Kaluza-Klein with constant scalar field


coordinate convention: $dx^0 = c dt, \partial_0 = \frac{1}{c} \partial_t$

$c = $ $\cdot \frac{m}{s} = 1 = $ the speed of light.
$G = $ $\cdot \frac{m^3}{kg \cdot s^2} = 1 = $ gravitational constant.
$k_e = $ $\cdot \frac{kg \cdot m^3}{C^2 \cdot s^2}$ = Coulomb's constant (typically $\frac{1}{4 \pi \epsilon_0}$).
$\sqrt{{\frac{1}{G}} {{k_e}}}$ = {=={ sqrt_Coulomb_constant_over_gravitational_constant_in_kg_per_C = Math.sqrt(Coulomb_constant_in_kg_m3_per_C2_s2 / gravitational_constant_in_m3_per_kg_s2) }==} $ \cdot \frac{kg}{C} = 1 =$ conversion from kg to C
$kg = $ {=={ 1 / sqrt_Coulomb_constant_over_gravitational_constant_in_kg_per_C }==} $C$.

${ A} _u$ = electromagnetic four-potential, in units $\frac{{{kg}} \cdot {{m}}}{{{C}} {{s}}}$
$A_5$ is constant in natural units, but to cancel the units of $\phi_K$ it is in units of $\frac{{{kg}} \cdot {{m}}}{{{C}} {{s}}}$ so $A_5$ is proportional to $c \sqrt{\frac{k_e}{G}} = $ ${{3.4789926447386\cdot{10^{18}}}} {{kg}} \cdot {{m}} {{\frac{1}{C}}} {{\frac{1}{s}}}$

${\phi_K}$ = scalar field, proportional to $\frac{1}{A_5}$, in units $\frac{{{C}} {{s}}}{{{kg}} \cdot {{m}}}$

${{ g} _{\mu}} _{\nu}$ = 4D metric tensor, with units $[g_{\mu\nu}] = 1$
${{ g} ^{\mu}} ^{\nu}$ = 4D metric inverse

${{ \tilde{g}} _a} _b$ = 5D metric tensor, with units $[\tilde{g}_{ab}] = 1$
${{{ \tilde{g}} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{ \tilde{g}} _{\alpha}} _{\beta} & {{ \tilde{g}} _{\alpha}} _5 \\ {{ \tilde{g}} _5} _{\beta} & {{ \tilde{g}} _5} _5\end{matrix} \right]}}$
${{{ \tilde{g}} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{{ g} _{\alpha}} _{\beta}} + {{{{{\phi_K}}^{2}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}}} & {{{{\phi_K}}^{2}}} {{{ A} _{\alpha}}} {{{ A} _5}} \\ {{{{\phi_K}}^{2}}} {{{ A} _{\beta}}} {{{ A} _5}} & {{{{\phi_K}}^{2}}} {{{{ A} _5}^{2}}}\end{matrix} \right]}}$

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What is $u_5$ in terms of $u^5$?
${{ u} _5} = {{{{{ \tilde{g}} _5} _a}} {{{ u} ^a}}}$
${{ u} _5} = {{{{{{ \tilde{g}} _5} _{\beta}}} {{{ u} ^{\beta}}}} + {{{{{ \tilde{g}} _5} _5}} {{{ u} ^5}}}}$
substitute definition of ${{ \tilde{g}} _a} _b$
${{ u} _5} = {{{{{{\phi_K}}^{2}}} {{{ A} _{\beta}}} {{{ A} _5}} {{{ u} ^{\beta}}}} + {{{{{\phi_K}}^{2}}} {{{{ A} _5}^{2}}} {{{ u} ^5}}}}$

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On a side note, later for the Lorentz force to arise we are going to set:
${{ u} ^5} = {{{\frac{1}{4}}} {{{\frac{1}{M}} {q}}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}$, ${{ A} _5} = {{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}$, ${{\phi_K}} = {{{\frac{2}{c}}} {{\sqrt{{\frac{1}{{k_e}}} {G}}}}}$

${{ A} _5} = {{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}$ = {=={ A5L = speed_of_light_in_m_per_s * Math.sqrt(Coulomb_constant_in_kg_m3_per_C2_s2 / gravitational_constant_in_m3_per_kg_s2) }==} $\frac{kg \cdot m}{C \cdot s} =$ fifth component of electromagnetic potential.
${{\phi_K}} = {{{\frac{2}{c}}} {{\sqrt{{\frac{1}{{k_e}}} {G}}}}}$ = {=={ phiK = 2 / A5L }==} $\frac{C \cdot s}{kg \cdot m} =$ Kaluza-Klein 5th dimension scalar cylinder radius.

For an electron this comes out to be...
$e =$ $\cdot C$ = electron charge
$m_e =$ $\cdot kg$ = electron mass
$u^5 = \frac{1}{4} \frac{q}{M} \sqrt{\frac{k_e}{G}} =$ {=={ u5U = .25 * electron_charge_in_C / electron_mass_in_kg * Math.sqrt(Coulomb_constant_in_kg_m3_per_C2_s2 / gravitational_constant_in_m3_per_kg_s2) }==} = {=={ u5U * speed_of_light_in_m_per_s }==} $\frac{m}{s}$.

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${{ \tilde{g}} ^u} ^v$ = 5D metric inverse
Notice, if you see a raised 4-index, it is being raised by the 4-metric and not the 5-metric.
${{{ \tilde{g}} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{ g} ^{\alpha}} ^{\beta} & \frac{-{{ A} ^{\alpha}}}{{ A} _5} \\ \frac{-{{ A} ^{\beta}}}{{ A} _5} & {{\left({{{{{ A} _{\mu}}} {{{ A} ^{\mu}}}} + {{{\phi_K}}^{-2}}}\right)}} {{{{ A} _5}^{-2}}}\end{matrix} \right]}}$

${{{{{{ \tilde{g}} _a} _c}} {{{{ \tilde{g}} ^c} ^b}}} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{ \delta} _{\alpha}} ^{\beta} & {{{{ A} _{\alpha}}} {{{{\phi_K}}^{2}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} {{{ A} _{\alpha}}} {{{{\phi_K}}^{2}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{\frac{1}{{ A} _5}}}} \\ 0 & {1} + {{{-1}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{\phi_K}}^{2}}}}\end{matrix} \right]}}} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{ \delta} _{\alpha}} ^{\beta} & 0 \\ 0 & 1\end{matrix} \right]}}$

The cylindrical constraint is
${{{{ \tilde{g}} _a} _b} _{,5}} = {0}$
Therefore
${\left[ \begin{matrix} {{{{ g} _{\alpha}} _{\beta}} _{,5}} + {{{{ A} _{\beta}}} {{{{ A} _{\alpha}} _{,5}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\alpha}}} {{{{ A} _{\beta}} _{,5}}} {{{{\phi_K}}^{2}}}} + {{{2}} {{{\phi_K}}} \cdot {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ {\phi_K}} _{,5}}}} & {{{\phi_K}}} \cdot {{\left({{{{{\phi_K}}} \cdot {{{ A} _5}} {{{{ A} _{\alpha}} _{,5}}}} + {{{{\phi_K}}} \cdot {{{ A} _{\alpha}}} {{{{ A} _5} _{,5}}}} + {{{2}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ {\phi_K}} _{,5}}}}}\right)}} \\ {{{\phi_K}}} \cdot {{\left({{{{{\phi_K}}} \cdot {{{ A} _5}} {{{{ A} _{\beta}} _{,5}}}} + {{{{\phi_K}}} \cdot {{{ A} _{\beta}}} {{{{ A} _5} _{,5}}}} + {{{2}} {{{ A} _5}} {{{ A} _{\beta}}} {{{ {\phi_K}} _{,5}}}}}\right)}} & {{2}} {{{\phi_K}}} \cdot {{{ A} _5}} {{\left({{{{{\phi_K}}} \cdot {{{{ A} _5} _{,5}}}} + {{{{ A} _5}} {{{ {\phi_K}} _{,5}}}}}\right)}}\end{matrix} \right]} = {0}$
Therefore, if $A_{5,5} = 0$ then we find:
${{ {\phi_K}} _{,5}} = {0}$
${{{ A} _{\mu}} _{,5}} = {0}$
${{{{ g} _{\alpha}} _{\beta}} _{,5}} = {0}$

For now I'll use a constant scalar as well
${{ {\phi_K}} _{,a}} = {0}$

metric partial:
${{{{ \tilde{g}} _a} _b} _{,c}} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{{{ g} _{\alpha}} _{\beta}} _{,c}} + {{{{ A} _{\beta}}} {{{{ A} _{\alpha}} _{,c}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\alpha}}} {{{{ A} _{\beta}} _{,c}}} {{{{\phi_K}}^{2}}}} & {{{{\phi_K}}^{2}}} {{\left({{{{{ A} _5}} {{{{ A} _{\alpha}} _{,c}}}} + {{{{ A} _{\alpha}}} {{{{ A} _5} _{,c}}}}}\right)}} \\ {{{{\phi_K}}^{2}}} {{\left({{{{{ A} _5}} {{{{ A} _{\beta}} _{,c}}}} + {{{{ A} _{\beta}}} {{{{ A} _5} _{,c}}}}}\right)}} & {{2}} {{{ A} _5}} {{{{ A} _5} _{,c}}} {{{{\phi_K}}^{2}}}\end{matrix} \right]}}$
${{{{ \tilde{g}} _a} _b} _{,c}} = {\overset{c\downarrow[{a\downarrow b\rightarrow}]}{\left[ \begin{matrix} \overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{{{ g} _{\alpha}} _{\beta}} _{,\gamma}} + {{{{ A} _{\beta}}} {{{{ A} _{\alpha}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\alpha}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} & {{{ A} _5}} {{{{ A} _{\alpha}} _{,\gamma}}} {{{{\phi_K}}^{2}}} \\ {{{ A} _5}} {{{{ A} _{\beta}} _{,\gamma}}} {{{{\phi_K}}^{2}}} & 0\end{matrix} \right]} \\ \overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} 0 & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$

lower connection
${{{{ \tilde{\Gamma}} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}}{\left({{{{{ g} _{\alpha}} _{\beta}} _{,\gamma}} + {{{{ A} _{\beta}}} {{{{ A} _{\alpha}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\alpha}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{{ g} _{\alpha}} _{\gamma}} _{,\beta}} + {{{{ A} _{\gamma}}} {{{{ A} _{\alpha}} _{,\beta}}} {{{{\phi_K}}^{2}}}} + {{{{{{{ A} _{\alpha}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{{\phi_K}}^{2}}}} - {{{{ g} _{\beta}} _{\gamma}} _{,\alpha}}} - {{{{ A} _{\gamma}}} {{{{ A} _{\beta}} _{,\alpha}}} {{{{\phi_K}}^{2}}}}} - {{{{ A} _{\beta}}} {{{{ A} _{\gamma}} _{,\alpha}}} {{{{\phi_K}}^{2}}}}}}\right)} & {\frac{1}{2}} {{{{ A} _5}} {{{{\phi_K}}^{2}}} {{\left({{{{ A} _{\alpha}} _{,\beta}} - {{{ A} _{\beta}} _{,\alpha}}}\right)}}} \\ {\frac{1}{2}} {{{{ A} _5}} {{{{\phi_K}}^{2}}} {{\left({{{{ A} _{\alpha}} _{,\gamma}} - {{{ A} _{\gamma}} _{,\alpha}}}\right)}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}} {{{{ A} _5}} {{{{\phi_K}}^{2}}} {{\left({{{{ A} _{\beta}} _{,\gamma}} + {{{ A} _{\gamma}} _{,\beta}}}\right)}}} & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{{ \tilde{\Gamma}} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}}{\left({{{{{ A} _{\beta}}} {{{{ F} _{\gamma}} _{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\alpha}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\gamma}}} {{{{ F} _{\beta}} _{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\alpha}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{{\phi_K}}^{2}}}} + {{{2}} {{{{{ \Gamma} _{\alpha}} _{\beta}} _{\gamma}}}}}\right)} & {\frac{1}{2}} {{{{ A} _5}} {{{{ F} _{\beta}} _{\alpha}}} {{{{\phi_K}}^{2}}}} \\ {\frac{1}{2}} {{{{ A} _5}} {{{{ F} _{\gamma}} _{\alpha}}} {{{{\phi_K}}^{2}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}} {{{{ A} _5}} {{{{\phi_K}}^{2}}} {{\left({{{{ A} _{\beta}} _{,\gamma}} + {{{ A} _{\gamma}} _{,\beta}}}\right)}}} & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$

upper connection
${{{{ \tilde{\Gamma}} ^a} _b} _c} = {{{\overset{a\downarrow e\rightarrow}{\left[ \begin{matrix} {{ g} ^{\alpha}} ^{\epsilon} & -{\frac{{ A} ^{\alpha}}{{ A} _5}} \\ -{\frac{{ A} ^{\epsilon}}{{ A} _5}} & \frac{{1} + {{{{ A} _{\mu}}} {{{ A} ^{\mu}}} {{{{\phi_K}}^{2}}}}}{{{{{\phi_K}}^{2}}} {{{{ A} _5}^{2}}}}\end{matrix} \right]}}} {{\overset{e\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}}{\left({{{{{ A} _{\beta}}} {{{{ F} _{\gamma}} _{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\epsilon}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\gamma}}} {{{{ F} _{\beta}} _{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\epsilon}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{{\phi_K}}^{2}}}} + {{{2}} {{{{{ \Gamma} _{\epsilon}} _{\beta}} _{\gamma}}}}}\right)} & {\frac{1}{2}} {{{{ A} _5}} {{{{ F} _{\beta}} _{\epsilon}}} {{{{\phi_K}}^{2}}}} \\ {\frac{1}{2}} {{{{ A} _5}} {{{{ F} _{\gamma}} _{\epsilon}}} {{{{\phi_K}}^{2}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}} {{{{ A} _5}} {{{{\phi_K}}^{2}}} {{\left({{{{ A} _{\beta}} _{,\gamma}} + {{{ A} _{\gamma}} _{,\beta}}}\right)}}} & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}}}$
${{{{ \tilde{\Gamma}} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} ^{\alpha}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} ^{\alpha}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ F} _{\gamma}} _{\epsilon}}} {{{{ g} ^{\alpha}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{{ g} ^{\alpha}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ F} _{\beta}} _{\epsilon}}} {{{{ g} ^{\alpha}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{{ g} ^{\alpha}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{{{ g} ^{\alpha}} ^{\epsilon}}} {{{{{ \Gamma} _{\epsilon}} _{\beta}} _{\gamma}}}} & {{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\beta}} _{\epsilon}}} {{{{ g} ^{\alpha}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} \\ {{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\gamma}} _{\epsilon}}} {{{{ g} ^{\alpha}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{2}}} {{{{ A} _{\beta}} _{,\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{ A} _{\gamma}} _{,\beta}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} ^{\mu}}} {{{ A} _{\mu}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} ^{\mu}}} {{{ A} _{\mu}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} ^{\epsilon}}} {{{{ F} _{\gamma}} _{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} ^{\epsilon}}} {{{ A} _{\epsilon}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} ^{\epsilon}}} {{{ A} _{\gamma}}} {{{{ F} _{\beta}} _{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} ^{\epsilon}}} {{{ A} _{\epsilon}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} {{{ A} ^{\epsilon}}} {{{{{ \Gamma} _{\epsilon}} _{\beta}} _{\gamma}}} {{\frac{1}{{ A} _5}}}} & {{-1}} \cdot {{\frac{1}{2}}} {{{ A} ^{\epsilon}}} {{{{ F} _{\beta}} _{\epsilon}}} {{{{\phi_K}}^{2}}} \\ {{-1}} \cdot {{\frac{1}{2}}} {{{ A} ^{\epsilon}}} {{{{ F} _{\gamma}} _{\epsilon}}} {{{{\phi_K}}^{2}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{{ \tilde{\Gamma}} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {{{{ \Gamma} ^{\alpha}} _{\beta}} _{\gamma}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} & {{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{\phi_K}}^{2}}} \\ {{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {{{-1}} {{{ A} _{\mu}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{ A} _{\gamma}} _{,\beta}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{ A} _{\beta}} _{,\gamma}}} {{\frac{1}{{ A} _5}}}} & {{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}} \\ {{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$

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geodesic equation:
${{ \dot{u}} ^a} = { {-{{{{ \tilde{\Gamma}} ^a} _b} _c}} {{{ u} ^b}} {{{ u} ^c}}}$
(Notice I am unifortunately denoting $\dot{u}^a = \partial_0 u^a = \frac{1}{c} \partial_t u^a$)

only look at spacetime components:
${{ \dot{u}} ^{\alpha}} = {{{{-1}} {{{{{ \tilde{\Gamma}} ^{\alpha}} _5} _5}} {{{{ u} ^5}^{2}}}} + {{{-2}} {{{ u} ^5}} {{{ u} ^{\beta}}} {{{{{ \tilde{\Gamma}} ^{\alpha}} _{\beta}} _5}}} + {{{-1}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \tilde{\Gamma}} ^{\alpha}} _{\beta}} _{\gamma}}}}}$
${{ \dot{u}} ^{\alpha}} = {{{{-1}} {{{ A} _5}} {{{ u} ^5}} {{{ u} ^{\beta}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{-1}} {{{ A} _{\beta}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{-1}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^{\alpha}} _{\beta}} _{\gamma}}}}}$
${{ \dot{u}} ^{\alpha}} = {{{{-1}} {{{ A} _5}} {{{ u} ^5}} {{{ u} ^{\beta}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{-1}} {{{ A} _{\beta}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{-1}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^{\alpha}} _{\beta}} _{\gamma}}}}}$
Substitute ${{ u} ^5} = {{{\frac{1}{4}}} {{{\frac{1}{M}} {q}}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}$, ${{ A} _5} = {{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}$, ${{\phi_K}} = {{{\frac{2}{c}}} {{\sqrt{{\frac{1}{{k_e}}} {G}}}}}$
${{ \dot{u}} ^{\alpha}} = {{{{-4}} {{G}} {{{ A} _{\beta}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^{\alpha}} _{\beta}} _{\gamma}}}} + {{{-1}} {{q}} {{{ u} ^{\beta}}} {{{{ F} _{\beta}} ^{\alpha}}} {{\frac{1}{M}}} {{\frac{1}{c}}}}}$

There you have gravitational force, Lorentz force, and an extra term.

Separate space and time, substitute spacetime geodesic with Newtonian gravity, etc:

Spatial evolution:
${{ \dot{u}} ^i} = {{{{-4}} {{G}} {{{ A} _{\beta}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\gamma}} ^i}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^i} _{\beta}} _{\gamma}}}} + {{{-1}} {{q}} {{{ u} ^{\beta}}} {{{{ F} _{\beta}} ^i}} {{\frac{1}{M}}} {{\frac{1}{c}}}}}$
Notice, if we assume $A_\mu u^\mu = 0$ then we are left only with terms for gravitational acceleration the and Lorentz force.
Splitting spacetime indexes into space+time
${{{{ \dot{u}} ^i}} {{{c}^{2}}}} = {{{{-4}} {{G}} {{{ A} _0}} {{{{ F} _0} ^i}} {{{{ u} ^0}^{2}}} {{\frac{1}{{k_e}}}}} + {{{-4}} {{G}} {{{ A} _0}} {{{ u} ^0}} {{{ u} ^k}} {{{{ F} _k} ^i}} {{\frac{1}{{k_e}}}}} + {{{-4}} {{G}} {{{ A} _j}} {{{ u} ^0}} {{{ u} ^j}} {{{{ F} _0} ^i}} {{\frac{1}{{k_e}}}}} + {{{-4}} {{G}} {{{ A} _j}} {{{ u} ^j}} {{{ u} ^k}} {{{{ F} _k} ^i}} {{\frac{1}{{k_e}}}}} + {{{-1}} {{{{{ \Gamma} ^i} _0} _0}} {{{c}^{2}}} {{{{ u} ^0}^{2}}}} + {{{-1}} {{{ u} ^0}} {{{ u} ^j}} {{{{{ \Gamma} ^i} _j} _0}} {{{c}^{2}}}} + {{{-1}} {{{ u} ^0}} {{{ u} ^k}} {{{{{ \Gamma} ^i} _0} _k}} {{{c}^{2}}}} + {{{-1}} {{{ u} ^j}} {{{ u} ^k}} {{{{{ \Gamma} ^i} _j} _k}} {{{c}^{2}}}} + {{{-1}} {{c}} {{q}} {{{ u} ^0}} {{{{ F} _0} ^i}} {{\frac{1}{M}}}} + {{{-1}} {{c}} {{q}} {{{ u} ^j}} {{{{ F} _j} ^i}} {{\frac{1}{M}}}}}$
Low-velocity approximation: ${{ u} ^0} = {1}$
${{{{ \dot{u}} ^i}} {{{c}^{2}}}} = {{{{-4}} {{G}} {{{ A} _0}} {{{{ F} _0} ^i}} {{\frac{1}{{k_e}}}}} + {{{-4}} {{G}} {{{ A} _0}} {{{ u} ^k}} {{{{ F} _k} ^i}} {{\frac{1}{{k_e}}}}} + {{{-4}} {{G}} {{{ A} _j}} {{{ u} ^j}} {{{{ F} _0} ^i}} {{\frac{1}{{k_e}}}}} + {{{-4}} {{G}} {{{ A} _j}} {{{ u} ^j}} {{{ u} ^k}} {{{{ F} _k} ^i}} {{\frac{1}{{k_e}}}}} + {{{-1}} {{{{{ \Gamma} ^i} _0} _0}} {{{c}^{2}}}} + {{{-1}} {{{ u} ^j}} {{{{{ \Gamma} ^i} _j} _0}} {{{c}^{2}}}} + {{{-1}} {{{ u} ^j}} {{{ u} ^k}} {{{{{ \Gamma} ^i} _j} _k}} {{{c}^{2}}}} + {{{-1}} {{{ u} ^k}} {{{{{ \Gamma} ^i} _0} _k}} {{{c}^{2}}}} + {{{-1}} {{c}} {{q}} {{{{ F} _0} ^i}} {{\frac{1}{M}}}} + {{{-1}} {{c}} {{q}} {{{ u} ^j}} {{{{ F} _j} ^i}} {{\frac{1}{M}}}}}$
Assume spacetime connection is only ${{{ \Gamma} ^i} _0} _0$
Low-velocity Faraday tensor:
Assume ${{{ F} _0} ^i} = { {-{\frac{1}{c}}} {{{ E} ^i}}}$ , ${{{ F} _i} ^j} = {{{{{{ \epsilon} _i} ^j} ^k}} {{{ B} _k}}}$
${{{{ \dot{u}} ^i}} {{{c}^{2}}}} = {{{{4}} {{G}} {{{ A} _0}} {{{ E} ^i}} {{\frac{1}{c}}} {{\frac{1}{{k_e}}}}} + {{{4}} {{G}} {{{ A} _0}} {{{ B} ^l}} {{{ u} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{\frac{1}{{k_e}}}}} + {{{4}} {{G}} {{{ A} _j}} {{{ E} ^i}} {{{ u} ^j}} {{\frac{1}{c}}} {{\frac{1}{{k_e}}}}} + {{{4}} {{G}} {{{ A} _j}} {{{ B} ^l}} {{{ u} ^j}} {{{ u} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{\frac{1}{{k_e}}}}} + {{{-1}} {{{{{ \Gamma} ^i} _0} _0}} {{{c}^{2}}}} + {{{c}} {{q}} {{{ B} ^k}} {{{ u} ^j}} {{{{{ \epsilon} ^i} _j} _k}} {{\frac{1}{M}}}} + {{{q}} {{{ E} ^i}} {{\frac{1}{M}}}}}$
Substitute ${{ A} _0} = {{{\frac{1}{c}}} {{{\phi_q}}}}$ is the electric field potential
${{{{ \dot{u}} ^i}} {{{c}^{2}}}} = {{{{4}} {{G}} {{{\phi_q}}} \cdot {{{ E} ^i}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{4}} {{G}} {{{\phi_q}}} \cdot {{{ B} ^l}} {{{ u} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{\frac{1}{c}}} {{\frac{1}{{k_e}}}}} + {{{4}} {{G}} {{{ A} _j}} {{{ E} ^i}} {{{ u} ^j}} {{\frac{1}{c}}} {{\frac{1}{{k_e}}}}} + {{{4}} {{G}} {{{ A} _j}} {{{ B} ^l}} {{{ u} ^j}} {{{ u} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{\frac{1}{{k_e}}}}} + {{{-1}} {{{{{ \Gamma} ^i} _0} _0}} {{{c}^{2}}}} + {{{q}} {{{ E} ^i}} {{\frac{1}{M}}}} + {{{c}} {{q}} {{{ B} ^k}} {{{ u} ^j}} {{{{{ \epsilon} ^i} _j} _k}} {{\frac{1}{M}}}}}$
Assume ${{{{ \Gamma} ^i} _0} _0} = {\frac{{{G}} {{{M_2}}} \cdot {{{ x} ^i}}}{{{{c}^{2}}} {{{r}^{3}}}}}$
${{{{ \dot{u}} ^i}} {{{c}^{2}}}} = {{{{4}} {{G}} {{{\phi_q}}} \cdot {{{ E} ^i}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{4}} {{G}} {{{\phi_q}}} \cdot {{{ B} ^l}} {{{ u} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{\frac{1}{c}}} {{\frac{1}{{k_e}}}}} + {{{4}} {{G}} {{{ A} _j}} {{{ E} ^i}} {{{ u} ^j}} {{\frac{1}{c}}} {{\frac{1}{{k_e}}}}} + {{{4}} {{G}} {{{ A} _j}} {{{ B} ^l}} {{{ u} ^j}} {{{ u} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{\frac{1}{{k_e}}}}} + {{{-1}} {{G}} {{{M_2}}} \cdot {{{ x} ^i}} {{\frac{1}{{r}^{3}}}}} + {{{c}} {{q}} {{{ B} ^k}} {{{ u} ^j}} {{{{{ \epsilon} ^i} _j} _k}} {{\frac{1}{M}}}} + {{{q}} {{{ E} ^i}} {{\frac{1}{M}}}}}$
Let ${{ u} ^i} = {{{\frac{1}{c}}} {{{ v} ^i}}}$ , ${{ \dot{u}} ^i} = {{{\frac{1}{{c}^{2}}}} {{\frac{\partial { v} ^i}{\partial t}}}}$
${0} = {{{{4}} {{G}} {{{\phi_q}}} \cdot {{{ E} ^i}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{4}} {{G}} {{{\phi_q}}} \cdot {{{ B} ^l}} {{{ v} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{4}} {{G}} {{{ A} _j}} {{{ E} ^i}} {{{ v} ^j}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{4}} {{G}} {{{ A} _j}} {{{ B} ^l}} {{{ v} ^j}} {{{ v} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{G}} {{{M_2}}} \cdot {{{ x} ^i}} {{\frac{1}{{r}^{3}}}}} + {{{q}} {{{ E} ^i}} {{\frac{1}{M}}}} + {{{q}} {{{ B} ^k}} {{{ v} ^j}} {{{{{ \epsilon} ^i} _j} _k}} {{\frac{1}{M}}}}}$
Using real-world values: ${G} = {{{6.67384\cdot{10^{-11}}}} \cdot {{\frac{{m}^{3}}{{{kg}} \cdot {{{s}^{2}}}}}}}$, ${{k_e}} = {{{8987551787.3682}} {{kg}} \cdot {{{m}^{3}}} {{\frac{1}{{C}^{2}}}} {{\frac{1}{{s}^{2}}}}}$, ${{M_2}} = {{{5.972\cdot{10^{24}}}} {{kg}}}$, ${r} = {{{6371000}} {{m}}}$, ${q} = {{{1.6021765654775\cdot{10^{-19}}}} {{C}}}$, ${M} = {{{9.1093835611\cdot{10^{-31}}}} {{kg}}}$, ${{ x} ^i} = {{{{ \hat{r}} ^i}} {{6371000}} {{m}}}$, ${{ E} ^i} = {{{3.2425382804713\cdot{10^{15}}}} {{{ \hat{E}} ^i}} {{\frac{{{kg}} \cdot {{m}}}{{{C}} {{{s}^{2}}}}}}}$, ${{ B} ^i} = {{{9.4500027932875\cdot{10^{-07}}}} {{{ \hat{B}} ^i}} {{\frac{kg}{{{C}} {{s}}}}}}$
...and looking at each term:
${{{4}} {{G}} {{{\phi_q}}} \cdot {{{ E} ^i}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} = {{{9.6311797426993\cdot{10^{-05}}}} {{C}} {{{\phi_q}}} \cdot {{m}} {{{ \hat{E}} ^i}} {{\frac{1}{kg}}} {{\frac{1}{{c}^{2}}}} {{\frac{1}{{s}^{2}}}}}$
${{{4}} {{G}} {{{\phi_q}}} \cdot {{{ B} ^l}} {{{ v} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} = {{{2.9702593800372\cdot{10^{-20}}}} {{{\phi_q}}} \cdot {{{ B} ^l}} {{{ v} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{{C}^{2}}} {{\frac{1}{{c}^{2}}}} {{\frac{1}{{kg}^{2}}}}}$
${{{4}} {{G}} {{{ A} _j}} {{{ E} ^i}} {{{ v} ^j}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} = {{{9.6311797426993\cdot{10^{-05}}}} {{C}} {{m}} {{{ A} _j}} {{{ \hat{E}} ^i}} {{{ v} ^j}} {{\frac{1}{kg}}} {{\frac{1}{{c}^{2}}}} {{\frac{1}{{s}^{2}}}}}$
${{{4}} {{G}} {{{ A} _j}} {{{ B} ^l}} {{{ v} ^j}} {{{ v} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} = {{{2.9702593800372\cdot{10^{-20}}}} {{{ A} _j}} {{{ B} ^l}} {{{ v} ^j}} {{{ v} ^k}} {{{{{ \epsilon} ^i} _k} _l}} {{{C}^{2}}} {{\frac{1}{{c}^{2}}}} {{\frac{1}{{kg}^{2}}}}}$
${{{-1}} {{G}} {{{M_2}}} \cdot {{{ x} ^i}} {{\frac{1}{{r}^{3}}}}} = {{{-9.819296622998}} {{m}} {{{ \hat{r}} ^i}} {{\frac{1}{{s}^{2}}}}}$
${{{q}} {{{ E} ^i}} {{\frac{1}{M}}}} = {{{5.7030410573769\cdot{10^{26}}}} {{m}} {{{ \hat{E}} ^i}} {{\frac{1}{{s}^{2}}}}}$
${{{q}} {{{ B} ^k}} {{{ v} ^j}} {{{{{ \epsilon} ^i} _j} _k}} {{\frac{1}{M}}}} = {{{175881996265.84}} {{C}} {{{ B} ^k}} {{{ v} ^j}} {{{{{ \epsilon} ^i} _j} _k}} {{\frac{1}{kg}}}}$


Time evolution:
${{ \dot{u}} ^0} = {{{{-4}} {{G}} {{{ A} _{\beta}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\gamma}} ^0}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^0} _{\beta}} _{\gamma}}}} + {{{-1}} {{q}} {{{ u} ^{\beta}}} {{{{ F} _{\beta}} ^0}} {{\frac{1}{M}}} {{\frac{1}{c}}}}}$
${{ \dot{u}} ^0} = {{{{-4}} {{G}} {{{ A} _0}} {{{ u} ^0}} {{{ u} ^0}} {{{{ F} _0} ^0}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-4}} {{G}} {{{ A} _j}} {{{ u} ^j}} {{{ u} ^0}} {{{{ F} _0} ^0}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-4}} {{G}} {{{ A} _0}} {{{ u} ^0}} {{{ u} ^k}} {{{{ F} _k} ^0}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-4}} {{G}} {{{ A} _j}} {{{ u} ^j}} {{{ u} ^k}} {{{{ F} _k} ^0}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{{ u} ^0}} {{{ u} ^0}} {{{{{ \Gamma} ^0} _0} _0}}} + {{{-1}} {{{ u} ^j}} {{{ u} ^0}} {{{{{ \Gamma} ^0} _j} _0}}} + {{{-1}} {{{ u} ^0}} {{{ u} ^k}} {{{{{ \Gamma} ^0} _0} _k}}} + {{{-1}} {{{ u} ^j}} {{{ u} ^k}} {{{{{ \Gamma} ^0} _j} _k}}} + {{{-1}} {{q}} {{{ u} ^0}} {{{{ F} _0} ^0}} {{\frac{1}{M}}} {{\frac{1}{c}}}} + {{{-1}} {{q}} {{{ u} ^j}} {{{{ F} _j} ^0}} {{\frac{1}{M}}} {{\frac{1}{c}}}}}$
Low-velocity approximation: ${{ u} ^0} = {1}$
${{ \dot{u}} ^0} = {{{{-4}} {{G}} {{M}} {{{ A} _0}} {{{{ F} _0} ^0}} {{\frac{1}{M}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-4}} {{G}} {{M}} {{{ A} _0}} {{{ u} ^k}} {{{{ F} _k} ^0}} {{\frac{1}{M}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-4}} {{G}} {{M}} {{{ A} _j}} {{{ u} ^j}} {{{{ F} _0} ^0}} {{\frac{1}{M}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-4}} {{G}} {{M}} {{{ A} _j}} {{{ u} ^j}} {{{ u} ^k}} {{{{ F} _k} ^0}} {{\frac{1}{M}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{M}} {{{k_e}}} \cdot {{{{{ \Gamma} ^0} _0} _0}} {{{c}^{2}}} {{\frac{1}{M}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{M}} {{{k_e}}} \cdot {{{ u} ^j}} {{{{{ \Gamma} ^0} _j} _0}} {{{c}^{2}}} {{\frac{1}{M}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{M}} {{{k_e}}} \cdot {{{ u} ^j}} {{{ u} ^k}} {{{{{ \Gamma} ^0} _j} _k}} {{{c}^{2}}} {{\frac{1}{M}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{M}} {{{k_e}}} \cdot {{{ u} ^k}} {{{{{ \Gamma} ^0} _0} _k}} {{{c}^{2}}} {{\frac{1}{M}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{c}} {{{k_e}}} \cdot {{q}} {{{{ F} _0} ^0}} {{\frac{1}{M}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{c}} {{{k_e}}} \cdot {{q}} {{{ u} ^j}} {{{{ F} _j} ^0}} {{\frac{1}{M}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}}}$
Assume spacetime connection is only ${{{ \Gamma} ^i} _0} _0$
Assume ${{{ F} _i} ^0} = { {-{\frac{1}{c}}} {{{ E} _i}}}$ , ${{{ F} _0} ^0} = {0}$
Substitute ${{ A} _0} = {{{\frac{1}{c}}} {{{\phi_q}}}}$ is the electric field potential
${{ \dot{u}} ^0} = {{{{4}} {{G}} {{{\phi_q}}} \cdot {{{ E} _k}} {{{ u} ^k}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{4}}}}} + {{{4}} {{G}} {{{ A} _j}} {{{ E} _k}} {{{ u} ^j}} {{{ u} ^k}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{3}}}}} + {{{-1}} {{{{{ \Gamma} ^0} _0} _0}}} + {{{-1}} {{{ u} ^j}} {{{{{ \Gamma} ^0} _j} _0}}} + {{{-1}} {{{ u} ^k}} {{{{{ \Gamma} ^0} _0} _k}}} + {{{q}} {{{ E} _j}} {{{ u} ^j}} {{\frac{1}{M}}} {{\frac{1}{{c}^{2}}}}}}$


Look at the 5th dimension evolution:
${{ \dot{u}} ^5} = { {-{{{{ \tilde{\Gamma}} ^5} _b} _c}} {{{ u} ^b}} {{{ u} ^c}}}$
${{ \dot{u}} ^5} = {{{ {-{{{{ \tilde{\Gamma}} ^5} _{\beta}} _{\gamma}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}}} - {{{2}} {{{{{ \tilde{\Gamma}} ^5} _5} _{\beta}}} {{{ u} ^5}} {{{ u} ^{\beta}}}}} - {{{{{{ \tilde{\Gamma}} ^5} _5} _5}} {{{{ u} ^5}^{2}}}}}$
${{ \dot{u}} ^5} = {{{{{ A} _{\mu}}} {{{ u} ^5}} {{{ u} ^{\beta}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\beta}} _{,\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\gamma}} _{,\beta}}} {{\frac{1}{{ A} _5}}}}}$
Substitute ${{ A} _5} = {{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}$
${{ \dot{u}} ^5} = {{{{{ A} _{\mu}}} {{{ u} ^5}} {{{ u} ^{\beta}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}}}} + {{{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}}} {{\frac{1}{{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\beta}} _{,\gamma}}} {{\frac{1}{{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\gamma}} _{,\beta}}} {{\frac{1}{{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}}}}}$
Substitute ${{\phi_K}} = {{{\frac{2}{c}}} {{\sqrt{{\frac{1}{{k_e}}} {G}}}}}$
${{ \dot{u}} ^5} = {{{{{ A} _{\mu}}} {{{ u} ^5}} {{{ u} ^{\beta}}} {{{{ F} _{\beta}} ^{\mu}}} {{{\left({{{\frac{2}{c}}} {{\sqrt{{\frac{1}{{k_e}}} {G}}}}}\right)}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{\left({{{\frac{2}{c}}} {{\sqrt{{\frac{1}{{k_e}}} {G}}}}}\right)}^{2}}} {{\frac{1}{{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\beta}} ^{\mu}}} {{{\left({{{\frac{2}{c}}} {{\sqrt{{\frac{1}{{k_e}}} {G}}}}}\right)}^{2}}} {{\frac{1}{{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}}}} + {{{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}}} {{\frac{1}{{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\beta}} _{,\gamma}}} {{\frac{1}{{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\gamma}} _{,\beta}}} {{\frac{1}{{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}}}}}$
${{ \dot{u}} ^5} = {{{{2}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{G}^{{{3}} \cdot {{\frac{1}{2}}}}}} {{\frac{1}{{c}^{3}}}} {{\frac{1}{{{k_e}}^{{{3}} \cdot {{\frac{1}{2}}}}}}}} + {{{2}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ F} _{\beta}} ^{\mu}}} {{{G}^{{{3}} \cdot {{\frac{1}{2}}}}}} {{\frac{1}{{c}^{3}}}} {{\frac{1}{{{k_e}}^{{{3}} \cdot {{\frac{1}{2}}}}}}}} + {{{4}} {{G}} {{{ A} _{\mu}}} {{{ u} ^5}} {{{ u} ^{\beta}}} {{{{ F} _{\beta}} ^{\mu}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}}}$
${{ \dot{u}} ^5} = {{{{-4}} {{{ A} ^{\gamma}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\mu}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{G}^{{{3}} \cdot {{\frac{1}{2}}}}}} {{\frac{1}{{c}^{3}}}} {{\frac{1}{{{k_e}}^{{{3}} \cdot {{\frac{1}{2}}}}}}}} + {{{4}} {{{ A} ^{\gamma}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\mu}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{G}^{{{3}} \cdot {{\frac{1}{2}}}}}} {{\frac{1}{{c}^{3}}}} {{\frac{1}{{{k_e}}^{{{3}} \cdot {{\frac{1}{2}}}}}}}} + {{{-4}} {{G}} {{{ A} ^{\gamma}}} {{{ u} ^5}} {{{ u} ^{\beta}}} {{{{ A} _{\beta}} _{,\gamma}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{4}} {{G}} {{{ A} ^{\gamma}}} {{{ u} ^5}} {{{ u} ^{\beta}}} {{{{ A} _{\gamma}} _{,\beta}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}}}$
${{ \dot{u}} ^5} = {{{{-4}} {{{ A} ^{\gamma}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\mu}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{G}^{{{3}} \cdot {{\frac{1}{2}}}}}} {{\frac{1}{{c}^{3}}}} {{\frac{1}{{{k_e}}^{{{3}} \cdot {{\frac{1}{2}}}}}}}} + {{{4}} {{{ A} ^{\gamma}}} {{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\mu}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{G}^{{{3}} \cdot {{\frac{1}{2}}}}}} {{\frac{1}{{c}^{3}}}} {{\frac{1}{{{k_e}}^{{{3}} \cdot {{\frac{1}{2}}}}}}}} + {{{{ A} _{\mu}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ u} ^{\beta}}} {{{ u} ^{\gamma}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{-1}} {{q}} {{{ A} ^{\gamma}}} {{{ u} ^{\beta}}} {{{{ A} _{\beta}} _{,\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{M}}} {{\frac{1}{{c}^{2}}}}} + {{{q}} {{{ A} ^{\gamma}}} {{{ u} ^{\beta}}} {{{{ A} _{\gamma}} _{,\beta}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{M}}} {{\frac{1}{{c}^{2}}}}}}$

---

connection partial:
${{{{{ \tilde{\Gamma}} ^a} _b} _c} _{,d}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {{{{{ \Gamma} ^{\alpha}} _{\beta}} _{\gamma}} _{,d}} + {{{\frac{1}{2}}} {{{{ A} _{\beta}} _{,d}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\gamma}} ^{\alpha}} _{,d}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{{ A} _{\gamma}} _{,d}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{{ F} _{\beta}} ^{\alpha}} _{,d}}} {{{{\phi_K}}^{2}}}} & {{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\beta}} ^{\alpha}} _{,d}}} {{{{\phi_K}}^{2}}} \\ {{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\gamma}} ^{\alpha}} _{,d}}} {{{{\phi_K}}^{2}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{2}}} {{{{{ A} _{\beta}} _{,\gamma}} _{,d}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ A} _{\mu}} _{,d}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{{ F} _{\gamma}} ^{\mu}} _{,d}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{{ A} _{\gamma}} _{,\beta}} _{,d}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ A} _{\mu}} _{,d}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{{ F} _{\beta}} ^{\mu}} _{,d}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} {{{{ A} _{\mu}} _{,d}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ A} _{\beta}} _{,d}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ A} _{\gamma}} _{,d}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} {{{ A} _{\mu}}} {{{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}} _{,d}}} {{\frac{1}{{ A} _5}}}} & {{{-1}} \cdot {{\frac{1}{2}}} {{{{ A} _{\mu}} _{,d}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{{ F} _{\beta}} ^{\mu}} _{,d}}} {{{{\phi_K}}^{2}}}} \\ {{{-1}} \cdot {{\frac{1}{2}}} {{{{ A} _{\mu}} _{,d}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{{ F} _{\gamma}} ^{\mu}} _{,d}}} {{{{\phi_K}}^{2}}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$
${{{{{ \tilde{\Gamma}} ^a} _b} _c} _{,d}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{{{ \Gamma} ^{\alpha}} _{\beta}} _{\gamma}} _{,\delta}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{{ F} _{\beta}} ^{\alpha}} _{,\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{{ A} _{\gamma}} _{,\delta}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\gamma}} ^{\alpha}} _{,\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{{ A} _{\beta}} _{,\delta}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} & 0 \\ {{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\beta}} ^{\alpha}} _{,\delta}}} {{{{\phi_K}}^{2}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\gamma}} ^{\alpha}} _{,\delta}}} {{{{\phi_K}}^{2}}} & 0 \\ 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{-1}} {{{{ A} _{\mu}} _{,\delta}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} {{{ A} _{\mu}}} {{{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}} _{,\delta}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ A} _{\gamma}} _{,\delta}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ A} _{\beta}} _{,\delta}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{{ F} _{\beta}} ^{\mu}} _{,\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ A} _{\mu}} _{,\delta}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{{ A} _{\gamma}} _{,\beta}} _{,\delta}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{{ F} _{\gamma}} ^{\mu}} _{,\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ A} _{\mu}} _{,\delta}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{{ A} _{\beta}} _{,\gamma}} _{,\delta}}} {{\frac{1}{{ A} _5}}}} & 0 \\ {{{-1}} \cdot {{\frac{1}{2}}} {{{{ A} _{\mu}} _{,\delta}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{{ F} _{\beta}} ^{\mu}} _{,\delta}}} {{{{\phi_K}}^{2}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{-1}} \cdot {{\frac{1}{2}}} {{{{ A} _{\mu}} _{,\delta}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{{ F} _{\gamma}} ^{\mu}} _{,\delta}}} {{{{\phi_K}}^{2}}}} & 0 \\ 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$

${{{{{{ \tilde{\Gamma}} ^a} _b} _e}} {{{{{ \tilde{\Gamma}} ^e} _c} _d}}} = {{{\overset{a\downarrow[{e\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}}{\left({{{{2}} {{{{{ \Gamma} ^{\alpha}} _{\epsilon}} _{\gamma}}}} + {{{{ A} _{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{2}}}}}\right)} & {\frac{1}{2}} {{{{ A} _5}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} \\ {\frac{1}{2}} {{{{ A} _5}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} \frac{{-{{{2}} {{{ A} _{\mu}}} {{{{{ \Gamma} ^{\mu}} _{\epsilon}} _{\gamma}}}}} + {{{ A} _{\epsilon}} _{,\gamma}} + {{{{{ A} _{\gamma}} _{,\epsilon}} - {{{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}}}} - {{{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{2}}}}}}{{{2}} {{{ A} _5}}} & -{{\frac{1}{2}} {{{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{2}}}}} \\ -{{\frac{1}{2}} {{{{ A} _{\mu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}}}} & 0\end{matrix} \right]}\end{matrix} \right]}}} {{\overset{e\downarrow[{b\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}}{\left({{{{2}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}}} + {{{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\delta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}}}\right)} & {\frac{1}{2}} {{{{ A} _5}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} \\ {\frac{1}{2}} {{{{ A} _5}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} \frac{{-{{{2}} {{{ A} _{\nu}}} {{{{{ \Gamma} ^{\nu}} _{\beta}} _{\delta}}}}} + {{{ A} _{\beta}} _{,\delta}} + {{{{{ A} _{\delta}} _{,\beta}} - {{{{ A} _{\beta}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\nu}}} {{{{\phi_K}}^{2}}}}} - {{{{ A} _{\delta}}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{\phi_K}}^{2}}}}}}{{{2}} {{{ A} _5}}} & -{{\frac{1}{2}} {{{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{\phi_K}}^{2}}}}} \\ -{{\frac{1}{2}} {{{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\nu}}} {{{{\phi_K}}^{2}}}}} & 0\end{matrix} \right]}\end{matrix} \right]}}}}$
${{{{{{ \tilde{\Gamma}} ^a} _b} _e}} {{{{{ \tilde{\Gamma}} ^e} _c} _d}}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\nu}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{{ \Gamma} ^{\nu}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{{ A} _{\beta}} _{,\delta}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{{ A} _{\delta}} _{,\beta}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{{{{ \Gamma} ^{\alpha}} _{\epsilon}} _{\gamma}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _{\delta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\epsilon}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} & {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\epsilon}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\gamma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} \\ {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\delta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}}} & {{\frac{1}{4}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\epsilon}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} & 0 \\ {{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{2}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{{ \Gamma} ^{\nu}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\mu}}} {{{{ A} _{\beta}} _{,\delta}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\mu}}} {{{{ A} _{\delta}} _{,\beta}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} {{{ A} _{\mu}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{{ \Gamma} ^{\mu}} _{\epsilon}} _{\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{ A} _{\epsilon}} _{,\gamma}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{ A} _{\gamma}} _{,\epsilon}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{{ A} _{\epsilon}} _{,\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{{ A} _{\gamma}} _{,\epsilon}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\delta}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{{ A} _{\epsilon}} _{,\gamma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{{ A} _{\gamma}} _{,\epsilon}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} & {{{\frac{1}{4}}} {{{{ A} _{\epsilon}} _{,\gamma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{{ A} _{\gamma}} _{,\epsilon}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} \\ {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}}} & {{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{4}}} {{{{ A} _{\epsilon}} _{,\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{{ A} _{\gamma}} _{,\epsilon}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} & 0 \\ {{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$

Riemann curvature tensor:
${{{{{ \tilde{R}} ^a} _b} _c} _d} = {{{{{{{{ \tilde{\Gamma}} ^a} _b} _d} _{,c}} - {{{{{ \tilde{\Gamma}} ^a} _b} _c} _{,d}}} + {{{{{{ \tilde{\Gamma}} ^a} _e} _c}} {{{{{ \tilde{\Gamma}} ^e} _b} _d}}}} - {{{{{{ \tilde{\Gamma}} ^a} _e} _d}} {{{{{ \tilde{\Gamma}} ^e} _b} _c}}}}$
${{{{{ \tilde{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} {{{\frac{1}{2}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{{ \Gamma} ^{\nu}} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\delta}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\delta}} _{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\delta}} _{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\gamma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\delta}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\gamma}}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\nu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{ F} _{\gamma}} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\gamma}} ^{\alpha}} _{,\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{{ F} _{\beta}} ^{\alpha}} _{,\delta}}} {{{{\phi_K}}^{2}}}} + {{{{{ R} ^{\alpha}} _{\beta}} _{\gamma}} _{\delta}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\delta}} ^{\alpha}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{ F} _{\gamma}} _{\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\delta}}} {{{{{ F} _{\beta}} ^{\alpha}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\nu}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{{ \Gamma} ^{\nu}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\delta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\epsilon}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{{ F} _{\beta}} _{\delta}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} & {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\epsilon}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\beta}} ^{\alpha}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} \\ {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\epsilon}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\beta}} ^{\alpha}} _{,\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\delta}} _{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\delta}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\epsilon}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\epsilon}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\nu}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\delta}} ^{\alpha}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\gamma}} ^{\alpha}} _{,\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{{ \Gamma} ^{\alpha}} _{\delta}} _{\epsilon}}} {{{{\phi_K}}^{2}}}} & {{-1}} \cdot {{\frac{1}{4}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}} \\ {{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{{ \Gamma} ^{\nu}} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} {{{{ A} _{\epsilon}} _{,\delta}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ F} _{\epsilon}} _{\delta}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\delta}}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\delta}} _{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\delta}} _{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\gamma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ A} _{\epsilon}} _{,\delta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\delta}}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{{ F} _{\epsilon}} _{\delta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ A} _{\epsilon}} _{,\delta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{ F} _{\gamma}} ^{\nu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{ F} _{\gamma}} _{\beta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{{{ A} _{\mu}} _{,\delta}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\gamma}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ A} _{\mu}} _{,\delta}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{{ F} _{\gamma}} ^{\mu}} _{,\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ A} _{\mu}} _{,\delta}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{{ F} _{\beta}} ^{\mu}} _{,\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} {{{{ A} _{\mu}} _{,\gamma}}} {{{{{ \Gamma} ^{\mu}} _{\beta}} _{\delta}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} {{{ A} _{\mu}}} {{{{{{ R} ^{\mu}} _{\beta}} _{\gamma}} _{\delta}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ A} _{\mu}} _{,\gamma}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{{ F} _{\delta}} ^{\mu}} _{,\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{{ F} _{\gamma}} _{\delta}} _{,\beta}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\delta}}} {{{{ A} _{\mu}} _{,\gamma}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\delta}}} {{{ A} _{\mu}}} {{{{{ F} _{\beta}} ^{\mu}} _{,\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{ F} _{\gamma}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{{ \Gamma} ^{\nu}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{{{ A} _{\epsilon}} _{,\gamma}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{ F} _{\epsilon}} _{\gamma}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\delta}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\delta}}} {{{{ A} _{\epsilon}} _{,\gamma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{ A} _{\epsilon}} _{,\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} _{\delta}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} & {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{{ A} _{\epsilon}} _{,\gamma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ A} _{\mu}} _{,\gamma}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{{ F} _{\beta}} ^{\mu}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} \\ {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ A} _{\epsilon}} _{,\delta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{{ A} _{\mu}} _{,\delta}}} {{{{ F} _{\beta}} ^{\mu}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\beta}} ^{\nu}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\delta}} _{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{{ F} _{\beta}} ^{\mu}} _{,\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\delta}}} {{{{\phi_K}}^{2}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ A} _{\epsilon}} _{,\delta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{{ A} _{\epsilon}} _{,\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\gamma}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{{ A} _{\mu}} _{,\delta}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ A} _{\mu}} _{,\gamma}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\mu}}} {{{{ F} _{\gamma}} ^{\nu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\delta}} ^{\nu}}} {{{{ F} _{\gamma}} ^{\mu}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{{ F} _{\delta}} ^{\mu}} _{,\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{{ \Gamma} ^{\mu}} _{\delta}} _{\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\mu}}} {{{{{ F} _{\gamma}} ^{\mu}} _{,\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ F} _{\epsilon}} _{\delta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} & {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\mu}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} \\ {{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\mu}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\mu}}} {{{{\phi_K}}^{4}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$

${{{{{ \tilde{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} {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\gamma}} ^{\alpha}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{{ F} _{\beta}} ^{\alpha}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\delta}} ^{\alpha}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{ F} _{\gamma}} _{\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\delta}}} {{{{{ F} _{\beta}} ^{\alpha}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{{{ R} ^{\alpha}} _{\beta}} _{\gamma}} _{\delta}} + {{{\frac{1}{4}}} {{{{ F} _{\beta}} _{\delta}}} {{{{ F} _{\gamma}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\alpha}}} {{{{ F} _{\gamma}} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\delta}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} & {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\beta}} ^{\alpha}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} \\ {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\beta}} ^{\alpha}} _{;\delta}}} {{{{\phi_K}}^{2}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\delta}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\delta}} ^{\alpha}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\gamma}} ^{\alpha}} _{;\delta}}} {{{{\phi_K}}^{2}}}} & {{-1}} \cdot {{\frac{1}{4}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}} \\ {{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\alpha}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{{ \Gamma} ^{\zeta}} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\delta}} ^{\zeta}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{{{ F} _{\gamma}} ^{\epsilon}} _{;\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{ A} _{\gamma}}} {{{{{ F} _{\beta}} ^{\epsilon}} _{;\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} {{{ A} _{\epsilon}}} {{{{{{ R} ^{\epsilon}} _{\beta}} _{\gamma}} _{\delta}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{{{ F} _{\gamma}} _{\delta}} _{;\beta}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\delta}}} {{{ A} _{\epsilon}}} {{{{{ F} _{\beta}} ^{\epsilon}} _{;\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{{ \Gamma} ^{\zeta}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\gamma}} ^{\zeta}}} {{{{{ \Gamma} ^{\epsilon}} _{\beta}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\delta}} ^{\zeta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\zeta}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{{ F} _{\beta}} _{\delta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\beta}} ^{\zeta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\zeta}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} _{\beta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\gamma}}} {{{ A} _{\zeta}}} {{{{ F} _{\beta}} ^{\zeta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{{ F} _{\epsilon}} _{\delta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\delta}}} {{{ A} _{\epsilon}}} {{{{ F} _{\gamma}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\gamma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\delta}}} {{{{\phi_K}}^{2}}} {{\frac{1}{{ A} _5}}}} + {{{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\gamma}}} {{{ A} _{\zeta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\delta}} ^{\zeta}}} {{{{\phi_K}}^{4}}} {{\frac{1}{{ A} _5}}}} & {{{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{{ F} _{\gamma}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\zeta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\beta}} ^{\zeta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{{{ F} _{\beta}} ^{\epsilon}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} \\ {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\beta}}} {{{ A} _{\epsilon}}} {{{{ F} _{\delta}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\delta}} ^{\zeta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\beta}} ^{\zeta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{{{ F} _{\beta}} ^{\epsilon}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\delta}}} {{{{\phi_K}}^{2}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\epsilon}}} {{{{ F} _{\gamma}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\epsilon}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\gamma}} ^{\zeta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\delta}} ^{\zeta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\epsilon}}} {{{{{ F} _{\gamma}} ^{\epsilon}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ F} _{\epsilon}} _{\delta}}} {{{{ F} _{\gamma}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}} & {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\epsilon}}} {{{{ F} _{\gamma}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} \\ {{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\epsilon}}} {{{{ F} _{\delta}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} & 0\end{matrix} \right]}\end{matrix} \right]}}$

Ricci tensor:
${{{ \tilde{R}} _a} _b} = {{{{{ \tilde{R}} ^c} _a} _c} _b}$

${{{ \tilde{R}} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {\frac{1}{4}}{\left({{{{{2}} {{{ A} _{\epsilon}}} {{{{{ F} _{\alpha}} ^{\epsilon}} _{;\beta}}} {{{{\phi_K}}^{2}}}} - {{{2}} {{{ A} _{\sigma}}} {{{{{ F} _{\alpha}} ^{\sigma}} _{;\beta}}} {{{{\phi_K}}^{2}}}}} + {{{2}} {{{ A} _{\alpha}}} {{{{{ F} _{\beta}} ^{\sigma}} _{;\sigma}}} {{{{\phi_K}}^{2}}}} + {{{2}} {{{{ F} _{\alpha}} ^{\sigma}}} {{{{ F} _{\sigma}} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{2}} {{{ A} _{\beta}}} {{{{{ F} _{\alpha}} ^{\sigma}} _{;\sigma}}} {{{{\phi_K}}^{2}}}} + {{{4}} {{{{ R} _{\alpha}} _{\beta}}}} + {{{{ A} _{\alpha}}} {{{ A} _{\sigma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\sigma}}} {{{{\phi_K}}^{4}}}} + {{{{{{{ F} _{\beta}} ^{\sigma}}} {{{{ F} _{\sigma}} _{\alpha}}} {{{{\phi_K}}^{2}}}} - {{{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\epsilon}} ^{\sigma}}} {{{{ F} _{\sigma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}}} - {{{{ A} _{\alpha}}} {{{ A} _{\epsilon}}} {{{{ F} _{\beta}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}}} + {{{{{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\alpha}} ^{\epsilon}}} {{{{ F} _{\beta}} ^{\zeta}}} {{{{\phi_K}}^{4}}}} - {{{{ A} _{\epsilon}}} {{{ A} _{\zeta}}} {{{{ F} _{\alpha}} ^{\zeta}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{\phi_K}}^{4}}}}} - {{{{{ F} _{\alpha}} ^{\epsilon}}} {{{{ F} _{\epsilon}} _{\beta}}} {{{{\phi_K}}^{2}}}}}}\right)} & {\frac{1}{4}} {{{{ A} _5}} {{{{\phi_K}}^{2}}} {{\left({{-{{{{ A} _{\alpha}}} {{{{ F} _{\epsilon}} ^{\sigma}}} {{{{ F} _{\sigma}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}}} + {{{2}} {{{{{ F} _{\alpha}} ^{\sigma}} _{;\sigma}}}}}\right)}}} \\ {\frac{1}{4}} {{{{ A} _5}} {{{{\phi_K}}^{2}}} {{\left({{{-{{{{ A} _{\beta}}} {{{{ F} _{\epsilon}} ^{\sigma}}} {{{{ F} _{\sigma}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}}} - {{{{ A} _{\epsilon}}} {{{{ F} _{\beta}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{\phi_K}}^{2}}}}} + {{{{ A} _{\sigma}}} {{{{ F} _{\beta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\sigma}}} {{{{\phi_K}}^{2}}}} + {{{2}} {{{{{ F} _{\beta}} ^{\sigma}} _{;\sigma}}}}}\right)}}} & -{{\frac{1}{4}} {{{{{ F} _{\epsilon}} ^{\sigma}}} {{{{ F} _{\sigma}} ^{\epsilon}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}}}\end{matrix} \right]}}$
${{{ \tilde{R}} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{{\phi_K}}^{2}}} {{ {-{{{ F} _{\beta}} _{\gamma}}} {{{{ F} _{\alpha}} ^{\gamma}}}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{ R} _{\alpha}} _{\beta}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{\phi_K}}^{4}}} {{{{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\alpha}} ^{\gamma}}} {{{{ F} _{\beta}} ^{\delta}}}}}} + {{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\alpha}} ^{\gamma}}} {{{{ F} _{\beta}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{{\phi_K}}^{2}}} {{ {-{{{ F} _{\alpha}} ^{\gamma}}} {{{{ F} _{\beta}} _{\gamma}}}}}} & {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} \\ {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} & {{-1}} \cdot {{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}\end{matrix} \right]}}$
${{{ \tilde{R}} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ F} _{\alpha}} ^{\gamma}}} {{{{ F} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{ R} _{\alpha}} _{\beta}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} & {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} \\ {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} & {{-1}} \cdot {{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}\end{matrix} \right]}}$

Gaussian curvature:
${\tilde{R}} = {{{{\frac{1}{4}}} {{{ A} ^{\alpha}}} {{{ A} _{\alpha}}} {{{{ F} _{\beta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\beta}}} {{{{\phi_K}}^{4}}}} + {{{{ A} _{\alpha}}} {{{{ g} ^{\alpha}} ^{\beta}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ F} _{\alpha}} ^{\beta}}} {{{{ F} _{\gamma}} _{\beta}}} {{{{ g} ^{\alpha}} ^{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} {{{ A} ^{\alpha}}} {{{{{ F} _{\alpha}} ^{\beta}} _{;\beta}}} {{{{\phi_K}}^{2}}}} + {R} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{ g} ^{\alpha}} ^{\beta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ F} _{\alpha}} ^{\beta}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{\phi_K}}^{2}}}}}$
${\tilde{R}} = {{\frac{1}{4}}{\left({{-{{{2}} {{{{ F} _{\alpha}} ^{\beta}}} {{{{ F} _{\gamma}} _{\beta}}} {{{{ g} ^{\alpha}} ^{\gamma}}} {{{{\phi_K}}^{2}}}}} + {{{{4}} {{{ A} _{\alpha}}} {{{{ g} ^{\alpha}} ^{\beta}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} - {{{4}} {{{ A} ^{\alpha}}} {{{{{ F} _{\alpha}} ^{\beta}} _{;\beta}}} {{{{\phi_K}}^{2}}}}} + {{{{{4}} {{R}}} - {{{{{ F} _{\alpha}} ^{\beta}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{\phi_K}}^{2}}}}} - {{{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{ g} ^{\alpha}} ^{\beta}}} {{{{\phi_K}}^{4}}}}} + {{{{ A} ^{\alpha}}} {{{ A} _{\alpha}}} {{{{ F} _{\beta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\beta}}} {{{{\phi_K}}^{4}}}}}\right)}}$
${\tilde{R}} = {{\frac{1}{4}}{\left({{{{{{ F} _{\alpha}} ^{\beta}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{{4}} {{{ A} _{\alpha}}} {{{{ g} ^{\alpha}} ^{\beta}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} - {{{4}} {{{ A} ^{\alpha}}} {{{{{ F} _{\alpha}} ^{\beta}} _{;\beta}}} {{{{\phi_K}}^{2}}}}} + {{{{4}} {{R}}} - {{{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{ g} ^{\alpha}} ^{\beta}}} {{{{\phi_K}}^{4}}}}} + {{{{ A} _{\alpha}}} {{{ A} ^{\alpha}}} {{{{ F} _{\beta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\beta}}} {{{{\phi_K}}^{4}}}}}\right)}}$
${\tilde{R}} = {{\frac{1}{4}}{\left({{{{{4}} {{{ A} _{\alpha}}} {{{{ g} ^{\alpha}} ^{\beta}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} - {{{4}} {{{ A} ^{\alpha}}} {{{{{ F} _{\alpha}} ^{\beta}} _{;\beta}}} {{{{\phi_K}}^{2}}}}} + {{{4}} {{R}}} + {{{{{ A} _{\alpha}}} {{{ A} ^{\alpha}}} {{{{ F} _{\beta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\beta}}} {{{{\phi_K}}^{4}}}} - {{{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{ g} ^{\alpha}} ^{\beta}}} {{{{\phi_K}}^{4}}}}} + {{{{{ F} _{\alpha}} ^{\beta}}} {{{{ F} _{\beta}} ^{\alpha}}} {{{{\phi_K}}^{2}}}}}\right)}}$
${\tilde{R}} = {{{{{ A} _{\mu}}} {{{{ g} ^{\mu}} ^{\nu}}} {{{{{ F} _{\nu}} ^{\rho}} _{;\rho}}} {{{{\phi_K}}^{2}}}} + {{{-1}} {{{ A} ^{\mu}}} {{{{{ F} _{\mu}} ^{\nu}} _{;\nu}}} {{{{\phi_K}}^{2}}}} + {R} + {{{\frac{1}{4}}} {{{ A} _{\mu}}} {{{ A} ^{\mu}}} {{{{ F} _{\nu}} ^{\rho}}} {{{{ F} _{\rho}} ^{\nu}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\mu}}} {{{ A} _{\nu}}} {{{{ F} _{\rho}} ^{\sigma}}} {{{{ F} _{\sigma}} ^{\rho}}} {{{{ g} ^{\mu}} ^{\nu}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{4}}} {{{{ F} _{\mu}} ^{\nu}}} {{{{ F} _{\nu}} ^{\mu}}} {{{{\phi_K}}^{2}}}}}$

Einstein curvature:
${{{ \tilde{G}} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ F} _{\alpha}} ^{\gamma}}} {{{{ F} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-3}} \cdot {{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} ^{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{{ g} _{\alpha}} _{\beta}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{ R} _{\alpha}} _{\beta}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{\phi_K}}^{2}}}} & {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-3}} \cdot {{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} \\ {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-3}} \cdot {{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} & {{{-3}} \cdot {{\frac{1}{8}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}} + {{{\frac{1}{2}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{{\phi_K}}^{2}}} {{{{ A} _5}^{2}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}} {{{{ A} _5}^{2}}}} + {{{\frac{1}{8}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}} {{{{ A} _5}^{2}}}}\end{matrix} \right]}}$

---

stress-energy tensor:
$\tilde{G}_{ab} = \frac{8 \pi G}{c^4} T_{ab}$
${\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ F} _{\alpha}} ^{\gamma}}} {{{{ F} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} ^{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{{ g} _{\alpha}} _{\beta}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{ R} _{\alpha}} _{\beta}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{8}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-3}} \cdot {{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} & {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-3}} \cdot {{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} \\ {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-3}} \cdot {{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} & {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}} + {{{\frac{1}{2}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{{\phi_K}}^{2}}} {{{{ A} _5}^{2}}}} + {{{\frac{1}{8}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}} {{{{ A} _5}^{2}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}} {{{{ A} _5}^{2}}}} + {{{-3}} \cdot {{\frac{1}{8}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}}\end{matrix} \right]}} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}} & {{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _5}} {{\frac{1}{{c}^{4}}}} \\ {{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _5} _{\beta}}} {{\frac{1}{{c}^{4}}}} & {{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _5} _5}} {{\frac{1}{{c}^{4}}}}\end{matrix} \right]}}$


Looking at the $\tilde{G}_{55}$ components:
${{{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}} + {{{\frac{1}{2}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{{\phi_K}}^{2}}} {{{{ A} _5}^{2}}}} + {{{\frac{1}{8}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}} {{{{ A} _5}^{2}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}} {{{{ A} _5}^{2}}}} + {{{-3}} \cdot {{\frac{1}{8}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}} {{{{ A} _5}^{2}}}}} = {{{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _5} _5}} {{\frac{1}{{c}^{4}}}}}$
Isolating the $\tilde{T}_{55}$ stress-energy term:
${{{ \tilde{T}} _5} _5} = {{{{-1}} \cdot {{\frac{1}{16}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}} + {{{\frac{1}{16}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}} + {{{-1}} \cdot {{\frac{1}{16}}} {{R}} {{{{\phi_K}}^{2}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}} + {{{-3}} \cdot {{\frac{1}{64}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}} + {{{-1}} \cdot {{\frac{1}{64}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}} + {{{\frac{1}{64}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}}}$
Isolating the spacetime scalar curvature R:
${R} = {{{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-3}} \cdot {{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{2}}}} + {{{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-16}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _5} _5}} {{\frac{1}{{{\phi_K}}^{2}}}} {{\frac{1}{{c}^{4}}}} {{\frac{1}{{{ A} _5}^{2}}}}}}$
Look at it with our substituted values:
Substitute ${{ A} _5} = {{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}$, ${{\phi_K}} = {{{\frac{2}{c}}} {{\sqrt{{\frac{1}{{k_e}}} {G}}}}}$, ${{k_e}} = {\frac{{{{\mu_0}}} \cdot {{{c}^{2}}}}{{{4}} {{\pi}}}}$
${R} = {{{{16}} {{G}} {{\pi}} \cdot {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-16}} {{G}} {{\pi}} \cdot {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-12}} {{G}} {{\pi}} \cdot {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-4}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _5} _5}} {{\frac{1}{{c}^{4}}}}} + {{{-64}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{64}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}}}$
It looks like $\tilde{T}_{55}$ provides the scalar curvature information ... with the exception of that extra term
What is the magnitude of that extra term?


Looking at the $\tilde{G}_{5\mu}$ components:
${{{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-3}} \cdot {{\frac{1}{8}}} {{{ A} _5}} {{{ A} _{\alpha}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _5}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}}} = {{{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _5}} {{\frac{1}{{c}^{4}}}}}$
Isolating the Faraday tensor divergence:
${{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}} = {{{{3}} \cdot {{\frac{1}{4}}} {{{ A} _{\alpha}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{2}}}} + {{{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} _{\alpha}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} {{{ A} _{\alpha}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{4}}} {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{R}} {{{ A} _{\alpha}}}} + {{{16}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _5}} {{\frac{1}{{ A} _5}}} {{\frac{1}{{{\phi_K}}^{2}}}} {{\frac{1}{{c}^{4}}}}}}$
Substitute ${R} = {{{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-3}} \cdot {{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{2}}}} + {{{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-16}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _5} _5}} {{\frac{1}{{{\phi_K}}^{2}}}} {{\frac{1}{{c}^{4}}}} {{\frac{1}{{{ A} _5}^{2}}}}}}$
${{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}} = {{{{16}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _5}} {{\frac{1}{{ A} _5}}} {{\frac{1}{{{\phi_K}}^{2}}}} {{\frac{1}{{c}^{4}}}}} + {{{-16}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{{ \tilde{T}} _5} _5}} {{\frac{1}{{{\phi_K}}^{2}}}} {{\frac{1}{{c}^{4}}}} {{\frac{1}{{{ A} _5}^{2}}}}}}$
And that conveniently cancelled a term. Now let's substitute the definition of $\tilde{T}_{55}$

Now to take a detour and write the stress-energy in terms of the four-current to see the Gauss-Ampere laws emerge:
Bring back the scalar curvature term R from $\tilde{T}_{55}$ and rewrite $\tilde{T}_{5\alpha}$ in terms of five-momentum:
Substitute ${{{ \tilde{T}} _5} _5} = {{{{-1}} \cdot {{\frac{1}{16}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}} + {{{\frac{1}{16}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}} + {{{-1}} \cdot {{\frac{1}{16}}} {{R}} {{{{\phi_K}}^{2}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}} + {{{-3}} \cdot {{\frac{1}{64}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}} + {{{-1}} \cdot {{\frac{1}{64}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}} + {{{\frac{1}{64}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}} {{{c}^{4}}} {{{{ A} _5}^{2}}} {{\frac{1}{G}}} {{\frac{1}{\pi}}}}}$, ${{{ \tilde{T}} _{\alpha}} _5} = {{{{c}^{2}}} {{\rho}} \cdot {{{ u} _5}} {{{ u} _{\alpha}}}}$, ${{ u} _5} = {{{{{{\phi_K}}^{2}}} {{{ A} _{\epsilon}}} {{{ A} _5}} {{{ u} ^{\epsilon}}}} + {{{{{\phi_K}}^{2}}} {{{{ A} _5}^{2}}} {{{ u} ^5}}}}$, ${{ u} ^5} = {{{\frac{1}{4}}} {{{\frac{1}{M}} {q}}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}$, ${{ A} _5} = {{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}$, ${{\phi_K}} = {{{\frac{2}{c}}} {{\sqrt{{\frac{1}{{k_e}}} {G}}}}}$, ${{k_e}} = {\frac{{{{\mu_0}}} \cdot {{{c}^{2}}}}{{{4}} {{\pi}}}}$
${{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}} = {{{{16}} {{G}} {{\pi}} \cdot {{\rho}} \cdot {{{ A} _{\epsilon}}} {{{ u} _{\alpha}}} {{{ u} ^{\epsilon}}} {{\frac{1}{{c}^{2}}}}} + {{{{\mu_0}}} \cdot {{\rho}} \cdot {{c}} {{q}} {{{ u} _{\alpha}}} {{\frac{1}{M}}}} + {{{16}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-16}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{R}} {{{ A} _{\alpha}}}} + {{{12}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-64}} {{{ A} _{\alpha}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{64}} {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}}}$
Define our four-current as ${{ J} _{\alpha}} = {{{c}} {{{\frac{1}{M}} {q}}} {{\rho}} \cdot {{{ u} _{\alpha}}}}$
${{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}} = {{{{-16}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{16}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{12}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{16}} {{G}} {{\pi}} \cdot {{\rho}} \cdot {{{ A} _{\epsilon}}} {{{ u} _{\alpha}}} {{{ u} ^{\epsilon}}} {{\frac{1}{{c}^{2}}}}} + {{{R}} {{{ A} _{\alpha}}}} + {{{-64}} {{{ A} _{\alpha}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{64}} {{{ A} _{\alpha}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{{\mu_0}}} \cdot {{{ J} _{\alpha}}}}}$
Move all but current to the left side:
${{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}} + {{{16}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-16}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-12}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-16}} {{G}} {{\pi}} \cdot {{\rho}} \cdot {{{ A} _{\epsilon}}} {{{ u} _{\alpha}}} {{{ u} ^{\epsilon}}} {{\frac{1}{{c}^{2}}}}} + {{{-1}} {{R}} {{{ A} _{\alpha}}}} + {{{-64}} {{{ A} _{\alpha}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{64}} {{{ A} _{\alpha}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}}} = {{{{\mu_0}}} \cdot {{{ J} _{\alpha}}}}$
Rewriting the right hand side as an operator
$ ( 12 \pi G \frac{1}{c^4 \mu_0} F^{\mu\nu} A_\alpha + \delta^\mu_\alpha \delta^\nu_\beta \nabla^\beta ) F_{\mu\nu} - ( 16 \frac{1}{c^2} \pi G \rho u_\alpha u^\beta + R \delta^\beta_\alpha ) A_\beta = \mu_0 J_\alpha $
In matter at macroscopic levels this becomes...
$\mu_0 \nabla^\beta ( {Z_{\alpha\beta}}^{\mu\nu} F_{\mu\nu} ) = \mu_0 J_\alpha$
...for some sort of operator $\nabla (Z \cdot ...)$...


Looking at the $\tilde{G}_{\mu\nu}$ components:
$ \tilde{G}_{\alpha\beta} = 8 \pi \frac{G}{c^4} \tilde{T}_{\alpha\beta}$
${{{{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ F} _{\alpha}} ^{\gamma}}} {{{{ F} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} ^{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{{ g} _{\alpha}} _{\beta}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{R}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{ R} _{\alpha}} _{\beta}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{8}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-3}} \cdot {{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}}} = {{{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}}}$
Isolating the spacetime Einstein tensor.
${{{ G} _{\alpha}} _{\beta}} = {{{{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{{ F} _{\alpha}} ^{\gamma}}} {{{{ F} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} ^{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} ^{\gamma}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{2}}} {{R}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{3}} \cdot {{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{6}}}} + {{{\frac{1}{8}}} {{{ A} _{\gamma}}} {{{ A} ^{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{\phi_K}}^{2}}}}}$
Substitute ${R} = {{{{\frac{1}{4}}} {{{ A} _{\delta}}} {{{ A} _{\rho}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} ^{\rho}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{ A} ^{\rho}}} {{{ A} _{\rho}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{\phi_K}}^{4}}}} + {{{-3}} \cdot {{\frac{1}{4}}} {{{{ F} _{\delta}} ^{\rho}}} {{{{ F} _{\rho}} ^{\delta}}} {{{{\phi_K}}^{2}}}} + {{{{ A} ^{\rho}}} {{{{{ F} _{\rho}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} {{{ A} _{\rho}}} {{{{ g} ^{\rho}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-16}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _5} _5}} {{\frac{1}{{{\phi_K}}^{2}}}} {{\frac{1}{{c}^{4}}}} {{\frac{1}{{{ A} _5}^{2}}}}}}$ , ${{ A} _5} = {{{c}} {{\sqrt{{\frac{1}{G}} {{k_e}}}}}}$
${{{ G} _{\alpha}} _{\beta}} = {{{{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\alpha}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{{ F} _{\alpha}} ^{\gamma}}} {{{{ F} _{\beta}} _{\gamma}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} _{\beta}}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{2}}} {{{ A} _{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{ A} ^{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{{{\phi_K}}^{2}}}} + {{{\frac{1}{8}}} {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{\phi_K}}^{2}}}} + {{{-1}} \cdot {{\frac{1}{8}}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\delta}} ^{\gamma}}} {{{{\phi_K}}^{4}}}} + {{{\frac{1}{8}}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{\phi_K}}^{4}}}} + {{{-8}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ \tilde{T}} _5} _5}} {{{G}^{2}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{6}}}}}}$
Notice that substituting R conveniently cancelled another of the terms on the r.h.s,
Substitute ${{\phi_K}} = {{{\frac{2}{c}}} {{\sqrt{{\frac{1}{{k_e}}} {G}}}}}$, ${{k_e}} = {\frac{{{{\mu_0}}} \cdot {{{c}^{2}}}}{{{4}} {{\pi}}}}$
${{{ G} _{\alpha}} _{\beta}} = {{{{-8}} {{G}} {{\pi}} \cdot {{{ A} ^{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-8}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-8}} {{G}} {{\pi}} \cdot {{{ A} _{\beta}}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{ A} _{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{{ F} _{\alpha}} ^{\gamma}}} {{{{ F} _{\beta}} _{\gamma}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{2}} {{G}} {{\pi}} \cdot {{{{ F} _{\delta}} ^{\gamma}}} {{{{ F} _{\gamma}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}}} + {{{32}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{-32}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\delta}} ^{\gamma}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{-32}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ \tilde{T}} _5} _5}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{8}}}}}}$
Let ${{{ {T_{EM}}} _{\alpha}} _{\beta}} = { {-{\frac{1}{{\mu_0}}}} {{\left({{{{{{ F} _{\alpha}} ^{\mu}}} {{{{ F} _{\mu}} _{\beta}}}} - {{{\frac{1}{4}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ F} _{\mu}} ^{\nu}}} {{{{ F} _{\nu}} ^{\mu}}}}}\right)}}}$
So ${{{{{ F} _{\alpha}} ^{\mu}}} {{{{ F} _{\mu}} _{\beta}}}} = {{\frac{1}{4}}{\left({{-{{{4}} {{{\mu_0}}} \cdot {{{{ {T_{EM}}} _{\alpha}} _{\beta}}}}} + {{{{{ F} _{\mu}} ^{\nu}}} {{{{ F} _{\nu}} ^{\mu}}} {{{{ g} _{\alpha}} _{\beta}}}}}\right)}}$
${{{ G} _{\alpha}} _{\beta}} = {{{{-8}} {{G}} {{\pi}} \cdot {{{ A} ^{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-8}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{{{ F} _{\beta}} ^{\gamma}} _{;\gamma}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-8}} {{G}} {{\pi}} \cdot {{{ A} _{\beta}}} {{{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{ A} _{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{{ {T_{EM}}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}}} + {{{32}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{-32}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{-32}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ \tilde{T}} _5} _5}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{8}}}}}}$
Substitute ${{{{ F} _{\alpha}} ^{\gamma}} _{;\gamma}} = {{{{16}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _5}} {{\frac{1}{{ A} _5}}} {{\frac{1}{{{\phi_K}}^{2}}}} {{\frac{1}{{c}^{4}}}}} + {{{-16}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{{ \tilde{T}} _5} _5}} {{\frac{1}{{{\phi_K}}^{2}}}} {{\frac{1}{{c}^{4}}}} {{\frac{1}{{{ A} _5}^{2}}}}}}$
${{{ G} _{\alpha}} _{\beta}} = {{{{-8}} {{G}} {{\pi}} \cdot {{{ A} ^{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{ A} _{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{{ {T_{EM}}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}}} + {{{32}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{-16}} {{{ A} _{\alpha}}} {{{{ \tilde{T}} _{\beta}} _5}} {{{G}^{{{3}} \cdot {{\frac{1}{2}}}}}} {{{\pi}^{{{3}} \cdot {{\frac{1}{2}}}}}} {{\frac{1}{{{\mu_0}}^{\frac{1}{2}}}}} {{\frac{1}{{c}^{6}}}}} + {{{-16}} {{{ A} _{\beta}}} {{{{ \tilde{T}} _{\alpha}} _5}} {{{G}^{{{3}} \cdot {{\frac{1}{2}}}}}} {{{\pi}^{{{3}} \cdot {{\frac{1}{2}}}}}} {{\frac{1}{{{\mu_0}}^{\frac{1}{2}}}}} {{\frac{1}{{c}^{6}}}}} + {{{-32}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{32}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ \tilde{T}} _5} _5}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{8}}}}}}$
Alternatively, substitute our specific stress-energy and four-current definitions:
${{{ G} _{\alpha}} _{\beta}} = {{{{-8}} {{G}} {{\pi}} \cdot {{{ A} ^{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{{ F} _{\gamma}} ^{\delta}} _{;\delta}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{ A} _{\gamma}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\gamma}} ^{\delta}}} {{{{{ F} _{\delta}} ^{\epsilon}} _{;\epsilon}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{4}}}}} + {{{-8}} {{G}} {{\pi}} \cdot {{{ A} _{\alpha}}} {{{ J} _{\beta}}} {{\frac{1}{{c}^{4}}}}} + {{{-8}} {{G}} {{\pi}} \cdot {{{ A} _{\beta}}} {{{ J} _{\alpha}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{{ {T_{EM}}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}}} + {{{8}} {{G}} {{\pi}} \cdot {{{{ \tilde{T}} _{\alpha}} _{\beta}}} {{\frac{1}{{c}^{4}}}}} + {{{32}} {{{ A} ^{\gamma}}} {{{ A} _{\gamma}}} {{{{ F} _{\delta}} ^{\epsilon}}} {{{{ F} _{\epsilon}} ^{\delta}}} {{{{ g} _{\alpha}} _{\beta}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{-32}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{{ F} _{\epsilon}} ^{\zeta}}} {{{{ F} _{\zeta}} ^{\epsilon}}} {{{{ g} _{\alpha}} _{\beta}}} {{{{ g} ^{\delta}} ^{\gamma}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{{\mu_0}}^{2}}}} {{\frac{1}{{c}^{8}}}}} + {{{32}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{{{ \tilde{T}} _5} _5}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{8}}}}} + {{{-128}} {{\rho}} \cdot {{{ A} _{\alpha}}} {{{ A} _{\gamma}}} {{{ u} _{\beta}}} {{{ u} ^{\gamma}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{6}}}}} + {{{-128}} {{\rho}} \cdot {{{ A} _{\beta}}} {{{ A} _{\gamma}}} {{{ u} _{\alpha}}} {{{ u} ^{\gamma}}} {{{G}^{2}}} {{{\pi}^{2}}} {{\frac{1}{{\mu_0}}}} {{\frac{1}{{c}^{6}}}}}}$


using a specific stress-energy tensor:
$\tilde{T}_{ab} = c^2 \rho u_a u_b + P (\tilde{g}_{ab} + u_a u_b)$

${{{ \tilde{T}} _a} _b} = {{{{{c}^{2}}} {{\rho}} \cdot {{{ u} _a}} {{{ u} _b}}} + {{{P}} {{{{ \tilde{g}} _a} _b}}}}$
${{{ \tilde{T}} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{{\left({{{{{c}^{2}}} {{\rho}}} + {P}}\right)}} {{{ u} _{\alpha}}} {{{ u} _{\beta}}}} + {{{P}} {{{{ \tilde{g}} _{\alpha}} _{\beta}}}} & {{{\left({{{{{c}^{2}}} {{\rho}}} + {P}}\right)}} {{{ u} _{\alpha}}} {{{ u} _5}}} + {{{P}} {{{{ \tilde{g}} _{\alpha}} _5}}} \\ {{{\left({{{{{c}^{2}}} {{\rho}}} + {P}}\right)}} {{{ u} _{\beta}}} {{{ u} _5}}} + {{{P}} {{{{ \tilde{g}} _{\beta}} _5}}} & {{{\left({{{{{c}^{2}}} {{\rho}}} + {P}}\right)}} {{{{ u} _5}^{2}}}} + {{{P}} {{{{ \tilde{g}} _5} _5}}}\end{matrix} \right]}}$
Substituting definitions for $\tilde{g}_{ab}, u_5, u^5, A_5, \phi_K$...
${{{ \tilde{T}} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} {{{\rho}} \cdot {{{ u} _{\alpha}}} {{{ u} _{\beta}}} {{{c}^{2}}}} + {{{P}} {{{ u} _{\alpha}}} {{{ u} _{\beta}}}} + {{{P}} {{{{ g} _{\alpha}} _{\beta}}}} + {{{4}} {{G}} {{P}} {{{ A} _{\alpha}}} {{{ A} _{\beta}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} & {{{4}} {{P}} {{{ A} _{\alpha}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{4}} {{P}} {{{ A} _{\gamma}}} {{{ u} _{\alpha}}} {{{ u} ^{\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{4}} {{\rho}} \cdot {{c}} {{{ A} _{\gamma}}} {{{ u} _{\alpha}}} {{{ u} ^{\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}}} + {{{P}} {{q}} {{{ u} _{\alpha}}} {{{{k_e}}^{\frac{1}{2}}}} {{\frac{1}{{G}^{\frac{1}{2}}}}} {{\frac{1}{M}}}} + {{{\rho}} \cdot {{q}} {{{ u} _{\alpha}}} {{{c}^{2}}} {{{{k_e}}^{\frac{1}{2}}}} {{\frac{1}{{G}^{\frac{1}{2}}}}} {{\frac{1}{M}}}} \\ {{{4}} {{P}} {{{ A} _{\beta}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{4}} {{P}} {{{ A} _{\gamma}}} {{{ u} _{\beta}}} {{{ u} ^{\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}} {{\frac{1}{c}}}} + {{{4}} {{\rho}} \cdot {{c}} {{{ A} _{\gamma}}} {{{ u} _{\beta}}} {{{ u} ^{\gamma}}} {{{G}^{\frac{1}{2}}}} {{\frac{1}{{{k_e}}^{\frac{1}{2}}}}}} + {{{P}} {{q}} {{{ u} _{\beta}}} {{{{k_e}}^{\frac{1}{2}}}} {{\frac{1}{{G}^{\frac{1}{2}}}}} {{\frac{1}{M}}}} + {{{\rho}} \cdot {{q}} {{{ u} _{\beta}}} {{{c}^{2}}} {{{{k_e}}^{\frac{1}{2}}}} {{\frac{1}{{G}^{\frac{1}{2}}}}} {{\frac{1}{M}}}} & {{{4}} {{P}}} + {{{4}} {{P}} {{q}} {{{ A} _{\delta}}} {{{ u} ^{\delta}}} {{\frac{1}{M}}} {{\frac{1}{c}}}} + {{{4}} {{\rho}} \cdot {{c}} {{q}} {{{ A} _{\delta}}} {{{ u} ^{\delta}}} {{\frac{1}{M}}}} + {{{16}} {{G}} {{P}} {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{ u} ^{\delta}}} {{{ u} ^{\gamma}}} {{\frac{1}{{k_e}}}} {{\frac{1}{{c}^{2}}}}} + {{{16}} {{G}} {{\rho}} \cdot {{{ A} _{\delta}}} {{{ A} _{\gamma}}} {{{ u} ^{\delta}}} {{{ u} ^{\gamma}}} {{\frac{1}{{k_e}}}}} + {{{4}} {{P}} {{q}} {{{ A} _{\gamma}}} {{{ u} ^{\gamma}}} {{\frac{1}{M}}} {{\frac{1}{c}}}} + {{{4}} {{\rho}} \cdot {{c}} {{q}} {{{ A} _{\gamma}}} {{{ u} ^{\gamma}}} {{\frac{1}{M}}}} + {{{P}} {{{k_e}}} \cdot {{{q}^{2}}} {{\frac{1}{G}}} {{\frac{1}{{M}^{2}}}}} + {{{\rho}} \cdot {{{k_e}}} \cdot {{{c}^{2}}} {{{q}^{2}}} {{\frac{1}{G}}} {{\frac{1}{{M}^{2}}}}}\end{matrix} \right]}}$