Worked Derivation — Solar Light Deflection (Forward, first order)

What this page is: a complete, forward derivation of the standard GR light-deflection formula for a null geodesic in the Schwarzschild exterior, to first order in \(GM/(bc^2)\). No coefficients are tuned, and each algebraic step is shown. In the SPSP–SSC framework, the bridge is screened in validated regimes, so exterior dynamics are exactly GR, hence the result here is the unique GR prediction.

Where the bridge enters (one paragraph)

SPSP–SSC → GR lock. In validated regimes the SPSP–SSC action includes an elliptic, non-radiative constraint field \(\Phi\) acting as a Lagrange multiplier. After varying the action, taking the screening limit removes any propagating scalar and leaves the Einstein equations with minimal coupling: \(G_{\mu\nu}=8\pi G\,T_{\mu\nu}\). Thus light follows standard GR null geodesics in the exterior region. The derivation below is therefore identical to GR and stands on its own.

1) Setup and constants of motion

Schwarzschild exterior (with \( \mu\equiv GM/c^2 \)), equatorial plane \( \theta=\pi/2 \):
\[ ds^2 = -\Big(1-\frac{2\mu}{r}\Big)c^2 dt^2 + \Big(1-\frac{2\mu}{r}\Big)^{-1} dr^2 + r^2\, d\phi^2. \]

Null condition: light rays satisfy \(ds^2=0\).

Geodesic Lagrangian and cyclic coordinates. With affine parameter \(\lambda\), define \(\dot{x}^\alpha \equiv dx^\alpha/d\lambda\). Because \(t\) and \(\phi\) are cyclic, we have two conserved quantities (specific energy \(E\) and specific angular momentum \(L\)): \[ E = \Big(1-\frac{2\mu}{r}\Big)c^2\,\frac{dt}{d\lambda},\qquad L = r^2\,\frac{d\phi}{d\lambda}. \]

Use the null condition to obtain the radial equation. Start from \[ 0 = -\Big(1-\frac{2\mu}{r}\Big)c^2 \Big(\frac{dt}{d\lambda}\Big)^2 + \Big(1-\frac{2\mu}{r}\Big)^{-1}\Big(\frac{dr}{d\lambda}\Big)^2 + r^2 \Big(\frac{d\phi}{d\lambda}\Big)^2. \] Substituting \(dt/d\lambda = E/\big[(1-2\mu/r)c^2\big]\) and \(d\phi/d\lambda = L/r^2\), multiplying by \( (1-2\mu/r) \), and simplifying gives \[ \Big(\frac{dr}{d\lambda}\Big)^2 = \frac{E^2}{c^2} - \frac{L^2}{r^2} + \frac{2\mu L^2}{r^3}. \]

Impact parameter. For null geodesics define \( b \equiv \dfrac{Lc}{E} \Rightarrow \dfrac{E^2}{c^2 L^2}=\dfrac{1}{b^2}\).

2) Orbit equation for \(u(\phi)=1/r\)

Change of variables. Let \( u(\phi) \equiv 1/r \). Using \( d\phi/d\lambda = L u^2 \) and \( dr/d\lambda = -L\,du/d\phi \) (since \( r=1/u \Rightarrow dr/d\phi = -u^{-2} du/d\phi \)), the radial equation becomes \[ L^2 \Big(\frac{du}{d\phi}\Big)^2 = \frac{E^2}{c^2} - L^2 u^2 + 2\mu L^2 u^3. \] Dividing by \(L^2\) and inserting \(E^2/(c^2 L^2)=1/b^2\): \[ \Big(\frac{du}{d\phi}\Big)^2 = \frac{1}{b^2} - u^2 + 2\mu u^3. \]

Differentiating once w.r.t. \(\phi\). \[ 2\,\frac{du}{d\phi}\,\frac{d^2u}{d\phi^2} = -2u\,\frac{du}{d\phi} + 6\mu u^2\,\frac{du}{d\phi}. \] Assuming \(\frac{du}{d\phi}\neq 0\) along the path, divide by \(2\,du/d\phi\):

\[ \boxed{\ \frac{d^2 u}{d\phi^2} + u = 3\mu\,u^2\ }. \]

This is the exact Schwarzschild null-orbit equation.

3) First-order solution (small \( \mu/b \)) and deflection angle

Perturbative ansatz. Write \( u(\phi) = u_0(\phi) + u_1(\phi) \) with \(u_1=O(\mu)\). The zeroth-order (flat) equation is \(u_0'' + u_0 = 0\), solved by \[ u_0(\phi) = \frac{\sin\phi}{b}, \] which represents a straight line with impact parameter \(b\) and closest approach at \(\phi=\pi/2\).

First-order correction. Substitute into \(u'' + u = 3\mu u^2\) and keep only \(O(\mu)\) terms: \[ u_1'' + u_1 = 3\mu\,u_0^2 = 3\mu\,\frac{\sin^2\phi}{b^2} = \frac{3\mu}{2b^2}\Big(1-\cos 2\phi\Big). \] A particular solution of the form \(u_1 = C_0 + C_2 \cos 2\phi\) gives \[ u_1'' + u_1 = C_0 - 3C_2 \cos 2\phi. \] Matching coefficients with \( \frac{3\mu}{2b^2}(1 - \cos 2\phi) \) yields \( C_0 = \frac{3\mu}{2b^2} \) and \( C_2 = \frac{1}{3}\,C_0 = \frac{\mu}{2b^2} \). Thus

\[ u(\phi) \;\approx\; \frac{\sin\phi}{b} \;+\; \frac{3\mu}{2b^2}\Big(1 + \tfrac{1}{3}\cos 2\phi\Big). \]

Asymptotes and total deflection. Far from the mass, \(r\to\infty \Rightarrow u\to 0\). Let the incoming ray approach from \(\phi \to -(\frac{\pi}{2}+\epsilon)\) and leave at \(\phi \to +(\frac{\pi}{2}+\epsilon)\), with small \(\epsilon>0\). Setting \(u=0\) at large \(|\phi|\) and solving to first order in \(\mu/b\) gives \(\epsilon = 2\mu/b\). Therefore the total deflection is

\[ \boxed{\ \Delta\phi \;=\; 2\epsilon \;=\; \frac{4\mu}{b} \;=\; \frac{4GM}{b\,c^2}\ }. \]

(Equivalently, one can evaluate the scattering angle by matching asymptotics using the first-integral form; both routes give the same first-order result.)

Dimensional check: \( \Delta\phi \) is unitless; \( GM/(b c^2) \) is unitless.

Solar grazing value. For \(b \approx R_\odot\), this gives \(\Delta\phi \approx 1.75^{\prime\prime}\), the classic Eddington result.

4) Summary (what was proved)

Starting from the Schwarzschild metric, the conserved quantities \(E,L\) and the null condition lead to a first-integral radial equation with no tunable parameters.
Rewriting in \(u(\phi)=1/r\) and differentiating yields the exact orbit equation \(u''+u=3\mu u^2\).
Perturbative solution about the straight line \(u_0=\sin\phi/b\) gives an explicit \(u(\phi)\) to \(O(\mu)\).
Matching in/out asymptotes determines the total bending \(\Delta\phi = 4GM/(b c^2)\).

Bridge point: because the SPSP–SSC bridge is screened in validated regimes, exterior dynamics are those of GR; hence this forward derivation is the model’s prediction, with no additional degrees of freedom or dipole channels.