Relativity Lite

74 | Relativity Lite BUT IS THE UNIVERSE A FOUR-DIMENSIONAL SPHERE? The constant k in equation (3) tells us about the shape of the universe. The balloon scenario is accurate only if k = +1 . Since this universe has no edge, if you waited long enough, you would eventually see light from the back of your head that had traveled all the way around the four-dimensional hypersphere to your eyes. If k = −1 , then space is infinite and boundless, part of which is the saddle shape shown in figure 3. (a) (b) Figure 3. A saddle-shaped universe also expands as the parameter R increases. For k = 0 , equation (3) looks very much like equation (1), with only the addition of the scale factor R that allows the universe to grow with time. Thus, in this case, as for special relativity, space is flat. For this to be the case, the density of the universe must be at a critical value given by the Hubble constant and Newton’s gravitational constant G as ρ π c H G = 3 8 0 2 . For a spacetime like that given by the Robertson-Walker metric equation (3) (and more generally, Friedmann–Lemaître spacetimes), something called the Friedmann equation gives the expansion velocity of the universe. In the matter-dominated era, it can be shown that the square of the ratio of the expansion velocity of the universe S to the speed of light is S 2 /c 2 = R max /R − k . (4a) When we look at the Hubble expansion in a k = 1 universe, we see that the expansion velocity S goes to zero at R = R max , (5)