Title: Linearized Einstein's field equations,
Abstract: From the Einstein field equations, in a weak-field approximation and for
speeds small compared to the speed of light in vacuum, the following system is
obtained \begin{align*}
\nabla \times \overrightarrow{E_g} & =
-\frac{1}{c} \frac{\partial \overrightarrow{B_g}}{\partial t},
\nabla \cdot \overrightarrow{E_g} \;\; & \approx -4\pi G\rho_g,
\nabla \times \overrightarrow{B_g} & \approx
-\frac{4\pi G}{c^{2}}\overrightarrow{J_g}+
\frac{1}{c}\frac{\partial \overrightarrow{E_g}}{\partial t},
\nabla \cdot \overrightarrow{B_g} \;\; & = 0, \end{align*} where
$\overrightarrow{E_g}$ is the gravitoelectric field, $\overrightarrow{B_g}$ is
the gravitomagnetic field, $\overrightarrow{J_g}$ is the space-time-mass
current density and $\rho_g$ is the space-time-mass density. This last
gravitoelectromagnetic field system is similar to the Maxwell equations, thus
showing an analogy between the electromagnetic theory and gravitation.