So, given this, how does one get BBT from GR? The basic equations for BBT come directly from Einstein's GR equation under two key assumptions: First, that the distribution of matter and energy in the universe is homogeneous and, second, that the distribution is isotropic. A simpler way to put this is that the universe looks the same everywhere and in every direction. The combination of these two assumptions is often termed the cosmological principle. Obviously, these assumptions do not describe the universe on all physical scales. Sitting in your chair, you have a density that is roughly 1000 000 000 000 000 000 000 000 000 000 times the mean density of the universe. Likewise, the densities of things like stars, galaxies and galaxy clusters are well above the mean (although not nearly as much as you). Instead, we find that these assumptions only apply on extremely large scales, on the order of several hundred million light years. However, even though we have good evidence that the cosmological principle is valid on these scales, we are limited to only a single vantage point and a finite volume of the universe to examine, so these assumptions must remain exactly that.
If we adopt these seemingly simple assumptions, the implications for the geometry of the universe are quite profound. First, one can demonstrate mathematically that there are only three possible curvatures to the universe: positive, negative or zero curvature (these are also commonly called "closed", "open" and "flat" models)...