On the observer dependence of the Hilbert space near the horizon of black holes

Goto, Kanato (Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902 Japan) ; Kazama, Yoichi (Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8902 Japan) (Research Center for Mathematical Physics, Rikkyo University, Toshima-ku, Tokyo 171-8501 Japan) (Quantum Hadron Physics Laboratory, RIKEN Nishina Center, Wako 351-0198, Japan)

12 February 2019

Abstract: One of the pronounced characteristics of gravity, distinct from other interactions, is that there are no local observables which are independent of the choice of the spacetime coordinates. This property acquires crucial importance in the quantum domain in that the structure of the Hilbert space pertinent to different observers can be drastically different. Such intriguing phenomena as Hawking radiation and the Unruh effect are all rooted in this feature. As in these examples, the quantum effect due to such observer dependence is most conspicuous in the presence of an event horizon and there are still many questions to be clarified in such a situation. In this paper we perform a comprehensive and explicit study of the observer dependence of the quantum Hilbert space of a massless scalar field in the vicinity of the horizon of Schwarzschild black holes in four dimensions, both in the eternal (two-sided) case and in the physical (one-sided) case created by collapsing matter. Specifically, we compare and relate the Hilbert spaces of three types of observers, namely (i) the freely falling observer, (ii) the observer who stays at a fixed proper distance outside of the horizon, and (iii) the natural observer inside of the horizon analytically continued from outside. The concrete results we obtain have a number of important implications on black hole complementarity pertinent to the quantum equivalence principle and the related firewall phenomenon, on the number of degrees of freedom seen by each type of observer, and on the “thermal-type” spectrum of particles realized in a pure state.

Published in: PTEP 2019 (2019) 023B01
Published by: Oxford University Press/Physical Society of Japan
DOI: 10.1093/ptep/pty146
arXiv: 1803.01672
License: CC-BY-3.0

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