Kaluza-Klein with varying scalar field

NOTICE this is nowhere as developed as my symmath-based worksheet on it found here:
https://thenumbernine.github.io/math/Kaluza-Klein%20-%20varying%20scalar.html

$\alpha - \omega$ = time+space.
$ijklmn$ = space.
the rest of $a - z$ = time+space+electromagnetism.

$A_u =$ electromagnetic potential, in units of $\frac{kg \cdot m}{C \cdot s}$
$\phi =$ scalar field, in units of $\frac{C \cdot s}{kg \cdot m}$
$\tilde{g}_{ab,5} = 0$

in 4+1:
$\tilde{g}_{ab} = {}_{\downarrow a(\mu)} \overset{\rightarrow b(\nu)}{\pmatrix{ \tilde{g}_{\mu\nu} & \tilde{g}_{\mu 5} \\ \tilde{g}_{5\nu} & \tilde{g}_{55} }} = \pmatrix{ g_{\mu\nu} + \phi^2 A_\mu A_\nu & \phi^2 A_\mu \\ \phi^2 A_\nu & \phi^2 }$
$\tilde{g}^{ab} = \pmatrix{ g^{\mu\nu} & -A^\mu \\ -A^\nu & g^{\alpha\beta} A_\alpha A_\beta + \frac{1}{\phi^2} }$

in ADM 1+3+1:
$\tilde{g}_{ab} = \pmatrix{ -\alpha^2 + \beta^k \beta_k + \phi^2 (A_t)^2 & \beta_j + \phi^2 A_t A_j & \phi^2 A_t \\ \beta_i + \phi^2 A_t A_i & \gamma_{ij} + \phi^2 A_i A_j & \phi^2 A_i \\ \phi^2 A_t & \phi^2 A_j & \phi^2 }$
$\tilde{g}^{ab} = \pmatrix{ -\alpha^2 & \beta^j / \alpha^2 & -A^t \\ \beta^i / \alpha^2 & \gamma^{ij} - \beta^i \beta^j / \alpha^2 & -A^i \\ -A^t & -A^j & A^2 + \phi^2 }$

metric partial, in 4+1:
$\tilde{g}_{ab,c} = \overset{\downarrow c[\downarrow a \rightarrow b]}{\pmatrix{ g_{\mu\nu,c} + 2 \phi \phi_{,c} A_\mu A_\nu + \phi^2 (A_{\mu,c} A_\nu + A_\mu A_{\nu,c}) & 2 \phi \phi_{,c} A_\mu + \phi^2 A_{\mu,c} \\ 2 \phi \phi_{,c} A_\nu + \phi^2 A_{\nu,c} & 0 }}$
$= {}_{\downarrow c(\rho)} \pmatrix{ {}_{\downarrow a(\mu)} \overset{\rightarrow b(\nu)}{ \pmatrix{ g_{\mu\nu,\rho} + 2 \phi \phi_{,\rho} A_\mu A_\nu + \phi^2 (A_{\mu,\rho} A_\nu + A_\mu A_{\nu,\rho}) & 2 \phi \phi_{,\rho} A_\mu + \phi^2 A_{\mu,\rho} \\ 2 \phi \phi_{,\rho} A_\nu + \phi^2 A_{\nu,\rho} & 0 }} \\ \pmatrix{ g_{\mu\nu,5} + 2 \phi \phi_{,5} A_\mu A_\nu + \phi^2 (A_{\mu,5} A_\nu + A_\mu A_{\nu,5}) & 2 \phi \phi_{,5} A_\mu + \phi^2 A_{\mu,5} \\ 2 \phi \phi_{,5} A_\nu + \phi^2 A_{\nu,5} & 0 } }$
$\tilde\Gamma_{abc} = \frac{1}{2} (g_{ab,c} + g_{ac,b} - g_{bc,a})$
$= \overset{\downarrow c[\downarrow a \rightarrow b]}{\pmatrix{ \pmatrix{ \Gamma_{uvw} + \phi^2 ( A_\mu A_{(\nu,\rho)} + A_\nu A_{[\mu,\rho]} + A_\rho A_{[\mu,\nu]} ) & \frac{1}{2} g_{\mu\rho,5} + \phi^2 A_{[\mu,\rho]} \\ -\frac{1}{2} g_{\nu\rho,5} + \phi^2 A_{(\nu,\rho)} & 0 } \\ \pmatrix{ \frac{1}{2} g_{\mu\nu,5} + \phi^2 A_{[\mu,\nu]} & \phi^2 A_{\mu,5} \\ 0 & 0 } }}$
${\tilde\Gamma^a}_{bc} = g^{ad} \cdot \tilde\Gamma_{dbc}$
$=\overset{\downarrow c [ \downarrow a \rightarrow b]}{\pmatrix{ \pmatrix{ {\Gamma^\mu}_{\nu\rho} + \frac{1}{2} g^{\mu\alpha} \phi^2 (A_\nu F_{\rho\alpha} + A_\rho F_{\nu\alpha}) + \frac{1}{2} A^\mu g_{\nu\rho,5} & \frac{1}{2} g^{\mu\alpha} (g_{\alpha\rho,5} + \phi^2 F_{\rho\alpha}) \\ -A^\alpha (\Gamma_{\alpha\nu\rho} + \frac{1}{2} \phi^2 (A_\nu F_{\rho\alpha} + A_\rho F_{\nu\alpha})) - \frac{1}{2} A^2 g_{\nu\rho,5} - \frac{1}{2\phi^2} g_{\nu\rho,5} + A_{(\nu,\rho)} & -\frac{1}{2} A^\alpha (g_{\alpha\rho,5} + \phi^2 F_{\rho\alpha}) } \\ \pmatrix{ \frac{1}{2} g^{\mu\alpha} (g_{\alpha\nu,5} + \phi^2 F_{\nu\alpha}) & \phi^2 g^{\mu\alpha} A_{\alpha,5} \\ -\frac{1}{2} A^\alpha (g_{\alpha\nu,5} + \phi^2 F_{\nu\alpha}) & -\phi^2 A^\alpha A_{\alpha,5} } }}$

Geodesic equation:
$\dot{\tilde{u}}^u + {\tilde{\Gamma}^u}_{ab} \tilde{u}^a \tilde{u}^b = 0$

spacetime:
$\dot{\tilde{u}}^\mu + {\tilde{\Gamma}^\mu}_{\alpha\beta} \tilde{u}^\alpha \tilde{u}^\beta + 2 {\tilde{\Gamma}^\mu}_{\alpha 5} \tilde{u}^\alpha \tilde{u}^5 + {\tilde{\Gamma}^\mu}_{55} (\tilde{u}^5)^2 = 0$
$\dot{\tilde{u}}^\mu + {\Gamma^\mu}_{\alpha\beta} \tilde{u}^\alpha \tilde{u}^\beta + (\phi^2 A_{(\alpha} {F_{\beta)}}^\mu + \frac{1}{2} A^\mu g_{\alpha\beta,5}) \tilde{u}^\alpha \tilde{u}^\beta + (g^{\mu\nu} g_{\nu\alpha,5} + \phi^2 {F_\alpha}^\mu) \tilde{u}^\alpha \tilde{u}^5 + \phi^2 g^{\mu\nu} A_{\nu,5} (\tilde{u}^5)^2 = 0$
if everything is constant in the 5th dimension:
$\dot{\tilde{u}}^\mu + {\Gamma^\mu}_{\alpha\beta} \tilde{u}^\alpha \tilde{u}^\beta + \phi^2 A_{(\alpha} {F_{\beta)}}^\mu \tilde{u}^\alpha \tilde{u}^\beta + \phi^2 {F_\alpha}^\mu \tilde{u}^\alpha \tilde{u}^5 = 0$
if $\tilde{u}^5 = \frac{q}{m}$
$\dot{\tilde{u}}^\mu + {\Gamma^\mu}_{\alpha\beta} \tilde{u}^\alpha \tilde{u}^\beta + \phi^2 A_{(\alpha} {F_{\beta)}}^\mu \tilde{u}^\alpha \tilde{u}^\beta + \phi^2 \frac{q}{m} {F_\alpha}^\mu \tilde{u}^\alpha = 0$

fifth term:
$\dot{\tilde{u}}^5 + {\tilde{\Gamma}^5}_{\alpha\beta} \tilde{u}^\alpha \tilde{u}^\beta + 2 {\tilde{\Gamma}^5}_{\alpha 5} \tilde{u}^\alpha \tilde{u}^5 + {\tilde{\Gamma}^5}_{55} (\tilde{u}^5)^2 = 0$
$\dot{\tilde{u}}^5 + (-A^\mu \Gamma_{\mu\alpha\beta} - \frac{1}{2} \phi^2 A_\mu A_{(\alpha} {F_{\beta)}}^\mu - \frac{1}{2} A^2 g_{\alpha\beta,5} - \frac{1}{2\phi^2} g_{\alpha\beta,5} + A_{(\alpha,\beta)} ) \tilde{u}^\alpha \tilde{u}^\beta - A^\mu (g_{\mu\alpha,5} + \phi^2 F_{\alpha\mu}) \tilde{u}^\alpha \tilde{u}^5 - \phi^2 A^\mu A_{\mu,5} (\tilde{u}^5)^2 = 0$