Physics:Mathematical Formulation of the Theory of Entropicity(ToE)

On the Mathematical Formulation of the Theory of Entropicity(ToE)

The Theory of Entropicity(ToE), first formulated and developed by John Onimisi Obidi, can be formalized by introducing a real scalar field S(x) — the entropy field — with its own kinetic term, potential, and universal coupling to matter. This section presents a minimal working model covering the nine basic elements outlined above.

1. Goals

• Treat entropy as a **real scalar field** S(x) with its own dynamics (“kinetics”).
• Allow S(x) to **backreact** on spacetime and matter to recover known physics in appropriate limits.
• Keep the model **minimal**, **universal**, and **testable**.

2. Field Content and Units

• S(x): dimensionless entropy field (S = s/s₀ if using thermodynamic units).
• g_μν: spacetime metric.
• ψ: generic matter fields.

3. Action Principle

[math]\displaystyle{ S_{\text{total}} = \int d^4x \,\sqrt{-g}\, \left[ \frac{M_{\text{Pl}}^2}{2}R + \frac{Z_S}{2}\,\nabla_\mu S \nabla^\mu S - V(S) \right] + S_{\text{matter}}[A^2(S)g_{\mu\nu},\psi] }[/math]

• M_Pl: reduced Planck mass, \(M_{\text{Pl}}^2=(8\pi G)^{-1}\).
• Z_S: kinetic normalization (can be set to 1 by rescaling S).
• V(S): potential of the entropy field.
• A(S): universal matter coupling (often chosen \(A(S)=e^{\beta S}\)).

4. Field Equations

Metric equation:

[math]\displaystyle{ G_{\mu\nu} = \frac{1}{M_{\text{Pl}}^2}\left[ T_{\mu\nu}^{(S)} + T_{\mu\nu}^{(m)} \right] }[/math]

with

[math]\displaystyle{ T_{\mu\nu}^{(S)} = \nabla_\mu S \nabla_\nu S - \frac{1}{2}g_{\mu\nu}(\nabla S)^2 - g_{\mu\nu}V(S) }[/math]

Entropy field equation:

[math]\displaystyle{ \Box S = \frac{dV}{dS} - \alpha(S)\,T^{(m)} }[/math]

where

[math]\displaystyle{ \alpha(S) \equiv \frac{d\ln A(S)}{dS}, \qquad T^{(m)} \equiv g^{\mu\nu}T_{\mu\nu}^{(m)} }[/math]

For \(A(S)=e^{\beta S}\), \(\alpha(S)=\beta\) constant.

5. Static Weak–Field Limit

For a static source of density ρ:

[math]\displaystyle{ \nabla^2 S - m_S^2 S = -\beta\,\rho }[/math]

Outside a point mass M:

[math]\displaystyle{ S(r)=\frac{\beta M}{4\pi r}\,e^{-m_S r} }[/math]

Metric potential in the Jordan frame:

[math]\displaystyle{ \Phi_{\text{eff}}(r) = -\frac{GM}{r}\left[1+\beta^2 e^{-m_S r}(1+m_S r)\right] }[/math]

This modifies gravitational forces, perihelion precession, and light deflection.

6. Working Modes

• Conservative mode: small |β|, massive S (short-range); solar-system safe.
• Entropic-dominance mode: shallow V(S), moderate β; long-range effects at galactic/cosmological scales with screening in dense environments.

7. Immediate Calculational Payoffs

• **Light deflection:** Photons follow null geodesics of \( \tilde{g}_{\mu\nu}=A^2(S)g_{\mu\nu}\); bending angle = GR value + δθ(β,m_S).
• **Perihelion precession:** Additional Yukawa-like potential from S modifies the precession formula.
• **Cosmology:** In FRW, S satisfies

 [math]\displaystyle{ S̈ + 3H Ṡ + \frac{dV}{dS} = \alpha(S)(\rho_m -3p_m) }[/math]
 acting as an effective dark component or entropy pressure.

8. Parameterization

Parameter	Meaning	Typical starting value/range
β	Universal entropic coupling to matter	β| ≤ 10⁻³ (solar system), up to O(1) (galaxies)
mS	Effective mass of S (inverse range)	mS ≥ 1/AU (lab/solar system), mS ~ (10–100 kpc)⁻¹ (halos)
V(S)	Potential of entropy field	Quadratic (mS²S²/2) or shallow runaways

9. Interpretation

Unlike previous entropic or informational approaches, the ToE postulates that S(x) is a genuine physical field mediating “entropic interactions.” Its gradients and time evolution produce finite interaction timescales (the “Entropic Time Limit”), redistribute energy, and potentially unify phenomena such as gravity, time flow, and quantum measurement.