A post-Newtonian diagnostic of a local-equivalence metric ansatz for the Schwarzschild field

Abstract Local arguments based on the equivalence principle are often used to motivate gravitational redshift, light bending, and other weak-field effects without invoking the full Einstein field equations. Such arguments are physically suggestive, but they do not, by themselves, determine the exterior metric of a gravitating body. This technical note provides a diagnostic of a local-equivalence ansatz closely related to the Mociuţchi metric. Starting from a local special-relativistic interpretation and the energy ansatz E ( 1 + Φ / c 2) = E ∞, with the ‘Newtonian potential’ proxy Φ = − G M / R, one obtains the clock and radial ruler factors 1 − G M / ( c 2 R). If these factors are promoted to a static, spherically symmetric line element, the result is d s 2 = ( 1 − G M c 2 R) 2 c 2 d T 2 − d R 2 ( 1 − G M c 2 R) 2 − R 2 d Ω 2. The ansatz agrees with the Schwarzschild metric at leading weak-field order and reproduces some familiar first-order intuition. Its post-Newtonian expansion gives γ = 1 and β = 3 / 2, in contrast with the general-relativistic values γ = β = 1 for an uncharged vacuum exterior. It follows that the perihelion correction is 5 / 6 of the Schwarzschild prediction; for Mercury this gives 35.82 arcsec per century instead of 42.98. The same ansatz shifts the horizon, photon sphere, and innermost stable circular orbit to smaller areal radii than in Schwarzschild, and gives a smaller critical impact parameter. These diagnostics suggest that local-equivalence heuristics can reconstruct first-order behavior, although they do not by themselves provide the Schwarzschild exterior metric unless further conditions recover the observed post-Newtonian structure.