We analyze the geometric response of spacetime to localized effective sources within a controlled leading-order expansion of general relativity. An exact analytic expression for the induced geometry is derived in terms of incomplete gamma functions, ensuring coefficient-level correctness. Beyond the explicit solution, we identify a universal scaling structure in which the geometric response factorizes into an amplitude parameter and a dimensionless profile. This factorization is independent of the specific functional form of the localized source and reflects a general feature of the underlying gravitational response. The asymptotic behavior defines an effective gravitational mass, providing a consistent physical interpretation of the backreaction.
DUC TAI NGUYEN (Sun,) studied this question.