Quantum time in near-horizon region of a black hole

Abstract: The understanding of time and dynamics can be elucidated by examining the concept of entanglement in quantum theory. This particular perspective on time is referred to as the timeless approach, which posits that the universe exists in a fixed state where two separate subsystems, namely the "clock" and the "rest," are entangled. By selecting an appropriate observable for the clock, the state of the rest of the universe evolves unitarily in relation to the variable that labels the clock observable's eigenstates, which is then interpreted as time. This intriguing model, initially introduced by Page and Wootters, has also been applied to the context of curved spacetime. In this study, we explore various uncertainties pertaining to the dynamics of the rest of the universe within a curved spacetime, including ambiguities related to the clock, the system's time evolution, the flow of time, and the recording of its history. Our investigation is primarily focused on the near horizon region of a black hole, as the peculiar behavior of quantum effects in this area allows for a thorough examination of the timeless depiction proposed by Page and Wootters in describing the system's dynamics within curved spacetime. This analysis may be valuable for quantum gravity projects that align with the approach put forth by Page and Wootters. It is worth noting that the application of the Page and Wootters approach in this particular region results in a distinct clock without any ambiguity. However, the other aforementioned issues, unlike those resolved in the realm of quantum mechanics, persist in this region.