Useful Demo — Part 2: Projection → Observables

What this page shows. We start from a single spin/flow projection element in SPSP–SSC and map it directly to concrete measured curves. In validated regimes, exterior physics is GR/QM, so observables follow the standard formulas. This page provides small, dependency-free calculators for four classic targets.

1) Lensing ring radius (Einstein angle)

For perfect alignment of source, lens, observer, the Einstein angle is \(\displaystyle \theta_E = \sqrt{\frac{4GM_{\rm enc}}{c^2}\,\frac{D_{ds}}{D_d D_s}}\). A single projection element sets \(M_{\rm enc}\) (via its spin/flow strength and location), then GR gives \(\theta_E\).

\[ \boxed{\;\theta_E^2 \;=\; \frac{4G}{c^2}\,\frac{M_{\rm enc}}{D_d}\,\frac{D_{ds}}{D_s}\;} \]

2) Interference fringe spacing (double-path)

A single element split into two projection directions \(\mathcal P_1,\mathcal P_2\) with amplitudes \(a_1=\sqrt{w_1}e^{i\phi_1}\), \(a_2=\sqrt{w_2}e^{i\phi_2}\) (with \(w_1+w_2=1\)) produces \(\displaystyle I(x)\propto w_1+w_2+2\sqrt{w_1 w_2}\cos(\Delta\phi + k\,\tfrac{d}{L}x)\). The measured fringe spacing is \(\Delta x = \lambda L / d\).

\[ \boxed{\;\Delta x \;=\; \frac{\lambda\,L}{d}\;} \]

3) Shapiro time delay (PPN with \(\gamma=1\))

Outside sources the metric is Schwarzschild/Kerr, so the PPN coefficient \(\gamma=1\). For superior conjunction a useful approximation is \(\displaystyle \Delta t \approx \frac{2GM}{c^3}\ln\!\frac{4 r_E r_R}{b^2}\), where \(r_E,r_R\) are distances from the mass to emitter/receiver, and \(b\) is the impact parameter.

\[ \boxed{\;\Delta t \;\approx\; \frac{2GM}{c^3}\,\ln\!\left(\frac{4 r_E r_R}{b^2}\right)\;} \]

4) Weak lensing: SIS convergence and shear

For a singular isothermal sphere (SIS) with velocity dispersion \(\sigma_v\), \(\displaystyle \theta_E^{\rm SIS}=4\pi(\sigma_v^2/c^2)\,(D_{ds}/D_s)\) and \(\kappa(\theta)=\gamma_t(\theta)=\theta_E^{\rm SIS}/(2|\theta|)\).

\[ \boxed{\;\theta_E^{\rm SIS} = 4\pi\frac{\sigma_v^2}{c^2}\frac{D_{ds}}{D_s},\qquad \kappa(\theta)=\gamma_t(\theta)=\frac{\theta_E^{\rm SIS}}{2|\theta|}\;} \]

Why this “projection → observable” map is clean

Key point: the projection element only fixes the source parameters (effective enclosed mass, or phase split/geometry). The propagation to observables then uses validated GR/QM relations: null geodesics for lensing and Shapiro, and standard interference for fringes. No extra degrees of freedom are introduced, so predictions match the canonical curves exactly in these tests.