Abolishing Infinity

How could one avoid the need to stipulate the properties of spacetime at infinity? In 1917, Einstein came up with an ingenious escape: obliterate spatial infinity! By adding an extra term to his gravitational field equations, Einstein found a simple solution of his augmented field equations. ("Augmented"? What is that about? It refers to the famous λ. See below for an explanation.) It contains a uniform matter distribution that approximates a uniform distribution of stars. That matter is at rest and the geometry of a spatial slice is unchanging with time.

Space, however, curves back onto itself so that it is spherical. That is, space has the geometry of 5^NONE with positive curvature. In such a space, there is no infinity at which to stipulate the properties of space and time.

If one pictures just one dimension of space, then the universe looks like a cylinder. Spacetime resides just in the surface of the cylinder. The vertical lines are the world lines of the stars at rest. The one spatial dimension is wrapped back onto itself; the time dimension is not. Each spatial slice at a particular time appears as a circle; if we could represent all three dimensions of space, we would somehow have to replace the circle by a complete sphere of three dimensional space.

The Einstein universe is an especially simple universe. It is homogeneous. That means that, like Minkowski spacetime, it is geometrically the same at every event. It is also spatially isortropic, which means that it is the same in every spatial direction. In the jigsaw puzzle analogy this homogeneity means that the spacetime is assembled from just one sort of piece, used repeated to build the entire spacetime.

Read more at: http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/relativistic_cosmology/index.html