Worked Derivations — Core Observables from the Bridge

Purpose. This page shows forward, step-by-step derivations of three cornerstone observables — light deflection, Shapiro delay, and weak lensing — exactly as they arise in GR. In the SPSP–SSC framework, these appear when the projection bridge is screened in validated regimes. For interactive compute-and-check with presets, see the Bridge Demo.

How the Bridge Forces the GR Forms (Minimal Reduction)

Text-only summary (no display equations in this blue box): In validated regimes, the SPSP–SSC action includes an elliptic (non-radiative) constraint field Φ acting as a Lagrange multiplier. Vary the action first; then take the screening limit. This removes any propagating scalar and leaves Einstein’s equations with universal minimal coupling. Hence light and matter follow GR geodesics, and the standard GR observables below follow directly.

Action & constraint (displayed separately):

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

Metric equations (validated limit):

\[ G_{\mu\nu}=8\pi G\,T_{\mu\nu},\qquad \text{no extra radiative DOF, no validated-regime anisotropic stress.} \]

Hamiltonian/DOF note: ADM scalar/vector constraints remain first-class (Dirac algebra); Φ enforces elliptic balance and adds no conjugate DOF. Propagating content stays the two TT graviton modes.

Micro-derivation (explicit variations) \[ S[g,\Phi,\psi] = \int d^4x\,\sqrt{-g}\Big[\tfrac{M_P^2}{2}R + \mathcal L_{\rm SM}(\psi,g) - \Phi(\rho-\varepsilon)\Big] \ (+\ \text{screened/boundary}). \] \[ \delta_\Phi S=0 \Rightarrow \rho-\varepsilon=0 \Rightarrow \nabla^2\Phi = 4\pi G\,\rho . \] \[ \delta_g S=0 \Rightarrow \tfrac{M_P^2}{2}G_{\mu\nu} = T^{\rm SM}_{\mu\nu} + T^{(\Phi)}_{\mu\nu} \stackrel{\text{validated}}{\Longrightarrow} G_{\mu\nu}=8\pi G\,T^{\rm SM}_{\mu\nu}. \] Hence standard GR geodesics/observables apply (no tuned coefficients).

PPN normalization & observational lock

Linearizing the validated-regime field equations about Minkowski with universal minimal coupling reproduces the GR Parametrized Post-Newtonian limits: γ = 1, β = 1, and no extra polarizations or −1PN (dipole) terms. Solar-system observables (deflection, Shapiro, perihelion) are therefore exactly GR at 1PN.

\[ g_{\mu\nu} = \eta_{\mu\nu}+h_{\mu\nu},\quad h_{00}= \tfrac{2U}{c^2}+O(c^{-4}),\quad h_{ij}= \tfrac{2\gamma U}{c^2}\delta_{ij}+O(c^{-4}),\ \gamma=1,\ \beta=1. \]

Assumptions used (consolidated):

Units & conventions

  • Signature: \((-++\,+)\); \(c\) kept explicit in PN counting.
  • Schwarzschild exterior for solar tests; motion in the equatorial plane (\(\theta=\pi/2\)).
  • Asymptotically flat boundary for PPN comparisons; screened/Neumann for bounded testbeds.

A) Solar Light Deflection

Setup. Null geodesic in Schwarzschild exterior of mass \(M\), impact parameter \(b\), weak-field/small-angle limit.

1) Metric and null condition

\( ds^2 = -\Big(1-\frac{2GM}{rc^2}\Big)c^2 dt^2 + \Big(1-\frac{2GM}{rc^2}\Big)^{-1} dr^2 + r^2(d\theta^2+\sin^2\theta\,d\phi^2),\quad ds^2=0 \) (light).

2) Orbit equation (first order in \(GM/(bc^2)\))

With \(u(\phi)=1/r\), the null geodesic equation reduces to

\( \dfrac{d^2u}{d\phi^2} + u = \dfrac{3GM}{c^2}\,u^2 \) \quad\(\Rightarrow\) solve perturbatively: \(u=u_0+\delta u\), with \(u_0=\sin\phi/b\).

3) Scattering angle

\[ \boxed{\,\Delta\phi \;=\; \frac{4GM}{bc^2}\,} \]

Dimensional check: \(\Delta\phi\) is dimensionless; \(GM/(bc^2)\) is dimensionless.

Convert to arcseconds by \( \Delta\phi_{\rm arcsec}=\Delta\phi\times 180/\pi\times 3600\). For grazing solar rays (\(b\!\approx\!R_\odot\)) this is \(\approx 1.75^{\prime\prime}\).

Interactive check (Demo §A) →

B) Shapiro Time Delay

Setup. Radar signal Earth→reflector→Earth, closest approach \(b\) to mass \(M\). First order in \(GM/(rc^2)\).

1) Coordinate time along a null path

For \(ds^2=0\) and \(\theta=\pi/2\), solve for \(dt\) in terms of \(dr,d\phi\) and integrate the coordinate time along the bent path.

2) Logarithmic excess due to curvature

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

Dimensional check: \(2GM/c^3\) is seconds; log is unitless ⇒ \(\Delta t\) in seconds.

Here \(r_E\) and \(r_R\) are Earth–Sun and reflector–Sun distances at superior conjunction. Matches Cassini/long-baseline tests with \(\gamma=1\).

Interactive check (Demo §B) →

C) Weak Lensing — SIS Einstein Radius

Setup. Singular isothermal sphere (SIS) lens, 1D velocity dispersion \(\sigma_v\), angular-diameter distances \(D_d,D_s,D_{ds}\).

1) Deflection angle of an SIS

\( \hat\alpha = 4\pi\,\dfrac{\sigma_v^2}{c^2}. \)

2) Einstein condition and radius

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

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

Convert to arcseconds via \( \theta_E({\rm arcsec})=\theta_E({\rm rad})\times 180/\pi\times 3600\). Typical scales: galaxies (\(\sigma_v\!\sim\!200\)–250 km/s) → arcseconds; clusters (\(\sim\)1000 km/s) → tens of arcseconds.

Interactive check (Demo §C) →

Linearized mode count (why only 2 TT modes)

In the validated limit the elliptic variable \(\Phi\) is a Lagrange multiplier with primary constraint \(\pi_\Phi\!\approx\!0\). Preservation of constraints removes any propagating scalar: the coupled set remains first-class (Dirac algebra), leaving 2 tensorial radiative DOF as in GR.

\[ \text{DOF} = \tfrac{1}{2}\big(\#g_{\mu\nu}\text{ comps} - 2\times \#\text{first-class constraints}\big) = 2\ \text{(TT)}. \]