Varying-G gravity: physical properties, asymptotic regimes, and Green’s functions, an event horizon, the vacuum energy density, and the external pressure that modifies Jeans stability

We study varying-G gravity and we add the necessary proofs (general force law, asymptotic forms, and Green’s functions, vacuum and external pressures, linearization of perturbations leading to a new Jeans stability criterion, and a physical origin) to elevate this novel idea to the status of a classical theory. The theory we lay out is not merely a correction to Newtonian gravity, it is a brand-new theory of gravity that encompasses the Newtonian framework and weak-field Weyl gravity in the limit of high accelerations, as well as Modified Newtonian Dynamics (MOND) in the opposite limit. In varying-G gravity, the source of the potential of a spherical mass distribution M(x) is σ(dG/dx) + (G/x2)(dM/dx), where x is the dimensionless radial coordinate and σ(x) = M(x)/x2 is the surface density away from the center x = 0. We calculate the potential $\Phi (x)=\int {G(x)\, \sigma (x)\, dx}$ from Poisson’s equation and the radial acceleration $a(x) = G(x)\, \sigma (x)$. Furthermore, a non-linear scaling transformation of the radial coordinate $x\in (0, \infty)\longmapsto \xi \in (0, 1)$ with scale factor ξ/x ∝ 1/G produces a finite space, in which the spherical surface ξ = 1 is an event horizon. In this classical context, it is the coupling of σ(x) to the gradient dG/dx in the above source that modifies the dynamics at all astrophysical scales, including empty space (where dG/dx ≠ 0). In vacuum, the source σ(dG/dx) supports an energy density distribution that supplies a repelling pressure gradient outside of discrete isolated massive systems. Surprisingly, the same source becomes attractive in linearized perturbations, and its linear pressure gradient opposes the kinetic terms in the Jeans stability criterion.