Summary: A geometrical analysis of the Hubble constant showed that it should not be constant but data from the most distant supernovas show that it is constant.. This is only possible if time is expanding. The Hubble constant is an acceleration constant conforming to the criteria of a fundamental force of nature. It corresponds to a previously suggested force that arises from mass. This force is responsible for the expansion of the universe as well as the acceleration of mass towards a gravitational source. Time is inversely related to mass. As time slows so mass reverts to energy. This fuels solar and all stellar radiation as well as gravity. Hydrogen fusion is endothermic and plays a key role in maintaining the stability of all stars. Helium is violently exothermic at very high temperatures destroying most of the mass in  supernovas. Time slowing releases energy at a constant rate and through its effect on the mass of the moon leads to a reasonably precise prediction of the height of the tides in mid Pacific Ocean. The time/mass effect causes excess heat for all the planets. It is responsible for the violent storms in Jupiter’s equatorial belt. It is probably responsible for Saturn’s ring system through destroying small icy moons. The time slowing effect on the dust surrounding the solar system explains of the characteristics of the Background Microwave Radiation. The universe’s system of time is an exponential oneso resolving a number of paradoxes relating to the universe’s age. The Big Bang theory does not take into account the role of gravity, and a replacement is suggested,

1.      Theoretical considerations

The physics underlying the Hubble constant is obscure. Dimensionally the Hubble constant, the ratio of velocity to distance, is describing a frequency, which is inappropriate. If velocity increases with distance this describes acceleration. The denominator of the Hubble constant should therefore be time. The only fixed relationship between distance and time is c, the velocity of light. It follows that the denominator in the Hubble constant should be the time for light to travel the distance quoted in the ratio.

There is disagreement as to the exact value of the constant.  If the age of the universe is 13.7 billion years,  theHubble  constant, Ho, should be 47.6 km.s-1Mparsec-1 (equations 1 & 2) giving an acceleration constant of 4.62 x 10-10 m.s-2.   If Ho is 51.8 km.s-1Mpc-1 (see later) the acceleration constant is 5.0 x 10-10 m.s-2 and the age of the universe is 12.58 billion years.At this acceleration for this length of time the velocity of the furthermost galaxy should be 0.66 of the velocity of light and the distance travelled by that galaxy should be 4.16 billion light years or 1273 Mpc.  Yet velocities from Supernovas have been observed that are much greater than 0.66c and distances have been deduced   that are much greater than 1273 Mpc (Tables 1 and 2).If 73.5 km.s-1Mpc-1 is the Hubble constant (NASA 2009) the acceleration is 7.14 x 10-10 m.s-2 and the age of the universe is only 9.13 billion years. In this time the furthermost galaxy should  have travelled  only 2.19 billion light years.  Clearly there must be another factor to be taken into account to correct these anomalies.

There is another problem. A geometrical analysis based on the Euclidean geometry of similar figures (Figure 1) shows that if the visible universe is expanding at a constant acceleration then the ratio of velocity to distance cannot be constant over time. The geometric analysis (Appendix 1) shows that the equation defining the Hubble value should be

Equation 1.                             Ho =  v/D = 2 x 100 x 3.26 x (Age –DT)-1

where v is the velocity in km/sec of the galaxy being examined, D is its distance in Mega parsecs, Age is the age of the universe in earth time (that is the system of time used on earth), DT is the time for light to travel the distance D, with both Age and DT being in units of billions of years. 3.26 is the conversion factor for parsecs to light years.

Thus the only way for Ho to be constant would be to multiply the value by the ratio of ages, (Age-DT)/Age. That is if Ho doubled then the ratio of ages would equal 0.5, i.e. the period of the second is halved and so on.The time corrected equation 1 is therefore

Equation 2                              Ho =  v/D = 2 x 100 x 3.26/Age

The key conclusion is if the Hubble constant is truly constant, that is it is constant throughout time and space, then time must be expanding and has been expanding since the beginning of time. Moreover the rate of time expansion is 1/Age where Age is the age of the universe incosmological seconds.  Equation 2 is testable.


2.      A new fundamental constant

Sandage (1993) found that the very outer stars of galaxies had accelerations of slightly less than 50 km.s-1Mpc-1, or slightly less   than 5 x 10-10 m.s-2. That is they were comparable to the Hubble constant. The two Pioneer probes also have accelerations of the same order of magnitude, Anderson (2002).Three bodies, of vastly different masses, galaxies, the outerstars, and the Pioneer probes, having more or less than same acceleration points to a system of acceleration which is independent of mass.  In this it is akin to gravity and suggests that the Hubble constant is describing a fundamental constant of nature. This independence of mass suggests that, like gravity, the force causing the acceleration must arise from the components that make up mass.  The theoretical nuclear physicist, the late Burkhard Heim (Leitz 2006, Hauser 2009) postulated that in addition to the four fundamental forces of nature, the weak and strong intra nuclear forces, the electromagnetic force and gravity there must exist two other forces. That is there is a family of three fundamental forces with gravity-like properties in that they affect the movement of mass. A Hubble constant fulfils Heim’s suggestion as being one of these three forces (he called them interactions, Hauser, 2009). The others are standard gravity as described by Newton, and a gravito-electro-magnetic force. A force with a constant value of between 4.62 and  7.14 x 10-10 m.s-2 that is independent of mass would be such a fundamental force if it can be shown to be constant in all time frames. This may be compared with the Gravitational constant G (6.67 x 10-11).

 If the radius of a proton is 10-15 m and the mass of a proton is 1.6 x 10-27 kg then the gravitational force at the surface of the proton is ~250 times that of the acceleration force also acting on that surface. It follows that if the radius of a spiral galaxy is proportional to its mass then, at the outer 1/15th of the radius (250-0.5), the gravitational force acting on any star will equal the acceleration force resulting in a velocity that is twice that predicted from the gravitational strength of that galaxy. Any star in this outer 1/15th fraction of the radius will have velocity even greater. All the observations that have been attributed to “dark matter” can be explained by this Heim force. The consequential distribution of velocities also explains the hitherto inexplicable shape of the spiral arms of our Galaxy with the edges of the spiral arms moving faster than expected.

Proof of the existence of this fundamental force, one that is constant in all time frames, and its rider, that time is expanding, will be by showing that even at the furthest limit of our observations the Hubble value is constant and not rising according to the prediction of the geometric analysis given in equation 1. Furthermore, proof that time is expanding, will be the demonstration of the existence of the consequences of that time expansion (see later).

3.      The observational data

The data of 60 supernovas were taken (Tables 1 and 2). This data consists of two populations of Type 1A supernovas, Hamuy et al, (1996). Perlmutter et al, (1999). The first population, the Calan/Tolelo set, consists of 18 supernovas that had estimated distances ranging from 64 to 654 Mpc. The second set, the Supernova Cosmology Project, (SCP), consisted of 42 Type 1A supernovas whose  distances ranged from 750 to 4500 Mpc.

The assumption behind Type 1A supernovas is that they emit a set amount of light that decays in a particular fashion and so can act as standard candles.  It follows that with uniform expansion of the universe all such supernovas with the same velocity should have the same magnitude as they will be at the same distance from their origin. But supernovas can occur anywhere in a galaxy including behind dust clouds which can obscure some of the light.The results, tables 1 and 2, show that this is the case and variations up to 0.5 of Magnitude have occurred between galaxies of similar velocities. 

The equation used to derive the distances is a refinement of that developed by Kayser et al (1997).It includes subtracting the values for Galactic extinction in the observed B band,  given in the original data, and for  relativistic time dilatation (Appendix 2)

 Equation 3                    Log Distance (Mpc) =  (Mag-(-19.7) – 25)5-1

Mag is the B-band effective peak magnitude, the -19.7 refers to the magnitude of a reference supernova type 1a at a distance of 10 parsecs, whilst the 25 refers to the ratio of Mega parsecs to ten parsecs adjusted for the conversion to log base 2.5 from log base 10.The 5 refers to converting back from log base 2.5 to log base 10 and for the inverse square law.

For the faster supernovas there is a second time frame superimposed on that due to universal time expansion. This is relativity induced time dilatation. The effective B band magnitude is similar to that described (Perlmutter et al. 1999) but must include an allowance for the relativistic effects.

Equation 4.    Effective Mag = observed Mag - Ax – K- a(s-1)

The red shift is also contaminated by the relativistic time dilatation so that the velocity will appear to be slower than it should be. The equation for calculating the Hubble constant (see appendix then becomes

Equation 5       Ho =  v x(1/(1-vc2))1.5 x ((antilog(Effective Mag + 5.3)/5)-1

The relativity factor is the standard time dilatation derived from special relativity theory. The velocity vc  is the velocity expressed as a fraction of the velocity of light.



 In the Calan Toledo data the velocities of the supernovas were sufficiently low that the relativity effect was insignificant. This was not the case for the the SCP data. Equally the brightness, or rate of photon emission and so magnitude, was affected from the same cause. The supernovas were dimmer than they should have been for their distances.

The results (tables 1 and 2) gave a mean Hubble value of 51.8 +/- 1.35 km.s-1Mpc-1 for the Calan/Tolelo data. For the SCP data the mean for the uncorrected Hubble value was 32.2 +/- 1.0 km.s-1Mpc-1. When the relativity time dilatation factor shown in Equation 5 was applied it was found that the mean Hubble value for the SCP data was 47.9 +/- 1.2 km.s-1Mpc-1. Such a value predicts that the age of the universe is 13.61 +/- 0.35 billion years compared with the generally acceptedvalue of 13.7 billion years. Allowing for the various uncertainties in the calculation of each of equation’s components it was concluded that the Hubble constant was indeed constantover the whole range of distances covered by the data with a value around 48-52 km.s-1Mpc-1.

 Gribbin .(1999) in his book describes  an analysis of 1388 galaxies at distances of up to 100 M.parsec and reported Ho at 52+/-6 km.s-1Mpc-1, but there is a plethora of reports to be found in both in print (Gribbin, 1999) and on the Internet giving different values.  The more recent reports show two populations of values. One population, based on the brightness from supernovas or Cepheid stars centres around 51-5 km.s-1Mpc-1. The other based on a variety of physics concepts, but relying on the anisometry of the microwave radiation, centres on a Hubble value of 73.5 km.s-1Mpc-1 (NASA.2009)


4.      Reconciling the different Hubble values.

If the Hubble value really is 73.5 km.s-1Mpc-1 then the lower Hubble values   (derived from the SCP optical data) implies that some of the light is being intercepted, falsely increasing the magnitude by a factor of 1.44.From the inverse  square law increasing the distance by x 1.44 reduces the brightness by 58%.More than half the light has apparently been intercepted by dust.    This interception would apply to all astronomical light sources outside our Galaxy and possibly outside the Solar system.That is their distances should be 2/3rds of their current calculated values.

For the M31 or Andromeda Galaxy this means that its distance is only 1.6 x 106 light years away from the Solar system. That galaxy has a diameter of 0.228 x 106 light years. It follows that M31 is only about 6 galaxy diameters from the edge of our Galaxy.  Photographs of colliding spiral galaxies show that at a separation distance of two galaxy diameters there is massive distortion of the structure of the galaxies due to mutual gravitational attraction.  At six galaxy diameters distance there should be some signs of this distortion but M31 show no sign of this. The inference then is that light is not being intercepted to anything like the extent suggested. But this loss of more than half the light from light absorption rests on the assumption that the Ho is 73.5 km.s-1.Mpc-1.This was derived from measurements of the strength of the incoming microwaves. The hidden assumption is that the microwaves originated solely from some kind of Big Bang. One particular anomaly is that this Hubble value predicts that the age of the universe is 8.87 billion years (Equation 2) but the 73.5 km.s-1Mpc-1  calculation relied on the age of the universe as being 13.7 billion years. Another assumption is if there is a dust cloud surrounding the solar system it does not absorb microwaves. But if there is a dust cloud intercepting more than 50% of the incoming light then it would be generating low energy microwaves as a direct consequence of the effects of time slowing on mass (see later). Time slowing must occur if the Hubble value if truly constant.It follows that the NASA’s WMap data may have been contaminated by locally produced microwaves with the anisometry (on which NASA’s calculation is based) caused by irregular clouds of dust.

 For comparison if 5% of the incoming light was absorbed by dust the effect would be a change in the magnitude of 0.055, which is well within the error of measurement.  A measure of uncertainty still exists as to the precise value of the Hubble constant.


5.       Theoretical conclusion

What is very clear is that the Hubble value does not increase with distance as it should according to the geometrical analysis, summarised in Equation 1.   The conclusion must be that the Hubble constant is constant throughout the universe and this only possible if time was and is expanding.That is the Hubble constant is indeed constant in all time frames and so fulfils the criteria necessaryto be considered as an expression of a fifth fundamentalforce of nature. The observations which led to the three hypotheses, dark matter, dark energy and inflation, are fully accounted for by this fundamental force and the expansion of time.  It also follows that the pace of time initially was very fast, by a factor of 106 and has slowed to become the present pace of time, and is continuing to slow .The expansion of the universe is slowing down.

6.                  Time and Mass

 There is a major additional factor. Time and mass are inversely related.This can be seen in various physics equations. Thus the pendulum equation shows that the period of the pendulum cycle is inversely related to the square root of the gravitational force experienced by the pendulum.That force in turn is proportional to the mathematical product of two masses. It follows that if time expands mass must be reduced.  Equally if time was quicker mass would be greater. 

A similar effect is seen in the energy equation of quantum mechanics.

Equation 6.                      Energy = h x frequency = mc2 = h x n/time

 where h is Planck’s constant, and n is the number of cycles of a wave in one second.

 Another example is in special relativity theory. The special relativity equation relating time to velocity, equation 7, means that at a velocity of 0.8c time would have expanded by another 0.66. Similarly the special relativity equation relating mass to velocity, equation 7, shows that at the same velocity mass would have decreased by a factor 0.6.

The two equations of special relativity are

Equation 7.                            Texpanded = tnormal(1/(1-vc2))0.5

Equation 8.                            Mreduced = mnormal(1- vc2)0.5

where vc is the velocity expressed as a fraction of the velocity of light. That is one equation is the inverse of the other.The change in mass with velocity in equation 8  is a direct effect of expanding time and its effect on mass.

7.                  The Age of Universe in cosmological time

The universe would appear to operate its own system of time, hereinafter called cosmological time. When time first started its pace was very fast but it has been slowing exponentially ever since converging to the pace of  earth time, that is the rate of time expansion has now become so small that within the period of human existence the pace of time seems to be constant leading to the assumption that the period of the second has always been unchanging.There is no evidence that justifies that assumption.The calculation of the age of the universe in earth time is based on that assumption. 

Fundamental  to calculating the cosmological age of the universe is the doubling time.This is the time taken for the period of the second to double. Given the exponential nature of time expansion this period is the same for all doublings.  It can be calculated from the expression Age/20.5 where Age is the Age derived from the Hubble constant, equation 2. The age of the universe is then the product of the doubling time multiplied by the number of doublings since time started.

Relativity theory shows that when gravitational acceleration equals the speed of light, time, the period of the second, is infinite. Time is at a standstill. That is the start of cosmological time occurred when a large primeval black hole began to disrupt. This is in contrast with the Big Bang model which posits that time started when a singularity burst and time has continued to accumulate at a uniform rate to reach its present earth time age.

The universe has been calculated as having some 1080 nucleons, (Hawking, 1988) giving a mass of 3.35 x 1053 kg. At the beginning this mass, or its equivalent, would have been concentrated into a black hole. The equation which defines the radius of a black hole is

  Equation 10                          c =  GMr-2

If the mass was 3.35 x 1053 kg the radius would have been 28.88 light years.But time was much faster then.Whenever time was twice as fast mass would have doubled.   The radius of the primeval black hole would then have been

 Equation 11                 r  =(GM2nc-1)0.5

where n is the number of occasions that time was twice as fast as its successor period. The value of n can be found from  how often(2n)  those 28.88 x 2n/2 years doubled  to become the age of the universe in earth time units.  This can be found from

Equation 12                                          28.88 x 2n/2 x 2n = earth time age

The cosmological age of the universeis then n times the doubling period.

Table 3 shows the results with three different values for Ho.