Renormalization group running of Newton’s constant $G$: The static isotropic case

Corrections are computed to the classical static isotropic solution of general relativity, arising from nonperturbative quantum gravity effects. A slow rise of the effective gravitational coupling with distance is shown to involve a genuinely nonperturbative scale, closely connected with the gravitational vacuum condensate, and thereby related to the observed effective cosmological constant. We argue that in contrast to phenomenological approaches, the underlying functional integral formulation of the theory severely constrains possible scenarios for the renormalization group evolution of couplings. The general analysis is extended here to a set of covariant nonlocal effective field equations, intended to incorporate the full scale dependence of $G$, and examined in the case of the static isotropic metric. We find that the existence of vacuum solutions to the effective field equations in general severely restricts the possible values of the scaling exponent $\nu$.

Figure 1

Schematic scale dependence of the gravitational coupling $G(r)$ from Eq. (3.8), here for $\nu =1/2$. The gravitational coupling rises initially like a power of $r$, and then approaches the asymptotic value $G_{\infty }=(1+a_0)G$ for large $r$. The behavior for other values of $\nu >1/3$ is similar.