Before discussing the question whether the assumption that the
universe is isotropic and homogeneous in the large scale is a good
approximation or not, we have to be aware of the important fact that the
general (global) curvature which is related to the shape , evolution and fate
of the universe is quite different concept from the average curvature which is associated
with the average density of the universes implied by Einstein's Field Equation.
Let us consider the application of the field equation in the
universe as a whole ( this is an example of the interior application of the
equation)
The average density of the universe is :
∑ 𝛒(𝔁) v(𝔁) /
V
Where 𝛒(𝔁) ≡ the density in the region 𝔁 , v(𝔁)≡ the
volume of the region 𝔁 , V
≡ the
total volume of the universe.
Similarly,
the Average Curvature of the universe is:
∑ R(𝔁) v(𝔁) / V
The General or Global Curvature of the universe is the curvature
which associated with the outline of the shape of the universe or the general
geometrical description in which we ignore the details when they occur in small
parts of the universe .
Interestingly , while wrongly and unconsciously assumed to be
similar , the average curvature and the global curvature describe very
different properties of the universe as a whole.
Let us clarify that this two terms are not equivalent by the
following example using two-dimensional space for simplicity and it is clear
that the same analysis can be generalized to any number of dimensions:
The average curvature of a uniform closed surface with large number
of polygons and rounded edges is very close to zero while the global curvature
of this surface can be thought of as the curvature of a sphere with the same
size . The general curvature depends
only on the size of this shape and has nothing to do with the average curvature
. Another example is the simple fact that the general curvature of the surface of the
earth depends only on the radius of the earth and is not related to the average
curvature which is affected by the topological details .
