Conformal transformations and conformal invariance in gravitation

TL;DR

This paper explores conformal transformations in gravity theories, detailing their effects on geometric quantities and matter conservation, and discusses constructing conformally invariant theories and their relation to string theory dualities.

Contribution

It provides detailed transformation rules for curvature invariants and matter tensors, and introduces a novel perspective on conformal invariance akin to superstring dualities.

Findings

Transformation rules for curvature invariants derived

Matter energy-momentum conservation law's subtlety explained

Conformal invariance related to superstring dualities

Abstract

Conformal transformations are frequently used tools in order to study relations between various theories of gravity and the Einstein relativity. In this paper we discuss the rules of these transformations for geometric quantities as well as for the matter energy-momentum tensor. We show the subtlety of the matter energy-momentum conservation law which refers to the fact that the conformal transformation "creates" an extra matter term composed of the conformal factor which enters the conservation law. In an extreme case of the flat original spacetime the matter is "created" due to work done by the conformal transformation to bend the spacetime which was originally flat. We discuss how to construct the conformally invariant gravity theories and also find the conformal transformation rules for the curvature invariants $R^2$, $R_{ab}R^{ab}$, $R_{abcd}R^{abcd}$ and the Gauss-Bonnet invariant in a spacetime of an arbitrary dimension. Finally, we present the conformal transformation rules in the fashion of the duality transformations of the superstring theory. In such a case the transitions between conformal frames reduce to a simple change of the sign of a redefined conformal factor.