This works just like Kaluza-Klein, except without the extra dimension needed.

$F_{ab} = \left[\matrix{ 0 & -E_j \\ E_i & \epsilon_{ijk} B_k }\right]$
$F_{ab} = 2 A_{[b,a]}$
$F_{it} = A_{t,i} - A_{i,t} = E_i$
$F_{ij} = A_{j,i} - A_{i,j} = \epsilon_{ijk} B_k$

$g_{ab} = \left[\matrix{ -1 + 2 A_t & A_j \\ A_i & \delta_{ij} }\right]$

$g_{ab} = \eta_{ab} + 2 A_{(a} \delta_{b)t}$
$g_{tt} = -1 + 2 A_t$
$g_{ti} = g_{it} = A_i$
$g_{ij} = \delta_{ij}$

$g_{tt,u} = 2 A_{t,u}$
$g_{ti,u} = g_{it,u} = A_{i,u}$
$g_{ij,u} = 0$

$\Gamma_{uvw} = \frac{1}{2} (g_{uv,w} + g_{uw,v} - g_{vw,u})$
$\Gamma_{ttt} = \frac{1}{2} g_{tt,t} = A_{t,t}$
$\Gamma_{tti} = \frac{1}{2} g_{tt,i} = A_{t,i}$
$\Gamma_{itt} = g_{it,t} - \frac{1}{2} g_{tt,i} = A_{i,t} - A_{t,i} = -E_i$
$\Gamma_{ijt} = \frac{1}{2} (g_{ij,t} + g_{it,j} - g_{jt,i}) = \frac{1}{2}( A_{i,j} - A_{j,i}) = -\frac{1}{2} \epsilon_{ijk} B_k$
$\Gamma_{tij} = \frac{1}{2} (g_{ti,j} + g_{tj,i} - g_{ij,t}) = \frac{1}{2}( A_{i,j} + A_{j,i})$
$\Gamma_{ijk} = \frac{1}{2} (g_{ij,k} + g_{ik,j} - g_{jk,i}) = 0$

From this, let $u^t = 1$, and solve the covariant geodesic:

$\dot{u}^u g_{uv} = -\Gamma_{iab} u^a u^b$
$= -\Gamma_{itt} - 2 \Gamma_{ijt} u^j - \Gamma_{ijk} u^j u^k$
$= E_i + \epsilon_{ijk} u^j B_k$

$\dot{u}^u g_{ut} = -\Gamma_{tab} u^a u^b$
$= -\Gamma_{ttt} - 2 \Gamma_{tjt} u^j - \Gamma_{tjk} u^j u^k$
$= -A_{t,t} - 2 A_{t,j} u^j - \frac{1}{2} (A_{j,k} + A_{k,j}) u^j u^k$

How about the appropriate, contravariant geodesic?

$g = -1 + 2 A_t - A_x^2 - A_y^2 - A_z^2$
$= -1 + 2 A_t - A_t^2 + A_t^2 - A_x^2 - A_y^2 - A_z^2$
$= -(1 - A_t^2) - \eta^{ab} A_a A_b$

$g^{ab} = \left[\matrix{ 1/g & -A_j/g \\ -A_i/g & A_i A_j / g + \delta^{ij} }\right]$

$g^{tt} = \frac{1}{g}$
$g^{ti} = g^{it} = -\frac{1}{g} A_i$
$g^{ij} = \frac{1}{g} A_i A_j + \delta^{ij}$

$\dot{u}^i = -{\Gamma^i}_{ab} u^a u^b = -(g^{it} \Gamma_{tab} + g^{ij} \Gamma_{jab}) u^a u^b$
$= \frac{1}{g} A_i (A_{t,t} + 2 A_{t,j} u^j + \frac{1}{2} (A_{j,k} + A_{k,j}) u^j u^k) + \frac{1}{g} (A_i A_j + \delta^{ij}) (E_j + \epsilon_{jkl} u^k B_l)$
$= \frac{1}{g} \left( A_i (A_{t,t} + 2 A_{t,j} u^j + \frac{1}{2} (A_{j,k} + A_{k,j}) u^j u^k + A_j (E_j + \epsilon_{jkl} u^k B_l)) + (E_i + \epsilon_{ijk} u^j B_k) \right)$

But of course the geodesic is missing charge...