Relativity Lite

Cosmology | 73 For a galaxy at the 12 o’clock position, exactly 60° around the curve from where we are at the 2 o’clock position in figure 2, the distance across the chord between the galaxies equals the radius of the universe, because they are two sides of an equilateral triangle. This is true for any time: C(t) = R(t) . Then the velocity U , the change in C with time, is precisely the expansion speed of the universe (the change in R with time) in this 60° case, or U(t) = S(t) . Let us define a Hubble constant (as it is called, though it changes with time) at the present time H(4) = H 0 as the ratio of the present expansion speed divided by the present radius: H 0 ≡ S(4)/R(4) = U(4)/C(4) for galaxies in this 60° case. The Hubble–Lemaître law is simply a rewriting of this relation, giving the recessional velocity as proportional to the distance, U(4) = H 0 C(4) , with H 0 as the constant of proportionality. The actual distances between, and velocities of, galaxies are along the circumference, not across the chord, but those are in proportion. We check this by using the arc lengths in figure 2, laid out in a different 60° pie slice for visual clarity, using a galaxy at the 4 o’clock position exactly 60° around the curve from where we are at the 2 o’clock position. It turns out that an arc of a circle has length R θ , where the angle θ is expressed in fractions of the circumference (2 π R ) of a circle that has radius 1, and thus as fractions of 2 π . For our case, 60° is 1/6 of the circumference so θ = π/3 = 1.05 to three decimal places. (As you eyeball figure 2, you might agree that the arcs could be about 5% larger than the chords.) As noted in the previous paragraph, the length of the chord across this 60° arc is precisely R , because they are two sides of an equilateral triangle. The adjacent arc has length 1.05 R for our 60° case. So if we just multiply all chord lengths C and their corresponding velocities U by 105%, we get the actual galactic distances D and recessional velocities V . But since we are taking ratios, this factor of 1.05 cancels out for any time: V/D = (1.05 U)/(1.05 C) = 1.05/1.05 U/C = 1 U/C = S/R = H 0 , or V = H 0 D . The arcs have the advantage of being able to be divided into smaller pie wedges whose arc lengths are easily seen. Suppose we imagine a galaxy at the 3 o’clock position at time 4 , only 30° around the curve from where we are at the 2 o’clock position. Its distance from us will be half of D(4) . We can also split the corresponding recessional velocity V(4) up into two parts, half associated with the 4 o’clock galaxy’s motion away from the 3 o’clock galaxy, and half associated with the 3 o’clock galaxy’s motion away from us at the 2 o’clock position. Looking at each half, we see that ( D(4) /2)/( V(4) /2) = D(4) / V(4) = H 0 yet again, so the Hubble–Lemaître law holds for galaxies at any distance from us. We are used to seeing the relationship between velocity and distance written as D = V T , so we see that the Hubble constant is the inverse of some time, H 0 = 1/T . T turns out to be an estimate of the age of the universe, good to about 60% accuracy.