Scaling Structure of Geometric Backreaction from Localized Scalar Sources in a Controlled Weak-Field Gravitational Expansion

We derive the complete leading-order geometric response of a static, spherically symmetric spacetime to a localized, minimally coupled massless scalar source within a rigorously controlled weak-field expansion of general relativity. Working at first order in the small coupling ϵ = 16πGs2 0 (dimensionless in natural units), we solve both independent Einstein equations and obtain closed form analytic expressions for the shape function b(r) and the redshift function Φ(r) in terms of the lower incomplete gamma function γ...