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.