The Angular-Momentum Spectrum of Kerr Black Holes: Basis of the Existence Criterion, Spin Evolution and Breaking

Abstract In the capsule-counting framework, the energy of a black hole is quantized in whole packs of the dark-energy capsule Ec, whereas angular momentum is not quantized on its own. This paper studies how angular momentum arises from this, how it is gained and lost, and where it ends. There are two starting points. The state of a black hole is fixed by two integers, the total capsule number N and the irreducible capsule number Nirr; the angular momentum is the geometric mean of the entropies of the two horizons, J = ħ√(S+S−)/(2πkB), belongs to the black hole as a whole and cannot be apportioned to individual capsules; a capsule that enters or leaves the black hole can only be registered as a whole pack, either in the irreducible part or in the rotational part. By the 2dm axiom, the lower limit 2ℓp on the separation of the two horizons is the minimum distinguishable span, and has the same origin as the coefficient 1/4 of the Bekenstein–Hawking entropy. This yields the angular-momentum spectrum: black holes with N ≤ 4 cannot rotate, the minimal rotating state is BH(5,4), and J is in general not an integer multiple of ħ. The step-by-step growth of small-N black holes is isomorphic to Ostwald's rule of stages; a rotating state breaks on absorbing a single rotational pack; in evaporation it must lose all of its angular momentum before falling to N = 4. In large-N black holes, the angular momenta of the two registrations lie on either side of the baseline 2χNħ, the angular momentum brought in by an accretion disk lies between them, and probabilistic registration recovers Bardeen's spin-evolution equation on average. If each emitted pack carries away the angular momentum brought in by accretion, evaporation retraces the accretion path: a black hole with initial spin 0.99 loses all of its angular momentum when N has fallen to about 0.49 of its initial value, close to Page's result (about one half); the two kinds of capsule are emitted together, only rotational packs in a proportion far above their share, so the angular momentum reaches zero before the mass does. Real evaporation is irreversible and follows Page's path. The first law gives T ΔS+ = ΩH(L − ΔJrot): when the spin is above a threshold χc (about 0.88–0.92), the outer-horizon area increases with each pack emitted, which appears as rotational capsules being re-registered as irreducible capsules, and this locates Page's result. The spin limit χmax = 1 − 1/N2 sits right at the breaking point: near the limit, absorbing one rotational pack causes breaking in a single step, and the temperature does not fall to zero before breaking. Going from the Thorne limit to the upper limit requires a mass increase of only about 11%, but the spin limits given by the various accretion models must be passed; ordinary black holes therefore remain near those limits as long-lived cosmic somatic cells, and only cosmic egg cells that meet the conditions go on to breaking. The framework agrees on average with general relativity and with semiclassical results, and gives discrete structure at small N and at the spin limit.