Scalar conformal invariants of weight zero

TL;DR

This paper investigates scalar conformal invariants of weight zero in pseudo-Riemannian conformal structures, introduces a conformal scalar curvature, and compares it with classical curvature scalars in known spacetimes.

Contribution

It introduces a family of scalar conformal invariants of weight zero and defines the conformal scalar curvature for generic conformal structures in higher dimensions.

Findings

Defined the conformal scalar curvature for conformal structures.

Calculated the conformal scalar curvature for well-known spacetimes.

Compared conformal scalar curvature with Ricci and Kretschmann scalars.

Abstract

In the class of metrics of a generic conformal structure there exists a distinguishing metric. This was noticed by Albert Einstein in a lesser-known paper of 1921 (Berl. Ber., 1921, pp. 261-264). We explore this finding from a geometrical point of view. Then, we obtain a family of scalar conformal invariants of weight 0 for generic pseudo-Riemannian conformal structures $[g]$ in more than three dimensions. In particular, we define the conformal scalar curvature of $[g]$ and calculate it for some well-known conformal spacetimes, comparing the results with the Ricci scalar and the Kretschmann scalar. In the cited paper, Einstein also announced that it is possible to add an scalar equation to the field equations of General Relativity.