# Calculating the Riemann Christoffel tensor for a diagonal metric

I am trying to calculate the entries of the Riemann curvature tensor $R^m_{\phantom{m}ijk}$ for the metric $g_{ij}$.

The Riemann-Christoffel tensor is given as \begin{align} R^m_{\phantom{m}ijk} = \frac{\partial}{\partial x^j}{m \brace i\,\,k} - \frac{\partial}{\partial x^k}{m \brace i\,\,j} + {n \brace i\,\,k}{m \brace n\,\,j} - {n \brace i\,\,j}{m \brace n\,\,k} \end{align} where the Christoffel symbol of second kind are given as \begin{align} {m\brace b\,\,c} = {m\brace c\,\,b} = g^{mb} \left[ac,b\right], \end{align} and the Christoffel symbol of first kind \begin{align} [ac, b] &= [ca,b] = \frac{1}{2} \left[\frac{\partial g_{ab}}{\partial x^c} + \frac{\partial g_{bc}}{\partial x^a} - \frac{\partial g_{ac}}{\partial x^b}\right]. \end{align}

I have to basically calculate all these symbols and insert them in the Riemann curvature tensor. Is there any way I can easily keep track of the elements of this fourth-order tensor ? (Visual tricks, or simply using simplifications to end up with trivial results.) The metric $g_{ij}$ is diagonal. So obviously there will be many trivial terms.

• First, ignore the assumption that the metric is diagonal. It doesn't help. Oct 2 '15 at 12:57
• Second, why are you doing this? Oct 2 '15 at 12:58
• I agree with Deane, except if the metric is conformal to the euclidean one, in which case the formulas get much simpler. Oct 2 '15 at 12:58
• I intend to contract the fourth order tensor and find the Ricci tensor. After that I intend to perform another contraction on the Ricci tensor. I want to visualize the contracted quantity for a given metric. Oct 2 '15 at 13:07
• You want to compute the scalar curvature of your metric $g$ if I understand correctly. Still the question stands, what use would you have of the formula you will get for the scalar curvature ? Oct 2 '15 at 13:32