# Free-fall Time

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
 (2.12)

and is written as
 (2.13)

This leads to the equation motion for a cold gas under a control of the self-gravity is written
 (2.14)

Analyzing the equation is straightforward, multiplying gives
 (2.15)

which leads to the conservation of mechanical energy as
 (2.16)

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
 (2.17)

because at , and .

The solutions of equation (2.16) are well known as follows:

1. the case of negative energy . Considering the case that the gas sphere is inflowing , equation (2.16) becomes
 (2.18)

where we assumed initially at . Using a parameter , the radius of the gas element at some epoch is written
 (2.19)

In this case, equation (2.18) reduces to
 (2.20)

This gives us the expression of as
 (2.21)

This corresponds to the closed universe in the cosmic expansion ().
2. if the energy is equal to zero, the solution of equation (2.16) is written as
 (2.22)

where .

Subsections
Kohji Tomisaka 2009-12-10