gravity

Open

Varying G — The M12 Problem

The M13 derivation of G contains H₀ explicitly: G ∝ H₀. This implies G varies as H₀ evolves: Ġ/G = Ḣ₀/H₀. The LLR (Lunar Laser Ranging) bound Ġ/G < 10⁻¹² yr⁻¹ conflicts with the M12 estimate Ġ/G ≈ H₀ ≈ 2×10⁻¹⁸ s⁻¹ ≈ 6×10⁻¹¹ yr⁻¹. This is an open tension.

The M12 Prediction

The M13 master equation predicts Newton's gravitational constant G directly from other fundamental constants, notably containing the Hubble parameter .

M13 Equation for G

Since , G varies as varies. In an expanding S³ universe, decreases slowly, meaning (G decreases with time).

The Rate of Variation

For a matter-dominated expansion where , the fractional rate of change of G is approximately the negative of the Hubble constant:

Fractional variation rate

Numerically, this evaluates to .

The LLR Bound and the Tension

Lunar Laser Ranging (LLR) uses Apollo retroreflectors to measure the Earth-Moon distance with millimetre precision. Any variation of G would change the lunar orbital period. The current observational bound is:

  • M12 prediction:
  • LLR bound:
  • Casini bound (pulsars):

The M12 prediction is ~300× larger than the LLR bound. This is a genuine, unresolved tension in the framework.

Possible Resolutions

Several avenues are currently under investigation to resolve this discrepancy:

  • The M13 equation is an approximation — the exact G derivation may not contain in a simple linear form.
  • The varying G may be screened locally; Milky Way halo dynamics could modify local .
  • The LLR measurement may probe a different effective G than the cosmological G in an S³ universe.
  • The M13 derivation may require refinement at the next order of approximation.

Status: Open Problem

This remains one of the most critical open problems in the CPT framework. A successful resolution would either require fixing the G derivation at higher orders or demonstrating a rigorous mechanism by which local G is screened from cosmic evolution.

Source Documents

The primary research documents underlying this page. All files are freely viewable and downloadable. They represent the working research as written — including integrity tags, equations, and finding IDs.

Documents open in the CPT Research Library — a formatted viewer for all source research files.