Mass-to-Horizon Entropy Relation

• Mass-to-horizon entropy relation is a framework that generalizes the Bekenstein–Hawking area law by linking horizon geometry to a horizon’s associated mass or energy.
• It governs the thermodynamic structure of gravitational field equations, enabling modified gravity models and entropic cosmologies through variations in mass–entropy dependencies.
• The relation underpins quantum corrections, mass-independent entropy products, and cosmological dark energy implications, driving observational tests in gravitational physics.

The mass-to-horizon entropy relation encapsulates the precise functional link between the mass (or energy) associated to a horizon and its geometric (areal) entropy, generalizing the canonical Bekenstein–Hawking area law to a unified class of mass–entropy dependencies across black hole and cosmological contexts. This relation, when implemented consistently with the Clausius law of thermodynamics and under the standard or modified horizon temperature, governs the thermodynamic structure of the gravitational field equations, determines the universality or violation of mass-independent entropy products, underpins extensions to dark energy cosmology, and mediates the quantum or microstructural corrections to classical gravitational entropy.

1. General Formalism of Mass-to-Horizon Entropy Relations

The prototypical context for the mass-to-horizon entropy relation is the identification of black hole horizon entropy as proportional to area, with mass and other conserved charges woven into the horizon structure. The classical Bekenstein–Hawking law for a Schwarzschild black hole is

$S_{\rm BH} = \frac{A}{4G} = 4\pi M^2,$

with $A$ the event horizon area and $M$ the asymptotic mass ($G = c = \hbar = k_B = 1$ units) (Banerjee et al., 2010).

In more general settings, including higher-dimensional, rotating, or charged spacetimes, the entropy of each horizon is derived from the first law

$dM = T_i dS_i + \cdots,$

where $T_i$ and $S_i$ denote the temperature and entropy associated to each horizon $i$ (Zhang et al., 2014). This allows for the algebraic and thermodynamic structure of the horizon functions $f(r)$ to control the detailed mass–entropy coupling.

A broad class of phenomenological and theoretical extensions now posits a generalized mass–to–horizon relation,

$M = \gamma \frac{c^2}{G} L^n,$

with $L$ the horizon radius, $n$ an index governing extensivity, and $\gamma$ a parameter with dimensions $[L]^{1-n}$ (Gohar et al., 2023, Basilakos et al., 31 Mar 2025, Sheykhi, 19 Dec 2025). Thermodynamic consistency, imposed via the Clausius relation $dE = T dS_h$ and the (generalized) Hawking temperature $T = \hbar c / (2\pi k_\mathrm{B} L)$, then leads to the universal entropy formula

$$S_n(L) = \gamma \frac{2\pi n c^3}{G \hbar (n+1)} L^{n+1}.$$

Special choices recover area law ($n=1$), volume law ($n=2$), and fractal or nonextensive forms (Barrow, Tsallis entropies), covering a wide landscape of both black hole and cosmological horizon applications (Gohar, 8 Oct 2025).

2. Mass Independence and Products of Horizon Entropies

A key diagnostic of the underlying mass–horizon algebra is the independence or dependence of certain combinations of horizon entropies on the mass parameter. For stationary black holes with $n$ horizons at radii $r_i$, entropy $S_i$, and temperature $T_i$, one defines the entropy product and sum: $$S_{\text{prod}} = \prod_{i=1}^n S_i, \quad S_{\text{sum}} = \sum_{i=1}^n S_i.$$

Zhang and Gao establish strict theorems from the first law and a Vandermonde-determinant lemma: $S_{\text{prod}}$ (resp.\ $S_{\text{sum}}$) is mass-independent if and only if $\sum_{i=1}^n \frac{1}{T_i S_i} = 0$ (resp.\ $\sum_{i=1}^n \frac{1}{T_i} = 0$) (Zhang et al., 2014). For spherically symmetric metrics $f(r)$ with Laurent expansion

$f(r) = a_{-m} r^{-m} + \dots + a_n r^n,$

these vanishing conditions are met if and only if $m \ge d-2$ and $n \ge 4-d$ (for $S_{\text{prod}}$) or $n \ge 2$ (for $S_{\text{sum}}$). Therefore, for higher-dimensional Myers–Perry black holes ($d>4$), both product and sum can be mass independent, but for 4D Kerr, only the product is, not the sum. Contrarily, in rapidly accelerating or non-spherically symmetric black holes, the mass-independent product fails due to irreducible $M$-dependence in the horizon structure (Pradhan, 2016).

This algebraic decoupling highlights that mass-independent combinations probe only intrinsic charges and angular momenta, not the total energy content, and it is tightly governed by the functional form of the mass–to–horizon relation and the symmetry of the horizon set.

3. Clausius Law, Horizon Thermodynamics, and Modified Gravity

The mass-to-horizon entropy law mediates gravity’s thermodynamic structure. Imposing the Clausius relation

$dE = T_h dS,$

where $dE$ is the energy change inside the horizon of radius $L$, $T_h$ the (Hayward, Kodama, or standard) horizon temperature, and $S$ the horizon entropy, one derives either the classical Einstein equations ($n=1$) or, for generalized $n\neq 1$, a modified Friedmann equation (Basilakos et al., 31 Mar 2025, Sheykhi, 19 Dec 2025): $$H^2 = \frac{8\pi G}{3}\left[\rho_m + \rho_{DE}\right], \quad \rho_{DE}= \frac{3}{8\pi G} \left[ \frac{\Lambda}{3} + H^2 - \frac{2\gamma n}{3-n} H^{3-n} \right],$$ with $H=\dot a/a$ the Hubble rate, and $\rho_{DE}$ an effective dark energy sector emerging from the entropy modification.

The generalized mass-to-horizon relation is necessary for thermodynamic consistency: under the standard temperature, substituting any non-area-law entropy $S(L)$ without modifying $M(L)$ leads to inconsistency or the collapse of all models back to the area law. Thus, only by generalizing $M(L)$ alongside $S(L)$ does one evade the “no-go” theorem and access the extended landscape of admissible entropic cosmologies (Luciano, 1 Oct 2025).

4. Quantum and Statistical Microstructure: Corrections to the Relation

Quantum corrections, microstructure, and statistical mechanics signal further refinements to the classical mass–horizon entropy law.

Polymer black hole models introduce two Dirac observables (black and white hole masses), with entropy corrections of the form

$$S = 4\pi(M_{BH}^2 - M_{ext}^2) - 2\pi (m\lambda_k)^{2/3} \ln (M_{BH}/M_{ext}) + \dots$$

for quadratic mass couplings (Mele et al., 2021), capturing both area and logarithmic behavior, and predicting extremal Planckian remnants with vanishing entropy—contrasting with the classical divergence as $M \to 0$. Similarly, statistical mechanics approaches for the black hole interior yield entropy–area relations growing as $S \propto \ln A$ (ultra-relativistic gas at fixed $N$) or $S \propto A^{3/4}$ (radiation), exposing the detailed quantum underpinnings of the mass–to–horizon map (Schürmann, 2018).

Generalizations also accommodate Tsallis–Cirto and Barrow (fractal) entropy corrections via master formulae such as (Gohar, 8 Oct 2025)

$$M(L) = \gamma \frac{c^2}{G} \ell_{Pl} \left[ \frac{L}{\ell_{Pl}} \mp \beta \left( \frac{L}{\ell_{Pl}} \right)^{3-\alpha} \right]^m,$$

recovering area, power law, and entanglement/loop-quantum corrections depending on the choice of exponents and correction terms.

5. Cosmological Implications and Observational Constraints

The mass-to-horizon entropy relation, when implemented in cosmological horizon thermodynamics, directly modifies the evolution of the cosmic scale factor and the dark energy sector. Generalized relations yield effective dark energy densities of the Hubble function, with equation-of-state parameters $w_{DE}$ that may cross the phantom divide or interpolate between quintessence and de Sitter-like asymptotics (Gohar et al., 2023, Basilakos et al., 31 Mar 2025). Specifically, $n < 1$ implies a dark energy equation-of-state evolving from phantom ($w<-1$) at high redshift to quintessence ($w>-1$) at late times, while $n > 1$ results in the converse.

Comprehensive MCMC and likelihood analyses with SNIa, cosmic chronometer, and BAO data constrain the parameter space. Presently, $n$ and $\gamma$ are tightly clustered near the area law values ($n = \gamma = 1$), but small deviations are permitted, with constraints such as $n = 0.945 \pm 0.070$, $\gamma = 1.70^{+0.86}_{-0.67}$ (Luciano et al., 18 Aug 2025). Cosmological baryogenesis in these models imposes $1-n \lesssim 10^{-2}$ at the inflation scale for compatibility with measured matter–antimatter asymmetries (Luciano et al., 3 Nov 2025).

Importantly, all mass-to-horizon entropy models with $n=3$ are fully equivalent at the background level to $\Lambda$CDM (cosmological constant), establishing a novel entropic foundation for dark energy (Gohar et al., 2023).

6. Extensions to Spacetime Emergence and Microstructure

Mass-to-horizon entropy serves as a cornerstone in emergent gravity programs. Two distinct derivations—(i) the first law $dE= T dS + W dV$ on cosmological horizons and (ii) Padmanabhan’s “emergence of space” proposal, $dV/dt \propto N_{sur} - N_{bulk}$—both converge to Friedmann equations that incorporate the modified entropy, illustrating the fundamental role of the mass-to-horizon relation as a bridge between microstructure/information and large-scale gravitational dynamics (Sheykhi, 19 Dec 2025).

The generalized second law is preserved, with an explicit demonstration that both horizon and matter entropy rates are nonnegative as long as the dominant energy condition $\rho + p \geq 0$ is maintained, regardless of the chosen $(n, \gamma)$.

7. Outlook and Fundamental Significance

Current and future observations—especially CMB, large-scale structure, growth of perturbations, and gravitational wave backgrounds—will further tighten constraints on deviations from the classical mass-to-horizon entropy law (Ali et al., 11 Jul 2025, Luciano, 1 Oct 2025). Provided the generalized relation is respected, the framework enables consistent extensions of gravity, unifies diverse entropy proposals, and structurally links quantum gravity, statistical mechanics, and cosmological phenomenology. The explicit requirement that any entropy modification must be accompanied by a concomitant deformation of the mass-horizon law is both a structural constraint and a guide to constructing physically viable models (Gohar, 8 Oct 2025).

In summary, the mass-to-horizon entropy relation provides the algebraic and thermodynamic backbone for black hole thermodynamics, entropic cosmology, and microstructural extensions of gravity, with physical content and observational signatures determined by the explicit parameterization of the mass–horizon map and the corresponding entropy functional.