Am I using the correct metric?

Question

I am trying to calculate the Christoffel symbol for the Schwarzschild metric, which I write:$$ds^2=-F(r) dt^2+F(r)^{-1}dr^2+r^2 d\theta^2 + r^2\, \sin^2 \theta d\phi^2$$ with $$F(r)=1-\dfrac{2m}{r}$$ (I am imposing $c=G=1$).

When calculating the Christoffel symbols using the formula $$\Gamma^\mu_{\nu\rho} = \dfrac{1}{2}g^{\mu\sigma}\left( g_{\nu\sigma,\rho}+g_{\rho\sigma,\nu} - g_{\nu\rho,\sigma} \right)$$ I get terms like: $$\Gamma^r_{\theta\theta}=2m-r.$$ Are these terms correct? I thought that the Christoffel symbols had unit of an inverse length. To check my result I have used the code of this answer and it gives the same Christoffel symbols that I have calculated; moreover, when calculating the Ricci tensor with the Mathematica code I get 0 and, analogously, calculating the Riemann tensor I get terms that do not have the dimension of an inverse squared length, so there is definetly something wrong in my calculations, in the code or in the metric which I use.

Since the code and my calculations give the same result, I think I am using the wrong metric, but I do not see any error.

Can anyone help?

Answer 1

The Christoffel symbols only have dimensions of length$^{-1}$ when the coordinates have units of length. In your case, you are working with spherical coordinates, which has two coordinates without units of length (they are angles), and so it doesn't always return dimensions of length$^{-1}$. For example, $$\Gamma^r_{rr}=-\frac{m}{r^2-2mr}$$ which has units of inverse length. However, $$\Gamma^\theta_{\phi\phi}=-\sin\theta\cos\theta$$ which has no units of length at all. It really depends on your coordinates and their dimension. The same thing occurs when you calculate the Christoffel symbols for Minkowski space in spherical coordinates: some will have dimensions of length$^{-1}$, others are dimensionless.