Entropy-Area Density in Horizon Thermodynamics: A Role Audit of Newton's Constant
Year: 2026
Type: preprint
Abstract: This manuscript argues that Newton's constant enters gravitational physics through distinct coefficient packages with distinct operational roles, and that in horizon and surface-entropy sectors the object being normalized is often not the bulk field-equation source-coupling package 8πG, but an entropy-area density, entropy functional, or area-response coefficient associated with geometric area. In four-dimensional Einstein gravity, 8πG = 1 is naturally adapted to the Einstein-equation source coupling, while 4πG = 1 is naturally adapted to the horizon entropy-area density. In this setting, the Bekenstein-Hawking/Wald entropy density is dS_grav/dA = k_B/(4Għ) in c = 1 units, so the normalization dS_grav/dA = πk_B/ħ is equivalent, after fixing the gravitational unit scale, to the convention 4πG = 1. Thus, in the four-dimensional Einstein specialization, dS_grav/dA = πk_B/ħ ↔ 4πG = 1. The paper's claim is not that Newton's constant has different meanings, nor that a new theory is proposed. Rather, the same Newton constant appears inside different coefficient structures that play different operational roles. In the Einstein equation it appears through 8πG. In the Einstein-Hilbert action it appears through 1/(16πG). In horizon thermodynamics it appears through the entropy-area density k_B/(4Għ), and in the stationary first law through the area-response coefficient κ/(8πG). The manuscript formulates this as a role audit: when a formula contains G, what coefficient package is G part of? For reversible horizon-response terms, the baseline object is the entropy-area density η_H = dS_rev/dA. For horizon settings admitting the standard semiclassical thermal factor T = ħα_H/(2πk_B) after setting c = 1, the half-scale response condition T(dS_rev/dA) = α_H/2 is equivalent, for nonzero α_H, to dS_rev/dA = πk_B/ħ. In four-dimensional Einstein gravity, this reproduces the equivalence with 4πG = 1. These are reduced-unit numerical statements after the relevant unit convention has been fixed; they are not invariant equalities between unreduced dimensionful quantities. The paper then audits this role distinction across standard thermodynamic-gravity settings. Jacobson's 1995 local-horizon derivation and later entanglement-equilibrium argument are read as examples in which an assumed local entropy-area density fixes the Einstein coupling. The Eling-Guedens-Jacobson non-equilibrium extension shows that beyond constant Einstein area density, the meaningful object is the full entropy density or entropy functional entering the horizon balance relation, together with the required internal entropy-production term, not the bare Newton constant alone. In stationary higher-curvature black-hole settings, Wald and Iyer-Wald entropy clarify that the disciplined extension of the Einstein result is: normalize the relevant entropy density, not the bare Newton constant. The paper also distinguishes entropy density from degree-count density using Padmanabhan's horizon degree-count picture: in Einstein gravity, the dimensionless entropy density and the Planck-cell counting density differ by a factor of four. Dynamical and cosmological horizon settings are discussed at the coefficient level: area-response coefficients of the form κ/(8πG) or their analogues reappear, but the normalization does not replace the underlying dynamical-horizon or cosmological machinery. The role audit is further extended to horizon transport through the KSS viscosity ratio, where in two-derivative Einstein bulk gravity the relevant bulk Newton coupling enters both the entropy density and the shear-response coefficient, but in different packages. Finally, the paper considers holographic and generalized gravitational entropy: in the classical Einstein-bulk Ryu-Takayanagi formula, 1/(4G_N^(bulk)) is again a surface-area entropy-density coefficient, while in higher-derivative cases the relevant object is the full entropy functional or effective density rather than the bare bulk Newton constant alone. These holographic entropy surfaces are not treated as ordinary thermal horizons satisfying the half-scale response relation. The conclusion is organizational rather than dynamical. In four-dimensional Einstein gravity, 8πG = 1 is bulk/source-coupling adapted, while 4πG = 1 is Einstein entropy-area-density adapted. More broadly, in thermodynamic, Noether-charge, transport, and holographic surface-entropy sectors, the relevant Newton coupling or effective entropy coupling often enters through an entropy functional, entropy density, or response coefficient associated with geometric area. The manuscript proposes no new dynamics, no modified field equations, no new entropy formula, no microscopic interpretation of horizon entropy, and no invariantly preferred unit system. Version 2 retains the central coefficient-role distinction of Version 1 while developing it into an explicit five-step operational procedure: identify the mathematical object, its defining operation, the data held fixed, the status of the normalization, and the inference justified by those conditions. The four-dimensional Einstein-gravity normalization and its reduced-unit interpretation have been aligned with the revised treatment in The 4πG = 1 Convention as an Entropy-Area Density Normalization. The treatment of Wald entropy and effective horizon coupling has been substantially refined to distinguish local entropy integrands, integrated entropy, cross-sectional mean density, and directional entropy-area response. In particular, the revision explicitly accounts for variations of the entropy density along a family of solutions. A five-dimensional Einstein–Gauss–Bonnet black-hole example, with the gravitational couplings held fixed, demonstrates that a density uniform on each horizon need not equal the derivative of total entropy with respect to area, while remaining consistent with the stationary first law. The assumptions and limitations governing the Jacobson constructions, non-equilibrium thermodynamics, dynamical and cosmological horizons, KSS transport, and holographic entropy have also been sharpened. The related-work discussion now acknowledges research on area-dependent effective gravitational couplings and multi-horizon gravitational susceptibility. These revisions strengthen the manuscript’s methodological and mathematical scope without introducing new gravitational dynamics, proposing modified field equations, deriving a new entropy formula, or selecting an invariantly preferred unit system.
Author: Enzo Cabrera Iglesias
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162188016
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