Worked Derivation — Shapiro Time Delay (via SPSP–SSC Bridge → GR)

Purpose. Forward derivation of the Shapiro time delay at first post-Newtonian order. In the SPSP–SSC framework, the bridge (elliptic constraint) is screened in validated regimes, so exterior dynamics reduce to GR. We show where that reduction happens and then derive the standard result without tuning.

Bridge → GR in Validated Regimes

Summary (text only): The SPSP–SSC action contains an elliptic, non-radiative constraint field \( \Phi \) that acts as a Lagrange multiplier. Vary first, take the screening limit: no new propagating scalar, Einstein equations with minimal coupling hold. Hence light follows GR null geodesics and Shapiro’s logarithmic time delay follows directly.

Action & constraint (shown separately):

\[ S=\!\int d^4x\,\sqrt{-g}\Big[\tfrac{M_P^2}{2}R+\mathcal L_{\rm SM}\;-\;\Phi(\rho-\varepsilon)\;+\;\text{screened/boundary}\Big], \qquad \mathcal C:\ \nabla^2\Phi = 4\pi G\,\rho . \]

Validated field equations:

\[ G_{\mu\nu}=8\pi G\,T_{\mu\nu},\qquad \text{(no extra radiative DOF in the exterior)}. \]

Setup and Metric (Schwarzschild Exterior)

Consider radar ranging (Earth → reflector → Earth) with closest approach \(b\) to a mass \(M\). Work to first order in \(GM/(rc^2)\) (1PN). The exterior line element is

\[ ds^2 = -\Big(1-\frac{2GM}{rc^2}\Big)c^2 dt^2 + \Big(1-\frac{2GM}{rc^2}\Big)^{-1} dr^2 + r^2(d\theta^2+\sin^2\theta\,d\phi^2). \]

Constrain motion to the equatorial plane, \( \theta=\pi/2 \), and impose the null condition \( ds^2=0 \) (light).

Coordinate Time Along a Null Path

For \(ds^2=0\) and \(\theta=\pi/2\), solve for \(dt\) in terms of \(dr\) and \(d\phi\). To 1PN order, parameterizing the trajectory by the flat-space asymptote (impact parameter \(b\)) one finds the coordinate time integral acquires a logarithmic excess over the Minkowski value. A standard route is to write

\[ dt \;=\; \frac{1}{c}\Big(1+\frac{2GM}{rc^2}\Big)\sqrt{dr^2+r^2 d\phi^2}\;+\;\mathcal O\!\left(\frac{G^2M^2}{c^6}\right), \]

and integrate along the bent path using the straight-line approximation for the zeroth-order geometry (valid at 1PN). Subtract the flat-space light-time to isolate the curvature-induced excess \(\Delta t\).

Logarithmic Excess (Shapiro Delay)

Carrying out the integral between Earth (distance \(r_E\)) and the reflector (distance \(r_R\)) at superior conjunction gives the classic first-order result:

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

Dimensional check: \(2GM/c^3\) has units of time; the logarithm is dimensionless, so \(\Delta t\) is in seconds.

This matches Cassini and long-baseline radar tests (PPN parameter \(\gamma=1\)).