in relativity

and in differential

P.G.L. Porta Mana

Western Norway University of Applied Sciences, Bergen

13 December 2019; updated 20 May 2021

This note provides a short guide to dimensional analysis in Lorentzian and and in . It tries to revive Dorgelo and

Schouten’s notion of ‘intrinsic’ or ‘absolute’ of a tensorial quantity.

The is independent of the of the coordinates and expresses the physical and operational meaning of a . The dimensional analysis of several important 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) tensors are dimensionless. The relation between dimension and operational meaning for the and stress-energy-momentum tensors is discussed; and the possible conventions for the dimensions of these two tensors and of Einstein’s constant κ , including the curious possibility κ = 8π 퐺 without 푐 factors, are reviewed.

per la piccola Emma arXiv:2007.14217v4 [gr-qc] 20 May 2021

1 Introduction paper

From the of view of dimensional analysis, do all components Letter or of a tensor need to have the same dimension? What happens to these A4 on components if we choose coordinates that don’t all have dimensions of or ? And if the components of a tensor have different printing dimensions, then does it make sense to speak of “the dimension of the

tensor”? What are the dimensions of the metric and of the curvature two-up and tensors? What is the dimension of the constant in the Einstein ?

A sense of insecurity gets hold of many students (and possibly of reading some researchers) in relativity, when they have to discuss and answer screen this kind of questions. This is evident in many question & answer web- for sites and wiki pages, where several incorrect or unfounded statements about dimensional analysis in relativity are in circulation. (For the re- designed is cord, the answers to the questions above are: 1. No, they don’t. 2. See document

1 This

P orta M ana Dimensional analysis, relativity,

1 1 2 1 eq. (12) below. 3. Yes. 4. They are dimensionless. 5. M− L− T , or M− L ,

1 3 2 or M− L T− , depending on the dimensions you assign to the metric and stress-energy-momentum tensors, see §§ 9–11. )

Several factors contribute to these misconceptions and insecurity.

Modern texts in Lorentzian and general relativity commonly use geomet- rized units. They say that, for finding the dimension of some constant in a tensorial , it’s sufficient to compare the dimensions of the terms in the equation. But the application of this procedure is sometimes not so immediate, because some tensors don’t have universally agreed dimensions – prime example the . Many texts use four coordinates with dimension of length, and base their dimensional ana- 1 lyses on that specific choice