Effective Matter Sector: Entropic Origins

• Effective Matter Sector of Entropic Origin is a framework where matter emerges from underlying thermodynamic and information-theoretic principles, redefining gravitational dynamics.
• The models employ modified entropy–area laws to derive anisotropic effective fluids, offering insights into energy conditions and singularity regularization via Rényi, Barrow, and Kaniadakis deformations.
• Cosmological implications include unifying dark sector explanations, with quantum gravitational corrections mimicking dark matter behavior and influencing emergent gravity phenomena.

An effective matter sector of entropic origin is a theoretical framework in which the apparent energy–momentum content of spacetime—traditionally attributed to fundamental fields or particles—arises instead from underlying thermodynamic or information-theoretic principles associated with spacetime entropy. In such constructions, the conventional matter action is not taken as fundamental but emerges from the response of geometry to entropy gradients, modified entropy-area laws, or information loss across horizons. This concept unifies and extends ideas from black hole thermodynamics, quantum statistical mechanics, holography, and emergent gravity, yielding a broad class of models in which both geometry and “matter” are dynamically generated from an entropic starting point.

1. Foundational Principles: Thermodynamics, Information, and Geometry

The core principle in all entropic-origin matter sector constructions is the identification of entropy—often horizon or boundary entropy—with geometrical data and the dynamics of matter and spacetime. Classical black hole thermodynamics assigns an entropy

$S_{BH} = \frac{c^3 k_B}{4\hbar G} A_H,$

to a horizon of area $A_H$. Generalizations, such as the extension to achronal Cauchy hypersurfaces with boundary area $\mathcal{A}_C$, yield the so-called trivial entropy

$S_{TR} = \frac{c^3 k_B}{4\hbar G} \mathcal{A}_C,$

which applies to any region enclosing matter (Chen, 2020).

Quantum statistical and information-theoretic approaches, as in the Rindler horizon analysis, derive the familiar partition function and von Neumann entropy of emergent field theory from maximal ignorance about phase-space beyond a causal horizon (Lee, 2010). These foundational results motivate the replacement of direct microscopic matter actions with entropic functionals, viewed as primary sources of gravitational backreaction and, in generalizations, the full semiclassical Einstein equations.

2. Entropic Actions and Emergent Matter Sectors

Formal entropic actions are constructed by identifying gravitational or thermodynamic actions with entropy functionals. One approach introduces a quantum relative entropy between a geometric density operator $\rho_g$ (encoding the metric) and a matter-induced density operator $\rho_f$ (derived from Dirac–Kähler matter fields) (Bianconi, 26 Aug 2024): $$S_{\text{ent}}(\rho_g \parallel \rho_f) = \text{Tr}[\rho_g \ln \rho_g - \rho_g \ln \rho_f].$$ On a Lorentzian manifold with Planck length $\ell_P$, the entropic action becomes

$$S[\rho_g, \rho_f] = \frac{1}{\ell_P^d} \int \sqrt{-g} \left\{ \mathrm{Tr}[\rho_g\ln\rho_g] - \mathrm{Tr}[\rho_g\ln\rho_f] \right\}\,d^dx.$$

By selecting $\rho_g$ as the identity operator, the action reduces to a quantum cross-entropy, and the variational principle yields coupled metric–matter equations, with auxiliary “G-fields” entering as Lagrange multipliers enforcing consistency between the induced and geometric metrics.

Similarly, in the “trivial entropy” approach, imposing that the Komar (entropic) mass derived from entropy and Unruh temperature matches the proper mass contained in a region leads directly to the Einstein field equations, with the emergent effective stress–energy tensor $T^{(e)}_{\mu\nu}$ identified by

$$G_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G\, T^{(e)}_{\mu\nu}, \qquad T^{(e)}_{\mu\nu} = \frac{1}{c^4} T_{\mu\nu},$$

where $T_{\mu\nu}$ is the conventional matter stress–energy (Chen, 2020).

3. Explicit Models: Modified Entropy Functions and Effective Fluids

A systematic framework for constructing effective matter sectors from entropic origin emerges by specifying a generalized entropy–area law $S = S(A)$ and deriving the corresponding static, spherically symmetric metric and its Einstein–tensor backreaction (Anand et al., 6 Nov 2025). The procedure starts from the black hole first law,

$$dM = T\,dS, \quad T = \frac{\kappa}{2\pi} = \frac{f'(r_+)}{4\pi},$$

with metric ansatz $$ds^2 = -f(r)\,dt^2 + f(r)^{-1}\,dr^2 + r^2 d\Omega^2$$ and horizon at $r_+$.

Given a deformation

$$f(r) = 1 - M\,g(r), \quad g(r) = \frac{4\pi}{S'(r)},$$

one finds

$$\rho(r) = -\frac{M[rS''(r) - S'(r)]}{2 r^2 S'(r)^2},\qquad p_r(r) = -\rho(r),\qquad p_t(r) = \frac{M\{S'(r)[r S'''(r) + 2 S''(r)] - 2r S''(r)^2\}}{4 r S'(r)^3},$$

with the stress–energy tensor $T^\mu_\nu=\mathrm{diag}(-\rho, p_r, p_t, p_t)$. Models based on Barrow, Tsallis-Cirto, Rényi, Kaniadakis, logarithmic, power-law, and exponential entropies yield a suite of effective anisotropic fluids, whose detailed energy conditions, anisotropy ratios, and singularity regularization properties depend on the specific entropy deformation.

For example, Rényi-modified entropies with $S_R=\lambda^{-1}\ln(1+\lambda\pi r^2)$ produce fluids with all energy conditions satisfied for $\lambda>0$, while Barrow and Kaniadakis deformations induce negative-energy cores and regularize central singularities, with varying degrees of energy condition violation near the core (Anand et al., 6 Nov 2025).

4. Information Loss, Quantum Fields, and Emergence

In the information-theoretic paradigm, the quantum field theory (QFT) matter sector emerges naturally as the thermal (Unruh) fluctuations observable to an accelerated observer with a Rindler horizon (Lee, 2010). Maximizing the Shannon/von Neumann entropy over inaccessible configurations leads to a Rindler Hamiltonian ensemble,

$$P[\phi] = \frac{1}{Z_R} \exp(-\beta H_R[\phi]),\quad Z_R = \operatorname{Tr} e^{-\beta H_R},$$

whose partition function matches that of standard Euclidean QFT. The first law of thermodynamics,

$dE = T_U\,dS_h,$

yields the standard semiclassical field equations for matter. From the action perspective, the emergent matter Lagrangian arises directly from information loss at causal horizons, explaining both the path-integral structure and the origin of entropic forces, inertia, and matter–gravity coupling.

5. Cosmological and Phenomenological Implications

The entropic-origin matter sector paradigm extends to cosmological dark sectors. In the “entropic force Schrödinger mechanism,” microscopic entropic corrections to gravity at sub-micron distances generate a family of quantum gravitational bound states whose collective energy mimics cold dark matter (Plastino et al., 2019). The mechanism relies on an entropic two-body potential,

$V(r) = V_{\text{ent}}(r) - \frac{G m M}{r},$

inserted into the Schrödinger equation, with boundary conditions yielding a discrete spectrum for bound pairs. The emergent energy density acts as pressureless dust ($w_{EE} = 0$), with all modifications to Newton’s law limited to scales below $\sim 25~\mu\mathrm{m}$. The model accounts for the cosmological dark matter budget through the occupation of quantized energy levels without introducing new fundamental particles.

In another direction, the entropic–gravity picture incorporating spacetime foam and the holographic principle yields Newton’s law, Einstein’s equations, and effective dark energy and dark matter sectors with “infinite statistics,” nonlocality, and modifications at galactic and cosmic scales, while maintaining consistency with classical energy conditions and galaxy rotation curves (Ng, 2019).

6. Energy Conditions, Anisotropy, and Physical Consistency

The emergent matter sectors detailed in these frameworks often correspond to anisotropic effective fluids whose physical viability is assessed through energy conditions. Mild entropy deformations, such as logarithmic or Rényi forms, typically produce fluids that satisfy the weak, dominant, and strong energy conditions over a broad range, resembling mildly perturbed vacuum. Stronger deformations (Barrow, Kaniadakis, exponential) yield negative energy density in central regions or cores, regularizing the Schwarzschild singularity but violating some classical energy conditions locally. Anisotropy, defined via $\Delta_p = (p_t - p_r)/|\rho|$, varies with the specific entropy function and is essential for singularity resolution and thermodynamic stability in many models (Anand et al., 6 Nov 2025).

7. Synthesis and Outlook

The entropic origin of effective matter sectors frames gravity and the energy–momentum of matter as a consequence of microscopic entropy and information-theoretic accounting of spacetime degrees of freedom. By promoting the thermodynamic first law and associated entropy–area connections to the bedrock of dynamics, one recovers not just Einstein’s equations, but the entire apparatus of semiclassical matter Lagrangians, anisotropic fluid dynamics, and dark sector phenomenology from a single unifying principle. This approach clarifies longstanding puzzles regarding horizon microstructure, regularization of singularities, and the nature of the dark sector and bridges diverse modified entropy proposals under a universal geometric–thermodynamic mapping (Anand et al., 6 Nov 2025, Bianconi, 26 Aug 2024, Chen, 2020, Lee, 2010, Ng, 2019, Plastino et al., 2019). Continuing development in this field may elucidate the microscopic origins of gravity, the role of nonlocality and statistics in dark matter, and the deep interplay between information, entropy, and the fabric of spacetime itself.