Relativity Lite

Cosmology | 69 work if we discard one of our three spatial dimensions, as usual, and look at a spacetime of two spatial dimensions wrapping around the surface of a balloon, with the time dimension along its radius. Figure 1 is a representation of the universe in two spatial dimensions lying on the sur- face of a sphere whose radius grows larger with time. As the universe doubles in size, the galaxies—originally arbitrarily colored yellow at the initial time and red at the later time to distinguish them—have moved outward on the expanding sphere and have simultaneously been dragged apart by the expanding universe without the galaxies changing size. A yellow galaxy sits at about the three o’clock position on the edge of our view of the universe at its initial size, with a second galaxy just above it, and a third just above that. As the universe doubles in size, the corresponding, now-red, galaxies move twice as far apart. A blue arrow adjacent to the yellow galaxies shows the center-to-center distance between the first and second yellow galaxies. Its length is the disk diameter, and we will use that as the unit of measure. A second blue arrow combined with the first shows that the center-­ to-center distance between the first and third yellow galaxies is two disk diameters. After the universe has doubled in size, the center-to-center distance between the first and second now-red galaxies has doubled to two disk diameters. Put another way, its po- sition relative to the first has gotten one disk diameter larger. This additional distance is shown by the green arrow above the lowest blue arrow adjacent to the red galaxies. Notice that after the universe has doubled in size, the center-to-center distance between the first and third now-red galaxies has also doubled, in this case to four disk diameters. Put another way, its position relative to the first has gotten two disk diameters larger. This additional distance is shown by the two green arrows in addition to the two blue arrows. Since we define a velocity as the change in distance divided by the time interval, the re- cessional velocity of the third galaxy from the first is therefore twice the recessional velocity of the second galaxy from the first, since we are dividing by the same time interval in both measurements. This is precisely the Hubble–Lemaître law. The scale of the universe is expanding and drags the galaxies apart with it. Note that in such a scheme, every galaxy would see every other galaxy receding, and we thus avoid the idea that we are at the center of the universe. To see this, just reverse the arrows in figure 1 to see that galaxy 1 is moving away from galaxy 3 as much as the reverse is true. So creatures living in galaxy 3 will also come up with the Hubble–Lemaître law—though of course named for one of their own.