Book reference: Ch 20 (PPN Tests), Ch 21 (Solar System Tests)
Test file: test_precession.py
Paper: 04 (Metric), 20 (Observational Tests)
Total measured precession of Mercury: 5600.73 arcsec/century
Contributions:
- Planetary perturbations (Jupiter, Venus, etc.): 5557.62 arcsec/century
- Earth's equatorial bulge (J2): 0.0254 arcsec/century
- Relativistic (GR/SSZ): 42.98 arcsec/century
- Observed relativistic: 43.11 ± 0.45 arcsec/century
In the weak field, both GR and SSZ give the same precession formula:
Delta_phi = 6*pi*G*M / (a * (1-e^2) * c^2) [per orbit]
where:
- M = M_sun = 1.989e30 kg
- a = 5.791e10 m (semi-major axis)
- e = 0.2056 (eccentricity)
- G = 6.674e-11, c = 3e8
Calculation:
Delta_phi = 6*pi * 6.674e-11 * 1.989e30 / (5.791e10 * (1-0.04228) * (3e8)^2)
= 6*pi * 1.327e20 / (5.550e10 * 9e16)
= 6*pi * 1.327e20 / 4.995e27
= 6*pi * 2.656e-8
= 5.012e-7 rad/orbit
Mercury orbital period: T = 87.97 days = 0.2408 years
Orbits per century: 100/0.2408 = 415.2
Precession per century:
5.012e-7 rad/orbit * 415.2 orbits * (3600 * 180/pi) arcsec/rad
= 5.012e-7 * 415.2 * 206265
= 42.98 arcsec/century
Observed: 43.11 ± 0.45 arcsec/century. Agreement within 0.3 sigma.
Mercury's orbit has r_s/a = 2953/5.791e10 = 5.1e-8 (extremely weak field). At this level:
Xi_SSZ(a) = r_s/(2a) = 2.55e-8
Xi_SSZ ≈ Xi_GR_equivalent
The precession formula Delta_phi ~ Xi is to first order in r_s/a, and both SSZ and GR give the same first-order result. SSZ deviates from GR only at order (r_s/a)^2 and higher, which is completely negligible for Mercury.
For objects with smaller semi-major axis (r_s/a not negligible), SSZ gives a correction:
Delta_phi_SSZ = Delta_phi_GR * [1 + f(Xi(a))]
For Mercury: correction < 1e-7 (unmeasurable).
For a hypothetical compact binary (a = 5*r_s): correction ~ 5%.
The general PPN precession formula:
Delta_phi = ((2+2*gamma-beta)/3) * 6*pi*G*M / (a*(1-e^2)*c^2)
For SSZ (gamma=1, beta=1): (2+2-1)/3 = 1, giving the standard formula.
Any theory with 2*gamma - beta != 1 would give a different rate. Mercury precession constrains 2*gamma - beta = 1.00 ± 0.003.
The 43 arcsec/century "anomalous" precession was unexplained by Newtonian mechanics for over 50 years before Einstein. SSZ reproduces this result because:
- The weak-field Xi formula matches GR to first PPN order
- The precession is entirely a first-PPN-order effect
- No higher-order SSZ effects are detectable at this precision
NASA's MESSENGER spacecraft (2004-2015) measured Mercury's perihelion with spacecraft tracking, improving the precession measurement to 1% precision. BepiColombo (2025 arrival at Mercury) will improve to 0.1%.
SSZ prediction for BepiColombo:
Delta_phi_SSZ = 42.98 arcsec/century (identical to GR to this precision)
- PPN Formulas — complete PPN context
- Worked Examples — full numerical calculation
- Geodesics — radial equation of motion
- GR vs SSZ Tables — comparison summary