${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{\exp\left({{{E}} {{z}}}\right)} & 0 & 0 & 0 \\ 0 & -{\exp\left({{{E}} {{z}}}\right)} & 0 & 0 \\ 0 & 0 & {-{{r}^{2}}} {{\exp\left({{{E}} {{z}}}\right)}} & 0 \\ 0 & 0 & 0 & {\frac{1}{2}} {\exp\left({{{E}} {{z}}}\right)}\end{matrix} \right]}}$ ${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{\frac{1}{\exp\left({{{E}} {{z}}}\right)}} & 0 & 0 & 0 \\ 0 & -{\frac{1}{\exp\left({{{E}} {{z}}}\right)}} & 0 & 0 \\ 0 & 0 & -{\frac{1}{{{\exp\left({{{E}} {{z}}}\right)}} {{{r}^{2}}}}} & 0 \\ 0 & 0 & 0 & \frac{2}{\exp\left({{{E}} {{z}}}\right)}\end{matrix} \right]}}$ ...produces...
conn from manual metric
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & {\frac{1}{2}} {E} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ {\frac{1}{2}} {E} & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{2}} {E} \\ 0 & 0 & -{r} & 0 \\ 0 & {\frac{1}{2}} {E} & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & \frac{1}{r} & 0 & {\frac{1}{2}} {E} \\ 0 & 0 & {\frac{1}{2}} {E} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} E & 0 & 0 & 0 \\ 0 & E & 0 & 0 \\ 0 & 0 & {{E}} {{{r}^{2}}} & 0 \\ 0 & 0 & 0 & {\frac{1}{2}} {E}\end{matrix} \right]}\end{matrix} \right]}}$ vs manual conn ${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & E \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ E & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -{r} & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & \frac{1}{r} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} -{E} & 0 & 0 & 0 \\ 0 & E & 0 & 0 \\ 0 & 0 & {{E}} {{{r}^{2}}} & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
Ricci from manual metric
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {E}^{2} & 0 & 0 & 0 \\ 0 & {E}^{2} & 0 & 0 \\ 0 & 0 & {{{E}^{2}}} {{{r}^{2}}} & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}}$
Ricci from manual conn
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {E}^{2} & 0 & 0 & 0 \\ 0 & {E}^{2} & 0 & 0 \\ 0 & 0 & {{{E}^{2}}} {{{r}^{2}}} & 0 \\ 0 & 0 & 0 & -{{E}^{2}}\end{matrix} \right]}}$
desired Ricci
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {E}^{2} & 0 & 0 & 0 \\ 0 & {E}^{2} & 0 & 0 \\ 0 & 0 & {{{r}^{2}}} {{{E}^{2}}} & 0 \\ 0 & 0 & 0 & -{{E}^{2}}\end{matrix} \right]}}$

manual metric Gaussian -- equal to zero according to EM stress-energy trace:
${0} = {-{\frac{{{3}} {{{E}^{2}}}}{\exp\left({{{E}} {{z}}}\right)}}}$
${\frac{{{3}} {{{E}^{2}}}}{\exp\left({{{E}} {{z}}}\right)}} = {0}$

${{{{{ R} ^a} _b} _c} _d} = {{{{{{{{{ \Gamma} ^a} _b} _d} _{,c}} - {{{{{ \Gamma} ^a} _b} _c} _{,d}}} + {{{{{{ \Gamma} ^a} _e} _c}} {{{{{ \Gamma} ^e} _b} _d}}}} - {{{{{{ \Gamma} ^a} _e} _d}} {{{{{ \Gamma} ^e} _b} _c}}}} - {{{{{{ \Gamma} ^a} _b} _e}} {{\left({{{{{ \Gamma} ^e} _d} _c} - {{{{ \Gamma} ^e} _c} _d}}\right)}}}}$
${{{ R} _a} _b} = {{{{{{{{{ \Gamma} ^c} _a} _b} _{,c}} - {{{{{ \Gamma} ^c} _a} _c} _{,b}}} + {{{{{{ \Gamma} ^c} _d} _c}} {{{{{ \Gamma} ^d} _a} _b}}}} - {{{{{{ \Gamma} ^c} _d} _b}} {{{{{ \Gamma} ^d} _a} _c}}}} - {{{{{{ \Gamma} ^c} _a} _d}} {{\left({{{{{ \Gamma} ^d} _b} _c} - {{{{ \Gamma} ^d} _c} _b}}\right)}}}}$