Relativity Lite

Cosmology | 71 Figure 1 also shows that the sizes of the galaxies (disks) do not change as their distance apart grows, nor do our local rulers grow with time, nor our noses. This is true for two rea- sons. First of all, the expansion equations showing an expanding universe are based on the idea that the universe is homogeneous, which holds true only on a large scale. The universe is decidedly lumpy on the scale of galaxies, solar systems, rulers, and atoms. Second, even if we could prove that a lumpy cosmology would expand, the gravitational attraction of nearby stars would overwhelm any cosmic tendency for galaxies to expand. And our bodies are held together by electrical forces that are (locally) enormously stronger than gravitational ones. (Otherwise we would fall through the chairs we sit on.) However, in an accelerating expansion that we will discuss in a minute, both local gravity and even electrical attraction may eventually be overwhelmed. Finally, the question always comes up, “What is the universe expanding into?” If we were living in the universe of figure 1 in the yellow-galaxy era, there would be no outer surface with red galaxies sitting on it. We could take a photograph inward from our surface into the past, but to photograph outward would be to photograph the future, and that is something no one I know is skilled at. From this perspective, one might say, “There is noth- ing there for the universe to expand into, and by ‘nothing’ I don’t mean emptiness of some sort but nonexistence. The universe must make the universe it is expanding into!” Remember that figure 1 is a representation of the universe in two spatial dimensions lying on the surface of a sphere whose radius grows larger with time. Put another way, in this picture, time is synonymous with the radius of the sphere. As time gets big, the sphere’s radius gets big in exact proportion. So for a spherical universe, one might answer, “The universe is expanding into the future!” If you are satisfied with the Hubble–Lemaître law as given above, you may skip the fol- lowing section that actually proves the Hubble–Lemaître law using some fairly complicated visual relationships. THE HUBBLE–LEMAÎTRE LAW We can show how the Hubble–Lemaître law works in more detail using figure 2, a cross sec- tion of the four-dimensional hypersphere of figure 1, but now with the added complication of a universe that is slowing its expansion velocity as time progresses. This slowing is what one would expect as the mass of all the rest of the universe tugs backward on the speed of every individual galaxy.