shapiro_delay.md

Shapiro Delay in SSZ

Book reference: Ch 9 (PPN), Ch 10 (Tests), Appendix B.5.2
Test file: test_shapiro_delay.py
Paper: 01 (Radial Scaling)

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CRITICAL: Use PPN Formula

Delta_t_Shapiro = (1+gamma) * (r_s/c) * ln(4*r1*r2/d^2)
               = 2 * (r_s/c) * ln(4*r1*r2/d^2)    [gamma=1 in SSZ]

Do NOT compute Shapiro delay as integral[Xi/c * dr]. That formula only captures g_tt. The full Shapiro delay includes spatial curvature (g_rr) and requires the PPN factor (1+gamma) = 2.

Setup

       Mass M
         |
   r1 *--+--* r2
         |  (d = closest approach)
   Emitter    Receiver

• r1: distance from mass to emitter
• r2: distance from mass to receiver
• d: closest approach distance to mass (impact parameter)
• r_s = 2GM/c^2

Derivation from Null Geodesics

For a radial null geodesic:

dt = dr / (D^2 * c) = (1+Xi)^2/c * dr

Expanding to first order in Xi:

Delta_t = (r2-r1)/c + (2/c) * integral[Xi(r) dr]   [first order]

However, the spatial curvature contributes a second integral of the same form, giving the total factor of 2 (= 1+gamma):

Delta_t_total = (r2-r1)/c + (2/c) * integral[r_s/(2r) dr]
             = (r2-r1)/c + (r_s/c) * [ln(r2) - ln(r1)]

For the two-way path (radar experiment), the round-trip excess delay is:

Delta_t_round_trip = 2 * (r_s/c) * ln(4*r1*r2/d^2)

Cassini Measurement (2003)

Setup: Cassini spacecraft at Saturn opposition, signal passing near Sun

• r1 = 1 AU = 1.496e11 m (Earth-Sun)
• r2 = 8.43 AU = 1.263e12 m (Saturn-Sun)
• d = 1.6 R_sun = 1.114e9 m (closest approach)
• r_s_sun = 2953 m

Calculated:

Delta_t = 2 * (2953/3e8) * ln(4 * 1.496e11 * 1.263e12 / (1.114e9)^2)
        = 1.97e-5 * ln(6.09e5)
        = 1.97e-5 * 13.32
        = 262 microseconds

Measured: 264 +/- 2 microseconds

PPN test result: gamma = 1.000021 +/- 0.000023
SSZ prediction (gamma=1) confirmed at 1 part per 50,000.

One-Way vs Two-Way

One-way Shapiro delay (signal from emitter to receiver):

Delta_t_one_way = (r_s/c) * ln(4*r1*r2/d^2)

Round-trip (radar echo, e.g. to planets):

Delta_t_round_trip = 2 * (r_s/c) * ln(4*r1*r2/d^2)

Gravitational vs SSZ Additive Delay

SSZ splits the Shapiro delay into two physically distinct contributions:

1. Temporal delay (from g_tt): Delta_t_xi = integral[Xi/c * dr]
2. Spatial delay (from g_rr): same magnitude = Delta_t_xi
3. Total: Delta_t = 2 * Delta_t_xi = (1+gamma) * Delta_t_xi

This is not two separate effects — it is one integral evaluated twice because both metric components contribute equally (gamma=1).

Relation to Other Sections

• PPN Formulas — complete PPN context
• Light Travel Time — additive form of travel time
• Gravitational Lensing — analogous factor-of-2
• Cassini Validation — measurement details
• Method Assignment — when to use PPN