PPN Sector Diagnostics from the ADM–Plebański Factorisation: β and γ Separation, Shapiro–Deflection Asymmetry, and Near-Horizon Geometry

The Plebański refractive index of the Schwarzschild geometry, n(r) = (1 + x)3 /(1 − x) with x ≡ GM/(2rc2 ) in isotropic coordinates, admits an exact factorisation n = nspatial × nlapse under the Arnowitt–Deser–Misner (ADM) decomposition. This factorisation is a mathematical isomorphism within General Relativity: it reexpresses known geometry in a form that makes the geometry’s parametrized post-Newtonian (PPN) sector structure transparent, without altering the physics or introducing any vacuum-medium ontology. All results are specific to isotropic Schwarzschild coordinates, the natural gauge for the opti-cal analogy; gauge dependence is discussed explicitly. Five results follow. (i) At first post-Newtonian order, light deflection and Shapiro delay split into equal spatial and lapse contributions — the algebraic consequence of γ = 1, transparent within the optical metric. (ii) The PPN refractive logarithm separates β and γ at each post-Newtonian order in isotropic gauge. (iii) The lapse sector carries 4(1 − β)x2 ; integrating along the straight-line path produces a 2PN lapse Shapiro excess ∆t2PN lapse = π(1 − β)G M /(bc ), which is geometry- independent — the logarithmic r1 , r2 dependence of the 1PN delay is absent at this order — and is absent from the deflection observable Φ at the same PN order, providing a separable β-sensitive channel within the optical decomposition. (iv) At second post-Newtonian order the refractive index contributes −(π/4)(GM/bc2 )2 to the deflection; path geometry contributes the remainder, +(16π/4)(GM/bc2 )2 , giving the known +15π/4 × (GM/bc2 )2 total.(v) Near the horizon, nlapse diverges while nspatial → 4; the qualitative distinction between bounded spatial conformal behaviour and divergent lapse behaviour persists across standard foliations and identifies a geometric limitation of theunified-index reading.