Worked Derivation — SIS Einstein Radius (via SPSP–SSC Bridge → GR)

Purpose. Forward derivation of the Einstein radius for a singular isothermal sphere (SIS) lens. In validated regimes the SPSP–SSC bridge locks the exterior metric to GR; the lensing calculation is therefore the standard one (no tuning).

Bridge → GR in Validated Regimes

Summary (text only): The elliptic constraint field \( \Phi \) is a Lagrange multiplier; after variation and screening there are no new propagating DOF. The exterior obeys Einstein’s equations, and matter (photons) couple minimally ⇒ standard GR lensing formulas apply.

Field equations in the validated limit:

\[ G_{\mu\nu}=8\pi G\,T_{\mu\nu},\qquad \text{(two TT graviton modes only; no scalar dipole channel)}. \]

SIS Mass Model and Surface Density

For a 3D isothermal sphere with one-dimensional velocity dispersion \( \sigma_v \), the density profile is

\[ \rho(r) \;=\; \frac{\sigma_v^2}{2\pi G\,r^2}. \]

Projecting along the line of sight gives the surface density at impact radius \( \xi \) in the lens plane,

\[ \Sigma(\xi) \;=\; \int_{-\infty}^{+\infty}\rho\!\left(\sqrt{\xi^2+z^2}\right)\,dz \;=\; \frac{\sigma_v^2}{2G\,\xi}. \]

Deflection Angle and Enclosed Mass

The mass enclosed within radius \( \xi \) in the lens plane is

\[ M(<\xi) \;=\; 2\pi\int_0^\xi \Sigma(\xi')\,\xi'\,d\xi' \;=\; 2\pi\int_0^\xi \frac{\sigma_v^2}{2G\,\xi'}\,\xi'\,d\xi' \;=\; \frac{\pi\sigma_v^2}{G}\,\xi. \]

The (unreduced) deflection angle for a light ray at impact parameter \( \xi \) is

\[ \hat\alpha(\xi) \;=\; \frac{4GM(<\xi)}{c^2\,\xi} \;=\; \frac{4G}{c^2\,\xi}\cdot \frac{\pi\sigma_v^2}{G}\,\xi \;=\; 4\pi\,\frac{\sigma_v^2}{c^2}, \]

which is independent of \( \xi \) for an SIS.

Einstein Radius

The lens equation in angular variables is \( \beta = \theta - \frac{D_{ds}}{D_s}\,\hat\alpha(\theta) \). For perfect alignment \( \beta=0 \), the Einstein ring radius satisfies

\[ \boxed{\;\theta_E \;=\; 4\pi\,\frac{\sigma_v^2}{c^2}\,\frac{D_{ds}}{D_s}\;}\,. \]

Dimensional check: \( \sigma_v^2/c^2 \) and \( D_{ds}/D_s \) are dimensionless ⇒ \( \theta_E \) is in radians.

Converting to arcseconds via \( \theta_E({\rm arcsec})=\theta_E({\rm rad})\times 180/\pi\times 3600 \) yields the usual arcsecond (galaxies) to tens-of-arcseconds (clusters) scales.