Vacuum Property Decomposition from the ADM Plebański Factorisation: β and γ Separation, Shapiro Deflection Asymmetry, and Near-Horizon Vacuum Structure

The effective refractive index obtained from Plebański’s constitutive relations for the Schwarzschild metric in isotropic coordinates — here termed the Plebański index — encodes the gravitational vacuum’s optical response as a single scalar 𝑛(𝑟). The Arnowitt–Deser–Misner (ADM) decomposition identifies two mathe-matically distinct factors within it, 𝑛(𝑟) = 𝑛spatial × 𝑛lapse = (1 + 𝑥)^2 × (1 + 𝑥)/(1 − 𝑥), where 𝑥 ≡ 𝐺𝑀/(𝑟𝑐^2 ); it is a hypothesis explored here, not a consequence of general relativity alone, that they also correspond to distinct physical properties of the vacuum. Within the Parametrized Post-Newtonian (PPN) expansion, these properties decouple: 𝛾 exclusively enters the spatial factor, while 𝛽 is confined to the temporal factor. At 1PN, the two factors already contribute equally to light deflection, a known result reframed here as arising from two separable vacuum properties. At 2PN, the deflection integrand itself splits into a straight-line index term and a second-order path-geometry residual, −𝜋/4 and 16𝜋/4 respectively; this split is independently confirmed by solving the second-order optical ray equation directly. Separately, retaining the (1 − 𝛽) term in the 2PN lapse expansion yields a cleanly isolated time-delay diagnostic, 𝒮 ≡ 𝜋(1 − 𝛽), free of endpoint dependence in the asymptotic source/receiver limit used here. The corresponding (1 − 𝛽)-proportional contribution to the deflection observable vanishes identically at the GR point (𝛽 = 1); away from it, 𝛽 remains coupled to 𝛾 in a single deflection coefficient, so this decoupling is not generic. 𝒮 ’s value is unchanged under the radial reparametrisations tested, including a direct re-derivation from scratch in areal (standard Schwarzschild) coordinates. Near the Schwarzschild horizon, the physical distinction becomes marked. The spatial factor remains bounded across the spacelike foliations examined, whereas 1the temporal factor—the static observer’s redshift—diverges. This asymmetric divergence persists across isotropic, Painlevé–Gullstrand, and Eddington–Finkelstein descriptions; the connection between this divergence and the standard Hawking/Unruh thermal result is made explicit via the horizon surface gravity 𝜅, without the decomposition itself constituting a quantum-field-theoretic derivation. Although measuring 𝒮 currently lies several orders of magnitude below solar-system ranging sensitivity, the paper sketches a potential experimental route using a Bose–Einstein condensate (BEC) analogue-gravity mapping to test this separated vacuum structure.