7. The Mathematical Behaviour of the ZPE

7.1 Behaviour of Atomic Quantities With Time
7.2 Mathematical Description of Atomic Behavior
7.3 Conclusion

7.1 Behaviour of Atomic Quantities With Time

Finally, the behaviour pattern can be found for those constants linked with the ZPE as in (2A), (6A), (16A), (20A), in section 3, and (26A) in section 5.  Taking their inverse, and ignoring the ripple on m, then

 U1/U2 = h1/h2 = c2/c1 =  √(m1/m2) = t1/t2 = (1+z)        (52)

If we define each of these quantities in terms of their current value, so this equals unity, then lightspeed, c, will be in terms of c1which is c (now) =1. Equation (52) then becomes:

1/U ~ 1/h ~ c/1 ~ √(1/m) ~ 1/t ~ (1+z)                 (53)

We notice the redshift term there. But astronomers who want to treat the redshift as a velocity phenomena demand that (1 + z) obeys the relativistic Doppler formula namely [128]:

(1+z) = [1 + (v/c)] /  √[1-(v2/c2)]                     (54)

Now equation (54) distinctly relates redshifts, z, to velocities, v, and, as such, has always drawn some criticism from astronomers with alternative explanations for the redshift. For this reason, and also because of some doubt about distances resulting from uncertainties in the Hubble constant, Ho, a different form of this equation can be employed. This form has no velocities implied.  The horizontal axis values can be plotted as distances, x, from x = 0 near our galaxy, to x = 1, at the origin of the cosmos. The quantity x is then dimensionless, being the ratio of any given distance to the furthest distance in space. The standard redshift/distance equation then takes the form [129]:

 (1+z) = (1+x) / √ (1-x2)                             (55)

In this form, we have both z and x as dimensionless ratios. In this case, the observational data does not require velocities at all since a given distance x will automatically give a redshift z. A given redshift, z, will also result in a set distance, x, as it can be established that

x  =  z ( z + 2 ) / ( z2 + 2 z + 2 )              (56)

Equation (56) can be verified by inserting this expression for x back in (55). These results therefore demonstrate that the equation is free from data interpretation problems, since the distance, x, has intrinsically has nothing to do with velocity, only distance. Any deduction about a presumed relationship between v/c and x is then entirely dependent upon an individual’s preferences in redshift interpretations. But that is different from raw data.

        However, as already noted above, equation (55) begins to break down at redshifts greater than about 0.8.  The brightness of Type Ia supernovae in distant galaxies indicate that something is amiss with (55). This is usually overcome by those favouring the velocity interpretation by stating that dark energy is forcing the universe to expand at an ever increasing rate. But the equation describing the behavior of the cosmos on that approach becomes unwieldy with too many parameters. One therefore suspects that the expansion scenario is not the correct way to go, especially in light of the quantized redshift. But (55) is nonetheless a fair approximation to the redshift z that can be found at a given distance x. It is at this juncture that the results of Appendix 1 and reference [25] assume some importance. Both these references derive the equation that describes the inverse of ZPE behaviour with orbital time, T. That equation was generated by modeling the physical conditions relating to turbulence and recombination of Planck particle pairs. That equation has the following form:

1/U  ~  [1 + T]/  [√ (1 – T2)]            (57)

Orbital time, T, is a ratio with T = 1 at the origin of the cosmos and T = 0 near the present.

7.2 Mathematical Description of Atomic Behavior

Equation (57) delineates the behaviour over time of those atomic quantities associated with the ZPE. Therefore, linking (57) and (53) describes the behaviour of these 6 quantities,

1/U ~ 1/h ~ c ~ √(1/m) ~ 1/t ~ (1+z) ~ [(1+T) / √(1-T2)]         (58)

This is graphed in Figure 7. This equation, and the observational data and physics behind it, provides a logical framework for the theoretical proposals of a high initial value for lightspeed at the inception of the cosmos [51-53]. It also provides support for the approach of those who suggested a decline in c over the lifetime of the cosmos [25, 50, 54]. It may be helpful to remember that when the curve is examined on a smaller scale for z and m, a ripple will be apparent. In addition, superimposed on this smooth function for the ZPE, are the effects of the oscillation of the universe on the ZPE’s energy density. Thus, even after T = 0 near the present epoch, these quantities will follow an oscillation which had a change in direction in 1970.

Note that this equation also describes the behavior of the redshift with time in a way that has nothing to do with either the expansion of space-time or the motion of galaxies. Instead it is more like (55) in which distance is independent of velocities. Importantly, one of the terms in this derivation has a degree of liberty that allows fine tuning to data from z = 0.8 and greater. This derivation reveals that the denominator term in (57), namely [√ (1 – T2)] is where the trouble lies. Instead of an exact square-root, which, if written in terms of exponents, would read ½, experimental evidence from turbulence suggests a range from about 0.3 to something in excess of 0.66. This resolves the redshift/distance problem caused by Type Ia supernovae, without the necessity for a cosmological constant or dark energy. The years elapsed, te, on the atomic clock, or its equivalent, namely the distance in light years that light has travelled, is then given by substituting for T in the integral of (58) which is formulated as follows:

te + K[arcsin T - √(1-T2) + 1]                          (59)

The final term of unity in (59) is included since this gives te = 0 when T = 0. At the origin of the cosmos, when T = 1, the terms within the square brackets in (59) total 2.5708.  Since the universe is about 10 to 14 billion atomic years old, and light has traveled 10 to 14 billion light years, then the numerical value of K must approximate to K = 4 x 109.

7.3 Conclusion

The evidence presented here indicates there is a link between five physical quantities and the Zero Point Energy. These quantities are Planck’s constant, h, the speed of light, c, atomic masses, m, the rate of ticking of atomic clocks, t, which includes radiometric clocks, and the redshift, z. Any ZPE variation over time will result in a variation in these quantities as has been discussed by a number of researchers and referenced here. It has been demonstrated mathematically in the Setterfield and Dzimano paper [25], and affirmed here, that the ZPE should increase with time. Once the maximum value of the ZPE was attained, the small oscillations inherent in a static cosmos, first predicted by Narliker and Arp, would add a variable component to the ZPE.  This occurs because the ZPE energy per unit volume is slightly greater when the cosmos was at its minimum size, and slightly less at the maximum. Data from the five main physical quantities and some other atomic constants support this. They indicate that the ZPE energy density was at a maximum around 1970, followed by a slow decline. This supports Narliker and Arp’s concepts. This approach with the ZPE permits an explanation of the trends in five different sets of data over a long period which many paradigms struggle to account for, or treat dismissively. It provides a physical reason behind Dirac’s idea of variations in the constants, and opens up a potentially fruitful line of enquiry.


[25]  B. Setterfield, D. Dzimano, ‘The Redshift and the Zero Point Energy’ Journal of Theoretics, (Dec. 2003).  Available online at http://www.journaloftheoretics.com/Links/Papers/Setter.pdf

[50]  V. S. Troitskii, Astrophys. & Space Science 139 (1987) 389.
[51]  J. Moffat, Int. J. Mod. Phys. D 2 (1993) 351.
[52]  J. Moffat, Int. J. Mod. Phys. D 23 (1993) 411.
[53]  A. Albrecht and J. Magueijo,Phys. Rev. D 59:4 (1999) 3515 .
[54]  J. D. Barrow, Phys. Rev. D 59:4 (1999) 043515-1.

[128] B. Lovell, T. O. Paine and P. Moore, “The Readers Digest Atlas of the Universe,” Mitchell Beazley Ltd., (1974) 214.
[129] J. Audouze and G. Israel, Cambridge Atlas of Astronomy, Cambridge/Newnes, (1985) 356, 382.

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