Dimensional analysis in relativity and in differential geometry

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) and Ricci (fully covariant) curvature tensors are dimensionless. The relation between dimension and operational meaning for the metric and stress–energy–momentum tensors is discussed; and the main conventions for the dimensions of these two tensors and of Einstein's constant are reviewed. A more thorough and updated analysis is available as a preprint.