The Gorkavyi–Kutschera Anti-Gravity Effect: Weak-Field Verification, Strong-Field Impossibility, and Irrelevance for Wormhole Stabilization — with a Note on the Alcubierre Warp Drive

The Gorkavyi–Kutschera metric, derived from a retarded potential with a time-dependent mass M (u) = M (t - r/c), produces a repulsive (anti-gravity) radial acceleration aᵣ = +α G M / (c r) in the weak-field, slow-motion limit of linearized general relativity. This effect has recently attracted attention as a possible mechanism for stabilizing traversable wormholes without exotic matter, as well as for powering Alcubierre warp drives. We present a comprehensive analysis. First, we confirm the original formula via explicit calculation in linearized GR, a Python/SymPy symbolic verification, and a Lean 4 formal proof. Second, we prove that the metric ansatz cannot be extended to strong fields: the full nonlinear vacuum Einstein equations force dM/du = 0 and then M = 0, so the anti-gravity term is a purely linearized artifact with no exact strong-field counterpart. Third, we show that for a macroscopic (r0 = 1 m) Hamada wormhole, the ratio aG/aN is either astronomically small (Hawking evaporation) or leads to instantaneous evaporation (quantum-gravity scale mass loss) ; a no-go argument demonstrates that the conditions aG/aN >= 1 and tau >= r0/c are mutually exclusive. Fourth, we analyse the Alcubierre warp drive and prove that the Gorkavyi term cannot replace exotic matter: the required energy density is parametrically larger by factors αR/c and GM/ (c² R), and the null energy condition violation is a geometric necessity independent of the matter model. We conclude that the only viable non-exotic-matter approach to wormhole stabilization remains the non-perturbative quantum gravity mechanism of Hamada (running gravitational coupling, BRST conformal invariance, and the Bach–Hamada equations).