The Constant Density Framework: A Volumetric Ontology of Space, Mass, Gravity and Cosmology

Constant Density Model of Gravity and Cosmology Chapter 7 — Weak-Field Limit and PPN Constraints Alain Kaczorowski Independent Researcher [email protected] November 26, 2025 CHAPTER 7 — Weak-Field Limit and PPN Constraints 7.1 Overview Any alternative gravitational model must reproduce the classical weak-field predictions tested in the Solar System. This chapter evaluates the weak-field limit of the Constant–Density Ontology (CDO), using the effective metric derived from the dilation field in Chapter 3 and the exterior solution of Chapter 4. We compare the resulting expressions to the standard Parametrized Post-Newtonian (PPN) framework, focusing on the three most restrictive observables: 1. light bending near the Sun, 2. perihelion precession of Mercury, 3. Shapiro time delay. The goal is to verify whether the geometric dilation field, together with the identification of the effective metric components, is naturally consistent with the tight constraints of weak-field tests. 7.2 Weak-Field Expansion of the Effective Metric From Chapters 3 and 4, the effective metric generated by the dilation field ϕ(r) takes the form: gtt(r) = −(1 −2ϕ(r)) , grr (r) = (1 + 2ϕ(r)) , with ϕ(r) = GM c2r. In the weak-field regime (ϕ≪1), these components may be written as: gtt(r)≈ −1 + 2GM c2r, grr(r)≈1 + 2GM c2r. This already displays the standard isotropic weak-field structure familiar from the Schwarzschild metric, suggesting that the CDO framework is naturally compatible with first-order relativistic corrections. 1 7.3 PPN Parameters γand β The PPN expansion of a static, spherically symmetric metric yields: gtt =−1+2U−2βU2+O(U3), grr = 1 + 2γU +O(U2), where U=GM c2r. Extracting γ.Comparing grr: grr = 1 + 2ϕ= 1 + 2U, immediately yields γ= 1. Extracting β.Since the CDO correction to gtt contains no quadratic term in ϕ, we find β= 1. Conclusion. The CDO model reproduces the GR values: γ= 1, β = 1. These values are mandatory to satisfy present-day Solar System tests. 7.4 Light Deflection The deflection angle for a null geodesic passing with impact parameter bis: ∆θ= (1 + γ)GM c2b. Using γ= 1, we obtain: ∆θCDO = 2GM c2b, identical to General Relativity. This agrees with historical Eddington and modern VLBI measurements to better than 10−4. 7.5 Shapiro Time Delay The PPN expression for the Shapiro delay of a radar signal grazing the Sun is: ∆t= (1 + γ)GM c3ln 4rErR b2. Again, with γ= 1: ∆tCDO = 2GM c3ln 4rErR b2. This saturates the Cassini constraint on γ: |γ−1|<2.3×10−5. Thus the CDO model passes the Shapiro test exactly as GR does. 2 7.6 Perihelion Precession The standard PPN expression for perihelion advance per orbit is: ∆φ=6πGM c2a(1 −e2)2+2γ−β 3. With (γ, β) = (1,1): ∆φCDO =6πGM c2a(1 −e2), identical to General Relativity. This matches the observed Mercury precession (43 arcsec/century). 7.7 Summary of Weak-Field Predictions Observable GR CDO Prediction Status PPN parameter γ1 1 Consistent PPN parameter β1 1 Consistent Light deflection OK OK Consistent Perihelion precession OK OK Consistent Shapiro delay OK OK Consistent 7.8 Interpretation in the CDO Framework In the Constant–Density Ontology, these successful weak-field predictions arise from two simple geometric principles: 1. The dilation field ϕbehaves as a Newtonian potential in the weak-field region, because dilation falls off as 1/r outside the signature volume. 2. The effective metric is not fundamental but emerges from the volumetric dilation of spatial axes. In the weak-field limit, this dilation reproduces the same first-order corrections to distances and signal propagation as in GR. This provides a striking result: The CDO model matches all weak-field tests of gravity not by imposing a metric structure, but as a geometric consequence of a constant-density vacuum with dilatable spatial axes. 7.9 Conclusion The CDO theory passes all classical weak-field tests. This is a crucial milestone: any viable gravitational model must first reproduce these empirical results before extending into the strongfield, quantum, or cosmological domains. The remaining chapters will therefore move into regimes where deviations from GR may appear: •light propagation in strong fields, •photon sphere and black-hole shadow, 3 •potential differences in higher-order PPN parameters, •cosmological expansion and the role of the volumetric beat. These domains offer the first possibility of experimental or observational falsification of the CDO framework. 4