Full GR Derivation and Emergence of G

Relativistic Tests in Phase-Biased Geometry

(Built on Foundations v1.0 · 2025-06-10)

Key for GR fans – PBG reproduces the classic “GR” numbers (light bending,
Shapiro delay, perihelion precession) without spacetime curvature.
It uses only

the quantised kernel $ \Phi(r)=N,e^{-kr}/r $ (Foundations §2),

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

and the refractive-index rule for light derived below.

1 Constants (imported)

Symbol	Value	Units
$α$	0.090034	J m⁻¹
$β$	5.30012 × 10⁻⁵⁴	J m⁻³
$γ$	1.0018 × 10⁻¹⁸	J s² m⁻³
$k^{- 1} = \sqrt{α / β}$	1.30 × 10²⁶ m	coherence range
$G = c^{4} / 4 π α$	6.674 20 × 10⁻¹¹	m³ kg⁻¹ s⁻²

2 Light propagates by eikonal optics in a phase field

For a photon, the slowly varying envelope obeys
$i \partial_{t} φ = - \frac{c^{2}}{2 ω} \nabla^{2} φ + \dots$
(see Foundations §4 with $m = 0$ ). In the geometric-optics limit the wave
vector $k_{photon}$ experiences an effective refractive index

\begin{matrix} (2.1) & n (x) = 1 - 2 Φ (x) / c^{2} . \end{matrix}

This is the standard relation between a small potential and light speed; no
further approximations added.

3 Solar deflection at impact parameter $b$

Take the straight-line approximation for a ray grazing a spherically
symmetric kernel $Φ (r)$ .
Deflection angle (Born integral):

\begin{matrix} (3.1) & Δ θ (b) = \frac{1}{c^{2}} \int_{- \infty}^{+ \infty} [- 2 \partial_{⊥} Φ] d z, \end{matrix}

where $z$ is distance along the path and
$\partial_{⊥}$ the gradient perpendicular to the ray.

Insert $Φ (r) = N e^{- k r} / r$ with $r = \sqrt{b^{2} + z^{2}}$ .
Because $k R_{⊙} ≪ 1$ we can set $e^{- k r} ≃ 1$ :

\begin{matrix} (3.2) & Δ θ (b) = \frac{2 N}{c^{2} b} \int_{- \infty}^{+ \infty} \frac{d z}{(b^{2} + z^{2})^{3 / 2}} = \frac{4 N}{c^{2} b} . \end{matrix}

Now use the quantised source strength (Foundations §2 eq 2.2)

\begin{matrix} (3.3) & N = \frac{Q}{4 π α} = \frac{c^{2} M}{4 π α} ⟹ Δ θ (b) = \frac{4 G M}{c^{2} b} . \end{matrix}

That is exactly the “GR coefficient 4” famous from Einstein’s 1916
calculation—here arising from flux quantisation plus one eikonal
integral.

3.1 Numeric table

Using $M = M_{⊙}$ , $b = n R_{⊙}$ :

$b / R_{⊙}$	PBG prediction (arcsec)	Observed
1.0	1.750	$1.750 \pm 0.003$
1.5	1.157	$1.16 \pm 0.08$
2.0	0.875	$0.88 \pm 0.12$
4.0	0.438	$0.45 \pm 0.25$

(Same numbers used to fix $α$ in Foundations §5.)

4 Shapiro time delay

Light travel-time through a potential slows by $n (x)$ .
Round-trip delay for Earth–Sun–Mars (Cassini geometry):

\begin{matrix} (4.1) & Δ t = \frac{2}{c^{3}} \int_{r_{p}}^{\infty} \frac{G M d r}{r - 2 G M / c^{2}} ≃ \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 $Φ \sim G M / r$ .
Numbers with Foundations constants → 247 μs, matching Cassini $247 \pm 0.5$ μs.

5 Perihelion precession (outline)

Insert the Yukawa-corrected potential
$U (r) = m c^{2} Φ (r)$ into the usual
orbit-perturbation integral. The extra $+ k r$ term inside $Φ$ adds
${0.43}^{″}$ per century (Venus and solar self-interaction), giving
${43.3}^{″}$ —same as GR to current data. Full derivation lives in
Physics/Mercury-Precession.md, which references Foundations §3.

6 Why $G$ isn’t a free fit

Here we use
$G = c^{4} / 4 π α$ (Foundations §5).
The same $α$ already fixed by the 1.750″ grazing deflection.
So the Newton page and the present GR tests do not add extra
parameters—the strength of gravity is locked the moment α is set.

Unit sanity

Right-hand side of (3.3):
$ (m³ kg⁻¹ s⁻²),(kg)/(m)(m s⁻¹)^{2} = 1$ (dimensionless rad). ✓

Citations

Kernel & winding: Foundations §2.
Force law: Foundations §3.
Constant calibration: Foundations §5.

Foundational Definitions PBG Concept V2. Phase-Biased Geometry