A Coefficient-Role Diagnostic for G in Four-Dimensional Einstein Black-Hole Thermodynamics

This manuscript argues that, in standard four-dimensional Einstein black-hole thermodynamics, the convention 4πG = 1 is best understood not as a field-equation normalization, but as a coefficient-role convention adapted to the Bekenstein-Hawking entropy-area density. The paper begins from the observation that Newton’s constant appears in several distinct coefficient packages with distinct operational roles. In c = 1 units, the convention 8πG = 1 is naturally adapted to the bulk source-coupling coefficient in the Einstein equation, G_ab + Λ g_ab = 8πG T_ab, whereas, in ordinary four-dimensional Einstein gravity, the black-hole entropy-area density is η_Ein := dS_BH/dA = k_B/(4Għ). In this setting, the convention 4πG = 1 gives η_Ein = πk_B/ħ. In this usage, “adapted to” does not mean “sets the coefficient to unity”; it means that the 1/(4G) entropy-area package is foregrounded. The manuscript’s central claim is therefore organizational rather than dynamical. It does not propose that 4πG = 1 is physically preferred, that it defines a new gravitational theory, or that it simplifies every horizon-related coefficient. Its claim is that, in the restricted four-dimensional Einstein black-hole setting, 8πG = 1 is bulk-source-coupling adapted, while 4πG = 1 is Einstein entropy-area-density adapted. The paper formulates this as a coefficient-role diagnostic: when a formula contains G, which coefficient package is G part of, and which numerical convention foregrounds that package? A coefficient package is understood as an expression involving Newton’s constant that plays a specified operational role in a specified formula, theory, dimension, and, where relevant, ensemble. The diagnostic procedure is: specify the theory, dimension, and ensemble; identify the relevant coefficient package containing G; ask which convention foregrounds that package; and then check whether other coefficient packages remain structurally distinct. The manuscript then distinguishes several standard four-dimensional Einstein packages involving G: the source-coupling coefficient 8πG in the Einstein equation; the Einstein-Hilbert and Gibbons-Hawking-York prefactors 1/(16πG) and 1/(8πG); the Einstein entropy-area density k_B/(4Għ); the stationary first-law area coefficient κ/(8πG); and the entropy and work packages appearing in Euclidean thermodynamic potentials. The point is not that Newton’s constant has different meanings, but that the same constant appears inside different coefficient structures with different operational roles. A central clarification of the paper concerns the first-law area term. For a stationary Einstein black hole, T_H = ħκ/(2πk_B), dS_BH = (k_B/(4Għ)) dA, so T_H dS_BH = (κ/(8πG)) dA. The paper emphasizes that this does not mean the convention 4πG = 1 is adapted to the full coefficient κ/(8πG). Rather, that coefficient is composite from the thermodynamic viewpoint: it combines the 1/(2π) factor in the Hawking-temperature relation with the 1/(4G) entropy-area factor. Thus 4πG = 1 is adapted to the entropy-area-density factor, while 8πG = 1 is the convention that simplifies the final first-law coefficient itself. In this sense, the occurrence of 1/(8πG) in the first law is composite horizon-response data, not source-coupling data. The manuscript next separates entropy-area adaptation from action-prefactor adaptation. In c = 1 units, the Einstein-Hilbert and Gibbons-Hawking-York prefactors are 1/(16πG) and 1/(8πG). The paper stresses that the convention 4πG = 1 does not remove these prefactors and is therefore not an action-prefactor convention. The safe statement is that 4πG = 1 adapts the entropy-area contribution, not the Euclidean action prefactors themselves. A sharper version of the same lesson is then given through a Euclidean/Smarr diagnostic. Writing Ŝ := S/k_B, β := 1/(k_B T), and considering a dimensionless thermodynamic potential of the form Ψ̂ = β(M - Σ_i n_i Y_i X_i) - Ŝ, the paper combines this with a verified Smarr relation M = λTS + Σ_i w_i Y_i X_i to obtain the algebraic identity Ψ̂ = (λ - 1)Ŝ + Σ_i (w_i - n_i) β Y_i X_i. This is presented as a Smarr-Legendre survival identity. Its purpose is diagnostic: it shows whether a work package cancels structurally or survives structurally in the chosen thermodynamic potential. The consequence is that adapting the entropy-area coefficient does not, in general, adapt the full thermodynamic potential. The manuscript uses β as inverse thermodynamic temperature; the Euclidean time period is ħβ, not β itself unless ħ = 1. The standard Kerr-Newman grand-canonical example makes this explicit. For the non-extremal four-dimensional asymptotically flat Kerr-Newman family, with the usual Einstein-Maxwell normalization, dM = T dS + Ω_H dJ + Φ_H dQ, M = 2TS + 2Ω_H J + Φ_H Q. In the fixed-(β, Ω_H, Φ_H) thermodynamic potential, Ψ̂ = β(M - Ω_H J - Φ_H Q) - Ŝ, the Smarr-Legendre identity gives Ψ̂ = Ŝ + βΩ_H J. Thus, when the corresponding Euclidean saddle interpretation is available, Î_E = Ŝ + βΩ_H J. Under 4πG = 1, the entropy package becomes Ŝ = πA/ħ, so the potential becomes Î_E = πA/ħ + βΩ_H J. The electric work package cancels structurally, while the rotational work package survives. The paper’s point is that entropy-area adaptation does not remove structurally surviving work packages. The manuscript also records the same bookkeeping for the standard four-dimensional asymptotically flat Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman families. In these examples, the resulting dimensionless potential is Ŝ for Schwarzschild and grand-canonical Reissner-Nordström, but Ŝ + βΩ_H J for Kerr and Kerr-Newman. This reinforces the claim that 4πG = 1 foregrounds the entropy package without adapting every other thermodynamic term. The paper also makes explicit that equations such as 8πG = 1 and 4πG = 1 are numerical unit conventions after c = 1 and the relevant gravitational scale have been fixed. They are not invariant equalities between dimensionful quantities before a unit convention is imposed. This caveat is part of the paper’s basic interpretive discipline. The manuscript then states the limit of export beyond four-dimensional Einstein gravity. In a general diffeomorphism-invariant theory, the relevant gravitational entropy is the appropriate Noether-charge or Wald entropy functional rather than the bare Einstein area law alone. Therefore 4πG = 1 is not automatically entropy-adapted outside the Einstein setting. The disciplined extension is: first identify the relevant entropy functional or local entropy density, and only then ask which coefficient convention is adapted to it. Likewise, in higher-dimensional, holographic, or effective-coupling settings, the relevant coupling may be G_D, G_bulk, or G_eff, rather than the four-dimensional Einstein G. The manuscript’s conclusion is therefore deliberately narrow. It proposes no new gravitational dynamics, no modified Einstein equation, no modified Bekenstein-Hawking entropy law, no microscopic interpretation of black-hole entropy, and no invariantly preferred unit system. Its claim is diagnostic: in standard four-dimensional Einstein black-hole thermodynamics, 8πG = 1 and 4πG = 1 are adapted to different coefficient roles. The former is bulk-source-coupling adapted. The latter is Bekenstein-Hawking entropy-area-density adapted. The usefulness of this distinction is that it prevents source coupling, action normalization, first-law packages, and Euclidean work-term survival from being conflated.