The derivation of the Freedman equation requires the mathematical result known as the shell theorem which can easily be proved. The theorem states that in a spherical symmetric distribution of matter, a particle at a distance (r = R) from the centre (r=0) feels no force at all from the material in the outside region (r<R).Then the calculation is carried out to lead to Friedmann equation:
The full derivation can be found in
any cosmology textbook and is usually followed by the discussion of the three
possible geometries for the universe according to the value of (ĸ) being zero,
positive or negative.
Unfortunately, this equation has been
derived by erroneous method making use of the incorrect assumption that the
shell theorem can be applied in the case of non-Euclidean geometry. Let us see the
case of spherical geometry. Such geometry might well apply to our universe. It
can easily be seen that in an isotropic universe with spherical geometry no
gravitational force exists at all because every force in any direction is
matched by a force of the same magnitude in the opposite direction because of
total symmetry of space in this case. This brings out the interesting fact that
the large scale geometry of an isotropic universe with spherical geometry is
independent of its average density. This fact seems to contradict general
theory of relativity at first glance but, interestingly, this contradiction
will disappear if we agree to define the
cosmological constant in such a way that the average density of the universe is
cancelled out from the global
application field equation. In fact, we are forced to do so because the
application of Newtonian gravity in the large scale (not like the case of local
application in which GR supersedes Newtonian gravity) should give the same
results as that given by general theory of relativity. The independence of the
large scale geometry of the universe from its average distance can be attained
by assuming that the average density of the universe is a part of the
cosmological constant and therefore it is cancelled out because it appears
twice in the field equation with
opposite sign, first as a part of the
cosmological constant and then as the material part of the equation ( full
details of this definition of the cosmological constant and a proposed cosmological model associated with it is found in (Alternative Cosmological Model
without Ad Hoc Elements and without Modifications in GR or QM : http://vixra.org/abs/1706.0027).
Claims of alternative methods of
derivation of the Friedmann equation such as Machian derivation (see for
example: Machian derivation of the Friedmann equation by Herman Telkamp)
also fail in front of the total symmetry of the isotropic spherical universe.
