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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 at a distance from the point source as

 (2.6)

where is the electric permittivity of the vacuum. On the other hand, the gravitational acceleration by the point mass of at the distance from the point mass is written down as
 (2.7)

where is the gravitational constant. Comparing these two, replacing with and at the same time to these equations (2.6) and (2.7) are identical with each other.

The Gauss theorem for electrostatic field as

 (2.8)

and another expression using the electrostatic potential as
 (2.9)

lead to the equations for the gravity as
 (2.10)

and
 (2.11)

where and 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 is determined from the mass density distribution .

Subsections

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Kohji Tomisaka 2007-07-08