Deriving Hubble's Constant, Etc.

Here is one way to derive Hubble's constant. First, let's define the variables:

We start with an arbitrary total distance (D) which is the product of a small coefficient (a) and distance (x). As x grows, so does D, and vice versa. At equation 2 below, we find the time derivative of D (which equals V).

At equation 3 above, we multiply the left side by x/x. At equations 4 through 7 we do some algebraic slight of hand to derive Hubble's constant (H). Equation 8 shows the approximate numerical value of Hubble's constant. Its units are s^-1--the reciprocal of time.

Next, let's see what its relationship is to proper or relativistic time. To accomplish this we use 1/H in lieu of time (t).

Equation 14 shows how Hubble's constant stays constant. If proper time t' changes, so does variable u. The two are proportionate. Equation 15 below is the velocity the universe expands at distance D. When D increases, so does t'. You would think this amounts to a constant velocity for any distance D, But variable u also increases and offsets t'. As a result, velocity increases as D increases.

Equations 16 through 20 express distance D in terms of time (ctD). When examining the relativistic consequences of Hubble's constant, it is important to recognize that the time that makes up distance (ct) is a function of D only. Whereas t' is a function of D, mass and energy. High mass, for example, will contract time t to time t', and will contract length from ct to ct'. But distance D equals ct or act'; i.e., it can be measured out even if the units shrink due to relativity.