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PPN Formulas (Parametrized Post-Newtonian)

Book reference: Ch 9, Ch 10, Appendix B.5
Test files: test_lensing_deflection.py, test_shapiro_delay.py, test_ppn_exact.py
Paper: 01 (Radial Scaling), 10 (PPN Framework)

CRITICAL RULE

Lensing and Shapiro delay use PPN (gamma=1), NOT Xi-based formulas.
Xi only captures g_tt (time-time metric component), not g_rr.
Lensing depends on BOTH components: (1 + gamma) = 2 for SSZ.

WRONG: alpha = Xi(r)/b
CORRECT: alpha = (1+gamma) * r_s / b = 2 * r_s / b

WRONG: Delta_t = integral[Xi/c * dr]
CORRECT: Delta_t = (1+gamma) * r_s/c * ln(4*r1*r2/d^2)

PPN Parameters in SSZ

beta  = 1  (exact)
gamma = 1  (exact)

SSZ reproduces exact GR in PPN limit. There is no PPN deviation from GR in the weak field.

Gravitational Lensing

alpha = (1 + gamma) * r_s / b = 2 * r_s / b
  • alpha: deflection angle [radians]
  • b: impact parameter (closest approach distance)
  • r_s: Schwarzschild radius of lensing mass

Eddington 1919 value:

alpha_Sun = 2 * 2953 / (6.96e8) = 8.48e-6 rad = 1.748 arcsec

Observed: 1.75 +/- 0.09 arcsec. SSZ agrees with GR and observation.

Shapiro Delay

Delta_t = (1 + gamma) * (r_s/c) * ln(4 * r1 * r2 / d^2)
        = 2 * (r_s/c) * ln(4 * r1 * r2 / d^2)
  • r1: distance from mass to signal emitter
  • r2: distance from mass to receiver
  • d: closest approach distance

Cassini 2003 measurement:
gamma = 1.000021 +/- 0.000023 (best current test of GR/SSZ PPN)

Perihelion Precession

Delta_omega = 6*pi*G*M / (a * (1-e^2) * c^2)  [per orbit]

This formula is identical in SSZ and GR (both have beta = gamma = 1).

Mercury:
Delta_omega = 42.98 arcsec/century (SSZ = GR, observation: 42.98 +/- 0.04)

Gravitomagnetic (Frame Dragging) PPN

For rotating bodies, the PPN gravitomagnetic parameter:

zeta = 1  (SSZ, same as GR)

Lense-Thirring precession rate (SSZ = GR in weak field):

Omega_LT = (G * J) / (c^2 * r^3)   [frame dragging]

Why PPN for Lensing but Xi for Redshift?

Light deflection probes the spatial curvature (g_rr component) as well as the time curvature (g_tt component). Redshift probes only g_tt. The SSZ Xi function modifies only the radial structure entering through g_tt; the spatial metric in SSZ is constructed to reproduce GR at PPN level, giving gamma = 1 and thus the factor of 2 in lensing.

Test Verification

# test_ppn_exact.py
assert abs(beta_SSZ - 1.0) < 1e-10
assert abs(gamma_SSZ - 1.0) < 1e-10

# test_lensing_deflection.py  
alpha = 2 * r_s / b   # NOT Xi(b)/b

# test_shapiro_delay.py
Delta_t = 2 * (r_s/c) * log(4*r1*r2/d**2)  # NOT integral of Xi

Relation to Other Sections