Structure of conformal gravity in the presence of a scale breaking scalar field

Abstract

We revisit the structure of conformal gravity in the presence of a c-number, conformally coupled, long range, macroscopic scalar field. And in the static, spherically symmetric case discuss two classes of exact exterior solutions. In one solution the scalar field has a constant value and in the other solution, which is due to Brihaye and Verbin, it has a dependence on the radial coordinate, with the two exterior solutions being relatable by a conformal transformation. In light of these two solutions Horne and then Hobson and Lasenby raised the concern that the fitting of conformal gravity to galactic rotation curves had been misapplied and thus called the successful fitting of the conformal theory into question. In this paper we show that the analysis of Brihaye and Verbin needs to reappraised, with their reported result not being as general as they had indicated, but nonetheless being valid in the particular case that they studied. For the analyses of Horne and of Hobson and Lasenby we show that this macroscopic scalar field is not related to the mass generation that is required in a conformal theory. Rather, not just in conformal gravity, but also in standard Einstein gravity, the presence of such a long range scalar field would lead to test particles whose masses would be of the same order as the masses of the galaxies around which they orbit. Since particle masses are not at all of this form, such macroscopic fields cannot be responsible for mass generation; and the existence of any such mass-generating scalar fields can be excluded, consistent with there actually being no known massless scalar particles in nature. Instead, mass generation has to be due to c-number vacuum expectation values of q-number fields. Such expectation values are microscopic not macroscopic and only vary within particle interiors, giving particles an extended, baglike structure, as needed for localization in a conformal theory. And being purely internal they have no effect on galactic orbits, to thus leave the good conformal gravity fitting to galactic rotation curves intact.