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.

Relation of the Theory of Entropicity(ToE) to the Metric Potential in the Jordan Frame

In the Theory of Entropicity (ToE), the entropy field S(x) is treated as a real scalar field whose gradients modify the spacetime metric seen by matter. This is naturally described using the Jordan frame, where matter fields couple minimally to the metric and test particles follow geodesics.

Jordan vs. Einstein Frame

In scalar–tensor type models, and in the ToE formulation, one starts with:

• Einstein frame: metric \(g_{\mu\nu}\), standard Einstein–Hilbert term, scalar field kinetic & potential terms.
• Jordan frame: the frame where matter fields are minimally coupled (the frame in which clocks, rods, and particle masses are defined). This is the frame for physical measurements.

With the action

[math]\displaystyle{ S_{\text{matter}}[A^2(S)g_{\mu\nu},\psi] }[/math]

one defines the Jordan-frame metric

[math]\displaystyle{ \tilde{g}_{\mu\nu}=A^2(S)g_{\mu\nu}. }[/math]

All test particles follow geodesics of \(\tilde{g}_{\mu\nu}\). Therefore, lensing, orbits, and laboratory experiments should be computed in this metric.

Effective Metric Potential

In the Newtonian limit (weak fields, static source), write the Jordan metric as:

[math]\displaystyle{ \tilde{g}_{00} \simeq -\bigl(1 + 2 \Phi_\text{eff}(r)\bigr),\quad \tilde{g}_{ij} \simeq \delta_{ij}\bigl(1 - 2\gamma \Phi_\text{eff}(r)\bigr). }[/math]

From the field equations with \(A(S)=e^{\beta S}\) and \(V(S)=\frac{1}{2}m_S^2S^2\), the entropy field satisfies:

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

whose point-mass solution is

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

The Jordan-frame potential felt by matter becomes:

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

This expression shows explicitly how the entropy field modifies the effective gravitational potential.

How the Entropy Appears in the Jordan Frame

1. Why the explicit “S” seems to disappear

In our derivation we defined

[math]\displaystyle{ \tilde{g}_{\mu\nu}=A^{2}(S)\,g_{\mu\nu} }[/math]

and then wrote the weak–field metric in the Jordan frame as

[math]\displaystyle{ \tilde{g}_{00}\simeq -\bigl(1+2\Phi_{\text{eff}}(r)\bigr). }[/math]

By construction, all of the matter coupling to "[math]\displaystyle{ S }[/math]" is hidden inside the metric factor .

So when we solve the "[math]\displaystyle{ S }[/math]"–equation and substitute it back into , we get an effective potential that looks like the usual Newtonian potential plus a correction. The correction still comes from "[math]\displaystyle{ S }[/math]" — it’s just no longer written as “[math]\displaystyle{ +β S }[/math]” but as an extra term in .

That’s why the Jordan-frame potential looks like

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

This entire bracketed factor is nothing but the effect of the entropy field [math]\displaystyle{ S(r) }[/math]. In other words:

• In the Einstein frame we see [math]\displaystyle{ S }[/math] explicitly as a separate field mediating a force.
• In the Jordan frame we see its influence encoded in the modified metric (the factor multiplying [math]\displaystyle{ GM/r }[/math]).

2. Where the entropy information lives in the Jordan frame

We can make the link explicit if we write

[math]\displaystyle{ A(S)=e^{\beta S}\quad \text{and}\quad S(r)=\frac{\beta M}{4\pi r}e^{-m_S r}. }[/math]

Then the conformal factor is

[math]\displaystyle{ A^{2}(S)=e^{2\beta S(r)}\approx 1+2\beta S(r)+\dots }[/math]

and plugging that into the above gives

[math]\displaystyle{ \tilde{g}_{00}\simeq -\Bigl[1+2\Phi_\text{GR}(r)+2\beta S(r)\Bigr], }[/math]

which is literally the GR potential plus the entropic correction. If you then expand it, you get the “no S” expression above because you have substituted the explicit S(r) solution.

So the entropy is still there — it determines the form of — but in the Jordan frame it appears only through the numerical modification of the potential, not as a standalone “entropy” symbol.

Although the symbol [math]\displaystyle{ S }[/math] does not appear explicitly in the final expression for \(\Phi_\text{eff}(r)\), its influence is encoded in the conformal factor \(A^{2}(S)\) and thus in the modified metric potential. Starting from

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

the conformal factor expands as:

[math]\displaystyle{ A^{2}(S)=e^{2\beta S(r)}\approx 1+2\beta S(r)+\dots }[/math]

and the Jordan-frame metric component becomes:

[math]\displaystyle{ \tilde{g}_{00}\simeq -\Bigl[1+2\Phi_\text{GR}(r)+2\beta S(r)\Bigr], }[/math]

which is the GR potential plus the entropic correction. After substituting the explicit solution for S(r), the correction shows up as the extra bracketed term in \(\Phi_\text{eff}(r)\). Therefore the entropy field is still present—its gradients and potential determine the form of the Jordan-frame metric potential, even though it is no longer written as “[math]\displaystyle{ +β S }[/math]” explicitly.

Thus:

• “In the Jordan frame the entropy field [math]\displaystyle{ S(x) }[/math] does not appear as an explicit separate term in the metric; its influence is encoded in the conformal factor [math]\displaystyle{ A²(S) }[/math] and thus in the modified metric potential [math]\displaystyle{ Φ_eff(r) }[/math]. This potential is entirely determined by the solution [math]\displaystyle{ S(r) }[/math] of the entropy field equation.”...

Physical Meaning in ToE

• The entropy field \(S(x)\) acts like an extra mediator whose gradients contribute to the gravitational-like potential.
• In the Einstein frame, \(S\) has its own energy–momentum and modifies the Einstein equations.
• In the Jordan frame, its effect appears as a modification of the metric potential \(\Phi_\text{eff}(r)\) that governs geodesics and redshifts — the quantities actually measured in experiments.

Thus, within the Theory of Entropicity:

• The metric potential in the Jordan frame is where the “entropic interaction” actually shows itself to observers and test particles.
• Predictions for light deflection, perihelion precession, or cosmology should be computed using this Jordan-frame potential.
• The parameters \(β\) and \(m_S\) of the ToE control the size and range of the extra term in \(\Phi_\text{eff}(r)\).

Summary

In the ToE, the metric potential in the Jordan frame is the “effective gravitational potential” after including the entropy field’s influence. All observable phenomena — bending of light, time delay, orbital shifts — are determined by this potential, making the Jordan-frame metric the natural arena for comparing ToE predictions with experiments.