Relational Structure of the Bekenstein–Hawking Entropy: Discrete Temperatures, Superadditivity, Two-Level Quantization and Last-In-First-Out Evaporation

Abstract. The Bekenstein–Hawking entropy S = kBA/(4ℓp2) of a Schwarzschild black hole has been established by several independent methods, but it gives only the total amount of entropy and says nothing about what the entropy consists of. In a previous paper, taking the capsule Ec as the counting unit, we showed that the entropy of a Schwarzschild black hole of energy NEP decomposes exactly as S/kB = 4πN2 = 4π[N + N(N − 1)]: one self term per capsule and two ordered relations per pair of capsules, each term carrying the weight 4π, with no many-body terms. Taking the Bekenstein–Hawking entropy as the basis and this decomposition as the starting point, and introducing beyond general relativity only one structural parameter—the inner-interface radius Ri = GE/c4 = Rs/2 given by the force balance of the spacetime interfacial tension, exactly half the Schwarzschild radius—we derive a series of results that the Bekenstein–Hawking formula itself does not display. (1) The Hawking temperature, heat capacity and Smarr relation are expressed in the integer N; the discrete temperatures for absorbing and emitting one capsule are TH·2N/(2N ± 1), and the Hawking temperature is exactly their harmonic mean. (2) The entropy is superadditive: when N1 and N2 capsules combine, the entropy increase 8πkBN1N2 is precisely the newly established ordered relations between the capsules of the two parts, and what is additive is √S; this gives counting interpretations of the area theorem, the upper bound on gravitational-wave energy and the “no-bifurcation” theorem, and shows that no sub-black-hole survives a merger, which is a counting statement of the no-hair theorem. (3) Additivity distinguishes relational entropy from thermal entropy; “black-hole entropy is maximal, hence most disordered” and “all relations are established, hence most ordered” are the external and internal readings of the same number. (4) Accretion and evaporation are quantized at two levels: energy changes in whole packs of EP, area in sub-steps of Ao = 16πℓp2. Each sub-step has tunnelling probability e−4π, independent of N, while emitting a whole pack at once is suppressed to e−4π(2N−1); the characteristic wavelength of a sub-step is 2π(Rs + ℓp), and it is emitted as about 4.65 thermal particles (counted as photons). By the hoop criterion a pack EP is self-bound while a sub-step share is not: the horizon accepts only unpacked energy and packing is completed at the inner interface; a single Ec neither packs nor unpacks, so the Ec in the cosmic background are stable. (5) Temperature matching requires unpacking to be last in, first out: in each state only the last pack to enter can be released at the current temperature, and the history returns with the radiation in the reverse order of packing. In a merger the smaller (hotter) black hole unpacks in turn in its own last-in-first-out order, and the larger one packs in turn. The smallest black hole forms from two combining Ec, which requires a contact density reached only in the early universe; evaporation ends with the unpacking 2 → 1, leaving one Ec. (6) Conservation of relational entropy gives the Page curve directly: SBH/S0 = (1 − x)2, Sout/S0 = x(2 − x), with the information Page point xP = 1 − 1/√2 ≈ 0.293; the thermodynamic Page point, defined with the coarse-grained radiation entropy, moves earlier, to about 0.23–0.24, because of the entropy production of evaporation. (7) The end stage of evaporation is defined by kBT exceeding the rest energy of the electron (N ≲ 9.5×1020), after which the emitted particles include massive ones; the temperature staircase is significant only for N ≲ 102, far beyond present detection ranges. For a Kerr black hole counted by its irreducible mass, the decomposition holds unchanged with the capsule number Nirr; the inner interface lies exactly at the midpoint of the two horizons, and an interface spacing of at least ℓp makes the extremal Kerr black hole unattainable.