Constraints on Gravitational Scaling Dimensions from Non-Local Effective Field Equations

Quantum corrections to the classical field equations, induced by a scale dependent gravitational constant, are analyzed in the case of the static isotropic metric. The requirement of general covariance for the resulting non-local effective field equations puts severe restrictions on the nature of the solutions that can be obtained. In general the existence of vacuum solutions to the effective field equations restricts the value of the gravitational scaling exponent $\nu^{-1}$ to be a positive integer greater than one. We give further arguments suggesting that in fact only for $\nu^{-1}=3$ consistent solutions seem to exist in four dimensions.