18 - Cosmology

Summary

Very quickly after Einstein published his general theory, a number of researchers attempted to apply Einstein's equations to the universe as a whole. This was a natural, if quite radical, move. In Einstein's theory, the distribution of energy and momentum in the universe determines the structure of space-time, and this applies as much to the universe as a whole as to the region of space, say, around a star. To get started these early researchers made an assumption which, while logical, may seem a bit bizarre. They took the principles enunciated by Copernicus to their logical extreme, and assumed that space-time was homogeneous and isotropic, i.e. that there is no special place or direction in the universe. They had virtually no evidence for this hypothesis at the time – definitive observations of galaxies outside of the Milky Way were only made a few years later. It was only decades later that evidence in support of this cosmological principle emerged. As we will discuss, we now know that the universe is extremely homogeneous, when viewed on sufficiently large scales.

To implement the principle, just as, for the Schwarzschild solution, we begin by writing the most general metric consistent with an assumed set of symmetries. In this case, the symmetries are homogeneity and isotropy in space. A metric of this form is called Friedmann–Robertson–Walker (FRW). We can derive this metric by imagining our three-dimensional space, at any instant, as a surface in a four-dimensional space. There should be no preferred direction on this surface; in this way, we will impose both homogeneity and isotropy. The surface will then be one of constant curvature.