Dimensional analysis in relativity and in differential geometry

TL;DR

This paper reviews dimensional analysis in relativity and differential geometry, emphasizing the intrinsic dimension of tensors and clarifying the dimensions of key tensors like Riemann, Ricci, and Einstein tensors.

Contribution

It revisits the concept of intrinsic tensor dimension and summarizes the dimensional properties of important tensors in relativity.

Findings

Riemann, Ricci, and Einstein tensors are dimensionless.

Tensor components can have different dimensions.

Discusses conventions for tensor dimensions and Einstein's constant.

Abstract

This note provides a short guide to dimensional analysis in Lorentzian and general relativity and in differential geometry. It tries to revive Dorgelo and Schouten's notion of 'intrinsic' or 'absolute' dimension of a tensorial quantity. The intrinsic dimension is independent of the dimensions of the coordinates and expresses the physical and operational meaning of a tensor. The dimensional analysis of several important tensors and tensor operations is summarized. In particular it is shown that the components of a tensor need not have all the same dimension, and that the Riemann (once contravariant and thrice covariant), Ricci (twice covariant), and Einstein (twice covariant) curvature tensors are dimensionless. The relation between dimension and operational meaning for the metric and stress-energy-momentum tensors is discussed; and the possible conventions for the dimensions of these two tensors and of Einstein's constant $\kappa$, including the curious possibility $\kappa = 8\pi G$ without $c$ factors, are reviewed.