Four gravitating clocks generically cannot share a perfect time: redshift couplings, consensus, and the preferred-time problem
Year: 2026
Type: preprint
Abstract: In the timeless (Page and Wootters) approach to quantum time, gravitating clocks couple through their energies, C = Σ_i H_i + Σ_{i<j} λ_ij H_i H_j with λ_ij = −G/(c⁴ r_ij), at first order in G. Physical states depend only on the zero set of C, so one may ask when the coupling can be rewritten away. For two subsystems this is known: the constraint is equivalent to a noninteracting one with rescaled clock generators. We show that for three subsystems with arbitrary couplings a local additive rewriting still exists, and give it in closed form; for four it fails generically, since near the expansion point it requires λ_12 + λ_34 = λ_13 + λ_24 = λ_14 + λ_23, a condition that must hold for every quadruple when there are more. We then address the preferred-time problem raised by Stoica: a single clock does not fix which of the observables T + f(H) is its time. We prove a consensus lemma for the idealised, non-normalisable synchronised states of ideal clocks: if three clocks, or two clocks and a world with continuous energy spread, are perfectly synchronised, the only smooth re-choice of each clock's time of the form T_j → T_j + f_j(H_j) that keeps them synchronised is a common translation. For gravitating clocks, exact synchronisation in this idealised sense would need clock energies far outside the validity of the first-order coupling, so we combine the results at first order: with independent clock energies, the relative dephasing of the clocks can be removed by a choice of clock times if and only if the couplings among all subsystems are additive, λ_ij = −(g_i + g_j). This always holds for up to three subsystems and fails for four or more at generic positions, where a positive minimum disagreement remains. The choice that minimises it depends on the state and on how disagreement is aggregated, so it does not single out a time.Version 2 adds a remark on thermal time (Section 6) and two references; no result, number or conclusion of version 1 changed.
Author: Nguyen, Dat
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225417418