Scale-covariant theory of gravitation and astrophysical applications

Abstract

By associating the mathematical operation of scale transformation with the physics of using different dynamical systems to measure space-time distances, we formulate a scale-covariant theory of gravitation. Corresponding to each dynamical system of units is a gauge condition which determines the otherwise arbitrary gauge function. For gravitational units, the gauge condition is chosen so that the standard Einstein equations are recovered. Assuming the atomic units, derivable from atomic dynamics, to be distinct from the gravitational units, a different gauge condition must be imposed. It is suggested that Dirac's large-number hypothesis be used for the determination of this condition so that gravitational phenomena can be described in atomic units. The result allows a natural interpretation of the possible variation of the gravitational constant without compromising the validity of general relativity. A geometrical interpretation of the scale-covariant theory is possible if the covariant tensors in Riemannian space are replaced by cocovariant cotensors in an integrable Weyl space. A scale-invariant action principle is constructed from the metrical potentials of the integrable Weyl space. Application of the dynamical equations in atomic units to cosmology yields a family of homogeneous solutions characterized by $R\sim t$ for large cosmological times. Equations of motion in atomic units are solved for spherically symmetric gravitational fields. Expressions for perihelion shift and light deflection are derived. They do not differ from the predictions of general relativity except for secular variations, having the age of the universe as a time scale. Similar variations of periods and radii for planetary orbits are also derived. The generalized hydrodynamic equations derived for atomic units are studied. It is found that the stellar structure equations are formally unchanged, except that $G$ and $M$ can now be functions of the cosmological time. This in turn would imply secular variations of the stellar luminosities. The effects of these results on the past climatology of the earth and other geological effects are discussed. None of the consequences of the theory investigated so far is found to be in disagreement with observations.