In this answer it is stated that the metric tensor elements have no physical unit, i.e. $[g_{\mu\nu}] = 1$. What is the convention to get the physical unit of the line element $ds = g_{\mu\nu}dx^\mu dx^\nu$ right. I assume that $[dx^0] = s$ (seconds), and $[dx^i] = m$ (meter) for $i=1,2,3$. The flat spacetime, $$ ds^2 = -c^2dt^2 + dx^2+dy^2+dz^2$$

would suggest that $[g_{00}] = [c^2] = (m/s)^2$.

What is a (the) convention to get the units right.

  • $\begingroup$ There is no universal convention on this. As an example of how conventions can vary, see Dicke, Phys Rev 125 (1962) 2163. He lets the metric have units of distance. I have a detailed discussion in section 5.11 of my GR book, lightandmatter.com/genrel . $\endgroup$ – Ben Crowell Feb 25 '18 at 16:23
  • $\begingroup$ @BenCrowell - have you considered publishing your book as a paperback using Kindle Direct Publishing? It's free and reasonably straightforward. You just need a content file pdf and a cover pdf. Apologies if this sort of comment should be in chat. $\endgroup$ – Peter4075 Apr 3 at 16:49

The typical convention is $x_0=ct$, so $[dx_0]=[dx_i]=~\rm m$.

Although people frequently set $c=1$ to simplify things, and then the whole point is moot.


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