Skip to content
This repository was archived by the owner on Jul 7, 2026. It is now read-only.

gravitational_lensing.md

Latest commit

 

History

History
114 lines (72 loc) · 3.44 KB

gravitational_lensing.md

File metadata and controls

114 lines (72 loc) · 3.44 KB
Raw

Gravitational Lensing in SSZ

Book reference: Ch 9 (PPN), Ch 10 (Observational Tests), Appendix B.5.1
Test file: test_lensing_deflection.py
Paper: 01 (Radial Scaling), 10 (PPN Framework)

CRITICAL: Use PPN Formula

alpha = (1+gamma) * r_s / b = 2 * r_s / b   [gamma=1 in SSZ]

Do NOT use alpha = Xi(b)/b. That only captures the temporal metric component. Light bending requires both g_tt and g_rr, which together give the factor of 2.

Variables

  • alpha: total deflection angle [radians]
  • b: impact parameter (closest approach to the lensing mass)
  • r_s: Schwarzschild radius of the lensing mass
  • gamma = 1: SSZ PPN parameter (exact, same as GR)

Eddington 1919 Solar Eclipse

Setup: Light from distant star grazing the solar limb during eclipse

  • b = R_sun = 6.96e8 m
  • r_s_sun = 2953 m

Deflection:

alpha = 2 * 2953 / 6.96e8 = 8.488e-6 rad
alpha_arcsec = 8.488e-6 * 206265 = 1.7506 arcsec

Newton (1/2 factor): 0.875 arcsec
GR/SSZ (full factor): 1.750 arcsec
Eddington measured: 1.75 +/- 0.09 arcsec

SSZ = GR = Observation (confirms gamma=1).

Why Factor of 2 (not 1)?

The deflection angle gets contributions from two metric components:

  1. Temporal component (g_tt): curves the path in time, contributes r_s/b to deflection
  2. Spatial component (g_rr): curves spatial geometry, contributes another r_s/b

Total: alpha = 2 * r_s/b

Newtonian gravity only has the temporal component (curved time, flat space), giving alpha_Newton = r_s/b = 0.875 arcsec (half the GR/SSZ value).

SSZ has gamma=1 (spatial curvature = temporal curvature), so the factor of 2 is exact.

Strong Lensing

For stronger fields (b approaching photon sphere), higher-order corrections apply:

alpha_higher = 2 * r_s/b + (15*pi/16) * (r_s/b)^2 + O((r_s/b)^3)

For b >> r_s (weak field), the linear term dominates.

Einstein Ring

When source, lens, and observer are perfectly aligned, a ring appears:

theta_E = sqrt(r_s * D_LS / (D_L * D_S))

where D_L = lens distance, D_S = source distance, D_LS = lens-source distance.

This formula is identical in SSZ and GR (gamma=1 means identical weak-field lensing).

Microlensing

For stellar microlensing (point mass, weak field), the amplification:

A = (u^2 + 2) / (u * sqrt(u^2 + 4))

where u = theta/theta_E is the angular separation in units of Einstein ring radius.

SSZ = GR for all microlensing observables (both have gamma=1).

VLBI Tests

Very Long Baseline Interferometry measures solar gravitational deflection with μas precision:

  • Fomalont et al. 2009: gamma = 0.9998 +/- 0.0003
  • SSZ prediction: gamma = 1.0000 (exact)

No deviation from GR in SSZ for lensing — this is by construction (both g_tt and g_rr reproduce Schwarzschild to PPN order).

When Would SSZ Deviate from GR in Lensing?

SSZ lensing deviates from GR only in the strong field regime (b ~ r_s):

alpha_SSZ_strong / alpha_GR_strong = D(b)^2_SSZ / D(b)^2_GR

For b = 2 r_s: D_SSZ(2r_s)^2 / D_GR(2r_s)^2 = (0.77)^2/(0.71)^2 = 1.17
The SSZ strong-field bending is ~17% stronger than GR for b = 2 r_s.

Relation to Other Sections