Units in general relativity

Question

My question is pretty straight-forward: what are the units of the tensors in General Relativity? This should sound easy, but I always studied those in natural units ($c=1$) so I can't figure it out. In particular, what are the units of

• $G_{\mu\nu}$
• $g_{\mu\nu}$
• $R^\rho_{\mu\sigma\nu}$
• $R_{\mu\nu}$
• $R$
• $T_{\mu\nu}$
• $\Gamma^\lambda_{\mu\nu}$

?

Answer 1

Here's the general rule I use in my own calculations.

In cartesian like coordinates $x^{\mu} \sim \mathrm{L}^1$ : \begin{align} g_{\mu \nu} &\sim \mathrm{L}^0, \\[12pt] \Gamma_{\mu \nu}^{\lambda} &\sim \mathrm{L}^{-1}, \\[12pt] R^{\lambda}_{\; \kappa \mu \nu} &\sim \mathrm{L}^{-2}, \\[12pt] R_{\mu \nu} &\sim \mathrm{L}^{-2}, \\[12pt] G_{\mu \nu} &\sim \mathrm{L}^{-2}, \\[12pt] R &\sim \mathrm{L}^{-2}, \\[12pt] T_{\mu \nu} &\sim \mathrm{L}^{-4}, \\[12pt] \kappa \equiv 8 \pi G &\sim \mathrm{L}^{2}, \\[12pt] \end{align}

Take note that the dimensions of these quantities heavily depend on the dimensions of your coordinates, which are totally arbitrary. However, whatever the coordinates $x^{\mu}$, you have \begin{equation} ds^2 = g_{\mu \nu} \, dx^{\mu} \, dx^{\nu} \sim \mathrm{L}^2. \end{equation} Also, tensor invariants do not depend on the coordinates and have the same dimensions in all coordinates systems. For example: \begin{align} R \equiv g^{\mu \nu} \, R_{\mu \nu} &\sim \mathrm{L}^{-2}, \\[12pt] R^{\mu \nu \lambda \kappa} \, R_{\mu \nu \lambda \kappa} &\sim \mathrm{L}^{-4}. \end{align}

Answer 2

If you choose coordinates with the units of length, such as $(ct, x, y, z)$, then the metric tensor and its inverse are dimensionless, the Christoffel symbols have the dimensions of inverse length, and the curvature tensors are inverse length squared. In these coordinates the energy-momentum tensor has the dimensions of energy density.