coordinate chart
${{ u} ^A} = {\overset{A\downarrow}{\left[ \begin{matrix} t \\ {{r}} {{\cos\left( \phi\right)}} \\ {{r}} {{\sin\left( \phi\right)}} \\ z\end{matrix} \right]}}$
${{{ u} ^A} _{,a}} = {\overset{A\downarrow a\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos\left( \phi\right) & -{{{r}} {{\sin\left( \phi\right)}}} & 0 \\ 0 & \sin\left( \phi\right) & {{r}} {{\cos\left( \phi\right)}} & 0 \\ 0 & 0 & 0 & 1\end{matrix} \right]}}$
cartesian to cylindrical
${{{ e} _a} ^A} = {\overset{a\downarrow A\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos\left( \phi\right) & \sin\left( \phi\right) & 0 \\ 0 & -{{{r}} {{\sin\left( \phi\right)}}} & {{r}} {{\cos\left( \phi\right)}} & 0 \\ 0 & 0 & 0 & 1\end{matrix} \right]}}$
cylindrical to cartesian
${{{ e} ^a} _A} = {\overset{a\downarrow A\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos\left( \phi\right) & \sin\left( \phi\right) & 0 \\ 0 & -{{\frac{1}{r}} {\sin\left( \phi\right)}} & {\frac{1}{r}} {\cos\left( \phi\right)} & 0 \\ 0 & 0 & 0 & 1\end{matrix} \right]}}$
${{{{{ e} _a} ^A}} {{{{ e} ^b} _A}}} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{matrix} \right]}}$
${{{{{ e} _a} ^A}} {{{{ e} ^a} _B}}} = {\overset{A\downarrow B\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{matrix} \right]}}$
cartesian Ricci tensor
${{{ R} _A} _B} = {\overset{A\downarrow B\rightarrow}{\left[ \begin{matrix} {E}^{2} & 0 & 0 & 0 \\ 0 & {E}^{2} & 0 & 0 \\ 0 & 0 & {E}^{2} & 0 \\ 0 & 0 & 0 & -{{E}^{2}}\end{matrix} \right]}}$
cartesian Ricci tensor transformed to cylindrical
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {E}^{2} & 0 & 0 & 0 \\ 0 & {E}^{2} & 0 & 0 \\ 0 & 0 & {{{E}^{2}}} {{{r}^{2}}} & 0 \\ 0 & 0 & 0 & -{{E}^{2}}\end{matrix} \right]}}$
cartesian connection that gives rise to cartesian Ricci of EM stress-energy tensor:
${{{{ \Gamma} ^t} _t} _z} = {E}$; ${{{{ \Gamma} ^t} _z} _t} = {E}$; ${{{{ \Gamma} ^z} _t} _t} = {-{E}}$; ${{{{ \Gamma} ^z} _x} _x} = {E}$; ${{{{ \Gamma} ^z} _y} _y} = {E}$
Ricci from Riemann from cartesian connection
${{{ R} _A} _B} = {\overset{A\downarrow B\rightarrow}{\left[ \begin{matrix} {E}^{2} & 0 & 0 & 0 \\ 0 & {E}^{2} & 0 & 0 \\ 0 & 0 & {E}^{2} & 0 \\ 0 & 0 & 0 & -{{E}^{2}}\end{matrix} \right]}}$
cartesian connection transformed to cylindrical
${{{{ \Gamma} ^t} _t} _z} = {E}$; ${{{{ \Gamma} ^t} _z} _t} = {E}$; ${{{{ \Gamma} ^r} _{\phi}} _{\phi}} = {-{r}}$; ${{{{ \Gamma} ^{\phi}} _r} _{\phi}} = {\frac{1}{r}}$; ${{{{ \Gamma} ^{\phi}} _{\phi}} _r} = {\frac{1}{r}}$; ${{{{ \Gamma} ^z} _t} _t} = {-{E}}$; ${{{{ \Gamma} ^z} _r} _r} = {E}$; ${{{{ \Gamma} ^z} _{\phi}} _{\phi}} = {{{E}} {{{r}^{2}}}}$
Ricci from connections transformed from cartesian to cylindrical
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {E}^{2} & 0 & 0 & 0 \\ 0 & {E}^{2} & 0 & 0 \\ 0 & 0 & {{{E}^{2}}} {{{r}^{2}}} & 0 \\ 0 & 0 & 0 & -{{E}^{2}}\end{matrix} \right]}}$ ...matches the Ricci transformed to cylindrical