Finding the correct Christoffel-symbols in a 2+1D space-time

Question

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.

Answer 1

Taking the coordinate system to be $\{t,x,y\}$, with metric,

$$ds^2 = N(\vec r)^2 dt^2 - \phi(\vec r)(dx^2+ dy^2)$$

I presume $N(\vec r)$ to mean that $N =N(x,y)$, i.e. there is only a spatial dependence. Computing the Christoffel symbols is straightforward and simply requires applying the formula:

$\Gamma^t_{tx} = N^{-1}\partial_x N, \Gamma^t_{ty} = N^{-1}\partial_y N$. $\Gamma^x_{tt} = \frac{N}{\phi}\partial_x N, \Gamma^x_{xx} = - \Gamma^x_{yy} = \frac{1}{2\phi}\partial_x\phi.$ $\Gamma^{x}_{xy} = \frac{1}{2\phi}\partial_y \phi$. Then for $\Gamma^y_{ab}$ the matrix is the same, but with all derivatives w.r.t. $x \leftrightarrow y$. The scalar curvature of the space is,

$$R = \frac{1}{N\phi} \left( 2\nabla^2 N\right) - N\left[(\partial_i\phi)(\partial^i \phi) - \phi\nabla^2 \phi\right]$$

where the Laplacian is on $\mathbb R^2$ and $i=x,y$. All other curvature tensors are quite complicated in terms of $N$ and $\phi$.