Substituting this into equation (20.15), we have 4k Gp0
We can learn a lot about the evolution of an expanding universe by applying Newtonian gravitation. In the next section we will see how the Newtonian results are modified by general relativity.
The assumption of isotropy is equivalent to saying that the universe appears to be spherically symmetric from any point. This means that any spherical volume evolves only under its own influence. The gravitational forces exerted on the volume by material outside the volume sum (vec-torally) to zero. If the volume in question has a radius r, and mass M(r), the equation of motion for a particle of mass m on the surface of the sphere at position r is
Note that if p0 is not zero then R cannot be zero. A universe with matter cannot be static. It must be expanding or contracting. This is like saying that if you throw a ball up, and the Earth's mass is not zero, then the ball must be moving up or down; it cannot be forever stationary.
To integrate the equation of motion (20.17), we first multiply both sides by R, to give
Noting that d(R2)
In this case, we let r be a unit vector in the r-direction (radially outward). The assumption of homogeneity means that the density p is the same everywhere (though it can change with time). The mass M(r) is given by this becomes
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