Kaluza-Klein
with constant scalar field
$\alpha - \omega$ = 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 $\frac{C \cdot s}{kg \cdot m}$
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:
$\tilde{g}_{ab,c} = \overset{\downarrow c(\rho)[\downarrow a(\mu) \rightarrow b(\nu)]}{\pmatrix{
\pmatrix{
\tilde{g}_{\mu\nu,\rho} &
\tilde{g}_{\mu 5, \rho} \\
\tilde{g}_{5 \nu,\rho} &
\tilde{g}_{5 5, \rho}
} \\
\pmatrix{
\tilde{g}_{\mu\nu, 5} &
\tilde{g}_{\mu 5, 5} \\
\tilde{g}_{5 \nu, 5} &
\tilde{g}_{5 5, 5}
}
}}$
$= \overset{\downarrow c[\downarrow a \rightarrow b]}{\pmatrix{
g_{\mu\nu,c} + 2 \phi \phi_{,c} A_\mu A_\nu + \phi^2 (A_{\mu,c} A_\nu + A_\mu A_{\nu,c}) &
2 \phi \phi_{,c} A_\mu + \phi^2 A_{\mu,c} \\
2 \phi \phi_{,c} A_\nu + \phi^2 A_{\nu,c} & 2 \phi \phi_{,c}
}}$
$= {}_{\downarrow c(\rho)}
\pmatrix{
{}_{\downarrow a(\mu)}
\overset{\rightarrow b(\nu)}{
\pmatrix{
g_{\mu\nu,\rho}
+ 2 \phi \phi_{,\rho} A_\mu A_\nu
+ \phi^2 (A_{\mu,\rho} A_\nu + A_\mu A_{\nu,\rho}) &
2 \phi \phi_{,\rho} A_\mu + \phi^2 A_{\mu,\rho} \\
2 \phi \phi_{,\rho} A_\nu + \phi^2 A_{\nu,\rho} &
2 \phi \phi_{,\rho}
}} \\
\pmatrix{
g_{\mu\nu,5} + 2 \phi \phi_{,5} A_\mu A_\nu + \phi^2 (A_{\mu,5} A_\nu + A_\mu A_{\nu,5}) &
2 \phi \phi_{,5} A_\mu + \phi^2 A_{\mu,5} \\
2 \phi \phi_{,5} A_\nu + \phi^2 A_{\nu,5} &
2 \phi \phi_{,5}
}
}$
$\tilde\Gamma_{abc} = \frac{1}{2} (\tilde{g}_{ab,c} + \tilde{g}_{ac,b} - \tilde{g}_{bc,a})$
$= \overset{\downarrow c(\rho)[\downarrow a(\mu) \rightarrow b(\nu)]}{\pmatrix{
\pmatrix{
\tilde\Gamma_{\mu\nu\rho} &
\tilde\Gamma_{\mu 5 \rho} \\
\tilde\Gamma_{5 \nu\rho} &
\tilde\Gamma_{5 5 \rho}
} \\
\pmatrix{
\tilde\Gamma_{\mu\nu 5} &
\tilde\Gamma_{\mu 5 5} \\
\tilde\Gamma_{5 \nu 5} &
\tilde\Gamma_{5 5 5}
}
}}$
$= \frac{1}{2} \overset{\downarrow c(\rho)[\downarrow a(\mu) \rightarrow b(\nu)]}{\pmatrix{
\pmatrix{
\tilde{g}_{\mu \nu, \rho}
+ \tilde{g}_{\mu \rho, \nu}
- \tilde{g}_{\nu \rho, \mu} &
\tilde{g}_{\mu 5, \rho}
+ \tilde{g}_{\mu \rho, 5}
- \tilde{g}_{\rho 5, \mu} \\
\tilde{g}_{\nu 5, \rho}
+ \tilde{g}_{\rho 5, \nu}
- \tilde{g}_{\nu \rho, 5} &
\tilde{g}_{5 5, \rho}
} \\
\pmatrix{
\tilde{g}_{\mu \nu, 5}
+ \tilde{g}_{\mu 5, \nu}
- \tilde{g}_{\nu 5, \mu} &
2 \tilde{g}_{\mu 5, 5} - \tilde{g}_{5 5, \mu} \\
\tilde{g}_{5 5, \nu} &
\tilde{g}_{5 5, 5}
}
}}$
TODO bottom half of this and more.
$= \overset{\downarrow c(\rho)[\downarrow a(\mu) \rightarrow b(\nu)]}{\pmatrix{
\pmatrix{
\Gamma_{\mu\nu\rho}
+ \phi (
A_\mu A_\nu \phi_{,\rho}
+ A_\mu A_\rho \phi_{,\nu}
- A_\nu A_\rho \phi_{,\mu}
)
+ \phi^2 (
A_\mu A_{(\nu,\rho)}
+ A_\nu A_{[\mu,\rho]}
+ A_\rho A_{[\mu,\nu]}
)
&
\frac{1}{2} g_{\mu\rho,5}
+ \phi (
2 A_{[\mu} \phi_{,\rho]}
+ \phi_{,5} A_\mu A_\rho
)
+ \phi^2 (
A_{(\rho} A_{\mu),5}
+ A_{[\mu,\rho]}
)
\\
\frac{1}{2} g_{\nu\rho,5}
+ \phi (
2 A_{[\nu} \phi_{,\rho]}
+ \phi_{,5} A_\nu A_\rho
)
+ \phi^2 (
A_{(\rho} A_{\nu),5}
+ A_{[\nu,\rho]}
)
&
\phi \phi_{,\rho}
} \\
\pmatrix{
\frac{1}{2} (
\tilde{g}_{\mu \nu, 5}
+ \tilde{g}_{\mu 5, \nu}
- \tilde{g}_{\nu 5, \mu}
)
&
\frac{1}{2} (
\tilde{g}_{\mu 5, 5}
+ \tilde{g}_{\mu 5, 5}
- \tilde{g}_{5 5, \mu}
)
\\
\frac{1}{2} (
\tilde{g}_{5 \nu, 5}
+ \tilde{g}_{5 5, \nu}
- \tilde{g}_{\nu 5, 5}
)
&
\frac{1}{2} (
\tilde{g}_{5 5, 5}
+ \tilde{g}_{5 5, 5}
- \tilde{g}_{5 5, 5}
)
}
}}$
$= \overset{\downarrow c(\rho)[\downarrow a(\mu) \rightarrow b(\nu)]}{\pmatrix{
\pmatrix{
\Gamma_{\mu\nu\rho}
+ \phi (
\phi_{,\rho} A_\mu A_\nu
+ \phi_{,\nu} A_\mu A_\rho
- \phi_{,\mu} A_\nu A_\rho
)
+ \phi^2 (
A_\mu A_{(\nu,\rho)}
+ A_\nu A_{[\mu,\rho]}
+ A_\rho A_{[\mu,\nu]}
)
&
\frac{1}{2} g_{\mu\rho,5} + \phi^2 A_{[\mu,\rho]} \\
-\frac{1}{2} g_{\nu\rho,5} + \phi^2 A_{(\nu,\rho)} &
0
} \\
\pmatrix{
\frac{1}{2} g_{\mu\nu,5} + \phi^2 A_{[\mu,\nu]} & \phi^2 A_{\mu,5} \\
0 & 0
}
}}$
${\tilde\Gamma^a}_{bc} = g^{ad} \cdot \tilde\Gamma_{dbc}$
$=\overset{\downarrow c [ \downarrow a \rightarrow b]}{\pmatrix{
\pmatrix{
{\Gamma^\mu}_{\nu\rho} + \frac{1}{2} g^{\mu\alpha} \phi^2 (A_\nu F_{\rho\alpha} + A_\rho F_{\nu\alpha}) + \frac{1}{2} A^\mu g_{\nu\rho,5} &
\frac{1}{2} g^{\mu\alpha} (g_{\alpha\rho,5} + \phi^2 F_{\rho\alpha}) \\
-A^\alpha (\Gamma_{\alpha\nu\rho} + \frac{1}{2} \phi^2 (A_\nu F_{\rho\alpha} + A_\rho F_{\nu\alpha})) - \frac{1}{2} A^2 g_{\nu\rho,5} - \frac{1}{2\phi^2} g_{\nu\rho,5} + A_{(\nu,\rho)} &
-\frac{1}{2} A^\alpha (g_{\alpha\rho,5} + \phi^2 F_{\rho\alpha})
} \\
\pmatrix{
\frac{1}{2} g^{\mu\alpha} (g_{\alpha\nu,5} + \phi^2 F_{\nu\alpha}) &
\phi^2 g^{\mu\alpha} A_{\alpha,5} \\
-\frac{1}{2} A^\alpha (g_{\alpha\nu,5} + \phi^2 F_{\nu\alpha}) &
-\phi^2 A^\alpha A_{\alpha,5}
}
}}$
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:
$\dot{\tilde{u}}^5 + {\tilde{\Gamma}^5}_{\alpha\beta} \tilde{u}^\alpha \tilde{u}^\beta
+ 2 {\tilde{\Gamma}^5}_{\alpha 5} \tilde{u}^\alpha \tilde{u}^5
+ {\tilde{\Gamma}^5}_{55} (\tilde{u}^5)^2 = 0$
$\dot{\tilde{u}}^5
+ (-A^\mu \Gamma_{\mu\alpha\beta} - \frac{1}{2} \phi^2 A_\mu A_{(\alpha} {F_{\beta)}}^\mu
- \frac{1}{2} A^2 g_{\alpha\beta,5}
- \frac{1}{2\phi^2} g_{\alpha\beta,5}
+ A_{(\alpha,\beta)}
) \tilde{u}^\alpha \tilde{u}^\beta
- A^\mu (g_{\mu\alpha,5} + \phi^2 F_{\alpha\mu}) \tilde{u}^\alpha \tilde{u}^5
- \phi^2 A^\mu A_{\mu,5} (\tilde{u}^5)^2 = 0$