Structure of conformal gravity in the presence of a scale breaking scalar field. (arXiv:2105.14679v3 [gr-qc] UPDATED)
<a href="http://arxiv.org/find/gr-qc/1/au:+Mannheim_P/0/1/0/all/0/1">Philip D. Mannheim</a>
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 of which the scalar field has a constant value and in the other, which is
due to Brihaye and Verbin, it has a radial dependence. 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. We show
that the analysis of Brihaye and Verbin is not actually general, but is
nonetheless 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.
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 of which the scalar field has a constant value and in the other, which is
due to Brihaye and Verbin, it has a radial dependence. 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. We show
that the analysis of Brihaye and Verbin is not actually general, but is
nonetheless 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.
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