Quantum model for black holes and clocks

TL;DR

The paper develops a two-system quantum model in which a quantum source $\\Xi$—encoded by a non-compact $\\mathfrak{su}(1,1)$ theory with Generalized Coherent States—induces the classical near-horizon dynamics of a test particle $\\Gamma$ around a Schwarzschild black hole via a quantum-to-classical crossover. A mapping between the quantum phase spaces generates the SBH-like infall Hamiltonian $h_{\\Gamma}(p,q)=\frac{p^2}{2m}+m\\kappa q$ with the surface gravity $\\kappa$, while a bosonic realization of the $\\Xi$ system yields Hawking-like radiation through a thermal reduced state, with the temperature matched to the Hawking temperature $T_H$. The framework naturally aligns with the Page-Wootters mechanism, identifying the SBH as a perfect clock whose conjugate time parameter emerges as the test particle’s proper time. These results connect microscopic quantum degrees of freedom to macroscopic black hole phenomenology and spacetime emergence, offering a non-field-theoretic route to black hole thermodynamics and clock-like dynamics, with potential links to holography and quantum gravity formalisms.

Abstract

We consider a stationary quantum system consisting of two non-interacting yet entangled subsystems, $Ξ$ and $Γ$. We identify a quantum theory characterizing $Ξ$ such that, in the quantum-to-classical crossover of the composite system, $Γ$ 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 $Ξ$ 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 $Ξ$. Finally, we recognize that the conditions used to characterize $Γ$ and $Ξ$ 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 $Ξ$ endows the SBH with all the characteristics of a "perfect" clock.