Book reference: Ch 20 (PPN Tests), Ch 21 (GPS and Atomic Clocks)
Test file: test_validation.py (ssz-qubits), test_gps_relativistic.py
Paper: 01 (Radial Scaling), 20 (Observational Tests)
GPS satellites orbit at h = 20,200 km altitude. Their onboard atomic clocks must be corrected for two relativistic effects:
- Gravitational blueshift (higher altitude = weaker gravity = faster clock): +45.7 μs/day
- Kinematic time dilation (orbital velocity = slower clock): -7.1 μs/day
Net correction: +38.4 to +38.6 μs/day
Without this correction, GPS position errors accumulate at ~10 km/day.
Earth parameters:
M_earth = 5.972e24 kg
R_earth = 6,371 km = 6.371e6 m
r_s_earth = 2*G*M_earth/c^2 = 2 * 6.674e-11 * 5.972e24 / (3e8)^2 = 8.87 mm = 0.00887 m
Xi at Earth's surface:
Xi_surface = r_s / (2*R_earth) = 0.00887 / (2 * 6.371e6) = 6.953e-10
Xi at GPS orbit altitude:
R_GPS = 6371 + 20200 = 26,571 km = 2.6571e7 m
Xi_GPS = r_s / (2*R_GPS) = 0.00887 / (2 * 2.6571e7) = 1.668e-10
Gravitational frequency shift (fractional):
Delta_f/f = Xi_surface - Xi_GPS = 6.953e-10 - 1.668e-10 = 5.285e-10
Per day:
5.285e-10 * 86400 s = 45.66 μs/day (gravitational)
Kinematic correction (special relativity):
v_GPS = sqrt(G*M/R_GPS) = sqrt(6.674e-11 * 5.972e24 / 2.657e7) = 3874 m/s
Delta_f/f = -v^2/(2c^2) = -(3874)^2 / (2*(3e8)^2) = -8.34e-11
Per day: -8.34e-11 * 86400 = -7.21 μs/day (kinematic)
Net SSZ prediction:
+45.66 - 7.21 = +38.45 μs/day
| Source | Value [μs/day] |
|---|---|
| Gravitational (SSZ) | +45.66 |
| Kinematic (SR) | -7.21 |
| SSZ Total | +38.45 |
| GPS Specification | +38.4 |
| GR Total | +38.4 |
| Deviation SSZ vs GR | < 0.1% |
Result: SSZ and GR agree with the GPS specification to <0.1%.
This confirms SSZ in the weak-field regime (r_s/R << 1).
For Earth, r_s/R = 1.39e-9 (extremely weak field). Both SSZ formulas give:
Xi_weak = r_s/(2r) ≈ Xi_strong = 1 - exp(-phi*r_s/r) for r >> r_s
The two formulas agree to better than 1 part per million at Earth's distance.
The GPS clock correction demonstrates that gravitational time dilation is real and precisely predictable. The formula used:
Delta_f/f = Xi(r_surface) - Xi(r_satellite)
This is the additive form of frequency shift, directly derivable from the SSZ segment density model: clocks tick faster where segment density is lower.
# From test_validation.py
def test_gps_correction():
M_earth = 5.972e24
R_surface = 6.371e6
R_orbit = 2.6571e7
r_s = 2 * G * M_earth / c**2
xi_surface = r_s / (2 * R_surface)
xi_orbit = r_s / (2 * R_orbit)
grav_shift_per_day = (xi_surface - xi_orbit) * 86400 * 1e6 # microseconds
kinematic_shift_per_day = -(3874**2) / (2 * c**2) * 86400 * 1e6
total = grav_shift_per_day + kinematic_shift_per_day
assert abs(total - 38.4) < 0.5 # within 0.5 μs/day of spec- Time Dilation D — D = 1/(1+Xi) used for redshift
- Segment Density Xi — Xi formula
- Worked Examples — full calculation with numbers
- PPN Formulas — SSZ weak-field = PPN = GR