A Conformally Invariant Theory of Gravitation in Metric Measure Space

TL;DR

This paper develops a conformally invariant gravitational theory within metric measure space, generalizing Einstein's equations while maintaining metricity and integrability, and explores its geometric properties and relations to Weyl space.

Contribution

It introduces a new conformally invariant gravitational framework in metric measure space, extending Einstein's equations and analyzing its geometric identities.

Findings

Generalized Einstein equations in metric measure space.

Invariance under diffeomorphism and conformal transformations.

Similarity of Bianchi identities with Weyl space.

Abstract

In this manuscript, a conformally invariant theory of gravitation in the context of metric measure space is studied. The proposed action is invariant under both diffeomorphism and conformal transformations. Using the variational method, a generalization of the Einstein equation is obtained, wherein the conventional tensors are replaced by their conformally invariant counterparts, living in metric measure space. The invariance of the geometrical part of the action under a diffeomorphism leads to a generalized contracted second Bianchi identity. In metric measure space, the covariant derivative is the same as it is in the Riemannian space. Hence, in contrast to the Weyl space, the metricity and integrability are maintained. However, it is worth noting that in metric measure space the divergence of a tensor is not simply the contraction of the covariant derivative operator with the tensor that it acts on. Despite the fact that metric measure space and integrable Weyl space, are constructed based on different assumptions, it is shown that some relations in these spaces, such as the contracted second Bianchi identity, are completely similar.