Relativity Lite

Figure 2. Consider the cross section of a spherical, expanding universe whose expansion rate S is slowing with time. R is the cosmic scale factor and, in the spherical case, is equal to the radius of the sphere. D is the arc of a circle, giving the distance between two galaxies exactly 60° around the curve from each other. V is the recessional speed of one galaxy, at the 2 o’clock position, from the one at the 4 o’clock position around the circumference of the circle—the change in D with time. The distance D at time 1 , D(1) , has been duplicated and shifted outward and placed next to D(2) . The additional distance that must be added to D(1) to make the full distance D(2) is one time unit multiplying V(2) , the recessional velocity at time 2 . As the universal expansion slows, the recessional velocities also get smaller at times 3 and 4 . It may be easier to see distances and their changes over time by looking at equilateral triangles rather than arcs of circles. Thus, we also show C , the chord of the arc D , adjacent to a different 60° pie slice for visual clarity. The corresponding change in chord length from C(1) to C(2) with time is one time unit multiplying the velocity U(2) . Since the chords are proportional to the arcs, U(2) is therefore proportional to the actual recessional velocity V(2) .