The Poisson Equation of the Self-Gravity

In this section, we will show the basic equation describing how the gravity works. First, compare the gravity and the static electric force. Consider the electric field formed by a point charge $Q$ at a distance $r$ from the point source as

$$E=\frac{1}{4\pi \epsilon_0}\frac{Q}{r^2},$$ (2.6)

where $\epsilon_0$ is the electric permittivity of the vacuum. On the other hand, the gravitational acceleration by the point mass of $M$ at the distance $r$ from the point mass is written down as

$$g=-G\frac{M}{r^2},$$ (2.7)

where $G=6.67\times 10^{-8}{\rm kg^{-1}\,m^{3}\,s^{-2}}$ is the gravitational constant. Comparing these two, replacing $Q$ with $M$ and at the same time $1/4\pi \epsilon_0$ to $-G$ these equations (2.6) and (2.7) are identical with each other.

The Gauss theorem for electrostatic field as

$${\rm div}\,{\bf E}=\frac{\rho_e}{\epsilon_0},$$ (2.8)

and another expression using the electrostatic potential $\phi_e$ as

$$\nabla^2\phi_e=-\frac{\rho_e}{\epsilon_0},$$ (2.9)

lead to the equations for the gravity as

$${\rm div}\,{\bf g}=-4\pi G\rho,$$ (2.10)

and

$$\nabla^2\phi=4\pi G\rho,$$ (2.11)

where $\rho_e$ and $\rho$ represent the electric charge density and the mass density. Equation (2.11) is called the Poisson equation for the gravitational potential and describes how the potential $\phi$ is determined from the mass density distribution $\rho$.