Even if the pressure force can be neglected in the equation of motion
(A.1), the gravitational force remains.
Assuming the spherical symmetry, consider the gravity at the position
of radial distance from the center being equal to .
Using the Gauss' theorem, is related to the mass inside of ,
which is expressed by the equation
and is written as
This leads to the equation motion for a cold gas under a control of
the self-gravity is written
Analyzing the equation is straightforward, multiplying gives
which leads to the conservation of mechanical energy as
in which represents the total energy of the pressureless
gas element and it is obtained from the initial condition.
If the gas is static initially at the distance , the total
energy is negative as
because at , and .
The solutions of equation (2.16) are well known as follows:
- the case of negative energy .
Considering the case that the gas sphere is inflowing ,
equation (2.16) becomes
where we assumed initially at .
Using a parameter , the radius of the gas element
at some epoch is written
In this case, equation (2.18) reduces to
This gives us the expression of as
This corresponds to the closed universe in the cosmic expansion
- if the energy is equal to zero,
the solution of equation (2.16) is written as