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A Short History of Big Bang Theory

The formulation of boundary conditions for an apparently infinite universe nearly confounded Einstein, who wrote that at some points he felt he should be confined to a madhouse.  What emerged from his application of general relativity theory, however, was a surprisingly simple and almost naiive model of gravitational behavior of matter at large over time.  His universe essentially had a boundary which remained fixed in time, and a center.
From Cosmological Considerations on the General Theory of Relativity, he first considers Possion's equation:

Where phi is the gravitational potential, K is a constant, and rho a particular density for a region of space.  Applied to a Newtonian universe with a universal force of gravity, the universe will collapse after a time dependant upon the density.  To force the universe to be in a steady state, an extra term can be added.  Einstein does this with no small consternation, commenting that it is simply 'a foil for what is to follow.'
This is a differential equation, where lambda denotes a universal constant, rho the density, modeled to decrease to zero as distance approaches infinity, so that the universe may have finite space according to Newton, but infinite mass.
A solution to the differential equation, if rho-naught is actually equal to the mean density of matter in the universe, is:

The resulting universe, with the gravitational potential balanced by the cosmological constant, is a bounded universe with no center with respect to the mean gravitational potential or the mean density.  If two of the spatial dimensions are supressed (so that the boundary appears to be finite in space and the 3-d + time sphere is projected on a 3-d surface), the model can be imagined like so:

The tensor form of the gravitational field equations is this:
where the sub-scripts are tensors, where m,n = 0,1,2,3, therefore ten second order differential equations exist.  Rmn is the Ricci curve tensor, and R is a curvature invariant derived from Rmn.  The constant k is related to the constant of gravity as k = 8piG/c^4.  The field equations relate geometrical and physical entities - on the left side, the geometry of space; on the right, Tmn is the Energy-momentum tensor, describing the physical contents of space.
Einstein commented that lambda "is not justified by our actual knowledge of gravitation...[but is] necessary for the purpose of making a quasi-static distribution of matter."
Nonetheless, he found that the cosmological constant related to the average density of the universe by the relation:

This equation describes the dependancy of the radius of the universe upon the density of matter in space.
Einstein was never satisfied with the cosmological constant, even though it allowed a dynamic universe -one which had a definite beginning and changed in scale over time.

The Beginning
Cylindrical Universe
Hubble & Humason
the CBR & Inflation
Timeline & the Future
Future of Big Bang Theory
Bibliography