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Energy Records and Relational Clocks: An Analytic Decomposition in a Black-Hole–Radiation Toy Model

Jérôme Turcotte

Published Sep 7, 2026• 0 citations
Publisher

Abstract

We study a stationary finite-dimensional bipartite quantum system composed of mass levels, labelled “black hole,” and energy levels, labelled “radiation.” Matched levels satisfy a fixed total-energy constraint. Analytically and across 128 paired numerical realizations, the diagonal part of the global state provides a perfect record between remaining mass and radiated energy, whereas off-diagonal coherence across energy branches supports Page–Wootters conditional evolution. Complete dephasing preserves stationarity, the energy shell, both imposed thermal local density matrices, and the classical energy record, while eliminating all nontrivial clock variation. Relational clocks are detected in 128/128 coherent states and 0/384 mixed controls. Within this finite family, diagonal correlations encode the record of change, whereas coherence supports relational change. This construction is a controlled toy model. It does not reproduce black-hole geometry, Hawking pair creation, backreaction, or the Page curve, and it does not claim to resolve the black-hole information paradox.

DOI
10.5281/zenodo.22642972

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We study a stationary finite-dimensional bipartite quantum system composed of mass levels, labelled “black hole,” and energy levels, labelled “radiation.” Matched levels satisfy a fixed total-energy constraint. Analytically and across 128 paired numerical realizations, the diagonal part of the global state provides a perfect record between remaining mass and radiated energy, whereas off-diagonal coherence across energy branches supports Page–Wootters conditional evolution. Complete dephasing preserves stationarity, the energy shell, both imposed thermal local density matrices, and the classical energy record, while eliminating all nontrivial clock variation. Relational clocks are detected in 128/128 coherent states and 0/384 mixed controls. Within this finite family, diagonal correlations encode the record of change, whereas coherence supports relational change. This construction is a controlled toy model. It does not reproduce black-hole geometry, Hawking pair creation, backreaction, or the Page curve, and it does not claim to resolve the black-hole information paradox.
This project explores a fully timeless formulation of quantum theory motivated by the Page–Wootters mechanism and canonical quantum gravity. The central assumption is that the universe is described by a single global quantum state ∣Ψ⟩ satisfying a Wheeler–DeWitt–type Hamiltonian constraint, with time appearing only as an internal, relational variable defined by “clock” subsystems. I formulate a simple multi–clock framework, introduce a clock neutrality conjecture, and test it in an explicit finite-dimensional toy model. The model consists of two three-tick internal clocks and a qubit system, with joint step relations that encode a static global state over a 3×3 grid of clock readings. Imposing multi-clock consistency selects a constrained family of states in which the two clock evolutions commute on the support of ∣Ψ⟩. Within the decoherent histories framework, I then show that for histories expressible with either clock, the probabilities for relational events agree whenever the two clocks implement the same underlying physical evolution. In this sense the toy model realises a discrete version of clock neutrality: decoherent-histories probabilities are independent of which internal clock is used as the parameter. The construction does not yet yield new empirical predictions, but it isolates a precise structural condition that any timeless, deterministic description of the universe would need to satisfy in order to be compatible with clock-independent low-energy physics.
We consider a stationary quantum system consisting of two non-interacting yet entangled subsystems, ΞΞΞ and ΓΓΓ. We identify a quantum theory characterizing ΞΞΞ such that, in the quantum-to-classical crossover of the composite system, ΓΓΓ behaves as a test particle within the gravitational field of a Schwarzschild Black Hole (SBH) near its event horizon. We then show that this same quantum theory naturally provides a representation of ΞΞΞ in terms of bosonic modes, whose features match those of the Hawking radiation; this facilitates the establishment of precise relations between the phenomenological parameters of the SBH and the microscopic details of the quantum model for ΞΞΞ. Finally, we recognize that the conditions used to characterize ΓΓΓ and ΞΞΞ coincide with those required by the Page and Wootters mechanism for identifying an evolving system and an associated clock. This leads us to discuss how the quantum model for ΞΞΞ endows the SBH with all the characteristics of a "perfect" clock.
We revisit the black-hole information problem from the viewpoint of a population-coherence decomposition of density-matrix purity. Building on a previously developed formalism for ndimensional density matrices, we characterize each state by a normalized global purity index and two complementary indices, which quantify the contributions of level populations and coherences. This yields a simple quadratic relation and a geometric representation in a population-coherence plane, where different routes to purity can be distinguished. In the two-level case we construct explicit families of states with identical spectra and global purity but opposite internal structure, realizing population-dominated and coherence-dominated routes. We then apply this framework to a standard Page-type evaporation model without an explicit Hamiltonian, in which a black hole and its Hawking radiation form a bipartite pure state with varying Hilbert-space dimensions. Using known results for typical reduced states in large dimensions, we analyze the behavior of population and coherence components of purity along the evaporation process. Under the physically motivated requirement that, in this energy-free setting, the radiation populations remain nearly uniform in the chosen basis, we show that the late-time recovery of purity must be coherence-dominated: the global purity of the radiation approaches unity while the population index stays small and the coherence index carries essentially all the purity.
We construct a finite operational realization of Page–Wootters clock conditioning using destructive Enter on a named clock register. The complete output consists of a classical clock code and the surviving system; the measured clock names are consumed. An explicit channel identity proves equality of the Born record law and the unnormalized conditional system states with their Page–Wootters counterparts, including arbitrary untouched references. Under a separately stated covariance condition for the clock basis, stationarity of the joint preparation implies unitary relational evolution at the calibrated readings. An exact finite spectral construction establishes that these hypotheses are jointly inhabited. The probability and update formulas have two distinct justifications: they follow algebraically from the specified instrument, and, when the full Enter characterization hypotheses are satisfied, they are the measurement semantics selected by the previously established operational characterization. We do not rederive that characterization from abbreviated axiom names. We also give a typed hybrid propagator, complete-channel stability bounds, and refinement independence within an explicitly matched class. The philosophy of Enter is expressed as an interface discipline: a destructive record is not a surviving clock, a decoder, a thermodynamic ledger, or a spacetime geometry. The construction realizes relational extraction, not clock-free quantum gravity, a dynamical causal cone, or spacetime emergence.
-dimensional density matrices, we characterize each state by a normalized global purity index and two complementary indices, which quantify the contributions of level populations and coherences. This yields a simple quadratic relation and a geometric representation in a "population-coherence plane", where different routes to purity can be distinguished. In the two-level case, we construct explicit families of states with identical spectra and global purity but opposite internal structure, realizing population-dominated and coherence-dominated routes. We then apply this framework to a standard Page-type evaporation model without an explicit Hamiltonian, in which a black hole and its Hawking radiation form a bipartite pure state with varying Hilbert-space dimensions. Using known results for typical reduced states in large dimensions, we analyze the behavior of population and coherence components of purity along the evaporation process. Under the physically motivated requirement that, in this energy-free setting, the radiation populations remain nearly uniform in the chosen basis, we show that the late-time recovery of purity must be coherence-dominated: the global purity of the radiation approaches unity while the population index stays small and the coherence index carries essentially all the purity.
We investigate a plausible route to resolving the black hole information paradox by examining the effects of decoherence on Hawking radiation. In particular, we show that a finite but nonzero rate of decoherence can lead to efficient extraction of information from the evaporating black hole. This effectively pushes the paradox from becoming manifest at the Page time when the black hole has evaporated to half its size, to a timescale solely determined by the rate of decoherence. If this rate is due to a putative interaction with gravitons, the black hole at this timescale can be expected to be Planck sized, but notably without an extensive amount of information packed inside. We justify our findings by numerically studying a toy model of stabilizer circuits that can efficiently model black hole evaporation in the presence of decoherence. The latter is found to be well described by an effective rate equation for the entanglement and which corroborates our findings.