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Though the author is quite familiar with the
more theoretical and mathematically rigorous treatments that many of modern cosmological models are based upon, this discussion
will be minimalist in its mathematical descriptions. This is done for two reasons.
The first reason is that CONCEPTS
precede and form the basis of hypotheses, not abstract mathematical relationships or derivatives. And everyone, even those
without advanced mathematical training, should be able to fully grasp the underlying concepts without the quantitative rigor
of a mathematically-based presentation.
The second reason is more esoteric. When theoretical physicists, cosmologists
and astrophysicists impart a sometimes false reality to the purely mathematical descriptions that they develop, they often
fall victim to either the emotional impact of their mathematical ruminations or become slavishly bound to the mathematical
laws and corollaries that they follow, and which the actual physical reality might not track with. It is good to remember
that we use mathematics to try to estimate or represent the actual quantitative behavior of any system. But the underlying
physical reality may still have no real congruence with the equations we scribble on our blackboards and computers. Thus we
should always keep our qualitative conceptualizations separate and somewhat independent of our subsequent mathematical models
that attempt to describe the more measurable aspects of that system.
Physical chemists have developed an extraordinary respect for what has become known
as "boundary conditions". Boundary conditions is something of a scientific hedge due to the persistent perversion of
Mother Nature to "not follow Man's rules about how it should behave"! When and electron falls into a pure theoretical
potential well, as basic quantum mechanics considers, it drops to some specific quantum level that can be described, measured
and calculated to the nth degree with our basic equations. In our basic theory, the electron will always remain in that
potential well as long as no other energy intrudes and transfers to the electron. But in reality, electrons in the potential
well of atoms and molecules frequently exit their wells through a process now known as "tunneling". This is one example
of a boundary condition. Another example would be the interface between two phases, such as liquid water and air.
In the air, molecules of various gases bounce around like billiard balls at normal temperatures, careening off of each other
and off of the surface of the water. In the interior of the liquid, water molecules move about but slower than the air
molecules overhead. They form transient weak hydrogen bonds and van der Waal bonds between each other in three dimensions
and thus behave in a fashion and as a somewhat cohesive ensemble that has physical properties like viscosity and temperature.
But at the uppermost layer of the surface of the water, the water molecules bond to each other differently than they do in
the liquid interior. At the surface, they form a two dimensional weakly bonded lattice that actually has even stronger
intermolecular attractive forces. This causes the water phase to behave as if it were covered by a "skin". This
surface layer of water molecules thus changes the properties of the bulk liquid water molecules and creates a new force or property
we measure as "surface tension". Surface tension does not exist within the liquid interior. It is another
type of boundary condition that is not derivable from first chemical or physical principles, but rather springs from the reality
of nature and her quirky ways.
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