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\begin{center}
{\large\bf From the Appendix of Rindler\\ 
Ricci tensor for a diagonal metric.}
\end{center} \large
\medskip

The appendix of Rindler (p.419-421) deals with the metric
\begin{eqnarray} \label{dmetric}
  d\vec{s}^{\,2} = a\,(dx^1)^2+b\,(dx^2)^2+c\,(dx^3)^2+d\,(dx^4)^2\,,
\end{eqnarray}
where $a,\,b,\,c,\,d$ are functions of the coordinates. Further notation
is
\begin{eqnarray} \label{alpha}
  \alpha = \frac{1}{2a}\,,~~~\beta=\frac{1}{2b}\,,~~~
  \gamma = \frac{1}{2c}\,,~~~\delta=\frac{1}{2d}\,,
\end{eqnarray}
and 
\begin{eqnarray} \label{derivatives}
  a_{i} = \frac{\partial a}{\partial x^i}\,,~~~~a_{ij} 
  = \frac{\partial^2 a}{\partial x^i\,\partial x^j}\,,
  ~~{\rm and\ so\ on,}
\end{eqnarray}
where we allow $i,j=1,2,3,4$.
From the definitions of the Christoffel symbols 
\begin{eqnarray} \label{Chris1}
  \Gamma_{ijk}=\frac{1}{2}\,\left(g_{ij,k}+g_{ik,j}-g_{jk,i}\right)\,,
  ~~~~\Gamma^i_{~jk}=g^{ih}\,\Gamma_{hjk} 
\end{eqnarray}
[Rindler (10.13) and (10.14), p.206] one finds
\begin{eqnarray} \label{Chris2}
  \Gamma^1_{~23}=0\,,~~~\Gamma^1_{~22}=-\alpha\,b_1\,,~~~
  \Gamma^1_{~1i}= \alpha\,a_i\,.
\end{eqnarray}
From these trheee typical $\Gamma$s all other can be obtained by 
the obvious permutations. For example, $\Gamma^2_{~33}=-\beta\,c_2$,
$\Gamma^4_{~44}=-\delta\,d_2$.

After calculating the Riemann curvature tensor first, the Ricci 
tensor $R_{ij}$ as defined by Rindler (10.68), p.219 becomes
\begin{eqnarray} \nonumber
  R_{11}\ =\ &+& \beta\,a_{22} + \gamma\,a_{33} + \delta\,a_{44}
  \\ \nonumber
             &+& \beta\,b_{11} + \gamma\,c_{11} + \delta\,d_{11}
  \\ \nonumber
             &-& \beta^2\,b_1^2 - \gamma^2\,c_1^2 - \delta^2\,d_1^2
  \\ \nonumber - \alpha\,a_1\ (
             &+& \beta\,b_1 + \gamma\,c_1 + \delta\,d_1~~)
  \\ \nonumber - \beta\,a_2\ (~~\alpha\,a_2
             &+& \beta\,b_2 - \gamma\,c_2 - \delta\,d_2~~)
  \\ \nonumber - \gamma\,a_3\ (~~\alpha\,a_3
             &-& \beta\,b_3 + \gamma\,c_3 - \delta\,d_3~~)\\ 
               - \delta\,a_4\ (~~\alpha\,a_4
             &-& \beta\,b_4 - \gamma\,c_4 + \delta\,d_4~~)
\end{eqnarray}
\begin{eqnarray} \nonumber
  R_{12}\ =\ &+& \gamma\,c_{12}~~~~~\,+~\ \delta\,d_{12} \\ \nonumber
  &-& \gamma^2\,c_1\,c_2~~\, -~\ \delta^2\,d_1\,d_2 \\ \nonumber
  &-& \alpha\,\beta\,a_2\,c_1\ -~\ \alpha\,\delta\,a_2\,d_1\\
  &-& \beta\,\gamma\,b_1\,c_2~\, -~\ \beta\,\delta\,b_1\,d_2
\end{eqnarray}

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