This addendum does not change the physics. It:
- Fixes global conventions (signs/units/frames) used implicitly in v3.0.
- Canonicalizes the scalar sector and matter coupling.
- Proves the boxed Newton–Yukawa and PPN results referenced in v3.0.
- Clarifies parameter identifications and normalizations for α and m_y.
Upstream reference: v2.3 mass law Ψ → y.
v3.0 geometrization: y sets masses and clock rates; gravity = Einstein–Hilbert + canonical scalar with potential.
Metric signature: (-,+,+,+). Natural units ℏ = c = 1 unless displayed.
Einstein frame action:
Matter coupling: Jordan metric g̃_μν = A²(φ)g_μν with A(φ) = e^(φ/√κ_y) ≡ y.
Background: φ₀ with y₀ = e^(φ₀/√κ_y). Scalar mass about background: m_y² = (V₀/κ_y)y₀.
Choosing φ₀ = 0 recovers y₀ = 1.
Stress-energy conventions: T_μν = (2/√(-g)) δS_m/δg^μν. Non-relativistic matter: T₀₀ ≈ ρ.
Define the canonical scalar χ by χ = ∫ α₀(φ) dφ with α₀(φ) ≡ d ln A/dφ.
Result: The Einstein-frame action is canonical in χ with potential Ṽ(χ) and universal conformal matter coupling ∂χ T.
Linearize around Minkowski and φ₀: g_μν = η_μν + h_μν, φ = φ₀ + δφ. Static fields: ∂₀ = 0.
4.1 Scalar Equation
Variation yields (details in App. A)
with our sign convention ∇² = ∂ᵢ∂ᵢ. When a normalization to G_topo is desired: α₀ρ/(4πG_topo).
Point source: ρ = m₁δ³(r) gives
4.2 Metric Potential
The 00 Einstein equation yields
Solution for a point source:
4.3 Effective Two-Body Potential and Force
A nonrelativistic test mass follows Jordan geodesics; to leading order its energy shift is
Substituting the solutions:
From the derivation, α = α₀²/(4πM*²G_topo), with α₀ = 1/√κ_y.
Option A (Measured-G form, recommended):
M²_Pl,eff = (8πG_topo)^(-1)
Option B (GR-matched identification):
α = 2α₀² = 2/κ_y
(Planck units with M*² = M²_Pl).
Note: v3.0's text "α ∼ 0.2" is consistent when not equating G_topo with G_N. This addendum records both choices to eliminate normalization ambiguity.
Use isotropic gauge g₀₀ = -(1+2Φ), gᵢⱼ = (1-2γΦ)δᵢⱼ. Solving the coupled linearized equations (App. C) yields metric potentials,
Light deflection (impact b): α̂(b) = (1+γ(b))2GM/(c²b)
Shapiro delay: Δt = (1+γ(b))GM/c³ ln(4r_E r_R/b²)
GW speed: c_T = 1 from the quadratic tensor action.
Note on scale choice: Evaluating γ(r) at r = b is the standard leading-order approximation for short-range Yukawa. A path-average reproduces the same result at this order.
No ghosts/gradients: Canonical scalar ⇒ c²_s = 1.
Mass positivity: m²_y ≥ 0.
Tensor sector: c_T = 1.
GR limit (screened): m_y r ≫ 1: γ → 1, pure Newton at large scales.
Massless-like limit: m_y → 0. With Option B and α = 2/κ_y, this matches DEF/Brans–Dicke ω = κ_y/2 - 3/2.
Measured-G calibration: Fix G_topo once; propagate predictions elsewhere without refit.
Given V(r), sampling ΔF/F_Newt on m_y^(-1) separated length scales uniquely recovers (α, m_y) except in degenerate corners: (i) α = 0, (ii) m_y r ≫ 1 for all sampled r. Spectral-ladder usage from v2.3 requires m_y ≲ n for nearest-integer n identifiability.
Appendix A — Variational Details (Sketch)
δS_m/δφ with A²(φ) conformal coupling gives the scalar source term. Linearization in the static limit reduces □φ → ∇²φ and T₀₀ → ρ.
Appendix B — Green's Function Solutions
Yukawa Green's function in 3D: G_Y(r) = e^(-m_y r)/(4πr). Convolution with point source yields δφ(r) as quoted.
Appendix C — Optics to Leading Order
Linearized field equations yield Φ ∝ +ρ, δφ ∝ +ρ with opposite scalar signs; hence the ratio γ(r). Shapiro and deflection follow by inserting γ(r) into the standard PPN expressions evaluated at r = b.
\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,geometry}
\geometry{margin=1in}
\title{UTMF v3.1 Addendum: Canonicalization, Newton--Yukawa Sector, and PPN Optics}
\author{(Redacted for review)}
\date{}
\begin{document}
\maketitle
\begin{abstract}
We lock conventions, give a canonical scalar--tensor map, derive the Poisson--Yukawa sector and PPN optics, state stability, and fix parameter normalizations for $\alpha$ and $m_y$, all within v3.0.
\end{abstract}
\section{Conventions}
Metric $(-+++)$, $c=1$. Action $S=\int\sqrt{-g}[\tfrac{M_*^2}{2}R-\tfrac12(\partial\varphi)^2-V_0 e^{\varphi/\sqrt{\kappa_y}}]+S_m[A^2(\varphi)g,\Psi]$, $A(\varphi)=e^{\varphi/\sqrt{\kappa_y}}\equiv y$. Background $y_0=e^{\varphi_0/\sqrt{\kappa_y}}$, $m_y^2=(V_0/\kappa_y)y_0$.
\section{Canonical map}
$\alpha_0\equiv d\ln A/d\varphi|_{\varphi_0}=1/\sqrt{\kappa_y}$.
\section{Weak-field}
$(\nabla^2-m_y^2)\delta\varphi=\alpha_0\rho$ (or $\alpha_0\rho/M_*^2$), $\nabla^2\Phi=4\pi G_{\rm topo}\rho$. For a point source, $\delta\varphi(r)=-(\alpha_0 m_1/4\pi M_*^2) e^{-m_y r}/r$, $\Phi(r)=-G_{\rm topo}m_1/r$. The two-body potential and fractional force shift:
\[ V(r)=-\frac{G_{\rm topo}m_1m_2}{r}(1+\alpha e^{-m_y r}),\quad \frac{\Delta F}{F_{\rm Newt}}=\alpha(1+m_y r)e^{-m_y r}.\]
\section{Normalization}
$\alpha=\alpha_0^2/(4\pi M_*^2 G_{\rm topo})$. Option A: $\alpha=2\alpha_0^2 M_{\rm Pl,eff}^2/M_*^2$, $M_{\rm Pl,eff}^2=(8\pi G_{\rm topo})^{-1}$. Option B (define $G_{\rm topo}=1/(8\pi M_*^2)$): $\alpha=2\alpha_0^2=2/\kappa_y$.
\section{PPN optics}
$g_{00}=-(1+2\Phi)$, $g_{ij}=(1-2\gamma(r)\Phi)\delta_{ij}$, $\gamma(r)=(1-\alpha e^{-m_y r})/(1+\alpha e^{-m_y r})$. Shapiro: $\Delta t=(1+\gamma(b))\,GM/c^3\,\ln(4r_E r_R/b^2)$. Deflection: $\hat\alpha(b)=(1+\gamma(b))2GM/(c^2 b)$. Tensors: $c_T=1$.
\section{Stability}
Canonical kinetic term $\Rightarrow...$ no ghosts, $c_s^2=1$. $m_y^2\ge0$. GW quadratic action standard $\Rightarrow$ $c_T=1$.
\section{Limits}
$m_y r\gg1\Rightarrow\gamma\to1$. $m_y\to0\Rightarrow\gamma=(1-\alpha)/(1+\alpha)$.
\end{document}