--- title: Generalized Mass-to-Horizon Entropy url: https://www.emergentmind.com/topics/generalized-mass-to-horizon-entropy type: topic --- # Generalized Mass-to-Horizon Entropy Searching arXiv for the most relevant GMHE papers and related thermodynamic-gravity context. Generalized mass-to-horizon entropy is a class of horizon-entropy constructions in which the entropy assigned to a cosmological or black-hole horizon is not postulated independently, but is derived from a generalized relation between the horizon-associated mass and the horizon scale, together with the Clausius relation and a Hawking-type temperature. In the simplest and most widely used formulation, the generalized mass-to-horizon relation is written as \(M=\gamma \frac{c^2}{G}L^n\), with \(L\) the horizon scale, \(n\) a real exponent, and \(\gamma\) a normalization parameter; the associated entropy then scales as \(S\propto L^{n+1}\), reproducing the Bekenstein–Hawking area law in the appropriate limit [2307.06239, 2503.24355]. Within cosmology, this construction has been developed as a thermodynamically consistent extension of horizon entropy that modifies Friedmann dynamics through the gravity-thermodynamics conjecture, often yielding an effective dark-energy sector without introducing a new fluid by hand [2508.13260]. Subsequent work has emphasized both its phenomenological viability near the standard limit and the strong restrictions imposed by background evolution, structure growth, baryogenesis, and thermodynamic consistency [2607.00133, 2507.08647, 2511.01693, 2605.12003]. ## 1. Conceptual definition and thermodynamic motivation Generalized mass-to-horizon entropy is founded on the claim that, in horizon thermodynamics, the primary object to generalize is the mass-to-horizon relation rather than the entropy functional alone. In the standard construction, one combines the Bekenstein–Hawking entropy \[ S_{BH}=\frac{A}{4}, \qquad A=4\pi \tilde r_A^2, \] or in conventional units \(S_{BH}=k_B c^3 A/(4\hbar G)\), with a Hawking-like temperature and a linear horizon-mass scaling. In the generalized framework, that linear scaling is replaced by a nonlinear one, and the entropy is then derived from thermodynamic consistency rather than imposed ad hoc [2508.13260, 2510.07587]. A recurring motivation in the literature is the observation that many generalized entropy proposals cease to generate genuinely new cosmological dynamics if one simultaneously insists on the Clausius relation and retains a linear mass-to-horizon relation. In that case, generalized entropic cosmologies can become effectively degenerate with the standard Bekenstein–Hawking setup, inheriting the same limitations for late-time acceleration [2508.13260, 2510.00673]. The generalized mass-to-horizon construction was introduced specifically to break that degeneracy. In this program, thermodynamic consistency means that the triplet \((M,T_h,S_h)\) is chosen so that the Clausius identity holds by construction. Depending on the formulation, this is written as \[ dE=c^2\,dM=T_h\,dS_h, \] or, in cosmological apparent-horizon thermodynamics, \[ dU=T_h\,dS-\mathcal W\,dV, \] with \(dU=-dE\), \(E=\rho V\), and \(\mathcal W\) the work density [2508.13260, 2607.00133]. This distinguishes generalized mass-to-horizon entropy from approaches that simply replace \(S_{BH}\) by some \(S(A)\) while leaving the other thermodynamic ingredients untouched. ## 2. Generalized mass-to-horizon relation and entropy formula The most studied version of the generalized mass-to-horizon relation is \[ M=\gamma \frac{c^2}{G}L^n, \] where \(M\) is the effective mass associated with the system, \(L\) is the horizon length scale, \(n\) is a non-negative real parameter controlling the power-law scaling, and \(\gamma\) is a constant with dimensions \([L]^{1-n}\) [2508.13260, 2503.24355]. In cosmological applications, the relevant horizon is usually the apparent horizon, \(L=\tilde r_A\), while some entropic-cosmology formulations use the Hubble horizon \(L=c/H\) in a flat universe [2512.22103, 2512.18054]. Combining this generalized mass law with the Hawking temperature \[ T_h=\frac{1}{2\pi \tilde r_A} \] or, in conventional units, \[ T_H=\frac{\hbar c}{2\pi k_B L}, \] and the Clausius relation yields the generalized entropy \[ S=\gamma\,\frac{2n}{n+1}\,\tilde r_A^{\,n-1}S_{BH}, \] equivalently \[ S_n=\gamma \frac{n}{1+n}\frac{2\pi k_B c^3}{G\hbar}L^{n+1}, \] so that \(S_n\propto L^{n+1}\) [2307.06239, 2508.13260]. In black-hole or apparent-horizon language this is also written as \[ S_{GS}=\gamma\,\frac{2\pi}{G}\,\frac{n}{n+1}\,r_{hor}^{\,n+1} =\gamma\,\frac{2n}{n+1}\,r_{hor}^{\,n-1}S_{BH} \] [2606.24495]. The standard Bekenstein–Hawking entropy is recovered at \[ n=1,\qquad \gamma=1, \] for which the generalized mass-to-horizon relation becomes linear and the entropy reduces exactly to \(S_{BH}\) [2508.13260, 2503.24355]. Several works assume physically viable departures satisfy \[ |n-1|,\;|\gamma-1|\ll 1 \] [2508.13260]. A more elaborate extension replaces the simple power law by \[ M = \gamma \frac{c^2}{G}\ell_p \left[ \frac{r_a}{\ell_p} \mp \beta \left(\frac{r_a}{\ell_p}\right)^{3-\alpha} \right]^m, \] which induces, under the approximation \[ \beta \left(\frac{r_a}{\ell_p}\right)^{2-\alpha}\ll 1, \] the generalized entropy \[ S_G= 2\pi\gamma k_B \left[ \frac{m}{m+1} \left(\frac{r_a}{\ell_p}\right)^{m+1} \mp m\beta \frac{\sigma-1}{\sigma} \left(\frac{r_a}{\ell_p}\right)^\sigma \right], \qquad \sigma=m+3-\alpha \] [2607.00133, 2510.07587]. This extended form is presented as a unifying framework for corrected Tsallis–Cirto, Barrow, entanglement, and logarithmic quantum-gravity entropies [2510.07587]. ## 3. Cosmological horizon thermodynamics and modified Friedmann dynamics The principal cosmological arena for generalized mass-to-horizon entropy is an FLRW universe bounded by the apparent horizon. In the spatially flat case, \[ \tilde r_A=\frac{1}{H}, \] and the cosmic fluid is modeled as a perfect fluid obeying \[ \dot\rho+3H(\rho+p)=0. \] The work density is \[ \mathcal W=\frac12(\rho-p), \] and the thermodynamic identity at the apparent horizon is taken as \[ dU=T_h\,dS-\mathcal W\,dV \] or, in related formulations, \[ \delta Q=-dE=T_h\,dS_h \] [2508.13260, 2607.00133]. With the Bekenstein–Hawking entropy, this reproduces the standard Friedmann equations. Replacing the entropy by the generalized mass-to-horizon entropy modifies the derivation. In one widely used formulation, the resulting cosmological dynamics can be recast as \[ H^2=\frac{8\pi}{3}\left(\rho+\rho_{DE}\right), \qquad \dot H=-4\pi\left(\rho+p+\rho_{DE}+p_{DE}\right), \] with effective dark-energy density and pressure \[ \rho_{DE}=\frac{3}{8\pi}\left[\frac{\Lambda}{3}+H^2-\frac{2\gamma n}{3-n}H^{3-n}\right], \] \[ p_{DE}=-\frac{1}{8\pi}\left[\Lambda+\left(2\dot H+3H^2\right)-2\gamma n H^{1-n}\left(\dot H+\frac{3}{3-n}H^2\right)\right] \] [2508.13260, 2503.24355]. In this interpretation, the generalized entropy acts as a source of effective dark energy; no extra physical dark-energy fluid is introduced by hand. The \(\Lambda\)CDM limit is explicitly recovered for \[ n=\gamma=1. \] At that point, the generalized MHR becomes linear, the entropy becomes exactly Bekenstein–Hawking, and the modified Friedmann equations reduce to standard \(\Lambda\)CDM [2508.13260]. A separate line of work formulates the dynamics directly as a modified Friedmann equation of the form \[ \left(H^2+\frac{k c^2}{a^2}\right)^{\frac{3-n}{2}} =\frac{8\pi G_{\rm eff}}{3}(\rho+\rho_\Lambda), \qquad G_{\rm eff}=\frac{(3-n)G}{2n\gamma}, \] derived from the first law on the apparent horizon and reproduced independently from Padmanabhan’s emergent-space proposal [2512.18054]. This suggests that the same generalized entropy can be embedded both in equilibrium horizon thermodynamics and in emergent-gravity-inspired cosmic space dynamics. ## 4. Relations to other generalized entropies and modified-gravity interpretations Generalized mass-to-horizon entropy is often presented as a unifying umbrella for several known entropy modifications. The mapping is typically expressed through the entropy’s power-law dependence on the horizon radius or area. For the simple \(M\propto L^n\) model, the entropy scales as \[ S_n\propto L^{n+1}\propto A^{(n+1)/2}, \] which reproduces several familiar cases: - \(n=1\): Bekenstein entropy [2307.06239]. - \(n=2\delta-1\): Tsallis–Cirto entropy [2307.06239, 2606.24495]. - \(n=1+\Delta\): Barrow entropy [2307.06239, 2606.24495]. - \(n=d-1\): Tsallis–Zamora entropy [2606.24495]. The extended \(m,\alpha,\beta,\gamma\) formulation was explicitly designed to encompass the standard area-law extension with \(\beta=0\), power-law entanglement corrections with \(\beta=1\) and \(2<\alpha1\) enhances growth relative to \(\Lambda\)CDM, whereas \(n<1\) suppresses it [2510.00673]. The same work analyzed primordial gravitational waves and found that \(n<1\) enhances the relic PGW spectrum while \(n>1\) suppresses it, leading to the approximate current lower bound \[ n>0.884^{+0.002}_{-0.001} \] from PTA exclusions in the nHz band [2510.00673]. Baryogenesis provides an independent early-universe constraint. In a radiation-dominated background, standard GR gives \(R=\dot R=0\), preventing the usual gravitational baryogenesis operator from generating an asymmetry. In the generalized mass-to-horizon entropy cosmology, the modified Friedmann equations make \(\dot R\neq 0\) even during radiation domination, so standard gravitational baryogenesis can operate [2511.01693]. Expanding near \(n=1\), the predicted asymmetry is \[ \eta_B\simeq 1.05\times 10^{-12}\,g_B\,\sqrt{g_*}\,(1-n)\,x^5 + \mathcal O((n-1)^2), \] with \(T_D/M_p=10^{-3}x\), yielding the bound \[ 0.98\lesssim n<1 \] for a benchmark decoupling scale \(T_D\simeq 10^{16}\,\mathrm{GeV}\) [2511.01693]. This points to a small sub-extensive deviation from the Bekenstein–Hawking area law. Thermodynamic viability has also been studied directly. One analysis derived the entropy balance relation \[ dS=\frac{\delta Q}{T}+d_iS \] for generalized mass-to-horizon entropy, separating the thermal entropy flux from an internal entropy-production term, and found that the generalized second law holds for \(\omega\ge -1\) while the entropy tends toward a stable maximum in the asymptotic de Sitter regime [2605.12003]. The same work studied horizon-energy fluctuations and concluded that the relative fluctuation becomes extremely small at late times, supporting thermal stability [2605.12003]. Another paper combining the first-law and emergent-space derivations also found that the generalized second law is fulfilled for the universe enveloped by the apparent horizon [2512.18054]. ## 7. Interpretive issues, caveats, and current outlook Several caveats recur throughout the literature. First, the generalized mass-to-horizon relation itself is usually introduced as a phenomenological ansatz rather than derived from first principles. Although special cases such as the linear relation can be tied to the Misner–Sharp mass, the deeper geometric origin of the nonlinear scaling remains open in most cosmological formulations [2512.18054, 2605.12003]. Second, the framework is often developed specifically for FLRW cosmology with the apparent horizon and an equilibrium or quasi-equilibrium Clausius relation. Results obtained in this setting are not automatically general non-FLRW statements [2607.00133]. Third, several studies emphasize that current evidence for deviations from the standard area law remains weak: late-time data generally allow the \(\Lambda\)CDM limit within \(1\sigma\), and background viability often collapses the admissible parameter space to a tiny neighborhood of the Bekenstein–Hawking case [2508.13260, 2607.00133]. There are also conceptual differences between thermodynamic and modified-gravity readings of the framework. One interpretation treats generalized mass-to-horizon entropy as an effective thermodynamic description of cosmic acceleration, with dark energy emergent from horizon thermodynamics [2503.24355]. Another interpretation holds that any generalized entropy must first imply modified gravitational dynamics, often encoded as an area-dependent or curvature-dependent effective coupling, before one discusses horizon thermodynamics [2407.00484, 2606.24495]. These are not necessarily incompatible, but they represent different emphases. A plausible implication is that generalized mass-to-horizon entropy is best viewed not as a single finished theory but as a thermodynamically constrained family of horizon-based modifications whose physically acceptable region is very close to the standard area law. This suggestion is consistent with the sharp background constraints found in the extended MHR framework and with the observational result that all viable models are effectively \(\Lambda\)CDM-like at the present epoch [2607.00133, 2508.13260]. At present, the most robust claims are the following. Generalized mass-to-horizon entropy provides a thermodynamically consistent route to generalized horizon entropies by promoting the mass-to-horizon relation to the primary object [2307.06239, 2510.07587]. It modifies Friedmann dynamics through the gravity-thermodynamics conjecture and can generate an effective dark-energy sector without postulating a new fluid [2508.13260]. It encompasses several known generalized entropies as limiting cases and may admit a modified-gravity foundation via Jacobson or Iyer–Wald constructions [2407.00484, 2606.24495]. But all currently viable realizations lie close to the Bekenstein–Hawking limit, and neither late-time observations nor early-universe constraints presently require a statistically significant departure from standard cosmology [2508.13260, 2607.00133]. Source: https://www.emergentmind.com/topics/generalized-mass-to-horizon-entropy