for a uniformly charged wire...
${{{{ \Gamma} ^t} _t} _t} = {1}$; ${{{{ \Gamma} ^t} _{\phi}} _{\phi}} = {{\frac{{{-4}} {{{I}^{2}}}}{{r}^{2}}} + {{{4}} {{{\lambda}^{2}}}}}$; ${{{{ \Gamma} ^t} _z} _z} = {{\frac{{{4}} {{{I}^{2}}}}{{r}^{4}}} + {\frac{{{4}} {{{\lambda}^{2}}}}{{r}^{2}}}}$; ${{{{ \Gamma} ^r} _t} _t} = {{\frac{ {-{\frac{4}{3}}} {{{I}^{2}}}}{{r}^{3}}} - {{\frac{1}{r}} {{{4}} {{{\lambda}^{2}}}}}}$; ${{{{ \Gamma} ^r} _t} _z} = {\frac{{{4}} {{I}} {{\lambda}}}{{r}^{2}}}$; ${{{{ \Gamma} ^r} _z} _t} = {\frac{{{4}} {{I}} {{\lambda}}}{{r}^{2}}}$; ${{{{ \Gamma} ^{\phi}} _r} _r} = {{{4}} {{\phi}} \cdot {{\left({{\frac{{I}^{2}}{{r}^{4}}} - {\frac{{\lambda}^{2}}{{r}^{2}}}}\right)}}}$
${{{{{ R} ^a} _b} _c} _d} = {{{{{{{{{ \Gamma} ^a} _b} _d} _{,c}} - {{{{{ \Gamma} ^a} _b} _c} _{,d}}} + {{{{{{ \Gamma} ^a} _e} _c}} {{{{{ \Gamma} ^e} _b} _d}}}} - {{{{{{ \Gamma} ^a} _e} _d}} {{{{{ \Gamma} ^e} _b} _c}}}} - {{{{{{ \Gamma} ^a} _b} _e}} {{\left({{{{{ \Gamma} ^e} _d} _c} - {{{{ \Gamma} ^e} _c} _d}}\right)}}}}$
${{{ R} _a} _b} = {{{{{{{{{ \Gamma} ^c} _a} _b} _{,c}} - {{{{{ \Gamma} ^c} _a} _c} _{,b}}} + {{{{{{ \Gamma} ^c} _d} _c}} {{{{{ \Gamma} ^d} _a} _b}}}} - {{{{{{ \Gamma} ^c} _d} _b}} {{{{{ \Gamma} ^d} _a} _c}}}} - {{{{{{ \Gamma} ^c} _a} _d}} {{\left({{{{{ \Gamma} ^d} _b} _c} - {{{{ \Gamma} ^d} _c} _b}}\right)}}}}$
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{\left({{{{{I}^{2}}} {{{r}^{2}}}} + {{{{\lambda}^{2}}} {{{r}^{4}}}}}\right)}}}{{r}^{6}} & 0 & 0 & -{\frac{{{8}} {{I}} {{\lambda}}}{{r}^{3}}} \\ 0 & \frac{{{4}} {{\left({{-{{{{\lambda}^{2}}} {{{r}^{2}}}}} + {{I}^{2}}}\right)}}}{{r}^{4}} & 0 & 0 \\ 0 & 0 & \frac{{{4}} {{\left({{{{{\lambda}^{2}}} {{{r}^{2}}}} - {{I}^{2}}}\right)}}}{{r}^{2}} & 0 \\ -{\frac{{{8}} {{I}} {{\lambda}}}{{r}^{3}}} & 0 & 0 & \frac{{{4}} {{\left({{{{{\lambda}^{2}}} {{{r}^{2}}}} + {{I}^{2}}}\right)}}}{{r}^{4}}\end{matrix} \right]}}$ ${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{\left({{{{{I}^{2}}} {{{r}^{2}}}} + {{{{\lambda}^{2}}} {{{r}^{4}}}}}\right)}}}{{r}^{6}} & 0 & 0 & -{\frac{{{8}} {{I}} {{\lambda}}}{{r}^{3}}} \\ 0 & \frac{{{4}} {{\left({{-{{{{\lambda}^{2}}} {{{r}^{2}}}}} + {{I}^{2}}}\right)}}}{{r}^{4}} & 0 & 0 \\ 0 & 0 & \frac{{{4}} {{\left({{{{{\lambda}^{2}}} {{{r}^{2}}}} - {{I}^{2}}}\right)}}}{{r}^{2}} & 0 \\ -{\frac{{{8}} {{I}} {{\lambda}}}{{r}^{3}}} & 0 & 0 & \frac{{{4}} {{\left({{{{{\lambda}^{2}}} {{{r}^{2}}}} + {{I}^{2}}}\right)}}}{{r}^{4}}\end{matrix} \right]}}$ vs Ricci of EM stress-energy tensor ${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{\left({{{I}^{2}} + {{\lambda}^{2}}}\right)}}}{{r}^{2}} & 0 & 0 & -{\frac{{{8}} {{I}} {{\lambda}}}{{r}^{2}}} \\ 0 & \frac{{{4}} {{\left({{{I}^{2}} - {{\lambda}^{2}}}\right)}}}{{r}^{2}} & 0 & 0 \\ 0 & 0 & {{4}} {{\left({{-{{I}^{2}}} + {{\lambda}^{2}}}\right)}} & 0 \\ -{\frac{{{8}} {{I}} {{\lambda}}}{{r}^{2}}} & 0 & 0 & \frac{{{4}} {{\left({{{I}^{2}} + {{\lambda}^{2}}}\right)}}}{{r}^{2}}\end{matrix} \right]}}$
EM with no current -- in frame of wire carrier drift velocity, right? How do you apply Lorentz boost to create this setup?
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{{\lambda}^{2}}}}{{r}^{2}} & 0 & 0 & 0 \\ 0 & -{\frac{{{4}} {{{\lambda}^{2}}}}{{r}^{2}}} & 0 & 0 \\ 0 & 0 & {{4}} {{{\lambda}^{2}}} & 0 \\ 0 & 0 & 0 & \frac{{{4}} {{{\lambda}^{2}}}}{{r}^{2}}\end{matrix} \right]}}$
EM with no charge -- in rest frame -- purely magnetic field
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{{I}^{2}}}}{{r}^{2}} & 0 & 0 & 0 \\ 0 & \frac{{{4}} {{{I}^{2}}}}{{r}^{2}} & 0 & 0 \\ 0 & 0 & -{{{4}} {{{I}^{2}}}} & 0 \\ 0 & 0 & 0 & \frac{{{4}} {{{I}^{2}}}}{{r}^{2}}\end{matrix} \right]}}$
If the wire has either no current or no charge then $R_{tz}$ will be zero. If $R_{tz}$ is zero then there is no way to apply a Lorentz transformation to create this term (right?). There would also be no way to transform a EM stress-energy of purely current (observer frame) into one of purely current (frame of charge carriers).
Lorentz boost
${{{ \Lambda} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \gamma & 0 & 0 & {-{\beta}} {{\gamma}} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ {-{\beta}} {{\gamma}} & 0 & 0 & \gamma\end{matrix} \right]}}$
EM boosted:
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{\left({{{{{I}^{2}}} {{{\beta}^{2}}} {{{\gamma}^{2}}}} + {{{{I}^{2}}} {{{\gamma}^{2}}}} + {{{{\beta}^{2}}} {{{\gamma}^{2}}} {{{\lambda}^{2}}}} + {{{{\gamma}^{2}}} {{{\lambda}^{2}}}} + {{{4}} {{I}} {{\beta}} \cdot {{\lambda}} \cdot {{{\gamma}^{2}}}}}\right)}}}{{r}^{2}} & 0 & 0 & -{\frac{{{8}} {{\left({{{{I}} {{\lambda}} \cdot {{{\beta}^{2}}} {{{\gamma}^{2}}}} + {{{I}} {{\lambda}} \cdot {{{\gamma}^{2}}}} + {{{\beta}} \cdot {{{I}^{2}}} {{{\gamma}^{2}}}} + {{{\beta}} \cdot {{{\gamma}^{2}}} {{{\lambda}^{2}}}}}\right)}}}{{r}^{2}}} \\ 0 & \frac{{{4}} {{\left({{{I}^{2}} - {{\lambda}^{2}}}\right)}}}{{r}^{2}} & 0 & 0 \\ 0 & 0 & {{4}} {{\left({{-{{I}^{2}}} + {{\lambda}^{2}}}\right)}} & 0 \\ -{\frac{{{8}} {{\left({{{{I}} {{\lambda}} \cdot {{{\beta}^{2}}} {{{\gamma}^{2}}}} + {{{I}} {{\lambda}} \cdot {{{\gamma}^{2}}}} + {{{\beta}} \cdot {{{I}^{2}}} {{{\gamma}^{2}}}} + {{{\beta}} \cdot {{{\gamma}^{2}}} {{{\lambda}^{2}}}}}\right)}}}{{r}^{2}}} & 0 & 0 & \frac{{{4}} {{\left({{{{{I}^{2}}} {{{\beta}^{2}}} {{{\gamma}^{2}}}} + {{{{I}^{2}}} {{{\gamma}^{2}}}} + {{{{\beta}^{2}}} {{{\gamma}^{2}}} {{{\lambda}^{2}}}} + {{{{\gamma}^{2}}} {{{\lambda}^{2}}}} + {{{4}} {{I}} {{\beta}} \cdot {{\lambda}} \cdot {{{\gamma}^{2}}}}}\right)}}}{{r}^{2}}\end{matrix} \right]}}$
EM, no charge, boosted:
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{\left({{{{{I}^{2}}} {{{\beta}^{2}}}} + {{I}^{2}}}\right)}}}{{{{r}^{2}}} {{\left({{1} + {\beta}}\right)}} {{\left({{1} - {\beta}}\right)}}} & 0 & 0 & -{\frac{{{8}} {{\beta}} \cdot {{{I}^{2}}}}{{{{r}^{2}}} {{\left({{1} - {\beta}}\right)}} {{\left({{1} + {\beta}}\right)}}}} \\ 0 & \frac{{{4}} {{{I}^{2}}}}{{r}^{2}} & 0 & 0 \\ 0 & 0 & -{{{4}} {{{I}^{2}}}} & 0 \\ -{\frac{{{8}} {{\beta}} \cdot {{{I}^{2}}}}{{{{r}^{2}}} {{\left({{1} - {\beta}}\right)}} {{\left({{1} + {\beta}}\right)}}}} & 0 & 0 & \frac{{{4}} {{\left({{{{{I}^{2}}} {{{\beta}^{2}}}} + {{I}^{2}}}\right)}}}{{{{r}^{2}}} {{\left({{1} + {\beta}}\right)}} {{\left({{1} - {\beta}}\right)}}}\end{matrix} \right]}}$
four-current in rest frame. no charge, only current:
${{ J} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} 0 \\ 0 \\ 0 \\ I\end{matrix} \right]}}$
four-current boosted
${{ J} _a} = {\overset{a\downarrow}{\left[ \begin{matrix} -{{{I}} {{\beta}} \cdot {{\gamma}}} \\ 0 \\ 0 \\ {{I}} {{\gamma}}\end{matrix} \right]}}$
EM, no charge, boosted, with four-current exchanged as well
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{\left({{{{{I}^{2}}} {{{\beta}^{2}}}} + {{{{I}^{2}}} {{{\beta}^{2}}} {{{\gamma}^{2}}}} + {{{{I}^{2}}} {{{\beta}^{4}}} {{{\gamma}^{2}}}} + {{{I}^{2}} - {{{4}} {{\gamma}} \cdot {{{I}^{2}}} {{{\beta}^{2}}}}}}\right)}}}{{{{r}^{2}} - {{{2}} {{{\beta}^{2}}} {{{r}^{2}}}}} + {{{{\beta}^{4}}} {{{r}^{2}}}}} & 0 & 0 & \frac{{{8}} {{\beta}} \cdot {{\left({{{-{{{{I}^{2}}} {{{\beta}^{2}}} {{{\gamma}^{2}}}}} - {{I}^{2}}} + {{{\gamma}} \cdot {{{I}^{2}}}} + {{{\gamma}} \cdot {{{I}^{2}}} {{{\beta}^{2}}}}}\right)}}}{{{{r}^{2}} - {{{2}} {{{\beta}^{2}}} {{{r}^{2}}}}} + {{{{\beta}^{4}}} {{{r}^{2}}}}} \\ 0 & \frac{{{4}} {{\left({{{-{{{{I}^{2}}} {{{\beta}^{2}}}}} - {{{2}} {{{I}^{2}}} {{{\beta}^{2}}} {{{\gamma}^{2}}}}} + {{{{I}^{2}}} {{{\gamma}^{2}}}} + {{{{I}^{2}}} {{{\beta}^{4}}} {{{\gamma}^{2}}}}}\right)}}}{{{{r}^{2}} - {{{2}} {{{\beta}^{2}}} {{{r}^{2}}}}} + {{{{\beta}^{4}}} {{{r}^{2}}}}} & 0 & 0 \\ 0 & 0 & \frac{{{4}} {{\left({{{{{I}^{2}}} {{{\beta}^{2}}}} + {{{{{2}} {{{I}^{2}}} {{{\beta}^{2}}} {{{\gamma}^{2}}}} - {{{{I}^{2}}} {{{\gamma}^{2}}}}} - {{{{I}^{2}}} {{{\beta}^{4}}} {{{\gamma}^{2}}}}}}\right)}}}{{{1} - {{{2}} {{{\beta}^{2}}}}} + {{\beta}^{4}}} & 0 \\ \frac{{{8}} {{\beta}} \cdot {{\left({{{-{{{{I}^{2}}} {{{\beta}^{2}}} {{{\gamma}^{2}}}}} - {{I}^{2}}} + {{{\gamma}} \cdot {{{I}^{2}}}} + {{{\gamma}} \cdot {{{I}^{2}}} {{{\beta}^{2}}}}}\right)}}}{{{{r}^{2}} - {{{2}} {{{\beta}^{2}}} {{{r}^{2}}}}} + {{{{\beta}^{4}}} {{{r}^{2}}}}} & 0 & 0 & \frac{{{4}} {{\left({{{{{I}^{2}}} {{{\beta}^{2}}}} + {{{{I}^{2}}} {{{\beta}^{2}}} {{{\gamma}^{2}}}} + {{{{I}^{2}}} {{{\beta}^{4}}} {{{\gamma}^{2}}}} + {{{I}^{2}} - {{{4}} {{\gamma}} \cdot {{{I}^{2}}} {{{\beta}^{2}}}}}}\right)}}}{{{{r}^{2}} - {{{2}} {{{\beta}^{2}}} {{{r}^{2}}}}} + {{{{\beta}^{4}}} {{{r}^{2}}}}}\end{matrix} \right]}}$
Ricci without charge, then boosted to recreate B
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{4}} {{{I}^{3}}} {{{{\left({{{I}^{2}} - {{{4}} {{{\lambda}^{2}}} {{{r}^{2}}}}}\right)}} {{\sqrt{{{I}^{2}} - {{{4}} {{{\lambda}^{2}}} {{{r}^{2}}}}}}}}}}{{{{r}^{2}}} {{\left({{-{{{8}} {{{I}^{2}}} {{{\lambda}^{2}}} {{{r}^{2}}}}} + {{I}^{4}} + {{{16}} {{{\lambda}^{4}}} {{{r}^{4}}}}}\right)}}} & 0 & 0 & {\frac{1}{r}} {{{8}} {{\lambda}} \cdot {{{I}^{2}}} {{\frac{1}{\sqrt{{{I}^{2}} - {{{4}} {{{\lambda}^{2}}} {{{r}^{2}}}}}}}}} \\ 0 & \frac{{{4}} {{{I}^{2}}}}{{r}^{2}} & 0 & 0 \\ 0 & 0 & -{{{4}} {{{I}^{2}}}} & 0 \\ {\frac{1}{r}} {{{8}} {{\lambda}} \cdot {{{I}^{2}}} {{\frac{1}{\sqrt{{{I}^{2}} - {{{4}} {{{\lambda}^{2}}} {{{r}^{2}}}}}}}}} & 0 & 0 & \frac{{{4}} {{{I}^{3}}} {{{{\left({{{I}^{2}} - {{{4}} {{{\lambda}^{2}}} {{{r}^{2}}}}}\right)}} {{\sqrt{{{I}^{2}} - {{{4}} {{{\lambda}^{2}}} {{{r}^{2}}}}}}}}}}{{{{r}^{2}}} {{\left({{-{{{8}} {{{I}^{2}}} {{{\lambda}^{2}}} {{{r}^{2}}}}} + {{I}^{4}} + {{{16}} {{{\lambda}^{4}}} {{{r}^{4}}}}}\right)}}}\end{matrix} \right]}}$