Appendix 

The derivation of two equations

A.    The Hubble constant anomaly.

This derivation of the Hubble constant equation rests on three assumptions. The first is that the universe is a regularly expanding sphere. The second is that light travels in straight lines. The third is that universe follows Euclidean geometry.

 

Let the velocity of an object at the edge of a universe be I. If the radius of the universe is R then the mean increase in the radius over the previous second is I/2.  From Euclidean geometry of similar figures the following ratios apply (Figure 1)

Equation A1.                     I/2R  = i/2r  =  v/2D

Where i is the velocity of an object that is r distance along that radius and v is the velocity of a galaxy that is D distance from earth. The mean volume of the universe during that next second is

Equation A2                 Volume = (4/3) (R + I/2)3

Equation A3                      I/2    =  vR/2D

Substituting in Equation A2 and expanding the equation, but discarding the third and fourth components of the expansion as being too small to be significant when D is very large, results in

Equation A4               Volume =  (4/3) (R + Rv/2D)3

                                                                 =  (4/3) (R3 + 3R2Rv/2D)

For a previous time, t,  the volume was

Equation A5                 Volume =  (4/3) R3(1-t3v/2D)

When the volume was insignificantly small compared with the present size of the universe, that is effectively zero, at the beginning of time

Equation A6                           0 = (4/3) R3 (1-t3v/2D)

                                                   = 1-t3v/2D

That is

Equation A7                               t = 2D/3v  = 2/3 x Ho

Equation A8                        3v/D = 2/t

But t is the age of the universe when the light from a distant galaxy that we now receive, was emitted.  It has taken Dt billions of years  for that light  to reach us. (See figure A1.) That is

Equation   A9                       3v/D = 2/(Age -Dt )   

When v is in fractions of the velocity of light and D is in billions of light years converting  this to km.s-1Mpc-1 results in

 Equation A10                          Ho = 3.26 x 200/(Age-Dt)

It follows that the Hubble value should increase with distance, albeit that when that distance is less than a thousand Mega parsecs the effect is so small as not to be noticeable.

 

 

B. The Relativity correction

The standard equation defining the effective magnitude has four components, Equation 4. Each of these four components is time sensitive  and each needs correcting by use of Special relativity’s basic equation, (Equation 7). The correction needs to be in the log format using the log base applicable to Magnitudes (i.e. log x 2.5). The result is

Equation B1.        Effective Mag = observed Mag - Ax – K- a(s-1)-10(log(1/(1-vc2 ))0.5

Where vc is the velocity expressed as a fraction of the velocity of light. When this is applied to determining the logarithm of distance ,(Equation 3) and simplifying this becomes

Equation B2.           Log distance (Mpc) = (Effective Mag -  5.3)/5  + log(1/(1- vc2)

For the determination of the Hubble constant v/D where v is in kilometres per second, the velocity also must be corrected for  relativity’s distortion of time i.e.

 Equation  B3               v = observed v x  (1/(1- vc2))0.5

This simplifies further to become Equation 5