Additional Worked Calculations (SPSP–SSC = GR in Validated Regimes)

Principle. In SPSP–SSC the sorting field is elliptic (non-radiative) and projection terms are screened locally. Outside sources the field equations reduce to vacuum Einstein, so the metric, geodesics, and PPN parameters match GR exactly. Below we show full working for several classic and near-classic tests.

1. Binary Pulsar Orbital Decay (Quadrupole Flux)

Because SPSP–SSC forbids a scalar dipole channel, energy loss is the GR quadrupole (Peters–Mathews) result. For semi-major axis \(a\), eccentricity \(e\), masses \(m_1,m_2\), total \(M=m_1+m_2\), reduced mass \(\mu=m_1 m_2/M\), the averaged orbital decay is

\[ \left\langle \frac{da}{dt} \right\rangle = -\frac{64}{5}\frac{G^3}{c^5}\frac{\mu M^2}{a^3}\, \frac{1+\tfrac{73}{24}e^2+\tfrac{37}{96}e^4}{(1-e^2)^{7/2}},\quad \left\langle \frac{dP}{dt} \right\rangle = \frac{3}{2}\frac{P}{a}\left\langle \frac{da}{dt} \right\rangle. \]

Worked example (Hulse–Taylor, schematic numbers)

With \(m_1\approx m_2\approx 1.4\,M_\odot\), \(e\approx 0.617\), \(P\approx 7.75\,\mathrm{h}\), the formula reproduces the classic GR value \(\dot P\simeq -2.4\times10^{-12}\) (after standard corrections). SPSP–SSC prediction is identical (no dipole term). Any persistent excess \(|\Delta_{\rm dip}|\ge 10^{-3}\) falsifies the model.

Matches GR SPSP–SSC curve overlays the GR quadrupole curve; the “−1PN” dipole coefficient \(\beta_{-2}\) is strictly zero in validated regimes.

2. Gravitational Redshift (Lab/GPS Scale)

In the weak field \(\Phi_N=-GM/r\), the GR (and SPSP–SSC) gravitational redshift between two clocks separated by \(\Delta \Phi_N\) is

\[ \frac{\Delta \nu}{\nu} \;=\; -\,\frac{\Delta \Phi_N}{c^2}. \]

Worked example (tower/GPS)

Height difference \(h=22.5\,\mathrm{m}\) near Earth’s surface with \(g\simeq 9.81\,\mathrm{m\,s^{-2}}\): \(\Delta\nu/\nu \simeq gh/c^2 \simeq 2.45\times10^{-15}\), consistent with Pound–Rebka and modern optical clocks. GPS corrections use the same formula. SPSP–SSC reproduces this exactly.

3. Geodetic (de Sitter) Precession

A gyroscope orbiting a central mass on a circular orbit of radius \(r\) with orbital angular frequency \(\Omega=\sqrt{GM/r^3}\) undergoes a geodetic precession

\[ \boldsymbol{\Omega}_{\rm dS} \;=\; \frac{3}{2}\,\frac{GM}{c^2 r}\,\boldsymbol{\Omega}. \]

Worked example (Gravity Probe B)

Insert Earth \(GM\), GP-B orbital radius \(r\simeq R_\oplus+642\,\mathrm{km}\), \(\Omega_{\rm dS}\) matches the GR prediction \(\sim 6.6\,\mathrm{arcsec/yr}\). SPSP–SSC matches GR because \(\gamma=\beta=1\) and no extra vector/tensor fields.

4. Lense–Thirring (Frame-Dragging)

For a central body with spin angular momentum \(\mathbf{J}\), the nodal LT precession of a test orbit with semi-major axis \(a\) and eccentricity \(e\) is

\[ \dot{\Omega}_{\rm LT} \;=\; \frac{2GJ}{c^2 a^3(1-e^2)^{3/2}}. \]

Worked example (LAGEOS Earth satellites)

Using Earth’s \(J\) and satellite orbital elements yields GR’s tens-of-mas/yr predictions; SPSP–SSC reproduces the same LT rate (no preferred-frame parameters, \(\alpha_i=0\)).

5. Weak Lensing: General Impact Parameter

With PPN \(\gamma=1\), the deflection for a point lens and impact parameter \(b\) is

\[ \alpha(b) \;=\; \frac{4GM}{b\,c^2}\,. \]

This underlies lensing shear/convergence in the weak-field limit; SPSP–SSC uses the same relation (no slip \(\Phi_g=\Psi\)).

6. Linear Growth of Structure \(D(a)\) (Cosmology)

In the validated linear regime with \(\Phi_g=\Psi\) and standard conservation, matter perturbations obey

\[ D''(a) + \left[\frac{3}{a} + \frac{H'(a)}{H(a)}\right] D'(a) - \frac{3}{2}\,\frac{\Omega_{m,0} H_0^2}{a^5 H(a)^2}\,D(a) = 0, \]

with \(H(a)\) the ΛCDM background. SPSP–SSC gives the same \(D(a)\) and growth rate \(f=\frac{d\ln D}{d\ln a}\approx \Omega_m(a)^{0.55}\) to current precision.

7. Gravitational-Wave Speed \(c_T\)

Linearizing the SPSP–SSC action in the validated regime yields the TT tensor equation

\[ \ddot h_{ij}^{\rm TT} + 3H\dot h_{ij}^{\rm TT} - \frac{\nabla^2}{a^2} h_{ij}^{\rm TT} \;=\; 0, \]

hence \(c_T=1\) (luminal). This matches GW170817/GRB170817A bounds; any \(c_T\neq 1\) would falsify SPSP–SSC.

Why These Calculations Are Identical to GR in SPSP–SSC

Elliptic constraint \(\Rightarrow\) no extra radiative DOF; no scalar dipole channel; no preferred frames.
Screening \(\Rightarrow\) projection/boundary terms vanish in local/high-density environments.
PPN lock \(\Rightarrow\) \(\gamma=\beta=1\), others \(0\) \(\Rightarrow\) same perihelion, deflection, Shapiro, redshift, geodetic/LT.
Cosmology \(\Rightarrow\) linear SVT system reduces to GR/ΛCDM; growth, lensing, CMB match within precision.

Falsifiability. Any confirmed dipole radiation, non-luminal GW speed, non-GR PPN values, or ΛCDM-inconsistent linear observables would decisively rule out the model.