Worked Notes — Perihelion Precession of Mercury (Forward 1PN Derivation)

Purpose. Show, step-by-step, how perihelion precession arises at first post-Newtonian (1PN) order. In validated regimes the SPSP–SSC framework reduces exactly to GR, so the derivation below is the standard GR computation (no coefficient tuning or “working backwards”).

Provenance (SPSP–SSC → observable).
  1. Action & constraint: In SPSP–SSC the sorting field \( \Phi \) is elliptic (a Lagrange multiplier). Variation enforces \( \nabla^2 \Phi = 4\pi G\,\rho \) and introduces no new propagating degrees of freedom.
  2. Validated reduction: Screening is on; projection functional is weak. The metric obeys Einstein’s equations with the GR Poisson limit: \( \nabla^2 \Phi_g = 4\pi G \rho \), \( g_{00} = -\big(1 + 2\Phi_g/c^2\big) \), \( g_{ij} = \big(1 - 2\Phi_g/c^2\big)\delta_{ij} \), and PPN \( \gamma=\beta=1 \).
  3. Observable: Compute using this weak-field metric (i.e., the standard GR 1PN derivation). Any slip or dipole radiation would contradict 1–2; SPSP–SSC predicts none in validated regimes.

0) What does “expand the metric to \(O(v^4)\)” mean?

In PN bookkeeping, the Newtonian equations are \(O(v^2)\). The first relativistic corrections are \(O(v^4)\) (called “1PN”). Expanding the metric and equations of motion consistently to 1PN adds terms of order \((v/c)^2\) beyond Newtonian dynamics and yields the observed perihelion shift.

\[ \text{Newtonian} \Rightarrow O(v^2), \qquad \text{1PN correction} \Rightarrow O(v^4). \]

1) Metric, symmetry, and conserved quantities

Work with the Schwarzschild exterior metric for a central mass \(M\) (the Sun), using \(\mu\equiv GM/c^2\):

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

Motion is planar; set \(\theta=\pi/2\). From cyclic coordinates \(t,\phi\) we get two constants of motion for a test particle (Mercury) with proper time \(\tau\):

\[ E = \Big(1-\frac{2\mu}{r}\Big)c^2\frac{dt}{d\tau}, \qquad L = r^2\frac{d\phi}{d\tau}. \]

The timelike normalization \(g_{\mu\nu}\dot x^\mu \dot x^\nu = -c^2\) gives the radial equation below.

Show full intermediate steps (radial → Binet)

A) Constants of motion

From the Schwarzschild metric with $\theta=\pi/2$, cyclic coordinates give:

\[ E = \Big(1-\tfrac{2\mu}{r}\Big)c^2 \frac{dt}{d\tau}, \qquad L = r^2\frac{d\phi}{d\tau}. \]

Timelike normalization $g_{\mu\nu}\dot x^\mu \dot x^\nu = -c^2$ yields the radial equation:

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

B) Change of variable

Let $u(\phi)=1/r$. Then $dr/d\tau = (dr/d\phi)(d\phi/d\tau) = -L\,du/d\phi$. Substituting:

\[ L^2\Big(\tfrac{du}{d\phi}\Big)^2 = \tfrac{E^2}{c^2} - (1-2\mu u)(c^2+L^2u^2). \]

C) Differentiate

Differentiate both sides w.r.t. $\phi$. Cancel $2u'$ (where $u'=du/d\phi$) and divide by $L^2$:

\[ \tfrac{d^2u}{d\phi^2} = \mu\,\tfrac{c^2+L^2u^2}{L^2} - (1-2\mu u)u. \]

Expand and keep terms up to 1PN ($O(\mu)$):

\[ \tfrac{d^2u}{d\phi^2} = -u + \tfrac{\mu c^2}{L^2} + 3\mu u^2. \]

D) Final Binet equation

Move $-u$ left, substitute $\mu c^2 = GM$:

\[ \tfrac{d^2u}{d\phi^2} + u = \tfrac{GM}{L^2} + \tfrac{3GM}{c^2}u^2. \]

The $+\,3GMu^2/c^2$ term is unique and fixed — this is the source of perihelion precession.

2) Radial equation and orbit equation

Combining the normalization with the constants of motion yields (after standard algebra):

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

Change variables to \(u(\phi)\equiv 1/r\) and use \(\frac{dr}{d\tau}=(dr/d\phi)\,(d\phi/d\tau)=-L\,du/d\phi\). Differentiating and keeping terms up to 1PN yields the classic Binet-type equation:

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

The first term on the right reproduces the Newtonian ellipse. The \(+\frac{3GM}{c^2}u^2\) term is the relativistic 1PN correction responsible for perihelion advance.

Sketch of the intermediate algebra (optional) Start from \[ \Big(\frac{dr}{d\tau}\Big)^2 = \frac{E^2}{c^2} - \Big(1-\frac{2\mu}{r}\Big)\Big(c^2 + \frac{L^2}{r^2}\Big). \] Substitute \(r=1/u\), so \(dr/d\tau = -L\,du/d\phi\). Differentiate w.r.t. \(\phi\), expand to first order in \(\mu\) (i.e. keep terms up to \(O(\mu)\sim O(v^2/c^2)\) in the equations of motion; the precession itself is \(O(v^2/c^2)\)), and reorganize to isolate \(d^2u/d\phi^2\). The \(u^2\) term arises from the curvature contribution in the Schwarzschild geometry and is unambiguously fixed; it is not a tunable coefficient.

3) Perturbative solution and precession per orbit

Solve perturbatively for small relativistic correction by writing \(u(\phi)=u_0(\phi)+\delta u(\phi)\), where \(u_0 = \frac{GM}{L^2}\big(1+e\cos\phi\big)\) is the Newtonian ellipse. The \(u^2\) term shifts the frequency slightly, producing a slow rotation of the ellipse:

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

where \(a\) is the semi-major axis and \(e\) the eccentricity. Using the Newtonian relation \(L^2=GMa(1-e^2)\) (to leading order) gives the compact result above.

4) Mercury: numeric evaluation (≈ 43″/century)

Insert Solar/Mercury parameters and convert radians per orbit to arcseconds per century. This calculator uses CODATA values and Mercury’s orbital elements.

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

With the default values you should see ≈ 43 arcseconds per century.

5) Why this is not “worked backwards”

Takeaway. The chain is forward: metric → constants of motion → orbit equation with a unique 1PN term → small frequency shift → perihelion advance. No tuning, no interpolation.

6) (Optional) Light bending & Shapiro delay at 1PN

For completeness (same philosophy, no tuning):

7) Where SPSP–SSC enters

In the SPSP–SSC framework, the elliptic constraint sector is screened in validated regimes, so exterior dynamics are identically those of GR. That is why the 1PN derivation above applies verbatim. Outside validated regimes (e.g., deep interiors), the elliptic constraints provide new, falsifiable structure without altering these classical Solar-system tests.

8) Light Bending (Worked 1PN)

Provenance (SPSP–SSC → observable). Same lock-in as above: elliptic constraint ⇒ no new DOF, zero slip, PPN γ=1 ⇒ GR weak-field metric. Compute null geodesic deflection with that metric.

Forward steps.

For a point mass \(M\), potential \(\Phi_g=-GM/r\). With \(\Phi_g=\Psi_g\) (γ=1), the first-order null geodesic gives \[ \alpha = \frac{2}{c^2}\!\int_{-\infty}^{+\infty}\!\nabla_\perp(\Phi_g+\Psi_g)\,dz = \frac{4GM}{b\,c^2}. \] At the solar limb (\(b\simeq R_\odot\)) this yields \(\alpha \simeq 1.75''\).

9) Shapiro Time Delay (Worked 1PN)

Provenance (SPSP–SSC → observable). Same lock-in: γ=1, weak-field metric ⇒ standard Shapiro formula in superior conjunction geometry.

Forward steps.

For emitter–receiver distances \(r_E,r_R\) and impact parameter \(b\), \[ \Delta t \;=\; \frac{2GM}{c^3}\,\ln\!\left(\frac{4\,r_E r_R}{b^2}\right). \]

10) SIS Lensing (Einstein Radius)

Provenance (SPSP–SSC → observable). Same lock-in: GR geodesics, γ=1. For a Singular Isothermal Sphere (SIS), the Einstein angle is fixed by velocity dispersion and distance ratios.

Forward steps.

For velocity dispersion \(\sigma_v\) and angular-diameter distances \(D_{ls}, D_s\), \[ \theta_E \;=\; 4\pi\,\frac{\sigma_v^2}{c^2}\,\frac{D_{ls}}{D_s}. \] Convergence \(\kappa(\theta)=\tfrac{1}{2}\theta_E/\theta\); shear \(\gamma(\theta)=\tfrac{1}{2}\theta_E/\theta\).