What's the connection coefficients, and therefore the gravity, that this would subject something to...
$\Gamma_{abc} = \frac{1}{2} (g_{ab,c} + g_{ac,b} - g_{bc,a})$
$\Gamma_{abc} = \frac{1}{2} (h_{ab,c} + h_{ac,b} - h_{bc,a})$
${\Gamma^a}_{bc} = \frac{1}{2} ({h^a}_{b,c} + {h^a}_{c,b} - {h_{bc}}^{,a})$
${\Gamma^i}_{00} = {h^i}_{0,0} - \frac{1}{2} {h_{00}}^{,i}$
${\Gamma^i}_{j0} = \frac{1}{2} ({h^i}_{j,0} + {h^i}_{0,j} - {h_{j0}}^{,i})$
${\Gamma^i}_{jk} = \frac{1}{2} ({h^i}_{j,k} + {h^i}_{k,j} - {h_{jk}}^{,i})$
Geodesics:
$[\frac{1}{m}] : {u^i}_{,0} = -{\Gamma^i}_{ab} u^a u^b$
$[\frac{1}{m}] : {u^i}_{,0} = -{\Gamma^i}_{00} (u^0)^2 - 2 {\Gamma^i}_{0j} u^0 u^j - {\Gamma^i}_{jk} u^j u^k$
$[\frac{1}{m}] : {u^i}_{,0} =
- ({h^i}_{0,0} - \frac{1}{2} {h_{00}}^{,i}) (u^0)^2
- ({h^i}_{0,j} + {h^i}_{j,0} - {h_{0j}}^{,i}) u^j u^0
- \frac{1}{2} ({h^i}_{j,k} + {h^i}_{k,j} - {h_{jk}}^{,i}) u^j u^k$
...using $(u^i)^2 \approx 0$, $u^0 \approx 1$, lower using $\eta_{ab}$
$[\frac{1}{m}] : {u^i}_{,0} =
- h_{i0,0} + \frac{1}{2} h_{00,i}
+ (-h_{i0,j} - h_{ij,0} + h_{0j,i}) u^j$
$[\frac{1}{m}] : {u^i}_{,0} =
4 \frac{1}{c} (A^g)_{i,0}
+ \frac{1}{c^2} (\Phi^g)_{,i}
+ (
4 \frac{1}{c} (A^g)_{i,j}
- 2 \frac{1}{c^2} \delta_{ij} (\Phi^g)_{,0}
- 4 \frac{1}{c} (A^g)_{j,i}
) u^j
$
$[\frac{1}{s}] : c {u^i}_{,0} = {u^i}_{,t} =
4 (A^g)_{i,0}
+ \frac{1}{c} (\Phi^g)_{,i}
+ (
4 (A^g)_{i,j}
- 2 \frac{1}{c} \delta_{ij} (\Phi^g)_{,0}
- 4 (A^g)_{j,i}
) u^j
$
$[\frac{m}{s^2}] : c {u^i}_{,t} = {v^i}_{,t} =
4 c (A^g)_{i,0}
+ (\Phi^g)_{,i}
+ (
4 (A^g)_{i,j}
- 2 \frac{1}{c} \delta_{ij} (\Phi^g)_{,0}
- 4 (A^g)_{j,i}
) v^j
$
using $(E^g)_i = (\Phi^g)_{,i} - c (A^g)_{i,0}$ and $(B^g)_i = {\epsilon_i}^{jk} (A^g)_{k,j}$
so $(\Phi^g)_{,i} = (E^g)_i + c (A^g)_{i,0}$ and $\epsilon^{ijk} (B^g)_k = 2 (A^g)_{[i,j]}$
$[\frac{m}{s^2}] : c {u^i}_{,t} = {v^i}_{,t} =
5 c (A^g)_{i,0}
+ (E^g)_i
+ 4 \epsilon^{ijk} v_j (B^g)_k
- 2 \frac{1}{c} v_i (\Phi^g)_{,0}
$
...let $5 (A^g)_{i,0} - 2 \frac{1}{c^2} (\Phi^g)_{,0} v_i \approx 0$...
$[\frac{m}{s^2}] : {v^i}_{,t} = (E^g)_{i} + 4 \epsilon^{ijk} v_j (B^g)_k$