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 $g_{munu}$ to $g^{munu}$.
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.

JamalS · Answer

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:

$Gamma^t_{tx} = N^{-1}partial_x N, Gamma^t_{ty} = N^{-1}partial_y N$. $Gamma^x_{tt} = frac{N}{phi}partial_x N, Gamma^x_{xx} = - Gamma^x_{yy} = frac{1}{2phi}partial_xphi.$ $Gamma^{x}_{xy} = frac{1}{2phi}partial_y phi$. 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,

$$R = frac{1}{Nphi} left( 2nabla^2 Nright) - Nleft[(partial_iphi)(partial^i phi) - phinabla^2 phiright]$$

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

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

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