Useful Demo • Light Bending, Shapiro Delay, Weak Lensing (SPSP–SSC = GR in validated regimes)

What this page does. It gives (i) compact derivations, (ii) simple interactive calculators, and (iii) tiny inline plots for three classic observables. In SPSP–SSC the exterior field equations reduce to vacuum Einstein (elliptic constraint + local screening), so the predictions exactly match GR in these tests. This page lets reviewers and readers verify numbers on the spot—no dependencies.

1) Solar Light Bending

1.1 Derivation (leading PN)

\[ ds^2 = -\Big(1-\frac{2GM}{rc^2}\Big)c^2dt^2 + \Big(1-\frac{2GM}{rc^2}\Big)^{-1}dr^2 + r^2 d\Omega^2 \] For an equatorial null geodesic with impact parameter \(b\), the leading deflection is \[ \boxed{\alpha(b)=\frac{4GM}{b\,c^2}}\quad (\text{PPN } \gamma=1). \]

Calculator: Deflection at the Sun

1.0 = grazing the solar limb
\(GM_\odot=1.32712440018\times10^{20}\,\mathrm{m^3/s^2}\)
\(R_\odot=6.957\times10^8\,\mathrm{m}\), \(c=299{,}792{,}458\,\mathrm{m/s}\)
Show a tiny reference table
\(b/R_\odot\)\(\alpha(b)\) (arcsec)
1.01.75
1.51.17
2.00.88
3.00.58
5.00.35

2) Shapiro Time Delay

2.1 PPN with \(\gamma=1\)

\[ \Delta t \;=\; (1+\gamma)\frac{GM}{c^3}\,\ln\!\left(\frac{(r_E+r_R+R)^2 - b^2}{(r_E+r_R-R)^2 - b^2}\right) \;\xrightarrow{\ \gamma=1\ }\; \frac{2GM}{c^3}\,\ln\!\left(\frac{(r_E+r_R+R)^2 - b^2}{(r_E+r_R-R)^2 - b^2}\right) \]

For far-separated Earth–receiver geometry with superior conjunction, a common approximation is \( \Delta t \approx \tfrac{2GM}{c^3}\ln\!\left(\tfrac{4 r_E r_R}{b^2}\right) \).

Calculator: Earth–Mars (schematic)

Use AU→m internally (1 AU = 1.495978707×1011 m); \(R\) is set to ≈\(r_E+r_R\) for superior conjunction in the full form.

3) Weak Lensing (Einstein angle, SIS profile)

3.1 Point mass Einstein angle

\[ \theta_E \;=\; \sqrt{\frac{4GM}{c^2}\,\frac{D_{ds}}{D_d D_s}}\,,\qquad \alpha(\theta)=\frac{\theta_E^2}{\theta}. \]

Calculator: Einstein angle (arcseconds)

3.2 SIS lens (galaxies, clusters)

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

Calculator: SIS Einstein angle & shear

Optional: Derivation toggles

Null-geodesic sketch for deflection integral

Start with the Schwarzschild metric and the conserved quantities \(E\) and \(L\). For null rays, set \(ds^2=0\), define \(u=1/r\), derive \(\left(\frac{du}{d\phi}\right)^2 = \frac{E^2}{L^2 c^2} - u^2\left(1-\frac{2GMu}{c^2}\right)\), expand to first order in \(GM/(c^2 b)\), and integrate across the trajectory. The result is \(\alpha=4GM/(b c^2)\) when \(\gamma=1\).

Shapiro delay integral (log form)

Integrate the coordinate travel time of a null ray through the Schwarzschild geometry. The leading potential term yields a path integral of the form \(\int \Phi_N\,dl\), producing the logarithm once the straight-line approximation is applied with closest approach \(b\). PPN with \(\gamma=1\) doubles the Newtonian potential contribution, giving the final \(2GM/c^3\) coefficient.

Weak lensing kernel

In the Born approximation, the lensing potential is \(\phi(\hat{\boldsymbol n}) = -2\int_0^{\chi_s} d\chi\, \frac{\chi_s-\chi}{\chi_s \chi}\, \Phi_g(\chi\hat{\boldsymbol n},\chi)\). With \(\Phi_g=\Psi\), convergence \(\kappa = \tfrac{1}{2}\nabla_\perp^2 \phi\) matches GR. For point mass and SIS, closed forms follow.

Why SPSP–SSC matches GR here. The elliptic sorting constraint and screening eliminate extra radiative/gravitational degrees of freedom in local/exterior domains, leaving vacuum Einstein. Hence the standard GR geodesics, delays, and lensing relations apply unchanged.

Constants used