${{{{{ R} ^a} _b} _c} _d} = {{{{{{{{{ Gamma} ^a} _b} _d} _{,c}} - {{{{{ Gamma} ^a} _b} _c} _{,d}}} + {{{{{{ Gamma} ^a} _e} _c}} {{{{{ Gamma} ^e} _b} _d}}}} - {{{{{{ Gamma} ^a} _e} _d}} {{{{{ Gamma} ^e} _b} _c}}}} - {{{{{{ Gamma} ^a} _b} _e}} {{\left({{{{{ Gamma} ^e} _d} _c} - {{{{ Gamma} ^e} _c} _d}}\right)}}}}$
manual metric:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {\frac{1}{r}}{\left({-{I}}\right)} & 0 & 0 & 0 \\ 0 & \frac{1}{{{I}} {{r}}} & 0 & 0 \\ 0 & 0 & {r}^{2} & 0 \\ 0 & 0 & 0 & \frac{1}{r}\end{matrix} \right]}}$
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{{\frac{1}{I}} {r}} & 0 & 0 & 0 \\ 0 & {{I}} {{r}} & 0 & 0 \\ 0 & 0 & \frac{1}{{r}^{2}} & 0 \\ 0 & 0 & 0 & r\end{matrix} \right]}}$
${{{{ \Gamma} _t} _t} _r} = {\frac{I}{{{2}} {{{r}^{2}}}}}$; ${{{{ \Gamma} _t} _r} _t} = {\frac{I}{{{2}} {{{r}^{2}}}}}$; ${{{{ \Gamma} _r} _t} _t} = {-{\frac{I}{{{2}} {{{r}^{2}}}}}}$; ${{{{ \Gamma} _r} _r} _r} = {-{\frac{1}{{{2}} {{I}} {{{r}^{2}}}}}}$; ${{{{ \Gamma} _r} _{phi}} _{phi}} = {-{r}}$; ${{{{ \Gamma} _r} _z} _z} = {\frac{1}{{{2}} {{{r}^{2}}}}}$; ${{{{ \Gamma} _{phi}} _r} _{phi}} = {r}$; ${{{{ \Gamma} _{phi}} _{phi}} _r} = {r}$; ${{{{ \Gamma} _z} _r} _z} = {-{\frac{1}{{{2}} {{{r}^{2}}}}}}$; ${{{{ \Gamma} _z} _z} _r} = {-{\frac{1}{{{2}} {{{r}^{2}}}}}}$
${{{{ \Gamma} ^t} _t} _r} = {-{\frac{1}{{{2}} {{r}}}}}$; ${{{{ \Gamma} ^t} _r} _t} = {-{\frac{1}{{{2}} {{r}}}}}$; ${{{{ \Gamma} ^r} _t} _t} = {-{\frac{{I}^{2}}{{{2}} {{r}}}}}$; ${{{{ \Gamma} ^r} _r} _r} = {-{\frac{1}{{{2}} {{r}}}}}$; ${{{{ \Gamma} ^r} _{phi}} _{phi}} = {-{{{I}} {{{r}^{2}}}}}$; ${{{{ \Gamma} ^r} _z} _z} = {\frac{I}{{{2}} {{r}}}}$; ${{{{ \Gamma} ^{phi}} _r} _{phi}} = {\frac{1}{r}}$; ${{{{ \Gamma} ^{phi}} _{phi}} _r} = {\frac{1}{r}}$; ${{{{ \Gamma} ^z} _r} _z} = {-{\frac{1}{{{2}} {{r}}}}}$; ${{{{ \Gamma} ^z} _z} _r} = {-{\frac{1}{{{2}} {{r}}}}}$
connection from manual metric:
${{{{ \Gamma} ^t} _t} _r} = {-{\frac{1}{{{2}} {{r}}}}}$; ${{{{ \Gamma} ^t} _r} _t} = {-{\frac{1}{{{2}} {{r}}}}}$; ${{{{ \Gamma} ^r} _t} _t} = {-{\frac{{I}^{2}}{{{2}} {{r}}}}}$; ${{{{ \Gamma} ^r} _r} _r} = {-{\frac{1}{{{2}} {{r}}}}}$; ${{{{ \Gamma} ^r} _{phi}} _{phi}} = {-{{{I}} {{{r}^{2}}}}}$; ${{{{ \Gamma} ^r} _z} _z} = {\frac{I}{{{2}} {{r}}}}$; ${{{{ \Gamma} ^{phi}} _r} _{phi}} = {\frac{1}{r}}$; ${{{{ \Gamma} ^{phi}} _{phi}} _r} = {\frac{1}{r}}$; ${{{{ \Gamma} ^z} _r} _z} = {-{\frac{1}{{{2}} {{r}}}}}$; ${{{{ \Gamma} ^z} _z} _r} = {-{\frac{1}{{{2}} {{r}}}}}$
manual connection:
${{{{ \Gamma} ^t} _t} _{phi}} = {{{2}} {{I}}}$; ${{{{ \Gamma} ^t} _{phi}} _t} = {{{2}} {{I}}}$; ${{{{ \Gamma} ^{phi}} _t} _t} = {\frac{{{-2}} {{I}}}{{r}^{2}}}$; ${{{{ \Gamma} ^{phi}} _r} _r} = {\frac{{{2}} {{I}}}{{r}^{2}}}$; ${{{{ \Gamma} ^{phi}} _z} _z} = {\frac{{{2}} {{I}}}{{r}^{2}}}$
Ricci from manual metric:
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{I}^{2}}{{{4}} {{{r}^{2}}}} & 0 & 0 & 0 \\ 0 & -{\frac{3}{{{2}} {{{r}^{2}}}}} & 0 & 0 \\ 0 & 0 & {\frac{1}{2}} {{{I}} {{r}}} & 0 \\ 0 & 0 & 0 & -{\frac{I}{{{4}} {{{r}^{2}}}}}\end{matrix} \right]}}$
Gaussian from manual metric:
${{R} = {-{\frac{{{3}} {{I}}}{{{2}} {{r}}}}}} = {0}$
${\frac{{{3}} {{I}}}{{{2}} {{r}}}} = {0}$
$R_{tt} - R_{rr}$:
${\frac{{6} + {{I}^{2}}}{{{4}} {{{r}^{2}}}}} = {0}$
$R_{rr} - R_{zz}$:
${\frac{{-{6}} + {I}}{{{4}} {{{r}^{2}}}}} = {0}$
$r^{-2} R_{\phi\phi} + R_{tt}$:
${\frac{{{I}} {{\left({{I} + {{{2}} {{r}}}}\right)}}}{{{4}} {{{r}^{2}}}}} = {0}$
Ricci from manual connection:
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{{I}^{2}}}}{{r}^{2}} & 0 & 0 & 0 \\ 0 & \frac{{{4}} {{{I}^{2}}}}{{r}^{2}} & 0 & 0 \\ 0 & 0 & -{{{4}} {{{I}^{2}}}} & 0 \\ 0 & 0 & 0 & \frac{{{4}} {{{I}^{2}}}}{{r}^{2}}\end{matrix} \right]}}$
vs $8 \pi \times$ EM stress-energy tensor = Ricci tensor ${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{{I}^{2}}}}{{r}^{2}} & 0 & 0 & 0 \\ 0 & \frac{{{4}} {{{I}^{2}}}}{{r}^{2}} & 0 & 0 \\ 0 & 0 & -{{{4}} {{{I}^{2}}}} & 0 \\ 0 & 0 & 0 & \frac{{{4}} {{{I}^{2}}}}{{r}^{2}}\end{matrix} \right]}}$
now the same thing for no-current only-charge