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

$c = $$ \cdot \frac{m}{s} = $ speed of light
$G = $$ \cdot \frac{m^3}{kg \cdot s^2} = $ gravitational constant

$\epsilon_g = \frac{1}{4 \pi G} =$ {{ gravitational_permittivity_in_kg_s2_per_m3 = 1 / (4 * Math.PI * gravitational_constant_in_m3_per_kg_s2) }} = gravitational permittivity, has units $[\frac{kg \cdot s^2}{m^3}]$
$\mu_g = \frac{4 \pi G}{c^2} =$ {{ gravitational_permeability_in_m_per_kg = 4 * Math.PI * gravitational_constant_in_m3_per_kg_s2 / Math.pow(speed_of_light_in_m_per_s, 2) }} = gravitational permeability, has units $[\frac{m}{kg}]$

cylindrical object material:
$\rho = $$ \frac{g}{cm^3} =$ {{object_density_in_kg_per_m3 = object_density_in_g_per_cm3 * 1e+3}} $\frac{kg}{m^3} = $ object material density.
$h = $$ \cdot m = $ height of cylindrical object. Object z coordinate spans $[-\frac{h}{2}, \frac{h}{2}]$.
$r = $$ \cdot m = $ radius of cylindrical object.
$V = \pi r^2 h = $ {{ object_volume_in_m3 = Math.PI * object_radius_in_m * object_radius_in_m * object_height_in_m }} $ \cdot m^3 = $ object volume.
$m = \rho V = $ {{ object_mass_in_kg = object_volume_in_m3 * object_density_in_kg_per_m3 }} $ \cdot kg = $ object mass.
$\omega = $$ \cdot \frac{rad}{s} = $ angular velocity of cylindrical object.
$I = \frac{1}{2} m r^2 = $ {{ object_inertia_in_kg_m2 = object_mass_in_kg * object_radius_in_m }} $ \cdot kg \cdot m^2 = $ object inertia.
$v_{\hat{\phi}} (r) = \omega \cdot r = $ linear velocity at a specific point, as a function of radius, has units $[\frac{m}{s}]$.
Velocity in Cartesian components, at cylindrical coordinates:
$v_x(r, \phi, z) = -\omega r sin(\phi)$
$v_y(r, \phi, z) = \omega r cos(\phi)$
$v_z(r, \phi, z) = 0$
Velocity in Cartesian components, at Cartesian coordinates:
$v_x(x, y, z) = -\omega y$
$v_y(x, y, z) = \omega x$
$v_z(x, y, z) = 0$
Velocity in vector notation:
$\vec{v}(\vec{x}) = \omega \vec{e}_z \times \vec{x}$

gravitational 4-potential:
$\Phi^g = \frac{1}{4 \pi \epsilon_g} \int_V \frac{T_{00}}{c^2 r} dV$ has units $[\frac{m^2}{s^2}]$
$(A^g)_i = -\frac{\mu_g}{4 \pi} \int_V \frac{T_{0i}}{c r} dV$ has units $[\frac{m}{s}]$

$\Phi^g = \frac{G}{c^2} \int_V \frac{1}{r} T_{00} dV$
$(A^g)_i = -\frac{G}{c^3} \int_V \frac{1}{r} T_{0i} dV$

For fluid stress-energy:
$T_{00} = c^2 \rho$ has units $[\frac{kg}{m \cdot s^2}]$ of energy density
$T_{0i} = c \rho v_i$ has units $[\frac{kg}{m \cdot s^2}]$ of energy density

$\Phi^g = G \int_V \frac{\rho}{r} dV$
$(A^g)_i = -\frac{G}{c^2} \int_V \frac{1}{r} \rho v_i dV$

For our object with constant density and angular velocity:
$\Phi^g(\vec{x}) = G \rho \int_V \frac{1}{|\vec{x}' - \vec{x}|} d\vec{x}'$
$(A^g)_{\hat{\phi}}(\vec{x}) = -\frac{G}{c^2} \rho \int_V \frac{1}{|\vec{x}' - \vec{x}|} v_{\hat{\phi}}(|\vec{x}'|) d\vec{x}'$

in Cartesian components for cylindrical coordinates:
$(A^g)_x(r, \phi, z) = \frac{G}{c^2} \rho \int_V \frac{1}{|\vec{x}' - \vec{x}|} \omega r' sin(\phi') \cdot r' dr' d\phi' dz'$
$(A^g)_y(r, \phi, z) = -\frac{G}{c^2} \rho \int_V \frac{1}{|\vec{x}' - \vec{x}|} \omega r' cos(\phi') \cdot r' dr' d\phi' dz'$
$(A^g)_z(r, \phi, z) = 0$
In cylindrical coodrinates, $|\vec{x}' - \vec{x}|^2 = r^2 - 2 r r' cos(\phi - \phi') + r'^2 + (z - z')^2$.

in Cartesian components for Cartesian coordinates:
$(A^g)_x(\vec{x}) = \frac{G }{c^2} \rho \omega \int_V \frac{y}{|\vec{x}' - \vec{x}|} d\vec{x}'$
$(A^g)_y(\vec{x}) = -\frac{G }{c^2} \rho \omega \int_V \frac{x}{|\vec{x}' - \vec{x}|} d\vec{x}'$
$(A^g)_z(\vec{x}) = 0$
In Cartesian coodrinates, $|\vec{x}' - \vec{x}|^2 = (x - x')^2 + (y - y')^2 + (z - z')^2$.

GEM acceleration:
${v^i}_{,t} = (E^g)^i + 4 \epsilon^{ijk} v_j (B^g)_k$

$(E^g)_i = (\Phi^g)_{,i} - c (A^g)_{i,0}$ has units $[\frac{m}{s^2}]$

For a steady state:
$(E^g)_i(\vec{x}) = \frac{\partial}{\partial x^i} (\Phi^g)(\vec{x})$
$(E^g)_i(\vec{x}) = \frac{\partial}{\partial x^i} G \rho \int_V \frac{1}{|\vec{x}' - \vec{x}|} d\vec{x}'$
$\vec{(E^g)}(\vec{x}) = G \rho \int_V \vec{\nabla} \left( \frac{1}{|\vec{x}' - \vec{x}|} \right) d\vec{x}'$
$\vec{(E^g)}(\vec{x}) = G \rho \int_V \frac{\vec{x}' - \vec{x}}{|\vec{x} - \vec{x}'|^3} d\vec{x}'$

For our object: TODO derive this.
$(E^g)_i(\vec{x}) = -\frac{G m}{|\vec{x}|^2} \frac{x^i}{|\vec{x}|}$

Integrating the gravitoelectric force across the object:
(Mind you, I shouldn't just integrate the gravitoelectric force, but I should be applying the impulse formula to consider force and torque produced at each point.)
I should be integraing the impulse equation, to consider torque as well as acceleration.)
$(E^g)_i$, the gravitoelectric force on the object by the object, which should cancel itself out. Verify this:

Integrating the gravitomagnetic force across the object:
Would I need to average the acceleration by dividing by the volume? I'm thinking otherwise the units might not work out ... ? But does it physically make sense? I guess that's what happen when you just add pointwise acceleration and don't consider the impulse equation, to include both force and torque generated from the force at each point of a rigid body.
$(F^E)_i = \int_V (E^g)_i(\vec{x}) dx$
$(F^E)_i = \int_{z=-\frac{h}{2}}^{z=\frac{h}{2}} \int_{\phi=0}^{\phi=2\pi} \int_{r=0}^{r=r} (E^g)_i(r,\phi,z) r dr d\phi dz $
$(F^E)_i = 0$

How about $(B^g)_i$?

$(A^g)_{\hat{\phi}}(\vec{x}) = -\frac{G}{c^2} \rho \int_V \frac{1}{|\vec{x}' - \vec{x}|} v_{\hat{\phi}}(|\vec{x}'|) d\vec{x}'$

$\frac{\partial}{\partial x^k} (A^g)_j(\vec{x}) = $

Divergence theorem: $\int_V {F^i}_{,i} dV = \int_S F^i n_i dS$

$(B^g)^i = \epsilon^{ijk} (A^g)_{k,j}$ has units $[\frac{1}{s}]$
$\vec{(B^g)} = \vec{\nabla} \times \vec{(A^g)}$.
$\vec{(B^g)}(\vec{x}) = \vec{\nabla} \times -\frac{G}{c^2} \rho \int_V \frac{1}{|\vec{x}' - \vec{x}|} \vec{v}(\vec{x}') d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \rho \int_V \vec{\nabla} \times \left( \frac{1}{|\vec{x}' - \vec{x}|} \vec{v}(\vec{x}') \right) d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \rho \int_V \left( \vec{\nabla} \frac{1}{|\vec{x}' - \vec{x}|} \right) \times \vec{v}(\vec{x}') + \frac{1}{|\vec{x}' - \vec{x}|} \cdot \vec{\nabla} \times \vec{v}(\vec{x}') d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \rho \int_V \left( \frac{\vec{x}' - \vec{x}}{|\vec{x} - \vec{x}'|^3} \right) \times \vec{v}(\vec{x}') d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \rho \int_V \omega \frac{1}{|\vec{x}' - \vec{x}|^3} \left[ \begin{matrix} x' - x \\ y' - y \\ z' - z \end{matrix} \right] \times \left[ \begin{matrix} -y' \\ x' \\ 0 \end{matrix} \right] d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = \frac{G}{c^2} \rho \omega \int_V \frac{1}{|\vec{x}' - \vec{x}|^3} \left[ \begin{matrix} x' (z' - z) \\ y' (z' - z) \\ -x' (x' - x) - y' (y' - y) \end{matrix} \right] d\vec{x}'$

$\vec{(B^g)}_x (\vec{x}) = -\frac{G}{c^2} \rho \omega \int_V \frac{x' (z' - z)}{ \left( (x' - x)^2 + (y' - y)^2 + (z' - z)^2 \right)^\frac{3}{2} } d\vec{x}'$

$\vec{(B^g)}_y (\vec{x}) = -\frac{G}{c^2} \rho \omega \int_V \frac{y' (z' - z)}{ \left( (x' - x)^2 + (y' - y)^2 + (z' - z)^2 \right)^\frac{3}{2} } d\vec{x}'$

$\vec{(B^g)}_z (\vec{x}) = -\frac{G}{c^2} \rho \omega \int_V \frac{-x' (x' - x) - y' (y' - y)}{ \left( (x' - x)^2 + (y' - y)^2 + (z' - z)^2 \right)^\frac{3}{2} } d\vec{x}'$

in cylindrical coordinates:

$\vec{(B^g)}_x (\vec{x}) = -\frac{G}{c^2} \rho \omega \int_{z'} \int_{\phi'} \int_{r'} (r')^2 \frac{cos(\phi') (z' - z)}{ \left( r'^2 - 2 r r' cos(\phi - \phi') + r^2 + (z' - z)^2 \right)^\frac{3}{2} } dr' d\phi' dz' $

$\vec{(B^g)}_y (\vec{x}) = -\frac{G}{c^2} \rho \omega \int_{z'} \int_{\phi'} \int_{r'} (r')^2 \frac{sin(\phi') (z' - z)}{ \left( r'^2 - 2 r r' cos(\phi - \phi') + r^2 + (z' - z)^2 \right)^\frac{3}{2} } dr' d\phi' dz' $

$\vec{(B^g)}_z (\vec{x}) = -\frac{G}{c^2} \rho \omega \int_{z'} \int_{\phi'} \int_{r'} (r')^2 \frac{-cos(\phi') (r' cos(\phi') - r cos(\phi)) - sin(\phi') (r' sin(\phi') - r sin(\phi))}{ \left( r'^2 - 2 r r' cos(\phi - \phi') + r^2 + (z' - z)^2 \right)^\frac{3}{2} } dr' d\phi' dz' $

...again...
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \rho \int_V \vec{\nabla} \times \left( \frac{1}{|\vec{x}' - \vec{x}|} \vec{v}(\vec{x}') \right) d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \rho \int_V \vec{\nabla} \times \left( \frac{1}{|\vec{x}' - \vec{x}|} \omega \vec{e}_z \times \vec{x}' \right) d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \omega \rho \int_V \left( \frac{\vec{x}' - \vec{x}}{|\vec{x} - \vec{x}'|^3} \right) \times (\vec{e}_z \times \vec{x}') d\vec{x}'$
using $a \times (b \times c) = (a \cdot c) b - (a \cdot b) c$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \omega \rho \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \cdot \left( ((\vec{x}' - \vec{x}) \cdot \vec{x}') \vec{e}_z - ((\vec{x}' - \vec{x}) \cdot \vec{e}_z) \vec{x}' \right) d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \omega \rho \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \cdot \left( (\vec{x}' \cdot \vec{x}' - \vec{x} \cdot \vec{x}') \vec{e}_z - (\vec{x}' \cdot \vec{e}_z - \vec{x} \cdot \vec{e}_z) \vec{x}' \right) d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \omega \rho \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \cdot \left( ( (x')^2 + (y')^2 + (z')^2 - x x' - y y' - z z') \vec{e}_z - (z' - z) \vec{x}' \right) d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \omega \rho \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \cdot \left[ \begin{matrix} - (z' - z) x' \\ - (z' - z) y' \\ x' ((x' - x) + y' (y' - y) \end{matrix} \right] d\vec{x}'$

...again using Jacobi's identity:
$a \times (b \times c) + b \times (c \times a) + c \times (a \times b) = 0$
$a \times (b \times c) = - b \times (c \times a) - c \times (a \times b)$

$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \omega \rho \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} ( \vec{x}' - \vec{x} ) \times (\vec{e}_z \times \vec{x}') d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \omega \rho \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \left( -\vec{e}_z \times (\vec{x}' \times (\vec{x}' - \vec{x})) - \vec{x}' \times ((\vec{x}' - \vec{x}) \times \vec{e}_z) \right) d\vec{x}'$
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \omega \rho \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \left( \vec{e}_z \times (\vec{x}' \times \vec{x}) - \vec{x}' \times ((\vec{x}' - \vec{x}) \times \vec{e}_z) \right) d\vec{x}'$

$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \omega \rho \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \cdot \left[ \begin{matrix} - (z' - z) x' \\ - (z' - z) y' \\ x' (x' - x) + y' (y' - y) \end{matrix} \right] d\vec{x}'$

$\vec{F}^B = \int_V 4 \vec{v}(\vec{x}) \times \vec{B}^g(\vec{x}) d\vec{x}$
$ = -4 \frac{G}{c^2} \omega \rho \int_V \vec{v}(\vec{x}) \times \left( \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \cdot \left[ \begin{matrix} - (z' - z) x' \\ - (z' - z) y' \\ x' (x' - x) + y' (y' - y) \end{matrix} \right] d\vec{x}' \right) d\vec{x}$
$ = -4 \frac{G}{c^2} \omega \rho \int_V \omega (\vec{e}_z \times \vec{x}) \times \left( \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \cdot \left[ \begin{matrix} - (z' - z) x' \\ - (z' - z) y' \\ x' (x' - x) + y' (y' - y) \end{matrix} \right] d\vec{x}' \right) d\vec{x}$
$ = -4 \frac{G}{c^2} \omega^2 \rho \int_V \left( \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \cdot \left[ \begin{matrix} -y \\ x \\ 0 \end{matrix} \right] \times \left[ \begin{matrix} - (z' - z) x' \\ - (z' - z) y' \\ x' (x' - x) + y' (y' - y) \end{matrix} \right] d\vec{x}' \right) d\vec{x}$
$ = -4 \frac{G}{c^2} \omega^2 \rho \int_V \left( \int_V \frac{1}{|\vec{x} - \vec{x}'|^3} \cdot \left[ \begin{matrix} x (x' (x' - x) + y' (y' - y)) \\ y (x' (x' - x) + y' (y' - y)) \\ y y' (z' - z) + x x' (z' - z) \end{matrix} \right] d\vec{x}' \right) d\vec{x}$

going back to:
$\vec{(B^g)}(\vec{x}) = -\frac{G}{c^2} \rho \omega \int_V \vec{\nabla} \times \left( \frac{1}{|\vec{x}' - \vec{x}|} \vec{e}_z \times \vec{x}' \right) d\vec{x}'$
and
$\vec{F}^B = \int_V 4 \vec{v}(\vec{x}) \times \vec{B}^g(\vec{x}) d\vec{x}$
$\vec{F}^B = \int_V 4 \omega (\vec{e}_z \times \vec{x}) \times \vec{B}^g(\vec{x}) d\vec{x}$
and using a consequence of the Divergence theorem: $\int_V \nabla \times F dV = \int_S n \times F dS$
$\vec{F}^B = \int_V 4 \omega (\vec{e}_z \times \vec{x}) \times \left( -\frac{G}{c^2} \rho \omega \int_V \vec{\nabla} \times \left( \frac{1}{|\vec{x}' - \vec{x}|} \vec{e}_z \times \vec{x}' \right) d\vec{x}' \right) d\vec{x}$
$\vec{F}^B = -4 \frac{G}{c^2} \rho \omega^2 \int_V \int_V (\vec{e}_z \times \vec{x}) \times \left( \vec{\nabla} \times \left( \frac{1}{|\vec{x}' - \vec{x}|} \vec{e}_z \times \vec{x}' \right) \right) d\vec{x}' d\vec{x}$