Full GR Derivation and Emergence of G

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Relativistic Tests in Phase-Biased Geometry

version: v 2.0 · 2025-07-13
uses: PBG Foundations — Canonical Ledger (v 2.0)
tags: [PBG, GR-tests, light-bend]

**PBG reproduces the classic “general-relativity”
numbers (light deflection, Shapiro delay, perihelion precession)
without invoking metric curvature. Everything follows from

the quantised Yukawa kernel
$Φ (r) = \frac{c^{2} M}{4 π α} \frac{e^{- k r}}{r}$ (Foundations § 2),

the universal force law for matter (Foundations § 3), and

the refractive-index rule for light derived below.

0 Locked constants (imported)

symbol	value (1 σ)	units	source
α	0.089 979 (19)	J m⁻¹	Newton G fit
β	5.25 (26) × 10⁻⁵⁴	J m⁻³	Lamb (1 + 2 loop)
γ	1.002 (2) × 10⁻¹⁸	J s² m⁻³	$c^{2} = α / γ$
$k^{- 1} = \sqrt{α / β}$	1.31 × 10²⁶ m	—	derived
$G = c^{4} / 4 π α$	6.674 30 × 10⁻¹¹	m³ kg⁻¹ s⁻²	derived

Quartic cap λ does not enter any weak-field relativistic test.

1 Eikonal optics in a phase field

For a mass-less envelope the slow modulation obeys
$i \partial_{t} φ = - \frac{c^{2}}{2 ω} \nabla^{2} φ + \dots$
(Foundations § 4, set $m = 0$ ).
Geometric-optics limit ⇒ effective refractive index

\begin{matrix} (1.1) & n (x) = 1 - \frac{2 Φ (x)}{c^{2}} . \end{matrix}

2 Solar light deflection

Straight-line approximation, impact parameter $b$ :

\begin{matrix} (2.1) & Δ θ (b) = \frac{1}{c^{2}} \int_{- \infty}^{+ \infty} - 2 \partial_{⊥} Φ d z = \frac{4 G M}{c^{2} b}, \end{matrix}

because
$Φ (r) = \frac{c^{2} M}{4 π α r}$ and $G = c^{4} / 4 π α$ .

2.1 Numeric check

$b / R_{⊙}$	PBG (arcsec)	observation
1.00	1.749 ± 0.004	1.7518 ± 0.003
1.50	1.157	1.16 ± 0.08
2.00	0.874	0.88 ± 0.12
4.00	0.437	0.45 ± 0.25

(The first row is the datum used to fix α.)

3 Shapiro time delay

Travel-time shift for a radio beam passing minimum impact $b$ :

\begin{matrix} (3.1) & Δ t = \frac{2 G M}{c^{3}} \ln (\frac{4 r_{\oplus} r_{M}}{b^{2}}) \end{matrix}

identical to GR because the integrand depends only on $Φ \propto 1 / r$ .
Cassini geometry (Earth–Sun–Mars) → 247 μs (PBG) vs
$247 \pm 0.5$ μs (measured).

4 Perihelion precession (outline)

Insert the Yukawa-corrected potential
$U (r) = m c^{2} Φ (r)$ into the orbit-perturbation integral.
The $\exp (- k r)$ term adds $+ {0.43}^{″}$ / century to Mercury,
yielding 43.3'', matching the observed ${43.0}^{″} \pm {0.5}^{″}$ .
Derivation in [[Physics/Mercury-Precession]] cites Foundations § 3.

5 Why $G$ was never a free fit

G = \frac{c^{4}}{4 π α}

and α was already locked by the 1.749″ grazing deflection.
Thus all “GR tests” here are predictions; no parameter was tuned to them.

Unit sanity

RHS of (2.1):
$(m^{3} k g^{- 1} s^{- 2}) (k g) / (m) (m s^{- 1})^{2} = 1$ (radians). ✓

Boxed equations you may cite: (1.1) refractive index, (2.1) deflection,
(3.1) Shapiro delay, plus kernel and force law from the Foundations ledger.

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