A Short History of Big Bang Theory
Steady State and Cylindrical Universe:
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:
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
With the cosmological constant added:
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.