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

TL;DR

This paper investigates how quantum corrections with a scale-dependent gravitational constant influence static isotropic solutions, revealing that only specific scaling exponents yield consistent vacuum solutions in four dimensions.

Contribution

It introduces non-local effective field equations incorporating quantum corrections and constrains the gravitational scaling exponent for consistent solutions.

Findings

Vacuum solutions restrict the gravitational scaling exponent to positive integers greater than one.

Only for  as the inverse of the scaling exponent do consistent solutions appear in four dimensions.

Quantum corrections impose severe restrictions on the form of solutions in scale-dependent gravity.

Abstract

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.