry but with second-order field equations and non-trivial mass scales, unlike the original higher-order construction by Weyl. We prove that the gravitational field equations are equivalent to the (trace-free) Einstein field equations, ensuring their compatibility with known tests of general relativity. As a corollary, the effective cosmological constant is rendered radiatively stable due to Weyl invariance. A novel phenomenological consequence characteristic of this construction, potentially relevant for cosmological observations, is the existence of an energy scale below which effects associated with the non-integrability of spacetime distances, and an effective mass for the electromagnetic field, appear simultaneously (as dual manifestations of the use of Weyl connections). We explain how former criticisms against Weyl's ideas lose most of their power in its present reincarnation, which we refer to as Weyl relativity, as it represents a Weyl-invariant, unified description of both the Einstein and Maxwell field equations.