Conformal invariance, microscopic physics, and the nature of gravitation

The question of whether the gravitational "constant" can vary in spacetime has been among the most vexing in physics. The thrust of this paper is that the issue may be fully resolved if one accepts the principle (first proposed by Weyl and lucidly discussed by Hoyle and Narlikar) that all the fundamental equations of physics should be invariant under local (spacetime-dependent) transformations of units (principle of conformal invariance). Theoretical arguments in favor of the principle are discussed. We then show that the presently accepted dynamics for the fundamental particles and their electromagnetic, weak, and strong interactions indeed satisfy the principle. Their conformal invariance is due not least to the indispensable transformation properties of rest masses. Thus in arbitrary units each type of rest mass is a spacetime field. The principle of conformal invariance then demands conformal invariance of the dynamics of each such "mass field." If all rest-mass ratios are strictly constant there is only one mass field. Its dynamics automatically induces dynamics for gravitation. In units defined by particle masses the gravitational action is manifestly that of general relativity, a fact discovered in different guises and independently by several workers. This would seem to forbid the construction of a conformally invariant theory of gravitation with "varying gravitational constant" $G$. Such theories have been proposed by Dirac, and later by Canuto and coworkers, who have argued that, a priori, gravitational (Einstein) units are distinct from those defined by matter (atomic units). We find that to implement such distinction while simultaneously avoiding undetermined elements in the theory, one must introduce conformally invariant dynamics for gravitation and for the mass field separately. We construct this theory; it is a "varying-$G$ theory." We then show that it is definitely ruled out by the solar-system gravitational experiments. We conclude that the principle of conformal invariance requires that gravitation be described by general relativity, and that the dimensionless gravitational constant $\gamma$ be strictly constant. We also consider the possibility that gravitation, or the mass field, explicitly break conformal invariance. The corresponding theory, the theory of variable rest masses (VMT), was developed earlier from a different viewpoint. Although it predicts variability of $\gamma$, we point out that for a vast majority of cosmological models, the temporal variability of $\gamma$ is well below experimental sensitivities.