APPENDIX 3: Missing Mass and the ZPE
An astronomical anomaly appeared in the mid 1970’s, and its prominence continues. In 1983, Vera Rubin noted the anomaly is that the rotational velocity of many galaxies is constant out to the edges of those galaxies . The rotational velocity, V, of stars appeared the same at all distances, r, from the galactic centres, except for the galactic nuclei themselves. The graph of rotational velocity against distance from the galactic centres was thus flat, and these flat rotation curves precipitated the missing mass scenario. Milgrom suggested our theories of gravity were incomplete and advanced a Modified Newtonian Dynamics (MOND) approach. It has produced good agreement with data since its introduction in 1983 . But the lack of a physical mechanism for this theory has been a major drawback to its general acceptance, so an alternative approach was pursued. This claimed that rotation curves could be accounted for if galaxy mass, m, increased linearly with distance, r, from their centers. But the mass distribution indicated by luminous matter gave a vastly different arrangement of matter. It was concluded there must be ‘dark’ matter existing within the extended reaches of galaxies in order for the rotation curves to match a suitable matter distribution. However, the search for such ‘dark’ or ‘missing’ matter since the mid 1980’s has been largely unsuccessful.
Now an approach to gravitation using the ZPE holds promise since it shows there is an additional term in the Newtonian equation for gravitational acceleration. This term’s effect increases with distance from the centre of a galaxy and may account for the anomalous rotation velocities. It may thus obviate the need for both the “missing mass” and MOND. To this end, we note that reference  shows that, where gravitating bodies are present, there is a local increase in the strength of the ZPE. This is due to the secondary fields of oscillating partons. Reference  demonstrated that these secondary fields change the values of some physical quantities by a factor A such that, if the present ZPE energy density is U0, and the greater energy density of the ZPE in the vicinity of massive objects is U, then
In gravitational fields we find 0 < A ≤1. If this approach is adopted, we have already seen that the parton’s rest-mass may be expressed by equation (14) which is reproduced here such that
The additional factor now emerges. In the 2003 paper, Setterfield showed that the results of General Relativity were reproduced using the ZPE approach provided one important condition was maintained . There, as here in (7A), it was noted that the quantity e2/ε is constant throughout the universe with cosmological ZPE changes. But, in the presence of large gravitational fields, the magnitude of e2/ε is required to increase locally so that
where e02/ε0 is the value of e2/ε in a vacuum away from significant gravitational potential, and A is defined in (91). The reason for the increase in e2/ε is due to the fact that the self energy of the system changes in response to changes in the local vacuum polarizability. This is analogous to the change in the stored energy of a charged air capacitor during transport to a region of different dielectric constant . This gives a new formulation not mentioned in the 2003 paper. It means that, at any given instant, when the influence of gravitational potentials on the ZPE are taken into account, mass m behaves uniquely in a gravitational potential so that (92) becomes
where mois the value of m away from gravitating bodies. Thus, (94) shows the increase in ZPE strength due to secondary fields of oscillating partons marginally increases their mass. For the reasons explained above, this is the SED counterpart to the gravitational self-interaction of masses used in Relativity as a result of Dicke’s analysis . Therefore, substituting (94) into (19) we obtain the result for gravitational fields that
This leads to a potential resolution of the anomaly since, for an orbiting object, the quantity Gm/r2 is equal to the gravitational acceleration, a . Therefore, from (95) we can write
Now the 2003 paper pointed out that Eddington’s factor
where Eddington’s quantity μ is given by
where k is a conversion parameter. So Eddington’s factor 2 μ/r is equivalent to a gravitational potential term 2kGm/r. In this case, G is again the Newtonian gravitational constant, and m as used by Eddington is the same as the quantity mohere. Therefore equation (97) becomes
Since 2kGm0/r is small, we use the algebraic approximation in equation (99) so that it reads
Substituting the results of (100) in (96), we find the gravitational acceleration becomes
So the additional factor, 1/A2, that appears with the ZPE approach, results in an additional gravitational acceleration term, 2kao2r, which increases in a manner proportional to distance, r, from the centre of any galaxy. Thus, the anomalous rotation curves appear to be explicable. This conclusion is reinforced by a considering galactic rotational velocities. The galaxy rotation data require that the rotation velocity be independent of the distance from the centre of the galaxy. Now the standard equation relating orbital velocity, V, to the distance, r, is
where Q is some constant in line with (19). Now from (96) we have
Therefore, by substituting (103) in (102), we obtain
By cancelling the a0 term in numerator and denominator in (104) we find that
As a consequence we can write
The rotational velocity is therefore independent of r. Vera Rubin’s comment that the rotational velocity, V, of galaxies is constant out to the limits of those galaxies is therefore verified by (106). It is true that V for individual galaxies depends on the mass distribution in that galaxy. But the additional acceleration term is itself dependent upon mass distribution. So there may not be any “missing mass” at all, which may be why the search for it has been frustrating. Another problem is also resolved. If every star in a galaxy rotates at the same speed, the formation of spiral arms is a natural result that avoids complicated mechanisms.
 B. Setterfield, ‘General Relativity & the Zero Point Energy,’ Journal of Theoretics, October, 2003 available (March 7, 2006) online at http://www.journaloftheoretics.com/Links/Papers/BS-GR.pdf.
 V. Rubin, Science, 220:4604 (1983) 1339.