Euclidean Black-Hole Thermodynamics and Entropy-Area Normalization: A Role Audit of 4πG = 1

This manuscript argues that, in four-dimensional Einstein gravity, the convention 4πG = 1 is naturally adapted to the Bekenstein-Hawking entropy-area density, not to the Einstein-Hilbert or Gibbons-Hawking-York action prefactors. The paper audits that distinction in Euclidean black-hole thermodynamics. With c = 1, the manuscript defines the dimensionless Euclidean action and entropy by Î_E = I_E/ħ, Ŝ = S/k_B. In the Einstein black-hole settings considered, the Bekenstein-Hawking entropy gives Ŝ = A/(4Għ), so the entropy-area density is dS/dA = k_B/(4Għ). Under the numerical convention 4πG = 1, this becomes dS/dA = πk_B/ħ, or equivalently Ŝ = πA/ħ. The paper’s central claim is that this is the coefficient package normalized by 4πG = 1. By contrast, the Einstein-Hilbert and Gibbons-Hawking-York action prefactors, 1/(16πG), 1/(8πG), do not disappear under that convention; they become 1/4 and 1/2, respectively. Thus 4πG = 1 is not action-prefactor adapted. It is entropy-area-density adapted. The Euclidean black-hole calculation is used as a diagnostic because it is the setting in which action normalization, entropy, and thermodynamic work terms appear together without collapsing into a single role. The manuscript emphasizes that “rationalizes” means that the chosen unit convention puts the relevant coefficient package into a simple π-normalized numerical form; it does not mean that the convention is dynamically preferred or that all appearances of G are simultaneously simplified. For Schwarzschild, where the on-shell bulk Ricci scalar vanishes, the regulated background-subtracted Euclidean action is determined by the Gibbons-Hawking-York boundary term and gives Î_E = βM/2 = A/(4Għ). Hence Î_E = Ŝ. Under 4πG = 1, this becomes Î_E = Ŝ = πA/ħ. The paper stresses, however, that this equality is not caused by the 4πG = 1 convention. It already holds before that convention is imposed. In the Schwarzschild case it follows from the Euclidean thermodynamic identity Î_E = βM - Ŝ together with the Smarr relation, which implies βM = 2Ŝ. So the convention changes the shared numerical form of the result, but it is not the reason the Euclidean action equals the entropy. The paper then turns to the non-extremal four-dimensional asymptotically flat Kerr-Newman family and its Schwarzschild, Kerr, and Reissner-Nordström limits, treated in the standard grand-canonical thermodynamic-potential sense. In that framework, Î_E = β(M - Ω_H J - Φ_H Q) - Ŝ. Using the Einstein-Maxwell Smarr relation M = κA/(4πG) + 2Ω_H J + Φ_H Q together with β = 2π/(ħκ), the manuscript shows that, for Kerr and Kerr-Newman, Î_E = Ŝ + βΩ_H J. Thus, under 4πG = 1, Î_E = πA/ħ + βΩ_H J. This is one of the paper’s main conclusions: the convention 4πG = 1 rationalizes the entropy-area contribution extracted from the Euclidean thermodynamic potential, but it does not eliminate the rotational work term. Rotation therefore exposes the separation between the entropy-area package and the full Euclidean thermodynamic potential. The charged cases are used to sharpen that point. For Reissner-Nordström in the fixed-potential grand-canonical ensemble, the manuscript finds again that Î_E = Ŝ, so under 4πG = 1, Î_E = πA/ħ. The paper explains this by noting that the electric term cancels after using the Einstein-Maxwell Smarr relation because its Smarr weight is one, matching the grand-canonical subtraction. By contrast, the rotational term survives because its Smarr weight is two while the grand-canonical subtraction removes only one copy. Hence the surviving term in Kerr and Kerr-Newman is βΩ_H J, not an electric contribution. The manuscript also gives an explicit Kerr diagnostic in full 4π-rationalized units c = ħ = k_B = 1, 4πG = 1. There the dimensionless Euclidean action becomes Î_E = (M^2/2)(1 + 1/s), with s = sqrt(1 - χ^2) and χ = J/(GM^2). This reproduces the Schwarzschild limit when χ = 0 and shows that the non-extremal Euclidean thermodynamic potential diverges as s → 0, i.e. along the non-extremal approach to extremality where β → ∞. The paper explicitly states that this is not a claim that extremal black-hole entropy diverges; extremal Euclidean saddles require separate treatment. A central theme of the note is therefore a role audit of coefficient packages. In the Euclidean Einstein black-hole setting, Newton’s constant appears in at least four distinct operational packages: the Einstein-Hilbert action prefactor 1/(16πG), the Gibbons-Hawking-York prefactor 1/(8πG), the entropy-area coefficient 1/(4Għ) or k_B/(4Għ), and the first-law area-response coefficient κ/(8πG). The convention 4πG = 1 acts differently on these packages. It converts the action prefactors to 1/4 and 1/2, the dimensionless entropy-area coefficient 1/(4Għ) to π/ħ, the physical entropy-area coefficient k_B/(4Għ) to πk_B/ħ, and the first-law area coefficient κ/(8πG) to κ/2. The conclusion is that 4πG = 1 is entropy-area-density adapted and induces the reduced first-law area coefficient κ/2, but it is not a universal coefficient-minimizing convention and it is not adapted to the full Euclidean action. The paper is careful about scope. The results concern the non-extremal outer horizon of the four-dimensional asymptotically flat Kerr-Newman family and its Schwarzschild, Kerr, and Reissner-Nordström limits, in the stated grand-canonical thermodynamic-potential treatment. The manuscript notes that the ensemble matters, the charge normalization matters, Euclidean Kerr must be understood in the standard thermodynamic-potential sense rather than as a naive real Euclidean section, and extremal limits require separate treatment. It also states that the analysis should not be exported directly to anti-de Sitter, de Sitter, higher-dimensional, higher-curvature, Wald-entropy, effective-coupling, or holographic settings without redoing the corresponding Smarr relations, boundary terms, entropy functionals, and ensemble definitions. The manuscript proposes no new black-hole thermodynamics, no new entropy formula, no modified field equation, no microscopic interpretation of horizon entropy, and no preferred invariant unit system. Its claim is organizational: in four-dimensional Einstein black-hole thermodynamics, 4πG = 1 rationalizes the entropy-area contribution extracted from the Euclidean thermodynamic potential, but not the full Euclidean action or the action prefactor itself.