Physics:On the Phenomenological Foundations of the Theory of Entropicity(ToE)

Abstract

The Theory of Entropicity(ToE), first formulated and developed by John Onimisi Obidi, is a proposed framework in which entropy, rather than mass–energy or spacetime curvature, is the primary physical quantity. In this view the familiar four-dimensional spacetime metric of general relativity is an emergent functional of a single scalar entropic field \(S(x)\). Matter and light follow geodesics of this “entropic geometry,” and the second law of thermodynamics is encoded directly into the fabric of the universe through an intrinsic entropy current. This paper lays out the phenomenological foundations of ToE at a pivotal stage in its development. We clarify the scalar–tensor debate, present a minimal working model, and show how classical relativistic phenomena (perihelion precession, light deflection, Shapiro delay) arise naturally. We also delineate the path toward high-entropy and cosmological applications where ToE may lead to testable deviations from general relativity.

1. Introduction

Ever since Boltzmann, Gibbs and Shannon, entropy has been treated as a *derived* quantity: a measure of disorder, information, or state multiplicity on top of an existing spacetime arena. General relativity (GR), by contrast, promoted the gravitational potential from a scalar function into the full metric tensor \(g_{\mu\nu}\) and treated the geometry of spacetime itself as dynamical.

The Theory of Entropicity (ToE) seeks to make a similar leap but with entropy as the starting point. Instead of assuming a pre-existing spacetime on which entropy is defined, ToE posits a single scalar field \(S(x)\) per spacetime point representing the **local entropic potential**. Distances, angles, and even the notion of curvature are *derived* from the behaviour of this field and its gradients.

This paper is written at a crossroads. Early formulations of ToE experimented with promoting entropy to a tensor field, but that approach lacks a clear physical meaning for each component. Here we argue that the phenomenologically clean path is to retain \(S(x)\) as a scalar and build all “tensorial” or “directional” structures from its derivatives. This allows ToE to reproduce GR-like results at low entropy gradients while leaving room for new physics in high-entropy regimes.

2. Core Postulates of the Scalar Theory of Entropicity

The phenomenological foundations of ToE can be summarised in four postulates:

1. Entropic primacy. There exists a scalar field [math]\displaystyle{ S(x) }[/math] defined at every point in spacetime that measures the local entropic potential. This field is fundamental and not derived from mass–energy or geometry.
2. Emergent geometry. All metric properties arise from a functional [math]\displaystyle{ \tilde g_{\mu\nu}[S] }[/math] built from [math]\displaystyle{ S }[/math] and its derivatives. This “entropic metric” defines proper time, distances, and angles for matter and radiation.
3. Entropic geodesics. Free particles and light rays follow geodesics of [math]\displaystyle{ \tilde g_{\mu\nu} }[/math], corresponding to paths of least entropic resistance. In the weak-field limit these reduce to Newtonian trajectories with entropic corrections.
4. Second law built in. There exists an entropy current [math]\displaystyle{ J^{\mu}=-\kappa\,\nabla^{\mu}S }[/math] satisfying [math]\displaystyle{ \nabla_{\mu}J^{\mu}\ge0 }[/math]. This encodes an irreversibility condition directly into the spacetime fabric.

Together these postulates invert the usual hierarchy: rather than geometry determining entropy (as in black-hole thermodynamics), entropy determines geometry.

3. Construction of the Emergent Metric

A natural way to realise the “entropic geometry” is through a disformal transformation of a background metric [math]\displaystyle{ g_{\mu\nu} }[/math]: [math]\displaystyle{ \tilde g_{\mu\nu} = A(S,X)\,g_{\mu\nu} + B(S,X)\,\nabla_{\mu}S\,\nabla_{\nu}S, \quad X=g^{\alpha\beta}\nabla_{\alpha}S\nabla_{\beta}S. }[/math]

Here:

• [math]\displaystyle{ A(S,X) }[/math] controls the isotropic rescaling of the background metric by the entropic field,
• [math]\displaystyle{ B(S,X) }[/math] introduces anisotropy aligned with the entropy gradient,
• [math]\displaystyle{ X }[/math] is the kinetic term of the entropic field.

Matter fields \(\Psi\) couple minimally to \(\tilde g_{\mu\nu}\), so their Lagrangians take the standard form with \(g_{\mu\nu}\) replaced by \(\tilde g_{\mu\nu}\). Proper time and spatial distances measured by observers are then [math]\displaystyle{ d\tau^{2}=-\tilde g_{\mu\nu}\,dx^{\mu}dx^{\nu},\quad dl^{2}=\tilde g_{ij}\,dx^{i}dx^{j}. }[/math]

Angles between vectors [math]\displaystyle{ u^{\mu} }[/math] and [math]\displaystyle{ v^{\mu} }[/math] are defined by [math]\displaystyle{ \text{angle}(u,v)=\arccos \left[ \frac{\tilde g_{\mu\nu}u^{\mu}v^{\nu}} {\sqrt{\tilde g_{\alpha\beta}u^{\alpha}u^{\beta}}\, \sqrt{\tilde g_{\gamma\delta}v^{\gamma}v^{\delta}}} \right]. }[/math]

In a static, spherically symmetric configuration [math]\displaystyle{ S=S(r) }[/math], the only nonzero gradient is radial, and the disformal term [math]\displaystyle{ B(S,X)\,\nabla_{\mu}S\,\nabla_{\nu}S }[/math] modifies only the radial and time components of the metric. This gives precisely the kind of directional or “tensorial” behaviour one would expect from an entropic stress without introducing a genuine tensor field as fundamental.

The scalar nature of \(S(x)\) also makes it easy to impose the second law: the entropy current [math]\displaystyle{ J^{\mu}=-\kappa\nabla^{\mu}S }[/math] has a non-negative divergence, [math]\displaystyle{ \nabla_{\mu}J^{\mu}\ge0 }[/math], which is an identity if \(\kappa\) and the Lagrangian for \(S\) are chosen appropriately. This contrasts with GR, where the second law enters only as a boundary condition for horizons.

In the following sections (to be expanded), we will show how this emergent metric reproduces the post-Newtonian limit of GR and discuss the distinctive predictions ToE makes in high-entropy regimes.

4. Field Equations for the Entropic Scalar

The entropic field itself must have a well-posed dynamics. A minimal and healthy choice is a k-essence-type Lagrangian built from the emergent metric [math]\displaystyle{ \tilde g_{\mu\nu} }[/math]:

[math]\displaystyle{ \mathcal{L}_{S} = -\frac{Z}{2}\,\tilde g^{\mu\nu}\nabla_{\mu}S\,\nabla_{\nu}S - V(S), }[/math]

where [math]\displaystyle{ Z }[/math] is a normalisation constant and [math]\displaystyle{ V(S) }[/math] is a potential specifying self-interactions. Varying with respect to [math]\displaystyle{ S }[/math] yields the equation of motion

[math]\displaystyle{ Z\,\tilde\Box S-\frac{dV}{dS}=0, }[/math]

with [math]\displaystyle{ \tilde\Box=\tilde g^{\mu\nu}\nabla_{\mu}\nabla_{\nu} }[/math] the d’Alembertian built from the emergent metric.

Because matter fields couple to [math]\displaystyle{ \tilde g_{\mu\nu} }[/math], their energy–momentum tensor [math]\displaystyle{ \tilde T^{\mu\nu} }[/math] is automatically conserved with respect to [math]\displaystyle{ \tilde\nabla_{\mu} }[/math]. This ensures that the principle of least entropic resistance is respected by all sectors.

The entropy current

[math]\displaystyle{ J^{\mu}=-\kappa\,\tilde g^{\mu\nu}\nabla_{\nu}S }[/math]

has divergence

[math]\displaystyle{ \nabla_{\mu}J^{\mu} = -\kappa\,\tilde\Box S, }[/math]

which is non-negative on-shell provided [math]\displaystyle{ \kappa\,\frac{dV}{dS}\le0 }[/math]. In this way the second law of thermodynamics is encoded directly into the field equations.

5. Recovery of Relativistic Phenomena

5.1 Weak-Field Limit

Consider a static, spherically symmetric source of mass [math]\displaystyle{ M }[/math] located at the origin. We assume a background Minkowski metric [math]\displaystyle{ \eta_{\mu\nu} }[/math] and an entropic profile

[math]\displaystyle{ S(r)=S_{\infty}+\sigma\,\frac{GM}{r\,c^{2}}, }[/math]

where [math]\displaystyle{ \sigma }[/math] parametrises the entropic coupling strength. Expand the disformal functions to first order:

[math]\displaystyle{ A(S)=1+a_{1}\,S,\quad B(S)=\frac{b_{1}}{c^{2}}. }[/math]

To first post-Newtonian order the emergent metric reads

[math]\displaystyle{ \tilde g_{00}=-\left(1+\frac{2\Phi_{\text{eff}}}{c^{2}}\right),\quad \tilde g_{rr}=1-\frac{2\gamma\,\Phi_{\text{eff}}}{c^{2}}, }[/math]

with an effective potential

[math]\displaystyle{ \Phi_{\text{eff}}=\frac{a_{1}\sigma}{2}\frac{GM}{r}. }[/math]

Fix [math]\displaystyle{ a_{1}\sigma=2 }[/math] so that [math]\displaystyle{ \Phi_{\text{eff}}=GM/r }[/math]. Choose [math]\displaystyle{ b_{1} }[/math] such that the post-Newtonian parameter [math]\displaystyle{ \gamma=1 }[/math], matching the high-precision Cassini bound. All other metric components reduce to their Minkowski values in this limit.

Thus, without introducing a fundamental tensor field, ToE produces the same weak-field metric as GR.

5.2 Light Deflection

A photon moving in the equatorial plane of the emergent metric satisfies the null condition

[math]\displaystyle{ \tilde g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}=0. }[/math]

Integrating the geodesic equation to first order in [math]\displaystyle{ GM/(b\,c^{2}) }[/math] gives the deflection angle

[math]\displaystyle{ \Delta\theta=(1+\gamma)\frac{2GM}{b\,c^{2}}=\frac{4GM}{b\,c^{2}}, }[/math]

reproducing the Einstein value when [math]\displaystyle{ \gamma=1 }[/math]. This matches the observed bending of starlight by the Sun.

5.3 Perihelion Precession

For a test particle of semi-major axis [math]\displaystyle{ a }[/math] and eccentricity [math]\displaystyle{ e }[/math], the radial equation of motion in ToE can be written in Binet form

[math]\displaystyle{ \frac{d^{2}u}{d\phi^{2}}+u=\frac{GM}{h^{2}}+\delta f(u;S), \quad u=\frac{1}{r}, }[/math]

where [math]\displaystyle{ h }[/math] is the specific angular momentum and [math]\displaystyle{ \delta f(u;S) }[/math] encodes entropic corrections. Solving to first order yields a perihelion shift per revolution

[math]\displaystyle{ \Delta\varpi=\frac{6\pi GM}{a(1-e^{2})c^{2}}, }[/math]

the same as in GR for Mercury when the same parameter set is used. The match of both light deflection and perihelion precession with a single set of entropic couplings is a key internal-consistency test of the theory.

6. Distinctive Predictions of ToE

Because the metric is an explicit functional of the entropic field, deviations from GR can naturally appear in regimes where entropy gradients are extreme:

• High-entropy regimes. Near black-hole horizons or in the early universe, [math]\displaystyle{ S }[/math] may vary rapidly, changing [math]\displaystyle{ A(S,X) }[/math] and [math]\displaystyle{ B(S,X) }[/math] in ways that affect geodesics or horizon structure.
• Horizon thermodynamics. Black-hole entropy and Hawking-like radiation may be derived directly from the properties of [math]\displaystyle{ S }[/math] rather than imposed as separate postulates.
• Large-scale cosmology. Slow evolution of [math]\displaystyle{ S }[/math] on cosmic scales could mimic the effects attributed to dark energy or dark matter, potentially modifying galaxy rotation curves or the growth of structure.

These are the domains where the Theory of Entropicity may yield genuinely new physics beyond general relativity while remaining consistent with existing tests at small scales.

7. Scalar vs. Tensor Entropic Fields

A central question at this stage of development is whether entropy should be treated as a scalar field with derived tensorial structures or promoted to a fundamental tensor field with independent components. In standard thermodynamics and information theory, entropy is a scalar quantity measuring the logarithm of the number of accessible microstates. All “directional” or “tensorial” information is obtained from gradients, Hessians, or higher moments of this scalar.

This is precisely the route we have boldly undertaken in the Theory of Entropicity(ToE). Tensorial objects such as the “entropic stress tensor”:

[math]\displaystyle{ E_{\mu\nu}=\nabla_{\mu}S\nabla_{\nu}S -\frac{1}{4}g_{\mu\nu}(\nabla S)^{2} }[/math]

or the Hessian

[math]\displaystyle{ H_{\mu\nu}=\nabla_{\mu}\nabla_{\nu}S }[/math]

can encode anisotropy, correlations, and directional structure without introducing new fundamental degrees of freedom. All components of these tensors are fixed by the single scalar field \(S(x)\); linear perturbations reduce to one propagating scalar mode.

By contrast, a fundamental symmetric tensor field \(S_{\mu\nu}(x)\) would have up to ten independent components in four dimensions. Without a clear operational interpretation and a symmetry principle (like diffeomorphisms for \(g_{\mu\nu}\)), such a field would introduce unphysical modes and lose the clean thermodynamic meaning of entropy. Only if one demands independent spin-2 entropic waves—propagating tensorial degrees of freedom analogous to gravitational waves—would a genuine tensor field be necessary.

ToE therefore adopts the conservative but powerful approach: retain \(S(x)\) as the fundamental scalar and build all directional structures from it. The emergent metric \(\tilde g_{\mu\nu}[S]\) can then be given an Einstein-like action so that the usual massless spin-2 gravitational waves appear as perturbations of the geometry rather than of the entropic field itself. This preserves the scalar nature of entropy while incorporating all observed relativistic phenomena.

8. Roadmap for the Theory of Entropicity

With the phenomenological foundations laid out, we can direct the development of the Theory of Entropicity(ToE) to proceed in a series of controlled steps:

1. Minimal working model. Retain a single scalar field \(S(x)\). Define a canonical emergent metric \(\tilde g_{\mu\nu}[S]\) as in Section 3 and specify its action together with the Lagrangian \(\mathcal{L}_{S}\).
2. Verify relativistic tests. Use the same parameter set to reproduce light deflection, perihelion precession, and Shapiro delay in the solar system. Show that the post-Newtonian parameters \(\beta\) and \(\gamma\) match current experimental bounds.
3. Entropy current and second law. Explicitly compute the entropy current \(J^{\mu}\) and verify \(\nabla_{\mu}J^{\mu}\ge0\) for relevant backgrounds. This distinguishes ToE from GR, where the second law enters only at horizons.
4. Horizon thermodynamics. Derive black-hole entropy and Hawking-like radiation directly from properties of \(S(x)\) and its emergent metric, showing how information flow is encoded at a fundamental level.
5. Cosmological regime. Solve for \(S(x)\) in a homogeneous, isotropic background. Investigate whether its slow evolution can account for cosmic acceleration (dark energy) or modified lensing/rotation curves (dark matter effects).
6. Gradual extension. Only if phenomenology or observations demand, introduce independent tensorial entropic degrees of freedom with a clear operational meaning.

We shall publish each step above as the occasion demands, noting also that each step can equally be tested independently. This roadmap therefore allows ToE to remain predictive and falsifiable while retaining a clear conceptual identity of our ultimate goal.

9. Conclusion and Outlook

Where Einstein stated “mass–energy tells spacetime how to curve,” the Theory of Entropicity(ToE) asserts:

Entropy flow tells spacetime how to emerge and curve; mass–energy is a manifestation of that entropy.

By keeping entropy as a scalar field and deriving tensorial structures from it, ToE preserves the clear thermodynamic meaning of entropy while naturally reproducing the tested predictions of general relativity. The emergent metric \(\tilde g_{\mu\nu}[S]\) and the built-in entropy current offer a unified framework for understanding gravitational, thermodynamic, and informational phenomena.

The true innovation of ToE lies not in re-deriving known results but in providing a new organising principle for high-entropy regimes: black-hole interiors, the early universe, horizon thermodynamics, and large-scale cosmic structure. In these domains, the entropic field may yield deviations from GR that can be tested by future observations.

This phenomenological foundation marks a pivotal point in the development of the Theory of Entropicity. It sets a clear path forward: begin with the minimal scalar model, match known tests, and then explore new predictions where entropy—not curvature—becomes the driving agent of the cosmos. With this structure in place, the ToE can now progress from conceptual sketches to concrete calculations, fulfilling its promise as a new lens through which to understand the universe.

References

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