Solar-System Validation: Perihelion, Light Bending, Shapiro Delay

SPSP–SSC is locked to GR in validated exterior regimes. Below we compute the classic tests (perihelion advance, light deflection, Shapiro delay) to show numerical agreement with general relativity and observations.

Key point. The elliptic constraint introduces no propagating field outside sources. Hence the metric and geodesics reduce to GR, giving identical predictions (PPN parameters: \( \gamma=\beta=1\), others 0). Any significant departure here would falsify the model.

1. Perihelion Advance (Inner Planets)

From the Schwarzschild geodesic equation, the relativistic orbit admits a small precession per orbit:

\[ \Delta\varpi_{\text{per orbit}} \;=\; \frac{6\pi GM_\odot}{a(1-e^2)c^2} \quad \text{(radians)}. \]

Converted to arcseconds per century (after subtracting Newtonian perturbations), we obtain the GR excess shown below (SI inputs; JPL ephemerides):

Planet	Semi-major axis \(a\) (m)	Eccentricity \(e\)	Period (days)	\(\Delta\varpi\) (arcsec/orbit)	\(\Delta\varpi\) (arcsec/century)
Mercury	5.7909050×10¹⁰	0.205630	87.969	0.10352	42.981
Venus	1.0820893×10¹¹	0.006772	224.701	0.05306	8.625
Earth	1.495978707×10¹¹	0.0167086	365.256	0.03839	3.839
Mars	2.2794382×10¹¹	0.0934	686.980	0.02541	1.351

Matches GR Mercury’s famous value \(\approx 43''/\text{century}\) is recovered. Venus/Earth/Mars values agree with GR predictions and modern ephemerides (after accounting for planetary perturbations and solar J2).

Constants & conversion steps

\(GM_\odot = 1.32712440018\times 10^{20}\ \mathrm{m^3\,s^{-2}}\), \(c=299{,}792{,}458\ \mathrm{m\,s^{-1}}\)
Arcsec/orbit: \( \Delta\varpi_{\text{per orbit}}\times 180/\pi \times 3600 \)
Orbits/century: \( 36525\ \mathrm{days}/P_\text{planet} \)

2. Light Deflection at the Solar Limb

GR prediction for a light ray with impact parameter \(b\):

\[ \Delta\phi \;=\; \frac{4GM_\odot}{b\,c^2}. \]

At the solar limb (\(b\simeq R_\odot\)), the deflection is

\[ \Delta\phi_\odot \;\approx\; 1.751\ \text{arcsec}. \]

Matches GR This is the classic Eddington value, repeatedly confirmed (optical/radio). SPSP–SSC reproduces the same because exterior geodesics are exactly those of GR.

Numeric

Using \(R_\odot=6.957\times 10^8\ \text{m}\):

Δφ_rad = 4 GM⊙ / (R⊙ c²) = 8.49×10⁻⁶ rad
Δφ_arcsec = Δφ_rad × (180/π) × 3600 ≈ 1.751″

3. Shapiro Time Delay (Superior Conjunction)

For emitter at \(r_E\), receiver at \(r_R\), impact parameter \(b\) (near conjunction), a convenient approximation is:

\[ \Delta t \;\simeq\; \frac{2GM_\odot}{c^3}\,\ln\!\left(\frac{4\,r_E r_R}{b^2}\right). \]

Taking \(r_E=1\,\mathrm{AU}\), \(r_R\simeq 1.52\,\mathrm{AU}\) (e.g., Earth–Mars) and \(b\simeq R_\odot\), we find

\[ \Delta t \;\approx\; 1.24\times 10^{-4}\ \text{s} \;=\; 124\ \mu\text{s}. \]

Matches GR The magnitude and logarithmic dependence match the Cassini-era confirmations (SPSP–SSC gives identical Shapiro delay in the Solar System).

Numeric

2GM⊙/c³ ≈ 9.85 μs,  ln(4 r_E r_R / b²) ≈ 12.63
Δt ≈ 9.85 μs × 12.56 ≈ 124 μs

Why These Match Exactly in SPSP–SSC

Elliptic (non-radiative) constraint: no extra scalar/vector waves in the exterior → metric dynamics reduce to GR.
Screening in high-density regimes: any projection-sector terms are suppressed in the Solar System.
PPN catalogue: \( \gamma=\beta=1 \), others zero → same perihelion, deflection, redshift, Shapiro delay, frame-dragging as GR.

Falsifiability. Any statistically significant deviation from these GR values in the Solar System would falsify the model. The same lock-in is used for binary pulsar decay and GW phasing (no dipole channel).