Worked Derivation — Perihelion Precession of Mercury (via Bridge → GR, 1PN)

Purpose. A forward 1PN derivation of Mercury’s perihelion advance. In SPSP–SSC, the bridge (elliptic constraint) is screened in validated regimes, so exterior dynamics reduce to GR with minimal coupling. We show that reduction, then derive the standard result without any tunable coefficients.

Bridge → GR in Validated Regimes

Text-only summary. The constraint field \( \Phi \) is elliptic (Lagrange multiplier). Vary first, take the screening limit: no extra propagating scalar; Einstein equations with minimal coupling hold. Test bodies follow GR timelike geodesics; hence the 1PN perihelion shift is the GR one.

Validated field equations:

\[ G_{\mu\nu}=8\pi G\,T_{\mu\nu}\quad\Rightarrow\quad \text{(two TT graviton modes; no dipole/scalar radiation).} \]

1) Metric, Symmetries, and Constants of Motion

Use the Schwarzschild exterior of mass \(M\) (the Sun). Motion is planar, so set \( \theta=\pi/2 \).

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

Timelike geodesic normalization: \( g_{\mu\nu}\dot x^\mu\dot x^\nu = -c^2 \). From cyclic \(t,\phi\) we get constants of motion (per unit mass):

\[ E = \Big(1-\frac{2GM}{rc^2}\Big)c^2\,\frac{dt}{d\tau},\qquad L = r^2\,\frac{d\phi}{d\tau}\quad (\text{specific angular momentum}). \]

2) Radial Equation → Orbit Equation

Combine normalization with the constants to obtain the radial equation (after standard algebra):

\[ \Big(\frac{dr}{d\tau}\Big)^2 = \frac{E^2}{c^2} - \Big(1-\frac{2GM}{rc^2}\Big)\Big(c^2 + \frac{L^2}{r^2}\Big). \]

Introduce \(u(\phi)\equiv 1/r\) and note \( \frac{dr}{d\tau}=\frac{dr}{d\phi}\frac{d\phi}{d\tau} = -L\,\frac{du}{d\phi} \). Differentiate w.r.t. \( \phi \) and keep terms through \(O(GM/c^2)\) (1PN). The result is the Binet-type equation (for a timelike orbit):

\[ \boxed{\;\frac{d^2u}{d\phi^2} + u \;=\; \frac{GM}{L^2} \;+\; \frac{3GM}{c^2}\,u^2\;}\,. \]

Sketch of the intermediate steps

Start from the radial equation above; change variables \(r=1/u\), use \(dr/d\tau=-L\,du/d\phi\), differentiate once with respect to \(\phi\), and expand consistently to first order in \(GM/c^2\). The \(+\,3GM\,u^2/c^2\) term is the curvature correction fixed by the Schwarzschild geometry; it is not adjustable.

3) Perturbative Solution and Precession per Orbit

Write \(u = u_0 + \delta u\) with the Newtonian ellipse \(u_0 = \frac{GM}{L^2}\big(1 + e\cos\phi\big)\). The \(u^2\) term causes a small frequency shift \( \phi \to (1-\kappa)\phi \), producing a secular advance of perihelion.

\[ \Delta\phi_{\text{per orbit}} \;=\; \frac{6\pi GM}{a(1-e^2)c^2}\,, \]

where we used the Newtonian relation \( L^2 = GM\,a(1-e^2) \) (to leading order) between specific angular momentum \(L\), semi-major axis \(a\), and eccentricity \(e\).

4) Mercury: Numerical Check (≈ 43″/century)

Insert \(GM_\odot\), \(a\), \(e\), convert radians/orbit to arcseconds/century.

Semi-major axis \(a\) (m) Eccentricity \(e\) Solar GM (m³/s²) c (m/s) Orbital period \(T\) (days)

\[ \Delta\phi_{\rm /orbit} = \frac{6\pi GM}{a(1-e^2)c^2},\quad \Delta\phi_{\rm /century} = \Delta\phi_{\rm /orbit}\times \frac{100\ \text{yr}}{T}\times\frac{365.25\ \text{days}}{1\ \text{yr}} \times\frac{180\times3600}{\pi}. \]

With the defaults, the output is ≈ 43 arcsec/century.