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

The propagation of electromagnetic radiation through the gravitational field of a spherically symmetric mass is governed by the position-dependent optical properties of the vacuum. In isotropic Schwarzschild coordinates the Plebański refractive index,

n(r) = (1+x)³ / (1−x), x ≡ GM / 2rc²,

encodes those properties as a single scalar. The Arnowitt–Deser–Misner (ADM) 3+1 decomposition factorises this index into two physically distinct vacuum contributions,

n(r) = n_spatial × n_lapse = (1+x)² × (1+x)/(1−x),

where the spatial factor governs field-mode compression and the temporal factor governs gravitational time dilation. These are two physically separable aspects of the same medium's response to gravity, distinguishable in their observable consequences, and the decomposition makes their separate contributions to observable phenomena directly readable.

Five consequences of the decomposition are examined. (i) At first post-Newtonian (1PN) order the spatial and temporal vacuum properties contribute equally to light deflection and Shapiro delay, the algebraic consequence of γ = 1 in the Plebański framework. (ii) The parametrized post-Newtonian (PPN) refractive logarithm separates β and γ at each post-Newtonian order: γ is carried entirely by the spatial vacuum property, β entirely by the temporal. (iii) The temporal sector carries 4(1−β)x²; integrating along the straight-line path yields an r₁,r₂-independent 2PN temporal Shapiro excess

Δt²ᴾᴺ_lapse = π(1−β)G²M² / bc⁵,

providing a β-sensitive channel with natural observable 𝒮 = π(1−β), shown to exceed the δγ-limited systematic floor. (iv) At 2PN the spatial vacuum property contributes −(π/4)(GM/bc²)² to the deflection; path geometry contributes the remainder, giving the known +15π/4 × (GM/bc²)² total, confirmed self-consistently within the Plebański formalism. (v) Near the horizon the spatial vacuum property saturates and the temporal property diverges, a behaviour confirmed in Painlevé–Gullstrand coordinates.

We discuss how analogue-gravity systems offer a practical route to measuring 𝒮 where solar-system suppression renders direct measurement impractical. The analysis provides ontological clarity on the internal structure of the gravitational vacuum within the polarisable-vacuum programme.