ADM Decomposition of the Plebański Refractive Index: Equal-Split Signatures, a PPN β Probe, and the Near-Horizon Breakdown of the Optical Medium

The Plebański refractive index n(r) = (1 + x)3/(1 − x), where x ≡ GM/(2rc2), is commonly treated as aunified spatial scaling of the gravitational vacuum [3, 4]. The Arnowitt–Deser–Misner (ADM)decomposition exposes this as a coordinate artefact: n = nspatial × nlapse, where the two factors encodegeometrically distinct physics. Five results follow. (i) At 1PN, light deflection and Shapiro delay each splitinto equal contributions from the spatial and lapse sectors—the direct algebraic signature of γ = 1, herederived transparently within the Plebański formalism rather than from the PPN metric directly. (ii) ThePPN form of ln n for Schwarzschild-class metrics in isotropic gauge, with β and γ dependences separatedat each post-Newtonian order, is derived. (iii) The vanishing of the lapse x2 term in ln n is the directalgebraic consequence of β = 1; any β ≠ 1 theory restores it and breaks the spatial–lapse deflectionsymmetry. (iv) The 2PN deflection index contribution is −π/4 × (GM/bc2)2; the dominant 2PN effect(+15π/4) is geometric, not refractive, exceeding the index term by a factor of 16. (v) Near the horizon,nlapse diverges while nspatial remains bounded at 4—the cubic appearance of n(r) is a lapse effect, and thepolarisable-vacuum unified-index picture is least defensible precisely where it is most often invoked.Together these results elevate the optical-metric decomposition from a descriptive analogue to aparameter-sensitive diagnostic framework within post-Newtonian gravity: deviations from GR in β mapdirectly onto a lapse-sector excess in time-delay observables while leaving the refractive contribution todeflection unchanged at this order.