The planckon problem: deriving stable Planck-mass black-hole remnants from quantum gravity

Aharonov, Casher and Nussinov (Phys. Lett. B 191 (1987) 51) proposed that the unitarity of quantum mechanics during black-hole evaporation forces the existence of a practically stable remnant of mass of order the Planck mass $M_{\mathrm{Pl}}\approx 10^{19}\,\mathrm{GeV}$, carrying an extremely large (practically infinite) internal degeneracy and neutral under all non-gravitational interactions — a "planckon". The paper posited the properties of these objects but did not derive them: no microscopic theory of quantum gravity was given, the effective couplings $C^{(N)}$ to $N$ light quanta were assumed not to grow factorially, the suppression of virtual planckon production at low timelike momentum transfer (conjectured "form factors") was postulated, and the cosmological mechanism fixing the planckon relic density was left unspecified. The open problem is to give this hypothesis a microscopic foundation: derive planckons (or establish that they cannot exist) in a consistent theory of quantum gravity, compute their mass spectrum, degeneracy and couplings, derive the production-suppression mechanism, and construct the early-universe production history with its dark-matter phenomenology. This core remains open: later work sharpened the criticism of infinitely degenerate remnants (the pair-production problem), shifted the mainstream towards unitary evaporation without remnants (entanglement-island computations), and revived Planck-scale relics as dark matter candidates, but no work derives or refutes the planckon construction itself.

Background

A black hole of mass $M$ radiates thermally (Hawking evaporation) at the Hawking temperature $T=M_{\mathrm{Pl}}^{2}/(8\pi M)$, so an evaporating hole heats up as it shrinks toward the Planck mass $M_{\mathrm{Pl}}\approx 10^{19}\,\mathrm{GeV}$, below which semi-classical gravity fails and no proper theoretical framework is available. Quantum mechanics postulates that the time evolution of a closed finite system is unitary, implemented by a unitary operator on the Hilbert space, so that a pure initial state (describable by a single wave function) can only evolve into another pure state. Hawking's computation instead produces thermal radiation: once the degrees of freedom of the evaporating hole are traced out, the emitted radiation is described by a density matrix (the mixed-state generalization of a wave function, encoding statistical ignorance). If the black hole disappeared completely into such uncorrelated thermal quanta, the evolution of the total system from a pure collapsing state to a mixed state would violate the unitarity postulate; Hawking argued that quantum gravity is fundamentally unpredictable in this sense, while Banks, Susskind and Peskin showed that dynamical pure-to-mixed evolution generically threatens energy–momentum conservation, and Unruh and Wald later confirmed severe difficulties for any such generalized quantum mechanics.

Aharonov, Casher and Nussinov (1987) proposed the opposite resolution: preserve unitarity by keeping the large reservoir of states. During evaporation the emitted quanta and the states retained near the horizon can be correlated into one overall pure wave function; if instead the residual hole decayed completely into ordinary particles, that reservoir would be lost and the final state would be mixed. They therefore conjectured a stable, practically infinitely degenerate particle at the Planck mass — the "planckon" (descended from Markov's 1967 "maximon" speculation — a hypothetical stable particle of about the Planck mass) — as the end state of evaporation. Two of their arguments fix its scale: the mass cannot greatly exceed $M_{\mathrm{Pl}}$ because Hawking evaporation is trustworthy down to a few $M_{\mathrm{Pl}}$; and it cannot be far below $M_{\mathrm{Pl}}$, because a lighter species of huge degeneracy would be violently produced at energies $E\sim 2M$ and drag the effective cutoff of all low-energy propagators (including the graviton's) below the Planck scale. They estimated the planckon lifetime by noting that a Planck-mass object decaying into the $N\sim (M/M_{\mathrm{Pl}})^{2}$ quanta emitted during evaporation is suppressed by a wave-function-overlap factor of order $N^{-3N}$, so remnants of holes heavier than a few $M_{\mathrm{Pl}}$ live longer than the age of the universe. In an effective local (low-energy) lagrangian $\mathcal{L}\sim G^{(N)}\,\phi\,\varphi^{N}/N!$ describing planckon $\phi$ decay into $N$ light particles $\varphi$, dimensional analysis gives $G^{(N)}=C^{(N)}/M_{\mathrm{Pl}}^{N-3}$, and the picture is consistent only if the dimensionless coefficients $C^{(N)}$ do not grow like $N!$ (otherwise the effective theory ceases to be local and the very notion of a Planck-sized object fails). They also conjectured that the underlying Planck-scale theory must supply planckon "form factors" that vanish for timelike momentum transfer, so that low-momentum gravitons cannot produce virtual planckon pairs whose infinite degeneracy would otherwise diverge the graviton propagator; and they noted that planckons, being massive, gravitating, and effectively undetectable by collider or scattering searches (cross sections of order the geometric Planck-area value $\sim 10^{-66}\,\mathrm{cm}^{2}$), are natural cold-dark-matter candidates — provided some dynamical mechanism at the end of the quantum era creates the required relic density and fluctuation spectrum.

The subsequent literature sharpened the obstacles. Giddings analyzed the "species problem"/infinite pair-production catastrophe: if remnants carry an infinite number of internal states and can be treated as point particles in an effective field theory with cutoff above the Planck length, the vacuum would be catastrophically unstable; he argued the production rate is governed by the thermodynamic trace over internal states and is apparently infinite unless the number of internal states is finite. The related entropy problem is that a Planck-area remnant storing the information of a macroscopic hole must contain far more states than the Bekenstein–Hawking entropy $S=A/4$ (the black-hole entropy, proportional to horizon area) suggests, in tension with holographic counting. In parallel, the mainstream moved away from remnants: the quantum-extremal-surface ("island") computations of Penington and of Almheiri and collaborators — entropy calculations using extremal quantum surfaces in the entanglement wedge, the spacetime region whose fields can be reconstructed from the radiation — yield unitary Page curves (the entanglement entropy of the emitted radiation as a function of time, which returns to zero only if evaporation is unitary) for evaporating black holes in specific holographic settings (gravitational systems dual to a non-gravitational quantum theory), attributing unitarity to subtle correlations in the Hawking radiation rather than to a remnant. Recent dedicated reviews nevertheless conclude that no argument conclusively rules remnants out, and a separate, active thread studies Planck-scale relics (Planck stars, memory-burdened primordial black holes, trans-Planckian-scattering relics) as dark matter. What none of this provides is the missing microscopic foundation: a derivation of the planckon itself.

Problem Statement

Is there a consistent microscopic theory of quantum gravity in which the end state of black-hole evaporation is a practically stable particle of mass of order the Planck mass $M_{\mathrm{Pl}}$ — a planckon — with the properties that the unitarity argument of Aharonov, Casher and Nussinov requires: effectively unlimited internal degeneracy (a large enough reservoir of states to purify the Hawking radiation), neutrality under all interactions of range larger than the Planck length, and consistency of the effective local description above the Planck length? The problem comprises three jointly required results:

(1) A microscopic derivation of the planckon (or a demonstration that no such object can exist consistently): its mass spectrum, whether its internal degeneracy is finite or infinite and how large, and its effective couplings $G^{(N)}\sim C^{(N)}/M_{\mathrm{Pl}}^{N-3}$ to $N$ light quanta — establishing in particular whether the coefficients $C^{(N)}$ can avoid factorial growth in $N$, so that treating the planckon as a local Planck-sized object does not destroy the locality of the effective theory.

(2) A derived, rather than conjectured, mechanism that suppresses the production of virtual planckon pairs at low timelike momentum transfer — the 'form factors' posited in the source, or an alternative regulator with the same effect — so that the propagators of low-energy fields, including the graviton, remain finite and the Planck scale stays the only ultraviolet cutoff of low-energy physics.

(3) A dynamical cosmological mechanism, operating at the end of the quantum era (the final, Planck-scale epoch of the early universe, where semiclassical gravity fails and quantum-gravity effects dominate), that produces the relic density and fluctuation spectrum of planckons, together with the resulting dark-matter phenomenology and its confrontation with observational constraints.

These three results are bound by a single scientific objective — a microscopic account of the Planck-scale end state of gravitational collapse and its observable consequences — and are jointly required as one coherent construction; a demonstration that no such object can exist consistently (for example, by proving the internal-state count is finite and too small, or that no consistent effective coupling hierarchy exists) is an equally valid resolution of the problem.

References

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