${{ A} _u} = {\overset{u\downarrow}{\left[ \begin{matrix} {A_t} \\ {A_r} \\ {A_\phi} \\ {A_z}\end{matrix} \right]}}$
${{ E} _i} = {\overset{i\downarrow}{\left[ \begin{matrix} {E_r} \\ {E_\phi} \\ {E_z}\end{matrix} \right]}}$
${{ B} _i} = {\overset{i\downarrow}{\left[ \begin{matrix} {B_r} \\ {B_\phi} \\ {B_z}\end{matrix} \right]}}$
${{ S} _i} = {\overset{i\downarrow}{\left[ \begin{matrix} {S_r} \\ {S_\phi} \\ {S_z}\end{matrix} \right]}}$
Looking for a Riemann (and then a Christoffel) that gives rise to the electromagnetism stress-energy tensor
using ${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{1} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {r}^{2} & 0 \\ 0 & 0 & 0 & 1\end{matrix} \right]}}$
${{{ \gamma} _i} _j} = {\overset{i\downarrow j\rightarrow}{\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & {r}^{2} & 0 \\ 0 & 0 & 1\end{matrix} \right]}}$
$\sqrt{-g} =$ $r$
$\sqrt{\gamma} =$ $r$
${{{{ \epsilon} _i} _j} _k} = {\overset{i\downarrow[{j\downarrow k\rightarrow}]}{\left[ \begin{matrix} \overset{j\downarrow k\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & r \\ 0 & -{r} & 0\end{matrix} \right]} \\ \overset{j\downarrow k\rightarrow}{\left[ \begin{matrix} 0 & 0 & -{r} \\ 0 & 0 & 0 \\ r & 0 & 0\end{matrix} \right]} \\ \overset{j\downarrow k\rightarrow}{\left[ \begin{matrix} 0 & r & 0 \\ -{r} & 0 & 0 \\ 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{{{ \epsilon} _a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & r \\ 0 & 0 & -{r} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -{r} \\ 0 & 0 & 0 & 0 \\ 0 & r & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & r & 0 \\ 0 & -{r} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -{r} \\ 0 & 0 & r & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & r \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ -{r} & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & -{r} & 0 \\ 0 & 0 & 0 & 0 \\ r & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & r \\ 0 & 0 & 0 & 0 \\ 0 & -{r} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & -{r} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ r & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & r & 0 & 0 \\ -{r} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -{r} & 0 \\ 0 & r & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & r & 0 \\ 0 & 0 & 0 & 0 \\ -{r} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{r} & 0 & 0 \\ r & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{ F} _a} _b} = {{{{ A} _b} _{,a}} - {{{ A} _a} _{,b}}}$
${{{ F} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & {\partial_ {{t}}\left( {A_r}\right)} - {\partial_ {{r}}\left( {A_t}\right)} & {\partial_ {{t}}\left( {A_\phi}\right)} - {\partial_ {{\phi}}\left( {A_t}\right)} & {\partial_ {{t}}\left( {A_z}\right)} - {\partial_ {{z}}\left( {A_t}\right)} \\ {\partial_ {{r}}\left( {A_t}\right)} - {\partial_ {{t}}\left( {A_r}\right)} & 0 & {\partial_ {{r}}\left( {A_\phi}\right)} - {\partial_ {{\phi}}\left( {A_r}\right)} & {\partial_ {{r}}\left( {A_z}\right)} - {\partial_ {{z}}\left( {A_r}\right)} \\ {\partial_ {{\phi}}\left( {A_t}\right)} - {\partial_ {{t}}\left( {A_\phi}\right)} & {\partial_ {{\phi}}\left( {A_r}\right)} - {\partial_ {{r}}\left( {A_\phi}\right)} & 0 & {\partial_ {{\phi}}\left( {A_z}\right)} - {\partial_ {{z}}\left( {A_\phi}\right)} \\ {\partial_ {{z}}\left( {A_t}\right)} - {\partial_ {{t}}\left( {A_z}\right)} & {\partial_ {{z}}\left( {A_r}\right)} - {\partial_ {{r}}\left( {A_z}\right)} & {\partial_ {{z}}\left( {A_\phi}\right)} - {\partial_ {{\phi}}\left( {A_z}\right)} & 0\end{matrix} \right]}}$
${{ B} _i} = {{{{{{ \epsilon} _i} ^j} ^k}} {{{{ A} _k} _{,j}}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}}$
${\overset{i\downarrow}{\left[ \begin{matrix} {B_r} \\ {B_\phi} \\ {B_z}\end{matrix} \right]}} = {\overset{i\downarrow}{\left[ \begin{matrix} {\frac{1}{r}}{\left({{-{\partial_ {{z}}\left( {A_\phi}\right)}} + {\partial_ {{\phi}}\left( {A_z}\right)}}\right)} \\ {{r}} {{\left({{\partial_ {{z}}\left( {A_r}\right)} - {\partial_ {{r}}\left( {A_z}\right)}}\right)}} \\ {\frac{1}{r}}{\left({{-{\partial_ {{\phi}}\left( {A_r}\right)}} + {\partial_ {{r}}\left( {A_\phi}\right)}}\right)}\end{matrix} \right]}}$
${\partial_ {{\phi}}\left( {A_z}\right)} = {{{{{B_r}}} \cdot {{r}}} + {\partial_ {{z}}\left( {A_\phi}\right)}}$
${\partial_ {{z}}\left( {A_r}\right)} = {{\frac{1}{r}}{\left({{{{r}} {{\partial_ {{r}}\left( {A_z}\right)}}} + {{B_\phi}}}\right)}}$
${\partial_ {{r}}\left( {A_\phi}\right)} = {{{{{B_z}}} \cdot {{r}}} + {\partial_ {{\phi}}\left( {A_r}\right)}}$
${{{ F} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & -{{E_r}} & -{{E_\phi}} & -{{E_z}} \\ {E_r} & 0 & {{{B_z}}} \cdot {{r}} & -{{\frac{1}{r}} {{B_\phi}}} \\ {E_\phi} & -{{{{B_z}}} \cdot {{r}}} & 0 & {{{B_r}}} \cdot {{r}} \\ {E_z} & {\frac{1}{r}} {{B_\phi}} & -{{{{B_r}}} \cdot {{r}}} & 0\end{matrix} \right]}}$
raising:
${{{ F} _a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & -{{E_r}} & -{\frac{{E_\phi}}{{r}^{2}}} & -{{E_z}} \\ -{{E_r}} & 0 & {\frac{1}{r}} {{B_z}} & -{{\frac{1}{r}} {{B_\phi}}} \\ -{{E_\phi}} & -{{{{B_z}}} \cdot {{r}}} & 0 & {{{B_r}}} \cdot {{r}} \\ -{{E_z}} & {\frac{1}{r}} {{B_\phi}} & -{{\frac{1}{r}} {{B_r}}} & 0\end{matrix} \right]}}$
${{{ F} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & {E_r} & \frac{{E_\phi}}{{r}^{2}} & {E_z} \\ -{{E_r}} & 0 & {\frac{1}{r}} {{B_z}} & -{{\frac{1}{r}} {{B_\phi}}} \\ -{\frac{{E_\phi}}{{r}^{2}}} & -{{\frac{1}{r}} {{B_z}}} & 0 & {\frac{1}{r}} {{B_r}} \\ -{{E_z}} & {\frac{1}{r}} {{B_\phi}} & -{{\frac{1}{r}} {{B_r}}} & 0\end{matrix} \right]}}$
dual:
${{{{ \star F} _u} _v} = {{{\frac{1}{2}}} {{{{ F} ^a} ^b}} {{{{{{ \epsilon} _a} _b} _u} _v}}}} = {\overset{u\downarrow v\rightarrow}{\left[ \begin{matrix} 0 & {B_r} & {B_\phi} & {B_z} \\ -{{B_r}} & 0 & {{{E_z}}} \cdot {{r}} & -{{\frac{1}{r}} {{E_\phi}}} \\ -{{B_\phi}} & -{{{{E_z}}} \cdot {{r}}} & 0 & {{{E_r}}} \cdot {{r}} \\ -{{B_z}} & {\frac{1}{r}} {{E_\phi}} & -{{{{E_r}}} \cdot {{r}}} & 0\end{matrix} \right]}}$
Maxwell's laws in flat space (i.e. excluding connection terms) in covariant form:
${{{{ F} _u} _v} ^{,v}} = {{{4}} {{\pi}} \cdot {{{ J} _u}}}$
${{{{ \star F} _u} _v} ^{,v}} = {0}$
${{{{{ F} _u} _v} ^{,v}} = {\overset{u\downarrow}{\left[ \begin{matrix} {{-{{{\partial_ {{\phi}}\left( {E_\phi}\right)}} {{\frac{1}{{r}^{2}}}}}} - {{{{r}^{2}}} {{\partial_ {{r}}\left( {E_r}\right)}} {{\frac{1}{{r}^{2}}}}}} - {{{{r}^{2}}} {{\partial_ {{z}}\left( {E_z}\right)}} {{\frac{1}{{r}^{2}}}}} \\ {{-{\partial_ {{t}}\left( {E_r}\right)}} - {{{\partial_ {{z}}\left( {B_\phi}\right)}} {{\frac{1}{r}}}}} + {{{\partial_ {{\phi}}\left( {B_z}\right)}} {{\frac{1}{r}}}} \\ {{{-{\partial_ {{t}}\left( {E_\phi}\right)}} - {{B_z}}} - {{{r}} {{\partial_ {{r}}\left( {B_z}\right)}}}} + {{{r}} {{\partial_ {{z}}\left( {B_r}\right)}}} \\ {{-{\partial_ {{t}}\left( {E_z}\right)}} - {{{\partial_ {{\phi}}\left( {B_r}\right)}} {{\frac{1}{r}}}}} + {{{{\partial_ {{r}}\left( {B_\phi}\right)}} {{\frac{1}{r}}}} - {{{{B_\phi}}} \cdot {{\frac{1}{{r}^{2}}}}}}\end{matrix} \right]}}} = {\overset{u\downarrow}{\left[ \begin{matrix} -{{{4}} {{\pi}} \cdot {{\rho}}} \\ {{4}} {{\pi}} \cdot {{{j^r}}} \\ {{4}} {{\pi}} \cdot {{{j^\phi}}} \cdot {{{r}^{2}}} \\ {{4}} {{\pi}} \cdot {{{j^z}}}\end{matrix} \right]}}$
${{{{{ \star F} _u} _v} ^{,v}} = {\overset{u\downarrow}{\left[ \begin{matrix} {\partial_ {{r}}\left( {B_r}\right)} + {\partial_ {{z}}\left( {B_z}\right)} + {{{\partial_ {{\phi}}\left( {B_\phi}\right)}} {{\frac{1}{{r}^{2}}}}} \\ {{\partial_ {{t}}\left( {B_r}\right)} - {{{\partial_ {{z}}\left( {E_\phi}\right)}} {{\frac{1}{r}}}}} + {{{\partial_ {{\phi}}\left( {E_z}\right)}} {{\frac{1}{r}}}} \\ {{{\partial_ {{t}}\left( {B_\phi}\right)} - {{E_z}}} - {{{r}} {{\partial_ {{r}}\left( {E_z}\right)}}}} + {{{r}} {{\partial_ {{z}}\left( {E_r}\right)}}} \\ {{\partial_ {{t}}\left( {B_z}\right)} - {{{\partial_ {{\phi}}\left( {E_r}\right)}} {{\frac{1}{r}}}}} + {{{{\partial_ {{r}}\left( {E_\phi}\right)}} {{\frac{1}{r}}}} - {{{{E_\phi}}} \cdot {{\frac{1}{{r}^{2}}}}}}\end{matrix} \right]}}} = {\overset{u\downarrow}{\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ 0\end{matrix} \right]}}$
Maxwell's laws in flat space (i.e. excluding connection terms) contravariant form:
${{{{ F} ^u} ^v} _{,v}} = {{{4}} {{\pi}} \cdot {{{ J} ^u}}}$
${{{{ \star F} ^u} ^v} _{,v}} = {0}$
Maxwell's laws in curved space in contravariant form:
${{{{{{{ F} ^u} ^v} _{;v}} = {{{{{ F} ^u} ^v} _{,v}} + {{{{{ F} ^u} ^v}} {{{{{ \Gamma} ^a} _v} _a}}}}} = {{{{{ F} ^u} ^v} _{,v}} + {{{{{ F} ^u} ^v}} {{{\left( \sqrt{\log\left( -{g}\right)}\right)} _{,v}}}}}} = {\overset{u\downarrow}{\left[ \begin{matrix} {{{{E_r}}} \cdot {{\frac{1}{r}}}} + {\partial_ {{r}}\left( {E_r}\right)} + {\partial_ {{z}}\left( {E_z}\right)} + {{{\partial_ {{\phi}}\left( {E_\phi}\right)}} {{\frac{1}{{r}^{2}}}}} \\ {{-{\partial_ {{t}}\left( {E_r}\right)}} - {{{\partial_ {{z}}\left( {B_\phi}\right)}} {{\frac{1}{r}}}}} + {{{\partial_ {{\phi}}\left( {B_z}\right)}} {{\frac{1}{r}}}} \\ {{{{\partial_ {{z}}\left( {B_r}\right)}} {{\frac{1}{r}}}} - {{{\partial_ {{r}}\left( {B_z}\right)}} {{\frac{1}{r}}}}} - {{{\partial_ {{t}}\left( {E_\phi}\right)}} {{\frac{1}{{r}^{2}}}}} \\ {{{{\partial_ {{r}}\left( {B_\phi}\right)}} {{\frac{1}{r}}}} - {{{\partial_ {{\phi}}\left( {B_r}\right)}} {{\frac{1}{r}}}}} - {\partial_ {{t}}\left( {E_z}\right)}\end{matrix} \right]}}} = {{{4}} {{\pi}} \cdot {{{ J} ^u}}}$
${{{{{ \star F} ^u} ^v} _{;v}} = {\overset{u\downarrow}{\left[ \begin{matrix} {{{-{{{{B_r}}} \cdot {{r}} {{\frac{1}{{r}^{2}}}}}} - {{{\partial_ {{\phi}}\left( {B_\phi}\right)}} {{\frac{1}{{r}^{2}}}}}} - {{{{r}^{2}}} {{\partial_ {{r}}\left( {B_r}\right)}} {{\frac{1}{{r}^{2}}}}}} - {{{{r}^{2}}} {{\partial_ {{z}}\left( {B_z}\right)}} {{\frac{1}{{r}^{2}}}}} \\ {{\partial_ {{t}}\left( {B_r}\right)} - {{{\partial_ {{z}}\left( {E_\phi}\right)}} {{\frac{1}{r}}}}} + {{{\partial_ {{\phi}}\left( {E_z}\right)}} {{\frac{1}{r}}}} \\ {{{{\partial_ {{z}}\left( {E_r}\right)}} {{\frac{1}{r}}}} - {{{\partial_ {{r}}\left( {E_z}\right)}} {{\frac{1}{r}}}}} + {{{\partial_ {{t}}\left( {B_\phi}\right)}} {{\frac{1}{{r}^{2}}}}} \\ {\partial_ {{t}}\left( {B_z}\right)} + {{{{\partial_ {{r}}\left( {E_\phi}\right)}} {{\frac{1}{r}}}} - {{{\partial_ {{\phi}}\left( {E_r}\right)}} {{\frac{1}{r}}}}}\end{matrix} \right]}}} = {\overset{u\downarrow}{\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ 0\end{matrix} \right]}}$
${{{{{ F} _a} _u}} {{{{ F} _b} ^u}}} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} + {{{E_\phi}}^{2}}}{{r}^{2}} & {\frac{1}{r}}{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)} & {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & {\frac{1}{r}}{\left({{{{{B_r}}} \cdot {{{E_\phi}}}} - {{{{B_\phi}}} \cdot {{{E_r}}}}}\right)} \\ {\frac{1}{r}}{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)} & \frac{{-{{{{{E_r}}^{2}}} {{{r}^{2}}}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{B_\phi}}^{2}}}{{r}^{2}} & -{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)} & -{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)} \\ {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & -{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)} & {-{{{E_\phi}}^{2}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} & -{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)} \\ {\frac{1}{r}}{\left({{{{{B_r}}} \cdot {{{E_\phi}}}} - {{{{B_\phi}}} \cdot {{{E_r}}}}}\right)} & -{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)} & -{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)} & \frac{{-{{{{{E_z}}^{2}}} {{{r}^{2}}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{B_\phi}}^{2}}}{{r}^{2}}\end{matrix} \right]}}$
${{{{{ F} _u} _v}} {{{{ F} ^u} ^v}}} = {\frac{{{2}} {{\left({{{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{{{B_z}}^{2}}} {{{r}^{2}}}} - {{{{{E_r}}^{2}}} {{{r}^{2}}}}} - {{{{{E_z}}^{2}}} {{{r}^{2}}}}} + {{{{B_\phi}}^{2}} - {{{E_\phi}}^{2}}}}\right)}}}{{r}^{2}}}$
${{{ T} _a} _b} = {{{\frac{1}{{{4}} {{\pi}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}}}} {{\left({{{{{{ \overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & {{{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_r}}} - {\left({{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & {{{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_\phi}}} - {\left({{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & {{{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_z}}} - {\left({{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} \\ {{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_r}}}\right)} & 0 & {{{{{B_z}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{\phi}}\left( {A_r}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\partial_ {{\phi}}\left( {A_r}\right)} & {\partial_ {{r}}\left( {A_z}\right)} - {{\frac{1}{r}}{\left({{{{r}} {{\partial_ {{r}}\left( {A_z}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {{B_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)}} \\ {{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_\phi}}}\right)} & {\partial_ {{\phi}}\left( {A_r}\right)} - {\left({{{{{B_z}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{\phi}}\left( {A_r}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & 0 & {{{{{B_r}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{z}}\left( {A_\phi}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\partial_ {{z}}\left( {A_\phi}\right)} \\ {{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_z}}}\right)} & {{\frac{1}{r}}{\left({{{{r}} {{\partial_ {{r}}\left( {A_z}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {{B_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)}} - {\partial_ {{r}}\left( {A_z}\right)} & {\partial_ {{z}}\left( {A_\phi}\right)} - {\left({{{{{B_r}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{z}}\left( {A_\phi}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & 0\end{matrix} \right]}} _a} _u}} {{{{ \overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & {{{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_r}}} - {\left({{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & {{{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_\phi}}} - {\left({{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & {{{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_z}}} - {\left({{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} \\ {{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_r}}}\right)} & 0 & {{{{{B_z}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{\phi}}\left( {A_r}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\partial_ {{\phi}}\left( {A_r}\right)} & {\partial_ {{r}}\left( {A_z}\right)} - {{\frac{1}{r}}{\left({{{{r}} {{\partial_ {{r}}\left( {A_z}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {{B_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)}} \\ {{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_\phi}}}\right)} & {\partial_ {{\phi}}\left( {A_r}\right)} - {\left({{{{{B_z}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{\phi}}\left( {A_r}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & 0 & {{{{{B_r}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{z}}\left( {A_\phi}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\partial_ {{z}}\left( {A_\phi}\right)} \\ {{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_z}}}\right)} & {{\frac{1}{r}}{\left({{{{r}} {{\partial_ {{r}}\left( {A_z}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {{B_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)}} - {\partial_ {{r}}\left( {A_z}\right)} & {\partial_ {{z}}\left( {A_\phi}\right)} - {\left({{{{{B_r}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{z}}\left( {A_\phi}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & 0\end{matrix} \right]}} _b} ^u}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} - {{{\frac{1}{4}}} {{{{ g} _a} _b}} {{{{ \overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & {{{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_r}}} - {\left({{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & {{{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_\phi}}} - {\left({{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & {{{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_z}}} - {\left({{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} \\ {{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_r}}}\right)} & 0 & {{{{{B_z}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{\phi}}\left( {A_r}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\partial_ {{\phi}}\left( {A_r}\right)} & {\partial_ {{r}}\left( {A_z}\right)} - {{\frac{1}{r}}{\left({{{{r}} {{\partial_ {{r}}\left( {A_z}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {{B_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)}} \\ {{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_\phi}}}\right)} & {\partial_ {{\phi}}\left( {A_r}\right)} - {\left({{{{{B_z}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{\phi}}\left( {A_r}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & 0 & {{{{{B_r}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{z}}\left( {A_\phi}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\partial_ {{z}}\left( {A_\phi}\right)} \\ {{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_z}}}\right)} & {{\frac{1}{r}}{\left({{{{r}} {{\partial_ {{r}}\left( {A_z}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {{B_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)}} - {\partial_ {{r}}\left( {A_z}\right)} & {\partial_ {{z}}\left( {A_\phi}\right)} - {\left({{{{{B_r}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{z}}\left( {A_\phi}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & 0\end{matrix} \right]}} _u} _v}} {{{{ \overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & {{{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_r}}} - {\left({{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & {{{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_\phi}}} - {\left({{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & {{{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_z}}} - {\left({{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} \\ {{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_r}\right)} + {{E_r}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_r}}}\right)} & 0 & {{{{{B_z}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{\phi}}\left( {A_r}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\partial_ {{\phi}}\left( {A_r}\right)} & {\partial_ {{r}}\left( {A_z}\right)} - {{\frac{1}{r}}{\left({{{{r}} {{\partial_ {{r}}\left( {A_z}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {{B_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)}} \\ {{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_\phi}\right)} + {{E_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_\phi}}}\right)} & {\partial_ {{\phi}}\left( {A_r}\right)} - {\left({{{{{B_z}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{\phi}}\left( {A_r}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & 0 & {{{{{B_r}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{z}}\left( {A_\phi}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\partial_ {{z}}\left( {A_\phi}\right)} \\ {{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {\left({{{\partial_ {{t}}\left( {A_z}\right)} + {{E_z}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}} - {{E_z}}}\right)} & {{\frac{1}{r}}{\left({{{{r}} {{\partial_ {{r}}\left( {A_z}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {{B_\phi}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)}} - {\partial_ {{r}}\left( {A_z}\right)} & {\partial_ {{z}}\left( {A_\phi}\right)} - {\left({{{{{B_r}}} \cdot {{r}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}} + {\partial_ {{z}}\left( {A_\phi}\right)} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R} + {R}}\right)} & 0\end{matrix} \right]}} ^u} ^v}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}}}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} {{R}}}$
${{{ T} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} + {{{E_\phi}}^{2}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{B_\phi}}^{2}}}{{{8}} {{\pi}} \cdot {{{r}^{2}}}} & \frac{{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}{{{4}} {{\pi}} \cdot {{r}}} & \frac{{{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}}}{{{4}} {{\pi}}} & \frac{{{{{B_r}}} \cdot {{{E_\phi}}}} - {{{{B_\phi}}} \cdot {{{E_r}}}}}{{{4}} {{\pi}} \cdot {{r}}} \\ \frac{{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}{{{4}} {{\pi}} \cdot {{r}}} & \frac{{-{{{{{E_r}}^{2}}} {{{r}^{2}}}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{B_\phi}}^{2}} - {{{{{B_r}}^{2}}} {{{r}^{2}}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} + {{{E_\phi}}^{2}}}{{{8}} {{\pi}} \cdot {{{r}^{2}}}} & -{\frac{{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}{{{4}} {{\pi}}}} & -{\frac{{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}{{{4}} {{\pi}}}} \\ \frac{{{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}}}{{{4}} {{\pi}}} & -{\frac{{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}{{{4}} {{\pi}}}} & \frac{{-{{{E_\phi}}^{2}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{{{{E_z}}^{2}}} {{{r}^{2}}}} - {{{B_\phi}}^{2}}}}{{{8}} {{\pi}}} & -{\frac{{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}{{{4}} {{\pi}}}} \\ \frac{{{{{B_r}}} \cdot {{{E_\phi}}}} - {{{{B_\phi}}} \cdot {{{E_r}}}}}{{{4}} {{\pi}} \cdot {{r}}} & -{\frac{{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}{{{4}} {{\pi}}}} & -{\frac{{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}{{{4}} {{\pi}}}} & \frac{{-{{{{{E_z}}^{2}}} {{{r}^{2}}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{B_\phi}}^{2}} - {{{{{B_z}}^{2}}} {{{r}^{2}}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{E_\phi}}^{2}}}{{{8}} {{\pi}} \cdot {{{r}^{2}}}}\end{matrix} \right]}}$
${{{ T} ^a} _a} = {0}$
${{{ G} _a} _b} = {{{8}} {{\pi}} \cdot {{{{ T} _a} _b}}}$
${{{ G} _a} _b} = {{{{ R} _a} _b} - {{{\frac{1}{2}}} {{{{ R} ^c} _c}} {{{{ g} _a} _b}}}}$
${{{ R} _a} _b} = {{{{ G} _a} _b} - {{{\frac{1}{2}}} {{{{ G} ^c} _c}} {{{{ g} _a} _b}}}}$
${{{ R} _a} _b} = {{{{8}} {{\pi}} \cdot {{{{ T} _a} _b}}} - {{{4}} {{\pi}} \cdot {{{{ T} ^c} _c}} {{{{ g} _a} _b}}}}$
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} + {{{E_\phi}}^{2}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{B_\phi}}^{2}}}{{r}^{2}} & {\frac{1}{r}} {{{2}} {{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}}} & {{2}} {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & {\frac{1}{r}} {{{2}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_r}}}}} + {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}}} \\ {\frac{1}{r}} {{{2}} {{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}}} & \frac{{-{{{{{E_r}}^{2}}} {{{r}^{2}}}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{B_\phi}}^{2}} - {{{{{B_r}}^{2}}} {{{r}^{2}}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} + {{{E_\phi}}^{2}}}{{r}^{2}} & -{{{2}} {{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)}}} & -{{{2}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}}} \\ {{2}} {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & -{{{2}} {{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)}}} & {-{{{E_\phi}}^{2}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{{{{E_z}}^{2}}} {{{r}^{2}}}} - {{{B_\phi}}^{2}}} & -{{{2}} {{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)}}} \\ {\frac{1}{r}} {{{2}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_r}}}}} + {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}}} & -{{{2}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}}} & -{{{2}} {{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)}}} & \frac{{-{{{{{E_z}}^{2}}} {{{r}^{2}}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{B_\phi}}^{2}} - {{{{{B_z}}^{2}}} {{{r}^{2}}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{E_\phi}}^{2}}}{{r}^{2}}\end{matrix} \right]}}$
here's the Ricci curvature tensor that matches the Einstein field equations for the electromagnetic stress-energy tensor (in Cartesian coordinates)
${{{ R} _t} _t} = {{{E}^{2}} + {{B}^{2}}}$
${{{{ R} _i} _t} = {{{-2}} {{{ S} _i}}}} = {{{-2}} {{{{{ \epsilon} _i} ^j} ^k}} {{{ E} _j}} {{{ B} _k}}}$
${{{ R} _i} _j} = {{{{{{ \gamma} _i} _j}} {{\left({{{E}^{2}} + {{B}^{2}}}\right)}}} - {{{2}} {{\left({{{{{ E} _i}} {{{ E} _j}}} + {{{{ B} _i}} {{{ B} _j}}}}\right)}}}}$
${{{{ \hat{R}} _a} _b} = {{{{8}} {{\pi}} \cdot {{{{ T} _a} _b}}} - {{{4}} {{\pi}} \cdot {{{{ T} ^c} _c}} {{{{ g} _a} _b}}}}} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{{E_r}}^{2}} + {{{E_z}}^{2}} + {{{{{E_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}} + {{{B_r}}^{2}} + {{{B_z}}^{2}} + {{{{{B_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}} & {{{2}} {{{B_\phi}}} \cdot {{{E_z}}} \cdot {{\frac{1}{r}}}} - {{{2}} {{{B_z}}} \cdot {{{E_\phi}}} \cdot {{\frac{1}{r}}}} & {-{{{2}} {{r}} {{{B_r}}} \cdot {{{E_z}}}}} + {{{2}} {{r}} {{{B_z}}} \cdot {{{E_r}}}} & {-{{{2}} {{{B_\phi}}} \cdot {{{E_r}}} \cdot {{\frac{1}{r}}}}} + {{{2}} {{{B_r}}} \cdot {{{E_\phi}}} \cdot {{\frac{1}{r}}}} \\ {{{2}} {{{B_\phi}}} \cdot {{{E_z}}} \cdot {{\frac{1}{r}}}} - {{{2}} {{{B_z}}} \cdot {{{E_\phi}}} \cdot {{\frac{1}{r}}}} & {-{{{E_r}}^{2}}} + {{{B_z}}^{2}} + {{{{{{B_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}} - {{{B_r}}^{2}}} + {{{E_z}}^{2}} + {{{{{E_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}} & {-{{{2}} {{{B_\phi}}} \cdot {{{B_r}}}}} - {{{2}} {{{E_\phi}}} \cdot {{{E_r}}}} & {-{{{2}} {{{B_r}}} \cdot {{{B_z}}}}} - {{{2}} {{{E_r}}} \cdot {{{E_z}}}} \\ {-{{{2}} {{r}} {{{B_r}}} \cdot {{{E_z}}}}} + {{{2}} {{r}} {{{B_z}}} \cdot {{{E_r}}}} & {-{{{2}} {{{B_\phi}}} \cdot {{{B_r}}}}} - {{{2}} {{{E_\phi}}} \cdot {{{E_r}}}} & {-{{{E_\phi}}^{2}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{{{{E_z}}^{2}}} {{{r}^{2}}}} - {{{B_\phi}}^{2}}} & {-{{{2}} {{{B_\phi}}} \cdot {{{B_z}}}}} - {{{2}} {{{E_\phi}}} \cdot {{{E_z}}}} \\ {-{{{2}} {{{B_\phi}}} \cdot {{{E_r}}} \cdot {{\frac{1}{r}}}}} + {{{2}} {{{B_r}}} \cdot {{{E_\phi}}} \cdot {{\frac{1}{r}}}} & {-{{{2}} {{{B_r}}} \cdot {{{B_z}}}}} - {{{2}} {{{E_r}}} \cdot {{{E_z}}}} & {-{{{2}} {{{B_\phi}}} \cdot {{{B_z}}}}} - {{{2}} {{{E_\phi}}} \cdot {{{E_z}}}} & {-{{{E_z}}^{2}}} + {{{B_r}}^{2}} + {{{{{{B_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}} - {{{B_z}}^{2}}} + {{{E_r}}^{2}} + {{{{{E_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}}\end{matrix} \right]}}$
here's the constraints that have to be satisfied for the Riemann curvature tensor:
${{{{{{{{ R} _t} _t} = {{{E}^{2}} + {{B}^{2}}}} = {{{{{ R} ^a} _t} _a} _t}} = {{{{{{ R} ^t} _t} _t} _t} + {{{{{ R} ^j} _t} _j} _t}}} = {{{{{ R} ^j} _t} _j} _t}} = {{{{{{ g} ^j} ^t}} {{{{{{ R} _t} _t} _j} _t}}} + {{{{{ g} ^j} ^k}} {{{{{{ R} _k} _t} _j} _t}}}}} = {{{{{ g} ^j} ^k}} {{{{{{ R} _k} _t} _j} _t}}}$
${{{{{{{{ R} _i} _t} = {{{-2}} {{{ S} _i}}}} = {{{-2}} {{{{{ \epsilon} _i} ^j} ^k}} {{{ E} _j}} {{{ B} _k}}}} = {{{{{ R} ^a} _i} _a} _t}} = {{{{{{ R} ^t} _i} _t} _t} + {{{{{ R} ^j} _i} _j} _t}}} = {{{{{ R} ^j} _i} _j} _t}} = {{{{{{ g} ^j} ^t}} {{{{{{ R} _t} _i} _j} _t}}} + {{{{{ g} ^j} ^k}} {{{{{{ R} _k} _i} _j} _t}}}}$
${{{{{{{ R} _i} _j} = {{{{{{ \gamma} _i} _j}} {{\left({{{E}^{2}} + {{B}^{2}}}\right)}}} - {{{2}} {{\left({{{{{ E} _i}} {{{ E} _j}}} + {{{{ B} _i}} {{{ B} _j}}}}\right)}}}}} = {{{{{ R} ^a} _i} _a} _j}} = {{{{{{ R} ^t} _i} _t} _j} + {{{{{ R} ^k} _i} _k} _j}}} = {{{{{{ g} ^t} ^t}} {{{{{{ R} _t} _i} _t} _j}}} + {{{{{ g} ^t} ^k}} {{{{{{ R} _k} _i} _t} _j}}} + {{{{{ g} ^k} ^t}} {{{{{{ R} _t} _i} _k} _j}}} + {{{{{ g} ^k} ^l}} {{{{{{ R} _l} _i} _k} _j}}}}} = {{{{{{ g} ^t} ^t}} {{{{{{ R} _t} _i} _t} _j}}} + {{{{{ g} ^k} ^l}} {{{{{{ R} _l} _i} _k} _j}}} + {{{{{ g} ^t} ^k}} {{\left({{{{{{ R} _t} _j} _k} _i} + {{{{{ R} _t} _i} _k} _j}}\right)}}}}$
here's a Riemann curvature tensor that gives rise to that Ricci curvature tensor, subject to $g^{ab} \approx \eta^{ab}$
${{{{{ R} _t} _i} _t} _j} = {{{{{ E} _i}} {{{ E} _j}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} + {{{{ B} _i}} {{{ B} _j}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R}}$
${{{{{ R} _t} _i} _j} _k} = {{{{{{ \gamma} _i} _j}} {{{ \overset{i\downarrow}{\left[ \begin{matrix} {\frac{1}{r}}{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}}} + {{{{B_z}}} \cdot {{{E_\phi}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R}}\right)} \\ {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} - {{{{B_z}}} \cdot {{{E_r}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}}}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} \\ {\frac{1}{r}}{\left({{-{{{{B_r}}} \cdot {{{E_\phi}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}}} + {{{{B_\phi}}} \cdot {{{E_r}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R}}\right)}\end{matrix} \right]}} _k}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} - {{{{{ \gamma} _i} _k}} {{{ \overset{i\downarrow}{\left[ \begin{matrix} {\frac{1}{r}}{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}}} + {{{{B_z}}} \cdot {{{E_\phi}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R}}\right)} \\ {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} - {{{{B_z}}} \cdot {{{E_r}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}}}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}} \\ {\frac{1}{r}}{\left({{-{{{{B_r}}} \cdot {{{E_\phi}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}}} + {{{{B_\phi}}} \cdot {{{E_r}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R}}\right)}\end{matrix} \right]}} _j}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}}}$
${{{{{ R} _i} _j} _k} _l} = {{{{{{ \epsilon} _i} _j} ^m}} {{{{{ \epsilon} _k} _l} ^n}} {{\left({{{{{ E} _m}} {{{ E} _n}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} + {{{{ B} _m}} {{{ B} _n}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R} + {R}}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} {{R}}}$
${{{{{ R} _a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {{{E_r}}^{2}} + {{{B_r}}^{2}} & {{{{E_\phi}}} \cdot {{{E_r}}}} + {{{{B_\phi}}} \cdot {{{B_r}}}} & {{{{E_r}}} \cdot {{{E_z}}}} + {{{{B_r}}} \cdot {{{B_z}}}} \\ -{\left({{{{B_r}}^{2}} + {{{E_r}}^{2}}}\right)} & 0 & {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & {\frac{1}{r}}{\left({{-{{{{B_r}}} \cdot {{{E_\phi}}}}} + {{{{B_\phi}}} \cdot {{{E_r}}}}}\right)} \\ -{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)} & {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & 0 & 0 \\ -{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)} & {\frac{1}{r}}{\left({{{{{B_r}}} \cdot {{{E_\phi}}}} - {{{{B_\phi}}} \cdot {{{E_r}}}}}\right)} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {{{{E_\phi}}} \cdot {{{E_r}}}} + {{{{B_\phi}}} \cdot {{{B_r}}}} & {{{E_\phi}}^{2}} + {{{B_\phi}}^{2}} & {{{{E_\phi}}} \cdot {{{E_z}}}} + {{{{B_\phi}}} \cdot {{{B_z}}}} \\ -{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)} & 0 & {{r}} {{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}} & 0 \\ -{\left({{{{B_\phi}}^{2}} + {{{E_\phi}}^{2}}}\right)} & {{r}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}} & 0 & {{r}} {{\left({{{{{B_\phi}}} \cdot {{{E_r}}}} - {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}} \\ -{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)} & 0 & {{r}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_r}}}}} + {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {{{{E_r}}} \cdot {{{E_z}}}} + {{{{B_r}}} \cdot {{{B_z}}}} & {{{{E_\phi}}} \cdot {{{E_z}}}} + {{{{B_\phi}}} \cdot {{{B_z}}}} & {{{E_z}}^{2}} + {{{B_z}}^{2}} \\ -{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)} & 0 & 0 & {\frac{1}{r}}{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)} \\ -{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)} & 0 & 0 & {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} \\ -{\left({{{{B_z}}^{2}} + {{{E_z}}^{2}}}\right)} & {\frac{1}{r}}{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)} & {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{\left({{{{B_r}}^{2}} + {{{E_r}}^{2}}}\right)} & -{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)} & -{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)} \\ {{{E_r}}^{2}} + {{{B_r}}^{2}} & 0 & {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & {\frac{1}{r}}{\left({{{{{B_r}}} \cdot {{{E_\phi}}}} - {{{{B_\phi}}} \cdot {{{E_r}}}}}\right)} \\ {{{{E_\phi}}} \cdot {{{E_r}}}} + {{{{B_\phi}}} \cdot {{{B_r}}}} & {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & 0 & 0 \\ {{{{E_r}}} \cdot {{{E_z}}}} + {{{{B_r}}} \cdot {{{B_z}}}} & {\frac{1}{r}}{\left({{-{{{{B_r}}} \cdot {{{E_\phi}}}}} + {{{{B_\phi}}} \cdot {{{E_r}}}}}\right)} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & {{r}} {{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}} & 0 \\ {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & 0 & {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} & -{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)} \\ {{r}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}} & -{\left({{{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}}}\right)} & 0 & {{{r}^{2}}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}} \\ 0 & {{{{E_\phi}}} \cdot {{{E_z}}}} + {{{{B_\phi}}} \cdot {{{B_z}}}} & -{{{{r}^{2}}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{r}}{\left({{-{{{{B_r}}} \cdot {{{E_\phi}}}}} + {{{{B_\phi}}} \cdot {{{E_r}}}}}\right)} & 0 & {\frac{1}{r}}{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)} \\ {\frac{1}{r}}{\left({{{{{B_r}}} \cdot {{{E_\phi}}}} - {{{{B_\phi}}} \cdot {{{E_r}}}}}\right)} & 0 & -{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)} & \frac{{{{B_\phi}}^{2}} + {{{E_\phi}}^{2}}}{{r}^{2}} \\ 0 & {{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}} & 0 & -{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)} \\ {\frac{1}{r}}{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)} & -{\frac{{{{B_\phi}}^{2}} + {{{E_\phi}}^{2}}}{{r}^{2}}} & {{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)} & -{\left({{{{B_\phi}}^{2}} + {{{E_\phi}}^{2}}}\right)} & -{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)} \\ {{{{E_\phi}}} \cdot {{{E_r}}}} + {{{{B_\phi}}} \cdot {{{B_r}}}} & 0 & {{r}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}} & 0 \\ {{{E_\phi}}^{2}} + {{{B_\phi}}^{2}} & {{r}} {{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}} & 0 & {{r}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_r}}}}} + {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}} \\ {{{{E_\phi}}} \cdot {{{E_z}}}} + {{{{B_\phi}}} \cdot {{{B_z}}}} & 0 & {{r}} {{\left({{{{{B_\phi}}} \cdot {{{E_r}}}} - {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & {{r}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}} & 0 \\ {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & 0 & -{\left({{{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}}}\right)} & {{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}} \\ {{r}} {{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}} & {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} & 0 & -{{{{r}^{2}}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}}} \\ 0 & -{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)} & {{{r}^{2}}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & {{r}} {{\left({{{{{B_\phi}}} \cdot {{{E_r}}}} - {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}} & {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} \\ 0 & 0 & {{{r}^{2}}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}} & -{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)} \\ {{r}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_r}}}}} + {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}} & -{{{{r}^{2}}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}}} & 0 & {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}} \\ {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & {{{{E_\phi}}} \cdot {{{E_r}}}} + {{{{B_\phi}}} \cdot {{{B_r}}}} & -{\left({{{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}}}\right)} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & -{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)} & -{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)} & -{\left({{{{B_z}}^{2}} + {{{E_z}}^{2}}}\right)} \\ {{{{E_r}}} \cdot {{{E_z}}}} + {{{{B_r}}} \cdot {{{B_z}}}} & 0 & 0 & {\frac{1}{r}}{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)} \\ {{{{E_\phi}}} \cdot {{{E_z}}}} + {{{{B_\phi}}} \cdot {{{B_z}}}} & 0 & 0 & {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} \\ {{{E_z}}^{2}} + {{{B_z}}^{2}} & {\frac{1}{r}}{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)} & {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & {\frac{1}{r}}{\left({{{{{B_r}}} \cdot {{{E_\phi}}}} - {{{{B_\phi}}} \cdot {{{E_r}}}}}\right)} & 0 & {\frac{1}{r}}{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)} \\ {\frac{1}{r}}{\left({{-{{{{B_r}}} \cdot {{{E_\phi}}}}} + {{{{B_\phi}}} \cdot {{{E_r}}}}}\right)} & 0 & {{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}} & -{\frac{{{{B_\phi}}^{2}} + {{{E_\phi}}^{2}}}{{r}^{2}}} \\ 0 & -{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)} & 0 & {{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}} \\ {\frac{1}{r}}{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)} & \frac{{{{E_\phi}}^{2}} + {{{B_\phi}}^{2}}}{{r}^{2}} & -{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & {{r}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_r}}}}} + {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}} & {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} \\ 0 & 0 & -{{{{r}^{2}}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}}} & {{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}} \\ {{r}} {{\left({{{{{B_\phi}}} \cdot {{{E_r}}}} - {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}} & {{{r}^{2}}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}} & 0 & -{\left({{{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}}}\right)} \\ {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & -{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)} & {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
identities...
${{{{{ R} ^a} _t} _a} _t} = {{{{E_r}}^{2}} + {{{B_r}}^{2}} + {{{E_z}}^{2}} + {{{B_z}}^{2}} + {{{{{E_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}} + {{{{{B_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}}}$
${{{{{ R} ^a} _i} _a} _t} = {\overset{i\downarrow t\rightarrow}{\left[ \begin{matrix} {{{2}} {{{B_\phi}}} \cdot {{{E_z}}} \cdot {{\frac{1}{r}}}} - {{{2}} {{{B_z}}} \cdot {{{E_\phi}}} \cdot {{\frac{1}{r}}}} \\ {-{{{2}} {{r}} {{{B_r}}} \cdot {{{E_z}}}}} + {{{2}} {{r}} {{{B_z}}} \cdot {{{E_r}}}} \\ {-{{{2}} {{{B_\phi}}} \cdot {{{E_r}}} \cdot {{\frac{1}{r}}}}} + {{{2}} {{{B_r}}} \cdot {{{E_\phi}}} \cdot {{\frac{1}{r}}}}\end{matrix} \right]}}$
${{{{{ R} ^a} _i} _a} _j} = {\overset{i\downarrow j\rightarrow}{\left[ \begin{matrix} {{-{{{B_r}}^{2}}} - {{{E_r}}^{2}}} + {{{B_z}}^{2}} + {{{E_z}}^{2}} + {{{{{E_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}} + {{{{{B_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}} & {-{{{2}} {{{B_\phi}}} \cdot {{{B_r}}}}} - {{{2}} {{{E_\phi}}} \cdot {{{E_r}}}} & {-{{{2}} {{{B_r}}} \cdot {{{B_z}}}}} - {{{2}} {{{E_r}}} \cdot {{{E_z}}}} \\ {-{{{2}} {{{B_\phi}}} \cdot {{{B_r}}}}} - {{{2}} {{{E_\phi}}} \cdot {{{E_r}}}} & {{-{{{B_\phi}}^{2}}} - {{{E_\phi}}^{2}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}} & {-{{{2}} {{{B_\phi}}} \cdot {{{B_z}}}}} - {{{2}} {{{E_\phi}}} \cdot {{{E_z}}}} \\ {-{{{2}} {{{B_r}}} \cdot {{{B_z}}}}} - {{{2}} {{{E_r}}} \cdot {{{E_z}}}} & {-{{{2}} {{{B_\phi}}} \cdot {{{B_z}}}}} - {{{2}} {{{E_\phi}}} \cdot {{{E_z}}}} & {{-{{{B_z}}^{2}}} - {{{E_z}}^{2}}} + {{{B_r}}^{2}} + {{{E_r}}^{2}} + {{{{{B_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}} + {{{{{E_\phi}}^{2}}} {{\frac{1}{{r}^{2}}}}}\end{matrix} \right]}}$
building Ricci from Riemann...
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{E_\phi}}^{2}} + {{{B_\phi}}^{2}}}{{r}^{2}} & {\frac{1}{r}} {{{2}} {{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}}} & {{2}} {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & {\frac{1}{r}} {{{2}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_r}}}}} + {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}}} \\ {\frac{1}{r}} {{{2}} {{\left({{{{{B_\phi}}} \cdot {{{E_z}}}} - {{{{B_z}}} \cdot {{{E_\phi}}}}}\right)}}} & \frac{{{-{{{{{B_r}}^{2}}} {{{r}^{2}}}}} - {{{{{E_r}}^{2}}} {{{r}^{2}}}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} + {{{E_\phi}}^{2}} + {{{B_\phi}}^{2}}}{{r}^{2}} & -{{{2}} {{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)}}} & -{{{2}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}}} \\ {{2}} {{r}} {{\left({{-{{{{B_r}}} \cdot {{{E_z}}}}} + {{{{B_z}}} \cdot {{{E_r}}}}}\right)}} & -{{{2}} {{\left({{{{{B_\phi}}} \cdot {{{B_r}}}} + {{{{E_\phi}}} \cdot {{{E_r}}}}}\right)}}} & {{-{{{B_\phi}}^{2}}} - {{{E_\phi}}^{2}}} + {{{{{B_z}}^{2}}} {{{r}^{2}}}} + {{{{{E_z}}^{2}}} {{{r}^{2}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}} & -{{{2}} {{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)}}} \\ {\frac{1}{r}} {{{2}} {{\left({{-{{{{B_\phi}}} \cdot {{{E_r}}}}} + {{{{B_r}}} \cdot {{{E_\phi}}}}}\right)}}} & -{{{2}} {{\left({{{{{B_r}}} \cdot {{{B_z}}}} + {{{{E_r}}} \cdot {{{E_z}}}}}\right)}}} & -{{{2}} {{\left({{{{{B_\phi}}} \cdot {{{B_z}}}} + {{{{E_\phi}}} \cdot {{{E_z}}}}}\right)}}} & \frac{{{-{{{{{B_z}}^{2}}} {{{r}^{2}}}}} - {{{{{E_z}}^{2}}} {{{r}^{2}}}}} + {{{{{B_r}}^{2}}} {{{r}^{2}}}} + {{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{B_\phi}}^{2}} + {{{E_\phi}}^{2}}}{{r}^{2}}\end{matrix} \right]}}$
differences with the desired Ricci:
${{{{ \hat{R}} _a} _b} - {{{ R} _a} _b}} = {0}$
Riemann tensor constraints that need to be fulfilled:
${{{{{{ R} _a} _b} _c} _d} + {{{{{ R} _a} _b} _d} _c}} = {0}$
${{{{{{ R} _a} _b} _c} _d} + {{{{{ R} _b} _a} _c} _d}} = {0}$
${{{{{{ R} _a} _b} _c} _d} - {{{{{ R} _c} _d} _a} _b}} = {0}$
removing $g^{ab} \approx \eta^{ab}$
${{{{{ R} ^t} _i} _t} _j} = {{ {-{{ E} _i}} {{{ E} _j}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} - {{{{ B} _i}} {{{ B} _j}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}}$
${{{{{ R} ^t} _i} _j} _k} = {{{{{{ \gamma} _i} _k}} {{{ \overset{i\downarrow}{\left[ \begin{matrix} {\frac{1}{r}}{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}} + {{{{B_z}}} \cdot {{{E_\phi}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R}}\right)} \\ {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} - {{{{B_z}}} \cdot {{{E_r}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} \\ {\frac{1}{r}}{\left({{-{{{{B_r}}} \cdot {{{E_\phi}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}} + {{{{B_\phi}}} \cdot {{{E_r}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R}}\right)}\end{matrix} \right]}} _j}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} - {{{{{ \gamma} _i} _j}} {{{ \overset{i\downarrow}{\left[ \begin{matrix} {\frac{1}{r}}{\left({{-{{{{B_\phi}}} \cdot {{{E_z}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}} + {{{{B_z}}} \cdot {{{E_\phi}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R}}\right)} \\ {{r}} {{\left({{{{{B_r}}} \cdot {{{E_z}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} - {{{{B_z}}} \cdot {{{E_r}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}} \\ {\frac{1}{r}}{\left({{-{{{{B_r}}} \cdot {{{E_\phi}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}} + {{{{B_\phi}}} \cdot {{{E_r}}} \cdot {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R}}\right)}\end{matrix} \right]}} _k}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}}$
${{{{{ R} ^i} _j} _k} _l} = {{{{{{ \epsilon} ^i} _j} ^m}} {{{{{ \epsilon} _k} _l} ^n}} {{\left({{{{{ E} _m}} {{{ E} _n}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} + {{{{ B} _m}} {{{ B} _n}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} + {P} + {\Gamma} + {c} + {\Gamma} + {R}}\right)}} {{P}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}$
Rewriting sum of norms as a product of tensor
${{{ P} _i} _j} = {\overset{i\downarrow j\rightarrow}{\left[ \begin{matrix} {E_r} & {B_r} & 0 \\ {E_\phi} & {B_\phi} & 0 \\ {E_z} & {B_z} & 0\end{matrix} \right]}}$
${{{{{ P} _i} ^k}} {{{{ P} _j} _k}}} = {\overset{i\downarrow j\rightarrow}{\left[ \begin{matrix} \frac{{{{{{E_r}}^{2}}} {{{r}^{2}}}} + {{{B_r}}^{2}}}{{r}^{2}} & \frac{{{{{E_\phi}}} \cdot {{{E_r}}} \cdot {{{r}^{2}}}} + {{{{B_\phi}}} \cdot {{{B_r}}}}}{{r}^{2}} & \frac{{{{{E_r}}} \cdot {{{E_z}}} \cdot {{{r}^{2}}}} + {{{{B_r}}} \cdot {{{B_z}}}}}{{r}^{2}} \\ \frac{{{{{E_\phi}}} \cdot {{{E_r}}} \cdot {{{r}^{2}}}} + {{{{B_\phi}}} \cdot {{{B_r}}}}}{{r}^{2}} & \frac{{{{{{E_\phi}}^{2}}} {{{r}^{2}}}} + {{{B_\phi}}^{2}}}{{r}^{2}} & \frac{{{{{E_\phi}}} \cdot {{{E_z}}} \cdot {{{r}^{2}}}} + {{{{B_\phi}}} \cdot {{{B_z}}}}}{{r}^{2}} \\ \frac{{{{{E_r}}} \cdot {{{E_z}}} \cdot {{{r}^{2}}}} + {{{{B_r}}} \cdot {{{B_z}}}}}{{r}^{2}} & \frac{{{{{E_\phi}}} \cdot {{{E_z}}} \cdot {{{r}^{2}}}} + {{{{B_\phi}}} \cdot {{{B_z}}}}}{{r}^{2}} & \frac{{{{{{E_z}}^{2}}} {{{r}^{2}}}} + {{{B_z}}^{2}}}{{r}^{2}}\end{matrix} \right]}}$
${{{{{ R} ^t} _i} _t} _j} = { {-{{{ P} _i} ^k}} {{{{ P} _j} _k}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}$
${{{{{ R} ^t} _i} _j} _k} = {{{{{{ \gamma} _i} _k}} {{{ S} _j}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}} - {{{{{ \gamma} _i} _j}} {{{ S} _k}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}}$
${{{{{ R} ^i} _j} _k} _l} = {{{{{{ \epsilon} ^i} _j} _m}} {{{{{ \epsilon} _k} _l} _n}} {{{{ P} ^m} ^r}} {{{{ P} ^n} _r}} {{\Gamma}} \cdot {{c}} {{\Gamma}} \cdot {{R}}}$
connections that give rise to Riemann tensor
${{{{{{{ R} ^t} _i} _t} _j} = {{ {-{{ E} _i}} {{{ E} _j}}} - {{{{ B} _i}} {{{ B} _j}}}}} = {{{{{{{{{ \Gamma} ^t} _i} _j} _{,t}} - {{{{{ \Gamma} ^t} _i} _t} _{,j}}} + {{{{{{ \Gamma} ^t} _a} _t}} {{{{{ \Gamma} ^a} _i} _j}}}} - {{{{{{ \Gamma} ^t} _a} _j}} {{{{{ \Gamma} ^a} _i} _t}}}} - {{{{{{ \Gamma} ^t} _i} _a}} {{{{{ c} _t} _j} ^a}}}}} = {{{{{{{{{{{ \Gamma} ^t} _i} _j} _{,t}} - {{{{{ \Gamma} ^t} _i} _t} _{,j}}} + {{{{{{ \Gamma} ^t} _t} _t}} {{{{{ \Gamma} ^t} _i} _j}}} + {{{{{{ \Gamma} ^t} _k} _t}} {{{{{ \Gamma} ^k} _i} _j}}}} - {{{{{{ \Gamma} ^t} _t} _j}} {{{{{ \Gamma} ^t} _i} _t}}}} - {{{{{{ \Gamma} ^t} _k} _j}} {{{{{ \Gamma} ^k} _i} _t}}}} - {{{{{{ \Gamma} ^t} _i} _t}} {{{{{ c} _t} _j} ^t}}}} - {{{{{{ \Gamma} ^t} _i} _k}} {{{{{ c} _t} _j} ^k}}}}$
${{{{{{{{ R} ^t} _i} _j} _k} = {{{{{{ \gamma} _i} _k}} {{{ S} _j}}} - {{{{{ \gamma} _i} _j}} {{{ S} _k}}}}} = { {-{{{{ \epsilon} _i} ^n} ^m}} {{{{{ \epsilon} _j} _k} _m}} {{{{{ \epsilon} _n} ^p} ^q}} {{{ E} _p}} {{{ B} _q}}}} = {{{{{{{{{ \Gamma} ^t} _i} _k} _{,j}} - {{{{{ \Gamma} ^t} _i} _j} _{,k}}} + {{{{{{ \Gamma} ^t} _a} _j}} {{{{{ \Gamma} ^a} _i} _k}}}} - {{{{{{ \Gamma} ^t} _a} _k}} {{{{{ \Gamma} ^a} _i} _j}}}} - {{{{{{ \Gamma} ^t} _i} _a}} {{{{{ c} _j} _k} ^a}}}}} = {{{{{{{{{{{ \Gamma} ^t} _i} _k} _{,j}} - {{{{{ \Gamma} ^t} _i} _j} _{,k}}} + {{{{{{ \Gamma} ^t} _t} _j}} {{{{{ \Gamma} ^t} _i} _k}}} + {{{{{{ \Gamma} ^t} _l} _j}} {{{{{ \Gamma} ^l} _i} _k}}}} - {{{{{{ \Gamma} ^t} _t} _k}} {{{{{ \Gamma} ^t} _i} _j}}}} - {{{{{{ \Gamma} ^t} _l} _k}} {{{{{ \Gamma} ^l} _i} _j}}}} - {{{{{{ \Gamma} ^t} _i} _t}} {{{{{ c} _j} _k} ^t}}}} - {{{{{{ \Gamma} ^t} _i} _l}} {{{{{ c} _j} _k} ^l}}}}$
${{{{{{{ R} ^i} _j} _k} _l} = {{{{{{ \epsilon} ^i} _j} ^m}} {{{{{ \epsilon} _k} _l} ^n}} {{\left({{{{{ E} _m}} {{{ E} _n}}} + {{{{ B} _m}} {{{ B} _n}}}}\right)}}}} = {{{{{{{{{ \Gamma} ^i} _j} _l} _{,k}} - {{{{{ \Gamma} ^i} _j} _k} _{,l}}} + {{{{{{ \Gamma} ^i} _a} _k}} {{{{{ \Gamma} ^a} _j} _l}}}} - {{{{{{ \Gamma} ^i} _a} _l}} {{{{{ \Gamma} ^a} _j} _k}}}} - {{{{{{ \Gamma} ^i} _j} _a}} {{{{{ c} _k} _l} ^a}}}}} = {{{{{{{{{{{ \Gamma} ^i} _j} _l} _{,k}} - {{{{{ \Gamma} ^i} _j} _k} _{,l}}} + {{{{{{ \Gamma} ^i} _t} _k}} {{{{{ \Gamma} ^t} _j} _l}}} + {{{{{{ \Gamma} ^i} _m} _k}} {{{{{ \Gamma} ^m} _j} _l}}}} - {{{{{{ \Gamma} ^i} _t} _l}} {{{{{ \Gamma} ^t} _j} _k}}}} - {{{{{{ \Gamma} ^i} _m} _l}} {{{{{ \Gamma} ^m} _j} _k}}}} - {{{{{{ \Gamma} ^i} _j} _t}} {{{{{ c} _k} _l} ^t}}}} - {{{{{{ \Gamma} ^i} _j} _m}} {{{{{ c} _k} _l} ^m}}}}$
two-form as matrix, as seen in Misner, Thorne, and Wheeler problem 14.14
${R^{ti}}_{tj} = -(E^i E_j + B^i B_j)$
${R^{ti}}_{jk} = 2 \delta^i_{[k} S_{j]}$
${R^{ij}}_{kl} = {\epsilon^{ij}}_m (E^m E_n + B^m B_n) {\epsilon^n}_{kl}$
Assuming ${R_{ti}}^{ij} = -{R^{ti}}_{jk}$
and $S_i = {\epsilon_{ij}}^k E^j B_k$
$S = E \wedge B$ in 3D.
${R^{ab}}_{cd} = ab\downarrow \overset{cd\rightarrow}{
\left[\matrix{
& tx & ty & tz & yz & zx & xy \\
tx & -E^x E_x - B^x B_x & -E^x E_y - B^x B_y & -E^x E_z - B^x B_z & 0 & -S_z & S_y \\
ty & -E^y E_x - B^y B_x & -E^y E_y - B^y B_y & -E^y E_z - B^y B_z & S_z & 0 & -S_x \\
tz & -E^z E_x - B^z B_x & -E^z E_y - B^z B_y & -E^z E_z - B^z B_z & -S_y & S_x & 0 \\
yz & 0 & -S_z & S_y & E^x E_x + B^x B_x & E^x E_y + B^x B_y & E^x E_z + B^x B_z \\
zx & S_z & 0 & -S_x & E^y E_x + B^y B_x & E^y E_y + B^y B_y & E^y E_z + B^y B_z \\
xy & -S_y & S_x & 0 & E^z E_x + B^z B_x & E^z E_y + B^z B_y & E^z E_z + B^z B_z \\
}\right]
}$
$= ab\downarrow \overset{cd\rightarrow}{
\left[\matrix{
& tx & ty & tz & yz & zx & xy \\
tx & -E^x E_x - B^x B_x & -E^x E_y - B^x B_y & -E^x E_z - B^x B_z & 0 & E^y B_x - E^x B_y & E^z B_x - E^x B_z \\
ty & -E^y E_x - B^y B_x & -E^y E_y - B^y B_y & -E^y E_z - B^y B_z & E^x B_y - E^y B_x & 0 & E^z B_y - E^y B_z \\
tz & -E^z E_x - B^z B_x & -E^z E_y - B^z B_y & -E^z E_z - B^z B_z & E^x B_z - E^z B_x & E^y B_z - E^z B_y & 0 \\
yz & 0 & E^y B_x - E^x B_y & E^z B_x - E^x B_z & E^x E_x + B^x B_x & E^x E_y + B^x B_y & E^x E_z + B^x B_z \\
zx & E^x B_y - E^y B_x & 0 & E^z B_y - E^y B_z & E^y E_x + B^y B_x & E^y E_y + B^y B_y & E^y E_z + B^y B_z \\
xy & E^x B_x - E^z B_x & E^y B_z - E^z B_y & 0 & E^z E_x + B^z B_x & E^z E_y + B^z B_y & E^z E_z + B^z B_z \\
}\right]
}$
${R^a}_b = d {w^a}_b + {w^a}_c \wedge {w^c}_b$
${R^t}_i = d {w^t}_i + {w^t}_t \wedge {w^t}_i + {w^t}_k \wedge {w^k}_i$
${R^i}_j = d {w^i}_j + {w^i}_t \wedge {w^t}_j + {w^i}_k \wedge {w^k}_j$
${R^t}_i = -(E_i E_j + B_i B_j) dt \wedge dx^j + 2 \gamma_{i[k} S_{j]} dx^j \wedge dx^k$
${R^i}_j = {\epsilon^i}_{jm} (E^m E_n + B^m B_n) {\epsilon^n}_{kl} dx^k \wedge dx^l$