I'm trying to calculate the Christoffel Symbols in a 2+1D space-time with the following metric:
$$ds^2 = N^2(\vec r)c^2dt^2-\phi(\vec r)(dx^1)^2-\phi(\vec r)(dx^2)^2$$
To find the Christoffel ymbols I need to invert the metric tensor $g_{\mu\nu}$ to $g^{\mu\nu}$.
Am I correct in assuming that this last tensor only has non-zero elements on the diagonal and that these are the corresponding elements from the covariant metric tensor inverted? (i.e. $g^{00} = \frac{1}{g_{00}}$ etc.)
Because if that's right then I don't know how I'm supposed to find the correct Christoffel symbols.
For example, when calculating $\Gamma^{2}_{\hphantom{2}12}$ I get $\frac{1}{2(-\phi)}\frac{\partial (-\phi)}{\partial x^1}$ which apparently has 1 minus too much.